A MARKOV CHAIN DYNAMIC SOCIAL NETWORK MODELING AND SIMULATION TO ASSESS THE
IMPACT OF OUT-OF-SCHOOL SUSPENSIONS ON DROPOUTS AND TRUANCY
by
Kim Y. Peters
A dissertation submitted to the Graduate Faculty of
Auburn University
in partial fulfillment of the
requirements for the Degree of
Doctor of Philosophy
Auburn, Alabama
August 9, 2010
Keywords: Stochastic Process, Markov Chain, Modeling, Social Networking, Learning
Coefficient, Dynamic Connectivity, Student Dropouts, Truancy, Juvenile Delinquency
Copyright 2010 by Kim Y. Peters
Approved by
Jorge Valenzuela, Chair, Associate Professor of Industrial and Systems Engineering
Chan S. Park, Samuel Ginn Distinguished Professor of Industrial and Systems Engineering
Robert Thomas, Emeritus Professor of Industrial and Systems Engineering
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ABSTRACT
Generally, a social network is mainly concerned with actors, the actions of actors, and the
interactivities among actors in a community. Employing graph theory to represent overall social
network structure, this study develops a Student School Attendance System (SSAS) evaluation
model to provide a frame work in order to understand and assess the impacts of unsupervised
Out-of-School (OOS) Suspensions on dropout, truancy, and the overall attendance performance
of students and proposes a new attendance category, supervised out-of-school suspension. In
recent years, there has been a dramatic increase in juvenile delinquency and student dropout
rates. This research examines the impacts of a current OOS suspension program which is devoid
of any form of supervision or monitoring system. In brief, the school transfers the problem
student from the school to the street without any support mechanism. To examine and assess
OOS, this research develops a model that represents current student attendance states paradigms
considering student interaction by assimilating learning coefficients and threshold values for
connectivity between peers. The study further examines student attendance for supervised
versus unsupervised OOS.
This research employs a Stochastic Markov Chain Process that considers the random
events of the student school attendance phenomenon. The SSAS model represents a
comprehensive state diagram of student school attendance categories based on graph theory and
defines relevant parameters and variables. The model process begins with formulation of the
initial transition probability and calculation of cumulative and steady state probabilities. The
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student school attendance states are determined based on random numbers and ranges of
cumulative probability distributions applying the Markov Chain Memory-Less property. The
next day student attendance probability computation considers the influence of friends by
integrating a threshold value for dynamic connectivity and the fitness of the learning coefficient
as the interaction influential factor among students.
In addition, the SSAS model explores in three ways the development of the initial
transition probability. The first model employs a pure random process by generating the initial
transition probability of the student school attendance in a random selection scheme. The second
model provides a controlling mechanism by providing the initial transition probability
distribution for student school attendance via a pre-arranged input value data table. The third
model provides a controlling mechanism as well as the stability in the system by implementing a
pre-arranged output value look-up table which is formulated based on attendance data to
generate the initial transition probability. Furthermore, the model also exploits the connectivity
between the peers in three different ways: Median Connectivity, Virtual Connectivity, and
Hybrid Connectivity.
As a result of the SSAS model simulation, the Pre-Arranged Output Value model
generated the most stable system among all three models with 1.4% error. In terms of
connectivity, Virtual Connectivity produced the most connections between peers. In the SSAS
model student school attendance assessment there was about 22% improvement in the number of
students promoted to the next grade/graduated from high school under supervised OOS
suspensions in comparison to unsupervised OOS suspensions. Among three types of
connectivity, Virtual Connectivity produced the most improvement in the number of students
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succeeding at school. In conclusion, the SSAS model successfully rejects the null hypothesis
?Implementation of Supervised Out-of-School Suspensions will have no effect on students?
attendance and graduation rate.?
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ACKNOWLEDGMENTS
It is my privilege to acknowledge the contributions in directing this research mainly to
Dr. Jorge Valenzuela who is Chair for providing the fundamental framework of the Stochastic
Process Markov Chain modeling techniques, Operations Management, and Information
Technology, Dr. Chan S. Park for assisting in quality assurance management and risk analysis in
Economic Engineering, Dr. Robert Thomas for supporting the overall perspectives of Human
Behavioral Theory of social networking and relevant issues involved in Ergonomics, and the Dr.
Cheryl Seals who served as an outsider reader for my dissertation. My sincere appreciation
extends to Dr. Alice Smith, the Department Chair who has been advocator and motivator
throughout my graduate study.
I have the utmost gratitude and admiration to my husband, Judge Bobby G. Peters,
Superior Court of Chattahoochee Circuit, Columbus Georgia, for his unconditional love and
encouragement in all aspects. In addition, my ultimate source of strength, my sons David and
Jonathan for their love and understanding during my study.
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TABLE OF CONTENTS
ABSTRACT .................................................................................................................................... ii
ACKNOWLEDGMENTS ................................................................................................................... v
LIST OF TABLES ............................................................................................................................ x
LIST OF FIGURES ......................................................................................................................... xi
LIST OF ABBREVIATIONS ............................................................................................................ xii
1. INTRODUCTION ....................................................................................................................... 1
1.1. Background ............................................................................................................ 4
1.2. Motivation .............................................................................................................. 5
2. STATEMENT OF RESEARCH OBJECTIVES ................................................................................ 8
2.1. Goals and Problem Statement ................................................................................ 9
2.2. Objectives and Contributions ............................................................................... 10
3. LITERATURE REVIEW ........................................................................................................... 13
3.1. Introduction .......................................................................................................... 13
3.2. Social Networking ............................................................................................... 14
3.3. Basic Graph Theory ............................................................................................. 17
3.4. Stochastic Procedures ........................................................................................... 19
3.4.1. Property of Random Event Process ......................................................... 20
3.4.2. Stochastic Process and Exponential Distribution .................................... 20
3.5. Markov Chain Phenomenon ................................................................................. 21
vii
3.5.1. Characteristics of the Markov Chain ....................................................... 22
3.5.2. The Long-Run Behavior of Finite-State Markov Chain .......................... 23
3.6. Chapter Remarks and Summary .......................................................................... 25
4. MODEL FORMULATION ......................................................................................................... 27
4.1. Introduction .......................................................................................................... 27
4.2. Model Assumptions ............................................................................................. 29
4.3. Student School Attendance Model ....................................................................... 31
4.3.1. Background Introduction ........................................................................ 31
4.3.2. State Diagram of the Student Attendance Paradigms ............................. 33
4.3.3. State Probability Distributions ................................................................ 34
4.4. The Student Interaction Model ............................................................................ 36
4.4.1. The Graph Representation of Student Interactions .................................. 37
4.4.2. Assessment of the Interaction between Students ..................................... 39
4.4.2.1. Initial Student Attendance State Probabilities .......................... 39
4.4.2.2. Distribution of the Student Attendance States .......................... 39
4.4.2.3. Interactivity of the Students and Learning Coefficient ............. 40
4.4.3. Three Dynamic Modes of the Connectivity between the Peers ............... 41
4.4.4. Interaction Probability Computation ........................................................ 43
4.4.4.1. Initial Student Attendance State Probabilities .......................... 43
4.4.4.2. Determination of the Student Attendance States ...................... 43
4.5. The Student School Attendance Simulation (SSAS) Model ................................ 44
4.5.1. Introduction .............................................................................................. 44
viii
4.5.2. The Model ............................................................................................... 45
4.6. Chapter Remarks and Summary .......................................................................... 53
5. METHODS AND PROCEDURES OF SSAS MODEL ................................................................... 55
5.1. Introduction .......................................................................................................... 55
5.2. Procedures, Algorithms, and Model Validations ................................................. 57
5.2.1. Development of Initial Transition Probability Distribution Table ......... 57
5.2.2. Cumulative Probability and the Range of Attendance States ................. 60
5.2.3. Transition from Current State to Next State ........................................... 63
5.2.4. Formation of the Connectivity between the Peers .................................. 65
5.2.5. Interaction Probability Computation ....................................................... 70
5.4. Chapter Remarks and Summary .......................................................................... 72
6. THREE DIFFERENT TYPES OF THE SSAS MODEL .................................................................. 74
6.1. Introduction .......................................................................................................... 74
6.2. Pure Input Output Value (PIOV) Dynamic Model .............................................. 75
6.2.1. Procedures and Methods of PIOV Model ............................................... 76
6.2.2. Algorithms of PIOV Model .................................................................... 77
6.3. Pre-Arranged Input Value (PAIV) Dynamic Model ............................................ 78
6.3.1. Procedures and Methods of PAIV Model ............................................... 79
6.3.2. Algorithms of PAIV Model .................................................................... 80
6.4. Pre-Arranged Output Value (PAOV) Model ....................................................... 82
6.4.1. Procedures and Methods of PAOV Model ............................................. 82
6.4.2. Algorithms of PAOV Model ................................................................... 85
6.5. Chapter Remarks and Summary .......................................................................... 87
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7. EXPLOITATION RESULTS AND ANALYSIS .............................................................................. 90
7.1. Introduction .......................................................................................................... 90
7.2. Simulation Results of the SSAS Models ............................................................. 90
7.3. The Connectivity in between Three Types of Connections ................................. 93
7.4. Attendance Assessment between Supervised and Unsupervised OOS ................ 96
7.5. Correlation of Supervised and Unsupervised OOS and Dropouts ....................... 99
7.6. Chapter Remarks and Summary ........................................................................ 100
8. CONCLUSION AND FUTURE RESEARCH ............................................................................... 102
8.1. Remarks and Recommendations ........................................................................ 103
8.2. Future Research ................................................................................................. 104
REFERENCES ............................................................................................................................ 108
Appendix 1. THE SSAS MATLAB CODE ............................................................................. 111
Appendix 2. THE ATTENDANCE STATES OF THE STUDENTS ................................................... 121
x
LIST OF TABLES
Table 1 The Matrix Representation of the Relationships of 6 Students ................................... 38
Table 2 Procedures of Transition Probability Computation ................................................... 49
Table 3 Example of Input Data Table for the Initial Transition Probability Distribution ...... 58
Table 4 Transition Probability Distribution of the Attendance States for Student 2 .............. 59
Table 5 Examples of Cumulative Probability Distribution Table .......................................... 60
Table 6 Cumulative Probability Distribution Table for Student 1 and 2 ................................ 61
Table 7 The Current State and Next State Allocations ........................................................... 64
Table 8 The Rationale in Formulation of the Connectivity among the Students .................... 66
Table 9 Pre-Arranged Output Probability Distribution Look-Up Table ................................... 83
Table 10 Simulation Results of the SSAS Models ................................................................. 92
Table 11 Measurements of the Errors and Correctness of the SSAS Models ........................ 92
Table 12 The Number of Connectivity in a Five-Day Interval for School Year .................... 94
Table 13 Student Attendance Assessment Using Median Connectivity ................................. 97
Table 14 Student Attendance Assessment Using Virtual Connectivity .................................. 97
Table 15 Student Attendance Assessment Using Hybrid Connectivity .................................. 98
Table 16 The Difference in Number of Student Graduates in Three Types of Connectivity . 100
xi
LIST OF FIGURES
Figure 1 Examples of Undirected Graphs of Social Network ................................................ 18
Figure 2 Overview of Inputs, Outputs, and Parameters of the SSAS ..................................... 28
Figure 3 State Diagram Representation of Student School Attendances ................................ 33
Figure 4 Undirected Graph Representation of 6 Students and their Relationships ................ 38
Figure 5 Procedures of the Next Day Transition Probability Computation ............................ 50
Figure 6 The Pseudocode in Determining the Attendance States ........................................... 65
Figure 7 Initial Iteration for the Connectivity among 6 Students ........................................... 67
Figure 8 Pseudocode for Median Connectivity ...................................................................... 69
Figure 9 Flow Chart in Computing the Interaction Probability among the Students ............. 71
Figure 10 Process Overview of SSAS Model ......................................................................... 75
Figure 11 Procedures of the PIOV SSAS Model .................................................................... 76
Figure 12 Pseudocode of the PIOV Model ............................................................................. 78
Figure 13 Procedures of the PAIV SSAS Model .................................................................... 79
Figure 14 Pseudocode of the PAIV Model ............................................................................. 81
Figure 15 Procedures of the PAOV SSAS Model .................................................................. 84
Figure 16 Pseudocode of the PAOV Model ........................................................................... 86
Figure 17 Comparison of the PIOV, PAIV, and PAOV Models ............................................ 88
Figure 18 The Average Number of Connectivity in SSAS Models ........................................ 96
Figure 19 The Number of Student Graduates of Supervised and Unsupervised OOS ........... 99
xii
List of Abbreviations
E (Exe) Excused Absence
HC Hybrid Connectivity
ITP Initial Transition Probability
IP Interaction Probability
IS In-School Suspension
MC Median Connectivity
MCP Markov Chain Process
MCS Monte Carlo Simulation
MCMCS Markov Chain Monte Carlo Simulation
OS Out-of-School Suspension
PAIV Pre-Arranged Input Value
PAOV Pre-Arranged Output Value
PIOV Pure Input Output Value
SOS Out-of-School Suspension with Supervision
SSAS Student School Attendance System
SSP Steady State Probability
T (Tru) Truancy, unexcused absence
TP Transition Probability
VC Virtual Connectivity
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1. INTRODUCTION
Nowadays, cutting edge technology and the pervasiveness and ubiquitous access to the Internet
affect all aspects of our society. The younger generation in particular is sculptured with new
revolutionary technologies, impetuously establishing a new era of culture. Quite often, it seems
having our young children in-tune with their cell phone or I-pod in a general public place is a
part of our every day life style. This new life style of our younger generation brings enormous
ongoing challenges to parents, educators, and lawmakers. Daily, each school district faces the
immense task of trying to meet those challenges. Overwhelmingly, in spite of efforts to
accommodate those challenges, according to the national statistics, the student attendance rate in
middle school and high school is at an all time low [37, 40]. Moreover, in recent years, there has
been a dramatic increase in juvenile crime and the high school dropout rate [38]. In response to
the alarming statistical data and intricate issues surrounding this young population, the U.S.
Department of Justice established the Office of Juvenile Justice and Delinquency Prevention
(OJJDP). Since 1974, it has provided a support system to decrease the incidence of juvenile
delinquency of young students [38]. The main mission of the OJJDP is to provide various
resources to protect children and prevent juvenile delinquency. The OJJDP website reads in
part: ?OJJDP supports states and communities in their efforts to develop and implement effective
and coordinated prevention and intervention programs and to improve the juvenile justice system
so that it protects public safety, holds offenders accountable, and provides treatment and
rehabilitative services tailored to the needs of juveniles and their families? [38]. In addition,
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national data based on the Statistical Briefing Book (SBB) reflects the magnitude of the problem
and the seriousness related to juvenile crime [39]. According to the SBB report, in the year
2004, there were 1,571,230 juvenile related crimes and 1,593,661 for year 2005. The SBB data
synopsis indicates that among the age group from 12 to 17 the most prevalent age for juvenile
crime is at age 16, which accounted for approximately 394,000 crimes, the quarter percentile of
the total number. Furthermore, the data from the Department of Education shows the national
level for the high school graduation rate is at a record low, which is around 73 percent [40, 41].
The high school graduation rate for the State of Georgia and the City of Columbus, Muscogee
County is around 70 percent [35, 36]. Consequently, across the board, each school district faces
the monumental responsibility to improve student behavior and academic performance of the
students in an increasingly complex and multi-faceted school environment.
Subsequently, those students identified within the at-risk ages of 12 to 17, were found to have a
high drop-out rate, low achievement in school performance, and a high incidence of juvenile
delinquency. This realization is evidenced by the SBB statistics, where the enormous numbers
of young children are committing crimes, and fail to graduate from high school according to the
Department of Education. The negative impact of students? unsuccessful school experience and
the consequences of students? not attending class extend much further than merely failing the
class or missing instruction. A student?s not attending class could result in his or her becoming a
dropout and increase in the possibility of involvement in juvenile delinquency. This research
examines the problem associated with the current out-of-school suspension program. In general,
the current out-of-school suspension program does not require monitoring of the out-of-school
suspended student to ascertain whether or not he or she keeps up with school assignments or the
class work during out-of-school suspension period [36]. Out-of-school suspension is usually
3
considered an unexcused absence and the student is not entitled to ?make up? the missed grades
during the suspension period. Therefore, school is neither obligated nor mandated to provide
academic materials to the students who are subject to out-of-school suspension. Consequently,
these at-risk students fall further behind in school as a result of an out-of-school suspension. To
examine the correlation between the high dropout rates and the number of out-of-school
suspensions, this research develops a model that describes the various student attendance states
and identifies relevant parameters and variables. In addition, the model integrates the threshold
value for the connectivity and learning coefficient in accommodation of the interactions with
their peers during the school day.
Comprehensively, this research examines the complicated dynamic student school attendance
phenomenon in regard to interactions among peer paradigms. The main objective of this
research is to develop an evaluation model for student school attendance performance
considering the social interactions with their friends at the school. The model distinguishes
student attendance categories and employs a stochastic process with the Markov Chain?s
memory-less property assuming that the possibility of the student?s coming to school tomorrow
is solely based on today?s activity. The interactions of the students with their peers are modeled
using social network and graph theory. Overall, this research presents a Student School
Attendance System (SSAS) model which can be used to evaluate the student school attendance
paradigms.
In particular, this research focuses on assessment of the impacts of the current out-of-school
suspension program which is devoid of any form of supervision or monitoring program by the
school system. Currently, the school transfers the problem student from the school to the street
4
without any support or monitoring mechanism. To address the inevitable and yet often ignored
catastrophic problem resulting from suspending the student from school without any support
system, the underlying impact must be identified and assessed. The main premise of the research
in order to address the complexity of the student school social network is: the state of the student
school attendance and the interactions with their peers in accommodation of the influential
factor.
The subsequent chapters are organized as follows: Chapter 2 presents the statement of research
objectives. Chapter 3 includes a review of the literature and back ground study. Chapter 4
introduces the formal representation of the SSAS model. Chapter 5 explains the methods and
procedures of the model. Chapter 6 illustrates the three different types of the SSAS models.
Chapter 7 provides the results and the data analysis. Chapter 8 draws the conclusions and future
research areas.
1.1. Background
In January 2009, the City of Columbus, Georgia, established the ?Mayor?s Commission on
Crime Prevention? in order to establish a baseline to reduce the crime rate with the goal of
making the city a safer place. The main initiatives of the commission were outlined by the Chief
Executive of the City to research the underlying issues associated with criminal activity, study
the possible enhancement of existing programs currently addressing these issues, and evaluate
new alternatives or new systems to reduce and/or prevent deviant and illegal activity throughout
the community. Members of the commission represented various segments of the city and had
diverse backgrounds and experiences, such as a notable State Representative in the General
Assembly, bankers, lawyers, educators, pastors, sports director, social workers, and law
5
enforcement officers. Such diversity allowed the commission to be more creative and to openly
brainstorm ideas and suggestions in order to reach a consensus on which ideas and objectives
should be adopted to meet the main goal to ?reduce the crime rate in the city.? The Mayor
mandated total cooperation from the Police Chief, the Sheriff, and the Marshall. Each
department provided factual statistical data to map out the areas of the city with the highest crime
rates, identify the most frequent times for particular crimes, and demographics of the offenders.
The Commission discovered that the age groups most at-risk in criminal activity are young
offenders between ages 16 to 23, with most deviant behavior being committed after school hours
between 3:00pm and 2:00am. Specifically, age 16 was the prevalent age of the offender. Based
on these statistics, each commission member was charged to submit a proposal and initiative that
would address the underlying causes of juvenile crimes and reduce the opportunities for young
people to become involved in criminal activity. Based on these statistics, the essential attention
in juvenile crime is substantial because one way to prevent future adult offenders is to prevent
juvenile crimes.
1.2. Motivation
As a member of the Mayor?s Commission on Crime Prevention, and in response to the Mayor?s
charge to the commission, I evaluated and submitted a proposal to aim toward the high-risk
group, which is closely linked to truancy and is also a high indicator of juvenile delinquency.
This group involves students enrolled in the public school system who are subject to discipline
through the ?Out-Of-School Suspension Program.? These students, subjected to out-of-school
suspension, are usually marginal students to begin with. In most cases, they are simply sent
home without supervision or any structured follow-up procedures by the school system. The
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student attendance data from the Muscogee County School District in Columbus, Georgia,
provided the appalling realization that there were about 28,000 out-of-school suspension student
days that took place during year 2008 between the months of January through December. Of
180 student school days, the maximum acceptable absentee days a student can miss per year is
15 [36]. The out-of-school suspension period is considered as an absence albeit an involuntary
action of the student not coming to school. In general, the privilege of coming to school is taken
from these students usually due to their misconduct. When the student reaches the magic
number of 15 absences, he or she fails the grade and is unable to be promoted. The statistics
from the School District of Muscogee County also indicated that the most out-of-school
suspensions took place with students enrolled in the 9th grade level at ages 15 and 16. This is the
transition period from middle to high school where a young person is most vulnerable. These
data correlate with national statistics, SBB of juvenile delinquency where the most crime took
place at age 16 [39]. As a result, the Commission, School Board, and Mayor?s Office became
very alarmed and motivated to provide a reinforcement system for the problems associated with
the out-of-school suspensions.
Generally, students who are subjected to out-of-school suspensions are at home without
supervision possibly wandering around the neighborhood or local mall, become idle, bored, and
easily influenced by others on the street. Perhaps, they may join gangs, become unruly at home,
run away, and eventually may commit deviant behavior or other serious juvenile offenses.
