BIBLIOGRAPHY
ZENAIDA U. GARAMBAS, May 2007, Markov Chain Modeling and
Forecasting of Saint Louis University Enrolment on Selected Degree Programs in the
Tertiary Level
Adviser: Dr. Salvacion Z. Beligan
ABSTRACT

This study aimed at building a Markov chain to model the enrolment of selected
colleges in Saint Louis University. The average of the enrolment of the past ten years
was used to form the initial state vector. In addition to the enrolment statistics, the total
number of drop-outs, shifters and transferees were used to complete the transition matrix.

The analysis and interpretation of data resulted in the projection of enrolment for
the next five years for the selected degree programs at Saint Louis University. Based on
the computations made, there will be an increase in enrolment per year from school year
2007-2008 to school year 2011-2012. The increase, however, is not constant or the same
for all the degree programs. The following degree programs will have a very little
increase in the number of incoming first years, that is, less than 40 students increase in
each school year: BS Computer Science, BS Social Work, Bachelor of Philosophy, BS
Psychology, BS Industrial Engineering, AB Political Science, and BS Chemical
Engineering. On the other hand, the following colleges have big number of additional
incoming first years in the succeeding years after 2007-2008: BS in Accountancy, BS in
Electronics and Communications Engineering, BS Architecture, BS Civil Engineering,

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and BS Information Technology. The other degree programs will just have an average of
additional one section per school year.

This would help administrators in their five-year plan pertaining, but not limited
to, the number of new faculty members to be hired, availability of classrooms and budget.

It is recommended that further studies be conducted on forecasting student
enrolment and include other variables such as number of students who have withdrawn,
transferred to another school, and other pertinent data. Also, the probability that a student
will graduate given that he is now in the fourth year could also be added in the transition
matrix.

Other related studies such as finding the number of graduates employed in their
respective field of specialization, employed in fields other than their specialization, or the
number of graduates who go abroad may also be good areas of research studies.

Also, it is recommended that the other degree programs not included here be
analyzed in the same manner.


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TABLE OF CONTENTS












Page


Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

i
Thesis Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

i
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

iii
INTRODUCTION

Background of the Study . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . .

4
Objectives of the Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4
Importance of the Study . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5
Scope and Delimitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

REVIEW OF RELATED LITERATURE

Forecasting and Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
Markov Chain Application . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
Theoretical Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
Definition of Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24
METHODS AND PROCEDURES
Research Design and Methodology . . . . . . . . . . . . . . . . . . .

28
Transition Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29
Population of the Study . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29
Data Gathering Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30
Statistical Tool . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30
iii

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Projection of the Number of Students . . . . . . . . . . . . . . . . .

31
RESULTS AND DISCUSSION
Preliminary Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
The Transition Probabilities . . . . . . . . . . . . . . . . . . . . . . . .

34
Projected Number of Students for the Next Five Years . . . .
46
SUMMARY, CONCLUSION, RECOMMENDATION
Summary of Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58
Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58
Recommendation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59
LITERATURE CITED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
APPENDICES
A. Letter request to the Management In formation Systems 62
B. Total Enrollment from School Year 1997-1998 to

2006-2007 by Degree Program . . . . . . . . . . . . . . . . . . . .
64
C. Total Drop-outs From School Year 1997-1998 to

2006-2007 by Degree Program . . . . . . . . . . . . . . . . . . .

71
D. Total Shifters From School Year 1997-1998 to

2006-2007 by Degree Program . . . . . . . . . . . . . . . . . . . .
78
iv

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INTRODUCTION

Background of the Study

Education is the best legacy of parents to their children. Parents desire that
their children choose a degree that would land them to a gainful employment. A
high school graduate, therefore, who is contemplating to go to college uses
forecasting in choosing a major field of specialization in line with what he wants
to do when he graduates and what his career opportunities will be. His choice of a
college major involves forecasts of how well he will do in his courses.

The desire to forecast the future is as old as the human race. In ancient
times, people relied on prophets, soothsayers, and crystal balls. Today there are
computers that have an impressive array of quantitative capabilities which are
powerful tools used in predicting the future.

Many decisions are based on forecasts of one sort or another. A politician
planning to run in a forthcoming election may forecast his probability of winning
by using the past and present data. The demands for blood types in a blood-bank
inventory model exhibit both seasonal and trend patterns which can be estimated
by using statistical forecasting models. An investor relies on the help of financial
analysts who forecast market trends. A businessman studies the history of demand
for his product and make some forecasts on his revenue and sales.

Every start of the semester, the enrolment statistics are posted on a big
bulletin board at the lobby of the school. The data are fascinating, yet they simply
Markov Chain Modeling and Forecasting of Saint Louis University Enrolment on
Selected Degree Programs in the Tertiary Level / Zenaida U. Garambas. 2007

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show numbers of enrollees for each college for the current semester. But as to the
growth or decline compared to the previous years, one cannot know. Had there
been a steady increase, say, for the last five years? Will this growth continue for
the next five years? Or will there be a time for a plateau or even a decline in the
enrolment for a particular college?

Certainly, it is the utmost concern of school administrators to attract
students since students are their primary clientele. Thus, they should try to find
out what courses attract high school graduates most, and thus serve them better.
Furthermore, they are trying their best to find ways and means to improve their
curriculum, to determine which courses are to be phased out and what new
courses are needed to keep abreast with the changing times and the
technologically-oriented society.

Saint Louis University is one of the big universities in Baguio City which
was founded by the CICM missionaries in 1927. At present, it offers elementary,
secondary, tertiary, and post-graduate education. In the tertiary level, there are
sixty-two undergraduate degree offerings in the seven colleges. The colleges are
as follows: College of Commerce and Accountancy, College of Engineering and
Architecture, College of Education, College of Human Sciences, College of
Information and Computing Sciences, College of Natural Sciences, and College
of Nursing.
Markov Chain Modeling and Forecasting of Saint Louis University Enrolment on
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The researcher selected only some of the degree programs to be used in
this study to analyze and forecast. College of Nursing, for instance, retains a
definite number of students for the third and fourth year levels regardless of the
number of enrollees accepted for the first year level. The College of Commerce
and Accountancy has just phased out some courses and replaced by new ones.
What other courses are likely to be phased out in the future due to eventual
decline in the enrolment or due to very little increase in the forthcoming
freshmen? Forecasting would help answer these questions.

At the time of this study, there are several new course offerings in Saint
Louis University such as BS in Mechatronics Engineering, BS in Resort,
Restaurant and Hotel Management, Bachelor in Library and Information Science,
BS in Information Management, and BS in Human Resource Development.
However, there are degree offerings being phased out.

There are many forecasting techniques but the researcher believes that
Markov chain analysis is the best forecasting tool for this particular study because
students can be classified as belonging to the different states (freshmen,
sophomores, juniors, seniors, drop-outs, shifters or transferees). These are referred
to as the current states or conditions and can be used to forecast the next state.
Levin, et al (1979) described Markov analysis as a method of analyzing the
current behavior of some variable in an effort to predict the future behavior of that
Markov Chain Modeling and Forecasting of Saint Louis University Enrolment on
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same variable. They added further that Markov analysis has been successfully
applied to a wide variety of decision situations.

This study utilized the past and the present enrolment data and used them
to build a model which was used in forecasting future enrolment.

Statement of the Problem

The study aims to analyze and assess the enrolment statistics on selected
degree programs in the tertiary level of Saint Louis University. Specifically, the
researcher would like to answer the following questions:
1. What is the Markov chain model that fits the enrolment statistics for
the last ten years on the selected degree programs?
2. Based on the enrolment data for the past ten years, what is the
forecasted or projected enrollment pattern for the next five years on
each of the selected degree programs?

Objectives of the Study

This study aims to determine the enrollment pattern on selected degree
programs of the different colleges in Saint Louis University. Specifically, the
study aims to:
1. Construct a Markov chain model that fits the enrolment data for the
last ten years from school year 1997–1998 to school year 2006–2007
on the selected degree programs.
Markov Chain Modeling and Forecasting of Saint Louis University Enrolment on
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2. Forecast or project the enrolment pattern for the next five years on
each of the selected degree programs based on the enrolment for the
last ten years using Markov Chain Analysis.

Importance of the Study

The lifeblood of any educational institution is the number of enrollees. It
is therefore imperative to study past and present enrolment trend so as to be able
to project students’ enrolment distribution for the next, say, five years for the
preparation of a five-year development plan by the administration. This is an
important aspect in the planning and allocation of annual operating and budget
resources.

A problem that faces school administrators across the country is predicting
student enrollments in the coming years. The ability to predict the number of
current students that will be enrolled in future years is a critical factor in the
determination of future enrolments. For that reason an administrator would like to
know the probability that a currently enrolled freshman will be enrolled at the
same university as a senior three years hence.

This study focused on determining the enrolment behavior of selected
colleges in a university and consequently made forecasts. Results from this study
may provide information to school administrators, planning officers, and
researchers. Senior high school students could also utilize this study to help them
in their decision-making and planning activities. This study will also help the
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administrators in deciding which offerings are to be phased out and how long will
others be still offered. The forecasts may be used as benchmark for the Admission
Office to regulate enrolment for new students, and also for faculty requirements.

Scope and Delimitation of the Study

Enrolment data on selected degree programs in Saint Louis University for
SY 1997–1998 to SY 2006–2007 were gathered and utilized in this study. It also
included the following: 1) the number of continuing students by year level, 2) the
number of students who shift to another course, and 3) the number of students
transferring to SLU from another school.

This study was limited to the enrolment data in Saint Louis University in
the College of Engineering and Architecture, College of Information and
Computing Sciences, the College of Human Sciences. There were eight degree
programs selected from the College of Engineering and Architecture (BSME,
BSChe, BSIE, BSCE, BSEE, BSECE, BSGE and BS Architecture), seven from
the College of Human Sciences (AB Econ, AB English, Bachelor of Philosophy,
AB Political Sciences, AB Communication, BS Psychology, and BS Social
Work), three from the College of Information and Computing Sciences (BS Math,
BSIT, BSCS), and one from the College of Accountancy and Commerce (BS
Accountancy).

The forecasts were made on each degree program using Markov Chain
Analysis. A chain was constructed by multiplying the initial state vector by the
Markov Chain Modeling and Forecasting of Saint Louis University Enrolment on
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formulated transition matrix, and the resulting row vector would again be
multiplied by the same transition matrix, and so on; thus forming a Markov chain.

This study was conducted from November 2006 to April 2007 at Saint
Louis University, Baguio City.
Markov Chain Modeling and Forecasting of Saint Louis University Enrolment on
Selected Degree Programs in the Tertiary Level / Zenaida U. Garambas. 2007

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REVIEW OF RELATED LITERATURE

Forecasting and Modeling

The selection and implementation of the proper forecast methodology
have always been important in planning and decision-making for most firms,
institutions, and agencies.

The need to obtain a forecast has to be identified at the appropriate
management level. The historical data required must be compiled and by studying
these data, an appropriate model can be structured. A forecasting procedure that
behaves well under the model should be selected.

Forecasting, as defined by Freud (1977), is the process of arriving at the
values that some variables may take on the specific future time. Anderson (1971)
considered forecasting as a process of predicting the future with the knowledge of
the past occurrence. Pindick (1981) stated that forecasting is the prediction of
future observations of time series. Lee (1971) stated that the individual business
man, the farmer, officers of a union responsible for negotiating appropriate labor-
management relations and all the rest of the community in one way or another
should have heavy stake in the making of economic forecasts. Anderson (1971)
stated that most companies can forecast total demand for all products with errors
of less than 5%.

Smith (1985) stated that businesses routinely make forecasts of their sales,
materials costs, interest rates, and a dizzying array of other things. He added that
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government policymakers forecast output, unemployment, inflation, interest rates,
and so on.


Hillier and Lieberman (1986) asserted that forecasting is an essential
component of a successful inventory system. They added that forecasting also
plays an important role in industry in the areas of marketing, financial planning,
and production. A forecast, however, is not the final product itself; it is to be used
as a tool in making a managerial decision.

Anderson, et al (1971), stated that an essential aspect of managing any
organization is planning for the future, and indeed, long-run success of an
organization is closely related to how well management is able to anticipate the
future and develop appropriate strategies. He added however, that a forecast is
simply a prediction of what will happen in the future. Managers must learn to
accept the fact that regardless of the technique used, they will not be able to
develop perfect forecasts. Good judgment, intuition, and an awareness of the state
of the economy may give a manager a rough idea or “feeling” of what is likely to
happen in the future.

According to Galliers (1987), descriptive information attempted to replace
the real world of objects of rules and events by a set of symbols which map the
real world on one-to-one basis information systems included on the basis of
assumptions of statistical distribution and behavior.

Kerliger (1973) pointed out that one can predict from an independent to a
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dependent variable. Further, the author stressed the existence or non-existence of
relation can be predicted and can even tell something that happened in the past.

Aczel (1989) also noted that many variables, such as sales and other
business variables, can be forecast, and the use of statistics plays an important
role in forecasting these variables.

