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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
- - - - - -
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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
1.4AIMS 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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