CHAPTER-I

INTRODUCTION

1.1 GENERAL INTRODUCTION

A mathematical model is a representation of a system that uses
mathematical terms and concepts. Mathematical Modelling is the process of creating a
mathematical model. The use of various mathematical models is a common phenomenon
across all the fields of knowledge. Their usages are found in all the natural sciences,
technical disciplines, and in social sciences too. Especially, Modelling of the growth is
considered the heart of almost of all the areas of applied statistics such as Econometrics,
Demo metrics, Business, Biometrics, Time series, Industrial Statistics, and forecasting as
well. Growth model methodology is used extensively in the modeling of various researches
in the fields. In the present scenario, several studies focus on either mathematical or
stochastic modelling of growth to establish functional correlations among distinct variables
by fitting various linear or nonlinear growth models in practically all applied domains of
statistics.
In literature, a wide number of mathematical and stochastic growth models have
been established and effectively used to a variety of real-world circumstances relating to a
variety of research topics in many domains of applied statistics. However, because the
situations may be complex or the models generated are mathematically or stochastically
intractable, there are still a substantial number of instances that have not yet been
mathematically or stochastically modelled.

The traditional mathematical modelling of growth proceeds along the following

lines:

(i) selection of mathematical growth model to the real world problem; like
 launching a satellite.

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  Controlling pollution to vehicles.
 Predicting the arrival of the monsoon
 Reducing traffic jams in big cities. etc.

(ii) Solving and obtaining solutions for the selected mathematical growth model; and
(iii) Examining or testing the effectiveness of the framed mathematical
growth model.

The five statements below are important on which the traditional stochastic modelling of
growth proceeds:

(i) It has Specification or Formulation of Stochastical growth model;

(ii) It has the estimation of parameters of the specified Stochastical growth model;
(iii) It tests hypotheses and constructs confidence intervals related to the parameters of the
Stochastical growth model;
(iv) It forecasts validity of the estimated statistical growth model;
(v) It uses the estimated stochastic growth model for control and making policy
decisions.

The main function of growth modelling is to forecast a system's future development.

A growth model for the economy can be used to forecast future trends and to help
policymakers make decisions.

1.2 STATEMENT OF THE RESEARCH PROBLEM

Growth models have several objectives. Applications of growth modelling are

essential to analyse the various research problems in Business, Economics and Social
Sciences. Business is a dynamic affair and dynamism is relating to time. There are many
factors which change with the passage of time, that is, as time passes, their values also
change. For example, the sales of a product, the population of a country, demand of
commodities, prices etc., may increase with time. 'Time Series Data' is an arrangement of

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statistical data on a research variable in chronological order (i.e., in line with the passage of
time). Time series data includes things like a country's population in different years, a
firm's production in different years, a product's sales in different months of the year, a
location's temperature on different days of the week, and so on.

Exponential growth and decay processes can be modelling with differential

equations. A linear first order differential equations model can be used in the context of the
rate of change of study variable y is supposed to be proportional to the difference between
the current value of y and fixed value L.

 k  L  y (t )  , that gives the solution , y (t )  L   y0  L  e  kt , with

dy (t )
i.e.,
dt
y(t )  L as t   .

Even though linear growth models are thought to be exceptionally beneficial, realistic

models typically incorporate differential equations in which the expression for is not a

simple linear function of y.Nonlinear differential equations models, such as Logistic

models, are common in the modelling of individual animal or plant growth, as well as a
community of individuals.

 k y (t ) 1  y (t )  .
dy (t )
dt  L 

When y(t) is low, as well as when y(t) is close to L, this model predicts a slow rate of
increase.

The mathematical solution of this model is given by

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 Several American and British statisticians have looked into the stochastic
characteristics of growth models. Despite the fact that American and British statisticians
have reported on a significant amount of research on growth models, nothing is done in
India. In India, a few statisticians, Econometricians and Geometricians have made attempts
in modelling growth aspects of various fields science.

The purpose of the present study is to apply modeling for the forecasting analysis is to
analyses rainfall data. To say all the climate changes refer to long-term shifts in
temperature and weather pattern.

In India an agro based economy, the changing of pattern of rainfall affects the success or
failure of crops. Many studies have showed to decide the trend in perception on both
globally and regional scales.

It is believed that, the full understanding of the rainfall pattern in the quickly changing
environmental conditions shall be of a great help to make a better decision. And, it is also
helpful in improving the adapting-capacity of various communities to cope with the
extreme weather conditions or situations around.

The collected rainfall data from Metrological department from 2014-2021 showed the
details of the rainfall of 13 districts in Andhra Pradesh. These rain fall data to take actual
vales and apply to Box-jenkins(ARIMA)Model and analyses the data using R software.