Nowadays, 47 percent of our children live within a single parent home according to Families and
Lives by the U.S. Census Bureau [41]. Many of these children living with a single parent are left
alone and vulnerable to mischievous conduct since the mother or father is working elsewhere for
long hours to support the family instead of being home with the children. In addition, some
7
parents lack parenting skills or the education to make sound decisions, especially for those
impressionable at-risk youth, much less make decisions for a whole family [41]. Regrettably,
there is a lack of mechanism to examine the living situation or follow-up the progress of the
students who are subjected to the current out-of-school suspension program. Therefore, the
students are sent home or to the street with no evaluation of their living environment or the
conditions thereof, or any method to measure the impacts of such suspensions on their future
school performance, or means to measure the percentage of juvenile delinquency cases while on
out-of-school suspensions.
Considering the current 70 percent high school graduation rate in the state of Georgia and the
staggering number of the days in out-of-school suspension, the underlying impact of Out-Of-
School suspension is detrimental [36]. How many of those students just give up and never come
back or fall so far behind in their studies that they fail to graduate? In basic terms, the 70 percent
high school graduation rate reflects the 30 percent failure rate, and the loss of 30 percent of the
resources [36]. This overwhelming number of a 30 percent high school failure rate causes
apprehension for parents, educators, and lawmakers. In society as a whole, the responsibility lies
upon everyone to identify the causes of such a high percentage of student school dropout rate and
juvenile delinquency. It is everyone?s obligation to examine the problem and endeavor to find
solutions to maximize the student school success level and reduce the student population
committing crimes at an early age, thereby preventing them from circulating through the judicial
system for the rest of their lives.
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2. STATEMENT OF RESEARCH OBJECTIVES
School absenteeism is detrimental to students? self-esteem and reduces the opportunity for
graduation or being promoted to the next grade. It significantly affects their overall achievement
in their personal and professional endeavors, in addition to hampering their future employment
potential. Subsequently, students will fall behind their peers in the classroom if they continue to
miss school voluntarily or involuntarily such as a truancy or out-of-school suspension. Each
absence could affect the students unconstructively damaging both their grade point average and
their self-esteem. In addition, such a situation could result in an antagonistic environment due to
lack of self-esteem or could cause the students to be skeptical about self-worth which will
increase the likelihood that students may eventually give up, drop out, and/or be expelled from
school entirely. The current school system is neither mandated nor required to evaluate the
consequences or the impacts on the students due to out-of-school suspensions. Therefore, the
main theme of this research is to develop a model that can be used to evaluate the impacts of the
current out-of-school suspension program.
The model examines student school attendance patterns to assess the impacts of out-of-school
suspensions in relation to the dropout rate and overall student failure rate. This model can be
used as an evaluation tool to examine and investigate the overall impacts of the existing out-of-
school suspension system and identify a more positive direction which would promote the
success of the student in the future. Furthermore, studying the impacts of out-of-school
suspensions is necessary and vital for the juvenile criminal justice system also, since there are so
9
many young students on the street during the school day without any supervision, possibly as a
result of the existing OOS suspension program. Many of those students suspended may have
committed crimes during their out-of-school suspension period. The model can estimate and
measure the associated impacts of the out-of-school suspension in an attempt to improve the
program for the young population and to reduce juvenile delinquency and/or enhance the
rehabilitation program for young offenders. This research develops the evaluation model that
measures the impacts of unsupervised out-of-school suspensions in terms of student attendance
and success at school in order to examine the ongoing challenges of having high dropout rates
and criminal activities among the young population.
2.1. Goals and Problem Statement
The main goal of this research is to develop a student school attendance model in an attempt to
examine the impact of out-of-school suspensions, for the existing suspension program in
particular. The model can be used as a decision support system to a) measure the student school
attendance paradigms such as the in-school suspensions, out-of-school suspensions, and truancy
and b) evaluate the relationship between the out-of-school suspensions, the dropout rate,
graduation rate, and truancy. The students with high absentee records will most likely fail to
graduate and dissipate all the resources allocated for that child. The ultimate goal of this
dissertation is to provide a tool that can be used to evaluate the student school attendance
performances and overall impacts of out-of-school suspensions. In addition, the output of the
model can be used to encourage programs that reduce the student failure rate and the number of
juvenile delinquents, and subsequently guide the students toward a positive and productive future
at an early age.
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2.2. Objectives and Contributions
The objective of this research is to develop an evaluation model which can be used to quantify
student school attendance paradigms and thereby examine the student failure rate as a result of
out-of-school suspensions and the related impact that may negatively affect society as a whole.
Significantly, the outcome of this research can be used to help prevent at-risk students from
falling into the exclusive category of low achievers and high involvement in criminal activity
which may lead them to become life-long repeat offenders circulating through the criminal
justice system. The main purpose for developing SSAS is to respond to the need for analytical
tools that can be used to evaluate student school attendance paradigms and therefore facilitate
amendments to the malfunctioning programs and, hopefully, to reduce the number of our young
student population that may become victims of this unproductive system.
An impact on one child can have an impact on the whole. Providing a practical model to
evaluate the student school attendance paradigms would be a significant asset for school
administrators in amending programs to promote a higher graduation rate, ensure a successful
school year, enhance education, and facilitate a brighter future for the child. The fundamental
objectives of this research include:
1) Develop a simulation model that represents the student school attendance phenomenon
and the interrelationships of attendance states which include: in-school suspensions, out-
of-school suspensions without supervision, out-of-school suspensions with supervision,
truancy, and excused absences. The main purpose of this model is to simulate and
quantify student school attendance categories and assess the performance correlation
11
between attendance states during the school year. The associated attendance states are
quantified through a transition probability matrix which is used in estimating the
distribution of the student attendance states. Each student school attendance state is
represented by probabilities associated with student school attendance paradigms.
2) Model the student interactions with their friends based on ad hoc mobile networking
concepts. The model considers the random environment and the influence among the
students in the development process. The friendships are exhibited as the connections
between the students, and the influences of the peers are implemented through a learning
coefficient. The model provides a learning coefficient to integrate the influential factor
resulting from the student interactions. The dynamics of the connections between the
students are assimilated based on a random number and threshold value.
3) Evaluate the effects of supervised out-of-school suspensions and unsupervised out-of-
school suspensions, and their impacts on dropout rates and overall student school
attendances. The SSAS model examines and analyzes the overall student dropout rate
due to a result of the unsupervised out-of-school suspensions. This model can assist
school officials and community leaders in measuring and assessing the relationship
between supervised out-of-school suspensions and unsupervised out-of-school
suspensions in terms of the student attendance rate and high school graduation rate. The
model can assist in re-evaluating the current suspension programs, particularly the out-of-
school suspension program, which may contribute to the high student drop out rate,
truancy, or juvenile criminal activity.
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4) Verify and validate the model. Sensitivity analysis and heuristic testing are used to
verify the procedures and validate the data. The verification method for this research
employs the normal standard error procedure. To validate the model, this research
applies heuristic normal testing which compares the simulation results with actual student
attendance data from the Muscogee County School District, Columbus Georgia. The
model validation and verification process considers a) the student school attendance
information such as normal attendances, in-school-suspensions, out-of-school
suspensions, unexcused and excused absences, and make up days if any, b) the internal
dropout rates, and c) the high school graduation rate.
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3. LITERATURE REVIEW
3.1. Introduction
The social network represents the interactive behavioral structure between numerous actors such
as individuals, organizations, or other social entities. The social network involves interactions
through friendships, kinships, professional associates, and many others. Generally, the social
network studies actors and their interactivities and relationships in a given community. The
network is equivalent to complexity theory and systems theory which are interdisciplinary areas
where many traditional fields correspond. Examples include: a) anthropology that accentuates
social relationships, b) mathematics which provides structured algorithms and c) psychology
which concentrates on our social structure based on an individual?s perception [21]. The typical
social network analysis is the study of social structure and its effects. It conceives social
structure as a social network, that is, a set of actors (nodes) and a set of relationships (edges)
connecting pairs of these actors. Its basic premise is that knowledge about the structure of social
relationships enriches explanations based only on knowledge about the attributes of actors.
Overall, the social network consists of two major components: actors and relationships between
them. In order to understand the social network, many researchers have studied a pattern of
social relationships in a graphical representation to analyze and quantify them. The graph or
mathematical representation was used to better understand and evaluate behavioral patterns of a
given community. The actors are described by the node which also represents various entities
such as groups, individuals, or organizations. The edge between the nodes, which is the line that
14
connects the node, represents the relationship between the actors where it can describe
dependence, information exchange platform, control, cooperation, and competition [6, 21, 30].
The students? school attendance performances are the result of daily influences from multiple
factors. Whether the student comes to school each day is a purely random and unpredictable
discrete event. Events, whether the student comes to school and is subject to suspension in-
school or out-of-school, are purely discrete outcomes. This research considers the unique
properties of the random events and employs the stochastic process which effectively represents
the characteristics of randomness in the given environment. In brief, stochastic is a synonym for
random which involves probabilistic behavior and conjecture or guesswork. The process of
stochastic is to index the unknown parameters into real numbers using a probability distribution.
Furthermore, the possibility of the student coming to school tomorrow is based on today?s
activities. Therefore, this study applies the Markov Chain memory-less process to present the
independence from past history of student school attendance paradigms during the school year.
The attempt to evaluate student school attendance is merely based on guess work, especially
concerning the complexity that is associated with influences from peers. This chapter introduces
the relevant topics in modeling techniques for the student school attendance phenomenon of the
SSAS. The topics included in this chapter are: 1) social networking, 2) a brief description of
graph theory, 3) the stochastic process, and 4) the Markov Chain memory-less properties.
3.2. Social Networking
Social network theory is based on the intuitive notion that the patterns of social interactions are
essential elements to each individual who exhibits certain behavior. It can be stated that the
influence and behavior of an individual heavily depends on how he or she interacts with others
15
and how each is tied to a specific social network. In addition to individual behavior, network
theorists believe that the success or failure of societies or organizations often depends on the
internal interaction patterns among actors. Besides the theoretical essence, social network theory
is also characterized as a distinctive methodology encircling the pattern of certain behavior [6].
Hence, the social network encompasses the world which consists of actors and the connections
between them. The actors can be represented as nodes and the connectivity from one node to
another is the relationship between them. This connectivity between the actors and the ties
between the nodes represent a type of relationship such as kinship or friendship between the
actors, and they describe a particular pattern of action that an actor performs. Subsequently, the
different types of ties will generate different degrees of social networks and represent the
different types of functionality [19, 21].
The study of social networking begins with an understanding of the structure which is the way
the society is organized and mapped into rather predictable relationships. Mainly, the aspect of a
social structure considers three components: a) actors, b) the actions of the actor, and c) the
interactions of the actors. Social actors represent the static concept relatively with emphasis on
issues such as position, roles, and status. The actors, as individuals, are intermingled in a
particular social platform, hence, their actions and activities are heavily influenced by the
connections among them. Generally, social interaction can be defined as the way in which
people respond to one another. To some degree, the patterns of these interactions are
independent of the individual who creates the pattern of certain behavior [19, 21, 30].
In the past, most research on social interaction mainly focused on how individuals actually
communicated with one another in a given specific environment. These studies concentrate on
16
such issues as the patterns of interaction, the rules and ethics to guide the interaction, the
motivation for the way of interaction, and the effect of those interaction patterns which impact
the behavior as an individual and the performance as the group. These issues begin by
questioning what might be the interaction pattern in a specific social setting, thereby addressing
one of the main themes of this research: understanding the dynamics of social interaction [19].
Social network activity analysis incorporates the social dimensions and the challenges of the
adequate behavioral underlying theory. The paradigm of the social network mainly concerns
understanding on how social structures facilitate and constrain opportunities, behaviors, and
cognitions. Therefore, it involves designing and implementing a reliable and quantifiable data
collection method, and performing an informative analysis of the main behavioral elements [6].
Theoretically, the core values of social networks are the attributes and the structural knowledge
of social interrelationships and extended rationalizations solely based on information about the
characteristics of actors. Behavior is defined and addressed by attributes, constraints, and
opportunities which have a personal as well as a social aspect. Hence, the structure of social
characteristics which represents the interactivities among the members in a given social network
is the core source of addressing behavioral patterns.
Furthermore, the social network analysis consists of the methods and techniques for data
collection, mathematical and statistical analysis, and a visual and graphical description. Usually,
this process is referred to as the social network analysis, an essential procedure when studying
the cohesiveness of the social network. Since the main characteristic of social activity
constitutes the social dimension, an analysis of its performance is based on the vitality of the
personal interactions and the one-on-one connections among actors. The complementary
17
perspective element for the social network analysis extends to the attractiveness between the
actors at their destined social activity patterns. Conceptually, this interaction among the actors
generates the accessibility to one another embedding the influential characteristics. There are
intrinsic challenges in terms of the social network analysis due to unspecified social dimensions
and intangible human behavior. The stochastic process effectively formalizes the social
interactions by its unique properties; therefore, the behavior or the certain structure of a
particular social network can be quantified and assessed [19, 21, 29].
3.3. Basic Graph Theory
In order to understand the social network, many researchers studied patterns of social
relationships in a graphical representation to analyze and quantify. The graph or mathematical
representation was used to better understand and forecast such patterns for unknown futuristic
probability. In most cases, the social network is described in either a directed or undirected
graph. Using the graph, G (Node, Edge) the node represents the social context or community
which has the embedded entities. The edge represents the contact, interaction, influence, or
communication from one entity to another entity within the given community. In turn, social
scientists study and analyze the corresponding graph to forecast and understand the social
patterns, property, and behavior of the community. The size of the social network, therefore, can
be determined by the number of nodes which can be very large representing a massive
community such as the Internet or very few representing a small circle of friends. Each social
network also has its own property embedded in the node that represents the application area or
the specific type of the social community. The first application of digraph to describe the social
community was proposed by Leinhardt in 1977. This idea of graphical representation became
18
popular in other research areas such as e-learning modelers and network designers [19, 29, 30,
34].
A) B) C)
Figure 1. Examples of Undirected Graphs of Social Network: A) Complete connections, B)
Partial connections, and C) Hybrid connections
Overall, a social network graphical representation provides specific fundamental structural
properties which distinguish it from other types of networks or graphs. Designing the graph
leads to an understanding of real life in a community which in turn can be used to study the
correlation and overall changes in a given period. The graphs included in Figure 1 are the basic
types of graphs which describe the connectivity among the nodes and the characteristics of the
given community through the line which reflects the linkage between the nodes: A) represents
the community that has complete ties among the actors, B) represents partial connections and the
partial separations between the actors, and C) the actor completely isolated from the community.
Graph C features the isolated node which could be considered in a situation where a new
member enters into a community social network. In this case, as presented in the graph, an actor
may not have any connection within the community and is noted as an isolated node.
19
By examining these structural properties of the graph, the overall patterns of the particular social
network can be analyzed and evaluated. Most often, the graph theory can be used to compare the
two distinct communities to predict and study the better performance. One of the main
properties is the type of connectedness between the actors such as centrality of actors having
dense degree, connectivity between the pair indicating reciprocity, links between the source and
actor stating reach-ability, and the sub-division of actor having a strongly connected component.
Mitra presented the extensive form structural definition and the study on cycles and nested cycles
in his paper ?Design of a Data Model for Social Network Application.? Mitra stated that to
study the degree of cohesiveness in a community, the sub-division of actors into a cycle of
nested cycles is very vital for a structure of the community because the size, number, and
connections among the sub-division in a community are key factors for understanding the
characteristics of the network as a whole and their likely behavior. In the paper ?The dynamics
of social balance,? Moody stated, ?While we revel in the diversity of social settings, a primary
motivation for social theory is to explain common features across settings and account for social
differentiation endogenously? [20]. Generally, social scientists focus on the connectivity of the
network and how the structure is molded into smaller groups over an identified period. This
fundamental idea has been applied to other settings including the referral system and web
community [21].
3.4. Stochastic Procedures
Stochastic is a synonym for random which involves probabilistic behavior and conjecture of
guesswork for the entity or the system. Therefore, the stochastic process is known as a random
process that deals with possibilities and probabilities to study how future process might be
20
revolutionized in a given time using a probability distribution. Application of statistical
probability models has been widely used by many researchers to understand the nature of social
network structures since the late 1930s [33]. The stochastic statistical model has been very
popular as a research method in computer science and the fields of engineering, healthcare, and
social study areas because of its unique characteristic which can be used to simulate the
performance of a dynamic system that is subject to uncertainty.
3.4.1. Property of Random Event Process
By definition, Heyman stated ?A stochastic process is a collection of random variables that are
all defined on the same probability space and indexed by a real parameter.? The similar random
variables that are defined in a given sample space are collected over time in stochastic process.
Stochastic process is formally described as {X(t); t ?T}, where the X(t) is X-process, the valued
random variables at time t which indicates time parameter and T indicates the range of times
parameter set. When T is finite or countable, T represents discrete-time stochastic process with
{0, 1, ?}; otherwise T is continuous-time stochastic process with [0, ?) and the formal
description is {X(t), t?0}. State space, S, takes all the possible values of the random variable that
the X(t) can undertake. The process is discrete stochastic when S contains countable finite
numbers S = {0, 1,?., i} and the T is a set of integers X1, X2,.?, Xn; otherwise the process is
continuous stochastic. If X(t) = i for any time t?T and for any state i?S, then the process is in
state i at time t [2, 9, 13, 18, 22]. The detailed properties of discrete and continuous stochastic
processes are outlined in the following sections.
3.4.2. Stochastic Process and Exponential Distribution
21
The student attendance events, whether or not the student coming to school, are discrete events.
In order to represent the school attendance state of a student over a given period, a Markov Chain
is suitable to model the process.
3.5. Markov Chain Phenomenon
Markov Chain was developed by a Russian mathematician Andrey Andreyevich Markov who is
well known for work on the theory of stochastic processes. Markov Chain method has been
widely used because it satisfies the property of a linear equation system which is amenable to
numerical solution methods based on the finite and limited probability distributions [4]. A
discrete Markov Chain is a stochastic process with a discrete state and discrete time that has a
probability distribution of the next state of the process which depends solely on the last state of
the process. Therefore, the future state only depends on the current state and the process does
not consider the past history. The stochastic process {X(t); t?T} is a Markov process with given
integer i with time intervals of T where t1 0 where n ? 0. For some instances, if n=0, every
state is accessible from itself where one can move into all other states. These phenomena
provide the ability for a state i to communicate with state j and if every pair of states in a given
sample domain communicates with each other, the states are called the communicating class.
There are three properties for communicating states: a) a self-communicate where P0ij = P{Xo =i
| Xo = i}=1; b) if state i communicates with j, then j also communicates with state i where the
exchange of information is possible from one to another; c) if state i communicates with j and j
communicates with k, then i communicates with k also creating triangular connectivity and
information exchange platform.
The state could be a recurrent state if the process can go back to the original state. For example,
if state i is accessible to state j and j is also accessible to i where the process can go back to state
i, then state i is a recurrent state that the process can revisit. However, the state is transient if
state i is accessible to j but j is not accessible to i where the process can not go back to state i,
23
and state i is transient because the process can not be revisited. The formal representation is to
set Ti as the random variable that first return time to state i then, T i = inf { n ? 1: Xn = i | X0 = i}
with Pr(Ti = ?) > 0, state i is the transient state. The state is called an absorbing state if the
process never leaves the state. The state is recurrent if
0n
?
=
? p(n)ii = ? and if state i is unable to
leave this state, then the state i is an absorbing state iif pii = 1 and pii = 0 for i ? j. In any instance,
every state is either transient or recurrent and not all states can be transient in the finite-state
Markov Chain process. In the finite-state Markov Chain, if all states are irreducible then all the
states are also recurrent. The sufficient scenario for the Markov Chain is to have all states to be
accessible to each other communicating where pij >0 for all i and j. However in the real world
there is no guarantee such a system exists.
A Markov Chain also provides the periodicity where state i has period t if any return to state i
takes place in t time steps. Periodicity is formally represented as t = gcd {n: Pr (Xn = i | X0 = i)
> 0} where gcd refers to ?greatest common denominator.? In a given period t, the process may
be unable to reach the state in t steps, thus the resulting state is called aperiodic which occurs
when t =1, otherwise the state is called periodic where t >1. Every state within a communicating
class should have the same period. Ergodicity occurs when state i is aperiodic and positively
recurrent. The finite state irreducible Markov Chain is ergodic if the states are aperiodic [1, 2,
18, 22, 31].
3.5.2. The Long-Run Behavior of Finite-State Markov Chain
In addition to those properties presented in the previous section, Markov Chain provides another
significant feature, steady state probability. Remarkably, the Markov Chain process produces an
24
identical probability distribution after a large number of iterations. This is also called limiting
probability where the system remains in a certain state after a large number of iterations which is
independent of the initial probability, p0. The unique property of a long-run finite state of
Markov Chain is formally stated as
?j =
0
M
i=
? ?jpij for j = 0, 1, ? M where
0
M
j=
? ?j = 1.
Here ?j are the Markov Chain steady-state probabilities which are also called the stationary
probabilities. After a rather large number of iterations of the transition probability, the system
will be in particular state producing the identical probability distributions, ?j that are independent
of the initial transition probability distribution.
The steady state has M + 2 equations for M + 1 unknowns in order to satisfy the unique solution.
Having one extra equation, the system can utilize it as redundant which can be disregarded
leaving the final equation to produce the appropriate probability distribution. A simple
illustration of steady state equation for four unknowns using five equations can be expressed as
follows:
0 0 00 1 10 2 20 3 30
1 0 01 1 11 2 21 3 31
2 0 02 1 12 2 22 3 32
3 0 03 1 13 2 23 3 33
0 1 2 3
? ? ? ? ?
? ? ? ? ?
? ? ? ? ?
? ? ? ? ?
1 ? ? ? ?
p p p p
p p p p
p p p p
p p p p
= + + +
= + + +
= + + +
= + + +
+ + +=
In this example, the ?0, ?1, ?2, and ?3 are the four unknowns having five equations. Solving the
above equations, one can obtain the values for ?0, ?1, ?2, and ?3 rather easily. One important
aspect of the steady-state probability is that if i and j are a recurrent state but belong to different
25
classes, the probability of finding the process in state j is zero; formally p(n)ij = 0 for all n [18, 22,
33].
