AN INSIGHT ON THE DESIGN AND CONSTRUCTION OF A POWER LINE VANDALISM MONITORING SYSTEM USING ELECTRONIC SYSTEM

CHAPTER ONE

INTRODUCTION

1.1     BACKGROUND OF THE STUDY

This project gives the insight on the design and construction of a transmission power line vandalism detector system. Vandalism means destructive action. It is a hateful and deliberate defacement/destruction of somebody else’s property or national assets like thehigh-voltage transmission power lines, transmission towers and distribution transformers, etc.The high-voltage transmission power lines strung from support towers form the backbone of the nation’s electric power grid. Many of those 158,000 miles of lines, supported by nearly 800,000 towers, run through isolated areas as they deliver electricity from generating plants to cities. These high-voltage transmission power lines around the world are vulnerable to terrorism, vandalism, physical deterioration and extreme weather. Every year in Nigeria bad citizens vandalize the power line, towers, transformers, generators and other power transmission/distribution facilities. This causes frequent power failure and power surge that seriously affects over 50 million business firms and manufacturing industries in the country. These companies depend on electricity power supply from the power line for their productions. If the power line is vandalized, some of their machines cannot be powered on a generator then income, investments and commodities worth billions of Naira will be lost. If electricity is available during the power line vandalism incident, the lives and properties of the people within the area can be affected as a result of electric shocks, power surge and fire outbreak, etc. Vandalism is now largely driven by the soaring values of copper and aluminum in the international metals market due to increased world market demand fueled by China and India’s growth in industrialization. Consequently, the vandals now operate in wellorganized syndicates complete with offices and hierarchy and accumulate large volumes of stolen materials which they eventually consign for export as scrap. Transformer oil, copper and aluminum are the main targets of these vandals. The uses of the stolen transformer oil are:

  • Mixed with diesel and sold as fuel,
  • Used as fuel for industrial furnaces and as cooling for welding sets,
  • Mixed with vegetable oil and sold as cooking oil.
  • Used as a cosmetic and treatment for wounds.
  • On the other hand, copper and aluminum have been in the past been used on a minimum scale by the informal sector for welding sets but are now mainly exported as scrap. It is this later use that is behind the recent escalation of vandalism.
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AN INSIGHT ON THE DESIGN AND CONSTRUCTION OF A POWER LINE VANDALISM MONITORING SYSTEM USING ELECTRONIC SYSTEM

A STATISTICAL ANALYSIS ON THE FERTILITY AND MORTALITY RATE IN NIGERIA (A CASE STUDY OF OSOGBO LOCAL GOVERNMENT IN OSUN STATE)

CHAPTER ONE

1.1       INTRODUCTION

In statistical analysis and inference statistics is playing an important role nearly in all phases of human life. Formally dealing only with affair of the state and this account for its name statistics.  

The influence of statistics is now spread to manufacturing companies, agricultural sectors, bank, business communication, economic, education, political science, psychology, sociology and other numerous held of life. It’s therefore, a body of scientific method and theory of collecting, organizing, presenting and analyzing of data as well as drawing valid conclusion and making a reasonable decision. The application is found in all discipline of numerical form.

Furthermore, this project is to explain and to analyze more on fertility and mortality rate of people in “Osogbo Local Government”. This project cover four fiscal year from 2006 – 2009 and data is being collected at Lautech Teaching HospitalOSOGBO. Also computation would be determined from the total number of fertility and mortality rate and this will be use for information required for the actualization of facts.

1.2       HISTORICAL BACKGROUND

Lautech Teaching Hospital is owned by both Osun State Government and Oyo State Government. The hospital is situated at Idi-seke area in Osogbo Local Government. LAUTECH Teaching Hospital is equiped to standard that no other hospital in the state can complete with it. LAUTECH Teaching Hospital has many professional doctors and well trained and very efficient nurses.

1.3       AIMS AND OBJECTIVES OF THE STUDY

  1. To know if the economy can be predicted from the fertility rate.
  2. To predict the economy of Osogbo Local Government using the fertility and mortality rate.
  3. To know the nature of the relationship that exists amongst fertility rate and mortality rate the economy of Osogbo Local Government.
  4. To recommend ways of ensuring adequate documentation of fertility and mortality rate.
  1. Research Questions
  2. What are the challenges of proper documentation of the fertility and mortality rate?
  3. Can the economy of the Nigeria be predicted using the fertility and mortality rate?
  4. What is the nature of relationship that exists between fertility, mortality rate and the economy of Osogbo Local Government.
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A STATISTICAL ANALYSIS ON THE FERTILITY AND MORTALITY RATE IN NIGERIA (A CASE STUDY OF OSOGBO LOCAL GOVERNMENT IN OSUN STATE)

WIND SPEED FORECASTING IN THE PRESENCE OF SOME METEOROLOGICAL VARIABLES USING VECTOR AUTOREGRESSIVE AND MULTINOMIAL LOGISTIC REGRESSION MODEL

WIND SPEED FORECASTING IN THE PRESENCE OF SOME METEOROLOGICAL VARIABLES USING VECTOR AUTOREGRESSIVE AND MULTINOMIAL LOGISTIC REGRESSION MODEL

 

CHAPTER ONE

INTRODUCTION

1.1 Background to the Study

With the advent of science and technology, the demand for electrical energy becomes inevitable. Nigeria is a country endowed with abundant energy resources like coal, solar, water (Dam), wind and so on which can be used as a form of electricity generation. Despite the abundance of these energy resources, there is inconsistent supply of electricity, which may be due to underutilization of the potentials.

Wind energy is the fastest growing renewable source of energy. With respect to this, the need for wind speed modelling and forecasting becomes paramount. Energy generation by wind is of great advantage because wind turbines do not produce any form of pollution when sited strategically. Moreover, it blends with the natural landscape. The utilization of wind energy will ensure the growth of socio-economic development and improvement in the quality of life of the citizens. The demand for more sustainable energy sources are on the increase in order to address the growing needs of humans. It is also in line with taking care of the environment and the minimal use of natural resources, which means there is an urgent need for developing renewable energy.

Wind energy is now becoming the current trend in renewable energy as it addresses rising energy demand while being nature friendly at the same time. In the long run, electricity generated from the wind turbines cost less than the conventional power plants since it does not consume fossil fuel. Researches on the potentials of wind energy in some major cities in Nigeria show high wind speed in Lagos, Maiduguri, Enugu, Jos, Kano, Funtua and Sokoto (Idris et al., 2012).

