# Documentation

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# cov

## Description

example

C = cov(A) returns the covariance.

• If A is a vector of observations, C is the scalar-valued variance.

• If A is a matrix whose columns represent random variables and whose rows represent observations, C is the covariance matrix with the corresponding column variances along the diagonal.

• C is normalized by the number of observations-1. If there is only one observation, it is normalized by 1.

• If A is a scalar, cov(A) returns 0. If A is an empty array, cov(A)returns NaN.

example

C = cov(A,B) returns the covariance between two random variables A and B.

• If A and B are vectors of observations with equal length, cov(A,B) is the 2-by-2 covariance matrix.

• If A and B are matrices of observations, cov(A,B) treats A and B as vectors and is equivalent to cov(A(:),B(:)). A and B must have equal size.

• If A and B are scalars, cov(A,B) returns a 2-by-2 block of zeros. If A and B are empty arrays, cov(A,B) returns a 2-by-2 block of NaN.

example

C = cov(___,w) specifies the normalization weight for any of the previous syntaxes. When w = 0 (default), C is normalized by the number of observations-1. When w = 1, it is normalized by the number of observations.

example

C = cov(___,nanflag) specifies a condition for omitting NaN values from the calculation for any of the previous syntaxes. For example, cov(A,'omitrows') will omit any rows of A with one or more NaN elements.

## Examples

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Create a 3-by-4 matrix and compute its covariance.

A = [5 0 3 7; 1 -5 7 3; 4 9 8 10];
C = cov(A)
C =

4.3333    8.8333   -3.0000    5.6667
8.8333   50.3333    6.5000   24.1667
-3.0000    6.5000    7.0000    1.0000
5.6667   24.1667    1.0000   12.3333

Since the number of columns of A is 4, the result is a 4-by-4 matrix.

Create two vectors and compute their 2-by-2 covariance matrix.

A = [3 6 4];
B = [7 12 -9];
cov(A,B)
ans =

2.3333    6.8333
6.8333  120.3333

Create two matrices of the same size and compute their 2-by-2 covariance.

A = [2 0 -9; 3 4 1];
B = [5 2 6; -4 4 9];
cov(A,B)
ans =

22.1667   -6.9333
-6.9333   19.4667

Create a matrix and compute the covariance normalized by the number of rows.

A = [1 3 -7; 3 9 2; -5 4 6];
C = cov(A,1)
C =

11.5556    5.1111  -10.2222
5.1111    6.8889    5.2222
-10.2222    5.2222   29.5556

Create a matrix and compute its covariance, excluding any rows containing NaN values.

A = [1.77 -0.005 3.98; NaN -2.95 NaN; 2.54 0.19 1.01]
A =

1.7700   -0.0050    3.9800
NaN   -2.9500       NaN
2.5400    0.1900    1.0100

C = cov(A,'omitrows')
C =

0.2964    0.0751   -1.1435
0.0751    0.0190   -0.2896
-1.1435   -0.2896    4.4104

## Input Arguments

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Input array, specified as a vector or matrix.

Data Types: single | double

Additional input matrix, specified as a vector or matrix. B must be the same size as A.

Data Types: single | double

Normalization weight, specified as one of these values:

• 0 — The output is normalized by the number of observations-1. If there is only one observation, it is normalized by 1.

• 1 — The output is normalized by the number of observations.

Data Types: single | double

NaN condition, specified as one of these values:

• 'includenan' — include all NaN values in the input prior to computing the covariance.

• 'omitrows' — omit any row of input containing one or more NaN values prior to computing the covariance.

• 'partialrows' — omit rows containing NaN only on a pairwise basis for each two-column covariance calculation.

Data Types: char

## Output Arguments

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Covariance, specified as a scalar or matrix.

• For single matrix input, C has size [size(A,2) size(A,2)] based on the number of random variables (columns) represented by A. The variances of the columns are along the diagonal. If A is a row or column vector, C is the scalar-valued variance.

• For two-vector or two-matrix input, C is the 2-by-2 covariance matrix between the two random variables. The variances are along the diagonal of C.

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### Covariance

For two random variable vectors A and B, the covariance is defined as

$\mathrm{cov}\left(A,B\right)=\frac{1}{N-1}\sum _{i=1}^{N}{\left({A}_{i}-{\mu }_{A}\right)}^{*}\left({B}_{i}-{\mu }_{B}\right)$

where μA is the mean of A, μB is the mean of B, and * denotes the complex conjugate.

The covariance matrix of two random variables is the matrix of pairwise covariance calculations between each variable,

$C=\left(\begin{array}{cc}\mathrm{cov}\left(A,A\right)& \mathrm{cov}\left(A,B\right)\\ \mathrm{cov}\left(B,A\right)& \mathrm{cov}\left(B,B\right)\end{array}\right).$

For a matrix A whose columns are each a random variable made up of observations, the covariance matrix is the pairwise covariance calculation between each column combination. In other words, $C\left(i,j\right)=\mathrm{cov}\left(A\left(:,i\right),A\left(:,j\right)\right)$.

### Variance

For a random variable vector A made up of N scalar observations, the variance is defined as

$V=\frac{1}{N-1}\sum _{i=1}^{N}|{A}_{i}-\mu {|}^{2}$

where μ is the mean of A,

$\mu =\frac{1}{N}\sum _{i=1}^{N}{A}_{i}.$

Some definitions of variance use a normalization factor of N instead of N-1, which can be specified by setting w to 1. In either case, the mean is assumed to have the usual normalization factor N.

### Tall Array Support

This function supports tall arrays with the limitations:

• For the syntax C = cov(X,Y), the inputs X and Y must have the same size, even if they are vectors.

• The option 'partialrows' is not supported.