Management needs a clear picture of the nature of trends and the sort of
thing that are likely to happen in the economy. The decision-maker seeks to shape
the future but he works in the context of a broad network of economic forces.
Forecasting is used in planning and decision–making that involve individual
products. The nature of the demand curve for a product can be vital for a firm to
know yet hard to calculate (Lynn, 1974).

Pankratz (1981) cited three examples of how forecasting can aid in
planning:
1. A business firm manufactures computerized television games for retail
sale. If the firm does not manufacture and keep in inventory enough units
of its product to meet demand, it could lose sales to a competitor and thus
have lower profits. On the other hand, keeping an inventory is costly. If
the inventory of finished goods is too large, the firm will have higher
carrying costs and lower profits than otherwise. This firm can maximize
profits ( other things equal) by properly balancing the benefits of holding
inventory (avoiding lost sales) against costs (interest charges). Clearly, the
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inventory level the firm should aim for depends partly on the anticipated
amount of future sales. Unfortunately, future sales can rarely be known
with certainty so decisions about production and inventory levels must be
based on sales forecasts.
2. A nonprofit organization provides temporary room and board for

indigent transients in a large city in the northern part of the United States.

The number of individuals requesting the aid each month follows a

complex seasonal pattern. The directors of the organization could better

plan their fund-raising efforts and their ordering of food and clothing if

they had reliable forecasts of the seasonal variation in aid requests.
3. A specialty foods wholesaler knows from experience that sales are usually
sufficient to warrant delivery runs into a given geographic region if
population density exceeds a critical minimum number. Forecasting the
exact amount of sales is not necessary for this decision. The wholesaler
uses census information about population density to choose which regions
to serve.


Pankratz added that forecasts can be formed in many different ways,
depending on the purpose and importance of the forecasts as well as the costs of
the alternative forecasting methods. The food wholesaler in the example above
combines his or her experience and judgment with a few minutes looking up
census data. But the television game manufacturer might employ a trained
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statistician or economist to develop sophisticated mathematical and statistical
models in an effort to achieve close control over inventory levels.

Santos et. al. (1975) applied forecasting method in projecting the student’s
enrollment to three secondary institutions through the use of linear regression
analysis which served as a basis for educational planners to provide
accommodation for school children who seek admission to the school system in
the future.

Carino (2002) conducted a study aimed to forecast fish prices in Benguet
to help fish producers in deciding on the right kind of fish to produce, when to
culture, and how to gain more yield. Gannapao (1997) made a study aimed to
forecast market retail prices of selected fruit vegetables using time series analysis
and was able to come up with the predicted market retail prices of selected fruit
vegetables for the next five years.

Hillier and Lieberman (1986) stated that forecasts can be obtained by
using qualitative and quantitative techniques. In the former case, a forecast is
usually the result of an expression of one or more experts’ personal judgment or
opinion, and it is often called a judgmental technique. They further cited the
following examples whereby forecasting is applied: (1) a major research
university calls in its leading economists every September to obtain their
judgment on what to expect as an inflation rate for the next academic year –
number crucial to the budgeting process. This number is generally arrived at by
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consensus after prolonged discussion by the economists, (2) the daily market
closing prices of a particular stock over the period of a year constitute a time
series. Time series analysis exploits techniques that utilize these data for
forecasting the values that the variable of interest will take on in a future time, (3)
the West Coast distributor of 10-speed bicycles wants to make quarterly sales
forecasts for planning purposes; he wants to forecast the sales that will occur
during the next quarter; and (4) forecasting the total sales of a textbook in a given
period may be functionally related to the mail order sales during the same period.
Data on mail order sales and total sales over previous periods may be used to
forecast total sales in a future period given the mail order sales for that period.

Anderson (1971) classified forecasting methods as quantitative or
qualitative. Quantitative forecasting methods can be used when (1) past
information about the variable being forecast is available, (2) the information can
be quantified, and (3) a reasonable assumption is that the pattern of the past will
continue into the future. Furthermore, Glichist (1976) enumerated three methods
of forecasting such as (1)intuitive methods, (2) the causal methods, and (3) the
extrapolative methods. These extrapolative methods are based on the
extrapolation into the future of the features shown by relevant data in the past;
these are statistical as well as mathematical in nature. Some widely used methods
under this are the moving averages, exponential smoothing, trend projection, and
the stepwise autoregressive.
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In decision-making problems we are often faced with making decisions
based on phenomena that have uncertainty associated with them. Hillier and
Lieberman (1986) pointed out that this uncertainty is caused by inherent variation
due to sources of variation that elude control or due to the inconsistency of natural
phenomena. They suggest that rather than treat this variability qualitatively, it can
be incorporated into the mathematical model and thus handle it quantitatively.
They added that this generally can be accomplished if the natural phenomena
exhibit some degree of regularity, so that their variation can be described by a
probability model.

Some basic models for forecasting are the following: 1) constant mean
models; 2) linear trend models; 3) regression models; 4) stochastic models; 5)
seasonal models; 6) growth models; and 7) multivariate models.

Stochastic models are used when random elements play dominant part in
determining the structure of the model. Time series models are classified into two
classes namely: stationary time series and non-stationary time series. A stationary
series is one whose graph exhibits no trend and the series remains in equilibrium.
The data cluster about a constant mean. A series not satisfying these conditions
are known to be non- stationary. A non-stationary series is usually transformed to
a stationary state before modeling because it is quite difficult to build models on
non-stationary series. The study on how a random variable evolves over time
includes stochastic processes.
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Dinkel, et al (1978) defined stochastic process as a collection of random
variables whose values are observed at certain points in time. Also, Hillier and
Lieberman (1986) defined stochastic process to be simply an indexed collection
of random variables (Xt) where the index t runs through a given set T, where T is
taken to be the set of non-negative integers, and Xt represents a measurable
characteristic of interest at time t. They added further that a consideration of the
behavior of a system operating for some period of time often leads to the analysis
of a stochastic process.

A stochastic process made up of an infinite sequence of trials is called a
Markov chain, named after the Russian mathematician Andrei Andreevich
Markov (1856 – 1922) who developed much of the modern theory of stochastic
processes. Levin, et al (1979) stated that Andrei Markov first used this process to
describe the behavior of particles of gas in a closed container.

Markov Chain Application



Markov chains have been applied in areas such as education, marketing,
health services, finance, accounting, and production. Markov processes present
one of the best-known and most useful classes of stochastic processes.

Budnick, Mojena, and Vollmann (1977) defined Markov chain as a
stochastic process with the following properties: (1) discrete state space, (2)
Markovian property, and (3) one-step transition probabilities which remain
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constant over time (termed as stationary transition probabilities). And they further
characterized Markovian property as follows:
Given that the present (or most recent) state is known, the conditional
probability of the next state is independent of states prior to the present (or most
recent) state, that is,



P(Xt+1 = xt+1 | X0 = x0, X1 = x1,….., Xt = xt) = P(Xt+1 = xt+1 | Xt = xt)

for t = 0, 1, …. and all possible sequences for state values. Note that an upper-
case letter represents the random variable and a lower-case letter represents a
specific value of the random variable (termed as random variate).


According to Hillier and Lieberman (1986), the Markovian property can
be shown to be equivalent to stating that the conditional probability of any future
“event”, given any past “event” and the present state Xt = i, is independent of the
past event and depends only upon the present state of the process.

Gallin (1984) paraphrased the Markov process, that is, if the system is in
state i at a given time n, there is a fixed probability Pij that it will be in the state j
at time n+1. In other words, Pij represents the probability of going from state to
state or unit time period. The fact that Pij does not depend on n means that we can
calculate the probability of the next state of the system, if we know the current
state.

According to Mizrahi and Sullivan (1976), the Markov chain model is a
theory that can be used to characterize a series of experiments, in which the result
of each experiment will depend only on the result of the immediately preceding
experiment and not on other prior experiments. Furthermore, Markov chains and
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other stochastic models can be applied to genetics, the science of heredity. They
added that Markov chains are useful in controlling breeding experiments wherein
we are interested in what happens to the genotype of the offspring after several
generations.

Budnick, et al (1977), cited many situations where Markov models have
been used. It has been used to study the behavior of gas particles in a container, to
model the development of biological populations, and to forecast weather patterns
in meteorology. They added that managerial applications include analyses of
inventory and queuing systems; replacement and maintenance policies for
machines; brand loyalty in marketing; time series of economic data such as price
movements of stocks; accounts receivable in accounting; hospital systems such as
the movements of coronary and geriatric patient; management of resources such
as water and wildlife; and expected payout of life insurance policies. Specific
examples cited by Budnick, et al are: (a) Markov chains of brand switching
behavior for 500 consumers used as diagnostic tools for suggesting marketing
strategies such as prediction of market shares at specific future points in time,
assessment of rates of change in market shares over time, prediction of market
share equilibriums (if they exist), assessment of the specific effects of marketing
strategies in changing undesirable market shares, and evaluation of the process for
introducing new products; (b) Markov model for analyzing the flow of patients in
the geriatric ward of a state hospital wherein the patients are classified in one of
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the four states in any given month., and (c) A simplified Markovian scenario
concerned with predicting the future behavior of prices of securities (stocks,
bonds, treasury bills, and the like) for the purpose of formulating investment
strategy. The closing daily prices and the differences in closing prices from one
day to the next are stochastic processes over time. Based on these price
movements, a discrete three-state and discrete parameter stochastic process {Xt}
can now be defined with respect to actual changes in price of a stock from one
day to the next.

Examination of the given data from the Government Secondary School in
Victoria revealed that prior to 1975, promotion and repeated rates were fairly
stable, whereas post 1974, the majority of the rates took on some specific patterns
(Johnston et. al., 1973). However Williams (1982) cited that in the analysis
undertaken, important matters should be borne in mind. First, the transition rates
themselves, pre-1975, had a substantially different pattern of behavior compared
to post 1974. There is no reason to suggest that the behavior will not occur again
in the future, hence the behavior of the transition rates should be carefully
monitored. Second, the small sample size for the calculation of the function forms
of rates although compensated for by increased tightness of significance tests,
cause some concern regarding the general applicability of the patterns of change
over time.
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Winston (1991) illustrated the use of Markov chain in his example “The
Gambler’s Ruin”. He asserted that since the amount of money one player has after
t + 1 plays of the game depends on the past history of the game only through the
amount of money he has after t plays, there is definitely a Markov chain. And
since the rules of the game don’t change over time, then it is a stationary Markov
chain. Moreover, he cited the Cola example where each person’s purchases are
viewed as a Markov chain with the state at any given time being the type of cola
the person last purchased. Hence, each person’s cola purchases may be
represented by a two-state Markov chain. Another example cited by Winston is
about the accounts receivable situation of a firm modeled as an absorbing Markov
chain. His example “Work Force Planning” used absorbing Markov chain to
answer the following questions regarding the law firm: (1) What is the probability
that a newly hired lawyer will leave the firm before becoming a partner? (2) On
the average, how long does a newly hired junior lawyer stay with the firm? (3)
What is the average length of time that a partner spends with the firm?

As cited by Williams (1982), Stone (1972) postulated a Markov model
that allowed for changing promotion and repeated rates over time, although
nowhere did the author indicate how this might be effected. A Markov chain
model capable of coping with changing promotion and repeated rates is developed
and applied to Victorian Secondary School system where separate account is
taken from Government and Non-Government schools, and males and females.
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Fraleigh (1990) applied Markov chain model through his example on
predicting population distribution at income categorized as poor, middle, and rich
consecutively; a Markov chain dealing with the distribution of a population
among states measured over evenly spaced time intervals wherein a n x n
transition matrix T describes the movement of the population (income) among
states (poor, middle, and rich).

Salda (1998) conducted a study on “Forecasting Benguet State University
Enrolment” using Markov Chain and forecasted the enrolment distribution for the
next five years.

These studies are related to the present study because they made use of
Markov Chain models. The researcher used a Markov Chain model in forecasting
future enrolment on selected degree programs in Saint Louis University.

Theoretical Framework
In every educational institution the trend of the mobility of the students’
enrolment seems not to be stable on the population growth. The assumptions are
as follows:
1. A student in the freshmen level could only be promoted to the
sophomore level, or could be a drop-out or a repeater.
2. A sophomore student could only be promoted to the junior level,
maybe retained in the level or could be dropped-out.
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3. A junior student could be promoted to the senior level, or could be
retained or could be a drop-out.
4. A senior student could be retained, dropped or moved out through
graduation.
5. Students retained in a certain year level do not leave the educational
system.
Harden and Tcheng (1971) illustrated the flow of students in Figure 1.

B(1)

1

C(1)







G(1)
D(1)







B(t)










t

C(t)






G(t) D(t)








B(n)








n




C(n)









G(n) D(n)

Figure 1: Model of the flow of students in an educational institution where t =1,

2, 3, …., n are the indices for the various academic years.
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The following legend was used for Figure 1:

B(t) – new students entering the school at the beginning of the
academic year t.

C(t) – students during the academic year (t-1) continuing to year t.

D(t) – students leaving the school as drop-outs.

G(t) – students leaving school as graduates.