The steady state probability ?j can be used to compute the long-run average value per unit of time
called expected average value per given unit. If there is the value function X associated with a
given example, the expected average value can be computed by using the equation
E(x) =
0
M
i=
? ?jX(j).
For more complicated problems where the state associated with a value that is also affected by
the random variables, then the equation must consider an additional parameter y where X(At-1,
Sti), At-1 represent the number of items, t for day, and S is the sale demand, for example. The
total value for day t is a function of At-1 and S where X(At-1, S i). Expected value can be computed
by using the following equation:
E(x) =
0
M
i=
? yjX(j) where yj = E[X(j, S t).
Therefore, the central concentration of the given distribution can be measured by computing the
expectation value E(x) to evaluate and analyze the problem and understand the characteristics for
the given domain or system [9, 10].
3.6. Chapter Remarks and Summary
In a social network, actors represent the static concept with emphasis on issues such as position,
roles, and status relatively. The actors are intermingled in the social network setting and their
actions and activities are heavily influenced by the characteristics of the connections and the
density of connectivity among them. Generally, social interaction can be identified as the way in
which people respond to one another and it describes the characteristics of influence upon their
26
neighbors. The dynamics of the interaction among the actors also needs to be considered, albeit
the environment encompasses randomness, by means of probability distribution with a learning
coefficient fitness value. Nowadays, the social network represents every day life from the small
circle of colleagues to the large group web community where there is a massive multiplayer
group thereby forming the connections together.
A stochastic process can effectively model a social network over time since it comprehends the
randomness of the system. Generally, a stochastic process describes the social network through
a mathematical representation formally. The attendance state whether the student comes to
school or not, is affected by random events. Therefore, a stochastic process can be used to
represent student school attendance over a given period. The connections between actors are
also affected by random events. The connectivity can be integrated dynamically based on a
fitness value and a random number. The stochastic Markov Chain considers the discrete state
and discrete time processing through linearly enabling remarkable properties such as
irreducibility, periodicity, and ergodicity. The Markov Chain process is one of the most
powerful tools available to construct a model that consists of a discrete stochastic process. The
Markov Chain properties are very useful because they could yield results for both the steady
state of the system and time dependent evolution of the system. Therefore, the Markov Chain
process effectively assimilates the random variation for student school attendance states.
Overall, this chapter provides supporting materials in rationalization of the modeling techniques
for the student school attendance phenomenon.
27
4. MODEL FORMULATION
In brief, the SSAS model considers the complexity of the social network and attempts to describe
the student school social network; it applies the properties of graph theory to represent the school
attendance states of the student. The modeling procedure considers the random environment and
therefore employs the stochastic characteristics and the memory-less property of the Markov
Chain to assimilate the student school attendance paradigms. In essence, the model rationalizes
the random environment and the intricate patterns of the student-attending-school phenomenon.
This chapter provides the modeling techniques and the formal representations of the student
school attendance paradigms that include: the model assumptions, parameters and variables,
attendance state diagrams, mathematical representations, computation method for the student
attendance probabilities, dynamic connectivity, and the interactivities of the students.
4.1. Introduction
In the development of the SSAS, the procedure mainly concentrated on two areas: a) the student
attendance state and b) the interactions with their peers. This research investigates the out-of-
school suspension state closely and further categorizes it into two distinct states: a) supervised
out-of-school suspensions and b) unsupervised out-of-school suspensions. Therefore, the SSAS
model provides six different students? school attendance classifications: 1) normal class
attendance, 2) in-school suspension, 3) supervised out-of-school suspension, 4) unsupervised
out-of-school suspension, 5) truancy the unexcused absence, and 6) an excused absence. Figure
28
2 below describes the general overview of associated input or output values and parameters of
SSAS model.
In order to model the multi-dimensional student school network, the SSAS considers undirected
edge where the direction of the influence is disregarded among the peers.
Inputs Parameters Outputs
Number of Students Failure Rates
Number of Attendances Success Rates
Figure 2. Overview of Inputs, Outputs, and Parameters of the SSAS
The model assumption in the subsequent chapter introduces the detailed reasoning behind this
concept. Applying the graph theory, the model presents nodes which describe each student with
his or her own attendance probability distributions coupled with learning coefficient and
threshold value for connectivity. The nodes are able to communicate with each other in the
school social network, since the students are not prohibited from having contacts with their peers
at their school, and therefore they have knowledge about each other. Friendships are described
in the model as the connections between the students.
Furthermore, the SSAS considers a learning coefficient in realization of the influences through
friendships among the students that may affect the overall behavior of the student including
school attendance. The SSAS model employs peers? influences by providing a fitness value for
the learning coefficient. The initial probability distributions of the student school attendance are
either provided or generated according to the input or the output probability distribution table. In
order to update the next day probability of the student attendance, the model calculates the next
Transition Probabilities
Interaction Probabilities
Steady States Probabilities
29
day attendance probability based on four factors: 1) the student?s current probability, 2) the
student?s learning coefficient, 3) the connected peers? probability, and 4) the learning coefficient
of the connected friends. Subsequently, the next day student attendance state is constructed
based on the memory-less property of the Markov Chain applying today?s student attendance
state as tomorrow?s student attendance state.
4.2. Model Assumptions
A multi-faceted social network encapsulates perplexing human behavior and therefore it is very
difficult to understand since the subject matter is intangible, whether it represents a small circle
of friends or the large Web community. In an attempt to understand and assess the student
school attendance paradigms, this research selects two main areas to overcome the complexity of
the student school social network: a) the state of the student school attendance and b) the
interaction with their peers accommodating the peers? influential factor. In general, young
children are more prone to develop friendships with other children without considering the other
person?s class, per se. They tend to look at other people as to who they are and not what they
are. The status and the background of the person seem not to matter. This can be seen easily on
any play ground or in any nursery. Most of the time, young children will play with other
children who are physically closer to them regardless of their status. In comparison, adults
typically pursue a relationship based on a personal agenda or some type of motivation for a
connection with others, mostly for their own benefit, comfort, similarity, and many other
reasons. Therefore, there will be an apparent direction between the relationships whether it is
one-way or both-ways and this direction can be described either positively or negatively.
However, generally, young children do not have an agenda in mind for pursuing a relationship.
30
Young children have a tendency to develop relationships with others in the surrounding area and
can be easily influenced by those people who are closer to them, especially from closer friends.
The direction of the relationship, therefore whether one way or both ways or negative or positive,
is rather ambiguous. For this rationalization, the development of SSAS applies the undirected
graph theory in describing the relationship between peers as presented in Figure 1. Nonetheless,
the relationship between the school attendance states, the direction from one attendance state to
the other, can be easily identified as to whether or not a student can go from normal class state to
in-school-suspension or vise versa. Therefore, the SSAS applies the directed graph theory to
describe the relationship of the student school attendance states and undirected graph theory for
the connectivity with their peers. The directed graph procedure between the attendance states is
presented in section 4.3.1.
As introduced previously, in order to model the state of student school attendance paradigms and
the random nature of the friendships among peers, this research employs the Markov Chain
process to generate the discrete attendance states and integrate learning coefficients to
accommodate the influence among peers. The model also employs the random number with a
threshold value in connection range to determine whether or not connectivity exists between the
peers. Overall, the development procedure of the SSAS model considers the unique
characteristics and properties of the Stochastic Markov Chain Process and assumes the
following:
? At the first day (t=1), every student starts at the current state S0.
? The student can only be at one state at a time considering the discrete state and discrete
time events.
? The length of student?s attendance or absence from the class is mutually independent.
31
? After an absence, the student returns to the normal class with a certain probability.
? The true underlying process of the length of student absences is exponentially distributed.
? The connectivity between the nodes is determined based on probability distribution and
the learning coefficient.
? The student school attendance states are governed by a Markov Chain.
? The students are connected with their peers through an undirected graph.
? The connectivity between students is determined based on probability distributions and
the learning coefficient.
? The student school attendance state and the interactions with their peers are the main
premise.
4.3. Student School Attendance Model
4.3.1. Background Introduction
Most of the current school systems categorize student school attendance and absentee
characteristics into five areas or classifications: 1) normal class attendance, 2) in-school
suspension, 3) out-of-school suspension, 4) unexcused absence, and 5) excused absence. The
unexcused absence represents truancy where the student willfully misses school without any
valid reason or excuse. As presented, the SSAS model further indentifies student school
attendance categories into six classifications (Si), dividing the current out-of-school suspensions
into two areas: supervised and unsupervised. The SSAS model classifies a total of six student
school attendance states as:
S0: student with a normal in-class status (Normal),
32
S1: student with in-school suspension (IS),
S2: student with out-of-school suspension with supervision (SOS),
S3: student with out-of-school suspension without supervision (OS),
S4: student with unexcused absence (Truancy, T), and
S5: student with an excused absence (Exe, E).
The existing out-of-school suspension program removes from the student the privilege of coming
to school without any support mechanism for their academic work or follow-up procedure by the
school or affiliations. Apparently, the students subjected to out-of-school suspension will fall
further behind under the current suspension program. In an attempt to evaluate the impacts of
this apparent problem, the SSAS model distinguishes between the unsupervised out-of-school
suspensions (OS) and proposes a new attendance category, the supervised out-of-school
suspensions (SOS).
On any given school day, an ordinary student has the possibility of being absent from school,
either voluntarily or involuntarily. In general, the most typical involuntary absence or excused
absence could be the result of natural causes such as sickness, personal issues, field trips with the
class, a death in the family, and many others. Conventional and habitual causes of voluntary
absences, students who have unexcused absences, might include students trying to avoid school
work, peer pressure concerning their appearance or relationships, conflicts with a teacher, or
simple laziness, which all fall into the category of truancy, being absent from school without a
valid excuse. The SSAS provides a state diagram in the following section that describes the
interrelationships of the student attendance states presenting the six distinct categories of student
school attendance paradigms.
33
S0
S1
S2
S3
S4
S5
4.3.2. State Transition Diagram of the Student Attendance Paradigms
In the graphical representation in Figure 3, the six numbers of nodes indicate six possible student
attendance states adapting the five states from the current school system and employing a new
proposed state, the supervised out-of-school suspensions.
P1,1
XP0 P1=1-P1,1 P2,2
X1,P0
X2,P0 P2=1-P2,2
P0=P0,0 P3,3
X3,P0
P3=1-P3,3
?4,P0
P4,4
?5,P0 P4=1-P4,4
P5,5
P1=1-P1,1
Figure 3. State Diagram Representation of Student School Attendances
34
As noted in the assumptions, any student can fall into any other state on any given day
throughout the school year. Application of the directed graph for attendance shows the overall
performance of the student attendance can be evaluated and understood by examining the
characteristics such as the patterns of the students? school attendance. In Figure 3 below, the
graph displays a corresponding directed graph that explains the interrelationship between the six
student school attendance states. Each node represents each attendance state and contains the
state probability for each student. The directed edge between the nodes describes the flow of
attendance between the states. In this figure, the process begins with state S0, the initial state
supporting the assumption that the students will come to school on the very first day of school.
Also this is the only state which has the access to all the other five states as the arrow line
illustrates.
4.3.3. State Probability Distribution
State S0 in Figure 3 represents a typical student attending a normal class at school. Today, the
students in this state attending class may also continue to be in this state tomorrow which is
represented by P0,0. In general, the students in state S0 attend classes and conduct a satisfactory
behavior by continuing the normal school performance. Nonetheless, students who attend school
on a regular basis possibly could also fall into one of the five other states on any given school
day. The interrelationships among six attendance states are illustrated in Figure 3 by the directed
arrow line going from state S0 to all the other five attendance states. The top three states S1, S2,
and S3 in Figure 3 that are enclosed in the small dotted box, are subject to suspensions either in-
school or out-of-school. In these three states, the student is subject to the decisions (X) by the
school administrators concerning his or her suspension status: the duration or the types of the
35
suspension. When students are suspended, they fall in among three states S1, S2, or S3 with
probability X1P0, X2P0, and X3P0 respectively. The Xi denotes the decisions that are made by the
school officials in determining the suspensions state Si where i= 1 (IS), 2 (SOS), and 3(OS). The
next day, as illustrated in Figure 3, students in these three categories can either remain in their
suspension state with probabilities of PIS,IS, PSOS,SOS, POS,OS, respectively, or go back to their
original class, the state S0 upon the completion of their suspensions. The value of Xi is dependent
on school officials to determine the suspensions: the X1 for the in-school suspensions to compute
X1PIS, X2 for the out-of-school suspensions with supervision to compute X2PSOS, and X3 for the
out-of-school suspensions without supervision to compute X3POS.
Furthermore, the students can also fall into the last two states, the S4 and S5 which refer to
truancy, the unexcused absence and excused absence. The S4 state is the truancy state where the
student is absent from school without any excuse. The S5 state is the state where the student has
a valid excuse for the absence. Most of these student school attendances depend on their own
circumstantial preferences where the ?4 represents the student making the decision for truancy
with the probability ?5PT. The ?5PE represents the probability that the student is absent from the
school with a valid excuses. The students subjected to states S4 and S5 also have the probability
of not returning to school for an extended period after a prolonged suspension period and make
excuses or simply avoid coming to school based on individual choices. Therefore the P4,4 (PT,T)
represents the probability of the student repeating the truancy the next day and P5,5 (PE,E) for the
student continuing the excused absence the next day.
In SSAS model, each student is represented by a transition probability matrix P. The transition
probability matrix P between the transient states is represented by setting state S to be a Markov
36
Chain process in the sample space X = {Si: S0, S1, S2, S3, S4, S5}. The stochastic process X=
{S(n): n ? 0} represents the discrete time process with discrete state spaces {Si: S0, S1, S2, S3, S4,
S5} describing the six distinct student attendance state paradigms. The discrete time process with
time index t = 0, 1, 2?, represents the number of days. The transition probability matrix P, is
P =
1,1 1,2 1,
2,1 2,2 2,
,1 ,2 ,
0
j
j
i i i j
p p p
p p p
p p p
? ?
? ?
? ? ?
? ?
? ?
? ?
L
L
M M ML M
L
, where the row i represents the current state and the
column j represents the state corresponding to the next day in time t. Each row has the property
j X?
? pij = 1.
The student?s attendance state for tomorrow is a consequence of today?s attendance state. The
matrix of transition probability at time t is P(t) which represents that the probability P is a
function of time t. For example, the students? out-of-school suspensions state will not remain for
more than a specific period. The probability of students? changing their state from out-of-school
suspension to attending class will vary everyday through the school year for all students. In
addition, a factor of influence from peers which affects the student attendance varies over the
time during school year. Therefore, the events of student school attendance are functions of the
time given school days.
4.4. The Student Interaction Model
As stated in the assumption, the model considers that all students will come to school on the first
day of the school year. In general, students will connect with one another developing friendships
that may affect the outlook of the student, including school attendance. For this reason, the
SSAS model implements the connections between the peers implying that the students will be in
37
contact with one another once they come to school. In general, once the students develop
friendships, those friendships will continue for some reasonable time which raises a concern in
applying the same value for the connectivity throughout school days from the very first day of
the school to the very last day of the school year. Instantiating the connection between peers on
the first day of the school will not be applicable to the remaining days of the school year, since
the students have developed friendships which may continue for sometime. Therefore, in this
chapter, the SSAS provides three different approaches to investigate this phenomenon.
Another concern is the interactivity between the peers and the resulting influences thereof. Once
students develop friendships with one another, there will be influence factoring between them.
As explained in the previous chapter, the direction of the influence between students has been
disregarded based on the assumption that children tend to develop friendships with closer people
regardless of their background. An attempt to accommodate the influence between the peers, the
SSAS presents mechanisms to compute the future transition probabilities of student school
attendance paradigms employing a learning coefficient.
4.4.1. The Graph Representation of Student Interactions
The SSAS model assumes the undirected graph theory for the student interactions, not specifying
the directions of the relationship based on the ambiguous nature of characterizing their
friendships. In addition, the student relationships are described using the matrix representation.
For example, the following table includes 6 students and explains their relationships.
Illustrations in Table 1 matrix, the values 1, 0, -1 indicate connections, no connections, and self-
connections respectively among 6 students. Referring to student number 4, this student is
38
connected with 3 other students, student number 1, 3, and 6 which is notated with 1 but not
connected with the other 2 students, student number 2 and 5 which is notated with 0.
Student 1 2 3 4 5 6
1 -1 1 1 1 0 1
2 1 -1 1 0 0 1
3 1 1 -1 1 0 1
4 1 0 1 -1 0 1
5 0 0 0 0 -1 0
6 1 1 1 1 0 -1
Table 1. The Matrix Representation of the Relationships of 6 Students
Student number 5 represents the isolated student who is not connected to any other students in
this example. The matrix is illustrated using the undirected graph in Figure 4 below.
Figure 4. Undirected Graph Representation of 6 Students and their Relationships
In Figure 4, x indicates the student and y is the corresponding probability P0 of the student.
0.8
0.6
0.4
0.2
1 2 3 4 5 6
Student
1
y
x
Pro
ba
bil
ity
39
4.4.2. Assessment of the Interaction between Students
4.4.2.1. Initial Student Attendance State Probabilities
The model formulates the initial probability distributions of the student school attendance states
in two ways: 1) utilize the pre-arranged input probability distribution table or 2) generate the
probability based on the pre-arranged output lookup table. The SSAS imports the external file
that contains the pre-arranged input probability distribution or to simulate the relevant attendance
probability distribution based on the pre-arranged probability output lookup table.
4.4.2.2. Distribution of the Student Attendance States
As stated, SSAS employs the one-way data table to store the subsequent values throughout the
process. In determining the attendance state of the student, the model generates the random
number and compares that number to the first row of the cumulative probability distribution. As
presented in Table 1 that explains how to compute the transition probabilities, the very first state
S0 is the only state that is reachable from any other state. The reasoning behind the assumption is
that students, whether they are suspended (S1, S2, S3) or absent (S3, S4, S5) from school, will go
back to their original class upon returning to school from suspension or absence. For example, if
the student is suspended, either in-school suspension or out-of-school suspension, the student has
the right to go back to the same class upon completion. Moreover, if a student is absent, he or
she will attend their original class once they decide to come back to school instead of going
straight to suspension. Therefore, the SSAS determines the student attendance state considering
the random number and the cumulative probability distribution of the student.
40
4.4.2.3. Interactivity of the Students and Learning Coefficient
In calculating the next day attendance probability, the model considers four significant areas: 1)
the student?s current probability, 2) the student?s learning coefficient, 3) the connected peers?
probability, and 4) the learning coefficient of the connected friends. Therefore, influential
factors between the peers are implemented throughout the simulation. The information about
each student, their current attending state probability distributions and the learning coefficients
of the students are assumed known. The following equation provides the mechanism to compute
the next day attendance probability of the student k where ? is the learning coefficient and
represents the realization of the influences through friendships among the students which may
affect the overall behavior of the students including their school attendance.
Pk0(t+1) = (?* Pk0 (t)) + (1- ?)* (Average of Pk0 (t) of connected peers)
Specifically, the next day probability P0 of the student attendance state S0 depends on the current
state probability with the learning coefficient subtracted (or added) by the average probability P0
of the linked friends. The model computes the weighted-average probability of linked friends as
(1- ?)* (Average of Pk0(t) of connected peers). And if their weighted-average probability is
greater than the student?s current probability ?*Pk0, which is interpreted as a positive influence,
then the student?s current state probability will increase. However, if the average probability of
peers is less than the student?s current state probability which reflects a negative influence, then
the student?s current probability will decrease.
Overall, student friendships are represented as undirected vertices and the links between the
nodes are instantiated dynamically based on a random selection. The node represents a student
41
and characteristics associated with that student?s school attendance. The undirected link between
the nodes represents the inter-relationship with other students in the school social network.
Formally, it is defined as Gt = (Nt, Lt) where G refers to graph, N for node (the actor) and L for
the link between the nodes. SSAS employs the learning coefficient (?) to model the level of
influence between peers.
4.4.3. Three Dynamic Modes of the Connectivity between the Peers
The linkages between students are instantiated based on the random selection with a threshold
value, ?= 0.5 specifically for this research. SSAS generates random numbers through the Monte
Carlo Simulation to determine the connectivity between students. In essence, the connections
are instantiated based on 1 or 0, either yes or no respectively. Taking into account the initial
connectivity between the students on the first day of school, SSAS generates the random number
and provides the threshold to instantiate the connectivity between the peers. The SSAS model
assumes that, if the random number is greater or equal to 0.5, there will be a connection between
the students which will be displayed in the matrix as 1 and 0 for the disconnection. Throughout
the model simulation process, the resulting connections of the students are presented in two
different ways: a) displayed on the chart and b) displayed in the matrix form to verify and
validate the correctness of the connectivity among students. Fundamentally, the link or
connectivity between students exists if the random number is greater than or equal to 0.5.
Otherwise they are disconnected and there is no link which will be displayed on the chart
between the nodes. SSAS produces the charts and matrices to illustrate and verify the
connections between the peers throughout the simulation process of the student school
attendance phenomenon.
42
The next day connection between the peers considers the stronghold and bond as a result of
friendships. In general, if students have a tight friendship today, to a greater extent they will
have that strong relationship tomorrow. Subsequently, the connectivity no longer can be
rationalized with a threshold value equal to 0.5. Therefore, the SSAS updates the connectivity in
three different modes. The three modes for those dynamics are: a) Median Connectivity, b)
Virtual Connectivity, and c) Hybrid Connectivity.
a) The median connectivity refers to the procedures introduced previously which is the main
linkage process of the SSAS model. This method generates random numbers and generates the
link between the nodes that are determined by the threshold value ? formally as,
1 if ? ? 0.5, ??.. the connectivity between student i and j exists
l(i, j) =
0 otherwise, ?...... the connectivity does not exist.
b) The virtual connectivity targets the connections among peers at ? = 0.95 from the very first
day of school to the end of the school year. Since the fitness point for this scheme is rather high
at 0.95, it virtually links everyone in the network. The virtual connectivity can be presented
formally as,
1 if ? ? 0.95, ??. the connectivity between student i and j exists
l(i, j) =
0 otherwise, .?? the connectivity does not exist.
c) The hybrid connectivity combines both methods, median and virtual connectivity.
