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22%20Principle%20Component%20Analysis%204_17_08

# 22%20Principle%20Component%20Analysis%204_17_08 - Principle...

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1 1 Principle Component Analysis Peng Liu 4/17/2008 2 Principle Component Analysis PCA is concerned with explaining the variance- covariance structure. The general objectives of PCA are: Data reduction Interpretation 3 Principal Components Principal components can be useful for providing low-dimensional views of high-dimensional data. 1 2 ... m 1 2 X = . . . n Data Matrix or Data Set x 11 x 12 . . . x 1m x 21 . . . x n1 x 2m . . . x nm x n2 . . . object variable number of variables number of observations 4 Principal Components (continued) Each principal component of a data set is a variable obtained by taking a linear combination of the original variables in the data set. A linear combination of m variables x 1 , x 2 , ..., x m is given by c 1 x 1 + c 2 x 2 + + c m x m . For the purpose of constructing principal components, the vector of coefficients is restricted to have unit length, i.e., c 1 + c 2 + + c m = 1. ... ... 2 2 2

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2 5 Principal Components (continued) The first principal component is the linear combination of the variables that has maximum variation across the observations in the data set.
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22%20Principle%20Component%20Analysis%204_17_08 - Principle...

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