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**Covariance Matrix Applications**

Dimensionality Reduction

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**Outline What is the covariance matrix? Example**

Properties of the covariance matrix Spectral Decomposition Principal Component Analysis

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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).

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

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Example COV(R)

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**Moral: Covariance can only capture linear relationships**

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**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?

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**Hidden Sources of Variance**

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

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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.

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**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

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

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**Covariance Matrix Properties**

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

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**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.

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**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

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**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

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**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

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**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?

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

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**SVD Definition An N x M matrix X can be expressed as**

Lambda is a diagonal r x r matrix.

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**SVD Definition More importantly X can be written as**

Where the eigenvalues are in decreasing order. k,<r

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Example

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Compression Where k <=r <= M

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**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.

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