# Factor Analysis of Cross-Classified Data

This thesis introduces a model hierarchy related to Principal Component Analysis and Factor Analysis, in which vector measurements are linearly decomposed into a relatively small set of hypothetical principal directions, for purposes of dimension reduction. The mathematical specification of unknown parameters in the models is unified…

Contents

1 Introduction
1.1 Principal Component Analysis
1.2 Factor Analysis
1.3 Overview of the Thesis
1.4 Some Definitions and Notations
2 Factor Analysis Models and Model Hierarchy
2.1 General Factor Analysis Models
2.1.1 Generality of Y produced by the model
2.1.2 Identifiability for (M0) model
2.1.3 Identifiability for (M0) (continued)
2.2 Factor Model (M1)
2.3 Factor Model (M1R)
2.4 Cross-Classified Factor Model (M2)
2.5 Cross-Classified Factor Model (M3)
2.6 Factor Model (M4)
2.7 PARAFAC Model (M4a)
2.8 PARAFAC Random Model (M4′)
2.9 Relationship among models in the model hierarchy
3 ML Estimates for Factor Analysis Models
3.1 Maximum likelihood estimate for (M1)
3.2 Maximum likelihood estimate for (M1R)
3.2.1 Simplifying the probability density function for model (M1R)
3.2.2 Likelihood function and ML equation
3.2.3 The profile log-likelihood
3.3 Profile likelihood optimization in (M0)
3.3.1 Why it is good to use the profile likelihood?
3.4 Condition to check the local maximum likelihood estimate
3.5 Score Test for H0 : ψj = 0 versus HA : ψj > 0
3.5.1 Score Test for H0 : ψj = 0 vs HA : ψj > 0 under (M0) with μ = 0
3.6 Test of Fit for the PARAFAC Model
3.6.1 The Likelihood Ratio Test
3.6.2 The LRT for (M4a) against (M3)
4 Computational Methods
4.1 EM Algorithm
4.1.1 Introduction
4.1.2 EM algorithm and (M0) model
4.2 Newton-Raphson method
4.2.1 Newton-Raphson method on the profile likelihood
4.3 Computational results on simulated data
4.3.1 Comparison of EM and Newton-Raphson algorithms
4.4 The LRT for (M4a) against (M3)
4.4.1 Maximize the likelihood under H0 : θ ∈ M4a
4.5 Recommendations based on computational results
5 Application to 2-D Coronal Tongue Surface
5.1 Data Set
5.2 Application of Factor Analysis Models…

Author: Tsou, Hsiao-Hui Sophie

Source: University of Maryland