Connections between peers are implemented using both processes. The first day of school
student connectivity proceeds through the median connectivity scheme. Subsequently, if there is
a connection between two students, the virtual connectivity scheme is applied. Hybrid
43
connectivity promotes the idea of preserving the existing friendships between the students. If
students are connected, the friendships will continue and there would be little possibility of
breaking up daily.
4.4.4. Interaction Probability Computation
4.4.4.1. Initial Student Attendance State Probabilities
As introduced, the SSAS considers the learning coefficient in realization of the influences
through friendships among the students which may affect the overall behavior of the students
including their school attendance. The SSAS model employs the peers? influences in regard to
attending school by providing the threshold value for the learning coefficient at a = 0.7. The
model formulates the initial probability distributions of the student school attendance states in
two ways: 1) read the pre-arranged input probability distribution table or 2) generate the
probability based on the pre-arranged output lookup table. The SSAS provides the algorithm to
read the external file that contains the pre-arranged input probability distribution or to simulate
the relevant attendance probability distribution based on the pre-arranged probability output
lookup table. The process stores those values in a separate data table for future reference.
4.4.4.2. Determination of the Student Attendance States
SSAS modeling employs the one-way data table to store the subsequent values throughout the
process. In determining the attendance state of the student, the model generates the random
number through the Monte Carlo Simulation, and compares that number to the first row of the
cumulative probability distribution. The very first state S0 is the only state that undergoes
revolution and the other five states, S1, S2, S3, S4, and S5, are fixed and, therefore, remain the
44
same. The reasoning behind this assumption is that students, whether they are suspended (S1, S2,
S3) or absent (S3, S4, S5) from school, will go back to their original class upon returning to school
from suspension or absence. For example, if the student is suspended either in-school
suspension or out-of-school suspension, the student has the right to go back to their same class
upon completion. Additionally, if a student is absent, he or she will attend the original class once
they decided to come back to school instead of going straight to suspension from absence. Table
2 explains how to compute the transition probabilities in detail. Overall, the SSAS determines
the student attendance state considering the random number and the cumulative probability
distribution of the student.
4.5. The Student School Attendance Simulation (SSAS) Model
4.5.1. Introduction
The SSAS model considers the pre-set initial connectivity which undergoes dynamic changes
during the simulation based on the value of ?. In SSAS, each student is represented as a node as
described previously. Briefly, the connections between the students are instantiated by
generating a random number during simulation to determine connectivity based on the value of
?. SSAS creates the connections between the students dynamically considering the random
number and the specified connection range within the interval [0, 1]. As stated in the model
assumption, the students can be subjected to any attendance states on any given school day.
Additionally, this research selects two main areas to overcome the complexity of the student
school social network: a) the state of the student school attendance and b) the interactions with
their peers to accommodate the influential factor. Overall, the SSAS model considers the
influence among peers by integrating the threshold value of the learning coefficient and random
45
selection connections. The next day student attendance probability is computed based on their
own probability and the connected peers? probability coupling with the learning coefficient.
4.5.2. The Model
In the development procedure in formalizing the stochastic process of the student school
attendance phenomenon, as introduced, the SSAS model rationalizes the complexity of the
problem by focusing on two main areas: a) the student school attendance states and b) the
interactions with their peers. This chapter introduces the relevant nomenclature of the model.
The SSAS model parameters include:
Nt is the set of nodes (students) at day t.
nt is the total number of nodes at day t, | Nt |.
Skt is the state school attendance of student k at day t.
Lt is the set of links (friendship) between nodes at day t.
l(i, j)t is the link between node i and j at day t (1 if the link exist, 0 otherwise).
? is the learning coefficient.
? is the threshold to determine the connectivity among peers. The SSAS model sets
the initial connectivity threshold at ? = 0.5 and ? = 0.95 for the subsequent days
of the connections between the students.
Pki,j,t is the transition probability of state i to j of the student k at day t where pijt =
P[S(t+1) =j | S(t)= i] for all i, and j?X.
PKIS is the probability of the state S1 where the student k has in-school suspension (IS).
46
PKSOS is the probability of the state S2 where the student k has out-of-school suspension
with supervision (SOS).
PKOS is the probability of the state S3 where the student k has out-of-school suspension
without supervision (OS).
PKT is the probability of the state S4 where the student k has an unexcused absence,
(Truancy, T).
PKE is the probability of the state S5 where the student k has an excused absence (E).
ek is an epoch, a particular day at which student k returned to attend a normal class
earning grades following an absence.
elK is the number of days that have elapsed since e for student k.
q is the number of school days the student must attend to be promoted or graduate
which is the number required by the current school system, q=165.
? is the possibility that a student is suspended from the school.
? is the possibility that a student is absent but not suspended from the school,
?= ?-1.
The SSAS model includes the following variables in order to evaluate and assess the student
school attendance states during the school year:
t is the time in days for school year (1, 2,?., 180).
nakt is the total number of days a student k was absent in time t.
mat is the summation of number of days of absence for all students in a given period t.
ntkn is the number of times student k started his/her absence period.
47
PK0 is the probability of the state S0 where a normal class attending student k is going
to other state j.
As introduced, the SSAS model considers 5 different transition probabilities (PK0,IS, PK0,SOS, PK0,OS,
PK0,T, PK0,E) where the state 0 (S0) is the only state that the student transits to the other 5 states
and undergoes changes during the simulation process. In Figure 3, there are 2 distinct areas
where the student transitions from the normal class attendance state S0 with probability P0. The
top portion of the dotted box represents all types of suspensions (PK0,IS, PK0,SOS, PK0,OS) and the
dotted box at the bottom represents the absent students (PK0,T, PK0,E) who decided to stay out of
school (?). It is assumed that the proportion of students who move form state 0 to any of the
suspended states is ?. When a student is suspended, the school officials decide whether the
student is suspended in state IS, SOS, or OS. The proportion of students that are suspended in
state IS, SOS, and OS is denoted by the variables X1, X2, and X3 respectively. Similarly, a
student who is absent could be in state T or E. The variables ?4 and ?5 denote the proportion of
students with an unexcused absence, truancy and excused absence respectively. SSAS
distinguishes the absence state from the suspension state because the sole dependency of the
suspension is by the school officials and considers the following decision variables:
X1 represents the probability that a suspended student is in state IS. This value is
determined by school officials.
X2 represents the probability that a suspended student is in state SOS. This value is
determined by school officials.
X3 represents the probability that a suspended student is in state OS. This value is
determined by school officials where X3= 1-X1+X2 which implies X1+X2+X3= 1.
48
Accordingly, in order to compute the next day probability of student school attendance, the
SSAS model implements the following equations:
? PK0,0(t) = Pk0,0(t+1) = (?* Pk0(t)) + (1- ?)* (Average of Pk0(t) of connected peers) .? (4.1)
Equation 4.1 computes the next day probability of the student if he or she remains in state S0
which indicates the student attending normal class continuously. Computation of this
equation applies the same as introduced in the previous section.
? PK0,IS(t) = (1-PK0,0(t))*?X1 ????.?. (4.2)
Equation 4.2 computes the probability where the student changes attendance state from
normal attendance S0 with probability P0 to in-school suspension SIS with probability PIS. X1
represents the value determined by the school official in terms of in-school suspension of the
students.
? PK0,SOS(t) = (1-PK0,0(t))*?X2 ????.?. (4.3)
Equation 4.3 computes the probability where the student changes attendance state from the
normal attendance S0 to an out-of-school suspension with supervision SSOS with probability
PSOS. X2 represents the value determined by the school official in terms of supervised out-of-
school suspension of the students.
? PK0,OS(t) = (1-PK0,0(t))*?X3 ????.?. (4.4)
Equation 4.4 computes the probability where the student changes attendance state from
normal attendance S0 to an out-of-school suspension without supervision SOS with probability
POS. X3 represents the value determined by the school official in terms of supervised out-of-
school suspension of the students.
? PK0,T(t) = (1-PK0,0(t))*(1-?)*?4 ????.?. (4.5)
49
Equation 4.5 computes the probability where the student changes attendance state from
normal attendance S0 to truancy which is an unexcused absent ST with probability PT. The ?4
represents the student?s preference in not attending school without any valid excuse. SSAS
provides fixed value for ?4 in order to provide coherency during the simulation.
? PK0,E(t) = (1-PK0,0(t))*(1-?)*?5 ????.?. (4.6)
Equation 4.6 computes the probability where the student changes attendance state from
normal attendance S0 to an excused absence SE: The ?4 represents the student being absent
from school with valid excuse. SSAS provides fixed value for ?5 in order to provide
coherency during the simulation.
The procedure of computing the transition probability distributions of the six school attendance
states are presented in Table 2 below. The state 0 is the only state which undergoes changes as
explained earlier. The state from 1 to 5 has two options for each student: either remain in the
same state or go back to a normal class.
State 0 (Po) 1 (IS) 2 (SOS) 3 (OS) 4 (T, Tru) 5 (E, Exe)
0 Pk0,0(t) (1-Pk0,0(t))?X1 (1-Pk0,0(t))?X2 (1-Pk0,0(t))?X3 (1-Pk0,0(t))(1-?)?4 (1-Pk0,0(t))(1-?)?4
1 1-PkSI PkIS,IS - - - -
2 1-PkSOS - PkSOS,SOS - - -
3 1-PkOS - - PkOS,OS - -
4 1-PkT - - - PkT,T -
5 1-PkE - - - - PkE,E
Table 2. Procedures of Transition Probability Computation
50
These values are no longer time dependent since the determination whether to return to their
class is made by the school official regardless of the school days or the peer?s influences.
Figure 5. Procedures of the Next Day Transition Probability Computation
To illustrate the computation procedures of the next day student school attendance probabilities,
the diagrams in Figure 5 present the relevance of the probability distribution between the states.
The SSAS process generates the relevant probability distributions of the student throughout the
given period for each student and records those values in a separate file for future reference.
As an example, consider the following one-step transition probability matrix P where the process
change from state i to state j in one time step has been introduced as pijt = P[X(t+1) =j | X(t)= i]
and the transition probability defines the row vector as p = (pi;i?X) where i?X, i
i S
p
?
? = 1.
Sequentially, the student?s attendance state for tomorrow is a consequence of today?s attendance
state where the matrix of transition probability P at stage t is Pi(t) = pijt with pijt ? 0.
PO
?
?
X1
X2
X3
?4
?5
PSOS
PIS
POS
PT
PE
51
0.90 0.03 0.015 0.015 0.03 0.01
0.70 0.3 0 0 0 0
0.70 0 0.3 0 0 0
0.70 0 0 0.3 0 0
0.70 0 0 0 0.3 0
0.99 0 0 0 0 0.01
p
? ?
? ?
? ?
? ?=
? ?
? ?
? ?
? ?? ?
? ?
If the next day probability PK0,0(t) changes from 0.9 to 0.85 (i.e. PK0,0(t+1) =0.85) with given
value of ?= 0.6, X1= 0.6, X2= 0.2, X3= 1- X1- X2 = 0.2, ?4= 0.6, and ?5= 1- ?4= 0.4, the next day
transition probabilities are updated as follows:
PK0,SI(t+1) = (1-PK0,0(t+1))*?X1 = (1- 0.85) x (0.6)*(0.6) = 0.054,
PK0,SOS(t+1) = (1-PK0,0(t+1))*?X2 = (1- 0.85) x (0.6)*(0.2) = 0.018,
PK0,SO(t+1) = (1-PK0,0(t+1))*?X3 = (1- 0.85) x (0.6)*(0.2) = 0.018,
PK0,T(t+1) = (1-Pk0,0(t+1))(1-?)?4= (1- 0.85) x (1- 0.6)*(0.6) = 0.036,
PK0,E(t+1) = (1-Pk0,0(t+1))(1-?)?5= (1- 0.85) x (1-0.6)*(0.4) = 0.024.
Notice that these probabilities are the sum to 0.15. A new one-step transition probability matrix
P of the next day, t+1 is
0.85 0.054 0.018 0.018 0.036 0.024
0.70 0.3 0 0 0 0
0.70 0 0.3 0 0 0
0.70 0 0 0.3 0 0
0.70 0 0 0 0.3 0
0.99 0 0 0 0 0.01
p
? ?
? ?
? ?
? ?=
? ?
? ?
? ?
? ?? ?
? ?
.
52
The simulation of SSAS will yield the following output matrices:
s represents the number of successful students during the school year. The model
computes s by adding promoted/graduated and makeup/repeating students as the
passing students during the school year.
f represents the number of failing students.
v represents the number of promoted students.
y represents the number of students who completed the makeup or repeat in order to
meet q=165.
z represents the number of dropout students.
As introduced, the student coming to school tomorrow is mainly based on today?s activity.
Therefore the model applies the memory-less property of the Markov Chain process to generate
the next day attendance state applying the concept that today?s attendance state is the tomorrow?s
attendance state. The model simulates the stochastic function of student attendance over time
and examines the correlations, especially the impacts of the supervised or unsupervised out-of-
school suspension, and measures the overall student failure as the result of the unsupervised out-
of-school suspension.
The following chapter introduces the procedure in determining the connections between the
students and computing the subsequent probability distribution of the student attendance. In
addition, in the SSAS development process, the model configures in several different ways to
implement the best fitted procedure for the interactions and the connections between the peers.
The relevant variations of the modeling techniques of the SSAS are presented in the Chapter 6.
53
4.6. Chapter Remarks and Summary
The SSAS model considers the complexity of the social networking aspects of school social
network and concentrates on two main areas in the modeling process: a) the student attendance
phenomenon and b) the interactions with their peers. The development of the SSAS model
begins with constructing the comprehensive state diagram that represents the current existing
school system?s five attendance states (Si).and the model proposes a new category, supervised
out-of-school suspensions. Students? social networking in regard to the school attendance is
strategized considering the fundamentals of graph theory properties: the node that represents
students as the actors and their relationships with others applying the edges between the nodes.
The state of students attending school are presented with six different options: 1) student with a
normal in-class status (Normal, Po), 2) student with in-school suspension (IS), 3) student with
out-of-school suspension with supervision (SOS), 4) student with out-of-school suspension
without supervision (OS), 5) student with unexcused absence (Truancy, T), and 6) student with
an excused absence (Exe, E).
The model considers the random environment and applies the stochastic process of Markov
Chain properties to integrate the random and discrete events of the student school attendance
phenomenon. The students? coming to school and participating in the attendance states are
discrete and mutually exclusive events. The implementation of random environment is the
rationalization of understanding the unpredictable students? school attendance from day to day.
The student school attendance events for each day are modeled as a discrete state and discrete
time process where the student can be present in only one state either absent or present at any
time. The SSAS model presents the attendance of each student in a distribution pattern by
54
computing the transition probability. Therefore, each state of the student school attendance is
organized into a transition probability and calculates the steady state probabilities accordingly.
The SSAS assumes that the students will develop friendships while attending school. The
friendships between students are considered through the implementation of the learning
coefficient. In addition, the model considers the dynamic interactions among students using
connectivity threshold values. Overall, each student attending school categories are presented in
distributions through the transition probability and the cumulative probability. The state of the
student attendance is determined based on those probabilities and the random number. The
SSAS employs the memory-less property of Markov Chain where the current state is based
entirely on yesterday?s value and not on the previous data.
Overall, the SSAS model presents the function of each student?s probability (Pki) and associated
influences of their friends that may affect overall student attendance. The model applies the
stochastic process to consider a randomness environment and employs the learning coefficient ?
to integrate the influential factors between the peers. In order to create the connectivity between
the students, the model generates a random number with a definite threshold range for
connectivity. Once the students exhibit connectivity, the probability of student attendance
considers differentiated threshold ?. This chapter provides the formal mathematical
representation of the SSAS and the relevant assumptions, parameters and equations.
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5. METHODS AND PROCEDURES OF SSAS MODEL
5.1. Introduction
This chapter provides detailed modeling procedures of SSAS implementation. Mainly, the SSAS
algorithm consists of two significant parts: 1) the simulation of the student school attendance
phenomenon and 2) the assessment of student school attendances. The first part of the model
algorithm includes multiple processes in simulating the overall procedures presented in the
previous chapters:
a) initialize the number of students and the number of school days,
b) set parameters such as learning coefficient and decision variables,
c) set matrix notation for the connectivity,
d) read the input or output pre-arranged file from external source,
e) get random number and evaluate against the pre-arranged table,
f) allocate appropriate probability distribution range accordingly,
g) calculate the transition probability distribution table for all Si for each student, and
save those values in a separate file,
h) calculate the cumulative probability distribution table for all Si for each student based
on the transition probability distribution, and save those values in a separate file,
i) determine the corresponding current student attendance states, Si accordingly, and
save those values in a separate file,
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j) formulate the connections between the peers, lij, using random number and fitness,
summarize those results, save the data in a separate file,
k) calculate steady state probability, save the values in a separate file,
l) display the connections in the Matrix and the Chart for all N, save those data in a
separate file,
m) get the random number and evaluate it against the first row of the cumulative
probability distributions,
n) calculate the next day attendance probability Pk0(t+1), considering correlated learning
coefficient and the average probability of the connected friends coupling with their
learning coefficient, save those values in a separate file,
o) determine the next day state of the student attendance, Si(t+1) based on today?s
attendance state, save those values in a separate file, and
p) repeat the process for the all school days for each student.
The second part of the SSAS algorithm is designed to assess attendance data from the first part
of the simulation which includes:
q) set the counters,
r) define the absent and present states,
s) set the makeup attendance parameters,
t) define the student status such as promoted or dropout,
u) count each attendance states for all N, save the values in a separate file,
v) summarize the probabilities for the students, save the values in a separate file,
w) calculate present and absent days, save the values in a separate file,
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x) calculate promoted, dropout, or makeup number of students, save the values in a
separate file, and
y) summarize the success and failure of the students, output the values in a separate file.
The corresponding pseudocode for the simulation is presented in this chapter.
5.2. Procedures, Algorithms, and Model Verifications
5.2.1. Development of Initial Transition Probability Distribution Table
Implementation of the SSAS model begins with the probability distribution table that describes
the probability for each student attendance state to generate the initial transition probability. For
example, the conceptual data and the probability distribution of the students are described in the
following Table 2. The purpose of constructing this distribution table is to provide the baseline
to compute the transition probabilities for each student for the remaining school days. Table 3
below describes six students with six distinct categories of their school attendance states and its
distribution of the probabilities for each student. The table presents six classifications of the
student school attendance paradigms: 1) normal class attendance, Po, 2) in-school suspension, IS,
3) supervised out-of-school suspension, SOS, 4) unsupervised out-of-school suspension, OS, 5)
unexcused absence, Truancy Tru, and 6) an excused absence, Exe. Examining the Student 1
probability distribution, the student coming to school and attending the normal class (N) is 90%
for this particular example.
Student 1 can also have a chance of being in school but suspended (IS) for 3% of the time,
suspended out-of-school with supervision for 1.5% of the time, suspended out-of-school with no
supervision for 1.5% of the time, not showing up at the school without any valid reason for 1%
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of the time, and excused absence for 3% of the time. Again, this is a conceptual distribution and
for this particular example, this student comes to school and attends class for 90% which is a
very good student compared to Student 6 who has a value of 65% for the same category.
Name of States States
Students
Student 1 Student 2 Student 3 Student 4 Student 5 Student 6
Po 0 0.900 0.850 0.800 0.750 0.700 0.650
IS 1 0.030 0.040 0.050 0.060 0.070 0.080
SOS 2 0.015 0.030 0.030 0.040 0.050 0.060
OS 3 0.015 0.030 0.030 0.040 0.050 0.060
Tru 4 0.010 0.020 0.030 0.040 0.050 0.060
Exe 5 0.030 0.030 0.060 0.070 0.080 0.090
Total 6 1 1 1 1 1 1
Table 3. Example of an Input Data Table for the Initial Transition Probability Distribution
Applying the probability distribution guidelines presented in Figure 2 in arranging the initial
transition probability, the distribution of the student attendance state probabilities are organized
based on the following assumptions:
? The state S0 is accessible to itself and is the only state that has the accessibility to all the
other five sates.
? The state S1 has the accessibility to itself (S1) and to the state S0.
? The state S2 has the accessibility to itself, (S2) and to the state S0.
? The state S3 has the accessibility to itself (S3) and to the state S0.
? The state S4 has the accessibility to itself (S4) and to the state S0.
? The state S5 has the accessibility to itself (S5) and to the state S0.
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For example, in reference to Student 2 from the Table 3, the probability distribution of Student
2?s school attendance can be arranged as in Table 4 below which is the exact scheme explained
in matrix P in the previous chapter.
States 0 (Po) 1 (IS) 2 (SOS) 3 (OS) 4 (T, Tru) 5 (E, Exe)
0 0.85 0.04 0.03 0.03 0.02 0.03
1 0.7 0.3 - - - -
2 0.7 - 0.3 - - -
3 0.7 - - 0.3 - -
4 0.7 - - - 0.3 -
5 0.9 - - - - 0.1
Table 4. Transition Probability Distribution of the Attendance States for Student 2
As illustrated in Figure 2 state diagram, the state 0 represents a Student 2 attending a normal
class which is 85% for this case, indicates the he or she is relatively a good student in terms of
attendance where the probability of P0,0 = 0.85. In reference to the state diagram, the state S0
could fall into five states and those five states can only come back to a normal class, S0 or repeat
their status. The four states, S1, S2, S3, and S4 which represent IS, SOS, OS, and Tru respectively
contain the same fixed probability. All four states are fixed with 0.7 probability of returning to a
normal classroom, S0 and 0.3 for repeating their state status, S1, S2, S3, and S4. A student with an
excused absence, the state S5 has 0.9 probability of returning to a normal classroom and 0.1
possibility of remaining an excused absence. The outcome of four state students returning to
their normal classroom upon the completion of suspensions or absence remains on these fixed
probabilities for each state. Therefore, the state from S1 to S5 applies the same probability
distributions and the state S0 is the only state to undergo changes throughout the simulation for
every student in this school social network.