The gathering of wind data is important for the wind farm beginning from its feasibility to its actual operation. Prior to the construction of a wind farm, at least one year of meteorological study is necessary and a detailed verification of the specific on-site wind conditions are necessary. Meteorological values, more specifically wind speed and wind direction are necessary for the calculation of the wind farm’s yearly electrical generation profile. The harnessing of Kinetic energy through the wind has been used for centuries, be it in form of powering sail boats, wind mills, or furnaces (Aliyu and Mohammed 2014). But, it was not until 1979 that the modern wind power industry began in earnest with the production of wind turbines. The use of wind energy as a form of renewable energy gained momentum in the 80s and 90s and there are now thousands of wind turbines operating all over the world (Minh et al., 2011). The modern and most commonly used wind turbine has a horizontal axis with two or more aerodynamic blades mounted on the shaft. These blades can travel at over several times the wind speed, generating electricity which is captured by a medium voltage power collection system and fed through the power transmission network (Garba and Al-Amin, 2014).

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WIND SPEED FORECASTING IN THE PRESENCE OF SOME METEOROLOGICAL VARIABLES USING VECTOR AUTOREGRESSIVE AND MULTINOMIAL LOGISTIC REGRESSION MODEL

TRANSMUTATION OF WEIBULL PARETO DISTRIBUTION

TRANSMUTATION OF WEIBULL PARETO DISTRIBUTION

 

CHAPTER ONE

INTRODUCTION

1.1 Background to the study

In the field of Statistics and reliability engineering, the quality of the procedures used in statistical analysis depends heavily on the assumed probability distribution. Due to this fact, significant efforts have been made by many researchers in the development of standard probability distributions which are different from the known classical probability distribution. The standard probability distributions are obtained by generalizing the classical probability distribution such as exponential, Weibull, Pareto and Beta distributions. Application of probability distributions in engineering, medicine, finance, ICT among others, have further shown that many data sets do not follow the existing classical distributions. As a result of this, there is the need for development of standard probability distributions by generalization of some well-known classical distributions.

These distributions are derived by adding one or more parameters to the baseline model of continuous distributions. These families provide more flexibility in modelling and in analyzing real life data in many applied areas. For instance, the generalized transmuted-G family proposed by Nofalet al(2015), Transmuted Weibull Distribution: A Generalization of the Weibull Probability Distribution proposed by Aryal and Tsokos (2011), the transmuted geometric-G family introduced by Afifyet al(2016), the transmuted exponentiated generalized-G class of distributions defined by Yousofet al(2015), transmuted exponential distribution proposed by Enahoroet al(2015)and the Kumaraswamy transmuted-G family introduced by Afifyet al (2016).

Over the years, several attempts have been made to generalize the Weibull distribution by adding new parameters into the distribution which has led to the development of new distributions. For instance theExponentiated Weibull distribution (Pal et al 2003), Transmuted Weibull distribution (Gokarnal and Chris, 2011), Lomax-Weibul distribution (Almheidat et al 2015) ,Beta Weibull Distribution (Cordeiro et al 2012), New Weibull-Pareto distribution ( Suleiman and Albert 2015).These distributions have been found to be more flexible than the Weibull distribution when applied to real life data sets.

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TRANSMUTATION OF WEIBULL PARETO DISTRIBUTION

STUDY ON INFERENCES AND APPLICATIONS OF ODD GENERALIZED EXPONENTIAL-RAYLEIGH DISTRIBUTION

STUDY ON INFERENCES AND APPLICATIONS OF ODD GENERALIZED EXPONENTIAL-RAYLEIGH DISTRIBUTION

 

CHAPTER ONE

INTRODUCTION

1.1    Background of the study

The Rayleigh distribution was named after Lord Rayleigh (1842-1919) a British physicist as well as mathematician also known as John William Strutt. In 1895 he discovered the inert gas Argon (Ar), the research that earned him the 1904 Nobel Prize in Physics Venkatesh and Manikandan (2016).

The Rayleigh distribution has wide range of applications in the field of applied sciences, especially in modeling the lifetime of an object or service time. Battjes (1969) stated some areas where the distribution can also be applied, these includes sea waves, harbor, coastal and ocean engineering, heights and periods of wind waves. Despite its applicability, the distribution suffer’s the same problem other classical distributions suffered from, which is lack of flexibility due to the fact that it has only one parameter. For instance, Tahir and Cordeiro (2016) stated that, the well-known classical/baseline distributions such as exponential, Rayleigh, Weibull and gamma are limited in their characteristics and are unable to show wide flexibility. Because of this and several other problems; several researchers have worked and some are still trying to overcome these challenges by generalizing some of these classical distributions to come up with compound distributions.

According to Eugene et al. (2002) generalization of distributions started in the year 1925; Also, Ahuja and Nash (1967) introduced the generalized Gompertz-Verhulst family of distributions to study growth curve mortality. Gupta et al. (1998) added one parameter to the cumulative distribution function of the baseline distribution to define the exponentiated-G class of distributions and several others follows.

Also Gupta and Kundu (1999) pioneered the study of two-parameter generalization, in which they studied two-parameter Generalized Exponential (GE) distribution also called Exponentiated Exponential (EE) distribution, after which several other authors worked on GE distribution due to its attractive features, among which are Gupta and Kundu (1999), Kundu et al. (2005), Nadarajah (2006) and so on.

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STUDY ON INFERENCES AND APPLICATIONS OF ODD GENERALIZED EXPONENTIAL-RAYLEIGH DISTRIBUTION

STATISTICAL POWER OF HYPOTHESIS TESTING USING PARAMETRIC AND NONPARAMETRIC METHODS

STATISTICAL POWER OF HYPOTHESIS TESTING USING PARAMETRIC AND NONPARAMETRIC METHODS

 

 

CHAPTER ONE

INTRODUCTION

1.1         Background

Nonparametric approaches are often utilized when the conditions for parametric approaches are not satisfied and in most cases when the scale of measurement is ordinal or nominal. Statistical procedure in which inferences are made about the population parameters are referred to as Parametric Statistics Cyprain (1990). Parametric approach follows certain assumptions which include samples that are randomly drawn from a normally distributed population,

  1. Consist of independent observations, except for paired values,
  2. Consist of values on an interval or ratio measurement scale,
  3. Have respective populations of approximately equal variances,
  4. Are adequately large, and
  5. Approximately resemble a normal distribution.