In order to analyze such problem in determining the enrolment
distribution, the Markov process is used. A Markov chain is a special type of
stochastic process. According to Larson et. al. (1991), at each transition, each
member in the given state must either stay in that state or change to another state.
Since the entries in the ith column of the matrix represents the probabilities that a
member of the population will change from the jth state, it follows that the sum of
the entries in each column must be one. In a Markov chain, the outcomes on the
ith trial maybe influenced by the outcome on the trial by the outcome on the (t-1)
trial. The outcome S1, S2, S3,…, Sn are called states. And the probability of being
in one state to another during an increment of time is termed as the transition
probability.
Let
Pij the transition probability that a member of a population will change
from the jth state to the ith state where 0< Pij <1.

In particular, Pij the probability that a member of the population will
remain in the ith state. A probability of Pij means that the members certain to the
change from the jth state to the ith state.
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Collection of all probabilities Pij where 1< i <n and 1< j <n, is represented
by an n x n transition matrix P as follows:



S1
S2
S3 ………Sn


S1
P11
P12
P13 .……...P1n


S2
P21
P22
P23 ………P2n


S3
P31
P32
P33 ………P3n


. . . .


.
. . .


. . . .


Sn
Pn1
Pn2
Pn3 ……….Pnn


The above matrix shows the flow from one state to another. For instance
Pn is the probability of moving from S1 to S2. It is square matrix P with non-
negative elements, since Pij = 0 for all i and j.
The matrix of transition probabilities is computed from the available data
and enrolment projections are determined by repetitive multiplication of given
enrolment distribution and the transition probability matrix.
Considering the Markov chain with n different states (S1, S2, S3, Sn), the ith
state is called absorbing Pij = 1. Moreover, the Markov chain is called absorbing if
it has at least an absorbing state, and it is possible for a member of the population
to move from any non-absorbing state to an absorbing one in a finite number of
transition. In other words Markov chain is called an absorbing state if it is
impossible to leave it once it is entered (Gallin, 1984).
Winston (1991) classified and defined the states in a Markov chain as
follows:
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1) Given two states i and j, a path from i to j is a sequence of transitions that
begins in i and ends in j, such that each transition in the sequence has a
positive probability of occurring.
2) A state j is reachable from a state i if there is a path leading from i to j.
3) Two states i and j are said to communicate if j is reachable from i, and i is
reachable from j.
4) A set of states S in a Markov chain is a closed set if no state outside of S
is reachable from any state in S.
5) A state i is an absorbing state if pii = i
6) A state i is a transient state if there exists a state j that is reachable from i,
but the state i is not reachable from state j.
7) If a state is not transient, it is called a recurrent state.
8) A state i is periodic with period k > i if k is the smallest number such that
all paths leading from state i back to state i have a length that is a multiple
of k. If a recurrent state is not periodic, it is referred to as aperiodic.

Definition of Terms

The following terms have been defined in their operational context.
Trials of the process are the events that trigger transitions of the system
from one state to another. The enrolling of a student to a particular year level is a
trial of the process.
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State of the system is the condition of the system at any particular time
period.
Freshmen
are the students in the first year level of college.
Sophomores are the students in the second year of college and has an
academic standing of passing the first year level.
Juniors are students in the third year of college with an academic standing
of passing in the first and in the second year level.
Seniors are students in the fourth year of college with an academic
standing of passing in the first, second, and third year of college.
Drop-out
or
Quitter refers to a student who leaves school before the end of
the school year.

Shifter refers to a student who changes a degree to another at a certain
year level.

Transferee refers to a student who comes from one school and enrolled to
another school.

State probability is the probability that a system will be in any particular
state. For instance, the probability that a student is a freshman, a sophomore, a
junior, or a senior is referred to as a state probability.

Transition is the movement from one state to another. When a student
moves from one year during the current school year to another year level in the
following school year, then a transition has occurred.
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Transition probabilities are probabilities of the movement from one state
to another, say, the probability that a freshman now will become a sophomore
next school year.

Absorbing state. A state is said to be absorbing if the probability of
making a transition out of that state is zero. Thus once the system has made a
transition into an absorbing state, it will remain there forever.

A Markov process is a stochastic system for which the occurrence of
future states depends on the immediate preceding state.
A
Markov
chain is a probabilistic model that describes the random
movement over time of some activity.

Markov analysis is a method of analyzing the current behavior of some
variable in an effort to predict the future behavior of that variable.

Fundamental matrix is a matrix necessary for the computation of
probabilities associated with absorbing states of a Markov process.

Degree offering refers to the degree programs offered by the different
colleges in Saint Louis University. The different degree offerings used in this
study are the eight degree programs in the College of Engineering and
Architecture namely: BSME, BSChe, BSIE, BSCE, BSEE, BSECE, BSGE, and
BS Arch; seven degree programs in the College of Human Sciences which are AB
Econ, AB Engl, Bachelor of Philosophy, AB Pol Sci, AB Comm, BS Psychology,
and BSSW, and three from the College of Information and Computing Sciences
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27
which are BS Math, BSIT, and BSCS; and BSAc in the College of Accountancy
and Commerce. The acronyms used for the degree offerings are as follows:

BSChe – Bachelor of Science in Chemical Engineering

BSCE – Bachelor of Science in Civil Engineering

BSEE – Bachelor of Science in Electrical Engineering

BSECE – Bachelor of Science in Electrical Engineering

BSIE – Bachelor of Science in Industrial Engineering

BSME – Bachelor of Science in Mechanical Engineering

BSGE – Bachelor of Science in Geodetic Engineering

BSArch – Bachelor of Science in Architecture

AB Econ – Bachelor of Arts major in Economics

AB Engl – Bachelor of Arts major in English

AB Pol. Sci. – Bachelor of Arts major in Political Science

AB Comm – Bachelor of Arts major in Mass Communication

BS Psycho – Bachelor of Science in Psychology

BSSW – Bachelor of Science in Social Work

BSAc – Bachelor of Science in Accountancy

BS Math – Bachelor of Science in Mathematics

BSIT – Bachelor of Science in Information Technology

BSCS – Bachelor of Science in Computer Science
Markov Chain Modeling and Forecasting of Saint Louis University Enrolment on
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METHODS AND PROCEDURES

Research Design and Methodology

The data on the tertiary enrolment at the Saint Louis University were gathered.
The enrolment data utilized in this study is a ten-year period which started from the
school year 1997 – 1998 to school year 2006 – 2007.
In this study, a discrete time Markov chain was considered. Karlin et. al.
(1975) defined a discrete time Markov chain Sn as a Markov process whose state space
is countable or finite set and for which t = 0, 1, 2, 3,…., n, that is, a jump of more than
one steps occurred.
Harshbarger (1989) and Larson (1991) added that Markov chain is a study of
repeated trials in which the outcome on any trial depends only on the outcome of the
previous trials. Typically, each experiment has a finite fixed number of outcomes
called states.

Specifically, five states were considered: the freshmen, sophomore, junior and
senior levels and quit or drop-out. For the sake of simplicity each level was coded
correspondingly as 1, 2, 3, 4 and Q for the four-year degree programs and 1, 2, 3, 4, 5,
and Q for the five-year engineering programs. These are referred to as the Sn states.
From the gathered data, the transition probabilities were computed and comprise the
transition matrix. Forecast values on the number of students for the next five years
were computed based on the average of the recent ten years, that is, from school-year
1997-1998 to school year 2006-2007.
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Transition Probabilities

The matrices which show the flow of students was computed in a one-year
interval, starting from school-year 1997–1998 to school-year 2006–2007 for the
following degree courses: seven from the College of Engineering and Architecture
namely: BSME, BSChe, BSIE, BSCE, BSEE, BSECE, and BS Architecture; seven
degree offerings in the College of Human Sciences which are: AB Econ, AB English,
Bachelor of Philosophy, AB Political Sciences, AB Communication, BS Psychology,
and BS Social Work; three degree offerings in the College of Information and
Computing Sciences, and BS Accountancy in the College of Accountancy and
Commerce.

In computing the transition probabilities, the number of students continuing to
the next level was derived by subtracting from each year level the number of students
coming from another course (shifter) or from another school (transferee).

Population of the Study


The population comprises the students from the College of Human Sciences,
College of Engineering and Architecture, College of Information and Computing
Sciences, and BS Accountancy. The exact enrolment statistics from the stated colleges
form school year 1997-98 to school year 2006-2007 was utilized. Specifically, those
taking up courses leading to the following degrees: BS in Chemical Engineering, BS
in Civil Engineering, BS in Electrical Engineering, BS in Electronics and
Communications Engineering, BS in Industrial Engineering, BS in Geodetic
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30
Engineering, BS in Mechanical Engineering, BS in Architecture, AB in Economics,
AB in English, AB in Political Science, Bachelor of Philosophy, BS in Psychology,
BS in Social Work, BS in Information Technology, BS in Mathematics, and BS in
Computer Science.

Data Gathering Tools


The researcher made use of secondary sources of data in gathering relevant
information for this study. These data were acquired from the Management
Information Systems Office of Saint Louis University.

Statistical Tool

The average of the ten-year enrolment data was computed and used as a basis
for the computation of the transition probabilities. It is also used to derive the initial
state vector. The average is computed as follows:
10
∑ y



1
x =
n
10
where x = average enrolment for the year level, yn= total enrolment for the nth
year (n = 1 to 10 years)


Markov Chain Analysis was used to assess and analyze the enrolment statistics
and other pertinent data. From the gathered data, a Markov Chain model was
constructed. Consequently, the enrolment distribution for the next five years was
forecasted using the resulting Markov Chain model.
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31
Projection of the Number of Students


Forecast values on the number of students for the next five years were
computed based on the average of the recent ten years, that is, from school-year 1997-
1998 to school-year 2006-2007.

The projection for each year level was arrived at by multiplying the matrix P
by the row vector representing the recent ten years of the students’ distribution.

Taking the average of school-year 1997-1998 to school-year 2006-2007 as t =
0, the row vector which is the initial enrolment was multiplied by the matrix P
(transition matix) to come up with the row vector of enrolment for the year t = 1.
Further, the enrolment distribution at t = 1 was multiplied by the matrix P and the
resulting vector denotes the enrolment distribution at t = 2. Repetitive use of the above
procedure enables one to project the number of students for the succeeding years.

In notation, let the row vector of enrolment be Sn. To determine the enrolment
distribution at t = n, the formula Sn= Sn-1*P will be used where Sn represents the
current or present state, Sn-1 the previous state vector, and P the transition matrix.

Farlow and Haggard (1987) suggested the following formula for finding the
new state vector:

Let P be the transition matrix of a Markov chain with present state vector



p = (p1, p2, p3, ….., pm)



The new state vector



p’ = (p1’, p2’,……, pm’) is found by computing the matrix

product

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p11
p12 ……p1m





p21
p22 …….p2m


(p1, p2, …., pm)

.
= (p1’, p2’,…., pm’)
Pesent State

.

New State


Vector, Sn-1
pm1
pm2 …….pmn Vector, Sn



Transition
Matrix


In matrix language the above relationship can be written as pP = p’ where P is
the transition matrix. The limiting vector, called the steady state vector of the Markov
chain, of the above sequence of state vectors will give the proportion or fraction of
students that will ultimately be in each of the levels or states.

The flowchart on the research design for the Markov Chain is shown in Figure
2 on the next page.






















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Phase 1




Phase 6
Initial State Vector


Projection continued
Form the initial state vector from
The process in phase 5 is repeated
the average of the enrolment for the
until the desired number of pro-
recent past, say, ten years.


jections or forecasts is attained.






Thus, it is referred to as a chain.





Phase 2
Transition Probabilities




Phase 5
Compute the conditional proba-

Projection of Enrollment
bilities of continuing to the next state

The new state vector obtained
or year level given that a student is

represents the enrolment distri-
now in the present state or year.


bution for each level for the
Represent the levels as 1, 2, 3, 4, and

next school year. This vector
Q representing the 1st, 2nd, 3rd, and

is again multiplied by the same
4th year levels and Q for quit or


transition matrix to get another

drop-out.



new state vector.













Phase 3




Phase 4
Transition Matrix



Markov Chain Model

Form the matrix of conditional

Construct a Markov chain
probabilities for each state. This

model by multiplying the
should be a square matrix, and the

present state vector in phase 1
sum of each row is equal to one,

by the transition matrix (phase
except for the last year level since

3) resulting to a new state
most will graduate.



vector.





Figure 2 Flowchart on the research design for the Markov Chain
Markov Chain Modeling and Forecasting of Saint Louis University Enrolment on
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RESULTS AND DISCUSSION

Preliminary Analysis

The following data were gathered from the Management Information
Systems Office of Saint Louis University: a) enrolment statistics for each year
level in each degree program for the last ten years, b) number of shifters in each
year level for each degree program for the last ten years, and c) transferees in each
year level for each degree program for the last ten years. From these data, the
initial state vector was constructed. The total enrolment per year level for each
degree program for the last ten years is shown in Table 1.

It was observed that among the 19 degree programs used in this study, BS
Accountancy has the biggest population in the first year level, whereas the least
populated program is BS Geodetic Engineering.

