# KarhunenLoeveDecomposition

KarhunenLoeveDecomposition[{a1,a2,}]

gives the KarhunenLoeve transform {{b1,b2,},m} of the numerical arrays {a1,a2,}, where m.aibi.

KarhunenLoeveDecomposition[{b1,b2,},m]

uses the inverse of the matrix m for transforming bi to ai.

# Details and Options • KarhunenLoeve decomposition is typically used to reduce the dimensionality of data and capture the most important variation in the first few components.
• The ai can be arbitrary rank arrays or images of the same dimensions.
• The inner product of m and {a1,a2,} gives {b1,b2,}.
• In KarhunenLoeveDecomposition[{a1,}], rows of the transformation matrix m are the eigenvectors of the covariance matrix formed from the arrays ai.
• The matrix m is a linear transformation of ai. The transformed arrays bi are uncorrelated, are given in order of decreasing variance, and have the same total variance as ai.
• KarhunenLoeveDecomposition[{b1,b2,},m] effectively computes the inverse KarhunenLoeve transformation. If the length of {b1,b2,} is less than the size of m, missing components are assumed to be zero.
• With an option setting , datasets ai are shifted so that their means are zero.

# Examples

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## Basic Examples(2)

KarhunenLoeve decomposition of two datasets:

 In:= Out= Principal component decomposition of RGB color channels:

 In:= Out= ## Properties & Relations(7)

Introduced in 2010
(8.0)
|
Updated in 2015
(10.1)