If any of the samples breaks one of these rules, then the assumptions of a parametric test are violated. The nature of the study might be changed to adhere to the rules. If an ordinal or nominal measurement scale is being used, the study might be redesigned to use an interval or ratio scale. Also, try to seek additional participants to enlarge the sample sizes. Unfortunately, there are times when one or neither of these changes is appropriate or even possible. There are three major parametric assumption, which are and will continue to be violated by researchers in health sciences; level of measurement, sample size and normal distribution of the dependent variable Pett (1992).

If samples do not resemble a normal distribution, you might have learned to modify them so that you can use the tests you know. There are several legitimate ways to modify your data, so you can use parametric tests. First, if the reasons can be justified, then the extreme values from samples called outlier might be removed. Second, you can apply a mathematical adjustment to each value in your samples called a transformation. That is you might square every value in a sample. Transformations do not always work, however. Third, there are more complicated methods that are so advanced. Fortunately, there is a family of statistical tests that does not demand all the parameters, or listed rules above. They are called nonparametric tests.

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STATISTICAL POWER OF HYPOTHESIS TESTING USING PARAMETRIC AND NONPARAMETRIC METHODS

SPATIAL ANALYSIS OF CHOLERA IN KADUNA STATE, NIGERIA

SPATIAL ANALYSIS OF CHOLERA IN KADUNA STATE, NIGERIA

 

CHAPTER ONE

INTRODUCTION

1.1 Background to the Study

Spatial statistics is the process of extracting or creating new information about a set of geographic features to perform routine examination, assessment, evaluation, analysis or modeling of data in a geographic area based on pre-established and computerized criteria and standards. It is a technique for analyzing spatial data mostly on human scale. Complex issues arise in spatial analysis, many of which are neither clearly defined nor completely resolved. The most fundamental of these are the problems of defining the spatial location of the entities being studied (Scott and Getis, 2008).

Nigeria is a prime area in studying spatial patterns associated with diseases because it is a country where millions of people live in close proximity not only to other people but also to open and unsafe water sources and refuse dumps. It is also a country that is actively engaged in alteration of its aquatic ecosystems, a process often associated with changed disease ecologies (WHO 1993). Cholera is one of the deadliest diseases in Africa (WHO 1993), within 2-3 hours of unset symptoms, a previously healthy person may severely become dehydrated and if not treated may die within 24hours (Sack et al 2004; WHO 2010). A link between cholera, phytoplankton blooms and copepod zooplankton has been demonstrated in Asia (Colwell et al., 1996). The African Great Lakes have been suspected to play a role as reservoirs of the bacteria Vibrio cholerae (v.cholerea), while human infection and movement are probably involved in the propagation of the disease inland. During the 19th century, cholera spread repeatedly from its original reservoir or source in the Ganges delta in India to the rest of the world, before receding to South Asia (WHO 2008). Six pandemics were recorded that killed millions of people across Europe, Africa and the Americas. The seventh pandemic, which is still on going, started in 1961 in South Asia, reached Africa in 1971 and the Americas in 1991.

The disease is now considered to be endemic in many countries and the pathogen causing cholera cannot currently be eliminated from the environment, WHO(2008). Regions of the world where cholera is currently prevalent are Africa, Asia and parts of the Middle East. Imported cases occasionally occur in richer countries in travellers returning from endemic areas, (National Travel Health Network and Centre (2007). The disease no longer poses a threat to countries with minimum standards of hygiene, but it remains a challenge to countries where access to safe drinking water and adequate sanitation cannot be guaranteed, (WHO, 2009). The mechanistic basis for a climate-cholera connection involves multiple pathways and the primary transmission from environmental reservoirs initiates seasonal outbreaks of cholera in endemic regions, (Pascualet al, 2002).

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SPATIAL ANALYSIS OF CHOLERA IN KADUNA STATE, NIGERIA

ON FOUR-PARAMETER ODD GENERALIZED EXPONENTIAL-PARETO DISTRIBUTION: ITS PROPERTIES AND APPLICATIONS

ON FOUR-PARAMETER ODD GENERALIZED EXPONENTIAL-PARETO DISTRIBUTION: ITS PROPERTIES AND APPLICATIONS

 

CHAPTER ONE

INTRODUCTION

1.1    Background of the study

The Pareto distribution is a widely known distribution in applied sciences as well as in Economics. It was introduced in order to explain the distribution of income in the society (Pareto, 1896). It was first proposed by a Professor of Economics, Vilfredo Pareto (1843-1923). The distribution was found while studying various distributions for modeling income in Switzerland. The various forms of the Pareto distribution are versatile and can usually be used to model uncertainties. Since that time its applicability spans diverse areas of human endeavour comprising Biology, Physics, Actuarial Science, Geography, etc. Pareto made several important contributions to Economics, mostly in the study of income distribution and in the analysis of individuals choices.

Pareto found out that income approximately follows a Pareto distribution, which is considered as power law probability distribution. The Pareto principle was named after him and noted that 80% of the land in Italy was owned by 20% of the popula-tion. One of Pareto’s equations attained special importance and argument. He was captivated by problems of power and wealth. How do people get it? How is it spread around society? How do those who have it use it? The gap between rich and poor has always been part of the human condition, but Pareto resolved to measure it. He collected piles of data on wealth and income through different centuries, across different countries: the tax records of Basel, Switzerland, from 1454 and from Augs-burg, Germany, in 1471, 1498 and 1512; contemporary rental income from Paris; personal income from Britain, Prussia, Saxony, Ireland, Italy, and Peru. What he discovered or thought he discovered was striking. When he plotted the data on a graph sheet, with income on one axis, and number of people with that income on the other, he observed similar scenario nearly everywhere in every era. Society was not a “social pyramid” with the percentage of rich to poor sloping gently from one class to the next. Instead it was more of a “social arrow” the bottom was very fat indicating where the mass of men live, and at the top was very thin indicating where the wealthy elite reside. Nor was this effect by chance; the data did not remotely fit a bell curve, as one would anticipate if wealth were randomly distributed. “It is a social law”, he wrote: something “in the nature of man”. At the bottom of the Wealth curve, he wrote, Men and Women starve and children die young. This reason makes Pareto to develop model for distribution of wealth.

