cov

Parameters

x

a nobs-by-1 or nobs-by-n matrix of doubles

y

a nobs-by-1 or nobs-by-m matrix of doubles

C

a square matrix of doubles, the empirical covariance or cross-covariance

Description

If x is a nobs-by-1 matrix, then cov(x) returns the sample variance of x, normalized by nobs-1.

If x is a nobs-by-n matrix, then cov(x) returns the n-by-n sample covariance matrix of the columns of x, normalized by nobs-1. Here, each column of x is a variable among (1 ... n) and each row of x is an observation.

If x and y are two nobs-by-1 matrices, then cov(x, y) returns the 2-by-2 sample covariance matrix of x and y, normalized by nobs-1, where nobs is the number of observations.

If x and y are respectively a nobs-by-n and a nobs-by-m matrix then cov(x, y) returns the n-by-m sample cross-covariance matrix of x and y, normalized by nobs-1, where nobs is the number of observations.

cov(x, 0) is the same as cov(x) and cov(x, y, 0) is the same as cov(x, y). In this case, if the population is from a normal distribution, then C is the best unbiased estimate of the covariance matrix or cross-covariance matrix.

cov(x, 1) and cov(x, y, 1) normalize by nobs. In this case, C is the second moment matrix of the observations about their mean.

The covariance of two random vectors X and Y is defined by:

Cov(X,Y) = E[ (X-E(X)).(Y-E(Y))^t]

where E is the expectation.

Examples

x = [1; 2];
y = [3; 4];
C = cov(x, y)
expected = [0.5, 0.5; 0.5, 0.5];
C = cov([x, y])

x = [230; 181; 165; 150; 97; 192; 181; 189; 172; 170];
y = [125; 99; 97; 115; 120; 100; 80; 90; 95; 125];
expected = [
1152.4556, -88.911111
-88.911111, 244.26667
];
C = cov(x, y)
C = cov([x, y])

// Source [3]
A = [
4.0 2.0 0.60
4.2 2.1 0.59
3.9 2.0 0.58
4.3 2.1 0.62
4.1 2.2 0.63
];
S = [
0.025 0.0075 0.00175
0.0075 0.007 0.00135
0.00175 0.00135 0.00043
];
C = cov(A)

x = rand(10,3);
y = sum(x,2);
C1 = cov(x,y)

// sample cross-covariance matrix is a submatrix of sample covariance matrix
// of concatenated [x,y]

C1 = cov([x,y])(1:3,4)

Bibliography

Wikipedia: Covariance matrix Wikipedia: Cross-covariance matrix

[3] NIST/SEMATECH e-Handbook of Statistical Methods, 6.5.4.1. Mean Vector and Covariance Matrix

"Introduction to probability and statistics for engineers and scientists", Sheldon Ross

History