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).

General case

---

A random vector $X\; =\backslash dots,\; X\_N$ follows a multivariate normal distribution if it satisfies the following equivalent conditions:

• every linear combination $Y\; =\; a\_1\; X\_1\; +\; \backslash cdots\; +\; a\_N\; X\_N$ is normally distributed

• there is a random vector $Z\; =\backslash dots,\; Z\_M$, whose components are independent standard normal random variables, a vector $\backslash mu\; =\backslash dots,\; \backslash mu\_N$ and an $N\; \backslash times\; M$ matrix $A$ such that $X\; =\; A\; Z\; +\; \backslash mu$.

• there is a vector $\backslash mu$ and a symmetric, positive semi-definite matrix $\backslash Sigma$ such that the characteristic function of $X$ is

$$

\phi_X(u;\mu,\Sigma) = \exp \left( i \mu^\top u - \frac{1}{2} u^\top \Sigma 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 $\backslash mu\; =\backslash dots,\; \backslash mu\_N$ and a symmetric, positive definite covariance matrix $\backslash Sigma$ ($N\; \backslash times\; N$ 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)

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

The vector $\backslash mu$ in these conditions is the expected value of $X$ and the matrix $\backslash Sigma\; =\; A\; A^T$ 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:

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

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

$X\; \backslash sim\; N\_N(\backslash mu,\; \backslash Sigma)$

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 example, see MVNDST under ^* (includes FORTRAN code) or ^* (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 $\backslash rho$ is the correlation between $X$ and $Y$.

Linear transformation

---

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

Corollary: any subset of the $X\_i\backslash ,$ has a marginal distribution that is also multivariate normal. To see this consider the following example: to extract the subset $(X\_1,\; X\_2,\; X\_4)^T\; \backslash ,$, 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.

Higher moments

---

The $k$-order moments of $X$ are defined by

$$

\mu _{1,...,N}(X)\equiv \mu _{r_{1},...,r_{N}}(X)\equiv E\left[ \prod\limits_{j=1}^{N}x^{r_{j}}\right] where $r\_\{1\}+r\_\{2\}+...+r\_\{N\}=k$

The central $k$-order moments are given as follows

(a) If $k$ is odd, $\backslash mu\; \_\{1,...,N\}(X-\backslash mu\; )=0$.

(b) If $k$ is even with $k=2\backslash lambda$, then

$$

\mu _{1,...,2\lambda }(X-\mu )=\sum \left( \sigma _{ij}\sigma _{kl}...\sigma _{xz}\right) where the sum is taken over all permutations of $\backslash left\backslash \{\; 1,...,2\backslash lambda\; \backslash right\backslash \}$ giving $(2\backslash lambda\; -1)!/(2^\{\backslash lambda\; -1\}(\backslash lambda\; -1)!)$ terms in the sum each being the product of $\backslash lambda$ covariances

In particular, the 4-order moments are

$E\backslash leftx\_\{i\}^\{4\}\backslash right=\; 3\backslash sigma\; \_\{ii\}$

$E\backslash leftx\_\{i\}^\{3\}x\_\{j\}\backslash right=\; 3\backslash sigma\; \_\{ii\}\backslash sigma\; \_\{ij\}$

$E\backslash leftx\_\{i\}^\{2\}x\_\{j\}^\{2\}\backslash right=\; \backslash sigma\; \_\{ii\}\backslash sigma\; \_\{jj\}+2\backslash left(\; \backslash sigma\; \_\{ij\}\backslash right)\; ^\{2\}$

$E\backslash leftx\_\{i\}^\{2\}x\_\{j\}x\_\{k\}\backslash right=\; \backslash sigma\; \_\{ii\}\backslash sigma\; \_\{jk\}+2\backslash sigma\; \_\{ij\}\backslash sigma\; \_\{ik\}$

$E\backslash leftx\_\{i\}x\_\{j\}x\_\{k\}x\_\{n\}\backslash right=\; \backslash sigma\; \_\{ij\}\backslash sigma\; \_\{kn\}+\backslash sigma\; \_\{ik\}\backslash sigma\; \_\{jn\}+\backslash sigma\; \_\{in\}\backslash sigma\; \_\{jk\}$

Conditional distributions

---

Then if $\backslash mu$ and $\backslash Sigma$ are partitioned as follows

$$

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

$$

\Sigma = \begin{bmatrix} \Sigma_{11} & \Sigma_{12} \\ \Sigma_{21} & \Sigma_{22} \end{bmatrix} \quad with sizes

then the distribution of $x\_1$ conditional on $x\_2=a$ is multivariate normal $X\_1|X\_2=a\; \backslash sim\; N(\backslash bar\{\backslash mu\},\; \backslash overline\{\backslash Sigma\})$ 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 $\{\backslash mathbf\backslash Sigma\_\{22\}\}$ in $\{\backslash mathbf\backslash Sigma\}$.

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

The matrix $\backslash Sigma\_\{12\}\; \backslash Sigma\_\{22\}^\{-1\}$ 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 $X\; \backslash sim\; N(\backslash mu(\backslash theta),\; \backslash Sigma(\backslash theta))$ 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}

• $\backslash mathrm\{tr\}$ is the trace function

Kullback-Leibler divergence

---

The Kullback-Leibler divergence from $N0\_N(\backslash mu\_0,\; \backslash Sigma\_0)$ to $N1\_N(\backslash mu\_1,\; \backslash Sigma\_1)$ 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

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

and the ML estimator of the covariance matrix is

$\backslash hat\backslash Sigma\; =\; \{1\; \backslash over\; n\}\backslash sum\_\{i=1\}^n\; (X\_i-\backslash overline\{X\})(X\_i-\backslash overline\{X\})^T$

which is simply the sample covariance matrix. This is a biased estimator whose expectation is

$E*=\; \{n-1\; \backslash over\; n\}\backslash Sigma.$

An unbiased sample covariance is

$\backslash hat\backslash Sigma\; =\; \{1\; \backslash over\; n-1\}\backslash sum\_\{i=1\}^n\; (X\_i-\backslash overline\{X\})(X\_i-\backslash overline\{X\})^T.$

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 and Smith and Jain's adaptation of the Friedman-Rafsky test .

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 $\backslash mu$ and covariance matrix $\backslash Sigma$ (required to be symmetric and positive definite) works as follows:

1. Compute the Cholesky decomposition (matrix square root) of $\backslash Sigma$, that is, find the unique lower triangular matrix $A$ such that $A\backslash ,A^T\; =\; \backslash Sigma$.
2. Let $Z=(z\_1,\backslash dots,z\_n)^T$ be a vector whose components are $n$ independent standard normal variates (which can be generated, for example, by using the Box-Muller transform).
3. Let $X$ be $\backslash mu\; +\; A\backslash ,Z$.

References

---

Continuous distributions

loi normale multidimensionnelle | Многомерное нормальное распределение | Multivariat normalfördelning