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ON FOUR-PARAMETER ODD GENERALIZED EXPONENTIAL-PARETO DISTRIBUTION: ITS PROPERTIES AND APPLICATIONS

ODD GENERALIZED EXPONENTIAL-INVERSE-EXPONENTIAL DISTRIBUTION: IT’S PROPERTIES AND APPLICATIONS

ODD GENERALIZED EXPONENTIAL-INVERSE-EXPONENTIAL DISTRIBUTION: IT’S PROPERTIES AND APPLICATIONS

 

CHAPTER ONE:

INTRODUCTION

1.1 Background of the Study

Statistical data modelling is an important aspect of statistics that attracts researcher’s wide attention. Modelling lifetime data in areas comprising reliability analysis, engineering, economics, biological studies, environmental and medical sciences, requires an appropriate statistical model for proper actualization of the data. However, there still remains problems where the real data does not follow any of the classical or standard probability models. To address this, there is strong need then to propose new models that can better capture real-life phenomenon inherent in a given dataset. Introducing new probability models or their classes is an old practice and has ever been considered as very valuable as many other practical problems in statistics.

The idea simply started with defining different mathematical functional forms, and then induction of location, scale or inequality parameters (Tahir and Cordeiro, 2016a). Addition of new shape parameter(s) expands a model into a larger family of distributions and can provides significantly skewed and heavy-tailed new distributions. It also provides greater flexibility in the form of new distributions. This induction of parameter(s) has been proved useful in exploring tail properties and also for improving the goodness-of-fit of the proposed generator family (Saboor et al., 2015).

The one parameter Inverse Exponential distribution otherwise known as the Inverted Exponential distribution was introduced by Keller and Kamath (1982). It has an inverted bathtub failure rate and it is a competitive model for the Exponential distribution. It is an important probability distribution for modelling lifetime data.

1.2 Statement of the Problem

There exist several probability distributions for modelling lifetime data.However, some of these dataset do not follow any of the existing and well known standard probability distributions or at least are inappropriately described by them. This brings increased interest in proposing new univariate continuous distributions by adding one or more new shape parameter(s) to the baseline model. In this dissertation, a new probability distribution called Odd Generalized Exponential-Inverse-Exponential distribution (OGE-IED) taking inverse-exponential as the baseline distribution and using Tahir et al., (2015) generator is being proposed, aimed to provide greater flexibility and create more weight to the tails of the new distribution.

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ODD GENERALIZED EXPONENTIAL-INVERSE-EXPONENTIAL DISTRIBUTION: IT’S PROPERTIES AND APPLICATIONS

MODELING BRONCHO-PNEUMONIA STATUS IN INFANTS USING DISCRIMINANT AND LOGISIC REGRESSION ANALYSES

MODELING BRONCHO-PNEUMONIA STATUS IN INFANTS USING DISCRIMINANT AND LOGISIC REGRESSION ANALYSES

 

CHAPTER ONE

INTRODUCTION

1.1 Background of the study

Discriminant analysis is a procedure that can be used to build Discriminant functions which are linear functions of p-variables that can be used to describe or elucidate the differences among two or more groups. The goals of discriminant analysis include identifying the relative contribution of the p variables to separation of the groups and finding the optimal plane on which the points can be projected to best illustrate the configuration of the groups. Another use of discriminant analysis is the prediction or allocation of observations to groups, in which linear functions of the variables are employed to assign an individual sampling unit to one of the groups. The measured values in the observation vector for an individual or object are evaluated by the classification function to find the particular group to which the individual most likely belongs.

Interest in human development before birth is widely spread because of the interest in knowing more about our beginning and the desire to improve the quality of life. The intricate process by which a baby develops from a single cell is miraculous and few events are more exciting than a mother‟s viewing of her embryo during an Ultrasound examination. Human development is a continuous process that begins when an Oocyt (ovum) from a female is fertilized by a sperm (spermatozoa) from the male. By accepting the shelter of uterus, the foetus also takes the risk of disease or malnutrition and of biochemical immunological and hormonal adjustment.

Until the beginning of the Nineteenth Century, far more attention was paid to the collection and presentation of data than to their interpretation. Large volume of data were usually collected and frequently misinterpreted if indeed interpretation was attempted. However, since that time, the importance of scientific approach in the interpretation of data has been realized and great steps have been achieved in the development of appropriate methods.

In modern days, statistics has played a significant role in Biological, Pharmaceutical and Medical Sciences (Cornfield,1952). The application of multivariate statistical techniques to biological and medical data has dominated the areas of evidence-based medicine. Multivariate methods are relevant in virtually every branch of applied medicine, pharmacy and public health. They come into play either when we have a medical theory to test or when we have a relationship in mind that has some importance for medical decision or policy analysis in public health. Multivariate methods are also used in other disciplines.

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MODELING BRONCHO-PNEUMONIA STATUS IN INFANTS USING DISCRIMINANT AND LOGISIC REGRESSION ANALYSES

EXTENSION OF BURR V DISTRIBUTION: ITS PROPERTIES AND APPLICATION TO REAL-LIFE DATA

EXTENSION OF BURR V DISTRIBUTION: ITS PROPERTIES AND APPLICATION TO REAL-LIFE DATA

 

 

CHAPTER ONE:

INTRODUCTION

1.1         Background to the Study

Extended or generalized distributions have been extensively studied in recent years. Amoroso (1925) was the pioneer researcher to start generalizing continuous distributions, discussing the generalized gamma distribution to fit observed distribution of income rate. Since then, numerous authors have developed various classes of generalized distributions. Well-known distributions have been generated or extended in many ways. Some of the well-established generators are Marshal-Olkin generated family (MO-G) by Marshall and Olkin (1997), the beta-G by Eugene et al. (2002), Jones (2004), gamma-G (type 1) by Zografos and Balakrishnan (2009), Kumaraswamy-G (Kw-G for short) by Cordeiro and De Castro (2011), gamma-G (type 2) by Ristic and Balakrishnan (2012), gamma-G (type 3) by Torabi and Hedesh (2012), McDonald-G (Mc-G) by Alexander et al. (2012), log-gamma-G by Amini et al. (2014), exponentiated generalized-G by Cordeiro et al. (2013), Weibull-G by Bourguignion, et al. (2014) among others. Recent developments have been geared to define new families by introducing shape parameters to control skewness, kurtosis and tail weights thus providing great flexibility in modeling skewed data in practice (Jones, 2004; Cordeiro et al., 2013).

