# Wolfram Language & System 10.0 (2014)|Legacy Documentation

This is documentation for an earlier version of the Wolfram Language.
BUILT-IN WOLFRAM LANGUAGE SYMBOL

# KarhunenLoeveDecomposition

KarhunenLoeveDecomposition[{a1,a2,}]
gives the KarhunenLoeve transform of the numerical arrays , where .

KarhunenLoeveDecomposition[{b1,b2,},m]
uses the inverse of the matrix m for transforming to .

## Details and OptionsDetails 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 can be arbitrary rank arrays or images of the same dimensions.
• The inner product of m and gives .
• In KarhunenLoeveDecomposition[{a1,}], rows of the transformation matrix m are the eigenvectors of the covariance matrix formed from the arrays .
• The matrix m is a linear transformation of . The transformed arrays are uncorrelated, are given in order of decreasing variance, and have the same total variance as .
• KarhunenLoeveDecomposition[{b1,b2,},m] effectively computes the inverse KarhunenLoeve transformation. If the length of is less than the size of m, missing components are assumed to be zero.
• With an option setting "Centered"->True, KarhunenLoeveDecomposition[{a1,a2,}] shifts the datasets so that their means are zero.

## ExamplesExamplesopen allclose all

### Basic Examples  (2)Basic Examples  (2)

KarhunenLoeve decomposition of two datasets:

 Out[3]=

Principal component decomposition of RGB color channels:

 Out[1]=