cov

Covariance

Introduction

Covariance can be used to measure the linear relationship between two random variables.

For the real random variables $X$ and $Y$, their covariance is defined as

where $\mu_X$ and $\mu_Y$ are the means of $X$ and $Y$, respectively. A positive $\sigma_{XY}$ indicates that samples of $(X,Y)$ tend to lie on a line with positive slope. Whereas, a negative $\sigma_{XY}$ indicates that samples of $(X,Y)$ tend to lie on a line with negative slope. It is obvious that $\sigma_{XY}=\sigma_{YX}$ for real $X$ and $Y$.

If either $X$ or $Y$ is complex, the covariance is defined as

where $Y^*$ denotes the complex conjugate of $Y$. Note that $\sigma_{XY}$ can be complex and $\sigma_{XY}=\sigma_{YX}^*$.

Usage

C = cov(A)

C = cov(A, w)

C = cov(A, nanflag)

C = cov(A, w, nanflag)

• A should be an array of real or complex numbers with ndims(A) == 2.

• If A is empty, it returns NaN.

• If A is scalar, it returns 0.

• If A is a vector, it returns the variance of A.

• If A is a matrix, each column of A should contain observations of a random variable. Then, C is the covariance matrix where

  • C(i,j) is the covarance of the $i$-th and $j$-th columns if $i\neq j$.
  • C(i,i) is the variance of the $i$-th column of A.

• w should be either 0 or 1.

  • If w == 0 (default), variance or covariance is normalised by length(A)-1.
  • If w == 1, variance or covariance is normalised by length(A).

• nanflag should be either 'includenan', 'omitrows' or 'partialrows'.

  • 'includenan' (default): all NaN values are included in the computation of C.
  • 'omitrows': When any row of A contains NaN, the whole row of A will be removed before computing any element of C. In other words, the number of rows of A will be reduced before computing C. (See Example 2 and Example 4 below.) The output C is NaN if the removal of NaN results in an empty matrix.
  • 'partialrows': When computing covariance C(i,j), the $i$-th and $j$-th columns are examined. Any NaN appearing in one of columns and the corresponding element in the other column will be removed. (See Example 2 and Example 4 below.) The covariance C(i,j) is NaN if the removal of NaN results in two empty columns.

C = cov(A, B)

C = cov(A, B, w)

C = cov(A, B, nanflag)

C = cov(A, B, w, nanflag)

• A and B should be real or complex array with identical sizes.

• If A or B is empty, it returns NaN.

• If A or B is a scalar, it returns 0.

• If A and B are vectors, they contains observations of two random variables. The output C is a $2\times 2$ covariance matrix of A and B.

  • 💡 If ndims(A)>=3 or ndims(B)>=3, the matrix will be reshaped to a vector.

• w should be either 0 (default) or 1.

  • If w == 0, variance or covariance is normalised by length(A)-1.
  • If w == 1, variance or covariance is normalised by length(A).

• nanflag should be either 'includenan', 'omitrows' or 'partialrows'.

  • 'includenan' (default): all NaN values are included in the computation of C.
  • 'omitrows': When the $i$-th element of A or B contains NaN, the $i$-th elements of both A and B will be removed before computing any element of C. (See Example 3 below.) The output C is NaN if the removal of NaN results in empty arrays.
  • 'partialrows': This is different from omitrows in such a way that the removal of NaN is deferred to the stage when individual elements of C is computed. (See See Example 3 below.)

Examples

Example 1

This example shows the covariance of $X$ and $Y$, where $Y=1+2X+\epsilon$ with noise $\epsilon\sim N(0,1)$ following the standard normal distribution. The variables $X$ and $Y$ show a strong linear relationship and $\sigma_{XY}\approx 16.788$ as shown in the output below.

Input

clear
x=sort(10*rand(1,100));
% y = 1 + 2x + noise.
y=1+2*x+randn(size(x));
plot(x,1+2*x);
hold('on')
scatter(x,y);
hold('off')
% Covariance of x and y is C(1,2) or C(2,1)
C=cov(x,y)

Output

ans = 
 8.5876   16.788
 16.788   33.679

Example 2

This example demonstrates the difference between cov(a,'omitrows')and cov(a,'partialrows').

