Applied multivariate statistics
Introduction
1
Review of linear algebra
1.1
Basics
1.1.1
Notation
1.1.2
Elementary matrix operations
1.1.3
Special matrices
1.1.4
Vector Differentiation
1.2
Vector spaces
1.2.1
Linear independence
1.2.2
Row and column spaces
1.2.3
Linear transformations
1.3
Inner product spaces
1.3.1
Distances, and angles
1.3.2
Orthogonal matrices
1.3.3
Projections
1.4
The Centering Matrix
1.5
Computer tasks
1.6
Exercises
2
Matrix decompositions
2.1
Matrix-matrix products
2.2
Spectral/eigen decomposition
2.2.1
Eigenvalues and eigenvectors
2.2.2
Spectral decomposition
2.2.3
Matrix square roots
2.3
Singular Value Decomposition (SVD)
2.3.1
Examples
2.4
SVD optimization results
2.5
Low-rank approximation
2.5.1
Matrix norms
2.5.2
Eckart-Young-Mirsky Theorem
2.5.3
Example: image compression
2.6
Computer tasks
2.7
Exercises
3
Canonical Correlation Analysis (CCA)
3.1
The first pair of canonical variables
3.1.1
The first canonical components
3.1.2
Example: Premier league football
3.2
The full set of canonical correlations
3.2.1
Example continued
3.3
Properties
3.3.1
Connection with linear regression when
\(q=1\)
3.3.2
Invariance/equivariance properties of CCA
3.4
Computer tasks
3.5
Exercises
4
Multidimensional Scaling (MDS)
4.1
Classical MDS
4.1.1
Non-Euclidean distance matrices
4.1.2
Principal Coordinate Analysis
4.2
Properties
4.3
Non-classical MDS
4.4
Similarity measures
4.4.1
Binary attributes
4.4.2
Example
4.5
Non-metric MDS
4.6
Exercises
4.7
Computer Tasks
University of Nottingham
Multivariate Statistics
4.2
Properties
Scaling, translation, uniqueness of solution? Mean of solution???