The present investigation is an attempt to describe the various mathematical and the
stochastic aspects of growth models. It also derives some new and significant growth
models by using logistic and exponential growth models.

1.3 REVIEW OF LITERATURE:

Wishart (1938), Bartlett (1947), Box (1950), Rao (1958), Rat Kowskay (1983), Seber
(1989), Edwards et.al (1996), Buruham and Anderson (1998), Sibly and Hone (2002),
Tsoularis and Wallace (2002), Fu et.al (2005), Tsoularis and Khamis et.al (2005), and
others made significant contributions to growth modelling.

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It has been observed that the application of Logistic Growth and Exponential Growth
modelling have been investigated during the last decade. It has been done because of their
first acceptance in the area of epidemiologic research. The research conducted by Bai, D.S.
et al. in1993s emphasized on Fre'chet Distribution with Type-I Censoring.To elaborate on
it, contemporary reliability engineering seeded up life tests (ALT). At the same time, it
also speeded life tests (PALT) that are used to decide the failed manners of components for
normal conditions by analyzing the data generated from the experiment.
The constant stress PALT strategy appears to operate well, as evidenced by statistical
features of the parameters. Bakshi, G., and Chen, Z. (1994) studied Population Aging,Baby
Boom, and Capital Market. They were studied as part of professional jargon. And, they
were marked with results that had demographic-results-effect on capital market and over
all distinctiveness marked in statistical controls and social as well.
There is a link between our work and a paper titled Adaptive Bayes Estimators of the
Gompertz Survival Model (ABEGSM) by Ananda, M.M. et al. (1996), even though the
exploration variates were taken advancements in dimension namely shape and statistical
adoptions. The researcher discusses different stages in Estimation of the Failure Rate in a
Partially Accelerated Life Test: A Sequential Approach by Tahir, M. (2003). The selected
units have been tested under normal stress up to time t. Here t is considered a stopping time
which can minimize the predicted loss and the cost of running the test.
For those units that did not fail by time t, the stress is increased to a greater degree in the
second stage and maintained until they all fail. The Bayes estimator is then used to
estimate the accumulated data.
The investigation included statistical and heuristic methods to study cellular automata
framework of urban growth modelling. The investigation was aconcluded with aneffective
unrecalibrated elsewhere to simulate dynamic urban growth and assess the resulting natural
and socioeconomic impacts, according to Feng, Y. and Tong, X. (2019). Our research
focused on a new attentive approach to current new era concepts and discovered that there
is a research gap in mathematical and growth models that urges us to perform some sort of
research.
"The greatest shortcoming of the human race is our inability to understand the exponential
function."-Albert Bartlett

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 Verhulst (1938)“considered that, for the population model, a stable population would
consequently have a saturation level characteristic; this is typically called the carrying
capacity, K, and forms a numerical upper bound on the growth size. To incorporate this
limiting form the logistic growth equation has been introduced the logistic growth
equation which has provided an extension to the exponential model”

Albert Bartlett(1969) in the breakthrough study The Limits to Growth offered no forecasts.
Rather, the study simulates resource consumption through time under a variety of
situations, including a resource endowment that was twice what anyone expected at the
time. The study's alarming conclusion was that nearly every scenario resulted in the
collapse of industrial civilisation at some point. The study's main finding is similar to
Bartlett's: exponential growth in the utilisation of finite resources is unsustainable. Growth
in the rate of extraction will eventually come to an end. And, given the economy's need on
constant resource input expansion, particularly energy, this leads to instability and,
eventually, decline.

“Herman and Montroll (1972) have shown that as basic an evolutionary process as the
industrial revolution may also be modelled by logistic dynamics. Here, as the industrial
revolution evolved, the fraction of the labour force in agriculture declined while the
fraction in industry grew.”

Fisher and Fry (1971) The logistic model has been effectively used to describe the
market penetration of numerous new products and technology. In this case, N stands for
the fraction of the market that has already been captured, and (KN)/K stands for the
fraction of the market that has still to be caught.

According to George Adomian(1986) Stochastic linear systems and stochastic nonlinear

systems, deterministic linear systems, and deterministic nonlinear systems are examples of
dynamical models that provide a precise solution for general technique in a variety of
fields of research and can also be treated in a unified framework. Stochastic instances
require merely one extra step in identifying appropriate statistical measures, while

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deterministic cases are not made more complex by embedding in a broader framework.
The stochastic technique emerged to obtain statistical separability and avoid truncations,
although it is also useful in deterministic situations.

