Multivariate normal distribution

In probability theory and statistics, a multivariate normal distribution, also sometimes called a multivariate Gaussian distribution, is a specific probability distribution, which can be thought of as a generalization to higher dimensions of the one-dimensional normal distribution (also called a Gaussian distribution).

1 General case
- 1.1 Cumulative distribution function
2 A counterexample
3 Bivariate case
4 Linear transformation
5 Correlations and independence
6 Conditional distributions
7 Fisher information matrix
8 Kullback-Leibler divergence
9 Estimation of parameters
10 Multivariate normality tests
11 Drawing values from the distribution
12 References

General case

A random vector $X = [X_1, \dots, X_N]$ follows a multivariate normal distribution if it satisfies the following equivalent conditions:

every linear combination $Y = a_1 X_1 + \cdots + a_N X_N$ is normally distributed

there is a random vector $Z = [Z_1, \dots, Z_M]$ , whose components are independent standard normal random variables, a vector $\mu = [\mu_1, \dots, \mu_N]$ and an $N \times M$ 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

$\phi_X(u) = \exp \left( i \mu^\top u - \frac{1}{2} u^\top \Gamma u \right) .$

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 $\mu = [\mu_1, \dots, \mu_N]$ and a symmetric, positive definite covariance matrix $Σ$ ( $N \times N$ matrix) such that $X$ has density

$f_X(x_1, \dots, x_N) = \frac {1} {(2\pi)^{N/2} \left|\Sigma\right|^{1/2}} \exp \left( -\frac{1}{2} ( x - \mu)^\top \Sigma^{-1} (x - \mu) \right)$

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where $\left| \Sigma \right|$ is the determinant of $Σ$ . Note how the equation above reduces to that of the univariate normal distribution if $Σ$ is a scalar (i.e., a real number).

The vector $μ$ in these conditions is the expected value of $X$ and the matrix $Σ = A A T$ 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 $Z$ .

The multivariate normal can be written in the following notation:

$X \sim N(\mu, \Sigma)$

or to make it explicitly known $X$ is N-dimensional

$X \sim N_N(\mu, \Sigma)$

Cumulative distribution function

The cumulative distribution function (cdf) $F (x)$ is defined as the probability that all values in a random vector $X$ are less than or equal to the corresponding values in vector $x$ . Though there is no closed form for $F (x)$ , there are a number of algorithms that estimate it numerically. For one such example, see [1] (includes MATLAB code).

A counterexample

The fact that two random variables X and Y are normally distributed does not imply that the pair (X, Y) has a bivariate normal distribution. A simple example is one in which Y = X if |X| > 1 and Y = −X if |X| < 1.

If X and Y are normally distributed and independent, then they are "jointly normally distributed", i.e., the pair (X, Y) does have a bivariate normal distribution. There are of course also many bivariate normal distributions in which the components are correlated.

Bivariate case

In the 2-dimensional nonsingular case, the probability density function (with mean (0,0)) is

$f(x,y) = \frac{1}{2 \pi \sigma_x \sigma_y \sqrt{1-\rho^2}} \exp \left( -\frac{1}{2 (1-\rho^2)} \left( \frac{x^2}{\sigma_x^2} + \frac{y^2}{\sigma_y^2} - \frac{2 \rho x y}{ (\sigma_x \sigma_y)} \right) \right)$

where $ρ$ is the correlation between $X$ and $Y$ .

Linear transformation

If $Y = B X$ is a linear transformation of $X$ where $B$ is an $m \times p$ matrix then $Y$ has a multivariate normal distribution with expected value $B μ$ and variance $B Σ B T$ (i.e., $Y \sim N \left(B \mu, B \Sigma B^T\right)$ .

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 $(X 1, X 2, X 4) T$ , use

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

which extracts the desired elements directly.

Correlations and independence

In general, random variables may be uncorrelated but highly dependent. But if a random vector has a multivariate normal distribution then any two or more of its components that are uncorrelated are independent. This implies that any two or more of its components that are pairwise independent are independent.

But it is not true that two random variables that are (separately, marginally) normally distributed and uncorrelated are independent. Two random variables that are normally distributed may fail to be jointly normally distributed, i.e., the vector whose components they are may fail to have a multivariate normal distribution. For an example of two normally distributed random variables that are uncorrelated but not independent, see normally distributed and uncorrelated does not imply independent.

Conditional distributions

Then if $μ$ and $Σ$ are partitioned as follows

$\mu = \begin{bmatrix} \mu_1 \\ \mu_2 \end{bmatrix} \quad$ with sizes $\begin{bmatrix} q \times 1 \\ (N-q) \times 1 \end{bmatrix}$

$\Sigma = \begin{bmatrix} \Sigma_{11} & \Sigma_{12} \\ \Sigma_{21} & \Sigma_{22} \end{bmatrix} \quad$ with sizes $\begin{bmatrix} q \times q & q \times (N-q) \\ (N-q) \times q & (N-q) \times (N-q) \end{bmatrix}$

then the distribution of $x 1$ conditional on $x 2 = a$ is multivariate normal $X_1|X_2=a \sim N(\bar{\mu}, \overline{\Sigma})$ where

$\bar{\mu} = \mu_1 + \Sigma_{12} \Sigma_{22}^{-1} \left( a - \mu_2 \right)$

and covariance matrix

$\overline{\Sigma} = \Sigma_{11} - \Sigma_{12} \Sigma_{22}^{-1} \Sigma_{21}.$

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

Note that knowing the value of $x 2$ to be $a$ alters the variance; perhaps more surprisingly, the mean is shifted by $\Sigma_{12} \Sigma_{22}^{-1} \left(a - \mu_2 \right)$ ; compare this with the situation of not knowing the value of $a$ , in which case $x 1$ would have distribution $N_q \left(\mu_1, \Sigma_{11} \right)$ .

