1. Covariance Matrix Applications
Dimensionality Reduction

2. Outline What is the covariance matrix? Example
Properties of the covariance matrix
Spectral Decomposition
Principal Component Analysis

3. 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).

4. Covariance Formula
Let Data = N x D matrix.
The Cov(Data)

5. Example
COV(R)

6. Moral: Covariance can only capture linear relationships

7. 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?

8. Hidden Sources of VarianceX1 X2 H1 X1 X2 X3 X4 D A T X3 H2 X4
Model: Hidden Sources are Linear Combinations of Original Variables

9. 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(Hi-i)(Hj-j)=0.

10. 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 <, x> is an eigenvalue-eigenvector pair if
Ax =  x

11. Explanation a
We have two axis, X1 and X2. We want to project the data along the direction
of maximum variance.

12. Covariance Matrix Properties
The Covariance matrix is symmetric.
Non-negative eigenvalues.
0 · 1 · 2  d
Corresponding eigenvectors
u1,u2,,ud

13. 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.

14. 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

15. 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

16. Example database We 7/10 Thr 7/11 Fri 7/12 Sat 7/13 Sun 7/14 ABC 1 DEF
DEF 2
GHI
KLM 5
smith
john 3
tom

17. 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?

18. Intuition behind SVD y x’ y’ x
Customer are 2-D points

19. SVD Definition An N x M matrix X can be expressed as
Lambda is a diagonal r x r matrix.

20. SVD Definition More importantly X can be written as
Where the eigenvalues are in decreasing order.
k,<r

21. Example

22. Compression
Where k <=r <= M

23. 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 aTa = 1
It can be shown that a is given by the solution of the equation (XTX -  I)a = 0
In other words a is the eigenvector of the covariance matrix and the  is the eigenvalue.