# mvnpdf

Multivariate normal probability density function

## Description

returns an `y`

= mvnpdf(`X`

)*n*-by-`1`

vector
`y`

containing the probability density function (pdf) values
for the *d*-dimensional multivariate normal distribution with zero
mean and identity covariance matrix, evaluated at each row of the
*n*-by-*d* matrix `X`

. For
more information, see Multivariate Normal Distribution.

## Examples

### Standard Multivariate Normal pdf

Evaluate the pdf of a standard five-dimensional normal distribution at a set of random points.

Randomly sample eight points from the standard five-dimensional normal distribution.

mu = zeros(1,5); Sigma = eye(5); rng('default') % For reproducibility X = mvnrnd(mu,Sigma,8)

`X = `*8×5*
0.5377 3.5784 -0.1241 0.4889 -1.0689
1.8339 2.7694 1.4897 1.0347 -0.8095
-2.2588 -1.3499 1.4090 0.7269 -2.9443
0.8622 3.0349 1.4172 -0.3034 1.4384
0.3188 0.7254 0.6715 0.2939 0.3252
-1.3077 -0.0631 -1.2075 -0.7873 -0.7549
-0.4336 0.7147 0.7172 0.8884 1.3703
0.3426 -0.2050 1.6302 -1.1471 -1.7115

Evaluate the pdf of the distribution at the points in `X`

.

y = mvnpdf(X)

`y = `*8×1*
0.0000
0.0000
0.0000
0.0000
0.0054
0.0011
0.0015
0.0003

Find the point in `X`

with the greatest pdf value.

[maxpdf,idx] = max(y)

maxpdf = 0.0054

idx = 5

maxPoint = X(idx,:)

`maxPoint = `*1×5*
0.3188 0.7254 0.6715 0.2939 0.3252

The fifth point in `X`

has a greater pdf value than any of the other randomly selected points.

### Multivariate Normal pdfs Evaluated at Different Points

Create six three-dimensional normal distributions, each with a distinct mean. Evaluate the pdf of each distribution at a different random point.

Specify the means `mu`

and covariances `Sigma`

of the distributions. Each distribution has the same covariance matrix—the identity matrix.

firstDim = (1:6)'; mu = repmat(firstDim,1,3)

`mu = `*6×3*
1 1 1
2 2 2
3 3 3
4 4 4
5 5 5
6 6 6

Sigma = eye(3)

`Sigma = `*3×3*
1 0 0
0 1 0
0 0 1

Randomly sample once from each of the six distributions.

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

`X = `*6×3*
1.5377 0.5664 1.7254
3.8339 2.3426 1.9369
0.7412 6.5784 3.7147
4.8622 6.7694 3.7950
5.3188 3.6501 4.8759
4.6923 9.0349 7.4897

Evaluate the pdfs of the distributions at the points in `X`

. The pdf of the first distribution is evaluated at the point `X(1,:)`

, the pdf of the second distribution is evaluated at the point `X(2,:)`

, and so on.

y = mvnpdf(X,mu)

`y = `*6×1*
0.0384
0.0111
0.0000
0.0009
0.0241
0.0001

### Multivariate Normal pdf

Evaluate the pdf of a two-dimensional normal distribution at a set of given points.

Specify the mean `mu`

and covariance `Sigma`

of the distribution.

mu = [1 -1]; Sigma = [0.9 0.4; 0.4 0.3];

Randomly sample from the distribution 100 times. Specify `X`

as the matrix of sampled points.

rng('default') % For reproducibility X = mvnrnd(mu,Sigma,100);

Evaluate the pdf of the distribution at the points in `X`

.

y = mvnpdf(X,mu,Sigma);

Plot the probability density values.

scatter3(X(:,1),X(:,2),y) xlabel('X1') ylabel('X2') zlabel('Probability Density')

### Multivariate Normal pdfs Evaluated at Same Point

Create ten different five-dimensional normal distributions, and compare the values of their pdfs at a specified point.

Set the dimensions `n`

and `d`

equal to 10 and 5, respectively.

n = 10; d = 5;

Specify the means `mu`

and the covariances `Sigma`

of the multivariate normal distributions. Let all the distributions have the same mean vector, but vary the covariance matrices.

mu = ones(1,d)

`mu = `*1×5*
1 1 1 1 1

mat = eye(d); nMat = repmat(mat,1,1,n); var = reshape(1:n,1,1,n); Sigma = nMat.*var;

Display the first two covariance matrices in `Sigma`

.

Sigma(:,:,1:2)

ans = ans(:,:,1) = 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 ans(:,:,2) = 2 0 0 0 0 0 2 0 0 0 0 0 2 0 0 0 0 0 2 0 0 0 0 0 2

Set `x`

to be a random point in five-dimensional space.

rng('default') % For reproducibility x = normrnd(0,1,1,5)

`x = `*1×5*
0.5377 1.8339 -2.2588 0.8622 0.3188

Evaluate the pdf at `x`

for each of the ten distributions.

y = mvnpdf(x,mu,Sigma)

y =10×110^{-4}× 0.2490 0.8867 0.8755 0.7035 0.5438 0.4211 0.3305 0.2635 0.2134 0.1753

Plot the results.

scatter(1:n,y,'filled') xlabel('Distribution Index') ylabel('Probability Density at x')

## Input Arguments

`X`

— Evaluation points

numeric vector | numeric matrix

Evaluation points, specified as a
`1`

-by-*d* numeric vector or an
*n*-by-*d* numeric matrix, where
*n* is a positive scalar integer and
*d* is the dimension of a single multivariate normal
distribution. The rows of `X`

correspond to observations
(or points), and the columns correspond to variables (or
coordinates).

If `X`

is a vector, then `mvnpdf`

replicates it to match the leading dimension of `mu`

or
the trailing dimension of `Sigma`

.

**Data Types: **`single`

| `double`

`mu`

— Means of multivariate normal distributions

vector of zeros (default) | numeric vector | numeric matrix

Means of multivariate normal distributions, specified as a
`1`

-by-*d* numeric vector or an
*n*-by-*d* numeric matrix.

If

`mu`

is a vector, then`mvnpdf`

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

identity matrix (default) | symmetric, positive definite matrix | numeric array

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

If

`Sigma`

is a matrix, then`mvnpdf`

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 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-*n*
array containing just the diagonal entries.

**Data Types: **`single`

| `double`

## Output Arguments

`y`

— pdf values

numeric vector

pdf values, returned as an *n*-by-`1`

numeric vector, where *n* is one of the following:

Number of rows in

`X`

if`X`

is a matrixNumber of times

`X`

is replicated if`X`

is a vector

If `X`

is a matrix, `mu`

is a
matrix, and `Sigma`

is an array, then
`mvnpdf`

computes `y(i)`

using
`X(i,:)`

, `mu(i,:)`

, and
`Sigma(:,:,i)`

.

**Data Types: **`double`

## 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

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

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

In the one-dimensional case,

`Sigma`

is the variance, not the standard deviation. For example,`mvnpdf(1,0,4)`

is the same as`normpdf(1,0,2)`

, 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).

**Introduced before R2006a**

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