Evidently, one of the most used generalized distribution generator is the beta-G. The earliest of the beta extended distributions is the class of distributions generated from the logit of a beta random variable with cumulative distribution function that involve employing two parameters whose role is to introduce skewness and to vary tail weights (Eugene et al., 2002). Jones (2004) discusses general beta family influenced by its order statistics and shows that it has beautiful distributional properties and potential for interesting statistical applications. The generalization method used is the logit of beta distribution.

In this Research, we will define and study a four-parameter beta-Burr type V distribution. To attain this, we will let F(x) be the CDF of the Burr type V distribution as given in Equation (1.7) such that the emerging distribution is being referred to as the beta-Burr type V distribution (BBV) using appropriate transformations and employing the logit of beta. We will then prove

p

that this is a genuine distribution, in such a way that ò2p g (x )dx =1, where  g ( x) is the PDF of the – proposed distribution. Then, we will also be define and discuss some properties of the extended distribution.

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EXTENSION OF BURR V DISTRIBUTION: ITS PROPERTIES AND APPLICATION TO REAL-LIFE DATA

EXPONENTIATED-EXPONENTIAL WEIBULL DISTRIBUTION: ITS PROPERTIES AND APPLICATIONS

EXPONENTIATED-EXPONENTIAL WEIBULL DISTRIBUTION: ITS PROPERTIES AND APPLICATIONS

 

CHAPTER ONE

INTRODUCTION

1.1    Background to the Study

Several classical distributions have been widely used over the past decades for mod-elling lifetime data in many areas such as reliability, engineering, economics, biolog-ical studies, environmental actuarial, environmental and medical sciences, demog-raphy, and insurance. However, in many applied areas such as lifetime analysis, finance, and insurance, there is a clear need for extended forms of these distribu-tions. This is because there still remain many important problems where the real data does not follow any of the classical or standard probability models. For that reason, numerous methods for generating new families of distributions have been considered (Bourguignon et al., 2014). To handle this, there is a strong need to propose useful models for the better study of the real-life marvel. Introducing new probability models or their classes is an old practice and has ever been considered as valuable as many other practical problems in statistics. According to Tahir and Cordeiro (2016), the idea simply started with defining different mathematical func-tional forms, and then adding of location, scale or shape parameter(s).

Inducing of a new shape parameter(s) introduces a model into greater family of distributions and can give significantly skewed and heavy-tailed distributions and also provides greater flexibility in the form of new distribution. Gupta et al., (1998) proposed a generator for defining a new univariate continuous family of distribu-tions by adding one shape parameter to the baseline (parent) distribution. Gupta and Kundu (1999) pioneered the study of two-parameter generalization, in which they studied two-parameter Generalized-Exponential (GE) distribution also called Exponentiated-Exponential (EE) distribution, after which several other authors worked on GE distribution due to its attractive features, among which are Gupta and Kundu (2001a, 2001b, 2002 and 2003), Kundu et al., (2005) among others.

The authors present two real life data sets, where it is observed that in one data set Ex-ponentiated exponential distribution has a better fit compared to Weibull or gamma distribution and in the other data set Weibull has a better fit than Exponentiated exponential or gamma distribution. The induction of parameter(s) has been proved useful in discovering tail properties and also for improving the goodness-of-fit of the proposed distribution (Saboor et al., 2015).

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EXPONENTIATED-EXPONENTIAL WEIBULL DISTRIBUTION: ITS PROPERTIES AND APPLICATIONS

COMPARISM OF THE PENALIZED REGRESSION TECHNIQUES WITH CLASSICAL LEAST SQURES IN MINIMIZING THE EFFECT OF MULTICOLLINEARITY

COMPARISM OF THE PENALIZED REGRESSION TECHNIQUES WITH CLASSICAL LEAST SQURES IN MINIMIZING THE EFFECT OF MULTICOLLINEARITY

 

CHAPTER ONE

INTRODUCTION

1.1 Background of the Study

In order to reduce possible biasness,large number of predictor variables was introduced in a model and that lead to a serious concern of multicollinearity among the predictor variables in multiple linear regressions, variable selection is an important issue.(Mathew and Yahaya, 2015)

Multicollinearity and high dimensionality are two problems and computational issue that bring challenges to regression analysis. To deal with these challenges, variables selection and shrinkage estimation are becoming important and useful. The traditional approach of automatic selection (such as forward selection, backward elimination and stepwise selection) and best subset selection are computationally expensive and may not necessarily produce the best model.

Multicollinearity problem is being dealt with by Penalized least square (PLS) method by putting some constraints on the values of the parameters estimated. The aftermath is that the entries of the variancecovariance matrix are significantly reduced.When multicollinearity exist that predictor’s variables that are highly correlated form some groups. One of the waycollinearity problem can be dealt with is to remove one or more of the predictor variables within the same group, by making decision which among the group variables is to be eliminated tend to be difficult and complicated. Theaftermath of multicollinearityis that the parameter estimator and their variance or standard error tends to be large and prediction may be very inaccurate.

In a situation where there exist correlated data or data where the number of predictors is much larger than the sample size, penalized regression methods have beenintroduced to deal with this challenge, because they produce more stable results, penalized regression methods do not clearly select the variables; instead they minimize the Regression Sum of Square by using a penalty on the size of the regression coefficients. This penalty causes the regression coefficients to shrink toward zeroand thismay result in biased estimates through these regression coefficient estimates will have smaller variance. This can improve the prediction accuracy because of the smaller mean squared error (Hastie et al., 2009). This is why penalized regression methods are also known as shrinkage or regularization methods. Some regression coefficients are set to zero exactly if the shrinkage is large enough,thus, penalized regression methods perform variable selection and coefficient estimation simultaneously. The Least Absolute Shrinkage Selection Operator (LASSO) enables selection such that only the important variable stays in the model (Szymeezak,et al., 2009).

