Multivariate Analysis

5.

Principal Component Analysis

5.1.

Conceptual Foundation

5.1.1.

Objectives of PCA

5.1.1.1.

Dimensionality Reduction

5.1.1.2.

Data Visualization

5.1.1.3.

Noise Reduction

5.1.2.

Geometric Interpretation

5.1.3.

Variance Maximization Principle

5.2.

Mathematical Derivation

5.2.1.

Covariance Matrix Approach

5.2.2.

Correlation Matrix Approach

5.2.3.

Eigenvalue Problem Formulation

5.2.4.

Lagrange Multiplier Method

5.3.

PCA Implementation

5.3.1.

Data Preparation

5.3.1.1.

5.3.1.2.

Standardization Decision

5.3.2.

Covariance vs. Correlation Matrix Choice

5.3.3.

Eigenvalue Decomposition

5.3.4.

Principal Component Extraction

5.3.5.

Component Score Calculation

5.4.

Determining Number of Components

5.4.1.

Kaiser's Rule (Eigenvalue > 1)

5.4.2.

Scree Plot Analysis

5.4.3.

Cumulative Variance Explained

5.4.4.

Parallel Analysis

5.4.5.

Cross-Validation Methods

5.5.

Interpretation of Results

5.5.1.

Component Loadings

5.5.1.1.

Loading Interpretation

5.5.1.2.

5.5.1.3.

5.5.2.

Component Scores

5.5.2.1.

Score Calculation

5.5.2.2.

5.5.3.

Biplot Construction

5.6.

Component Rotation

5.6.1.

Orthogonal Rotations

5.6.1.1.

Varimax Rotation

5.6.1.2.

Quartimax Rotation

5.6.1.3.

Equamax Rotation

5.6.2.

Oblique Rotations

5.6.2.1.

Promax Rotation

5.6.2.2.

5.6.3.

Rotation Interpretation

5.7.

PCA Diagnostics and Validation

5.7.1.

Adequacy Measures

5.7.2.

Stability Assessment

5.7.3.

Cross-Validation

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4. Data Preparation and Exploration

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6. Factor Analysis