Transition Probabilities

The conditional probabilities or transition probabilities indicate the chance
or likelihood of moving from one state to another, or stated in another way, from
one year level to the next level. These probabilities were used to obtain the
transition matrices. Each matrix was used as a multiplier to the present state row
vector to arrive at the new state vector.



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Table 1. Total enrolment from school year 1997-98 to 2006-07 by degree program

Year
Level
Degree Program
First
Second
Third
Fourth
AB English
667
450
470 490
AB Economics
447
296
260
270
Bachelor of Philosophy
631
570
640
680
AB Political Science
3522
3319
3063
2899
AB Communication
3638
3215
2594
2289
BS Psychology
3593
3570
3096
2148
BS Social Work
1528
1527
1550
1418
BC Computer Science*
363
99
9
5
BS Information Technology** 5258
4117
3352
2592
BS Mathematics
526
275
289
256

For the five-year degree
programs


First Second Third Fourth Fifth
BS Accountancy*
5798
4682
1797
2009
1928
BS Chemical Engineering
1770
1640
1650
1640
1775
BS Civil Engineering
5027
3779
3606
2961
2725
BS Electrical Engineering
2009
1743
2711
2247
2301
BS ECE
13429
12220
9113
7730
7700
BS Industrial Engineering
1314
1217
1280
1330
1420
BS Mechanical Engineering
3127
2537
2533
2170
2160
BS Geodetic Engineering
206
156
174
156
228
BS Architecture
3347
2252
2161
1998
1685
* For the previous three years
** For the previous seven years




Table 2 shows the computed transition probabilities for AB English
corresponding to the conditional probabilities of the first, second, third and fourth
years as they move from one state (year level) to another state (next year level)
given that they are now in the present state. Also presented in the same matrix of
conditional probabilities are the probabilities that a student will not move on to
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36
the next state given that he is in the present state, that is, he may have dropped
out or retained in the same year level.

Using the past ten years enrolment statistics for AB English, the
probability that a student will be promoted to second year (t + 1) given that he is
presently enrolled as a first year is 0.60, and the probability of being retained in
the first year level is 0.35, while the probability that a freshman is likely to drop
or quit is 0.05. Likewise, the probability of the continuing second year students to
be promoted to the third year level (t + 2) is 0.97 and the probability of being
retained in the second year is 0.01, whereas the probability of quitting or leaving
the system is 0.02. Furthermore, the probability of the continuing third students to
be promoted to the fourth year level (t + 3) is 0.98 and the probability of quitting
is 0.02. Moreover, there is a probability of 0.01 that a fourth year student quits or
drops. It is plainly observed that the highest retention rate is in the first year level,
that is, 35%, and it has also the highest probability of students who drop or quit.
Furthermore, 1% of the seniors are likely to quit, thus it is presumed that 99% will
graduate.

Table 2. Transition probabilities for AB English

Level 1 2 3 4 Q
1 0.35
0.60
0 0
0.05
2
0 0.01
0.97 0 0.02
3
0
0 0.00 0.98 0.02
4 0 0 0 0
0.01
Q 0 0 0 0 1

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Table 3. Transition probabilities for Bachelor of Philosophy

Level 1 2 3 4 Q
1 0.09
0.87
0
0
0.04
2
0 0 0.98 0 0.02
3 0
0
0.01
0.97
0.02
4
0 0 0 0
0.02
Q
0 0 0 0 1



Table 3, on the other hand, shows the conditional probabilities for
Bachelor of Philosophy. There is a probability of 0.87 that a freshman will be
moving to the second year level given that he is now in the first year, 0.09 chance
of being retained, and 0.04 probability of quitting. Moreover, there is a 0.98
probability that a sophomore will be promoted to the third year level and a 0.97
probability for a junior to be promoted to the fourth year.

Table 4 shows the transition probabilities for AB Economics. It is
observed that there is a 62% chance of a continuing student to be promoted to the
second year level given that he is now in the first year level and the percentage
that a student will be retained in the first year is 34%, and about 4% of the
freshmen are likely to drop or quit. Moreover, there is an 87% chance that a
second year student will move to the third year and 96% chance that a third year

Table 4. Transition probabilities for AB Economics


Level
1
2
3
4
Q
1 0.34
0.62
0 0
0.04
2
0 0.09
0.87 0 0.04
3
0
0 0.01 0.96 0.03
4 0 0 0 0
0.01
Q 0 0 0 0 1
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Table 5. Transition probabilities for AB Political Science

Level 1 2 3 4 Q
1 0.07
0.89
0
0
0.03
2
0 0.06
0.92 0 0.03
3
0
0 0.04 0.94 0.01
4
0 0 0 0
0.01
Q
0 0 0 0 1

will move to the fourth years.

Table 5 shows the transition matrix for AB Political Science. It depicts an
89% chance that a first year student moves to the second year level and a 92%
chance of moving to the third year level given that a student is now in the second
year. Whereas, a junior has a 94% chance of being promoted to the fourth year
level.
Likewise,
the
conditional
probabilities for AB Communications in Table 6
show that there is an 84% chance that a first year student will move to the second
year level and an 80% chance of moving to the third year level given that a
student is now in the second year. Whereas, a junior has an 88% chance of being
promoted to the fourth year level.

Table 6. Transition probabilities for AB Communication

Level
1 2 3 4 Q
1 0.12
0.84
0
0
0.04
2
0 0.18
0.80 0 0.02
3
0
0 0.09 0.88 0.03
4
0 0 0 0
0.02
Q
0 0 0 0 1


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Table 7. Transition probabilities for BS Psychology

Level 1 2 3 4 Q
1 0.01
0.95
0 0
0.04
2
0 0.12
0.85 0 0.03
3
0
0 0.29 0.69 0.02
4 0 0 0 0
0.01
Q 0 0 0 0 1


Shown in Table 7 are the transition probabilities for BS Psychology where
it is observed that there is a probability of 0.95 that a student will be moving to
the second year level given that he is now in the first year; a 0.85 probability that
a sophomore will be promoted to the third year level; a 0.69 probability that a
junior will be promoted to the fourth year level. The highest retention rate is in the
third year level.

Based on the computed conditional probabilities shown in Table 8 for BS
Social Work, there is a probability of 0.95 that a student will be moving to the
second year level given that he is now in the first year; a 0.98 probability that a
sophomore will be promoted to the third year level; a 0.93 probability for a junior
to be promoted to the fourth year level.

Table 8. Transition probabilities for BS Social Work

Level 1 2 3 4 Q
1 0.01
0.95
0
0
0.04
2
0 0.01
0.98 0 0.01
3 0
0
0.06
0.93
0.01
4
0 0 0 0
0.02
Q
0 0 0 0 1

Markov Chain Modeling and Forecasting of Saint Louis University Enrolment on
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Table 9. Transition probabilities for BS Computer Science

Level
1 2 3 4 Q
1 0.02
0.92 0 0 0.06
2 0 0.05
0.93 0 0.02
3 0 0 0.03
0.94
0.03
4 0 0 0 0
0.03
Q 0 0 0 0 1



As for BS Computer Science, there is a probability of 0.92 that a student
will be moving to the second year level given that he is now in the first year; a
0.93 probability that a second year student will be promoted to the third year; a
0.94 probability for a junior to be promoted to the fourth year level. Table 9
shows these results.

For BSIT shown in Table 10, there is a 76% chance that a first year
student moves to the second year level and an 81% chance of moving to the third
year level given that a student is now in the second year. Whereas, a junior has a
77% chance of being promoted to the fourth year level. It is also observed that the
highest rate of retention is in the third year, and the highest percentage of
dropping students is in the first year.

Table 10. Transition probabilities for BS Information Technology

Level 1 2 3 4 Q
1 0.17
0.76
0 0
0.07
2
0 0.16
0.81 0 0.03
3
0
0 0.21 0.77 0.02
4 0 0 0 0
0.01
Q 0 0 0 0 1

Markov Chain Modeling and Forecasting of Saint Louis University Enrolment on
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41
Table 11. Transition probabilities for BS Math

Level 1 2
3
4
Q
1 0.33
0.62 0 0 0.05
2 0 0.01
0.96 0 0.03
3 0 0 0.07
0.90
0.03
4 0 0 0 0 0.01
Q 0 0 0 0 1



Table 11 shows the conditional probabilities for BS Math. There is a 62%
chance that a first year student is promoted to the second year level and a 96%
chance to be promoted to the third year level given that a student is now in the
second year. Whereas, a junior has a 90% chance of being promoted to the fourth
year level.

For BS Accountancy, there is a probability of 0.80 that a student will be
moving to the second year level given that he is now in the first year; a 0.44
probability for a sophomore to be promoted to the third year level, 0.88
probability for a junior to be promoted to the fourth year, and 0.99 probability for
a fourth year student to be promoted to the fifth year. Table 12 further shows that
the retention rate in the second year level is very high.

Table 12. Transition probabilities for BS Accountancy


Level 1 2 3 4 5 Q
1 0.18
0.80
0 0 0
0.04
2
0 0.55
0.44 0 0 0.01
3 0
0
0.11
0.88
0
0.01
4
0 0 0 0
0.99
0.01
5
0 0 0 0 0
0.01
Q 0 0 0 0 0 1
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42
This is due to the cut-off grade prescribed by the college in order for a sophomore
to move to the third level in the same course.

For BS Chemical Engineering, there is a 91% chance that a first year
student moves to the second year level and a 98% chance of moving to the third
year level given that a student is now in the second year. Whereas, a junior has a
98% chance of being promoted to the fourth year level, and also 98% to be
promoted to the fifth year level given that a student is now in the fourth year.
Table 13 shows the above-stated probabilities.

As for BSCE, there is a probability of 0.26 that a freshman will be retained
or will not move to the second year level, a probability of 0.74 that a student will
be moving to the second year level given that he is now in the first year; 0.95
probability for a sophomore to be promoted to the third year level, 0.82
probability for a junior to be promoted to the fourth year, and 0.92 probability for
a fourth year to be promoted to the fifth year. It is further observed that a
freshman is not likely to drop or quit. These results are shown in table 14.

Table 13. Transition probabilities for BS Chemical Engineering

Level
1 2 3 4 5 Q
1 0.09
0.91 0 0 0.00
0.004
2 0
0.01
0.98
0
0.00
0.01
3 0 0
0.01
0.98
0.00
0.01
4 0 0 0
0.00
0.98
0.02
5 0 0 0 0 0
0.01
Q 0 0 0 0
0.00
1.00


Markov Chain Modeling and Forecasting of Saint Louis University Enrolment on
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Table 14. Transition probabilities for BS Civil Engineering

Level
1 2 3 4 5 Q
1 0.26 0.74 0 0 0.00
0.004
2 0 0.02
0.95 0 0.00
0.03
3 0 0 0.15
0.82
0.00
0.03
4 0 0 0 0.07
0.92
0.01
5 0 0 0 0 0
0.01
Q 0 0 0 0
0.00
1.00



Table 15 shows the transition probabilities for BSEE. There is an 86%
chance that a first year student moves to the second year level and a 96% chance
of moving to the third year level given that a student is now in the second year.
Whereas, a junior has a 97% chance of being promoted to the fourth year level,
and also 98% to be promoted to the fifth year level given that a student is now in
the fourth year.

For BSECE, Table 16 shows that there is a 90% chance that a first year
student moves to the second year level and a 74% chance of moving to the third
year level given that a student is now in the second year. Whereas, a junior has a
85% chance of being promoted to the fourth year level, and also 99% to be
promoted to the fifth year level given that a student is now in the fourth year.

Table 15. Transition probabilities for BS Electrical Engineering

Level
1 2 3 4 5 Q
1 0.14
0.86 0 0 0.00
0.004
2 0 0.00
0.96 0 0.00
0.04
3
0
0 0.01 0.97 0.00 0.02
4 0 0 0
0.00
0.98
0.02
5 0 0 0 0 0
0.01
Q 0 0 0 0 0
1.00
Markov Chain Modeling and Forecasting of Saint Louis University Enrolment on
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44
Table 16. Transition probabilities for BS Electronics and Communications
Engineering

Level 1 2 3 4 5 Q
1 0.10
0.90 0 0 0.00
0.003
2 0
0.23
0.74
0
0.00
0.03
3
0
0 0.13 0.85 0.00 0.02
4 0 0 0
0.00
0.990
0.01
5 0 0 0 0 0
0.01
Q 0 0 0 0
0.00
1.00



As for BS Industrial Engineering, there is a probability of 0.83 that a
student will be moving to the second year level given that he is now in the first
year; 0.96 probability for a sophomore to be promoted to the third year level, 0.96
probability for a junior to be promoted to the fourth year, and 0.97 probability for
a fourth year to be promoted to the fifth year. These are shown in Table 17.