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The SSAS model presents three different ways to construct the initial transition probability: 1)
PIOV model constructs transition probability based on random number purely, 2) PAIV model
employs the pre-arranged input data table to generate the transition probability, and 3) PAOV
mode which composes the transition probability according to the pre-arranged output table.
Chapter 6 is devoted to explaining these three methods in detail including the algorithms for each
type.
5.2.2. Cumulative Probability and the Range of Attendance States
In Table 3, the initial transition probability distributions for students can be transformed into the
cumulative probability distribution for each student. In particular, Table 5 describes the
cumulative distribution probabilities for student 1, 2, 3, and 4.
States Student1 Student 2 Student 3 Student 4
Po, 0 0 0.9 0 0.85 0 0.8 0 0.75
IS, 1 0.9 0.93 0.85 0.89 0.8 0.85 0.75 0.81
SOS, 2 0.93 0.945 0.89 0.92 0.85 0.88 0.81 0.85
OS, 3 0.945 0.96 0.92 0.95 0.88 0.91 0.85 0.89
Tru, 4 0.96 0.97 0.95 0.97 0.91 0.94 0.89 0.93
Exe, 5 0.97 1 0.97 1 0.94 1 0.93 1
Table 5. Examples of Cumulative Probability Distribution Table
The same procedure and method in formulating the cumulative probability according to the
transition probability applies for every student in the network. Table 6 includes the cumulative
probabilities that are derived from Table 3 for Student 1 and 2. As the table illustrates, for each
student, the probability begins with 0 and ends with 1 and reveals the range for each attendance
state between the interval [0, 1] and organizes the subsequent attendance state for each interval.
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The SSAS simulates this procedure of constructing the cumulative probability distribution
according to the transition probability distribution for each student throughout the process. The
following tables explain the corresponding format of the cumulative probability computation that
the SSAS generates.
Student 1
States 0 1 2 3 4 5
0 0.9 0.93 0.945 0.96 0.97 1
1 0.7 1 1 1 1 1
2 0.7 0.7 1 1 1 1
3 0.7 0.7 0.7 1 1 1
4 0.7 0.7 0.7 0.7 1 1
5 0.99 0.99 0.99 0.99 0.99 1
Table 6. Cumulative Probability Distribution Table for Student 1 and 2
The examples of Student 1 and 2 in Table 6 show the cumulative probability distribution format
that the SSAS produces. The table clearly defines the states S1, S2, S3, S4, and S5 according to
Table 2 that illustrates how to compute transition probabilities for each state during the
simulation process. Overall, the transition probability from state S0 is the only probability that
changes throughout the process presented on the very first row of the matrix in Table 6.
Student 2
States 0 1 2 3 4 5
0 0.85 0.89 0.92 0.95 0.97 1
1 0.7 1 1 1 1 1
2 0.7 0.7 1 1 1 1
3 0.7 0.7 0.7 1 1 1
4 0.7 0.7 0.7 0.7 1 1
5 0.99 0.99 0.99 0.99 0.99 1
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For this research, constructing the appropriate cumulative probability distribution is a very
significant procedure. It provides the definite probability distribution range, and therefore, the
interval for each attendance state. The main functionality of the cumulative probability is to
provide the guideline to allocate the appropriate attendance state throughout the iterations. The
stochastic process effectively models the recognized random variation within the given
environment sample space. This random variation is included in the modeling process by
generating the random numbers. The generated random numbers are evaluated against the
cumulative distribution as the lookup table and allocate its proper range or interval in order to
determine the corresponding state. Specifically, the random number is measured up to the very
first row of the cumulative distribution as introduced in Table 5. Generally, in determining the
attendance state for students, the process selects the very first row of the cumulative probability
distributions of each student. Overall, the SSAS constructs a cumulative probability distribution
table according to the transition probability using the computation procedures of Table 2 that
defines how to calculate the transition probability of the student during the simulation process.
The determination process of student attendance state is accomplished through random number
generation and those random numbers are evaluated against the first row of the cumulative
probability distributions. For instance, if the random number is 0.87, the corresponding
attendance state for Student 1 is 0 which refers to the student who attends a normal class (Po), S0
and attendance state for the Student 2 is 1 which refers to a student who is in the in-school-
suspension (IS) the S2 category. Determination of the next day student-school-attendance state is
a more complicated process than the initial day of the school. The following chapter explains the
detailed procedures.
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5.2.3. Transition from Current State to Next State
The unique property of the Markov Chain features discrete state and discrete time processes.
Applying this concept, the student can only be in one state at a time and the attendance states are
mutually exclusive. Basically, the student can either be absent or present at any time and the
student only can participate in one attendance state at a time. Therefore, student attendances are
mutually exclusive and discrete events. Furthermore, the Markov Chain provides the memory-
less property. The past history of the process is no longer considered nor has any effect on the
next event. The only event that affects the next event is the current event and not the events from
a week ago or month before. This notion allows the next day student attendance state to be
dependent of just today?s current attendance state.
An example for this concept is illustrated in Table 6. In this table, ?CState? refers to current
state (today?s attendance state) and ?NState? refers to the next state (next day attendance state)
for each student. Additionally, the ?Rand ( )? is for random number and ?Days? expresses the
number of school days. In determining a future attendance state, the model generates the random
numbers which can be seen in the second column in the table and selects the first row of the
cumulative probability distribution table for each student and allocates the corresponding
attendance state. In examining Table 7, consider Table 5 that describes cumulative probability
distribution for four students; Student 1, 2, 3, and 4 (?CState? = 0) which supports the
assumption of having every student coming to school on the very first day of school. The next
day state (?NState?) of student attendance state is derived from today?s current attendance state.
To explain in detail, for Student 1 on school day 16, the ?NState? is determined as 1 and
examining the cumulative distribution table of this student, the range of the probability
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distribution between 0.9 (90%) to 0.93 (93%) falls in this state, the in-school suspension state
(SI). The generated random number for this day is 0.916 therefore, on day 16, the Student 1 is
determined to be at state ?1? which verifies the procedure.
Days Rand ( )
Student 1 Student 2 Student 3 Student 4
CState NState CState NState CState NState CState NState
1 0.2537232 0 0 0 0 0 0 0 0
2 0.7249950 0 0 0 0 0 0 0 0
3 0.1142639 0 0 0 0 0 0 0 0
4 0.7962289 0 0 0 0 0 0 0 1
5 0.5666119 0 0 0 0 0 0 1 0
6 0.8353732 0 0 0 1 0 1 0 2
7 0.4690057 0 0 1 0 1 0 2 0
8 0.7372688 0 0 0 0 0 0 0 0
9 0.3610402 0 0 0 0 0 0 0 0
10 0.7353362 0 0 0 0 0 0 0 0
11 0.8518764 0 0 0 1 0 2 0 3
12 0.1922290 0 0 1 0 2 0 3 0
13 0.5153358 0 0 0 0 0 0 0 0
14 0.8400221 0 0 0 0 0 1 0 2
15 0.5534624 0 0 0 0 1 0 2 0
16 0.9160283 0 1 0 2 0 4 0 4
17 0.0046494 1 0 2 0 4 0 4 0
18 0.1388602 0 0 0 0 0 0 0 0
19 0.3402275 0 0 0 0 0 0 0 0
20 0.6339770 0 0 0 0 0 0 0 0
21 0.1430599 0 0 0 0 0 0 0 0
22 0.0597933 0 0 0 0 0 0 0 0
23 0.4045422 0 0 0 0 0 0 0 0
24 0.9356192 0 2 0 3 0 5 0 5
25 0.7857365 2 0 3 0 5 0 5 1
Table 7. The Current State and Next State Allocations
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The exact methods and procedures apply to the rest of the entire student population throughout
the simulation process of the SSAS. The SSAS uses the transition probabilities and cumulative
probabilities to determine the next day state of the student attendance. Table 7 clearly illustrates
that each student starts with state S0. Figure 6 provides the pseudocode to determine the next day
of the student school attendance state.
Figure 6. The Pseudocode in Determining the Attendance States
The following chapter provides the relevant procedures in computing the next day attendance
probability distributions for the students.
5.2.4. Formation of the Connectivity between the Peers
The Pseudocode in Determining the Attendance States
?????? //upon calculation of cumulative probability
for set the first current state to 0;
end
get randNum = ( ); //get random number to be evaluated
set tempRow = ( ); //select the row to be used to
for i = 6 to 1
{ if randNum < cumStudent
state 0;
else randNum < cumStudent(i)
state i, i= 6-1;
} end for loop
write current state to an external file ( );
//continue remaining simulation
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Up to this point, the computation process has been rather simple because the majority of the
procedure was focused on how to initialize the process appropriately and compute the first day
outlook of the student attendance paradigms correspondingly. Once the students come to school,
the SSAS model assumes that they will intermingle with their peers and therefore make friends
which are described as the connections between the students. In addition, the model assumes
there will be an influential factor ? playing a roll in the phenomenon of the student school
attendance. Therefore, the model attempts to create the connections between the students that
may affect the overall aspects of the student school attendance.
Students Random # Logic Matrix Connectivity
1 0.702646 0.703>0.5 1 Yes
2 0.934999 0.935>0.5 1 Yes
3 0.881752 0.882>0.5 1 Yes
4 0.500214 0.500=0.5 1 Yes
5 0.355652 0.356<0.5 0 No
6 0.786445 0.786>0.5 1 Yes
Table 8. The Rationale in Formulation of the Connectivity among the Students
The next day connection between the peers considers the stronghold and bond as a result of
friendships. Initially, the SSAS generates the random numbers and provides the threshold value
for the connectivity. The model sets the connection threshold value to ?= 0.5. Fundamentally,
the SSAS forms the connections between the students if the random number is greater than or
equal to 0.5; otherwise they are disconnected and there is no link exhibited between the nodes.
Table 8 describes the rationale in how to formulate the connectivity between the students. In
Table 8, the second column consists of the random numbers that are assigned to each of six
students. At third column in the table, the ?Logic? presents the reasoning that the SSAS employs
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in the modeling process which provides the threshold value of the connectivity at ?= 0.5 and the
process evaluates the random numbers accordingly. The model generates the matrix for the
connectivity and exhibit 1 for the student who is eligible, for instance since 0.7 > 0.5, to be
connected with other students in the network illustrated in the fourth column of Table 8. In
addition, the model forms the line on the chart between the students who exhibit 1 in the matrix.
Refer to Table 8 and Figure 6 for 6 students? connectivity analysis. In this table and figure,
Student 1, 2, 3, 4, and 6 are connected but not student 5; this student is isolated from the
network. The SSAS provides the corresponding chart to verify the matrix of the connectivity
between the students. This chart displays the connections between the students described in
Table 8 and presented in Figure 6 below. For chart in Figure 6, x-axis represents the student
number and y-axis the attendance probability. As stated, the connection is exhibited between
Student 1, 2, 3, 4, and 6 but not with Student 5. Student 5 is isolated from the network in this
example.
Figure 7. Initial Iteration for the Connectivity among 6 Students
0.8
0.6
0.4
0.2
1 2 3 4 5 6
1
y
x
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The edges between the nodes are undirected assuming everyone is influencing one another
through their connectivity. Nonetheless, ?= 0.5 is only applicable for the connectivity between
the students on the first day of school because the SSAS integrates the influential factor ?
between the students the next day of school. SSAS model provides modification in creating the
connection between peers and several schemes in applying the connectivity at different parts of
the simulation.
Subsequently, the next day connection between peers considers the stronghold and bond as a
result of the friendships between the students. In general, if students have a tight friendship
today, they will probably continue that relationship tomorrow also. Therefore, the connectivity
no longer can be rationalized with ? ? 0.5 which may create instant breakup daily. Overall, the
SSAS provides a dynamic connectivity in three different ways: a) Median Connectivity, b)
Virtual Connectivity, and c) Hybrid Connectivity.
First, the median connectivity refers to the procedures introduced previously where the
connectivity is determined by the following guidelines:
1 if ? ? 0.5, ???. the connectivity between the student i to j exists
l(i, j) =
0 otherwise, .?...? the connectivity does not exist.
The median connectivity name comes from having the threshold value at half point between the
interval [0, 1]. The pseudocode for median connectivity in Figure 7 contains the only part that
differs from the other connectivity.
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Figure 8. Pseudocode for Median Connectivity
Second, virtual connectivity forms the connections at ? ? 0.95. Since the threshold point for this
mode is much higher than the median mode with ? ? 0.5, it virtually connects everyone in the
network. This virtual connectivity can be presented formally as:
The Pseudocode for Median Connectivity
?????????????????.. //upon the determination of the next state
def peers;
sn = s+1;
int matrix average = 0;
int sum of neighbor student?s probability = 0;
int number of neighbor students = 0;
for (counterSN = 1: NumberOfStudents -1) //
{ { if sn > NumberOfStudents
sn = sn -NumberofStudents;
} end
tempRand = rand( ); // get random number for connectivity
{ if n = = 1
{ if conn(s, sn) < 0 && tempRand >= 0.5 //initial day connection
conn (s, sn) = 1; // connection sets to 1 in the matrix
conn (sn, s) = 1;
elseif conn (s, sn) conn(s, sn) < 0 &&< tempRand < 0.5
conn (s, sn) = 0; // connection sets to 0 in the matrix
conn (sn, s) = 0;
} end
elseif n > 1
{ if conn(s, sn) < 0 && tempRand < 0.5 //following days connection
conn (s, sn) = 1; //connection sets to 1 in the matrix
conn (sn, s) = 1;
elseif conn (s, sn) conn(s, sn) < 0 &&< tempRand < 0.0
conn (s, sn) = 0; //connection sets to 0 in the matrix
conn (sn, s) = 0;
} end
} end
sn = sn +1;
} end for loop ??????????????.... //continue remaining simulation
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1 if ? ? 0.95 ??. the connectivity between the student i to j exists
l(i, j) =
0 otherwise, .?? the connectivity does not exist.
For example referencing Table 8, all 6 students are connected in this connectivity, whereas, there
were only 5 students connected in the median mode. Therefore, this mode virtually connects
everyone in the network and has only 0.05 possibility of disconnecting the students.
Finally, the hybrid connectivity method combines both the median and virtual connectivity
methods. Therefore the connections between the peers are implemented using both modes.
Initially, the hybrid connectivity mode applies the median connectivity which is on the first day
of school. After the first day of school, if connections exist between the students, the virtual
connectivity scheme is applied. Overall, the hybrid connectivity recognizes the existing
friendships between the students and the influential factors thereof. In essence, if the students
are connected and are friends today, they will be friends tomorrow also and there would be little
possibility of breaking up daily. The SSAS model considers this theory and applies the hybrid
connectivity mode to accommodate the ongoing friendships between the students.
5.2.5. Interaction Probability Computation
In calculating the next day attendance probability, the SSAS simulation integrates the influential
factor to compute interaction probability once the students are connected to each other which it
applies from the second day of the school year. As presented earlier, the next day probability
consists of four elements: 1) the student?s current probability, 2) the student?s learning
coefficient, 3) the connected peers? probability, and 4) the learning coefficient of the connected
friends. The model assumes the information about each student; the probabilities and the
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learning coefficients of the students are known. The mechanism to compute interaction
probability is presented in equation 5.1.
Figure 9. Flow Chart in Computing the Interaction Probability among the Students
The next day probability of the student, Pk0(t+1) is the difference of the probability between the
student probability and the average of the linked peers? probability coupled with the learning
coefficient. If linked peers? probability is greater than the student?s current probability, ?*Pk0 (t)
then there will be an increase in the student?s current state probability, Pk0(t+1) = (?* Pk0 (t)) +
(1- ?)* (Average of Pk0 (t) of connected peers). However, if the average probability of peers is
less than the student?s current state probability, then the student?s current probability will
decrease. The flow chart in Figure 9 illustrates the procedure using four students (N4) in
estimating the next day attendance probability of the students. As stated, the SSAS uses the
fitness value of the learning coefficient at ? = 0.7. For example using the data in Figure 9, in
calculating the next day probability for the Student 1, the first step is to find the difference
between Student 1 and other students where (- 0.1 -0.2 -0.25) = - 0.55, (0.8 -0.9 = -0.1, 0.7 -0.9 =
Student 1
S0: 0.9 & a
Pk0(t+1) = (?* Pk0 (t)) + (1- ?)* (Average of Pk0 (t) of connected peers)
if r = 0.823, all connected
Student 2
S0: 0.8 & a
Student 3
S0: 0.7 & a
Student 4
S0: 0.65 & a
Day 1
Day 2
Compute new 1st row of Matrix P
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- 0.2, and 0.65 -0.9 = - 0.25). Since the calculation results in a negative number, the process will
decrease Student 1?s probability applying (?* Pk0 (t)) + (1- ?))* (Average of Pk0 (t) of connected
peers) where (0.9*0.7) + ((0.8 + 0.7 + 0.65)/3)*0.3 = 0.63 + 0.2155 = 0.8455. Applying the
same method, the next day probability calculation for Student 4, the difference between Student
4 and other students is 0.45, therefore the process will increase the peers? probability of Student
4 which results in 0.695, (0.65*0.7) + ((0.9 + 0.8 +0.7)/3)*0.3 = 0.695.
The detailed development procedures are included in the previous section throughout the model
simulation process. The transition probability and the cumulative probability are presented in the
tables. The SSAS produces the charts and matrices to illustrate and verify the connections
between the peers throughout the process. The model verifies the connectivity between the
students by comparing the connectivity in the chart and the matrix notations represented by 1 or
0. Initially, the link or connectivity between the students exists if the random number is greater
than or equal to 0.5; otherwise they are disconnected and no link will be displayed on the chart
between the nodes.
5.4. Chapter Remarks and Summary
School attendance states are discrete and mutually exclusive events. The SSAS model considers
two main areas in the modeling process: the student attendance phenomenon and interactions
with their peers. The model constructs the comprehensive state diagram to represent the current
existing school system?s five attendance states (Si).and a new category, the supervised out-of-
school suspensions. The model presents the graph with the node that represents students and
their relationship with others with edges between the nodes. In addition, the model considers the
random environment and generates the random numbers through the Monte Carlo Simulation.
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The SSAS model presents the attendance of the each student in a distribution pattern by
computing the transition probability. Therefore, each state of the student school attendance is
organized into a transition probability and calculates the cumulative transition probability
distribution and steady state probabilities accordingly. Furthermore, the model considers the
friendships between the students and implements a learning coefficient. The connections
between the students are created dynamically using connectivity fitness values. Each student-
attending-school category is presented in distributions through the transition probability and the
cumulative probability. The state of the student attendance is determined based on those
probabilities and the random number. The SSAS employs the Memory-less property of the
Markov Chain where the current state is a result based solely on yesterday?s value and not on the
previous data.
Overall, the SSAS model presents the function of each student?s probability (Pkt) and associated
influences of their friends (?Pk) that may affect overall student attendance. The model applies
the stochastic process to consider randomness environment and employs the learning coefficient
to integrate the influential factors between the peers. In order to create the connectivity between
the students, the model generates a random number with a definite threshold range for the
connectivity. Once the students exhibit the connectivity, the probability of student attendance
considers the influential factor. This chapter provides the mathematical formal representation of
the SSAS and the relevant assumptions, parameters and constraints.
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6. THREE DIFFERENT TYPES OF THE SSAS MODEL
6.1. Introduction
The stochastic process comprehends random events, thereby selecting an actor, action of the
actor, and the interactivities of the actors in an arbitrary method. In the previous chapter
introduction of the procedures and methods, the model begins with the computation of the initial
transition probability distribution table. The SSAS explores the alternative ways to develop the
evaluation model and presents three different approaches: 1) PIOV that applies a pure random
selection, 2) PAIV that uses a pre-arranged input table, and 3) the PAOV that uses a pre-arranged
output look-up table. A Pure Input Output Value model considers the pure randomness of the
given community where students? relationships with their peers are random events combined
with unknown outlook in regard to school attendance. The PAIV model incorporates the pre-
arranged probability distribution table that can be used as the lookup table to allocate the range
of the distribution, thereby determining the corresponding student attendance state categories. In
pre-arranged probability distributions for each of the student attendance states, the range of the
state can be organized through the cumulative distribution table. The random events of students?
attending states and the interaction with their peers are considered in the model as explained in
the introduction of SSAS. The overall processes in relationship to probabilities are presented in
Figure 10.
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Figure 10. Process Overview of SSAS Model
In the PAOV model, the formulation of probability distributions look-up table is based on the
actual students? attendance data set. The previous PIOV model implementation is solely based
on random selection, and the PAIV model provides stability through the pre-arranged lookup
table of the probability distribution as the input values. The PAOV model differs in that the
initial probability distribution and depiction of the current student attendance state are based on
the pre-arranged output look-up table designed from the actual existing school attendance data
set. Therefore, the PAOV is most useful and feasible if the student attendance data are known
and available. However, the pre-arranged output probability distribution look-up table also can
be designed arbitrarily if the actual student attendance data set is unavailable. The following
three sections of this chapter describe the detailed procedures for each model.
6.2. Pure Input Output Value (PIOV) Dynamic Model
The SSAS PIOV model precedes the probability of the student and the interactions with peers by
purely a random selection process without a preset value of the student attendance probability
thereby selecting the transition probability randomly. Note that in the probability computation of
state 0, S0 is the only state which undergoes changes throughout the simulation and the state S1,
Initial Transition Probability
Cumulative Probability
Attendance State Determination
Interactivity
Learning Coefficient, ?
