Statistics.Covariance Method

Statistics

Overload List

#SignatureDescription
1procedure Covariance(const X: TDenseMtxVec; const Y: TDenseMtxVec; const aResult: TMtx; NormN: Boolean);Calculate the variance-covariance matrix (Result), assuming vectors X and Y are two variable and their elements are the observations.
2procedure Covariance(const X: TMtx; const aResult: TMtx; NormN: Boolean);Calculate the covariance matrix (Result), assuming matrix X columns are variables and its rows are observations.
3procedure Covariance(const X: TVec; out aResult: Double; NormN: Boolean);Covariance/variance.

Overload 1: procedure Covariance(const X: TDenseMtxVec; const Y: TDenseMtxVec; const aResult: TMtx; NormN: Boolean);

Calculate the variance-covariance matrix (Result), assuming vectors X and Y are two variable and their elements are the observations.

#NameTypeDescription
1XTDenseMtxVec
2YTDenseMtxVec
3aResultTMtx
4NormNBoolean

Result: stored in self (calling object)

Remarks:

For column-vector valued random variables X and Y with respective expected values mu and nu, and respective scalar components m and n, the covariance is defined to be the m-by-n matrix called the covariance matrix:

Cov(X,Y)=E((X−μ)(Y−ν)T).\text{Cov}(X,Y)= E\left( (X-\mu) (Y-\nu)^T \right) .

Overload 2: procedure Covariance(const X: TMtx; const aResult: TMtx; NormN: Boolean);

Calculate the covariance matrix (Result), assuming matrix X columns are variables and its rows are observations.

#NameTypeDescription
1XTMtx
2aResultTMtx
3NormNBoolean

Result: stored in self (calling object)

Remarks:

By definition the covariance matrix is a matrix of covariances between elements of a vector. It is the natural generalization to higher dimensions of the concept of the variance of a scalar-valued random variable.

If X columns represent observation samples (variables), it's rows sample(s) values (observables), muj Xj j-th column average value, then the covariance matrix is defined as:

Σi,j=E((Xi−μi)(Xj−μj)).\Sigma_{i,j} = E\left((X_i-\mu_i)(X_j-\mu_j)\right) \qquad.

or in matrix form:

Σ=E((X−I⋅μ)T(X−I⋅μ)).\Sigma = E \left((X-I\cdot\mu)^T (X-I\cdot\mu)\right) \qquad.

where E is the expected value. The inverse of this matrix, is called the inverse covariance matrix or the precision matrix.

Note
This version does all necessary calculations to calculate covariance matrix.

Overload 3: procedure Covariance(const X: TVec; out aResult: Double; NormN: Boolean);

Covariance/variance.

#NameDescription
1XDefines sample (variable) values (observables). In this case X is treated as row and not (as normally) column vector.
2aResultReturns the covariance (in this case equal to variance) for X vector elements. Because in this case X is represented as row vectro, the the result is simply scalar value E(X(T)*X)-E(X(T))E(X) = Var(X).
3NormNIf true (default value), the result will be normalized with number of observations (N), otherwise it will be normalized with N-1.

Result: stored in self (calling object)

Remarks:

The covariance between two real-valued random variables x and y,with expected values E(x)=mu and E(y)=nu is defined as:

Cov(x,y)=E((x−μ)(x−ν))=E(x⋅y)−E(x)E(y).\text{Cov}(x,y)= E\left( (x-\mu) (x-\nu) \right) = E \left( x\cdot y \right) - E(x)E(y)\qquad .

where E(x), E(y) are x and y expected values.

For more info about covariance definition and properties check thd following links:

1. http://mathworld.wolfram.com/Covariance.html

2. http://en.wikipedia.org/wiki/Covariance

Examples
var Data1, Data2: Vector;
CovMtx : Matrix;
begin
    Data1.SetIt(false,[1.2,3]);
    Data2.SetIt(false,[5,5.5]);
    Covariance(Data1,Data2,CovMtx,False);
    // cov = [1.62, 0.45,
    //        0.45, 0.125]
end;