In Table 18 are the conditional probabilities for BS Mechanical
Engineering. It is shown that there is a probability of 0.80 that a student will be
moving to the second year level given that he is now in the first year; 0.97
probability for a sophomore to be promoted to the third year level, 0.86
probability for a junior to be promoted to the fourth year, and 0.98 probability for

Table 17. Transition probabilities for BS Industrial Engineering

Level 1 2 3 4 5 Q
1 0.17
0.83
0
0
0.00
0.04
2 0
0.01
0.96
0
0.00
0.03
3
0
0 0.01 0.96 0.00 0.03
4
0 0 0
0.00
0.972
0.03
5
0 0 0 0 0
0.01
Q
0 0 0 0
0.00
1.00
Markov Chain Modeling and Forecasting of Saint Louis University Enrolment on
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Table 18. Transition probabilities for BS Mechanical Engineering

Level 1 2 3 4 5 Q
1 0.20
0.80
0 0
0.00
0.04
2 0
0.01
0.97
0
0.00
0.03
3 0
0
0.12
0.86
0.00
0.02
4 0
0
0
0.01
0.979
0.01
5 0
0
0
0
0
0.01
Q 0
0
0
0
0.00
1.00


a fourth year to be promoted to the fifth year.


As for BS Geodetic Engineering, in Table 19 there is a 67% chance that a
student will be moving to the second year level given that he is now in the first
year, 97% chance for a sophomore to be promoted to the third year level, 89%
chance for a junior to be promoted to the fourth year, and 96% for a fourth year to
be promoted to the fifth year. It is observed further that a freshman is less likely to
drop or quit. However, as compared to the other engineering courses, BS
Geodetic Engineering has the highest retention rate in the first year level which is
33%, leaving only 67% of the freshmen to continue to the second year level.

Table 20 shows that there is a 0.66 probability that a current freshman in
BS Architecture will continue to the second year level, and 34% are likely to be
retained. Furthermore, there is a probability of 0.95 for a sophomore to be
promoted to the third year level, a probability of 0.92 for a third year student to
move to the fourth year level, and a probability of 0.84 for a student in the fourth
year to move to the fifth year.


Markov Chain Modeling and Forecasting of Saint Louis University Enrolment on
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46
Table 19. Transition probabilities for BS Geodetic Engineering

Level 1
2 3 4
5
Q
1 0.33
0.67
0
0
0.00
0.003
2 0
0.01
0.97
0
0.00
0.02
3 0
0
0.09
0.89
0.00
0.02
4 0
0
0
0.03
0.962
0.01
5 0
0
0
0
0
0.02
Q 0
0
0
0
0
1.00

Table 20. Transition probabilities for BS Architecture

Level 1
2 3 4
5
Q
1 0.34
0.66
0
0
0.00
0.003
2 0
0.02
0.95
0
0.00
0.02
3 0
0
0.06
0.92
0.00
0.02
4 0
0
0
0.14
0.84
0.01
5 0
0
0
0
0
0.02
Q 0
0
0
0
0
1.00


Projected Number of Students for the Next
Five Years (School Year 2007-2008 to SY
2011-2012)

Corresponding to the derived transition matrix, projections on the
enrolment pattern for the year were made for each degree program by multiplying
the current state vector with the transition matrix. This process was repeated for
four times (iterations) to arrive at the forecasted number of enrollees for each
school-year on the next five years. This process results to a chain which is
referred to as the Markov chain model.


Markov Chain Modeling and Forecasting of Saint Louis University Enrolment on
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47
Table 21. Projected number of students for AB English for the next five years




Year Level
School Year
First
Second
Third
Fourth
2007-2008 67
45 47 49
2008-2009 90 41 44 46
2009-2010 122 55 39 43
2010-2011 165 74 53 39
2011-2012 223 100 72 52


For the program AB English, it is projected that there will be around 67,
90, 122, 165, and 223 incoming freshmen for the school years 2007-2008, 2008-
2009, 2009-2010, 2010-2011, and 2011-2012 respectively. The computed values
for each year level are shown in Table 21.

Likewise, a projection on the enrolment distribution for the next five years
for Bachelor of Philosophy is summarized in Table 22. Based on the computed
results, there is much lower additional number of incoming freshmen (ten or less)
for each school year in the course Bachelor of Philosophy as compared to that of
AB English. There will be around 63, 69, 75, 82, and 89 enrollees for the first
year in the next five school years.

Table 22. Projected number of students for Bachelor of Philosophy






Year Level
School Year
First
Second
Third
Fourth
2007-2008 63
57 64 68
2008-2009 69 55 57 62
2009-2010 75 60 54 55
2010-2011 82 65 59 53
2011-2012 89 71 64 57

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Table 23. Projected number of students for AB Economics




Year Level
School Year
First
Second
Third
Fourth
2007-2008 45 30 26 27
2008-2009 60 31 26 25
2009-2010 81 40 27 25
2010-2011 108 54 35 26
2011-2012 145 72 47 34


Projections on the enrolment pattern for the next five years in AB
Economics are shown in Table 23. Based on the results, it is observed that there is
a less increase in the number of incoming freshmen in AB Economics than in AB
English, but a little higher than in Bachelor of Philosophy. Furthermore, a
significant decrease is noted in the number of continuing second year students
(from the first year level) which means that many students may be retained,
dropped out, shift to another course or transfer.

As observed in Table 24 the projections are quite big but a closer look at
the first year level conveys just an average of less than thirty students added every
year for the next five years in AB Political Science.

Table 24. Projected number of students for AB Political Science






Year Level
School Year
First
Second
Third
Fourth
2007-2008 352
332 306 290
2008-2009 377 333 318 288
2009-2010 403 355 319 299
2010-2011 431 380 340 300
2011-2012 461 407 363 319

Markov Chain Modeling and Forecasting of Saint Louis University Enrolment on
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Table 25. Projected number of students for AB Communication






Year Level
School Year
First
Second
Third
Fourth
2007-2008 364
322 259 229
2008-2009 408 364 281 228
2009-2010 457 408 316 247
2010-2011 511 457 335 278
2011-2012 573 512 398 312


Likewise, Table 25 shows the projected enrolment distribution in the next
five years for Bachelor of Arts in Communications. About one section of
freshmen will be added each year.

Projections made on the enrolment distribution for the next five years for
BS Psychology are shown in Table 26. Based on the results, big numbers are seen
but the projected yearly increase in the incoming freshmen is just about 3 to 4
students; there is no significant increase.

Table 27 shows the projected enrolment distribution for BS Social Work
for the next five years. It is observed that there is a very little increase, about one
or two students, in the number of incoming freshmen for the next five years.
There is, however, a high rate of continuing students to the next year level.
Table 26. Projected number of students for BS Psychology






Year Level
School Year
First
Second
Third
Fourth
2007-2008 359
357 310 215
2008-2009 363 384 393 214
2009-2010 366 391 440 271
2010-2011 370 395 460 304
2011-2012 374 399 469 317
Markov Chain Modeling and Forecasting of Saint Louis University Enrolment on
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Table 27. Projected number of students for BS Social Work






Year Level
School Year
First
Second
Third
Fourth
2007-2008 153
153 155 142
2008-2009 155 147 159 144
2009-2010 156 148 153 148
2010-2011 158 150 155 143
2011-2012 159 151 156 144


The corresponding projected enrolment distribution for BSCS for the next
five years is shown in Table 2. The number of students in the third and fourth year
levels in school year 2007-2008 is very alarming. Most of the students most
probably shifted to BSIT or any related new course. Based on the present demand,
BS Information Technology and BS Information Management (BSIM) have
higher demand than BSCS. Despite the projected increase in the enrolment for the
next five years, there is a great possibility of being phased out since BSIT and
BSIM are already offered. Saint Louis University just started to offer BSIM this
school year 2006-2007. This could be a good alternative program.

Table 28. Projected number of students for BS Computer Science






Year Level
School Year
First
Second
Third
Fourth
2007-2008 121 33 3 2
2008-2009 123 113 31 3
2009-2010 126 119 106 29
2010-2011 128 122 114 100
2011-2012 131 124 117 107


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Table 29. Projected number of students for BSIT






Year Level
School Year
First
Second
Third
Fourth
2007-2008 751 588 479 370
2008-2009 879 665 577 369
2009-2010 1028 774 660 444
2010-2011 1203 905 766 508
2011-2012 1407 1059 894 590


Table 29 shows large populations indicating that many students are
attracted to the course BSIT. It is known for a fact that our society is becoming
technology-based and industries, companies, large firms and institutions cannot
survive without computer technology since large array of information could be
stored in them.

Table 30 shows the projected enrolment distribution for BS Math for the
next five years. A perusal of the results reveals that there is a high mortality rate
in the first year level. There is a very low chance for a freshman to continue to the
second year level. Furthermore, there is a gradual increase in the number of
forthcoming freshmen.

Table 30. Projected number of students for BS Math






Year Level
School Year
First
Second
Third
Fourth
2007-2008 53 28 29 26
2008-2009 70 33 29 26
2009-2010 94 44 34 26
2010-2011 125 59 45 30
2011-2012 166 78 59 40

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Table 31. Projected number of students for BS Accountancy






Year Level
School Year
First
Second
Third
Fourth
Fifth
2007-2008 1933 1561 599 670 643
2008-2009 2281 2405 753 527 663
2009-2010 2692 3147 1141 662 522
2010-2011 3176 3884 1510 1004 656
2011-2012 3748 4677 1875 1329 994


In Table 31, it is observed that BS Accountancy has a very large
population, almost 40 sections of freshmen. However, the chance of continuing
until the third level is quite low. The college conducts departmental examinations
for major (accounting) subjects so that the screening will be impartial. In this
regard, many students obtain deficiencies, thus, they cannot go to the next year
level. Many of them shift to another major or course.

Table 32 summarizes the enrolment pattern for the next five years in BS
Chemical Engineering. It is observed that there are more or less twenty students
added each year in the first year for the next five years.

Table 32. Projected number of students for BS Chemical Engineering






Year Level
School Year
First
Second
Third
Fourth
Fifth
2007-2008 177 164 165 164 162
2008-2009 193 163 162 162 161
2009-2010 210 177 161 159 159
2010-2011 229 193 175 158 156
2011-2012 250 211 191 172 155


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Table 33. Projected number of students for BS Civil Engineering






Year Level
School Year
First
Second
Third
Fourth
Fifth
2007-2008 503 378 361 296 273
2008-2009 634 380 413 317 272
2009-2010 799 477 423 361 291
2010-2011 1006 600 516 372 332
2011-2012 1268 757 648 449 342


Likewise, the projected enrolment distribution for the next five years in
BS Civil Engineering is shown on Table 33. It is observed that the number of
students who reach the last year level is very few compared to the initial number
of freshmen. This may be due to difficult major subjects that may lead students to
shift to another course or be retained. This, however, is a general observation for
all the engineering programs.

Table 34 shows the projected enrolment distribution for the next five years
for BSEE. About one section is added each year for the next five years in the first
year level.

Table 34. Projected number of students for BS Electrical Engineering






Year Level
School Year
First
Second
Third
Fourth
Fifth
2007-2008 201 174 271 225 230
2008-2009 229 173 170 263 221
2009-2010 261 197 168 165 258
2010-2011 298 225 191 163 161
2011-2012 339 256 218 185 159


Markov Chain Modeling and Forecasting of Saint Louis University Enrolment on
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Table 35. Projected number of students for BS Electronics and Communications

Engineering






Year Level
School Year
First
Second
Third
Fourth
Fifth
2007-2008 1343 1222 911 773 770
2008-2009 1477 1490 1023 774 765
2009-2010 1625 1672 1235 869 767
2010-2011 1788 1847 1398 1050 861
2011-2012 1966 2034 1549 1188 1040


It is noted that BSECE has the highest number of enrollees in the College
of Engineering and Architecture. Consequently, it produces the most number of
graduates in the same college. In the projections made as shown in Table 35, it
will still be the most populated department among the Engineering departments,
with about three sections of freshmen added each year for the next five years.
Saint Louis University addressed this concern by creating a related course, BS
Mechatronics Engineering, which is now in its first year of operation.

Table 36 displays the projected enrolment distribution for the next five
years in BSIE. In contrast to BSECE, there will be a very little increase in the

Table 36. Projected number of students for BS Industrial Engineering.






Year Level
School Year
First
Second
Third
Fourth
Fifth
2007-2008 131 122 128 133 142
2008-2009 153 110 118 123 129
2009-2010 179 128 107 114 119
2010-2011 210 150 124 102 110
2011-2012 245 176 145 119 100

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Table 37. Projected number of students for BS Mechanical Engineering






Year Level
School Year
First
Second
Third
Fourth
Fifth
2007-2008 313 254 253 217 216
2008-2009 376 253 277 220 212
2009-2010 451 303 279 240 215
2010-2011 541 364 327 242 235
2011-2012 649 436 392 284 237

number of incoming freshmen for the next five years in BSIE.

Table 37, likewise, shows the projected enrolment distribution for the next
five years in BS Mechanical Engineering. It is projected that one to two sections
of freshmen will be added for the next five years.

Table 38 shows the projection on the enrolment pattern for the next five
years in BS Geodetic Engineering. It is observed that at present, the number of
freshmen is below the normal one section and that the increase in the incoming
freshmen for the next five years is an average of ten students.