McArdle (1988) proposed The factor-of-curves and curve-of-factors models are two
alternate ways for undertaking a multivariate examination of the relationships among
various behaviours. In the factor-of-curves approach, researchers look at whether a higher-
order factor (such as drug use) appropriately characterises interactions between lower-
order developmental functions (such as alcohol, cigarettes, and marijuana). The curve-of-
factors concept involves fitting a growth curve to factor scores that describe what the
various behaviours have in common at any given period. Factor analysis is performed on
the observable variables at each time point to create factor scores, which are then used to
predict growth curves. In practise, these multivariate extensions provide different chances
for analysing the dynamic structure of intra- and inter-individual change, and they
represent a natural progression in evaluating the suitability of latent growth curve
representations of behavioural dynamic.

Terry.E.Ducan Susan.C.Ducan(2004)“An suitable developmental model reflects

individual variances in developmental trajectories through time as well as describes a
single individual's developmental trajectory. Individual differences in the slopes and
intercepts of those lines should be reflected in the developmental model if, for example,
trajectories created a collection of straight lines for a sample of individuals. Another key
feature of the developmental model is its capacity to investigate predictors of individual
differences in order to answer questions regarding which variables have a significant
impact on development rate. At the same time, the model should be able to capture
important group statistics in such a way that the researcher can examine group
development.”

According to Venkatesh.P;et,al;(2017) “Mathematical Modelling is a broad

interdisciplinary science that uses mathematical and computational techniques to model
and elucidate the phenomena arising in life sciences which is created in the hope that the

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behaviour it predicts will resemble the real behaviour on which it is based. It involves the
following processes. 1) The mimicking of a real-world problem in mathematical terms:
thus, the construction of mathematical model. 2) The analysis or solution of the resulting
mathematical problem. 3) The interpretation of the mathematical results in the context of
the original situation”

M. Piazzesi, M. Schneider (2016) “Stated that Stochastic growth model is an example of

a high-dimensional problem, we consider a stylized stochastic growth
model with N heterogeneous agents (interpreted as countries). Each country is
characterized by a capital stock and a productivity level, so that there are 2N state
variables. By varying N, we can control the dimensionality of the problem. In addition to a
potentially large number of state variables, the model features elastic labor supply,
heterogeneity in fundamentals, and adjustment cost for capital”.

Mir, Youness; Dubeau, François (2019) have successfully introduce a novel sigmoid
growth model with the same qualities as traditional inflection point growth models plus the
capacity to describe data with changing forms over time The proposed model is
characterised as a continuous piecewise combination of exponential functions with
unknown breakpoint that can be used to model two-phase growth events. This model could
be combined with other models that can describe, predict, and assess biological growth.

For forecasting Irish inflation, Meyler et al (1998) developed a de for ARIMA time series
models. In their study, they placed a strong emphasis on forecast performance. They had
emphasis on decreasing out-of-sample forecast errors in their study. They had hardly any
focus on maximizing in-sample 'goodness of fit.' S.C.Hillmer presents a model-based
strategy. This model is proposed to decompose a time series exclusively into mutually
independent additive seasonal, trend, and irregular noise components. S.C.Hillmer
discussed an Arima model based approach to seasonal adjustment in his research.
According to him, Gaussian ARIMA model is believed to apply to the data.

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,Langu (1993),One of the prominent scholars in the efield, quoted by Nail and Momani
(2009), in their research about time series analysis. According to them he employed time
series analysis to figure out the changes in rainfall and runoff patterns as well. Langu
attempted to find out the substantial changes in the components of a number of rainfall
time series.

1.4 FUNDAMENTAL CONCEPTS OF GROWTH MODELS

The term "Mathematical Model" refers to a collection of equations that connect the
features of many occurrences. Mathematical modelling will be examined from a practical
standpoint as a tool for forecasting the evolution of phenomena or their attributes. This
point of view is somewhat limited at the moment, but it will be expanded upon later.
Endogenous (or prognostic, or inner, or phase variables) characteristics are unknown
qualities of mathematical models in terms of which predictions are formulated and which
are meant to expose the mechanisms of mathematical modelling.

Several fundamental concepts of independence are frequently used to form hypotheses

regarding a mathematical model's independence. Newton's gravitational rule, for example,
is at the heart of one of the most fundamental concepts of independence found in
mathematical models of cosmic object motion. The composition of mathematical models
had already been studied. However, putting together mathematical models is merely the
first stage in the process of predicting phenomena attributes in practice (process
development, systems behavior).

