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cov

Covariance matrix

Calling Sequence

C=cov(x)
C=cov(x,0)
C=cov(x,1)
C=cov(x,y)
C=cov(x,y,0)
C=cov(x,y,1)

Parameters

x:

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

y:

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

C:

a square matrix of doubles, the empirical covariance

Description

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

If x is a nobs-by-nvar matrix, then cov(x) returns the nvar-by-nvar covariance matrix of the columns of x, normalized by nobs-1. Here, each column of x is a variable 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 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.

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 X and Y is defined by

where E is the expectation.

This function is compatible with Matlab.

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)

Authors

Bibliography

[1] http://en.wikipedia.org/wiki/Covariance_matrix

[2] "Introduction to probability and statistics for engineers and scientists.", Sheldon Ross

[3] NIST/SEMATECH e-Handbook of Statistical Methods, 6.5.4.1. Mean Vector and Covariance Matrix, http://www.itl.nist.gov/div898/handbook/pmc/section5/pmc541.htm


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