     # Techniques and Applications of Multivariate 2016. 9. 9.¢  Techniques and Applications of Multivariate

• View
0

0

Embed Size (px)

### Text of Techniques and Applications of Multivariate 2016. 9. 9.¢  Techniques and Applications of...

• 3. Factor Analysis

Techniques and Applications of Multivariate Analysis

• Lecture 3. Factor Analysis (FA)

Lecture3-1 3.1 Introduction 3.2 Orthogonal Factor Model 3.3 Methods of Estimation: PCM vs MLM Lecture3-2 3.4 Factor Rotation 3.5 Factor Scores: WLSM vs RM 3.6 Example : Air-pollution Data 3.7 Strategy for Factor Analysis : Example 9.14

• Gaelic English History Arithmetic Algebra Geometry

Gaelic 1.00 . . . . .

English 0.44 1.00 . . . .

History 0.41 0.35 1.00 . . .

Arithmetic 0.29 0.35 0.16 1.00 . .

Algebra 0.33 0.32 0.19 0.59 1.00 .

Geometry 0.25 0.33 0.18 0.47 0.46 1.00

3.1 Introduction of FA

 Definition

FA: technique for describing the covariance relationship among many variables in terms of a few factors which are underlying, but unobservable random quantities.

 Example (Ex. 9.8, p.502)

Correlation matrix of 6 subjects

Mathematical-ability factor

Verbal-ability factor

• 3.1 Introduction of FA

History :  K. Pearson and Charles Spearman provided beginnings of FA in the early 20th century.

 Charles Spearman is known for being the one who coined the term factor analysis

and actually used it to measure children’s cognitive performance.

 Spearman, C. (1904). “General intelligence” objectively determined and measured.

"American Journal of Psychology", 15, 201–293.

• 3.2 Orthogonal Factor Model

 Model with m common factors

 Properties

Vector of specific factors

 Assumptions

Common factors decomposition

communality

Specific variance

• 3.3 Methods of Estimation: PCM vs MLM

: Common factor decomposition

with

• 3.3 Methods of Estimation

• 3.3 Methods of Estimation: PCM

 How do we select the number of factors m in PCM?

for S, for R

• 3.3 Methods of Estimation: PCM

 Example (Ex. 9.8, p.502)

Correlation matrix of 6 subjects

 Program

Gaelic English History Arithmetic Algebra Geometry Gaelic 1.00 . . . . . English 0.44 1.00 . . . . History 0.41 0.35 1.00 . . .

Arithmetic 0.29 0.35 0.16 1.00 . . Algebra 0.33 0.32 0.19 0.59 1.00 .

Geometry 0.25 0.33 0.18 0.47 0.46 1.00

• 3.3 Methods of Estimation : Results of SAS

They are the only eigenvalues greater than 1.

2 factors account for a cumulative proportion of the total sample variance.

General intelligence factor

Bipolar factor: Half + and half -

All communalities are nearly about 1

All elements are small

• 3.3 Methods of Estimation: MLM

2) Algorithm for Maximum Likelihood Method

[step 1] Given , consider the likelihood function

[step 3] We have the ml estimators and mles of the communalities

[step 2]With and

obtain the maximization of the likelihood function subject to uniqueness condition

• 3.3 Methods of Estimation: MLM

 How do we select the number of factors m in MLM?

 Test the hypotheses with an appropriate m.

: Bartlett’s test statistic based on the chi-square approximation

 Residual matrix:

The diagonal elements are zero and the other elements are small: m factor model is appropriate !

• 3.3 Methods of Estimation: MLM

 Program

• 3.3 Methods of Estimation: Comparison

Recommended ##### Multivariate phase type distributions - Applications and parameter Multivariate phase type distributions
Documents ##### APPLICATIONS OF STATISTICAL TECHNIQUES IN QUALITY .APPLICATIONS OF STATISTICAL TECHNIQUES IN QUALITY
Documents ##### RAPID MANUFACTURING Techniques and Applications for .RAPID MANUFACTURING Techniques and Applications
Documents ##### Exploratory Multivariate Data Analysis - Title: Exploratory Multivariate Data Analysis with Applications
Documents ##### Applications of Artificial Intelligence Techniques in ... Applications of Artificial Intelligence Techniques
Documents ##### Multivariate Analysis in Ecology and Systematics: Panacea ... The first applications of multivariate
Documents ##### Assessing Landsat Classification Accuracy Using Discrete ... ... DISCRETE MULTIVARIATE ANALYSIS TECHNIQUES
Documents ##### Characterization, Classification and Authentication of ... multivariate classification techniques such
Documents ##### Application of multivariate statistical techniques in ... oleg.paliy/Papers/Paliy_ REVIEWS AND SYNTHESES Application of multivariate statistical techniques in microbial ecology O
Documents Documents