BORANG PENGESAHAN STATUS TESIS
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1 UNIVERSITI TEKNOLOGI MALAYSIA PSZ 19:16 (Pind. 1/97) BORANG PENGESAHAN STATUS TESIS JUDUL: STUDIES ON PRINCIPAL COMPONENTS ANALYSIS AND ITS APPLICATION IN MULTIVARIATE DATA SESI PENGAJIAN: 2005/2006 Saya WONG SOON MING (HURUF BESAR) mengaku membenarkan tesis (PSM/ SARJANA/Doktor Falsafah)* ini disimpan di Perpustakaan Universiti Teknologi Malaysia dengan syarat-syarat kegunaan seperti berikut: 1. Tesis adalah hakmilik Universiti Teknologi Malaysia. 2. Perpustakaan Universiti Teknologi Malaysia dibenarkan membuat salinan untuk tujuan pengajian sahaja. 3. Perpustakaan dibenarkan membuat salinan tesis ini sebagai bahan pertukaran antara institusi pengajian tinggi. 4. **Sila tandakan ( 4 ) SULIT TERHAD (Mengandungi maklumat yang berdarjah keselamatan atau kepentingan Malaysia seperti yang termaktub di dalam AKTA RAHSIA RASMI 1972) (Mengandungi maklumat TERHAD yang telah ditentukan oleh organisasi/badan di mana penyelidikan dijalankan) TIDAK TERHAD Disahkan oleh (TANDATANGAN PENULIS) (TANDATANGAN PENYELIA) Alamat Tetap: No. 54, Lane 10 Taman Rejang, Sibu, Sarawak. Pn. Siti Roslindar Yaziz Nama Penyelia Tarikh: 28 April 2006 Tarikh: 28 April 2006 CATATAN: * Potong yang tidak berkenaan. ** Jika tesis ini SULIT atau TERHAD, sila lampirkan surat daripada pihak berkuasa/organisasi berkenaan dengan menyatakan sekali sebab dan tempoh tesis ini perlu dikelaskan sebagai SULIT atau TERHAD. Tesis dimaksudkan sebagai tesis bagi Ijazah Doktor Falsafah dan Sarjana secara penyelidikan, atau disertasi bagi pengajian secara kerja kursus dan penyelidikan, atau Laporan Projek Sarjana Muda (PSM).
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3 STUDIES ON PRINCIPAL COMPONENTS ANALYSIS AND ITS APPLICATION IN MULTIVARIATE DATA WONG SOON MING A thesis submitted in fulfilment of the requirements for the award of the degree of Bachelor of Science and Education (Mathematics) Faculty of Science Universiti Teknologi Malaysia APRIL 2006
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5 iii To The LORD Jesus, whose grace and mercy is unfailing. My family, who loved me as who I am.
6 iv ACKNOWLEDGEMENT My greatest gratitude goes out to my supervisor and mentor, Pn Siti Roslindar Yaziz, for the invaluable advice, guidance and support given to me throughout the progress of this project. I also want to take this opportunity to thank my entire dearest friends and my housemates, Dolly Sii Tien Ching, Annie Su Fung and Lau Ching Kou who always support me and provide me with helpful suggestions in completing this project. My appreciation also goes out to proof readers, Ong Lai Mun and Tang See Guan. In addition, I would like to acknowledge my family support, encouragement and care. If not for their sacrifice and understanding, I would not be able to pursue this degree wholeheartedly. Last but not least, I would like to thank all those who are involved directly or indirectly in helping me to complete this project. THANK YOU.
7 v ABSTRACT Multivariate statistics is an extension of univariate and bivariate statistics. It is one of the statistical technique developed to analyze two or more variables simultaneously. Principal component analysis (PCA) is one of the techniques of multivariate statistics to analyse data. Fundamentally, PCA is a method to reduce the large numbers of original input variables of a generic system examined to a smaller number of variables in principal components form, that is a linear combination set of original variables. PCA can be performed by using either a covariance matrix or correlation matrix. For the purpose of this study, PCA is being applied to a data in social science regarding Malaysian Quality of Life ( ). The analysis of data is done manually using a statistical software SPSS (Statistical Package for the Social Science) version Number of principal components need to be extracted is determined by eigenvalue or scree plot. In this study, PC1 and PC2 are extracted from eleven variables and those two principal components account for % of the total variances, a great deal of the total variances in the original variables. Therefore, these two principal components are good representations of the original data with little lost of information.