Table 38. Projected number of students for BS Geodetic Engineering






Year Level
School Year
First
Second
Third
Fourth
Fifth
2007-2008 21 16 17 16 23
2008-2009 28 14 17 16 15
2009-2010 37 19 15 16 15
2010-2011 49 25 20 14 15
2011-2012 66 33 26 18 14




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Table 39. Projected number of students for BS Architecture






Year Level
School Year
First
Second
Third
Fourth
Fifth
2007-2008 335 225 216 200 169
2008-2009 449 226 227 227 168
2009-2010 602 301 228 240 190
2010-2011 806 403 299 243 202
2011-2012 1080 540 401 310 204


The projections on the enrolment pattern for BS Architecture are shown in
Table 39. It is projected that two to three sections are added each year for the next
five years.

Based on the over-all projected values, it has been observed that the
following degree programs will have a very little increase each year in the next
five years on the number of incoming first years, that is, less than 40 students for
each school year: BS Computer Science, BS Social Work, Bachelor of
Philosophy, BS Psychology and BS Industrial Engineering. On the other hand, the
following colleges have big number of additional incoming first years in the
succeeding years after 2007-2008: AB Economics, AB English, BS Mathematics,
BS in Accountancy, BS in Electronics and Communications Engineering, BS
Architecture, BS Civil Engineering, and BS Information Technology. The other
degree programs have an average of just one additional section of incoming first
years per school year for the next five years.
Markov Chain Modeling and Forecasting of Saint Louis University Enrolment on
Selected Degree Programs in the Tertiary Level / Zenaida U. Garambas. 2007

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SUMMARY, CONCLUSION, AND RECOMMENDATIONS



School administrators regularly prepare, among others, a budget plan for the
coming years, say, for the next five years. The forecasted number of enrolment for
each year level and each school year is of utmost importance in the plan.

The researcher was determined to pursue this study because of the relevance
in today’s competitive condition in schools. The assessment and analysis of the
present enrolment statistics would assist school administrators in the planning and
decision-making.

The enrolment statistics for the past ten years was used to form transition
matrices for the conditional probabilities of the movement of students from one year
level to the next. The average number of drop-outs, transferees and shifters were also
gathered and subtracted from the average enrolment to derive the conditional
probabilities. Nineteen matrices were formed to correspond to the nineteen degree
programs. Furthermore, an initial state row vector was derived from the average
enrolment for the past ten years.

The projected numbers of students for the next school year (SY 2007-2008)
were arrived at by multiplying the initial state vector by the transition matrix; the
resulting row matrix would again be multiplied by the same transition matrix to
arrive at the next set of projected number of students (for SY 2008-2009); and this
process went on until five groups of data were gathered corresponding to the next
five school years.

Markov Chain Modeling and Forecasting of Saint Louis University Enrolment on
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58
Summary of Findings


The analysis and interpretation of data resulted in the projection of enrolment
for the next five years for the selected degree programs at Saint Louis University.
Based on the computations made, there will be an increase in enrolment per year
from school year 2007-2008 to SY 20111-2012. The increase, however, is not
constant or the same for all the degree programs. The following degree programs
will have a very little increase in the number of incoming first years, that is, less than
40 students increase in each school year: BS Computer Science, BS Social Work,
Bachelor of Philosophy, BS Psychology and BS Industrial Engineering, AB Political
Science, and BS Chemical Engineering. On the other hand, the following colleges
have big number of additional incoming first years in the succeeding years after
2007-2008: BS in Accountancy, BS in Electronics and Communications
Engineering, BS Architecture, BS Civil Engineering, and BS Information
Technology. The other degree programs have just an average of additional one
section per school year.

This would help administrators in their five-year plan pertaining to, but not
limited to, number of new faculty members to be hired, availability of classrooms
and budget.

Conclusion


The Markov Chain Analysis is one of the best tools used in predicting
behavior of future events on the condition that prior events have occurred. This
process does not need too much historical data. It deals with the present state to
Markov Chain Modeling and Forecasting of Saint Louis University Enrolment on
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59
predict behavior of the future state. There are many applications in industry, in
education, in business, in government, and in medical fields.

From this study, it is concluded that the degree with the highest projected
increase in enrolment is BS Electronics and Communications Engineering.

The values obtained in this study, however, should not be taken as final and
accurate since there are still other intervening factors that affect enrolment like
parents’ income, inflation, demand for a certain skill, employability rate, line of
interest of the student, accessibility of school and the like.

Recommendation

It is recommended that further studies be conducted on forecasting student
enrolment and include other variables such as number of students who have
withdrawn, transferred to another school, and other pertinent data. Also, the
probability that a student will graduate given that he is now in the fourth year could
be added in the transition matrix. Further studies such as the employability of
graduates from each degree program could also be made using the same statistical
analysis. One could project the answers to the following questions: (a) how many of
the graduates get employed in their own field, (b) How many get employed in fields
other than their field of specialization? (c) How many of them leave the country?

Also, it is recommended that the other degree programs not included here be
analyzed in the same manner.

Markov Chain Modeling and Forecasting of Saint Louis University Enrolment on
Selected Degree Programs in the Tertiary Level / Zenaida U. Garambas. 2007

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60




LITERATURE CITED


ACZEL, A.D., 1989. Complete Business Statistics. Richard D. Irwin, Inc. Illinois.

pp.595 – 618

ANDERSON, A., 1971. Statistics for Business and Economics. CBS College

Publishing Co. pp. 745 – 746

ANDERSON, SWEENEY, and WILLIAMS, 1979, An Introduction to

Management Science: Quantitative Approaches to Decision-Making, 2nd

edition West Publishing Company, Minnesota. pp 656–658, 673–674

ANDERSON, M. and LIEVANO, R., 1986, Quantitative Management: An
Introduction,
2nd edition, Kent Publishing Company, Boston. p.463

BUDNICK, F., MOJENA, R., VOLLMANN, T., 1977. Principles of Operations

Research for Management. Richard D. Irwin Inc., Illinois. pp.587-605

CARINO, E.N. (2002) Forecasting of Fish Prices in Benguet. Unpublished
Undergraduate
Thesis.
B.S.U. at La Trinidad, Benguet.

COZZEN, B.M., R.D. PORTER, 1987. Mathematics and Its Application North
Eastern
University.pp.450-451

DINKEL, KOCHENBERGER, and PLANE, 1979. Management Science Text
and

Applications. Irwin-Dorsey Ltd, Georgetown, Ontario. Pp 228-229,247-
249, 254

FRALIEGH, B.J., R.A. BEAUREGARD, 1990. Linear Algebra. University of

Rhode Island. Massachusetts: Addison Wesly Publishing Co.pp.66-68

GALLIERS, R., 1987. Information Analysis Selected Readings. Curtin University
of Technology. Addison Wesly Publishing Company pp.67

GALLIN, D. 1984. Finite Mathematics. Glenview Illinois. Scott Foresman and
Company.pp.339-344.

GANNAPAO, J.S. 1997. Times Series Analysis of Market Retail Prices of
Selected Fruit Vegetables. Unpublished Undergraduate Thesis. B.S.U. La
Trinidad, Benguet.
Markov Chain Modeling and Forecasting of Saint Louis University Enrolment on
Selected Degree Programs in the Tertiary Level / Zenaida U. Garambas. 2007

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61





GILBERT, G.G., D.O. KOELER, 1984. Applied Mathematics. New York:
McGraw Hill Book Co. pp.485.

HARDEN, W.R., M.T. TCENG, 1971. Projecting of Enrolment Distribution.
Socio-
Economic Plan . p.5.

HILLIER, F. and LIEBERMAN, G. 1986. Introduction to Operations Research,
4th

edition. Mc-Graw Hill Publishing Company, New York. pp. 501-517,678-
698

LARSON, E.R., B.H. EDWARDS, 1991. Finite Mathematics. Lexington,
Massachusetts.
pp.398-399.

LEVIN, RUBIN, and STINSON, 1976, Quantitative Approaches to Management,
6th edition. Mc Graw-Hill Book Company. Pp 691-692, 714-717

PANGKRATZ, A. Forecasting With Univariate Box – Jenkins Models. John
Wiley
& Sons, New York pp.3–4

PINDICK, S.R., D.L. RUBINFIELD, 1981. Econometric Model and Economic

Forecasts. Mc Graw-Hill. International Book Company.N.Y. pp.203.

SALDA, CHARLIE A. 1998. Unpublished Undergraduate Thesis, BSU, La
Trinidad,
Benguet.

SMITH, G.1985. Statistical Reasoning. Mc Graw-Hill International Book
Company. N.Y p.655.

WILLIAMS, E.J. 1982. The Australian Journal. Volume 24. pp.178-190.

WINSTON, W.L. 1991. Operations Research Applications and Algorithms, 2nd

edition. PWS-Kent Publishing Company, Boston. pp. 909-939






Markov Chain Modeling and Forecasting of Saint Louis University Enrolment on
Selected Degree Programs in the Tertiary Level / Zenaida U. Garambas. 2007

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62















APPENDIX A


LETTER REQUEST TO THE MANAGEMENT
INFORMATION SYSTEMS
SAINT LOUIS UNIVERSITY
BAGUIO CITY





















Markov Chain Modeling and Forecasting of Saint Louis University Enrolment on
Selected Degree Programs in the Tertiary Level / Zenaida U. Garambas. 2007

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63




APPENDIX A


Saint Louis University
College of Information and Computing Sciences
Department of Mathematics
Bonifacio Street, Baguio City


March 26, 2007


MR. ANGELITO PERALTA
Director, MIS Office
Saint Louis University


Thru: Ms Cynthia Arceo

OIC

Dear Sir;

Greetings!



I, the undersigned, am presently conducting a study on MARKOV MODELING
AND FORECASTING OF SAINT LOUIS UNIVERSITY ENROLMENT IN THE
TERTIARY LEVEL as a partial fulfillment for the degree Master of Arts in Applied
Statistics. My study is limited to the data from school year 1997-1998 to school year
2006-2007 pertaining to the following colleges: College of Engineering and
Architecture, College of Human Sciences, College of Information and Computing
Sciences, College of Accountancy and Commerce (BS in Accountancy only).


In this connection, may I respectfully request from your office the following data:
a. Enrolment statistics per semester by degree program.
b. Number of drop-outs per semester by degree program.
c. Number of students who shifted from one program to another.
d. Number of transferees from other schools by year and by degree program.


Your favorable action on this request is highly appreciated.



Thank you very much.


Truly yours,

Zenaida Garambas
Markov Chain Modeling and Forecasting of Saint Louis University Enrolment on
Selected Degree Programs in the Tertiary Level / Zenaida U. Garambas. 2007

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64
















APPENDIX B


TOTAL ENROLLMENT FROM SCHOOL YEAR 1997-1998

TO 2006-2007 BY DEGREE PROGRAM


















Markov Chain Modeling and Forecasting of Saint Louis University Enrolment on
Selected Degree Programs in the Tertiary Level / Zenaida U. Garambas. 2007

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65




Appendix B-1. College of Human Sciences










YEAR LEVEL
COURSE SCHOOL YR
1

2

3

4
AB
Economics
1997-1998
105
30
47
42
1998-1999
84
59
23
38
1999-2000
44
48
34
27
2000-2001
79
34
31
37
2001-2002
34
33
33
32
2002-2003
45
32
21
30
2003-2004
24
25
30
17
2004-2005
18
25
20
25
2005-2006
14
10
22
17
2006-2007
- -

AB
English
1997-1998
56
` 37
42
51
1998-1999
30
38
51
65
1999-2000
79
13
43
62
2000-2001
51
63
32
27
2001-2002
40
31
52
60
2002-2003
50
32
44
52
2003-2004
61
50
50
61
2004-2005
52
48
32
40
2005-2006
123
49
49
40
2006-2007
125
89
77
32

Bachelor
of
Philo
1997-1998
62
54
93
91
1998-1999
25
34
60
70
1999-2000
49
37
44
53
2000-2001
101
42
43
52
2001-2002
80
102
52
57
2002-2003
82
66
77
61
2003-2004
48
82
85
85
2004-2005
79
42
75
80
2005-2006
49
65
52
65
2006-2007
56
46
61
63










Markov Chain Modeling and Forecasting of Saint Louis University Enrolment on
Selected Degree Programs in the Tertiary Level / Zenaida U. Garambas. 2007

---

66





Appendix B-1. Continued …










YEAR LEVEL
COURSE SCHOOL YR

1

2

3

4
AB
Pol
Sci
1997-1998
432
397
267
222
1998-1999
431
452
335
224
1999-2000
471
353
442
318
2000-2001
383
438
330
400
2001-2002
364
347
413
339
2002-2003
305
316
357
407
2003-2004
344
259
263
340
2004-2005
283
283
206
264
2005-2006
294
248
228
187
2006-2007
215
226
222
198

AB
Comm
1997-1998
499
519
289
234
1998-1999
462
452
335
224
1999-2000
369
331
490
264
2000-2001
354
268
288
359
2001-2002
356
249
255
272
2002-2003
301
326
232
223
2003-2004
256
257
225
206
2004-2005
345
197
130
187
2005-2006
406
291
149
176
2006-2007
290
325
201
144

BS
Psychology
1997-1998
453
514
258
133
1998-1999
411
582
273
197
1999-2000
436
423
364
233
2000-2001
425
398
388
243
2001-2002
374
325
443
227
2002-2003
343
365
356
272
2003-2004
279
370
270
209
2004-2005
253
311
261
227
2005-2006
349
276
234
215
2006-2007
270
322
249
192