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COMPARISM OF THE PENALIZED REGRESSION TECHNIQUES WITH CLASSICAL LEAST SQURES IN MINIMIZING THE EFFECT OF MULTICOLLINEARITY

ASSESSMENT OF THE RELATIONSHIP BETWEEN FOETAL HAEMOGLOBIN LEVELS, TISSUE DESTRUCTION AND DISEASE SEVERITY IN INDIVIDUALS WITH SICKLE CELL ANAEMIA

ASSESSMENT OF THE RELATIONSHIP BETWEEN FOETAL HAEMOGLOBIN LEVELS, TISSUE DESTRUCTION AND DISEASE SEVERITY IN INDIVIDUALS WITH SICKLE CELL ANAEMIA

 

CHAPTER ONE

1.0.        Introduction

Sickle cell anaemia (SCA) is a disorder of the Haemoglobin caused by inheritance of a single abnormal haemoglobin (termed sickle haemoglobin- HbS) which results in physical and chemical modifications of the haemoglobin molecule (Damanhouriet al., 2015). Under conditions of low oxygen concentration, the sickle haemoglobin causes the red blood cell to become fragile and prone to rupture. This condition causes the aggregation of haemoglobin (polymerization) and reduces the overall capacity of the haemoglobin to convey oxygen to tissues by impeding free passage of blood in the vessels (vaso-occlusion) to tissues. Increased rupturing of red blood cells leads to anaemia (reduction in number of red blood cells) (Elias et al., 2012; Damanhouriet al., 2015).

Lactate dehydrogenase (LDH), an enzyme of the glycolytic pathway that catalyzes the inter conversion of pyruvate to lactate is found in abundance in all tissues. It consist of five (5) isoenzymes forms classified according to their electrophoretic mobility. The concentration of these isoenzymes varies from one organ to the other, with LDH1 and LDH2 found primarily in RBCs and heart muscle (Ballas, 2013). There is increased level of LDH in the general blood circulation of individuals with SCA. This is due to the increased breakdown of diseased RBCs and other tissues and has been used as a marker of hemolysis and tissue damage. The haemolysis/tissue damage increases further in individuals with Sickle cell anaemia in crises state (Kato et al., 2006; Ballas 2013). Numerous studies have associated SCA individuals with high LDH level to high prevalence of pulmonary hypertension, renal failure and early death (Kato et al., 2013). High LDH observed in this individuals is as a result of haemolysis and tissue destruction (Mehdiet al., 2013; Alzahriet al., 2015). Despite the importance of LDH as a gold standard marker for tissue damage information about LDH in population suffering from SCA in Nigeria is scarce.

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ASSESSMENT OF THE RELATIONSHIP BETWEEN FOETAL HAEMOGLOBIN LEVELS, TISSUE DESTRUCTION AND DISEASE SEVERITY IN INDIVIDUALS WITH SICKLE CELL ANAEMIA

APPLICATION OF PORTFOLIO OPTIMIZATION: A STATISTICAL APPROACH

APPLICATION OF PORTFOLIO OPTIMIZATION: A STATISTICAL APPROACH

 

CHAPTER ONE

INTRODUCTION

1.1 Background of the Study

Nigeria stock market is a place where investors buyshares of assets or securities mostly for them to earn reward (returns) at a given level of „tolerable‟ risk. The word „portfolio‟ is used to mean a mix or combinatory pool of assets, securities or investment of financial or physical nature, in which an investor holds to meet his/her set objectives. On the other hand, portfolio selection is choosing the most suitable combination of assets by risk averse or rational investor to maximize return while at the same time minimizes risk.

Due toabrupt unpleasant surprises of global financial crisis, and inferior market downturn experienced by financial markets in different times, investors are becoming highly concerned about the risk of their investments coupled seeking ways to achieve more attractive risk/return characteristics and better capital protection in difficult environments. The main plan of portfolio management is to form diverse securities in a portfolio that meets the needs of investors and thereafter manage the portfolio in other to obtain required goals (Vigdiset al., 2011). Many players in the financial market have been frequently looking for strategic ways of investing which is capable of meeting this aforesaid protest of the market.

Modern Portfolio Theory(MPT) pioneered by Markowitz (1952, 1970, 1987) seminal work specifically solves the tradeoff between risk and return using formed curve in graph called efficient frontier. This frontier solves the problem by considering the risk, return of invested assets and the correlation that exist between the asset return. The curve identifies those that are at maximum return for a given level of risk, or at their minimum risk for a given level of return.

Markowitz‟s work to a great extent changes the behavior of investors and financial managers by providing more insight on the issue of assets selection. Nowadays, players in the market are embracing the framework viewed as the most standard model for today‟s investment management even with the model‟s unrealistic questionable assumptions.

Markowitz (1991), as well asElton and Gruber (1997) talk more on the main issue an investor faces when investing, of them is how to distribute resources among different assets alternatives. Almost all investors are being posed with this same problem, with the added sufferings and complications needed to clearly include the properties of the liabilities in the analysis (Bodieet al. 2004). The problems are different in terms of structure, but that can be categorized to the portfolio theory.

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APPLICATION OF PORTFOLIO OPTIMIZATION: A STATISTICAL APPROACH

A STUDY OF PROPERTIES AND APPLICATIONS OF WEIBULL-BURR XII DISTRIBUTION

A STUDY OF PROPERTIES AND APPLICATIONS OF WEIBULL-BURR XII DISTRIBUTION

 

CHAPTER ONE

INTRODUCTION

1.0.1    Background of the study

Probability distributions are recently receiving alot of attention with regards to in-troducing new generators for univariate continuous type of probability distributions by introducing additional parameter(s) to the base line distribution. This seemed necessary to reflect current realities that are not captured by the conventional prob-ability distributions since it has been proven to be useful in exploring tail properties of the distribution under study (Tahir, et.al; 2016).

This idea of adding one or more parameter(s) to the baseline distribution has been in practice for a quite long time. Several distributions have been proposed in the literature to model lifetime data. Some of these distributions include: a two-parameter exponential-geometric distribution introduced by Adamidis and Loukas in 1998 which has a decreasing failure rate. Following the same idea of the exponen-tial geometric distribution, the exponential-Poisson distribution was introduced by Kus (2007) with also a decreasing failure rate and discussed some of its properties. Marshall and Olkin (1997) presented a simpler technique for adding a parameter to a family of distributions with application to the exponential and Weibull families. Adamidis et al. (2005) suggested the extended exponential-geometric (EEG) distri-bution which generalizes the exponential geometric distribution and discussed some of its statistical properties along with its hazard rate and survival functions.

