mvnpdf
Multivariate normal probability density function
Description
returns an n-by-y
= mvnpdf(X
)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 = 10×1
10-4 ×
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, thenmvnpdf
replicates the vector to match the trailing dimension ofSigma
.If
mu
is a matrix, then each row ofmu
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, thenmvnpdf
replicates the matrix to match the number of rows inmu
.If
Sigma
is an array, then each page ofSigma
,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
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
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:
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 asnormpdf(1,0,2)
, where4
is the variance and2
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
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
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).
Version History
Introduced before R2006a
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