Multivariate normal distribution

In probability theory and statistics, a random vector X = (X₁, ..., X_n) follows a multivariate normal distribution, also sometimes called a multivariate Gaussian distribution (in honor of Carl Friedrich Gauss, who was not the first to write about the normal distribution), if it satisfies the following equivalent conditions:

every linear combination Y=a₁X₁ + ... + a_nX_n is normally distributed;

there is a random vector Z=(Z₁, ..., Z_m), whose components are independent standard normal random variables, a vector μ = (μ₁, ..., μ_n) and an n×m matrix A such that X = A Z + μ.

there is a vector μ and a symmetric, positive semi-definite matrix Γ such that the characteristic function of X is

φ_X(u)=exp(iμ^Tu − (½) u^T Γ u).

The following is not quite equivalent to the conditions above, since it fails to allow for a singular matrix as the variance:

there is a vector μ=(μ₁, ..., μ_n) and a symmetric, positive definite matrix Σ such that X has density

$f_X(x_1,\ldots,x_n)\, dx_1\ldots dx_n= \frac{1}{(2\pi)^{n/2}|\Sigma|^{1/2}} \exp\left(-\frac{1}{2}({\mathbf x}-{\mathbf\mu})^T{\mathbf\Sigma}^{-1}({\mathbf x}-{\mathbf\mu}) \right)dx_1\ldots dx_n$

where $\left|A\right|$ is the determinant of A. Note how the equation above reduces to that of the univariate normal distribution if Σ is a $1\times 1$ matrix (ie a real number).

The vector μ in these conditions is the expected value of X and the matrix ${\mathbf\Sigma}={\mathbf A}{\mathbf A}^T$ is the covariance matrix of the components X_i. It is important to realize that the covariance matrix must be allowed to be singular. That case arises frequently in statistics; for example, in the distribution of the vector of residuals in ordinary linear regression problems. Note also that the X_i are in general not independent; they can be seen as the result of applying the linear transformation A to a collection of independent Gaussian variables Z.

Contents

1 Linear transformation

2 Marginal distributions

3 Conditional distributions

4 Estimation of parameters

Linear transformation

If ${\mathbf y}={\mathbf B}{\mathbf x}$ is a linear transformation of ${\mathbf x}$ where ${\mathbf B}$ is a rank m $m\times p$ matrix with $m\leq p$ then ${\mathbf y}$ has a multivariate normal distribution with a mean of ${\mathbf B}{\mathbf\mu}$ and a covariance matrix ${\mathbf B}{\mathbf\Sigma}{\mathbf B}^T$ .

Corollary: any subset of the x_i has a marginal distribution that is also multivariate normal. To see this consider the following example: to extract the subset (x₁,x2,x₄)^T, use

${\mathbf B}= \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & \ldots & 0\\ 0 & 1 & 0 & 0 & 0 & \ldots & 0\\ 0 & 0 & 0 & 1 & 0 & \ldots & 0 \end{bmatrix}$

which extracts the desired elements directly.

Marginal distributions

If ${\mathbf x}$ is partitioned into ${\mathbf x}_1$ and ${\mathbf x}_2$ (so ${\mathbf x}=({\mathbf x}_1,{\mathbf x}_2)^T$ (note that vectors are column vectors by default). Say ${\mathbf x}_1$ has q elements, so ${\mathbf x}_2$ has p - q elements.

Conditional distributions

Then if ${\mathbf\mu}$ and ${\mathbf\Sigma}$ are partitioned as follows

${\mathbf\mu}=\left(\begin{matrix} {\mathbf\mu}_1\\ {\mathbf\mu}_2 \end{matrix} \right) \qquad {\mathbf\Sigma}= \begin{bmatrix} {\mathbf\Sigma}_{11} & {\mathbf\Sigma}_{12} \\ {\mathbf\Sigma}_{21} & {\mathbf\Sigma}_{22} \end{bmatrix}$

then the distribution of ${\mathbf x}_1$ conditional on ${\mathbf x}_2={\mathbf a}$ is multivariate normal with mean

${\mathbf\mu}_1+{\mathbf\Sigma}_{12}{\mathbf\Sigma}_{22}^{-1}\left({\mathbf a}-{\mathbf\mu}_2\right)$

and covariance matrix

${\mathbf\Sigma}_{11}- {\mathbf\Sigma}_{12} {\mathbf\Sigma}_{22}^{-1} {\mathbf\Sigma}_{21}.$

This matrix is the Schur complement of ${\mathbf\Sigma_{22}}$ in ${\mathbf\Sigma}$ .

Note that knowing the value of ${\mathbf x}_2$ to be ${\mathbf a}$ alters the variance; perhaps more suprisingly, the mean is shifted by ${\mathbf\Sigma}_{12}{\mathbf\Sigma}_{22}^{-1}\left({\mathbf a}-{\mathbf\mu}_2\right)$ ; compare this with the situation of not knowing the value of ${\mathbf a}$ , in which case ${\mathbf x}_1$ would have distribution $N_q\left({\mathbf\mu}_1,{\mathbf\Sigma}_{11}\right)$ .

The matrix ${\mathbf\Sigma}_{12}{\mathbf\Sigma}_{22}^{-1}$ is known as the matrix of regression coefficients.

Estimation of parameters

The derivation of the maximum-likelihood estimator of the covariance matrix of a multivariate normal distribution is perhaps surprisingly subtle and elegant. See estimation of covariance matrices.

Categories: Probability distributions

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