How to Calculate Covariance in MATLAB

Covariance is the measure of the strength of correlation between two or more random variables. Covariance of two random variables X and Y can be defined as:

{\displaystyle \operatorname {cov} (X,Y)={\frac {1}{n}}\sum _{i=1}^{n}(x_{i}-E(X))(y_{i}-E(Y))}

Where E(X) and E(Y) are expectation or mean of random variables X and Y respectively.

The covariance matrix of two random variables A and B is defined as

{\displaystyle \mathbf {c } ={\begin{bmatrix}\operatorname {cov(A,A)}&\operatorname {cov(A,B)}\\\operatorname {cov(B,A) }&\operatorname {cov(B,B) }\end{bmatrix}}}

MATLAB language allows users to calculate the covariance of random variables using cov() method. Different syntax of cov() method are:

1. C = cov(A)
2. C = cov(A,B)
3. C = cov(___,w)
4. C = cov(___,nanflag)

C = cov(A)

• It returns the covariance of array A.
• If A is a scalar, then it returns 0.
• If A is a vector, then it returns the variance of vector A.

                             {\displaystyle \operatorname {Var} (X)={\frac {1}{n}}\sum _{i=1}^{n}(x_{i}-\mu )^{2}} \\where \; \mu \; is \; the \; mean\;.         

• If A is a matrix, then it considers each column as a random variable and returns the covariance matrix of matrix A.

Note: disp (x) displays the value of variable X without printing the variable name. Another way to display a variable is to type its name, which displays a leading “X =” before the value. If a variable contains an empty array, disp returns without displaying anything.

Example 1:

% Input vector
A = [1 3 4];
disp("Vector :");
disp(A);

% Variance of vector A
C = cov(A);
disp("Variance :");
disp(C);

Output :

Example 2:

% Input vector
A = [2 7 1;
     3 5 1
     4 1 2];
disp("Matrix :");
disp(A);

% Covariance of matrix A
C = cov(A);
disp("Covariance matrix :");
disp(C);

Output :

C = cov(A,B)

• It returns the covariance matrix of arrays A and B.
• If A and B vectors, then it returns the covariance matrix of A and B.
• If A and B are matrices, then it considers them as vectors themselves by expanding the dimensions and returns the covariance matrix.

Example:

% Input vector
A = [3 5 7];
B = [-1 3 9];
disp("Vector A:");
disp(A);
disp("Vector B:");
disp(B);

% Covariance of vectors A,B
C = cov(A,B);
disp("Covariance matrix :");
disp(C);

Output :

C = cov(___,w)

• It returns the covariance of the input array by normalizing it to w.
• If w = 1, then covariance is normalized by the number of rows in the input matrix.
• If w = 0, then covariance is normalized by the number of rows in the input matrix - 1.

Example:

% Input vector
A = [2 4 6;
     3 5 7
     8 10 12];
disp("Matrix :");
disp(A);

% Variance of matrix A
C = cov(A,1);
disp("Variance matrix:");
disp(C);

Output :

C = cov(___,nanflag)

• It returns the covariance of the input array by considering the nanflag.
• If nanflag = 'includenan', then it considers NaN values in array.
• If nanflag = 'omitrows', then it omits the rows with at least one NaN value in the array.

Example:

% Input vector
A = [3.2 -1.005 2.98;
     NaN  -6.75  NaN; 
     5.37  0.19  1]
disp("Matrix :");
disp(A);

% Variance of matrix A
C = cov(A,'includenan');
disp("Variance matrix including NaN:");
disp(C);


% Variance of matrix A
C = cov(A,'omitrows');
disp("Variance matrix omitting NaN:");
disp(C);

Output :