8 vi ABSTRAK Analisis multivariat adalah perlanjutan daripada statistik univariat dan bivariat. Ia merupakan antara kaedah statistik yang menganalisis dua atau lebih pembolehubah secara serentak. Analisis komponen utama (PCA) merupakan antara teknik statistik multivariat untuk penganalisisan data. PCA memberikan kaedah mengurangkan satu set pembolehubah asal yang besar dalam suatu sistem yang dikaji kepada satu set pembolehubah yang kecil dalam bentuk komponen utama, iaitu gabungan linear set pembolehubah asal. Untuk kajian ini, PCA telah diaplikasikan ke atas satu set data Kualiti Hidup Malaysia ( ). Data kajian dianalisis menggunakan perisian computer SPSS (Statistical Package for the Social Science) versi Dua komponen utama yang diperoleh dapat memberi jumlah variasi yang tinggi, iaitu sebanyak % daripada pembolehubah asal. Oleh itu, kedua-dua komponen utama yang diperoleh dapat mewakili set pembolehubah asal dengan kehilangan maklumat sekecil mungkin.
9 vii TABLE OF CONTENTS CHAPTER TITLE PAGE TITLE DECLARATION DEDICATION ACKNOWLEDGEMENT ABSTRACT ABSTRAK TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES LIST OF SYMBOLS i ii iii iv v vi vii x xi xii 1.0 INTRODUCTION 1.1 Introduction Objectives Scope of Study Outline 5
10 viii 2.0 LITERATURE REVIEW INTRODUCTION OF PCA 3.1 Introduction of PCA Basic Concepts on PCA Assumptions Test of Significance Fundamental Theories on PCA PCA on Covariance Matrix PCA on Correlation Matrix Determining the Number of Principal Components Eigenvalue and Eigenvector Correlation Coefficients Data St andardization Example of Calculation Correlation Matrix Eigenvalue and Eigenvector on Correlation Matrix Eigenvalue Eigenvector Matrix A Principal Components Correlation Coefficients Conclusion DATA ANALYSIS USING PCA 4.1 Description About Data of Study Analyze Data Using SPSS Introduction of SPSS Data Analysis Using SPSS Results and Discussions 74
11 ix Correlation Matrix Total Variance Explained The Scree Plot Component Matrix Component Score Coefficients Matrix CONCLUSION AND RECOMMENDATIONS 5.1 Conclusion Recommendations 85 REFERENCES 86
12 x LIST OF TABLES TABLE NO TITLE PAGE 3.1 English and Greek Teachers Components of the Malaysian Quality of Life Composite Index and Indices by Area ( ) Reference Index Table SPSS Output on Correlation Matrix SPSS Output on Total Variance Explained SPSS Output on Component Matrix SPSS Output on Component Score Coefficient Matrix 80
13 xi LIST OF FIGURES FIGURES NO. TITLE PAGE 4.1 SPSS Data Editor Window SPSS Variable View Cell SPSS Data View Cell SPSS Factor Analysis Cell SPSS Variable List Checkbox SPSS Descriptives Checkbox SPSS Extraction Checkbox SPSS Rotation Checkbox SPSS Factor Scores Checkbox SPSS Data Output Window Scree Plot on Eigenvalues 77
14 xii LIST OF SYMBOLS a - Eigenvector Α - Eigenvector matrix Α - Transpose of matrix Α P - Number of original variable X - Original variable Y - Principal component Ι - Identity matrix λ - Eigen values σ - Variance of original variable n - Number of data 2 r x - Correlation x with x r xy - Correlation x with y r yx - Correlation y with x 2 r y - Correlation y with y R - Correlation matrix s - Standard deviation of original variable 2 s x - Variance for x 2 s y - Variance for y s xy - Covariance x and y s yx - Covariance y and x
15 xiii S - Covariance matrix X - Mean for X Z - Value for new data Λ - Diagonal matrix l - Loading r - Correlation coefficient
16 CHAPTER 1 INTRODUCTION 1.1 Introduction on Multivariate Statistics Multivariate statistics is an extension of univariate and bivariate statistics. Multivariate statistics is the complete or general case, while univariate and bivariate statistics are some special cases of the multivariate model. As the name implies, multivariate statistics refers to an assortment of descriptive and inferential techniques or a set of increasingly popular statistical techniques that are useful for analyzing more complicated data that have been developed to handle situations where sets of variables are involved either as predictors or as measures of performance. They provide a system for analysis under conditions in which there may be several independent variables and many dependent variables all correlated with one another to varying degrees.