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

Fisher information matrix

The Fisher information matrix (FIM) for a normal distribution takes a special formulation. The $(m, n)$ element of the FIM for $X \sim N(\mu(\theta), \Sigma(\theta))$ is

$\mathcal{I}_{m,n} = \frac{\partial \mu}{\partial \theta_m} \Sigma^{-1} \frac{\partial \mu^\top}{\partial \theta_n} + \frac{1}{2} \mathrm{tr} \left( \Sigma^{-1} \frac{\partial \Sigma}{\partial \theta_m} \Sigma^{-1} \frac{\partial \Sigma}{\partial \theta_n} \right)$

where

$\frac{\partial \mu}{\partial \theta_m} = \begin{bmatrix} \frac{\partial \mu_1}{\partial \theta_m} & \frac{\partial \mu_2}{\partial \theta_m} & \cdots & \frac{\partial \mu_N}{\partial \theta_m} & \end{bmatrix}$
$\frac{\partial \mu^\top}{\partial \theta_m} = \left( \frac{\partial \mu}{\partial \theta_m} \right)^\top = \begin{bmatrix} \frac{\partial \mu_1}{\partial \theta_m} \\ \\ \frac{\partial \mu_2}{\partial \theta_m} \\ \\ \vdots \\ \\ \frac{\partial \mu_N}{\partial \theta_m} \\ \\ \end{bmatrix}$
$\frac{\partial \Sigma}{\partial \theta_m} = \begin{bmatrix} \frac{\partial \Sigma_{1,1}}{\partial \theta_m} & \frac{\partial \Sigma_{1,2}}{\partial \theta_m} & \cdots & \frac{\partial \Sigma_{1,N}}{\partial \theta_m} \\ \\ \frac{\partial \Sigma_{2,1}}{\partial \theta_m} & \frac{\partial \Sigma_{2,2}}{\partial \theta_m} & \cdots & \frac{\partial \Sigma_{2,N}}{\partial \theta_m} \\ \\ \vdots & \vdots & \ddots & \vdots \\ \\ \frac{\partial \Sigma_{N,1}}{\partial \theta_m} & \frac{\partial \Sigma_{N,2}}{\partial \theta_m} & \cdots & \frac{\partial \Sigma_{N,N}}{\partial \theta_m} \end{bmatrix}$
$tr$ is the trace function

Kullback-Leibler divergence

The Kullback-Leibler divergence from $N 0 N (μ 0,Σ 0)$ to $N 1 N (μ 1,Σ 1)$ is:

$KL(N0, N1) = { 1 \over 2 } \left( \log \left( { \det \Sigma_1 \over \det \Sigma_0 } \right) + \mathrm{tr} \left( \Sigma_1^{-1} \Sigma_0 \right) + \left( \mu_1 - \mu_0\right)^\top \Sigma_1^{-1} ( \mu_1 - \mu_0 ) - N\right).$

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.

In short, the pdf is

$f(x)=(2 \pi)^{-p/2} \det(\Sigma)^{-1/2} \exp\left({-1 \over 2} (x-\mu)^T \Sigma^{-1} (x-\mu)\right)$

and the ML estimator of the covariance matrix is

$\Sigma = {1 \over n}\sum_{i=1}^n (X_i-\overline{X})(X_i-\overline{X})^T$

which is simply the sample covariance matrix.

Multivariate normality tests

Multivariate normality tests check a given set of data for similarity to the multivariate normal distribution. The null hypothesis is that the data set is similar to the normal distribution, therefore a sufficiently small p-value indicates non-normal data. Multivariate normality tests include the Cox-Small test ^[1] and Smith and Jain's adaptation of the Friedman-Rafsky test ^[2].

Drawing values from the distribution

A widely used method for drawing a random vector $X$ from the $n$ -dimensional multivariate normal distribution with mean vector $μ$ and covariance matrix $Σ$ (required to be symmetric and positive definite) works as follows:

Compute the Cholesky decomposition (matrix square root) of $Σ$ , that is, find the unique lower triangular matrix $A$ such that $A\,A^T = \Sigma$ .
Let $Z=(z_1,\dots,z_n)$ be a vector whose components are $n$ independent standard normal variates (which can be generated, for example, by using the Box-Muller transform).
Let $X$ be $\mu + A\,Z$ .

References

↑ Cox, D. R., N. J. H. Small (August 1978). "Testing multivariate normality". Biometrika 65 (2): 263–272.
↑ Smith, Stephen P., Anil K. Jain (September 1988). "A test to determine the multivariate normality of a dataset". IEEE Transactions on Pattern Analysis and Machine Intelligence 10 (5): 757–761. DOI:10.1109/34.6789.

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