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PRINCIPAL
COMPONENT ANALYSIS OF STUDENTS PERFORMANCE
ABSTRACT
Often in
multivariate analysis, fairly large numbers are be used but the objectives of
principal component analysis are data reduction and data interpretation.
Principal
component analysis can reveal relationships that were not ordinarily result.
Principle
component analysis as reduction technique to reduce the variable beings
considered in the performance of students in Anambra State and to know many
variables to keep and how many to discard. The use of correlation matrix in the
data analysis which takes care of the correlation of the principal components
not being invariant under separate scaling of the original set of variable. In
this work principle component analyses data form the past two WAEC result 2007
and 2008 sheet contain the entire senior secondary school three that took part
in (8) different courses. In analyzing the data, statistic package known as
state view and SPSS where used.
In
conclusion it has been established that four linear combination for the first
set of data can be replace by the original eight variables without loss of
information.
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
– – – – – –
– – – – – – – – – – – – –
– – – – – –
– – – – – – – – – – – – – –
– – – – – –
– – – – – – – – – – – – – –
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
importance 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
1.4 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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