17 2 Multivariate analyses are often concerned with finding relationship among the response variables, the experimental units, or both response variables and experimental units. Multivariate refers to data with many variables while multivariable refers to a system that varies in many ways. To begin discussion of multivariate data analysis methods, the concept of an experimental unit must be defined. An experimental unit is any object or item that can be measured or evaluated in some way. Measuring and evaluating experimental units are principal activities of most researchers. Examples of experimental units include people, animals, insects, fields, plots of land, companies and countries. Multivariate data resulted whenever a researcher measures or evaluates more than one attribute or characteristic of each experimental unit. These attributes or characteristics are usually called variables by statisticians. Variables are roughly dichotomized into two major types: Independent and dependent. Independent variables are the differing conditions (treatments) to which we expose our subjects, or characteristics (tall or short) the subjects themselves bring into the research situation. Independent variables are usually considered either predictor or causal variables because they predict or cause the dependent variables. Independent variables and dependent variables are defined within a research context; dependent variables in one research setting may be independent variables in another. Additional term for independent variables and dependent variables are predictor-criterion, stimulus-response, task-performance, or simply input-output. With multivariate statistics, multiple dependent and multiple independent variables are simultaneously analyzed. There are various multivariate methods that could be used to simplify and reduce the complexity of the problem. Theoretical and practical reasons are put into
18 3 due consideration when applying the various multivariate methods. Theoretically, the purpose of using multivariate methods is to reduce the dimension number, even at the expense of sacrificing some information or getting rid of number of nuisance parameters. However, on the practical side, multivariate methods are used to reduce the number of variables in order to save computational effort. It is also to avoid variables which are expensive to observe or those that involve a lot of delay in measurement, provided that nothing serious is lost for the purpose of the inquiry. A fundamental distinction that can be made between multivariate methods is that some methods are classified as variable-directed techniques, while others are classified as individual-directed techniques. Variable-directed techniques are those that are primarily concerned with relationships that might exist among the response variables being measured. Some examples of this type of technique are principal components analysis, factor analysis, regression analysis and canonical correlation analysis. Individual-directed techniques are those that are primarily concerned with relationships that might exist among the experimental units and/or individuals being measured. Some examples of this type of technique are discriminant analysis, cluster analysis, and multivariate analysis of variance (MANOVA). The new variables derived from the original ones by PCA are simply linear combinations of the original variables. PCA can be performed on either a sample covariance matrix or correlation matrix. The type of matrix used depends on the variables being measured.
19 4 1.2 Objectives The first objective is to study and understand the PCA in dimensionality reduction of variables by the extraction of the combination of variables (principal components) that explain the greatest amount of the total variance. The second objective is to apply PCA in multivariate social science data. In this study, Components of Malaysian Quality of Life adapted from the Economic Planning Unit, Department of Statistics, Malaysia is used as the multivariate social science data. The third objective of this report is to study and to use the Statistical Package for the Social Science (SPSS) computing system for the extraction of principal components. 1.3 Scope of Study This study will focus on one of the variable-directed techniques, Principal Component Analysis (PCA). PCA is applied to data in social science regarding Components of the Malaysian Quality of Life ( ). Study is done on correlation matrix, eigenvalues, eigenvectors, correlation coefficients and methods in deterimining number of principal components to be retained. Test of normality and null PCA is solved using SPSS computing system. The SPSS output will be analyzed.
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