Some of the well-known class of generators include the following: Kumaraswamy-G (Kw-G) proposed by Cordeiro and de Castro (2011), McDonald-G (Mc- G) intro-duced by Alexander et al. (2012), gamma-G type 1 presented by Zografos and Balakrishanan (2009), exponentiated generalized (exp-G) which was derived by Cordeiro et al. (2013), others are weibull-power function by Tahir et. al. (2010), ex-ponentiated T-X proposed by Alzaghal et al.(2013). Most recently, a New Weibull-G Family of Distributions by Tahir, (2016), The Weibull–G family of probability dis-tributions by Bourguignon et al. (2014). This research is motivated by the work done by Bourguignon et al. (2014) – The Weibull–G family of probability distri-butions who introduced a generator based on the Weibull random variable called a Weibull-G family. In this research, we propose an extension of the Burr XII pdf called the Weibull-Burr XII distribution based on the Weibull-G class of distribu-tions defined by Bourguignon et al (2014). i.e. we propose a new distribution with five parameters, referred to as the Weibull-Burr XII (Wei-BXII) distribution, which contains as special sub-models the Weibull and Burr XII distributions.

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A STUDY OF PROPERTIES AND APPLICATIONS OF WEIBULL-BURR XII DISTRIBUTION

TIME SERIES ANALYSIS OF SALES OF PETROLEUM PRODUCTS IN NIGERIA (1988-2011)

TIME SERIES ANALYSIS OF SALES OF PETROLEUM PRODUCTS IN NIGERIA (1988-2011)

ABSTRACT

This project is based on time series analysis of yearly sales of petroleum at Balewa fuel station Nigeria limited, Oshogbo, Osun State from 1988-2011. Various trend models of time series analysis were discussed with a view of showing understanding to the appropriate method to be used for forecasts. The time plot of yearly sales of petroleum shows that, there is low sales of petroleum within year 1994 and higher sales of petroleum occur during year 2011. Moreover, evidence from trend analysis shows that the quadratic model trend gives the best model out of all the models considered. This is due to the model which has the least Mean Absolute Deviation. The study adopted quadratic trend model, which was concluded that quadratic model was the best model that best fit the data and was used to forecast yearly sales of petroleum from year 1998 to 2023.

TABLE OF CONTENTS

TITLE                                                                                                                          PAGE

Title Page                                                                                                                          i

Certification                                                                                                                      ii

Dedication                                                                                                                        iii

Acknowledgement                                                                                                            iv

Abstract                                                                                                                             v

Table of Contents                                                                                                             vi-vii

List of Tables                                                                                                                       ix

List of Figures                                                                                                                     x

CHAPTER ONE

1.0    Introduction                                                                                                               1

1.1    Discovery of crude oil in Nigeria                                                                             1-3

1.2    The performance of oil sector in Nigeria                                                                 3- 5

1.3    Role in sales of petroleum product                                                                          6

1.4    Aim and Objective                                                                                                   6

1.5    Description of Data                                                                                                  6

Conclusion of chapter                                                                                              6

CHAPTER TWO: LITERATURE REVIEV

  • Introduction                                                                                                            7

2.1     Definition of time series                                                                                          7

2.1.1  Type of time series                                                                                                  7

2.1.2  Component of time series                                                                                        8

2.1.3  Time series Trend                                                                                                    8

2.1.4  Objective of time series                                                                                          9

2.1.5  Importance of time series                                                                                        9

  • Time Plot                                                                                                                 9
  •  Estimation of Trend                                                                                                9-10

2.3.1  Estimation of Seasonal Variation                                                                           10

2.4    Deseasonalisation of Data                                                                                      11

2.5   Estimation of Cyclical Variation                                                                             11

2.5.1 Forecasting Accuracy                                                                                              11

2.6    Stationary Time Series                                                                                            12

2.6.1 Strict Stationary                                                                                                      12

2.6.2 Weak Stationary                                                                                                     12

2.6.3 Causes of Stationary                                                                                               12

  • Non Stationary                                                                                                        13

2.8    Test of Stationarity                                                                                                13

2.8.1 Graphical Analysis                                                                                                 13

2.8.2 Correlogram                                                                                                           14

2.8.3 The unit Root Test                                                                                                 14

2.8.4 Models of Time Series                                                                                           15

2.8.4 The Augmented Dickey-Fuller (ADF) Test                                                           16

CHAPTER THREE

3.0 Introduction                                                                                                               17

3.1 Time series plot and Component of Time series                                                       17-20

3.2   Forecast                                                                                                                    21

 CHAPTER FOUR:

CONCLUSIONS AND RECOMMENDATIONS

4.0  Introduction                                                                                                              23

4.1  Summary and Conclusions                                                                                       23

4.2  Recommendations                                                                                                    23

References                                                                                                                24

Appendix                                                                                                                  25

LIST OF TABLES

Table 1: Summary for Trend Analysis                                                                            19

Table 2: Forecast Value of Quadratic                                                                              21

Table 3: Appendix                                                                                                           25

LIST OF FIGURES

Fig 1: Time series plot of petroleum                                                                                17

Fig 2: Linear Trend Analysis                                                                                           18

Fig 3: Quadratic Trend Analysis                                                                                      19

Fig 4: Exponential Trend Analysis                                                                                  20

Fig 5: Forecast plot                                                                                                          22

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TIME SERIES ANALYSIS OF SALES OF PETROLEUM PRODUCTS IN NIGERIA (1988-2011)

STATISTICAL ANALYSIS OF THE FEDERAL GOVERNMENT’S EXPENDITURE AND REVENUE

STATISTICAL ANALYSIS OF THE FEDERAL GOVERNMENT’S EXPENDITURE AND REVENUE

ABSTRACT

This research work was aimed at carrying out statistical analysis of federal government’s revenue and expenditure 2003-2008. Secondary data was obtained from National Bureau of Statistics. The statistical package used is Mintab. The result of the analysis shows that there is positive and strong relationship between expenditure and revenue 0.938 and the regression equation is expenditure = 123 + 0.367 revenue. The regression equation shows that when the revenue increase, the expenditure also increases.

TABLE OF CONTENTS

Title page

Declaration

Certification

Dedication

Acknowledgement

Table of Contents

Abstracts

 

CHAPTER ONE: INTRODUCTION

1.0     Introduction

1.1     Historical Background of the Study

1.2     Aims of the Study

1.3     Objectives of the Study

1.4     Scope of the Study

1.5     Definition of terms

 

CHAPTER TWO: LITERATURE REVIEW AND STATISCAL TOOL(S)

2.0     Introduction

2.1     Nigerian Economy and oil ….

2.2.    Inflation in Nigerian economy

2.3     Effect of the global economic meltdown on the Nigeria economy

2.4     Consolidation in the banking system

2.5     Capital base and bank soundness

2.6     Statistical tools

 

 CHAPTER THREE; METHODOLOGY

3.0     Introduction

3.1     Methods of data collection

3.2     Problems encountered in data collection

3.3     Data presentation

 

CHAPTER FOUR: DATA ANALYSIS AND DISCUSSION OF THE RESULTS

4.0     Introduction

4.1     Data analysis

4.2     Discussion of result

 

CHAPTER FIVE: SUMMARY, CONCLUSION AND RECOMMENDATION

5.0     Introduction

5.1     Summary

5.2     Conclusion

5.3     Recommendation

 

CHAPTER ONE

1.0     INTRODUCTION

Public finance is a field of economics concerned with how government raises money, how that money is spent and the effect of these activities on the economy and on the society.