BS
Social
Work
1997-1998
146
148
195
146
1998-1999
115
188
139
153
1999-2000
139
179
199
91
2000-2001
165
193
168
167
2001-2002
127
163
217
164
2002-2003
132
137
180
181
2003-2004
145
119
156
177
2004-2005
189
126
117
144
2005-2006
214
116
117
100
2006-2007
156
158
112
95

Markov Chain Modeling and Forecasting of Saint Louis University Enrolment on
Selected Degree Programs in the Tertiary Level / Zenaida U. Garambas. 2007

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67




Appendix B-2 College of Engineering and Architecture










YEAR LEVEL
COURSE SCH YR
1

2

3

4
5
BS
Che 1997-1998
197
232
193
153
154
1998-1999
213
174
190
169
167
1999-2000
196
211
188
152
170
2000-2001
226
180
214
197
167
2001-2002
224
199
213
161
207
2002-2003
195
194
233
164
199
2003-2004
121
162
194
219
173
2004-2005
150
87
205
179
202
2005-2006
120
108
138
165
176
2006-2007
128
93
130
139
160

BSCE
1997-1998
499
304
359
224
233
1998-1999
440
361
312
269
245
1999-2000
511
375
385
344
260
2000-2001
514
451
368
357
235
001-2002
393
385
357
333
321
2002-2003
476
342
364
308
355
2003-2004
487
374
340
284
277
2004-2005
577
367
385
230
310
2005-2006
600
403
372
284
256
2006-2007
530
417
364
328
233

BSEE
1997-1998
226
229
269
212
206
1998-1999
155
194
220
241
230
1999-2000
212
144
232
190
266
2000-2001
191
207
236
156
207
2001-2002
148
160
300
239
170
2002-2003
182
121
260
297
225
2003-2004
186
145
273
230
297
2004-2005
242
168
276
224
276
2005-2006
247
192
336
216
202
2006-2007
220
183
309
242
222

BSECE 1997-1998
1897
1483
1093
726
528
1998-1999
1434
1584
1314
852
660
1999-2000
1418
1406
1329
972
793


2000-2001
1575
1266
1093
876 1011


2001-2002
1452
1379
929
819 1010
2002-2003
1251
1445
888
848
909
2003-2004
1071
1126
840
791
937
2004-2005
1092
889
648
759
925
2005-2006
1109
801
477
610
868
2006-2007
1130
841
502
477
659

Markov Chain Modeling and Forecasting of Saint Louis University Enrolment on
Selected Degree Programs in the Tertiary Level / Zenaida U. Garambas. 2007

---

68




Appendix B-2. Continued…










YEAR LEVEL
COURSE SCH YR
1

2

3

4
5
BSIE
1997-1998
206
197
252
172
78
1998-1999
184
180
234
194
153
1999-2000
170
156
257
208
198
2000-2001
160
163
238
257
207
2001-2002
125
121
233
229
243
2002-2003
97
129
152
233
220
2003-2004
93
72
159
183
216
2004-2005
90
77
58
153
189
2005-2006
88
55
79
53
165
2006-2007
101
67
119
62
62

BSME
1997-1998
297
377
354
252
301
1998-1999
265
240
319
290
309
1999-2000
315
225
240
241
373
2000-2001
239
273
229
189
355
2001-2002
257
196
270
186
243
2002-2003
251
243
194
233
220
2003-2004
286
212
266
158
259
2004-2005
338
206
210
195
186
2005-2006
512
222
210
225
146
2006-2007
367
343
241
201
200

BSGE
1997-1998
16
9 18
15
17
1998-1999
22
9 10
17
24
1999-2000
23
18
15
16
21
2000-2001
23
20
18
18
21
2001-2002
18
18
11
15
30
2002-2003
35
11
24
6 24
2003-2004
24
27
22
20
13
2004-2005
15
24
22
20
23
2005-2006
18
11
20
18
31
2006-2007
12
9 14
11
24

BS
Arch
1997-1998
395
223
232
171
134
1998-1999
306
285
176
250
134
1999-2000
319
234
232
206
171
2000-2001
279
218
227
232
187
2001-2002
252
201
203
206
188
2002-2003
259
141
188
205
173
2003-2004
269
183
168
159
193
2004-2005
324
149
190
163
137
2005-2006
333
169
181
141
134
2006-2007
277
175
193
139
136

Markov Chain Modeling and Forecasting of Saint Louis University Enrolment on
Selected Degree Programs in the Tertiary Level / Zenaida U. Garambas. 2007

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69




Appendix B-3 College of Information and Computing Sciences










YEAR LEVEL
COURSE SCHOOL YR

1

2

3

4
BSCS
1996-1997
933
500
391
447
1997-1998
8 659
473
416
1998-1999
1 16
499
526
1999-2000
- - 90
487
2000-2001
- - 10
260
2001-2002
- - 20
14
2002-2003
- - 1 1
2003-2004
- - 1 4


2004-2005

89

1

1

1
2005-2006
178
52
2 2
2006-2007
95
46
15
1

BSMath
1996-1997
67
35
12
7
1997-1998
64
47
28
15
1998-1999
60
31
48
23
1999-2000
68
31
29
41
2000-2001
76
37
34
29
2001-2002
56
39
46
26
2002-2003
38
17
38
35
2003-2004
26
11
16
35
2004-2005
28
5 17
20
2005-2006
26
11
16
15
2006-2007
17
12
17
10

BSIT
1996-1997
- - - -
1997-1998
- - - -
1998-1999
- - - -
1999-2000
- - - -
2000-2001
1006
688
598
431
2001-2002
1175
711
621
474
2002-2003





776
692
565
492
2003-2004
596
461
518
366
2004-2005
578
323
437
391
2005-2006
527
365
321
230
2006-2007
600
877
292
208







Markov Chain Modeling and Forecasting of Saint Louis University Enrolment on
Selected Degree Programs in the Tertiary Level / Zenaida U. Garambas. 2007

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70




Appendix B-4 College of Commerce and Accountancy










YEAR LEVEL
COURSE SCH YR
1

2

3

4

5
BS
Ac 1996-1997
1262
1346
486
360
-
1997-
1998
1454
1150
580
354
-
1998-1999
1467
1086
528
422
-
1999-2000
2184
1246
543
463
-
2000-2001
2079
1742
536
501
-
2001-2002
1969
1781
862
459
-
2002-2003
1844
1613
951
824
-
2003-2004
1983
1467
818
894
-
2004-2005
1940
1528
571
744
794
2005-2006
2104
1511
695
605
817
2006-2007
1754
1643
531
660
767





























Markov Chain Modeling and Forecasting of Saint Louis University Enrolment on
Selected Degree Programs in the Tertiary Level / Zenaida U. Garambas. 2007

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71



















APPENDIX C


TOTAL DROP-OUTS FROM SCH00L YEAR 1997-1998

BY DEGREE PROGRAM





















Markov Chain Modeling and Forecasting of Saint Louis University Enrolment on
Selected Degree Programs in the Tertiary Level / Zenaida U. Garambas. 2007

---

72




Appendix C-1 College of Human Sciences










YEAR LEVEL
COURSE SCHOOL YR

1

2

3

4
AB
Econ
1997-1998
5 3 4 0
1998-1999
5 2 0 0
1999-2000
0 1 0 0
2000-2001
2 0 1 0
2001-2002
1 1 1 1
2002-2003
1 2 0 1
2003-2004
0 1 0 0
2004-2005
1 1 1 0
2005-2006
1 0 0 0
2006-2007
0 0 0 0

AB
English
1997-1998
1 1 1 1
1998-1999
0 0 0 0
1999-2000
3 0 1 0
2000-2001
3 0 0 1
2001-2002
3 0 1 0
2002-2003
6 1 2 0
2003-2004
4 2 4 0
2004-2005
2 2 0 0
2005-2006
4 2 0 0
2006-2007
12
1 2 2

Bachelor
of
Philo
1997-1998
3 3 2 1
1998-1999
2 0 1 3
1999-2000
1 0 2 0
2000-2001
4 1 1 1
2001-2002
3 1 0 1
2002-2003
1 4 3 0
2003-2004
1 0 5 1
2004-2005
5 2 0 4
2005-2006
2 2 1 0
2006-2007
3 1 0 1










Markov Chain Modeling and Forecasting of Saint Louis University Enrolment on
Selected Degree Programs in the Tertiary Level / Zenaida U. Garambas. 2007

---

73




Appendix C-1. Continued…










YEAR LEVEL
COURSE SCHOOL YR

1

2

3

4
AB
Pol.
Sci.
1997-1998
11
12
7 3
1998-1999
13
9 5 4
1999-2000
8 10
5 1
2000-2001
11
5 1 3
2001-2002
10
3 6 2
2002-2003
9 10
6 4
2003-2004
19
20
4 4
2004-2005
18
10
4 4
2005-2006
11
10
3 4
2006-2007
7 6 2 0

AB
Comm
1997-1998
19
15
9 8
1998-1999
11
12
9 8
1999-2000
11
4 9 3
2000-2001
20
10
8 6
2001-2002
20
7 5 3
2002-2003
12
10
7 6
2003-2004
11
6 5 1
2004-2005
14
5 7 2
2005-2006
7 6 5 3
2006-2007
8 5 3 1

BS
Psychology
1997-1998
12
7 2 1
1998-1999
7 12
3 2
1999-2000
6 9 3 3
2000-2001
23
5 6 1
2001-2002
21
5 6 1
2002-2003
16
7 5 1
2003-2004
16
11
5 1
2004-2005
23
10
7 0
2005-2006
11
11
4 3
2006-2007
9 9 2 1

BS
Social
Work
1997-1998
5 3 2 3
1998-1999
3 4 3 1
1999-2000
6 1 4 6
2000-2001
3 1 1 1
2001-2002
2 4 1 3
2002-2003
4 2 2 3
2003-2004
6 0 2 3
2004-2005
11
1 0 2
2005-2006
8 4 1 2
2006-2007
6 6 1 0

Markov Chain Modeling and Forecasting of Saint Louis University Enrolment on
Selected Degree Programs in the Tertiary Level / Zenaida U. Garambas. 2007

---

74




Appendix C-2. College of Engineering and Architecture










YEAR LEVEL
COURSE SCH YR
1

2

3

4

5
BS
ChE 1997-1998
4 4 1 0 1
1998-1999
7 6 6 3 2
1999-2000
8 1 0 0 2
2000-2001
2 2 4 4 2
2001-2002
5 4 2 4 1
2002-2003
7 5 3 4 1
2003-2004
0 3 5 3 0
2004-2005
4 1 4 4 0
2005-2006
4 0 2 1 1
2006-2007
2 1 1 3 3

BSCE
1997-1998
9 6 7 1 5
1998-1999
16
13
11
3 7
1999-2000
14
6 5 1 3
2000-2001
15
14
3 5 3
2001-2002
18
10
13
3 1
2002-2003
30
14
13
5 4
2003-2004
22
10
9 8 2
2004-2005
22
11
10
4 4
2005-2006
20
12
7 3 3
2006-2007
19
12
13
3 2

BSEE
1997-1998
13
5 4 1 2
1998-1999
2 8 1 3 0
1999-2000
4 5 4 2 5
2000-2001
5 6 2 3 2
2001-2002
5 6 7 4 2
2002-2003
13
7 1 0 3
2003-2004
11
6 3 6 0
2004-2005
9 7 5 4 6
2005-2006
11
10
7 4 0
2006-2007
6 7 6 2 1

BSECE 1997-1998
50
33
19
4 5
1998-1999
47
37
20
8 3
1999-2000
40
39
25
5 4
2000-2001
53
39
18
6 8
2001-2002
39
47
45
11
4
2002-2003
53
45
21
7 6
2003-2004
61
67
17
12
9
2004-2005
46
41
12
8 9
2005-2006
40
22
6 9 6
2006-2007
27
20
10
3 6

Markov Chain Modeling and Forecasting of Saint Louis University Enrolment on
Selected Degree Programs in the Tertiary Level / Zenaida U. Garambas. 2007

---

75




Appendix C-2. Continued…










YEAR LEVEL
COURSE SCH YR
1

2

3

4

5
BSIE
1997-1998
4 5 6 5 1
1998-1999
11
7 5 5 1
1999-2000
5 6 6 2 0
2000-2001
5 4 10
6 1
2001-2002
4 3 5 8 1
2002-2003
6 3 2 1 1
2003-2004
5 1 2 2 1
2004-2005
8 5 1 2 3
2005-2006
0 1 2 2 1
2006-2007
2 1 5 3 1

BSME
1997-1998
14
8 8 0 2
1998-1999
14
6 7 3 7
1999-2000
10
4 6 2 4
2000-2001
5 7 7 1 3
2001-2002
10
6 7 3 2
2002-2003
6 6 8 1 1
2003-2004
11
9 3 2 0
2004-2005
18
15
6 3 5
2005-2006
17
3 6 4 0
2006-2007
13
7 3 3 3

