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Cov

Cov

Cov

Compute the covariance matrix.

matrix matrix::Cov(
  const bool    rowvar=true  // rows or cols vectors of observations
);

matrix matrix::Cov(
  const bool    rowvar,      // rows or cols vectors of observations
  const int     ddof         // delta degrees of freedom
);

matrix vector::Cov(
  const vector& b            // second vector
);

matrix vector::Cov(
  const vector& b,           // second vector
  const int     ddof         // delta degrees of freedom
);

Parameters

rowwar

[in] If rowvar is true (default), then each row represents a variable, with observations in the columns. Otherwise, the relationship is transposed: each column represents a variable, while the rows contain observations.

b

[in] Second vector of observations.

ddof

[in] “Delta Degrees of Freedom”: the divisor used in the calculation is N - ddof, where N represents the number of elements. By default ddof is 1.

Note

Compute the covariance matrix.

A simple algorithm for calculating the covariance matrix of two vectors using MQL5:

bool VectorCovariation(const vector& vector_a,const vector& vector_b,matrix& matrix_c)
  {
   int i,j;
   int m=2;
   int n=(int)(vector_a.Size()<vector_b.Size()?vector_a.Size():vector_b.Size());
//--- checks
   if(n<=1)
      return(false);
   for(i=0; i<n; i++)
     {
      if(!MathIsValidNumber(vector_a[i]))
         return(false);
      if(!MathIsValidNumber(vector_b[i]))
         return(false);
     }
//---
   matrix matrix_x(2,n);
   matrix_x.Row(vector_a,0);
   matrix_x.Row(vector_b,1);
   vector t=vector::Zeros(m);
//--- calculation
   for(i=0; i<m; i++)
      for(j=0; j<n; j++)
         t[i]+=matrix_x[i][j]/double(n);
   for(i=0; i<m; i++)
      for(j=0; j<n; j++)
         matrix_x[i][j]-=t[i];
//--- syrk C=alpha*A^H*A+beta*C (beta=0 and not considered)
   matrix_c=matrix::Zeros(m,m);
   for(i=0; i<m; i++)
     {
      for(j=0; j<n; j++)
        {
         double v=matrix_x[i][j]/(n-1);
         for(int i_=i; i_<m; i_++)
            matrix_c[i][i_]+=v*matrix_x[i_][j];
        }
     }
//--- force symmetricity
   for(i=0; i<m-1; i++)
      for(j=i+1; j<m; j++)
         matrix_c[j][i]=matrix_c[i][j];
//---
   return(true);
  }

MQL5 example:

matrix matrix_a={{3,-2.1},{1.1,-1},{0.12,4.3}};
   Print("covariation cols\n",matrix_a.Cov(false));
   Print("covariation rows\n",matrix_a.Cov());

   vector vector_a=matrix_a.Col(0);
   vector vector_b=matrix_a.Col(1);
   Print("covariation vectors\n",vector_a.Cov(vector_b));

  /*
   covariation cols
   [[2.144133333333333,-4.286]
    [-4.286,11.71]]
   covariation rows
   [[13.005,5.355,-10.659]
    [5.355,2.205,-4.389]
    [-10.659,-4.389,8.736199999999998]]
   covariation vectors
   [[2.144133333333333,-4.286]
    [-4.286,11.71]]
  */

Python example:

import numpy as np
matrix_a=np.array([[3,-2.1],[1.1,-1],[0.12,4.3]])
matrix_c=np.cov(matrix_a,rowvar=False)
print("covariation cols\n",matrix_c)
matrix_c2=np.cov(matrix_a)
print("covariation rows\n",matrix_c2)

vector_a=matrix_a[:,0]
vector_b=matrix_a[:,1]
matrix_c3=np.cov(vector_a,vector_b)
print("covariation vectors\n",matrix_c3)

covariation cols
 [[ 2.14413333 -4.286     ]
 [-4.286      11.71      ]]
covariation rows
 [[ 13.005    5.355  -10.659 ]
 [  5.355    2.205   -4.389 ]
 [-10.659   -4.389    8.7362]]
covariation vectors
 [[ 2.14413333 -4.286     ]
 [-4.286      11.71      ]]
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