Connectivity Threshold, ?
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S2, S3, S4, and S5 remains the same as presented in Table 4, following the procedure introduced in
Table 2. The process of PIOV is introduced in detail in the following section.
6.2.1. Procedures and Methods of PIOV Model
Initially, the PIOV model generates the random number using rand ( ) function and the generated
random number becomes the value of current S0 probability of the student.
Figure 11. Procedures of the PIOV SSAS Model
get P0, rand ( )
determine Si
select lij randomly (0 or 1)
compute Pk0(t+1)
determine Si(t+1)
exit
calculate transition & cumulative probabilities
enter
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For example, if the random number is 0.8521, the student has a current probability of 0.852
(85%) for state S0 of the student attendance. This number is used as the first day, the current
probability of the student, P0,0. The initial attendance transition distribution probability is then
arranged based on the input file that defines the ratio for the five remaining states.
The cumulative probability and the next day transition probability computations are the same as
presented in the introduction. The learning coefficient also uses the exact value, a= 0.7. Briefly,
the flow chart in Figure 11 describes the overall procedures of the PIOV simulation method.
6.2.2. Algorithms of PIOV Model
In order to simplify, the algorithm included in this chapter is limited to the section that has been
modified from the other types of the model and the subsequent algorithms can be found in the
basic algorithm for the SSAS mode. The process introduced here includes the main areas where
the PIOV model is differentiated and the subsequent relevant 7 processes:
a) generate a random number and set it as the first day probability, P0,
b) read the fixed state distribution input file that sets the range for the other states,
c) calculate transition probability and cumulative probability for all N,
d) determine the current states based on first row of cumulative probability, Si,
e) create the linkage among peers, lij, based on threshold, ? ? 0.5,
f) compute the Pk0(t+1) with correlated learning coefficient ?=0.7, and
g) determine the next day attendance state, Si(t+1).
Mainly, the PIOV future outcome is solely based on a random selection process.
78
Figure 12. Pseudocode of the PIOV Model
6.3. Pre-Arranged Input Value (PAIV) Dynamic Model
The PAIV model generates the student attendance state based on the input data set which is the
pre-arranged probability distribution table describing the pre-set attendance data range for the
student attendances. The students? interactions with peers are determined by a similar method as
described in the previous chapters. The calculation procedure includes the corresponding
The Modified Portion of Pseudocode for PIOV Model
def school days, number of student, parameters, and chart style;
set external input file to be read and external file to output;
for each student n
write studentID into the external file;
end
read fixed state probabilities;
read preset range for the state;
for each day
get random number;
set random number as initial probability;
determine the appropriate state based on the preset range for the state;
for each num in cumulative probability (1-nth)
if rand< first num in cumprob
get state 0
else compare to the nth # in cumprob n: 1-6
if rand > nth # in cumprob
state = n
end
end
end
?????????????? //continue remaining simulation
79
transition probability and cumulative probability distributions coupled with the associated
learning coefficient.
6.3.1. Procedures and Methods of PAIV Model
Figure 13. Procedures of the PAIV SSAS Model
calculate transition & cumulative probability distribution
get r, rand ( )
predict Si using p0 of cumulative probability & r
determine lij based on fitness
compute Pk0(t+1)
determine Si(t+1)
exit
set probabilities, Pi
enter
80
Unlike the PIOV model, the PAIV model generates the student attendance status by the pre-
arranged input probability distribution table for every student. This is a similar procedure
presented in the introduction of the SSAS implementation procedures. In this development
process, the distribution table is pre-composed in a separate file and the processor reads those
pre-arranged values during the simulation in generating the initial probability distribution of the
student?s school attendance. Similarly, the PAIV model also considers random events and
generates the random number which compares with a pre-arranged probability distribution to
allocate the appropriate probability distribution range, thereby determining the current state of
the student attendance.
Therefore, the PAIV model exploits the student school attendance category based on the pre-
arranged probability distribution table as input data. This model is designed to provide a
mechanism to control the input probability distribution and the range of probability dispersion.
The flow chart in Figure 13 presents the overall process of the PAIV dynamic procedures. PAIV
is a more constructive model in that the pre-arranged probability distribution for the attendance
state provides some stability in determining the current state of the student. The next day
probability, Pk0(t+1) of the student is computed using a similar method as described in the
introduction. The exact value of learning coefficient also applies in the PAIV model.
6.3.2. Algorithms of PAIV Model
The procedures and algorithm of the PAIV model presented in this section are the portions that
differ from other models and the process related to the modification which includes:
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a) set the probability distribution table for all Si for each student, and save it in a
separate file,
b) generate the random number,
c) read the file,
d) calculate the transition probability and cumulative probability for all N,
d) determine the current states based on first row of cumulative probability, Si,
Figure 14. Pseudocode of the PAIV Model
e) create the linkage among peers, lij, based on threshold, ? ? 0.5,
The Modified Portion of Pseudocode for PAIV Model
def school days, number of student, parameters, and chart style;
set external input file to be read and external file to output;
for each student n
write studentID into the external file;
end
set the attendance probabilities for the external file ;
read the probability from input file;
for each student
generate random #;
for each num in cumprob (1-nth)
if rand < first num in cumprob
get state 0;
else compare to the nth# in cumpob
if rand > nth# in cumprob
state = n;
end
end
end
end
//continue remaining simulation
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f) compute the Pk0(t+1) with correlated learning coefficient ?=0.7, and
g) determine the next day attendance state, Si(t+1).
The modified area of the PAIV algorithm is presented in Figure 14.
6.4. Pre-Arranged Output Value (PAOV) Dynamic Model
The PAOV process calculates the transition probability and cumulative probability distributions
corresponding to the pre-arranged output probability distribution as a look-up table. The
interactivities of the students are calculated in a similar manner as in the POIV and PAIV models
using ? = 0.5 and a = 0.7 as the threshold value and the learning coefficient respectively.
6.4.1. Procedures and Methods of the PAOV Model
The PAOV process begins with defining the attendance probability distribution look-up table
based on the student attendance data set from the specific school district. Therefore, the initial
probability, S0 is determined by the pre-arranged probability distribution look-up table. The
process estimates the probability of student school attendance by evaluating the random number
against this pre-arranged look-up table which is designed based on the attendance data set. The
student attendance state is also allocated accordingly as the pre-arranged output look-up table
distribution range. See Table 9 for the pre-arranged probability distributions as the output look-
up values. For instance, if the student comes to school for 100% to 95% of the time and attends
a normal class, the student can be in the state of truancy for 1%, an excused absence for 4%, and
0% for in-school suspension or out-of-school suspension. Based on Table 9, in case the student
comes to school 70% of the school days and attends class, the student possibly could be in in-
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school suspension for 5%, out-of-school suspension for 4%, truancy for 7%, and excused
absence for 14%.
P0 IS OS Truancy Excused
0 1 3 4 5
0.95 0.00 0.00 0.01 0.04
0.90 0.01 0.00 0.03 0.06
0.85 0.02 0.01 0.04 0.08
0.80 0.03 0.02 0.05 0.10
0.75 0.04 0.03 0.06 0.12
0.70 0.05 0.04 0.07 0.14
0.65 0.06 0.05 0.08 0.16
0.60 0.07 0.06 0.09 0.18
0.55 0.08 0.07 0.10 0.20
0.50 0.09 0.08 0.11 0.22
0.45 0.1 0.09 0.12 0.24
0.40 0.11 0.10 0.13 0.26
0.35 0.12 0.11 0.14 0.28
0.30 0.13 0.12 0.15 0.30
0.25 0.14 0.13 0.16 0.32
0.20 0.15 0.14 0.17 0.34
Table 9. Pre-Arranged Output Probability Distribution Look-Up Table
Notice that Table 9 omits student school attendance state 2, the supervised out-of-school
suspension (SOS), because this category does not exist in the current school system. The
supervised out-of-school suspension state is a new category which this study proposes in order to
reduce the failure rate of students due to excessive absences. The attendance assessment
between supervised and unsupervised out-of-school suspensions and their impacts on students is
presented in Chapter 7.4.4 in detail.
As described in Figure 15, the process begins with formulating the output pre-arranged
probability distribution table. Using the actual students? attendance data set, the table is
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organized for each student attendance category. This table is used as a look-up table in setting
the initial transition probability and allocating the corresponding attendance state.
Figure 15. Procedures of the PAOV SSAS Model
calculate transition & cumulative probability distribution
generate r, rand ( )
determine Si using p0 of cumulative probability & r
create lij based on threshold ?
compute Pk0(t+1) with ?
determine Si(t+1)
exit
set probabilities, Pi
enter
formulate look-up probability distribution table, Pi via data set
85
The computations of interactivities among peers proceed as the procedure in the PIOV and PAIV
models. In addition, the PAOV model applies the same value as the other models for the
learning coefficient, a= 0.7, and threshold value, ?= 0.5 for connectivity. The transition
probability and the cumulative probability computation procedures are also the same as in the
PIOV and PAIV models.
6.4.2. Algorithms of the PAOV Model
The modified part of the PAOV model algorithm and the relevant calculation thereof consists of
8 major processes:
a) formulate the probability distribution look-up table for all Si for each student based on
the data set, and save it in a separate file,
b) read the file,
c) generate random number and allocate appropriate range,
d) calculate initial transition probability and cumulative probability for all N,
e) determine the corresponding current student attendance states, Si accordingly based
on first row of cumulative probability and output probability distribution look-up
table,
e) create the linkage among peers, lij, based on threshold, ? ? 0.5,
f) compute the Pk0(t+1) with correlated learning coefficient ?= 0.7, and
g) determine the next day attendance state, Si(t+1).
The pseudocode for PAOV model is explained in Figure 16.
86
Figure 15. Pseudocode of the PAOV SSAS Model
Figure 16. Pseudocode of the PAOV Model
The Modified Portion of Pseudocode for PAOV Model
def school days, number of student, parameters, and chart style;
set external input file to be read and external file to output;
for each student n
write studentID into the external file;
end
formulate the attendance probability look-up table AttendDistribution;
read external file, AttendDistribution file;
get the size for the AttendDistribution;
read the FixedState from input file;
for each student
generate random #;
for each num in cumprob (1-nth)
if rand < first num in cumprob
get state 0;
else compare to the nth# in cumpob
if rand > nth# in cumprob
state = n;
for (s=1:NumOfStudents)
{ InitialProb{s}=0;
InClassProb=NumOfAttend(s)/Period;
for (i=1:row )
get other initial probabilities;
end
end
end
end
for each student n
write studentID into the external file;
end //continue simulation
87
6.5. Chapter Remarks and Summary
The SSAS model provides three modifications in simulating the student school attendance
phenomenon: PIOV, PAIV, and PAOV. Considering the random environment of the student
school attendance and the interaction with peers at school, the PIOV implements the unique
attributes of the stochastic process very efficiently by using a random number to generate the
initial transition probability distribution. In the PIOV development process, the simulation
focuses on the main concept, the random selection process for the initial probability, P0 for the
state S0 of the student school attendance. The connections between the peers, l(i, j) are selected
randomly also with threshold value, ?= 0.5. The next day probability and the attendance state of
the students are then determined by considering the learning coefficient, ?= 0.7 and the
calculation procedures introduced in Table 2 and Table 4. The overall procedure and algorithms
of the PIOV model are presented in Figure 11 and Figure 12.
Understanding the lack of stability in the random environment of the student school social
networking, the PAIV model considers a controlling mechanism by providing the pre-arranged
initial transition probability distribution for the student school attendance states. The pre-
arranged initial transition probability distribution table provides significant stability in
determining the attendance state of students since it sets the pre-arranged attendance range for
each state for every student. A similar procedure applies in calculating the transition probability
and cumulative probability as described in Table 2 and Table 4. The next day probability and the
application of threshold value, ?= 0.5 for the connectivity and the learning coefficient, ?= 0.7 for
the interactivity, are also applied in a similar manner as in the previous model. The PAIV model
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flow chart in Figure 13 presents the overall procedure and Figure 14 describes the algorithms for
the model.
The main concept of the PAOV features the use of the pre-arranged output probability
distribution look-up table that is formulated based on student school attendance data. The initial
transition probability is generated based on the distributions in the look-up table and calculates
the cumulative probability thereof. Subsequently, the corresponding student attendance state is
determined accordingly also. The learning coefficient, ?= 0.7 for interactions and threshold
value, and ?= 0.5 for connectivity among peers, are considered in the same manner as in the
PIOV and PAIV models.
Models Similarities Differences
PIOV Learning coefficient threshold, a = 0.7 randomly selects initial P0 &
Transition & cumulative computation
States determination of Si
Interaction computation with ? ? 0.5
PAIV same as above pre-arranged input table for P0 &
controlling mechanism for ITP
PAOV same as above output look-up table for P0 &
stability for ITP
Figure 17. Comparison of the PIOV, PAIV, and PAOV Models
The PAOV model is most useful when the student school attendance data set is available. The
PAOV model provides significant stability by employing a pre-arranged output probability
89
distributions look-up table for each student attendance state. Furthermore, this modeling is
applicable in the case where the attendance data set is unavailable by composing an arbitrary
output probability distribution look-up table and generating initial transition probability based on
the range that look-up table provides. An overall summary is provided in Figure 16 comparing
all three models of the SSAS: PIOV, PAIV, and PAOV.
Exploiting the different procedures to accommodate the random environment, the stochastic
process provides an effective tool to index the parameters using statistical probability
distributions. Chapter 6 introduces three distinct implementations in arrangement of the initial
transition probability for the SSAS. It begins with the PIOV model, the most vulnerable method
where the initial transition probability is determined purely based on random events, followed by
the PAIV model that provides a controlling mechanism by employing an input table which
consists of pre-arranged initial transition probability distributions. Lastly, in the PAOV model,
the formulation of a pre-arranged output look-up table has the probability distribution based on a
data set which provides a controlling mechanism as well as significant stability in estimating the
student school attendance disseminations. In brief, Figure 17 describes the similarities and
differences of three models of the SSAS.
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7. EXPLOITATION RESULTS AND ANALYSIS
7.1. Introduction
As stated previously, in the development of the SSAS, the model consists of multiple processes:
random selection attachment; computation of transition, cumulative, and steady state probability
distribution; determination of the students? school attendance states; creation of dynamic
connectivity, learning coefficient integration, and attendance assessments. This research also
presents three models of the SSAS: PIOV, PAIV, and PAOV which are differentiated in
generation of the initial transition probabilities for the student school attendance states. For all
three models, the next day probability, Pk0(t+1) of the student attendance state is determined by
computing the difference of probability between the current student and the average of the linked
peers probabilities considering positive and negative influences as well as the threshold value of
the connectivity and learning coefficient for interactions.
The connectivity between students is constructed based on the threshold value of the
connectivity, ? = 0.5. In addition, the SSAS model also provides three methods for creating the
connectivity between students: Median Connectivity, Virtual Connectivity, and Hybrid
Connectivity. This chapter presents results from the three SSAS models and the three dynamic
connections between the students.
7.2. Simulation Results of the SSAS Models
91
The simulation of the SSAS model evaluates 1000 students for the 180-day school year. Table
10 includes overall results from all three SSAS models: PIOV, PAIV, and PAOV. Examining
the results according to data in Table 10 indicates that the PIOV model generated the least
number of students for the attendance category N, the student attending normal class with 642
students out of 1000 students.
Attendance
States
PIOV PAIV PAOV
Number % Number % Number %
N 642 0.64 711 0.71 702 0.70
IS 82 0.08 67 0.07 68 0.07
OS 67 0.07 54 0.05 56 0.06
Tru 101 0.10 89 0.09 91 0.09
Exe 108 0.11 79 0.08 83 0.08
Total 1000 1 1000 1 1000 1
Table 10. Simulation Results of the SSAS Models
However, the PIOV model generated the highest number of students for all the other categories:
82 students for in-school-suspension (IS), 67 students for out-of-school suspension (OS), 101
students for unexcused absences (Tru), and 108 students for excused absence (Exe). On the
other hand, the PAIV generated the highest number of students attending class, category N with
711 students and the least number of students for all the other five categories: 67 students for in-
school-suspension (IS), 54 students for out-of-school suspension (OS), 89 students for unexcused
absences (Tru), and 79 students for excused absences (Exe). Interestingly, the PAOV model
generated the middle range between the POIV and PAIV models for all five categories: 702
students for normal attendance (N), 68 students for in-school-suspension (IS), 56 students for
out-of-school suspension (OS), 91 students for unexcused absences (Tru), and 83 students for
excused absences (Exe).
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Further examination of the results in Table 11 presents an assessment in terms of error or the
correctness of the data. Standard errors are calculated by subtracting the resulting data from the
original data and dividing it by the original data formally, m-n/m. For example, where the N
original data was 699 and the resulting data was 642, the error can be measured by (699-
642)/699. Based on this table, the resultant errors are the PIOV model with 16.97 %, PAIV
model with 3.57%, and PAOV model with 1.4%. Consequently, the correctness of the system is
83.03% for the PIOV model, 96.43% for the PAIV model, and 98.6% for the PAOV model.
This research applies the standard acceptable range for error from 0 to 0.5 which is used by most
testing procedures. In other words, the acceptable system should not produce error at higher than
5% and the correctness of the system should be more than 95%.
Attendance States,
Number of Students
PIOV PAIV PAOV
Error % Error % Error %
N, 699 8.155 1.717 0.429
IS, 69 18.841 2.899 1.449
OS, 57 17.544 5.263 1.754
Tru, 93 8.602 4.301 2.151
Exe, 82 31.707 3.659 1.220
Average Error, % 16.970 3.568 1.401
Correctness, % 83.030 96.432 98.599
Table 11. Measurements of the Errors and the Correctness of the SSAS Models
Applying this range, the POIV model is an unacceptable model since it produced the most error
out of all three models at 0.1697%. This model is only feasible at about 83% which is much less
than the acceptable norm of 95%. PAIV and PAOV resulted in 96.4% and 98.6% correctness
respectively which are better than the 95% of acceptable correctness of the system. Therefore,
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the data results and the assessments thereof are focused on only two models, PAIV and PAOV of
the SSAS in the following chapters.
7.3. The Connectivity in between Three Types of Connections
As introduced, the SSAS simulation exploits three different ways to implement the dynamic
connectivity between the peers: a) Median Connectivity, b) Virtual Connectivity, and c) Hybrid
Connectivity. These three types of connectivity are tested using the PAOV model since this
model resulted in significant stability and produced the best result with 98.6% correctness. The
results of three connectivity types are included in the table below. In this table, the school days
are organized in intervals of five days for the 180-day school year. For easy assessment of
connections among the students, the resulting data in the table are also divided into two areas:
the number of students per class size of 30 and the number of students per grade size of 1000.
These numbers for the size of class and grade are arbitrary figures and apply to this table only.
The table also includes the average number, median of connections and the mode, the most
frequently appearing numbers of connections.
In examining the results, the following reasoning is based on a class size of 30 students for
simplicity. To begin, the second column of Table 12 depicts the number of connections of the
median connectivity (MC). In this connectivity, the process generates the random number and
the connection between the students is determined at ? ? 0.5 as introduced earlier. The median
connectivity forms the link between the students with this premise throughout the process. The
very first day of the link created starts out with 16 connections and ends with 15 connections.
The mean number of connections was 14, with 15 connections appearing most frequently.
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Days
Class Size of 30 Grade Size of 1000
Number of Friends Number of Friends
MC VC HC MC VC HC
1 16 28 17 533 933 567
5 17 28 19 567 932 633
10 13 27 18 432 899 601
15 19 25 20 632 831 667
20 13 27 19 433 901 634
25 15 29 20 501 967 667
30 15 29 20 499 966 665
35 12 27 22 399 901 733
40 15 29 21 500 967 700
45 15 29 20 499 967 667
50 18 25 20 602 833 665
55 11 29 21 367 967 701
60 11 29 22 365 967 733
75 11 27 22 366 900 732
80 14 27 21 467 901 701
85 10 29 21 333 967 700
90 15 27 23 500 900 767
95 9 28 21 300 933 700
100 13 27 20 433 900 667
105 15 27 20 501 901 666
110 15 27 24 503 900 800
115 12 28 25 401 933 833
120 17 29 25 567 967 833
125 19 29 23 633 967 767
130 16 27 21 533 900 701
140 16 28 19 532 933 633
145 11 27 19 367 900 632
150 16 28 20 533 933 667
155 17 26 20 567 867 666
160 16 26 18 531 867 599
165 16 25 17 533 833 567
170 15 27 17 531 901 565
175 13 27 18 530 897 606
180 15 26 19 532 933 533
Median 14 27 20 486 917 676
Mode 15 27 20 533 967 667
Table 12. The Number of Connectivity in a Five-Day Interval for School Year
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The third column of the table depicts the result from the virtual connectivity simulation which
targets the connections among peers at ? ? 0.95 throughout the process. This connectivity began
with 28 connections initially and ended with 26, 2 connections less than initial connections. The
average and the mode for this connectivity were 27 out of 30 students virtually having all
students connected. Lastly, the hybrid connectivity combines both methods of the median and
virtual connectivity. Connections between peers are implemented using both processes. The
first school day of student connectivity proceeds through the median connectivity scheme which
uses ? ? 0.5 and ? ? 0.95 for those students who are already connected. Hybrid connectivity
promotes the idea of continuing existing friendships. If students are connected and are friends,
more than likely the friendships will continue with very little possibility of an immediate
breakup. The hybrid connectivity started with 17 connections and ended with 19 connections.
The median and mode of the connectivity were each 20 students.