BSGE
1997-1998
1 0 3 0 0
1998-1999
1 0 0 0 0
1999-2000
2 0 0 0 0
2000-2001
2 2 0 0 1
2001-2002
0 0 1 0 1
2002-2003
0 0 0 0 0


2003-2004
0

0

0

0
0
2004-2005
0 1 0 0 0
2005-2006
0 0 1 0 0
2006-2007
0 0 0 1 1

BS
Arch
1997-1998
7 3 3 3 1
1998-1999
13
13
7 10
3
1999-2000
15
0 9 2 1
2000-2001
11
13
8 2 3
2001-2002
14
4 2 7 5
2002-2003
11
6 4 5 3
2003-2004
13
10
5 5 3
2004-2005
15
5 7 2 6
2005-2006
19
0 4 0 7
2006-2007
14
5 3 2 0

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Appendix C-3. College of Information and Computing Sciences










YEAR LEVEL
COURSE SCHOOL YR

1

2

3

4
BSCS
1997-1998
0 4 4 4
1998-1999
0 0 8 3
1999-2000
0 0 1 3
2000-2001
0 0 0 4
2001-2002

2002-2003
0 0 0 1
2003-2004
2004-2005
6 0 0 0
2005-2006
11
1 0 0
2006-2007
13
2 1 0

BS
Math
1997-1998
0 1 0 2
1998-1999
4 0 1 0
1999-2000
3 1 1 0
2000-2001
2 1 1 0
2001-2002
4 0 2 0
2002-2003
1 1 3 0
2003-2004
2 0 1 0
2004-2005
2 0 1 0
2005-2006
1 1 0 0
2006-2007
0 1 0 0

BSIT
1997-1998
15
0 0 0
1998-1999
24
7 0 0
1999-2000
42
12
5 0
2000-2001
49
17
5 4
2001-2002
45
23
3 2
2002-2003
41
26
12
3
2003-2004
46
21
5 4
2004-2005
63
8 6 1
2005-2006
35
6 3 1
2006-2007
25
14
6 3











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Appendix C-4. College of Commerce and Accountancy










YEAR LEVEL
COURSE SCHOOL YR

1

2

3

4
BS
Accountancy

1997-1998
15
6 6 3
1998-1999
18
10
3 1
1999-2000
31
11
6 4
2000-2001
38
13
3 2
2001-2002
35
16
7 2
2002-2003
49
16
6 2
2003-2004
48
26
9 2
2004-2005
55
14
0 3
2005-2006
50
22
7 5
2006-2007
28
23
8






























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APPENDIX D


TOTAL SHIFTERS FROM SCHOOL YEAR 1997-1998

TO 2006-2007 BY DEGREE PROGRAM





















Markov Chain Modeling and Forecasting of Saint Louis University Enrolment on
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Appendix D-1. College of Human Sciences




YEAR LEVEL


COURSE SCHOOL YR

1
2
3
4
ABEcon 1997-1998
11 2 1 0


1998-1999
3 1 1 0


1999-2000
3 1 1 0


2000-2001
2 1 0 1


2001-2002
2 1 0 1


2002-2003
1 0 0 0


2003-2004
1 0 0 0


2004-2005
0 0 0 0


2005-2006
2 1 0 0


2006-2007
0 0 0 0


AB
English
1997-1998
8 5 1 0


1998-1999
5 6 1 1


1999-2000
3 2 0 0


2000-2001
4 4 0 0


2001-2002
4 4 0 0


2002-2003
5 4 0 1


2003-2004
8 2 2 0


2004-2005
8 5 0 0


2005-2006
8 2 1 0


2006-2007
11 5 0 0

Bachelor of Philosophy




1997-1998
9 0 0 0


1998-1999
2 2 0 0


1999-2000
3 1 1 0


2000-2001
1 2 0 0


2001-2002
1 2 0 0


2002-2003
2 1 1 1


2003-2004
1 1 0 0


2004-2005
3 3 2 0


2005-2006
3 1 1 0


2006-2007
1 2 0 0


















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80




Appendix D-1. Continued




YEAR LEVEL


COURSE SCHOOL YR

1
2
3
4
AB Pol. Sci. 1997-1998

49
17 2
0

1998-1999
34 13 1 0


1999-2000
36
9 1
1


2000-2001
31 19 2 0


2001-2002
31 19 2 0


2002-2003
14 10 0 0


2003-2004
25
7 1
0


2004-2005
18 17 1 0


2005-2006
20 11 3 1


2006-2007
9 10
0
0


AB Comm
1997-1998

71
12 3
0

1998-1999
31 16 0 0


1999-2000
21
9 0
0


2000-2001
38
7 2
0


2001-2002
38
7 2
0


2002-2003
19 23 1 0


2003-2004
11 10 0 0


2004-2005
21
5 0
0


2005-2006
20
7 0
0


2006-2007
17
6 0
0


BS Psycho 1997-1998

46
17 1
0

1998-1999
23
7 0
0


1999-2000
20
3 0
0


2000-2001
60
9 0
0


2001-2002
60
9 0
0


2002-2003
17 26 1 0


2003-2004
18 15 1 0


2004-2005
15
8 0
0


2005-2006
27 25 0 0


2006-2007
19 10 6 1


BSSW 1997-1998

14
10 0
0

1998-1999
14
7 0
0


1999-2000
14
5 0
0


2000-2001
11
7 1
0


2001-2002
15
5 1
0


2002-2003
11
5 0
0


2003-2004
9 6 0
1


2004-2005
6 5 0
0


2005-2006
9 4 1
1


2006-2007
9 8 1
0


Markov Chain Modeling and Forecasting of Saint Louis University Enrolment on
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Appendix D-2. College of Engineering and Architecture




YEAR LEVEL

COURSE SCHOOL YR
1
2
3 4 5


BS ChE 1997-1998 2 6
0 0
0

1998-1999 2
1 0 0
0

1999-2000 4
1 0 0
0

2000-2001 3
1 1 0
0

2001-2002 5
3 1 0
0

2002-2003 5
1 0 0
0

2003-2004 1
1 1 0
0

2004-2005 4
0 1 0
0

2005-2006 4
0 0 0
0

2006-2007 3
2 0 0
0

BSCE 1997-1998 14 1 0 0
0

1998-1999 9
2 1 0
0

1999-2000 11 3 1 0
0

2000-2001 11 7 0 0
0

2001-2002 5
4 0 0
0

2002-2003 9
2 0 0
0

2003-2004 9
5 1 0
0

2004-2005 9
1 0 0
0

2005-2006 13 3 0 0
0

2006-2007 8
4 0 0
0

BSEE 1997-1998 8
2 0 0
0

1998-1999 5
1 0 0
0

1999-2000 2
1 0 0
0

2000-2001 4
0 0 0
0

2001-2002 1
1 0 0
0

2002-2003 4
2 0 0
0

2003-2004 6
0 1 1
0

2004-2005 3
3 0 0
0

2005-2006 2
2 0 0
0

2006-2007 0
1 0 0
0

BSECE 1997-1998 31 6 2 0
1

1998-1999 22 10
3 0
0

1999-2000 25 16
0 0
0

2000-2001 18 5 3 0
0

2001-2002 20 10
3 0
0

2002-2003 20 17
4 0
0

2003-2004 11 5 1 0
0

2004-2005 16 3 0 0
0

2005-2006 11 3 1 0
0

2006-2007 16 4 0 0
0

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Appendix D-2. Continued…








YEAR LEVEL

COURSE SCHOOL YR
1
2
3 4 5
BSIE 1997-1998 7
1 1 1
0

1998-1999 7
1 1 0
0

1999-2000 9
1 0 0
0

2000-2001 7
4 0 0
0

2001-2002 5
2 0 1
0

2002-2003 3
4 2 0
0

2003-2004 2
2 2 0
0

2004-2005 1
1 0 0
0

2005-2006 2
1 0 0
0

2006-2007 5
1 4 0
0

BSME 1997-1998 6
0 0 0
0

1998-1999 3
1 0 0
0

1999-2000 4
2 0 0
0

2000-2001 7
0 1 0
0

2001-2002 6
0 0 0
0

2002-2003 3
5 1 0
0

2003-2004 2
1 1 0
0

2004-2005 3
0 0 0
0

2005-2006 5
0 0 0
0

2006-2007 5
5 1 0
0

BSGE 1997-1998 2
1 0 0
0

1998-1999 0
2 0 0
0

1999-2000 0
2 0 0
0

2000-2001 0
1 0 0
0

2001-2002 0
1 0 0
0

2002-2003 1
1 0 0
0

2003-2004 0
0 0 0
0

2004-2005 0
0 0 0
0

2005-2006 0
0 0 0
0

2006-2007 0
0 0 0
0

BSArch 1997-1998 9
1 0 0
1

1998-1999 12 0 0 0
0

1999-2000 9
1 0 0
0

2000-2001 7
0 2 0
0

2001-2002 7
1 0 0
0

2002-2003 14 1 0 0
0

2003-2004 7
0 1 0
0

2004-2005 6
0 1 0
0

2005-2006 9
0 0 0
0

2006-2007 8
0 1 0
0






Markov Chain Modeling and Forecasting of Saint Louis University Enrolment on
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Appendix D-3. College of Information and Computing Sciences




YEAR LEVEL

COURSE SCHOOL YR

1
2
3 4
BSCS 1997-1998
2 6 0 0


1998-1999
0 0 0 0


1999-2000
0 0 0 0


2000-2001
0 0 0 0


2001-2002
0 0 0 0


2002-2003
0 0 0 0


2003-2004
0 0 0 0


2004-2005
7 0 0 0


2005-2006
3 2 0 0


2006-2007
8 8 0 0

BS
Math
1997-1998
0 1 1 1


1998-1999
0 1 1 0


1999-2000
1 1 5 0


2000-2001
0 1 0 1


2001-2002
2 0 1 0


2002-2003
0 0 0 0


2003-2004
0 0 0 0


2004-2005
2 0 2 0


2005-2006
3 0 2 0


2006-2007
1 0 1 0


BSIT 1997-1998
25
1 0
0

1998-1999
31 3 0 0


1999-2000
26 12 1 0


2000-2001
43 5 2 0


2001-2002
35 6 1 0


2002-2003
12 5 0 0


2003-2004
24 10 0 0


2004-2005
37 12 0 0


2005-2006
29 12 2 0


2006-2007
37 12 1 0











Markov Chain Modeling and Forecasting of Saint Louis University Enrolment on
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Appendix D-4. College of Commerce and Accountancy




YEAR LEVEL

COURSE SCHOOL YR

1
2
3 4
BS Ac 1997-1998

13
5
0
0

1998-1999 19
11
0
0


1999-2000 6
1
0
0


2000-2001 0
0
0
0


2001-2002 1
2
1
0


2002-2003 1
0
0
0


2003-2004 2
1
0
0


2004-2005 1
1
0
1


2005-2006 1
0
0
0


2006-2007
1 0 0 0
































Markov Chain Modeling and Forecasting of Saint Louis University Enrolment on
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BIOGRAPHICAL SKETCH


The author, Zenaida Ulpindo Garambas, is a native of Tagudin, the first
town of Ilocos Sur. Born on the eighth day of the eighth month in the year 1960 to
Mauricio Somera Ulpindo and Avelina Luque Fajardo, both from Tagudin, Ilocos
Sur, she happens to be their fifth child among seven.

The author studied at Saint Augustine’s School at her hometown from
kindergarten to second year high school, then moved to San Jose High School, La
Trinidad, Benguet to finish her high school studies where she graduated in 1976
as the class salutatorian.

For her tertiary education, she studied at Saint Louis University, Baguio
City and took up a course leading to the degree Bachelor of Science in
Mathematics (minor in Physics), which she successfully hurdled as the lone
graduate in BS Math in 1981.

After graduation, she was hired to teach Mathematics and Physics at Saint
Louis University Laboratory High School and while teaching there, she was
invited by SLU-RSTC/RSDC-DOST to be one of the trainers to train teachers
from Regions I, III, and CAR in the fields of Mathematics and Physics, which she
considers to be the best test of her career. These were conducted during Saturdays
and during the summer.
Markov Chain Modeling and Forecasting of Saint Louis University Enrolment on
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Married to Alfonso Lang-ay Garambas of Sablan, Benguet, they were
blessed with six wonderful children: Alny Gerald, Alfonso Sammy, Angelicus
Christopher, Avelmar Renan, Audrey Bella, and Adralin Phil.

At present, the author teaches at Saint Louis University in the College of
Information and Computing Science, Mathematics Department.










Markov Chain Modeling and Forecasting of Saint Louis University Enrolment on
Selected Degree Programs in the Tertiary Level / Zenaida U. Garambas. 2007

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Document Outline

• Markov Chain Modeling and Forecasting of Saint Louis University Enrolment on Selected Degree Programs in the Tertiary Level
  • BIBLIOGRAPHY
  • ABSTRACT
  • TABLE OF CONTENTS
  • INTRODUCTION
  • REVIEW OF RELATED LITERATURE
  • METHODS AND PROCEDURES
  • RESULTS AND DISCUSSION
  • SUMMARY
  • LITERATURE CITED
  • APPENDIX

---