mvnrnd

Multivariate normal random numbers

Syntax

R = mvnrnd(mu,Sigma,n)

R = mvnrnd(mu,Sigma)

Description

R = mvnrnd(mu,Sigma,n) returns a matrix R of n random vectors chosen from the same multivariate normal distribution, with mean vector mu and covariance matrix Sigma. For more information, see Multivariate Normal Distribution.

R = mvnrnd(mu,Sigma) returns an m-by-d matrix R of random vectors sampled from m separate d-dimensional multivariate normal distributions, with means and covariances specified by mu and Sigma, respectively. Each row of R is a single multivariate normal random vector.

Examples

Generate Multivariate Normal Random Numbers

Generate random numbers from the same multivariate normal distribution.

Define mu and Sigma, and generate 100 random numbers.

mu = [2 3];
Sigma = [1 1.5; 1.5 3];
rng('default')  % For reproducibility
R = mvnrnd(mu,Sigma,100);

Plot the random numbers.

plot(R(:,1),R(:,2),'+')

Sample from Different Multivariate Normal Distributions

Randomly sample from five different three-dimensional normal distributions.

Specify the means mu and the covariances Sigma of the distributions. Let all the distributions share the same covariance matrix, but vary the mean vectors.

firstDim = (1:5)';
mu = repmat(firstDim,1,3)

mu = 5×3

     1     1     1
     2     2     2
     3     3     3
     4     4     4
     5     5     5

Sigma = eye(3)

Sigma = 3×3

     1     0     0
     0     1     0
     0     0     1

Randomly sample once from each of the five distributions.

rng('default')  % For reproducibility
R = mvnrnd(mu,Sigma)

R = 5×3

    1.5377   -0.3077   -0.3499
    3.8339    1.5664    5.0349
    0.7412    3.3426    3.7254
    4.8622    7.5784    3.9369
    5.3188    7.7694    5.7147

Plot the results.

scatter3(R(:,1),R(:,2),R(:,3))

Input Arguments

mu — Means of multivariate normal distributions

numeric vector | numeric matrix

Means of multivariate normal distributions, specified as a 1-by-d numeric vector or an m-by-d numeric matrix.

• If mu is a vector, then mvnrnd replicates the vector to match the trailing dimension of Sigma.

• If mu is a matrix, then each row of mu is the mean vector of a single multivariate normal distribution.

Data Types: single | double

Sigma — Covariances of multivariate normal distributions

symmetric, positive semi-definite matrix | numeric array

Covariances of multivariate normal distributions, specified as a d-by-d symmetric, positive semi-definite matrix or a d-by-d-by-m numeric array.

• If Sigma is a matrix, then mvnrnd replicates the matrix to match the number of rows in mu.

• If Sigma is an array, then each page of Sigma, Sigma(:,:,i), is the covariance matrix of a single multivariate normal distribution and, therefore, is a symmetric, positive semi-definite matrix.

If the covariance matrices are diagonal, containing variances along the diagonal and zero covariances off it, then you can also specify Sigma as a 1-by-d vector or a 1-by-d-by-m array containing just the diagonal entries.

Data Types: single | double

n — Number of multivariate random numbers

positive scalar integer

Number of multivariate random numbers, specified as a positive scalar integer. n specifies the number of rows in R.

Data Types: single | double

Output Arguments

R — Multivariate normal random numbers

numeric matrix

Multivariate normal random numbers, returned as one of the following:

• m-by-d numeric matrix, where m and d are the dimensions specified by mu and Sigma

• n-by-d numeric matrix, where n is the specified input argument and d is the dimension specified by mu and Sigma

If mu is a matrix and Sigma is an array, then mvnrnd computes R(i,:) using mu(i,:) and Sigma(:,:,i).

More About

Multivariate Normal Distribution

The multivariate normal distribution is a generalization of the univariate normal distribution to two or more variables. It has two parameters, a mean vector μ and a covariance matrix Σ, that are analogous to the mean and variance parameters of a univariate normal distribution. The diagonal elements of Σ contain the variances for each variable, and the off-diagonal elements of Σ contain the covariances between variables.

The probability density function (pdf) of the d-dimensional multivariate normal distribution is

$$y\text{ = }f(x,\mu ,\Sigma )\text{ = }\frac{\text{1}}{\sqrt{\left|\Sigma \right|{\text{(2}\pi \text{)}}^d}}\exp \left(-\frac{\text{1}}{\text{2}}\text{(}x\text{-}\mu \text{) }{\Sigma }^{\text{-1}}\text{(}x\text{-}\mu \text{)'}\right)$$

where x and μ are 1-by-d vectors and Σ is a d-by-d symmetric, positive definite matrix. Only mvnrnd allows positive semi-definite Σ matrices, which can be singular. The pdf cannot have the same form when Σ is singular.

The multivariate normal cumulative distribution function (cdf) evaluated at x is the probability that a random vector v, distributed as multivariate normal, lies within the semi-infinite rectangle with upper limits defined by x:

$$\mathrm{Pr}\{v(1)\le x(1),v(2)\le x(2),\mathrm{...},v(d)\le x(d)\}.$$

Although the multivariate normal cdf does not have a closed form, mvncdf can compute cdf values numerically.

Tips

• mvnrnd requires the matrix Sigma to be symmetric. If Sigma has only minor asymmetry, you can use (Sigma + Sigma')/2 instead to resolve the asymmetry.

• In the one-dimensional case, Sigma is the variance, not the standard deviation. For example, mvnrnd(0,4) is the same as normrnd(0,2), where 4 is the variance and 2 is the standard deviation.

References

[1] Kotz, S., N. Balakrishnan, and N. L. Johnson. Continuous Multivariate Distributions: Volume 1: Models and Applications. 2nd ed. New York: John Wiley & Sons, Inc., 2000.

Extended Capabilities

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).