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Multivariate Statistics

  • Understand the concept of multivariate statistics.
  • Analyze relationships between multiple variables.
  • Learn about multivariate probability distributions.
  • Compute covariance and correlation matrices.
  • Understand the multivariate normal distribution.
  • Explore Principal Component Analysis (PCA).

Definition of Multivariate Statistics

Multivariate statistics involves analyzing multiple variables simultaneously to study their relationships.

  • Univariate analysis: Single variable (e.g., height distribution).
  • Bivariate analysis: Two variables (e.g., height vs. weight).
  • Multivariate analysis: Three or more variables (e.g., height, weight, and age).

Covariance and Correlation Matrices

The covariance matrix describes the relationships between multiple variables:

\[ \Sigma = \begin{bmatrix} \text{Var}(X_1) & \text{Cov}(X_1, X_2) & \dots & \text{Cov}(X_1, X_n) \\ \text{Cov}(X_2, X_1) & \text{Var}(X_2) & \dots & \text{Cov}(X_2, X_n) \\ \vdots & \vdots & \ddots & \vdots \\ \text{Cov}(X_n, X_1) & \text{Cov}(X_n, X_2) & \dots & \text{Var}(X_n) \end{bmatrix} \]

The correlation matrix standardizes the relationships:

\[ R = \begin{bmatrix} 1 & \rho_{12} & \dots & \rho_{1n} \\ \rho_{21} & 1 & \dots & \rho_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ \rho_{n1} & \rho_{n2} & \dots & 1 \end{bmatrix} \]

Multivariate Probability Distributions

Multivariate probability distributions describe joint behavior of multiple variables.

  • Joint Probability Function: \( P(X_1, X_2, \dots, X_n) \).
  • Marginal Distributions: Individual distributions obtained by summing/integrating over other variables.
  • Conditional Distributions: Probability of one variable given values of others.

Multivariate Normal Distribution

The multivariate normal distribution is an extension of the normal distribution to multiple variables:

\[ f(\mathbf{x}) = \frac{1}{(2\pi)^{n/2} |\Sigma|^{1/2}} \exp \left( -\frac{1}{2} (\mathbf{x} - \boldsymbol{\mu})^T \Sigma^{-1} (\mathbf{x} - \boldsymbol{\mu}) \right) \]

where:

  • \( \boldsymbol{\mu} \) = mean vector
  • \( \Sigma \) = covariance matrix
  • \( \mathbf{x} \) = multivariate variable

Derivation

The multivariate normal distribution generalizes the univariate normal:

\[ P(X) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(X - \mu)^2}{2\sigma^2}} \]

Extending this to \( n \) dimensions leads to the matrix form:

\[ f(\mathbf{x}) = \frac{1}{(2\pi)^{n/2} |\Sigma|^{1/2}} \exp \left( -\frac{1}{2} (\mathbf{x} - \boldsymbol{\mu})^T \Sigma^{-1} (\mathbf{x} - \boldsymbol{\mu}) \right) \]

Principal Component Analysis (PCA)

PCA is a technique for reducing dimensionality while preserving variance.

  • Step 1: Compute the covariance matrix.
  • Step 2: Find eigenvalues and eigenvectors.
  • Step 3: Select top \( k \) principal components.
  • Step 4: Transform data to new coordinate system.

Examples

Example 1: Compute the covariance matrix for the dataset:

X Y Z
2 4 3
3 5 4
4 6 5

Exercises

  • Question 1: Compute the correlation matrix for the dataset:
    • \( (2,3), (3,4), (5,6) \).
  • Question 2: Find the first principal component of:
    • \( X_1 = (1,2,3), X_2 = (4,5,6) \).
  • Question 3: Compute the eigenvalues of the covariance matrix:
  • Answer 1: The correlation matrix is:
    • \( \begin{bmatrix}1 & 0.9 \\ 0.9 & 1\end{bmatrix} \).
  • Answer 2: First principal component: \( (0.71, 0.71) \).
  • Answer 3: Eigenvalues: \( \lambda_1 = 2.5, \lambda_2 = 0.5 \).

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