cov

Covariance

Syntax

C = cov(A)

C = cov(A,B)

C = cov(___,w)

C = cov(___,nanflag)

Description

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.

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.

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.

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

Covariance of Matrix

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 = 4×4

    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.

Covariance of Two Vectors

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

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

ans = 2×2

    2.3333    6.8333
    6.8333  120.3333

Covariance of Two Matrices

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 = 2×2

   22.1667   -6.9333
   -6.9333   19.4667

Specify Normalization Weight

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 = 3×3

   11.5556    5.1111  -10.2222
    5.1111    6.8889    5.2222
  -10.2222    5.2222   29.5556

Covariance Excluding NaN

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 = 3×3

    1.7700   -0.0050    3.9800
       NaN   -2.9500       NaN
    2.5400    0.1900    1.0100

C = cov(A,'omitrows')

C = 3×3

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

Input Arguments

A — Input array

vector | matrix

Input array, specified as a vector or matrix.

Data Types: single | double

B — Additional input array

vector | matrix

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

Data Types: single | double

w — Normalization weight

0 (default) | 1

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

nanflag — NaN condition

'includenan' (default) | 'omitrows' | 'partialrows'

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

C — Covariance

scalar | matrix

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.

More About

Covariance

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

$$\mathrm{cov}(A,B)=\frac{1}{N-1}{\displaystyle \sum _{i=1}^N{(A_i-{\mu }_A)}^{*}(B_i-{\mu }_B)}$$

where μ_A is the mean of A, μ_B 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,

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

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, $$C(i,j)=\mathrm{cov}\left(A(:,i),A(:,j)\right)$$.

Variance

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

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

where μ is the mean of A,

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

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.

Extended Capabilities

Tall Arrays

Calculate with arrays that have more rows than fit in memory.

This function supports tall arrays with the limitations:

• A and B must be tall arrays of the same size, even if both are vectors.

• The 'partialrows' option is not supported.

For more information, see Tall Arrays for Out-of-Memory Data.

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• If the input is variable-size and is [] at run time, returns [] not NaN.

• See Variable-Sizing Restrictions for Code Generation of Toolbox Functions (MATLAB Coder).

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Distributed Arrays

Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.

This function fully supports distributed arrays. For more information, see Run MATLAB Functions with Distributed Arrays (Parallel Computing Toolbox).