Outline What is the covariance matrix? Example Properties of the covariance matrix Spectral Decomposition –Principal Component Analysis
Covariance Matrix Covariance matrix captures the variance and linear correlation in multivariate/multidimensional data. If data is an N x D matrix, the Covariance Matrix is a d x d square matrix.Think of N as the number of data instances (rows) and D the number of attributes (columns).
Covariance Formula Let Data = N x D matrix. The Cov(Data)
Moral: Covariance can only capture linear relationships
Dimensionality Reduction If you work in “data analytics” it is common these days to be handed a data set which has lots of variables (dimensions). The information in these variables is often redundant – there are only a few sources of genuine information. Question: How can be identify these sources automatically?
Hidden Sources of Variance X1 X2 X3 X4 H2 H1 X1X2X3X4 DATA DATA DATA DATA Model: Hidden Sources are Linear Combinations of Original Variables
Hidden Sources If the information that the known variables provided was different then the covariance matrix between the variables should be a diagonal matrix – i.e, the non-zero entries only appear on the diagonal. In particular, if Hi and Hj are independent then E(H i - i )(H j - j )=0.
Hidden Sources So the question is what should be the hidden sources. It turns out that the “best” hidden sources are the eigenvectors of the covariance matrix. If A is a d x d matrix, then is an eigenvalue-eigenvector pair if Ax = x
Explanation We have two axis, X1 and X2. We want to project the data along the direction of maximum variance. a
Covariance Matrix Properties The Covariance matrix is symmetric. Non-negative eigenvalues. –0 · 1 · 2 d Corresponding eigenvectors –u 1,u 2, ,u d
Principal Component Analysis Also known as –Singular Value Decomposition –Latent Semantic Indexing Technique for data reduction. Essentially reduce the number of columns while losing minimal information Also think in terms of lossy compression.
Motivation Bulk of data has a time component For example, retail transactions, stock prices Data set can be organized as N x M table N customers and the price of the calls they made in 365 days M << N
Objective Compress the data matrix X into Xc, such that –The compression ratio is high and the average error between the original and the compressed matrix is low –N could be in the order of millions and M in the order of hundreds
Example database We 7/10 Thr 7/11 Fri 7/12 Sat 7/13 Sun 7/14 ABC11100 DEF22200 GHI11100 KLM55500 smit h john00033 tom00011
Decision Support Queries What was the amount of sales to GHI on July 11? Find the total sales to business customers for the week ending July 12th?
Intuition behind SVD x y x’ y’ Customer are 2-D points
SVD Definition An N x M matrix X can be expressed as Lambda is a diagonal r x r matrix.
SVD Definition More importantly X can be written as Where the eigenvalues are in decreasing order. k,
Compression Where k <=r <= M
Explanation Let X be a mean-centered N x d matrix. Let a be an arbitrary d x 1 unit vector (initially). The projection of X onto a is given by Xa We want to maximize the variance of Xa. The constraint is that a T a = 1 It can be shown that a is given by the solution of the equation (X T X - I)a = 0 In other words a is the eigenvector of the covariance matrix and the is the eigenvalue.