Covariance Matrix Applications The Uniersity of Sydney协方差矩阵的应用悉尼大学.pptVIP

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Covariance Matrix Applications The Uniersity of Sydney协方差矩阵的应用悉尼大学.ppt

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Covariance Matrix Applications The Uniersity of Sydney协方差矩阵的应用悉尼大学

* * * * * * * * * * * * * * * * * * * Covariance Matrix Applications Dimensionality Reduction 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) Example COV(R) 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 X1 X2 X3 X4 D A T A D A T A D A T A D A T A 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(Hi-?i)(Hj-?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 ?, x 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 u1,u2,?,ud Principal Component Analysis Also known as Singular Value Decomposition Latent Semant