Overall, there were fewer connections resulting in the median and virtual of connectivity
compared to the initial connection: 1 connection less resulted in median connectivity (16-15 = 1)
and 2 connections less resulted in virtual connectivity (28-26 = 2) and the hybrid connectivity
generated 2 connections more (17-19 = -2). The initial expectation of the median connectivity
was that it would produce about half the number of connections, virtual connectivity would
connect almost everyone, and hybrid would have more than half the connections but fewer than
virtual connectivity. In conclusion, the SSAS simulation resulted as expected having MC with
14 connections out of 30 and 486 connections out of 1000 which is almost half the connectivity
among the population; VC with 27 connections out of 30 and 917 connections out of 1000 which
is almost all of the population; and HC with 20 connections out of 30 and 676 connections out of
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1000 which is more than half but fewer than in Virtual Connectivity. Figure 18 below illustrates
the difference in average number of connections among the three types of connectivity.
0
200
400
600
800
1000
MC VC HC
Connection Type
Nu
mb
er
Figure18. The Average Number of Connectivity in SSAS Models
7.4. Attendance Assessment between Supervised and Unsupervised OOS
The SSAS model attempts to assess the overall student school attendance in all three types of
connectivity using the PAOV model which includes: Table 13 with Median Connectivity, Table
14 with Virtual Connectivity, and Table 15 with Hybrid Connectivity. For the first simulation,
we consider the student attendance assessment using Median Connectivity. Under the
unsupervised out-of-school suspension program, there were 637 students promoted, 160
repeated, and 203 dropped out from the group of 1000 students. Hypothetically, if we use this
1000 student group as the senior year in high school and consider promoted and repeated
categories as successful students, there were 203 students failed and 797 students graduated from
high school. Under the supervised out-of-school suspension program, there were 897 students
promoted, 47 repeated, and 57 dropped out. Interpreting this data using the group of 1000
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students in high school, 943 students graduated and 57 students were unable to graduate from
high school.
Status of
Attendance
Unsupervised OOS Supervised OOS
Number % Number %
Promoted 637 0.6367 870 0.8700
Repeated 160 0.1600 47 0.0467
Drop-Out 203 0.2033 57 0.0567
Total 1000 1 1000 1
Status of
Student
Unsupervised OOS Supervised OOS
Number % Number %
Graduate 797 0.7967 943 0.9433
Failed 203 0.2033 57 0.0567
Total 1000 1 1000 1
Table 13. Student Attendance Assessment Using Median Connectivity
Second, considering student attendance assessment using Virtual Connectivity, under the
unsupervised out-of-school suspensions program, there were 493 students promoted, 71
repeated, and 436 dropped out from group of 1000 students.
Status of
Attendance
Unsupervised OOS Supervised OOS
Number % Number %
Promoted 493 0.4933 844 0.8443
Repeated 71 0.0700 53 0.0533
Drop-Out 436 0.4367 102 0.1023
Total 1000 1 1000 1
Status of
Student
Unsupervised OOS Supervised OOS
Number % Number %
Graduate 564 0.5643 898 0.8977
Failed 436 0.4357 102 0.1023
Total 1000 1 1000 1
Table 14. Student Attendance Assessment Using Virtual Connectivity
98
Examining this data for the senior year with 1000 students, 564 students graduated and 436
students were unable to graduate from high school. Under the supervised out-of-school
suspension program, there were 844 students promoted, 53 repeated, and 102 dropped out.
Applying this data to the senior year of high school with 1000 students, 898 students graduated
and 102 students were unable to graduate from high school.
Third, the simulation considers the student attendance assessment using Hybrid Connectivity.
Under the unsupervised out-of-school suspension program, there were 528 students promoted,
113 repeated, and 359 dropped out from the total of 1000 students. Considering this data for the
senior year of high school with 1000 students, 641 students graduated and 359 students were
unable to graduate. Under the supervised out-of-school suspension program, there were 786
students promoted, 61 repeated, and 153 dropped out. Once more, applying this data for the
senior year of high school with a group of 1000 students, 847 students graduated and 153
students were unable to graduate.
Status of
Attendance
Unsupervised OOS Supervised OOS
Number % Number %
Promoted 528 0.5267 786 0.7857
Repeated 113 0.1133 61 0.0610
Drop-Out 359 0.3600 153 0.1533
Total 1000 1 1000 1
Status of
Student
Unsupervised OOS Supervised OOS
Number % Number %
Graduate 641 0.6400 847 0.8467
Failed 359 0.3600 153 0.1533
Total 1000 1 1000 1
Table 15. Student Attendance Assessment Using Hybrid Connectivity
99
The difference in graduates between unsupervised OOS and supervised OOS in three types of
connectivity is illustrated in Figure 19.
0
200
400
600
800
1000
MC VC HC MC VC HC
Unsupervised OOS Supervised OOS
Figure 19. The Number of Student Graduates of Supervised and Unsupervised OOS
7.5. Correlation of Supervised and Unsupervised Out-of-School Suspensions and Dropouts
In an attempt to evaluate the success and failure rates of the student school attendance, Table 16
includes the difference in the number of students having supervised out-of-school suspension
that is considered as class attendance N status instead of absence, as the current school system
applies. The difference in the table below is the number of students in unsupervised out-of-
school suspension subtracted from the number in supervised out-of-school suspension. The
negative number indicates the decrease in the number and the positive number indicates an
increase.
Under supervised out-of-school suspension, there was an average of 284 more students
promoted, 56 fewer that repeated, and 229 fewer that dropped out. In addition, Table 16
indicates that there were 229 more students graduated on average. Therefore, the highest student
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success rate was achieved with supervision being provided during an out-of-school suspension
period.
Status of
Attendance
Difference in Number of Students
Median Co. Virtual Co. Hybrid Co. Average % Average
Promoted 241 351 260 284 0.284
Repeated -36 -18 -113 -56 -0.056
Drop-Out -206 -333 -147 -229 -0.229
Change in Graduate 147 333 206 229 0.229
Table 16. The Difference in Number of Student Graduates in Three Types of Connectivity
7.6. Chapter Remarks and Summary
This chapter presents the results of three types of SSAS model simulations. The simulation
consists of random selection attachment, computation of transition, cumulative, and steady state
probability distribution, determination of the students? school attendance states, connectivity
integration, learning coefficient integration, and attendance assessments. In addition, this
research presents three different models, PIOV, PAIV, and PAOV, in generating the initial
probabilities for student school attendance. The next day probability, Pk0(t+1) of the student
attendance state is determined by computing the difference of probability between students and
the average probabilities of the linked peers which reflect a positive and negative influence
coupled with the threshold value for the connectivity and the learning coefficient for interactions
between peers.
Furthermore, the SSAS model provides three different ways to create the connection
dynamically between students. The SSAS simulation forms the connectivity between the
students based on the random number evaluating against the threshold value of the connectivity
101
at ? ? 0.5 for the initial connectivity. The subsequent connectivity considers ? ? 0.5 and/or ? ?
0.95. The Median Connectivity uses threshold of ? ? 0.5 throughout the process, Virtual
Connectivity uses ? ? 0.95 throughout the process, and Hybrid Connectivity uses ? ? 0.5 for the
initial connections and ? ? 0.95 for the connected peers thereof.
This chapter also provides an assessment of student attendance. Assessments between the
unsupervised and supervised out-of-school suspensions are compared to estimate the overall
success or failure of the student in reference to graduating from high school.
102
8. CONCLUSIONS AND FUTURE RESEARCH
Based on the low percentage of 70% to 73% national and local high school graduation rates, the
notion is that if action is not taken for our younger generation and/or their educational
environment, especially for at-risk children, history will continue to repeat itself with higher
dropout, truancy, and juvenile delinquency rates, causing a negative impact on the student
individually and on society as a whole. Considering this appalling circumstance, this research
presents the SSAS model that can be used to evaluate the student school attendance
phenomenon. In particular, this research rigorously investigates the option of implementing a
better program to decrease the dropout and truancy rates. In doing so, the SSAS presents a new
approach, ?Supervised Out-of-School Suspension,? instead of continuing the unsupervised out-
of-school suspension program which is currently administered by the school system. The SSAS
model presents the student attendance phenomenon in order to assess the student school success
rates, which have been low as a result of having only unsupervised out-of-school suspensions.
The model presents three different ways to compose the transition probability distributions of the
student school attendance phenomenon.
The development of the SSAS explored the different methods of organizing and calculating the
initial transition probability: 1) The PIOV model employs the pure random selection by
generating the initial transition probability of the student school attendance in a random process.
2) The PAIV model provides a controlling mechanism by providing the initial transition
probability of the student school attendance through a pre-arranged input data table. 3) The
103
PAOV model provides controlling mechanism as well as the stability of the system by
implementing the pre-arranged output look-up table to generate the initial transition probability.
The model forms the connectivity between the students both dynamically and randomly.
Furthermore, the connectivity between the students is explored in three different ways: 1)
Median Connectivity that creates the link between the peers with ? ? 0.5, 2) Virtual Connectivity
which forms the connection at ? ? 0.95, and 3) Hybrid Connectivity that applies ? ? 0.5 for
initial connection and ? ? 0.95 for connected students thereof. The simulation results are
described in the following section.
8.1. Remarks and Recommendations
In the results of the SSAS model simulation, the PAOV model generated the most stable system
among all three models with 1.4% error; the POIV was the most unstable which resulted in 17%
error; and PAIV model exhibited 3.6% error. In terms of connectivity, first Median Connectivity
created 486 connections, Virtual Connectivity created the 917 links, and Hybrid Connectivity
created 676 connections out of 1000. The simulation explores the connectivity for a class size of
30 where Median Connectivity resulted in 14, Virtual Connectivity 27, and Hybrid Connectivity
20 connections out of 30. In terms of student school attendance assessment, under supervised
out-of-school suspensions, Virtual Connectivity produced improvement in number of students
graduating from high school with 333 students, 206 students from Hybrid Connectivity, and 147
students from Median Connectivity. Overall, there were 229 more students graduated with
supervised OOS in comparison to unsupervised OOS.
There are several major issues directly resulting from the current unsupervised out-of-school
suspension. First, unsupervised suspension leads to student failure with students misbehaving
104
further in an attempt to mask their inability to keep up, and finally convinced they will never be
able to catch up. Obviously, this is a vicious cycle where too many of our at-risk students find
themselves caught at a very young age. Then, the lack of supervision leads the child to the
streets. As the national report from police department statements, juvenile crime spikes when
students are not in school, making a case for finding new and creative ways to engage our youth
in constructive activities. Additionally, drop outs lead to complete loss of financial resources
and negative impact on the economy. In the current school system, the student failure rate is not
taken into consideration during the budget process. The budget is based on the total number of
students at full enrollment. Economic analysis related to the out-of-school suspension is
addressed in a following section.
8.2. Future Research
First, this research can be further advanced by optimizing the use of the SSAS model. Young
children are pre-assigned to a specific class or activities. The location of the student is well
defined at school or at home. Therefore, the SSAS model can be explored further by
concentrating on the position of the student. The particle swarm optimization algorithm is a
stochastic process used to optimize the population-based discrete and continuous problem. It is a
method of swarm intelligence techniques which is based on social and psychological principles
in order to provide the social behavior insights in terms of engineer perspectives [15]. In a given
situation, the particle swarm simulates social problems to evaluate a proposed solution
dynamically. A social network is defined by assigning interaction with neighbors for each
individual. Groups of these individuals form the population and they are defined as a random
probability in a given sample space with positions. The optimal candidate solution is achieved
105
by studying these individuals. These individuals are known as the particles. Then, the motion is
specified as best fitness throughout the iterative process to improve these particles candidate
solutions. During the iterative evaluation process, the particles remember the point at which they
had the best of the candidate successful solutions which refers to particle best or local best. This
information is available to their neighbors and thus each particle has access to neighbors?
information whether they have a successful solution or not. Ultimately, the number of these
successful solutions controls the flow of the movement in the search space of the social setting.
Each particle refers to an individual containing its position and velocity creating the swarm
population and moving dynamically through out the hyperspace, R. A particle contains two
pieces of critical information: 1) its best position memory and 2) the neighbor?s or global best
position. Therefore, each particle searches for better positions in the swarm hyperspace by
communicating good positions to each other and adjusting their own position and velocity based
on that information. This optimization is applicable for the SSAS model since it deals with a
similar stochastic process environment and the discrete nature of student school attendance
phenomena.
Second, the SSAS model can be expanded to analyze the economic impact of students failing
from school. The current school system formulates its annual budget based on the total number
of school age children available to enroll into the public school system [8, 11, 26]. However, the
student?s school attendance performance is not factored into the budgeting process or used as any
indicator of future resource requirements or those which might be misdirected to failing
academic procedures. The decision variables, X1, X2, and X3 and the output variables that SSAS
model provides in this study along with the steady state probability that enable us to compute the
expected cost can be used to study the expenditure correlation between the supervised and
106
unsupervised Out-of-School suspensions. Providing additional expenditures for alternatives to
putting the student unsupervised out-of-school and to maintaining some type of educational
setting during suspension is still a positive approach economically. This process will also
decrease juvenile crime if the supervised out-of-school suspension is in place. The student is
then able to return to the school current on course work without any difficulties. In addition,
more resources for court related expenditures are in the future if the students end up in trouble on
the street due to out-of-school suspension. A model can be used to evaluate the financial loss
due to unsupervised out-of-school suspension for the city leaders and school system officials in
examining the current unsupervised out-of-school suspension instead of continuing the program
without supervision, structure, and substance. Therefore, an economic cost-effect analysis would
be very useful in estimating and measuring the impact of the out-of-suspension on the school
budget and local economy.
There is no silver bullet to eradicate student truancy or dropouts. However, supervised out-of-
school suspension will reduce the drop-out rate and the number of juvenile crimes which in turn
will save innumerable dollars. Understanding the consequences of unsupervised out-of-school
suspension used for years throughout the U.S. and the subsequent financial loss, this research
provides a baseline model to evaluate the impact of unsupervised out-of-school suspension. The
results of the SSAS model indicated that there would be a significant improvement and impacts
on the problem related to student school attendance if the OOS is structured and supervised.
Also, the results of the model simulation indicated that the current unsupervised out-of-school
suspensions produce a higher dropout rate and failure rate in comparison to a ?Supervised Out-
of-School Suspension.? Therefore, SSAS provides an evaluation model for school official,
administrators, and others involved in the decision making process in our schools and
107
communities in improving the student success rate in terms of attendance. Overall, the SSAS
model provide a frame work for understanding and assessing the impacts of unsupervised out-of-
school suspensions and to propose a new attendance category, supervised out-of-school
suspensions, which would reduce the dropout and overall student failure rate. In addition, failure
from school at an early age greatly diminishes the potential earning capacity in later years
resulting in society as a whole being affected by the early failures of the few. Optimistically, the
SSAS model would provide opportunities to encourage a different approach to address the issue
related to OOS, thereby bringing new action to an ongoing problem, to ensure the success of
each student instead of continuing the past failing policies and practices.
108
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111
Appendix 1. SSAS MATLAB CODE WITH HYBRID CONNECTIVITY
clear all
clc
aMin=0; //define parameters
aMax=0.7;
afix=0.999;
g=0.6;
m=1-g;
m4=0.6;
m5=1-m4;
x1=0.6;
x2=1-x1;
MatrixLine=1;
Year=180;
Semester=90;
Week=5;
Period=180;
startCol=2;
iterCol=3;
color=['b','g','r','c','m','y','w']; //set colors for first seven students to be display on the chart
NumOfStudents=input('Number of students: ');
mycell={'Unsupervised'};
for i=1:NumOfStudents
xlswrite('result',mycell,'Present&Absent',[char(xlsColNum2Str(4*(i-
1)+2)),char(num2str(2))]);
end
mycell={'Supervised'};
for i=1:NumOfStudents
xlswrite('result',mycell,'Present&Absent',[char(xlsColNum2Str(4*(i-
1)+4)),char(num2str(2))]);
end
112
mycell={'Present'};
xlswrite('result',mycell,'Present&Absent',[char('A'),char(num2str(2))]);
mycell={'Absent'};
xlswrite('result',mycell,'Present&Absent',[char('A'),char(num2str(8))]);
mycell={'Total'};
xlswrite('result',mycell,'Present&Absent',[char('A'),char(num2str(7))]);
xlswrite('result',mycell,'Present&Absent',[char('A'),char(num2str(11))]);
mycell={'In Class'};
xlswrite('result',mycell,'Present&Absent',[char('A'),char(num2str(3))]);
mycell={'ISS'};
xlswrite('result',mycell,'Present&Absent',[char('A'),char(num2str(4))]);
mycell={'OOS'};
xlswrite('result',mycell,'Present&Absent',[char('A'),char(num2str(6))]);
xlswrite('result',mycell,'Present&Absent',[char('A'),char(num2str(9))]);
mycell={'Truancy'};
xlswrite('result',mycell,'Present&Absent',[char('A'),char(num2str(10))]);
mycell={'Excused'};
xlswrite('result',mycell,'Present&Absent',[char('A'),char(num2str(5))]);
%user inputs attendance
for i=1:NumOfStudents
NumOfAttend(i)=input(['Number of attendances for student ',num2str(i),': ']);
mycell={['Student',num2str(i)]};
xlswrite('result',mycell,'mresult',[char(xlsColNum2Str(3*(i-1)+2)),char(num2str(1))]);
xlswrite('result',mycell,'Attendances',[char(xlsColNum2Str(3*(i-
1)+1)),char(num2str(1))]);
xlswrite('result',mycell,'Present&Absent',[char(xlsColNum2Str(4*(i-
1)+2)),char(num2str(1))]);
end
AttenDist=xlsread('ModelStudent.xls','AttendDistribution','B8:F27');
[row,column]=size(AttenDist);
FixState=xlsread('ModelStudent.xls','DataInOutput','F28:F38');
for s=1:NumOfStudents
113
AbsencesProb{s}=0; %AbsencesProb is the Interaction Probability
InClassProb=NumOfAttend(s)/Period;
for i=1:row %get the first line of the Probability
if InClassProb-AttenDist(i,1)<0.0501&&InClassProb-AttenDist(i,1)>0
AbsencesProb{s}(1,1)=InClassProb;
AbsencesProb{s}(1,2)=(1-InClassProb)*AttenDist(row,2)/sum(AttenDist(row,2:5));
AbsencesProb{s}(1,3)=(1-InClassProb)*AttenDist(row,3)/sum(AttenDist(row,2:5));
AbsencesProb{s}(1,4)=(1-InClassProb)*AttenDist(row,4)/sum(AttenDist(row,2:5));
