Correlation/Covariance Matrix

Definition
Correlation is a statistical relationship between two random variables. The correlation matrix of nnn
 random variables X1,…,XnX_1, \ldots, X_nX1​,…,Xn​
 is the n×nn \times nn×n
 matrix whose (i,j)(i,j)(i,j)
 entry is corr(Xi,Xj)corr(X_i,X_j)corr(Xi​,Xj​)
. Thus the diagonal entries are all identically unity, i.e. 1.
Covariance is a measure of the joint variability of two random variables, in other words, the directional relationship. The covariance matrix of nnn
 random variables X1,…,XnX_1, \ldots, X_nX1​,…,Xn​
 is the n×nn \times nn×n
 matrix whose (i,j)(i,j)(i,j)
 entry is cov(Xi,Xj)cov(X_i,X_j)cov(Xi​,Xj​)
. Thus the diagonal entries are variance of XiX_iXi​
.
Calculation
The most common measure of the correlation is the Pearson's correlation coefficient. It is obtained by taking the ratio of the covariance of the two variables to the square root of their variances.
ρX,Y=corr(X,Y)=cov(X,Y)σXσY=E[(X−μX)(Y−μY)]σXσY\rho_{X,Y} =corr(X,Y) = {cov(X,Y) \over {\sigma_X \sigma_Y}}={E[(X-\mu_X)(Y-\mu_Y)] \over {\sigma_X \sigma_Y}}ρX,Y​=corr(X,Y)=σX​σY​cov(X,Y)​=σX​σY​E[(X−μX​)(Y−μY​)]​
Correlation matrix is 
R=[ρ11⋯ρ1n⋮⋮ρn1⋯ρnn]R=\begin{bmatrix}
\rho_{11} & \cdots & \rho_{1n}\\ 
\vdots &  & \vdots \\ 
\rho_{n1} & \cdots  & \rho_{nn}
\end{bmatrix}R=⎣⎡​ρ11​⋮ρn1​​⋯⋯​ρ1n​⋮ρnn​​⎦⎤​
Covariance matrix is
V=[σ11⋯σ1n⋮⋮σn1⋯σnn]V = \begin{bmatrix}
\sigma_{11} & \cdots & \sigma_{1n}\\ 
\vdots &  & \vdots \\ 
\sigma_{n1} & \cdots  & \sigma_{nn}
\end{bmatrix}V=⎣⎡​σ11​⋮σn1​​⋯⋯​σ1n​⋮σnn​​⎦⎤​
The conversion between them is
D=Diag(σi)V=D⋅R⋅DR=inv(D)⋅V⋅inv(D)\begin{align*}
D &= Diag(\sigma_i) \\
V &= D \cdot R \cdot D \\
R &= inv(D) \cdot V \cdot inv(D)
\end{align*}DVR​=Diag(σi​)=D⋅R⋅D=inv(D)⋅V⋅inv(D)​
Fixing Correlation / Covariance matrix
The correlation / covariance matrix is calculated using historical data and pairwise, and then put into the form of a symmetric matrix. For jointly normal random variables, this single number summarizes all there is to know about the relationship between the two.
In ideal cases, where observations are complete, we can produce the correlation / covariance matrix with by using the observed sample. However, the real world data poses several challenges. 
Data might not exist till recent one year, or
poor quality when contract was still undergoing acceptance by the communication, or
outliers due to market stress event.
There is no guarantee that this matrix satisfies that it is a positive definite matrix to enable decomposition. We may employ below procedure:
1.Try eigen decomposition of the matrix, setting any negative eigenvalues to zero, and then reconstructing the resulting matrix.
2.Update the "fixed" matrix such that it is the closest to the original matrix ([1], [2]).
Reference
[1] Nicholas J. Higham https://nhigham.com/2013/02/13/the-nearest-correlation-matrix/​
[2] Craig Lucas, Computing Nearest Covariance and Correlation Matrices, M.Sc. Thesis, University of Manchester, 2001.
Contributors
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kunlun#8324
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Original version

Discord Handle	ETH Address	Reward	Contribution
kunlun#8324	0x109B3C39d675A2FF16354E116d080B94d238a7c9	0 $CMK (internal)	Original version

Last modified 1yr ago