lhsnorm

Latin hypercube sample from multivariate normal distribution

Syntax

X = lhsnorm(mu,sigma,n)

X = lhsnorm(mu,sigma,n,Smooth)

[X,Z] = lhsnorm(___)

Description

X = lhsnorm(mu,sigma,n) returns a numeric matrix X containing a Latin hypercube sample of size n from a multivariate normal distribution with mean vector mu and covariance matrix sigma. The size of X is n-by-d, where d is the size of mu. X is similar to a random sample generated from the multivariate normal distribution (see mvnrnd), but lhsnorm adjusts the marginal distribution of each column so that its sample marginal distribution is close to its theoretical normal distribution.

X = lhsnorm(mu,sigma,n,Smooth) additionally sets the type of sample smoothing.

[X,Z] = lhsnorm(___) also returns the original multivariate normal sample Z, using any of the input argument combinations in the previous syntaxes. lhsnorm generates z with mvnrnd before it adjusts the marginal distributions to obtain X.

Examples

Create Latin Hypercube Sample from Multivariate Normal Distribution

Define the mean vector and covariance matrix of a two-dimensional normal distribution.

mu = [0 1];
sigma = [1 0.5; 0.5 1];

Verify that the covariance matrix is symmetric and positive semi-definite.

issymmetric(sigma)

ans = logical
   1

eigenvalues = eig(sigma);
all(eigenvalues >= 0)

ans = logical
   1

The eigenvalues of sigma are all nonnegative, indicating that the covariance matrix is positive semi-definite.

Create a Latin hypercube sample of size 100 from the two-dimensional normal distribution.

X = lhsnorm(mu,sigma,100);

Plot a histogram of the values in the first column of X.

hist(X(:,1))

Input Arguments

mu — Mean vector of multivariate normal distribution

numeric vector | numeric scalar

Mean vector of a multivariate normal distribution, specified as a 1-by-d numeric vector where d is the dimension of the multivariate normal distribution, or as a numeric scalar.

Data Types: single | double

sigma — Covariance matrix of multivariate normal distribution

symmetric, positive semi-definite matrix | numeric scalar

Covariance matrix of a multivariate normal distribution, specified as a d-by-d symmetric, positive semi-definite matrix, where d is the dimension of the multivariate normal distribution, or as a numeric scalar. If you specify sigma as a numeric scalar, it represents the variance of a one-dimensional normal distribution.

Data Types: single | double

n — Number of returned samples

nonnegative integer

Number of returned samples in X, specified as a nonnegative integer.

Data Types: single | double

Smooth — Flag for sample smoothing

"on" (default) | "off"

Flag for sample smoothing, specified as "on" or "off".

• If Smooth is "off", then each column of X has points equally spaced on the probability scale. Therefore, each column is a permutation of the values G(0.5/n), G(1.5/n), ..., G(1-0.5/n), where G is the inverse normal cumulative distribution for that column's marginal distribution.

• If Smooth is "on" (the default), then each column of X has points uniformly distributed on the probability scale. Therefore, lhsnorm samples G at n uniformly distributed random values between 0 and 1.

Data Types: char | string

Output Arguments

X — Latin hypercube sample from multivariate normal distribution

numeric matrix

Latin hypercube sample from a multivariate normal distribution, returned as an n-by-d numeric matrix, where d is the size of mu. X is similar to a random sample generated from the multivariate normal distribution (see mvnrnd), but lhsnorm adjusts the marginal distribution of each column so that its sample marginal distribution is close to its theoretical normal distribution.

Z — Original multivariate normal sample

numeric matrix

Original multivariate normal sample, returned as an n-by-d numeric matrix, where d is the size of mu. lhsnorm calls mvnrnd to create Z before it adjusts the marginal distributions to obtain X.

Tips

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

References

[1] Stein, M. “Large Sample Properties of Simulations Using Latin Hypercube Sampling.” Technometrics 29, no. 2 (May 1987): 143–151.

Version History

Introduced before R2006a