AbsencesProb{s}(1,5)=(1-InClassProb)*AttenDist(row,5)/sum(AttenDist(row,2:5));
break;
end
end
for i=2:5 %get the other lines probabilities
for j=[1,i]
if j==1
AbsencesProb{s}(i,j)=FixState((i-2)*3+1);
else
AbsencesProb{s}(i,j)=FixState((i-2)*3+2);
end
end
end
end
for s=1:NumOfStudents %set the first current state to 0
currentState(s)=0;
end
student=AbsencesProb; %get a copy of AbsencesProb
for i=1:NumOfStudents %calculate the steady state probability
steadyState{i}=student{i}^100;
xlswrite('result',steadyState{i},'Steady State',[char(xlsColNum2Str(6*(i-
1)+1)),char(num2str(2))]);
end
%get user input for the days, NumOfDays=input('Number of School days: ');
NumOfDays=180;
for n=1:NumOfDays
result(n,1)=n; %set the # of Day in the result set
for s=1:NumOfStudents
graphProbability(s)=student{s}(1,1);
end
114
if n<=20||mod(n,5)==0
figure();
hold on;
c=1;
title(['Day ',num2str(n)]);
xlabel('Student');
ylabel('Probability');
for s=1:NumOfStudents
if c>7
text(s,graphProbability(s),num2str(s));
else
plot(s,graphProbability(s),'rs','LineWidth',2,'MarkerEdgeColor','k','MarkerFaceColor',color(c),'M
arkerSize',5);
c=c+1;
end
end
end
%initialize the connectivity of students to -1 which means not calculated yet
for i=1:NumOfStudents
for j=1:NumOfStudents
conn(i,j)=-1;
end
end
%every student using the studentCopy{s}(1,1)same rand number in the result set
randNum=rand(1);
for s=1:NumOfStudents
result(n,startCol+iterCol*(s-1))=randNum;
result(n,startCol+iterCol*(s-1)+1)=currentState(s);
xlswrite('result',student{s},'InteractionProbability',[char(xlsColNum2Str((s-
1)*6+1)),char(num2str(MatrixLine))]);
cumStudent=student{s}; % the cumulative probability
for i=1:5
for cumState=2:5
cumStudent(i,cumState)=cumStudent(i,cumState-1)+student{s}(i,cumState);
end
end
xlswrite('result',cumStudent,'CumulativeProbability',[char(xlsColNum2Str((s-
1)*6+1)),char(num2str(MatrixLine))]);
%determine the next state and change the current state according to the matrix
tempRow=result(n,startCol+iterCol*(s-1)+1)+1;
for i=1:5
if randNumcumStudent(tempRow,6-i)
if i==1
result(n,startCol+iterCol*(s-1)+2)=4;
else
result(n,startCol+iterCol*(s-1)+2)=6-i;
end
currentState(s)=result(n,startCol+iterCol*(s-1)+2);
break;
end
end
sn=s+1; %s neighbor student: sn
matrixAverage=0;
sumNP=0;
numOfNP=0;
for counterSN=1:NumOfStudents-1
if sn>NumOfStudents
sn=sn-NumOfStudents;
end
tempRand=rand(1);
if n==1
if conn(s,sn)<0&&tempRand>=0.5
conn(s,sn)=1;
conn(sn,s)=1;
if n<=20||mod(n,5)==0
plot([s,sn],[graphProbability(s),graphProbability(sn)]);
end
elseif conn(s,sn)<0&&tempRand<0.5
conn(s,sn)=0;
conn(sn,s)=0;
end
elseif n>1
if conn(s,sn)<0&&tempRand>0.05
conn(s,sn)=1;
conn(sn,s)=1;
if n<=20||mod(n,5)==0
plot([s,sn],[graphProbability(s),graphProbability(sn)]);
end
elseif conn(s,sn)<0&&tempRand<=0.05
conn(s,sn)=0;
conn(sn,s)=0;
end
116
end
if conn(s,sn)==1
matrixAverage=matrixAverage+student{sn}(1,1)-student{s}(1,1);
sumNP=sumNP+student{sn}(1,1);
numOfNP=numOfNP+1;
end
sn=sn+1;
end
studentCopy{s}=student{s};
if numOfNP==0
studentCopy{s}(1,1)=afix*student{s}(1,1);
elseif matrixAverage>=0
studentCopy{s}(1,1)=afix*student{s}(1,1)+(1-afix)*sumNP/numOfNP/n;
else
studentCopy{s}(1,1)=afix*student{s}(1,1)-(1-afix)*sumNP/numOfNP/n;
end
TempStudent1=studentCopy{s}(1,1); % using look up table
for i=1:row %get the first line of the Probability
if TempStudent1-AttenDist(i,1)<0.05
studentCopy{s}(1,2)=(1-TempStudent1)*g*x1;
studentCopy{s}(1,3)=(1-TempStudent1)*g*x2;
studentCopy{s}(1,4)=(1-TempStudent1)*(1-g)*m4;
studentCopy{s}(1,5)=(1-TempStudent1)*1-sum(studentCopy{s}(1,1:4);
break;
end
end
disp([num2str(s),' ',num2str(n),' sumNP numOfNP MatrixAverage ']);
disp([num2str(sumNP),' ',num2str(numOfNP),' ',num2str(matrixAverage)]);
end
disp(conn); %print the connectivity table for each day
student=studentCopy;
%close the figure configuration
if n<=20||mod(n,5)==0
hold off;
end
MatrixLine=MatrixLine+8;
end
%output the total counts for each state
117
%start from column 3 which is the current state
for s=3:iterCol:startCol+iterCol*(NumOfStudents-1)+3
counter0=0;
counter1=0;
counter2=0;
counter3=0;
counter4=0;
for j=1:NumOfDays
if result(j,s)==0
counter0=counter0+1;
elseif result(j,s)==1
counter1=counter1+1;
elseif result(j,s)==2
counter2=counter2+1;
elseif result(j,s)==3
counter3=counter3+1;
else
counter4=counter4+1;
end
end
studentID=(s-3)/iterCol+1;
StateCouter(1,(studentID-1)*3+1)=counter0;
StateCouter(2,(studentID-1)*3+1)=counter1;
StateCouter(3,(studentID-1)*3+1)=counter2;
StateCouter(4,(studentID-1)*3+1)=counter3;
StateCouter(5,(studentID-1)*3+1)=counter4;
StateCouter(6,(studentID-1)*3+1)=sum(StateCouter(1:5,(studentID-1)*3+1));
StateCouter(1,(studentID-1)*3+2)=counter0/NumOfDays;
StateCouter(2,(studentID-1)*3+2)=counter1/NumOfDays;
StateCouter(3,(studentID-1)*3+2)=counter2/NumOfDays;
StateCouter(4,(studentID-1)*3+2)=counter3/NumOfDays;
StateCouter(5,(studentID-1)*3+2)=counter4/NumOfDays;
StateCouter(6,(studentID-1)*3+2)=sum(StateCouter(1:5,(studentID-1)*3+2));
StateCouter(1,(studentID-1)*3+3)=AbsencesProb{studentID}(1,1);
StateCouter(2,(studentID-1)*3+3)=AbsencesProb{studentID}(1,2);
StateCouter(3,(studentID-1)*3+3)=AbsencesProb{studentID}(1,3);
StateCouter(4,(studentID-1)*3+3)=AbsencesProb{studentID}(1,4);
StateCouter(5,(studentID-1)*3+3)=AbsencesProb{studentID}(1,5);
StateCouter(6,(studentID-1)*3+3)=sum(StateCouter(1:5,(studentID-1)*3+3));
end
%Calculate Present and Absent number of days as below
for s=1:NumOfStudents
118
Present(1,(s-1)*4+1)=StateCouter(1,(s-1)*3+1);
Present(1,(s-1)*4+2)=StateCouter(1,(s-1)*3+2);
Present(1,(s-1)*4+3)=StateCouter(1,(s-1)*3+1);
Present(1,(s-1)*4+4)=StateCouter(1,(s-1)*3+2);
Present(2,(s-1)*4+1)=StateCouter(2,(s-1)*3+1);
Present(2,(s-1)*4+2)=StateCouter(2,(s-1)*3+2);
Present(2,(s-1)*4+3)=StateCouter(2,(s-1)*3+1);
Present(2,(s-1)*4+4)=StateCouter(2,(s-1)*3+2);
Present(3,(s-1)*4+1)=StateCouter(5,(s-1)*3+1);
Present(3,(s-1)*4+2)=StateCouter(5,(s-1)*3+2);
Present(3,(s-1)*4+3)=StateCouter(5,(s-1)*3+1);
Present(3,(s-1)*4+4)=StateCouter(5,(s-1)*3+2);
Present(4,(s-1)*4+1)=0;
Present(4,(s-1)*4+2)=0;
Present(4,(s-1)*4+3)=StateCouter(3,(s-1)*3+1);
Present(4,(s-1)*4+4)=StateCouter(3,(s-1)*3+2);
Present(5,(s-1)*4+1)=sum(Present(1:4,(s-1)*4+1));
Present(5,(s-1)*4+2)=sum(Present(1:4,(s-1)*4+2));
Present(5,(s-1)*4+3)=sum(Present(1:4,(s-1)*4+3));
Present(5,(s-1)*4+4)=sum(Present(1:4,(s-1)*4+4));
Absent(1,(s-1)*4+1)=StateCouter(3,(s-1)*3+1);
Absent(1,(s-1)*4+2)=StateCouter(3,(s-1)*3+2);
Absent(1,(s-1)*4+3)=0;
Absent(1,(s-1)*4+4)=0;
Absent(2,(s-1)*4+1)=StateCouter(4,(s-1)*3+1);
Absent(2,(s-1)*4+2)=StateCouter(4,(s-1)*3+2);
Absent(2,(s-1)*4+3)=StateCouter(4,(s-1)*3+1);
Absent(2,(s-1)*4+4)=StateCouter(4,(s-1)*3+2);
Absent(3,(s-1)*4+1)=sum(Absent(1:2,(s-1)*4+1));
Absent(3,(s-1)*4+2)=sum(Absent(1:2,(s-1)*4+2));
Absent(3,(s-1)*4+3)=sum(Absent(1:2,(s-1)*4+3));
Absent(3,(s-1)*4+4)=sum(Absent(1:2,(s-1)*4+4));
end
if NumOfDays==180
%read the Make-Up lookup table
MakeUpTable=xlsread('ModelStudent','AttendDistribution','D59:D108');
mycell={'Unsupervised'};
xlswrite('result',mycell,'AttendAssmt','B1');
xlswrite('result',mycell,'AttendAssmt','H1');
mycell={'Supervised'};
xlswrite('result',mycell,'AttendAssmt','D1');
xlswrite('result',mycell,'AttendAssmt','J1');
119
mycell={'student ID'};
xlswrite('result',mycell,'AttendAssmt','A2');
mycell={'status'};
xlswrite('result',mycell,'AttendAssmt','B2');
xlswrite('result',mycell,'AttendAssmt','D2');
xlswrite('result',mycell,'AttendAssmt','G2');
mycell={'Promoted'};
xlswrite('result',mycell,'AttendAssmt','C2');
xlswrite('result',mycell,'AttendAssmt','E2');
xlswrite('result',{'Number'},'AttendAssmt','H2');
xlswrite('result','%','AttendAssmt','I2');
xlswrite('result',{'Number'},'AttendAssmt','J2');
xlswrite('result','%','AttendAssmt','K2');
xlswrite('result',{'Promoted'},'AttendAssmt','G3');
xlswrite('result',{'Repeated'},'AttendAssmt','G4');
xlswrite('result',{'Drop-Out'},'AttendAssmt','G5');
xlswrite('result',{'Total Promoted'},'AttendAssmt','G6');
%initial the counter for both unsupervised and supervised
UnsuperPromoted=0;
UnsuperMakeup=0;
UnsuperDropout=0;
SuperPromoted=0;
SuperMakeup=0;
SuperDropout=0;
for i=1:NumOfStudents
xlswrite('result',{['Student ',char(num2str(i))]},'AttendAssmt',['A',char(num2str(i+2))]);
TempRandom=rand(1);
if Present(5,(i-1)*4+1)>=165
UnsuperPromoted=UnsuperPromoted+1;
xlswrite('result',{'Promoted'},'AttendAssmt',['B',char(num2str(i+2))]);
xlswrite('result',{'Yes'},'AttendAssmt',['C',char(num2str(i+2))]);
else
if TempRandom<=MakeUpTable(Absent(3,(i-1)*4+1)-15,1)*7/40
UnsuperMakeup=UnsuperMakeup+1;
xlswrite('result',{'Make Up'},'AttendAssmt',['B',char(num2str(i+2))]);
xlswrite('result',{'Yes'},'AttendAssmt',['C',char(num2str(i+2))]);
else
UnsuperDropout=UnsuperDropout+1;
xlswrite('result',{'Drop Out'},'AttendAssmt',['B',char(num2str(i+2))]);
120
xlswrite('result',{'No'},'AttendAssmt',['C',char(num2str(i+2))]);
end
end
if Present(5,(i-1)*4+3)>=165
SuperPromoted=SuperPromoted+1;
xlswrite('result',{'Promoted'},'AttendAssmt',['D',char(num2str(i+2))]);
xlswrite('result',{'Yes'},'AttendAssmt',['E',char(num2str(i+2))]);
else
if TempRandom<=MakeUpTable(Absent(3,(i-1)*4+3)-15,1)*7/40
SuperMakeup=SuperMakeup+1;
xlswrite('result',{'Make Up'},'AttendAssmt',['D',char(num2str(i+2))]);
xlswrite('result',{'Yes'},'AttendAssmt',['E',char(num2str(i+2))]);
else
SuperDropout=SuperDropout+1;
xlswrite('result',{'Drop Out'},'AttendAssmt',['D',char(num2str(i+2))]);
xlswrite('result',{'No'},'AttendAssmt',['E',char(num2str(i+2))]);
end
end
end
UnsuperGraduate=UnsuperPromoted+UnsuperMakeup;
SuperGraduate=SuperPromoted+SuperMakeup;
AssmtVector(1,1)=UnsuperPromoted;
AssmtVector(1,2)=UnsuperPromoted/NumOfStudents;
AssmtVector(2,1)=UnsuperMakeup;
AssmtVector(2,2)=UnsuperMakeup/NumOfStudents;
AssmtVector(3,1)=UnsuperDropout;
AssmtVector(3,2)=UnsuperDropout/NumOfStudents;
AssmtVector(4,1)=UnsuperGraduate;
AssmtVector(4,2)=UnsuperGraduate/NumOfStudents;
AssmtVector(1,3)=SuperPromoted;
AssmtVector(1,4)=SuperPromoted/NumOfStudents;
AssmtVector(2,3)=SuperMakeup;
AssmtVector(2,4)=SuperMakeup/NumOfStudents;
AssmtVector(3,3)=SuperDropout;
AssmtVector(3,4)=SuperDropout/NumOfStudents;
AssmtVector(4,3)=SuperGraduate;
AssmtVector(4,4)=SuperGraduate/NumOfStudents;
xlswrite('result',AssmtVector,'AttendAssmt','H3');
end
xlswrite('result',result,'mresult','A2');
xlswrite('result',StateCouter,'Attendances','A2');
xlswrite('result',Present,'Present&Absent','B3');
xlswrite('result',Absent,'Present&Absent','B9'); ?????????????????..end
121
Appendix 2. THE ATTENDANCE STATES OF THE STUDENTS
Days Random # Stud1 Stud2 Stud3 Stud4 Stud5 Stud6 Stud7 Stud8 Stud9
1 0.814724 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
2 0.00782 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
3 0.186571 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
4 0.74849 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
5 0.377512 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
6 0.962959 0 4 0 0 0 1 0 3 0 4 0 0 0 2 0 3 0 4
7 0.789256 4 0 0 0 1 1 3 3 4 0 0 0 2 2 3 3 4 0
8 0.87345 0 4 0 0 1 1 3 3 0 0 0 0 2 2 3 3 0 1
9 0.935234 4 0 0 0 1 1 3 3 0 2 0 0 2 2 3 3 1 1
10 0.971035 0 4 0 0 1 1 3 3 2 2 0 1 2 2 3 3 1 1
11 0.944107 4 0 0 0 1 1 3 3 2 2 1 1 2 2 3 3 1 1
12 0.981283 0 4 0 3 1 1 3 3 2 2 1 1 2 2 3 3 1 1
13 0.194648 4 0 3 0 1 0 3 0 2 0 1 0 2 0 3 0 1 0
14 0.884512 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
15 0.323787 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0
16 0.781287 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
17 0.874201 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
18 0.250516 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0
19 0.499074 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
20 0.364042 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
21 0.242964 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
22 0.501671 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
23 0.642184 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
24 0.171214 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
25 0.813498 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
26 0.102277 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
27 0.178597 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
28 0.342104 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
29 0.61555 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
30 0.594426 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
31 0.114232 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
32 0.293887 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
33 0.700893 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
34 0.407163 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
35 0.972325 0 4 0 3 0 4 0 4 0 4 0 3 0 4 0 4 0 4
122
Days Random # Stud1 Stud2 Stud3 Stud4 Stud5 Stud6 Stud7 Stud8 Stud9
36 0.574937 4 0 3 0 4 0 4 0 4 0 3 0 4 0 4 0 4 0
37 0.743245 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
38 0.333447 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
39 0.06737 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
40 0.09843 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
41 0.92828 0 4 0 0 0 1 0 2 0 3 0 0 0 2 0 2 0 3
42 0.125912 4 0 0 0 1 0 2 0 3 0 0 0 2 0 2 0 3 0
43 0.718208 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
44 0.281619 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
45 0.800691 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
46 0.009753 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
47 0.075959 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
48 0.28207 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
49 0.147308 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
50 0.691719 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
51 0.231021 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
52 0.311761 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
53 0.970524 0 4 0 4 0 4 0 4 0 4 0 4 0 4 0 4 0 4
54 0.823455 4 0 4 0 4 0 4 0 4 0 4 0 4 0 4 0 4 0
55 0.974895 0 4 0 4 0 4 0 4 0 4 0 4 0 4 0 4 0 4
56 0.5405 4 0 4 0 4 0 4 0 4 0 4 0 4 0 4 0 4 0
57 0.963649 0 4 0 3 0 4 0 4 0 4 0 3 0 4 0 4 0 4
58 0.835781 4 0 3 3 4 0 4 0 4 0 3 3 4 0 4 0 4 0
59 0.644164 0 4 3 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0
60 0.929142 4 0 0 1 0 2 0 3 0 3 0 1 0 2 0 3 0 4
61 0.999258 0 4 1 1 2 2 3 3 3 3 1 1 2 2 3 3 4 4
62 0.199395 4 0 1 0 2 0 3 0 3 0 1 0 2 0 3 0 4 0
63 0.595685 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
64 0.734683 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
65 0.061514 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
66 0.132409 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
67 0.500203 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
68 0.656837 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
69 0.962317 0 4 0 3 0 4 0 4 0 4 0 4 0 4 0 4 0 4
70 0.718891 4 0 3 3 4 0 4 0 4 0 4 0 4 0 4 0 4 0
123
Days Random # Stud1 Stud2 Stud3 Stud4 Stud5 Stud6 Stud7 Stud8 Stud9
71 0.972249 0 4 3 3 0 4 0 4 0 4 0 4 0 4 0 4 0 4
72 0.100181 4 0 3 0 4 0 4 0 4 0 4 0 4 0 4 0 4 0
73 0.74716 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
74 0.566045 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
75 0.134025 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
76 0.803425 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
77 0.954566 1 1 0 3 0 4 0 4 0 4 0 3 0 4 0 4 0 4
78 0.255384 1 0 3 0 4 0 4 0 4 0 3 0 4 0 4 0 4 0
79 0.421467 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
80 0.343712 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
81 0.178318 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
82 0.94905 1 1 0 3 0 3 0 4 0 4 0 3 0 4 0 4 0 4
83 0.117084 1 0 3 0 3 0 4 0 4 0 3 0 4 0 4 0 4 0
84 0.971946 0 4 0 4 0 4 0 4 0 4 0 4 0 4 0 4 0 4
85 0.360879 4 0 4 0 4 0 4 0 4 0 4 0 4 0 4 0 4 0
86 0.917657 0 4 0 1 0 2 0 3 0 3 0 2 0 3 0 3 0 3
87 0.132365 4 0 1 0 2 0 3 0 3 0 2 0 3 0 3 0 3 0
88 0.471583 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
89 0.382117 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
90 0.970866 0 4 0 4 0 4 0 4 0 4 0 4 0 4 0 4 0 4
91 0.136015 4 0 4 0 4 0 4 0 4 0 4 0 4 0 4 0 4 0
92 0.712946 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
93 0.204704 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
94 0.81021 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
95 0.437507 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0
96 0.043786 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
97 0.03557 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
98 0.925134 0 4 0 2 0 3 0 3 0 3 0 2 0 3 0 3 0 4
99 0.79837 4 0 2 2 3 3 3 3 3 3 2 2 3 3 3 3 4 0
100 0.039324 0 1 2 0 3 0 3 0 3 0 2 0 3 0 3 0 0 0
101 0.858034 1 1 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 2
102 0.889536 1 1 0 1 0 1 1 1 1 1 0 1 1 1 1 1 2 2
103 0.611741 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 2 0
104 0.636304 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
105 0.477704 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
124
Days Random # Stud1 Stud2 Stud3 Stud4 Stud5 Stud6 Stud7 Stud8 Stud9
106 0.951487 0 4 0 4 0 4 0 4 0 4 0 4 0 4 0 4 0 4
107 0.991735 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
108 0.747589 4 0 4 0 4 0 4 0 4 0 4 0 4 0 4 0 4 0
109 0.740532 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
110 0.329241 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
111 0.921097 0 4 0 2 0 3 0 3 0 3 0 3 0 3 0 3 0 4
112 0.630423 4 0 2 0 3 0 3 0 3 0 3 0 3 0 3 0 4 0
113 0.36277 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
114 0.512032 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
115 0.781006 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
116 0.957533 4 0 0 4 0 4 0 4 0 4 0 4 0 4 0 4 0 4
117 0.718802 0 4 4 0 4 0 4 0 4 0 4 0 4 0 4 0 4 0
118 0.001877 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
119 0.422026 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
120 0.346543 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
121 0.099366 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
122 0.685273 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
123 0.891401 0 4 0 1 0 2 0 3 0 3 0 2 0 2 0 3 0 3
124 0.456547 4 0 1 0 2 0 3 0 3 0 2 0 2 0 3 0 3 0
125 0.174784 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
126 0.3021 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
127 0.202489 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
128 0.049169 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
129 0.993881 0 4 0 4 0 4 0 4 0 4 0 4 0 4 0 4 0 4
130 0.537953 4 0 4 0 4 0 4 0 4 0 4 0 4 0 4 0 4 0
131 0.10516 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
132 0.15506 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
133 0.228617 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
134 0.526155 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
135 0.920498 0 4 0 3 0 3 0 3 0 4 0 3 0 3 0 4 0 4
136 0.397261 4 0 3 0 3 0 3 0 4 0 3 0 3 0 4 0 4 0
137 0.287911 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
138 0.237313 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
139 0.054675 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
140 0.940114 1 1 0 4 0 4 0 4 0 4 0 4 0 4 0 4 0 4
125
Days Random # Stud1 Stud2 Stud3 Stud4 Stud5 Stud6 Stud7 Stud8 Stud9
141 0.029825 1 0 4 0 4 0 4 0 4 0 4 0 4 0 4 0 4 0
142 0.470704 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
143 0.82201 3 3 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 2
144 0.441015 3 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 2 0
145 0.676189 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
146 0.512393 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
147 0.245535 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
148 0.151448 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
149 0.573305 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
150 0.102441 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
151 0.070791 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
152 0.447446 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
153 0.060653 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
154 0.110286 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
155 0.956313 0 4 0 4 0 4 0 4 0 4 0 4 0 4 0 4 0 4
156 0.446454 4 0 4 0 4 0 4 0 4 0 4 0 4 0 4 0 4 0
157 0.48195 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
158 0.14739 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
159 0.621687 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
160 0.773824 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
161 0.455521 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0
162 0.09305 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
163 0.540615 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
164 0.974811 3 3 0 4 0 4 0 4 0 4 0 4 0 4 0 4 0 4
165 0.387845 3 0 4 0 4 0 4 0 4 0 4 0 4 0 4 0 4 0
166 0.766563 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
167 0.297282 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0
168 0.388117 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
169 0.25183 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
170 0.957281 0 4 0 4 0 4 0 4 0 4 0 4 0 4 0 4 0 4
171 0.091749 4 0 4 0 4 0 4 0 4 0 4 0 4 0 4 0 4 0
172 0.490297 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
173 0.58907 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
174 0.494992 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
175 0.792579 3 3 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 2
126
Days Random # Stud1 Stud2 Stud3 Stud4 Stud5 Stud6 Stud7 Stud8 Stud9
176 0.866647 3 3 0 2 0 2 1 1 1 1 0 2 0 2 1 1 2 2
177 0.678189 3 0 2 0 2 0 1 0 1 0 2 0 2 0 1 0 2 0
178 0.589848 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
179 0.220915 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
180 0.167721 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0