Expenditure and revenue of the country fall under the topic, public finance. However, in a developing economy like Nigeria, management of moderate deficit financing is tailored toward useful and development oriented projects. This necessitated me to focus attention on the amount of expenditure and revenue generated in Nigeria over the past years.

Government generates revenue from various economic sectors: these are divided into oil and non-oil revenue:

  1. Oil Revenue: This is the revenue generated from oil sectors of the economy which comprise:
    1. Petroleum profit tax and royalties
    2. Others which include revenue from export sales, domestics sales, tax on petroleum products, rents etc.

 

  1. Non Oil Revenue: This is revenue generated from other sectors of the economy other than the oil sector which comprises of:
  2. Company income tax
  3. Custom and exercise duties
  4. Value added tax (V.A.T)
  5. Federal government independent revenue which comprises revenue from interest payments rents on government properties, personal income tax of armed forces, police, external affair and federal capital residents
  6. Other which include custom levies, education tax etc.

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STATISTICAL ANALYSIS OF THE FEDERAL GOVERNMENT’S EXPENDITURE AND REVENUE

PERFORMANCE EVALUATION OF NNPC (1999-2013) USING REGRESSION ANALYSIS TECHNIQUES

PERFORMANCE EVALUATION OF NNPC (1999-2013) USING REGRESSION ANALYSIS TECHNIQUES

ABSTRACT

The oil industry which is the leading sector of the economy should have some spill over into the other sectors of the economy.

The Nigerian economy has become dependent oil revenues over the past decades. During the 1986-92 periods, oil export revenues increased at an average of 13 percent per annum which GDP measure in current US Dollars, decrease by an average while oil export revenues alongside the continuing decline of the non-oil economy implies higher dependency.

Over the years, the contributions of the oil industry to the growth of Nigeria economy are great. On this promise, the researchers want to evaluate the performance of the NNPC on the Economic development of Nigeria.

The aim of this study is to evaluate the contribution of NNPC in Nigeria economic development, to know if there is a relationship between NNPC’s performance and the economic development of Nigeria and to point out the negative roles of NNPC and the oil industry. This research work would be of importance to policy makers, researchers and the Nigerian government in improving the economy of the country.

In the methodology, the data used for this study is secondary data from the central bank of Nigeria 2012 statistical bulletin. The data are been analyzed using regression.

The findings from the survey revealed that:

-NNPC contributes massively towards the economic development of Nigeria.

-There is a significant relationship between the performance of NNPC and the economic development of Nigeria

 CHAPTER ONE

INTRODUCTION

  • Background of the study

The Nigerian national petroleum corporation (NNPC) was established on April 1, 1977 as a merger of the Nigerian National Oil Corporation and the Federal Ministry of Mines and Steel. NNPC by law manages the joint venture between the Nigerian federal government and a number of foreign multinational corporations, which include Royal Dutch Shell, Agip, Exxon Mobil, Chevron, and Texaco (now merged with Chevron). Through collaboration with these companies, the Nigerian government conducts petroleum exploration and production. The NNPC Towers in Abuja is the headquarters of NNPC Consisting of four identical towers. NNPC also has zonal offices in Lagos, Kaduna, Port Harcourt and Warri. It has an international office located in London, United Kingdom.

In addition to its exploration activities, the Corporation was given powers and operational interests in refining, petrochemicals and products transportation as well as marketing. Between 1978 and 1989, NNPC constructed refineries in Warri, Kaduna and Port

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PERFORMANCE EVALUATION OF NNPC (1999-2013) USING REGRESSION ANALYSIS TECHNIQUES

MATRIX AND ITS APPLICATIONS

MATRIX AND ITS APPLICATIONS

TABLE OF CONTENTS

 

 CHAPTER ONE

1.1   Background of the study

1.2   Statement the Problem

1.3   Types of Matrix

1.4   Definitions of Matrix

 

CHAPTER ONE

INTRODUCTION AND LITERATURE REVIEW

1.1   BACKGROUND OF THE STUDY

The introduction and development of the notion of a matrix and the subject of linear algebra followed the development of determinants, which arose from the study of coefficients of systems of linear equations. Leibnitz, one of the founder of calculus, used determinant in 1963 and Cramer presented his determinant based formula for solving systems of linear equation (today known as Cramer’s rule) in 1750.

The first implicit use of matrices occurred in Lagrange’s work on bilinear form in late 1700. Lagrange desired to characterize the maxima and minima of multi-variant functions. His method is now known as the method of Lagrange multipliers. In order to do this he first required the first order partial derivation to be 0 and additionally required that a condition on the matrix of second order partial derivatives holds; this condition is today called positive or negative definiteness, although Lagrange did not use matrices explicitly.

Gauss developed elimination around 1800 and used it to solve least square problem in celestial computations and later in computations to measure the earth and it’s surface (the branch of applied mathematics concerned with measuring or determining the shape of the earth or with locating exactly points on the earth’s surface is called Geodesy). Even though Gauss name is associated with this technique eliminating variable from system of linear equations there were earlier work on this subject.

Chinese manuscripts from  several centuries earlier have been found that explains how to solve a system of three equations in three unknown by “Guassian” elimination. For years Gaussian elimination was considered part of the development of geodgesy, not mathematics. The first appearance of Gaussian-Jordan elimination in print was in a handbook on geodesy written by Wihelm Jordan. Many people incorrectly assume that the famous mathematician, Camille Jordan is the Jordan in “Gauss-Jordan elimination”.

For matrix algebra to fruitfully develop one needed both proper notation and proper definition of matrix multiplication. Both needs were met at about the same time in the same place. In 1848 in England, J.J Sylvester first introduced the term “matrix”, which was the Latin word for “womb” as a name for an array of numbers.

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MATRIX AND ITS APPLICATIONS