CHAPTER ONE

INTRODUCTION

1. 0   INTRODUCTION                            

When the casual relationship between dependent variable and independent variable have to be explained and interpreted an x variable (independent variables) are highly correlated, multiple regression analysis becomes unsatisfactory.

Principal component analysis (P C A) is concerned with explaining the variance – covariance structure through a few linear combinations of the original variable with it general objective in data education and interpretation.

The general objective (according to S.I. Onyeagu) principal component analysis are data reduction and data interpretation techniques. A principal component analysis can reveal relationship that were not previous suspected thereby allowing interpretations that would not ordinarily result.

Principal component analysis is an advanced method and techniques by which a set of observed x variable can be expressed or transformed as a linear combination of smaller set of principal component which are linear independent.

Although principal component are required to reproduce the total system variability, often much of this variability can be counted by smaller number k of principal component. If so, there is almost as much information in the k components as there is in the original p variables the k principal components are more of a means   because they frequently serve as intermediate steps in much larger investigations. Principal components often reveal relationship that were not previously suspected and there by allows interpretations that would not ordinary result.

Principal component analysis examine whether the joint variable in p observable random variable x1,x2…..xp can describe approximately in terms of the joint variation of a fewer number, say K < P of hypothetical  variable. In other words, we want to replace a set of P variable by K linear function, K < P, without much loss of information. To do this we usually seek for linear transportation of this type yi = ∑a і = I, 2 ….p which we describe the original variable in lesser number of uncorrected dimensions. This is accomplished by the analysis of all the correlations among the variables. The success of the method depends on obtained two or three new uncorrelated variables, which account for as much of the variation as possible. If the first two or three of these new variables account for nearly the whole of the variation and the contribution of the other p2 a ps  is small, we may say that the total variation is approximately accounted for the first two or three of the new variables, and we may therefore neglect the remainder suppose, λ1, λ2,….. λm    be characteristic root of A we can find a vector pi (m xi), such that A PI = xipi, the p is called the latent  the vectors. Given the latent root  λ1 ≥λ2 ≥  …≥λm and the corresponding latent vectors pi,

P2,…..pm, the linear function

Y1 = pix corresponding to  λ1

y2 = P2x     corresponding to λ2

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ym  =  p1mx corresponding to λm

is called the principal component of x.

1.1    WHAT IS PRINCIPAL COMPONENT ANALYSIS

A principal component analysis is concern with explaining it’s method and technique by which a set of observe variables can be expressed or transformed as linear combination of smaller set of principal component which are linearly independent in other words, principal component analysis which aims to resolve the total variation of a set of variable into linear independent variability in data.

          Principal component is also concerned with explaining the variance – covariance structure through a few linear combination of the original variables. Its general objectives are.

1.           Data reduction and

2.           Interpretation.

The basic idea in carrying out principal component analysis is that the back of observation will be very close to a linear sub-space and hence one can use a new coordinates along the data explain great variability.

Most time we are dealing with a falsely large number pk of correlated random variable, it would be useful if we could reduce it to a smaller number of random variable in such away that

i         The random number of variable account for the large parts of the total variability.

ii        The remaining number of variable are interpretable in terms of original problem.

          In conclusion, a lot of useful information can be deprived and advice given using principal component analysis on normal data, provided the multivariate data have good structure to be extracted from it. In this word, the variables are assumed to have multivariate normal distribution.

1.2    IMPORTANCE OF PRINCIPAL COMPONENT ANALYSIS

Some of the importances are:

1.   An analysis of principal components are more of means to an end rather than an end in themselves because they frequently serve as intermediate step in much large investigations.

2.   Principal component analysis may be a solution of inputs to multiple regressions.

3.   An analysis of principal component often reveals relationship that not previously suspected and thereby allows interpretations that would not ordinarily result.

4.   Principal component analysis provides a statically method for detecting and interpreting linear singular reties in a set of data.

5.   Principal  components analysis are one factoring of the covariance matrices for the factor analysis model    

1.3        AIMS AND OBJECTIVES

Aims and objective of this work include. 

1.           To determine the relative correlation between the various subjects.

2.           To determine the relatives contribution of each course to the performance of students.

3.           To investigate the effect of some subject over the other.

4.           To show principal component potentials as a   means to explain the performance of student on some course.

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