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Multivariate Normal Distribution & Multivariate Regression

1. Multivariate Normal Distribution (MVN)

The Multivariate Normal Distribution (MVN) is a generalization of the univariate normal distribution to multiple variables. It describes a set of correlated Gaussian-distributed variables.

Definition:

A random vector X = $[X_1, X_2, ..., X_n]$ follows an MVN distribution if any linear combination of its components is also normally distributed. It is defined as:

$$X \sim \mathcal{N}(\mu, \Sigma)$$

where:

• $\mu$: Mean vector $[ \mu_1, \mu_2, ..., \mu_n ]$ (expected values of each variable).

• $\Sigma$: Covariance matrix (captures relationships between variables).

Probability Density Function (PDF):

$$P(X) = \frac{1}{(2\pi)^{n/2} |\Sigma|^{1/2}} \exp \left( -\frac{1}{2} (X - \mu)^T \Sigma^{-1} (X - \mu) \right)$$

Applications:

• Pattern Recognition: Face recognition, speech processing.

• Data Science: Modeling correlated features.

• Finance: Portfolio risk analysis.

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2. Multivariate Regression

Multivariate Regression is an extension of linear regression where multiple dependent variables ($Y$) are predicted using multiple independent variables ($X$).

Model Equation:

$$Y = X B + \epsilon$$

where:

• $Y$: Matrix of dependent variables.

• $X$: Matrix of independent variables.

• $B$: Coefficient matrix.

• $\epsilon$: Error term.

Applications:

• Economics: Predicting multiple economic indicators.

• Healthcare: Disease diagnosis based on multiple symptoms.

• Engineering: Predicting material properties based on experimental conditions.

Key Difference:

• Multivariate Normal Distribution models joint probability distributions.

• Multivariate Regression predicts multiple outcomes using independent features.