Multivariate Statistical Analysis
Lecture Notes
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\[f(\mathbf {x}) = \frac {1}{(2\pi )^{p/2}\left |\Sigma \right |^{1/2}} \exp \left \{-\tfrac {1}{2}(\mathbf {x}-\boldsymbol {\mu })'\,\Sigma ^{-1} (\mathbf {x}-\boldsymbol {\mu })\right \}\]
Contents
1 Review of Matrix Algebra
1.1 Matrix
1.2 Vector
1.3 Some Particular Matrices
1.4 Basic Matrix Operations
1.5 Further Matrices
1.6 Orthogonal Matrices
1.7 Centering Matrix
1.8 Vectors and Matrices
1.9 Spectral Decomposition
1.10 Practice Problems
2 Basic Properties of Random Vectors
2.1 Marginal and Conditional Distribution Functions
2.2 Independence
2.3 Population Moments
2.4 Conditional Moments
2.5 Correlation Matrix
2.6 Practice Problems
3 Introduction To Multivariate Data
3.1 Introduction
3.2 Data Matrix
3.3 Summary Statistics
3.4 Measures of Multivariate Scatter
3.5 Geometrical Ideals
3.6 Practice Problems
4 Principal Component Analysis
4.1 Population Principal Components
4.2 How Much Variation Each Component Explains
4.3 Components from the Correlation Matrix
4.4 Sample Principal Components
4.5 How Many Components to Keep
4.6 Practice Problems
5 The Multivariate Normal Distribution
5.1 Definition and Basic Properties
5.2 Sampling from a Multivariate Normal Population
5.3 The Wishart Distribution
5.4 Hotellings
5.5 Wilks Lambda
5.6 Confidence Regions
6 Comparison of Two Multivariate Means
6.1 Paired Comparisons
6.2 Two Independent Samples
6.3 Simultaneous Confidence Statements
6.4 Practice Problems
7 Classification and Discrimination
7.1 Fisher’s Discriminant Function
7.2 Fishers Classification Rule
7.3 Fishers Rule Through Mahalanobie’s Distance
7.4 More Than Two Population
7.5 Sample Values
7.6 Fishers Sample Linear Discriminate
7.7 Fishers Classification Procedure Based of On Sample Discriminate
7.8 Practice Problems
8 Multivariate Analysis of Variance (MANOVA)
8.1 Incorporating Data
8.2 Two-Way MANOVA
8.3 Practice Problems
9 Multivariate Multiple Regression
9.1 The Model and Its Notation
9.2 Least Squares Estimation
9.3 Fitted Values and Residuals
References
References
9.4 Practice Problems
10 Factor Analysis
10.1 The Orthogonal Factor Model
10.2 Non-uniqueness and Rotation
10.3 Estimation
10.4 Practice Problems
11 Canonical Correlation Analysis
11.1 Canonical Variates
11.2 Solution by Eigenvalues
11.3 Special Cases and Cautions
11.4 Practice Problems
12 Cluster Analysis
12.1 Distance
12.2 Hierarchical Methods
12.3 \(k\)-Means
12.4 Practice Problems
1 Review of Matrix Algebra
1.1 Matrix
1.2 Vector
1.3 Some Particular Matrices
1.4 Basic Matrix Operations
1.5 Further Matrices
1.6 Orthogonal Matrices
1.7 Centering Matrix
1.8 Vectors and Matrices
1.9 Spectral Decomposition
1.10 Practice Problems
2 Basic Properties of Random Vectors
2.1 Marginal and Conditional Distribution Functions
2.2 Independence
2.3 Population Moments
2.4 Conditional Moments
2.5 Correlation Matrix
2.6 Practice Problems
3 Introduction To Multivariate Data
3.1 Introduction
3.2 Data Matrix
3.3 Summary Statistics
3.4 Measures of Multivariate Scatter
3.5 Geometrical Ideals
3.6 Practice Problems
4 Principal Component Analysis
4.1 Population Principal Components
4.2 How Much Variation Each Component Explains
4.3 Components from the Correlation Matrix
4.4 Sample Principal Components
4.5 How Many Components to Keep
4.6 Practice Problems
5 The Multivariate Normal Distribution
5.1 Definition and Basic Properties
5.2 Sampling from a Multivariate Normal Population
5.3 The Wishart Distribution
5.4 Hotellings
5.5 Wilks Lambda
5.6 Confidence Regions
6 Comparison of Two Multivariate Means
6.1 Paired Comparisons
6.2 Two Independent Samples
6.3 Simultaneous Confidence Statements
6.4 Practice Problems
7 Classification and Discrimination
7.1 Fisher’s Discriminant Function
7.2 Fishers Classification Rule
7.3 Fishers Rule Through Mahalanobie’s Distance
7.4 More Than Two Population
7.5 Sample Values
7.6 Fishers Sample Linear Discriminate
7.7 Fishers Classification Procedure Based of On Sample Discriminate
7.8 Practice Problems
8 Multivariate Analysis of Variance (MANOVA)
8.1 Incorporating Data
8.2 Two-Way MANOVA
8.3 Practice Problems
9 Multivariate Multiple Regression
9.1 The Model and Its Notation
9.2 Least Squares Estimation
9.3 Fitted Values and Residuals
References
References
9.4 Practice Problems
10 Factor Analysis
10.1 The Orthogonal Factor Model
10.2 Non-uniqueness and Rotation
10.3 Estimation
10.4 Practice Problems
11 Canonical Correlation Analysis
11.1 Canonical Variates
11.2 Solution by Eigenvalues
11.3 Special Cases and Cautions
11.4 Practice Problems
12 Cluster Analysis
12.1 Distance
12.2 Hierarchical Methods
12.3 \(k\)-Means
12.4 Practice Problems