• When 'omitrows' is used, NaN elements in a(:,1) and the corresponding elements in a(:,2) are removed before the computation of C1. Therefore, when C1(2,2) is computed, the shortened vector [11 12 13 14 16 17 19 20] is used although a(:,2) has no NaN elements.
• When 'partialrows' is used, the removal of NaN is performed when individual elements of C2 is computed. Therefore, the computation of C2(2,2) uses the whole vector [11 12 13 14 15 16 17 18 19 20].

Input

% Matrix of two columns
a=[
1 11;
2 12;
3 13;
4 14;
NaN 15;
6 16;
7 17;
NaN 18;
9 19;
10 20];
% Use 'omitrows'
C1=cov(a,'omitrows')
% Use 'partialrows'
C2=cov(a,'partialrows')

Output

C1 = 
 10.786   10.786
 10.786   10.786

C2 = 
 10.786   10.786
 10.786   9.1667

Example 3

This example demonstrates the difference of cov(a,b,'omitrows') and cov(a,b,'partialrows').

• When 'omitrows' is used, NaN elements in a and the corresponding elements in b are removed before the computation of C1. Therefore, when C1(2,2) is computed, the shortened vector [11 12 13 14 16 17 19 20] is used although b has no NaN elements.
• When 'partialrows' is used, the removal of NaN is performed when individual elements of C2 is computed. Therefore, the computation of C2(2,2) uses the whole vector [11 12 13 14 15 16 17 18 19 20].

Input

% Vector with some NaN elements.
a=[1 2 3 4 nan 6 7 nan 9 10];
% Another vector
b=[11 12 13 14 15 16 17 18 19 20];
% Use 'omitrows'
C1=cov(a,b,'omitrows')
% Use 'partialrows'
C2=cov(a,b,'partialrows')

Output

C1 = 
 10.786   10.786
 10.786   10.786

C2 = 
 10.786   10.786
 10.786   9.1667

Example 4

In this example, the diagonal elements of a are all NaN.

• When includenan is used, all of the computed covariances are NaN.
• When omitrows is used, all rows of a are removed, and hence a becomes empty, when obtaining the covariances. As a result, the output covariances are all NaN.
• When paritalrows is used, only some of the rows are removed. For example, when the covariance of a(:,1) and a(:,2) are computed, only the first two rows are removed. Hence, the covariance obtained is not NaN.

Input

% 5x5 random matrix with NaN diagonal elements.
a=rand(5,5);
a(1,1)=nan;
a(2,2)=nan;
a(3,3)=nan;
a(4,4)=nan;
a(5,5)=nan;
a
% Computation includes NaN. The output contains all NaN.
cov(a)
cov(a,'includenan')
% Matrix a becomes an empty matrix,
% since all rows are removed before
% computing the variances.
cov(a,'omitrows')
% Only some elements are removed.
cov(a,'partialrows')

Output

a = 1e-1 × 
 NaN      6.0066   8.1434   6.5663   8.6598
 0.4499   NaN      9.3268   6.3138   3.7444
 1.0603   0.9255   NaN      6.0063   2.4672
 0.9516   4.9147   2.5800   NaN      9.7731
 7.2988   2.6474   8.3618   7.2957   NaN   

ans = 
 NaN      NaN      NaN      NaN      NaN   
 NaN      NaN      NaN      NaN      NaN   
 NaN      NaN      NaN      NaN      NaN   
 NaN      NaN      NaN      NaN      NaN   
 NaN      NaN      NaN      NaN      NaN   

ans = 
 NaN      NaN      NaN      NaN      NaN   
 NaN      NaN      NaN      NaN      NaN   
 NaN      NaN      NaN      NaN      NaN   
 NaN      NaN      NaN      NaN      NaN   
 NaN      NaN      NaN      NaN      NaN   

ans = 
 NaN      NaN      NaN      NaN      NaN   
 NaN      NaN      NaN      NaN      NaN   
 NaN      NaN      NaN      NaN      NaN   
 NaN      NaN      NaN      NaN      NaN   
 NaN      NaN      NaN      NaN      NaN   

ans = 1e-1 × 
 1.0562  -0.0680   0.4451   0.2430   0.0242
-0.0680   0.5193  -0.1295   0.0436   0.9899
 0.4451  -0.1295   0.9357  -0.0181  -0.8781
 0.2430   0.0436  -0.0181   0.0303   0.0850
 0.0242   0.9899  -0.8781   0.0850   1.2925