How can I generate data with a given sample covariance matrix?

How can I generate data with a given sample covariance matrix?

This is easy to do by generating samples from a standard Gaussian and multiplying them by a square root of the covariance matrix, e.g. by c h o l ( Σ). This is covered in many threads on CV, e.g. here: How can I generate data with a prespecified correlation matrix?

What to look for in a multivariate normal distribution?

For variables with a multivariate normal distribution with mean vector μ and covariance matrix Σ, some useful facts are: Each single variable has a univariate normal distribution. Thus we can look at univariate tests of normality for each variable when assessing multivariate normality.

Which is the determinant of the variance-covariance matrix?

This covariance is equal to the correlation times the product of the two standard deviations. The determinant of the variance-covariance matrix is simply equal to the product of the variances times 1 minus the squared correlation. The inverse of the variance-covariance matrix takes the form below:

How to understand the bivariate normal distribution in ESC?

ESC Bivariate Normal Distribution Section To further understand the multivariate normal distribution it is helpful to look at the bivariate normal distribution. Here our understanding is facilitated by being able to draw pictures of what this distribution looks like.

How to generate a random variable with a defined correlation?

Pick a vector X, of the same length as Y, according to any distribution you like. Let Y ⊥ be the residuals of the least squares regression of X against Y: this removes the Y component from X, producing a vector orthogonal to Y.

How to generate a sample with a correlation of ρ?

In contrast to caracal’s solution it does not produce a sample with the exact correlation of ρ, but two vectors whose population correlation is equal to ρ. Following function can compute a bivariate sample distribution drawn from a population with a given ρ.

How to generate data with a prespecified matrix?

Let’s say you start with (mean zero) uncorrelated data (i.e. the covariance matrix is diagonal) – since we’re talking about the correlation matrix, let’s just take Σ = I. You can transform this to data with a given covariance matrix by choosing A to be the cholesky square root of Ω – then A x would have the desired covariance matrix Ω.

How to simulate a multivariate normal distribution?

Suppose that you want to simulate k samples (each with N observations) from a multivariate normal distribution with a given mean vector and covariance matrix.

Can you generate 10 samples from a normal distribution?

Now suppose that you want to generate 10 samples, where each sample contains five observations from a trivariate normal distribution. You can generate 5 x 10 = 50 observations as follows:

How to generate a Gaussian random variable in MATLAB?

Not really sure how to tackle this, usually you need a covariance AND mean to generate a Gaussian random variable. I think sqrtm and chol could be useful. If you have access to the MATLAB statistics toolbox you can type edit mvnrnd in MATLAB to see their solution.