The next steps are:

 measurement (calculation, specific forms of characterization) of the values of the

exogenous features of models; identification of models
 development, calculation, and processing (typically for computation) of the values of
endogenous qualities under the condition that the values of external variables are
known; evaluating the mathematical model's closure
 Verification of models, i.e. cleaning up conditions to ensure that the model's predictions
are valid;

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 exploitation of the model, i.e. extraction of implications from the equations of the
model, in particular realization of endogenous characteristic values calculation

1.5 AIMS OF THE RESEARCH

The prim aim of the investigation is to describe the possible mathematical and
Stochastic aspects of growth models.

The objectives of the present enquiry are:

(i) to describe the Modelling of Growth by using Mathematical aspects of difference

equations, differential equations;
(ii) to review the various Stochastic aspects of growth models.
(iii) to discuss about the various models of Linear and Nonlinear Growth;
(iv) to analyse the rain fall data by using Box-Jenkin’s Method using R software in time
series analysis; and
(v) to develop some more new growth models through the implementation of Logistic
and Poisson regressions.

1.6 ORGANIZATION OF THE PRESENT RESEARCH WORK

The current research study's structure demonstrates how the study's objectives are
met within the given framework.

Chapter-I is an introduction that contains general introduction to Mathematical

and Stochastic Modellings of the Growth and fundamental concepts of Growth models. It
states clearly the research problem besides, the several important objectives of the
investigation. The chapter also discusses in detail about the organization and scheme of the
study.

Chapter-II Deals with several important mathematical aspects of Growth Models.

Mathematical Models building through different aspects such as geometry, differential
equations, analysis of population dynamics, Bio Mathematical Model, types of Models,

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Individual Growth Curve Models have been demonstrated in this chapter. Different
dimensional Mathematical Growth models are also discussed with their important
characteristics and limitations.

Chapter-III Describes the Stochastic Growth Model, stochastic analysis of

economic growth, Lineazation around the balanced growth path. It describesthe Logistic
curve, Exponencial Growth and decay. Further it talks about the fitting of some special
types of growth curves,they are: Gompertz Curve, Logistic Growth Curve,Modified
Exponencial growth curve.Exponential growth with continuous breeding, Geometric
Growth with discrete generations have been explained in this chapter.

Chapter-IV Discuss Latent Growth Curve Model and Measures of Linear and
Compound Growth Rates. Statistical estimation of linear and Compound Growth Rates.
Discuss about the Gompertz Curve as a growth curve. Determine nonlinear models.
Discuss about some important Nonlinear Growth models. The fitting of Nonlinear models.
Describes Non-Linear Growth and decay Models. Non-Linear method of estimation have
been discussed in this chapter. It describes three prominent methods, namely 1) Maximum
likelihood Estimation (MLE), 2) Method of moments(MOM), and 3) Least squares
method(LSM).

Chapter-V“Introduction and application of Time Serie Analysis. It describes the

types of time series analysis, define time series and analysis in R discuss. There is a
substantial discussion 1) Autoregressive Integrated Moving Average Method (ARIMA),2)
Box-Jenkin’s model, 3) Autoregressive Conditional Heteroscedasticity (ARCH) and 4)
Generalized Autoregressive Conditional Heteroscedasticity (GARCH) in the chapter. The
chapter also emphasizes on Time series Forecosts, ARIMA Forecosting work, Analyse the
rainfall data by using R software.”

Chapter-VI proposes some new Inferencial Stochastical growth model using

Logistic and Poisson regressions. Discusess Bass Growth Model ana fitting of Yule model.
The Logistic and Poisson regression models have been specified besides criteria for

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detecting and functional forms for growth models. The methods of estimating the
parameters of Logistic and Poisson regression models have been described along with
assessing the fit of a multiple Logistic regression model.This chapter provides the details
of how two dominant growth models; 1) Multiple Logistic and 2) Poisson were developed
as the part of the study.

Chapter-VII is the conclusions of the study. This chapter discusses various

suggestions for further research as an extension in future, in the lines of the present
research work.

1.7 CHAPTER SCHEME

The following headings have been used to organize the contents of the current
research project:

 CHAPTER-I : INTRODUCTION
 CHAPTER-II : SOME ASPECTS OF MATHEMATICAL GROWTH
MODELS
 CHAPTER-III : SPECIFICATION AND ESTIMATION OF SOME
STOCHASTIC GROWTH MODELS
 CHAPTER-IV : ADVANCED CRITERIA FOR SELECTION OF LINEAR
AND NON-LINEAR GROWTH MODELS
 CHAPTER-V : APPLICATIONS OF GROWTH MODELS IN TIMESERIES
ANALYSIS AND FORECASTING
 CHAPTER-VI : SOME NEW STOCHASTIC GROWH MODELS USING
LOGISTIC AND POISSON REGRESSIONS
 CHAPTER-VII : CONCLUSIONS

BIBLIOGRAPHY

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