Principal Component Analysis & Singular Value Decomposition in Matrix Dimensionality Reduction & Covariance/Correlation Estimation

Introduction: Measuring and managing risk has been of greater concern to investors and fund managers especially after the financial crisis 2007. Many mathematical and statistical methods have been developed and improved to aim at providing a more accurate and better control over risk and efficient asset allocation. While financial professionals refer to risk as standard deviation of securities, yet often in real life investors hold a pool of assets in their portfolios. Therefore, risk is not simply measured as previously, but it involves the correlations between securities in the portfolio. For a large holding which contains hundreds or thousands of assets, the estimation and analysis of correlation matrix becomes harder and trickier as the matrix is much dominated by noise and becomes more random. Take an example of S&P500 whose 500 constituents are spread out among all nine industries, the correlation matrix of this portfolio will comprise of 125000 different entries (because the matrix is symmetric and therefore the number of total entries are halved to obtain 1 side of the matrix). With a large number of entries like this, most of the information in the correlation matrix might be of much noise rather than useful information. The need of a method to reduce the dimensionality of matrix to take into account real information thus comes into play.