Wednesday, May 17, 2017

correlation - How do I graphically represent the evolution of a covariance matrix over time?


I am working with a set of covariance matrices evaluated at various points in time over some history. Each covariance matrix is $N\times N$ for $N$ financial time-series over $T$ periods. I would like to explore some of the properties of this matrix's evolution over time, particularly whether correlation as a whole is increasing or decreasing, and whether certain series become more or less correlated with the whole. I am looking for suggestions as to the kinds of analysis to perform on this data-set, and particularly graphical/pictorial analysis. Ideally, I would like to avoid having to look in depth into each series as $N$ is rather large.


Update



The following graphs were generated based on the accepted answer from @Quant-Guy. PC = principal component = eigenvector. The analysis was done on correlations rather than covariances in order to account for vastly different variances of the $N$ series. Principal Component Angle with Previous Month Principal Component Angle with Present Principal Component Angle with Initial



Answer



I would consider a motion chart that plots the eigenvalues of the covariance matrix over time.


For a static view you can create a table: rows represent dates, and columns represent eigenvectors. The entries of the table represent changes in the angle of the eigenvector from the previous row. This will show how stable your covariance structure is.


You can also create a second table this time with eigenvalues as the columns sorted from high to low (and the corresponding values below for each date). This shows the variance described by each eigenvector so you can see whether correlation as whole is increasing or decreasing


Update: You can also measure the distance between the two covariance matrices via some distance measure metric such as Kullback-Leibler divergence, euclidean distance, Mahalanabois, etc.


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