BUILT-IN MATHEMATICA SYMBOL

# KarhunenLoeveDecomposition

KarhunenLoeveDecomposition[{a1, a2, ...}]
gives the Karhunen-Loeve transform of the numerical arrays and the transformation matrix m, returning the result in the form .

KarhunenLoeveDecomposition[{a1, a2, ...}, m]
uses the inverse of the matrix m for transforming the .

## Details and OptionsDetails and Options

• KarhunenLoeveDecomposition works with lists of arbitrary numerical commensurate arrays.
• KarhunenLoeveDecomposition also works with arbitrary images.
• The inner product of m and gives .
• The total variance of the is the same as the total variance of the .
• The are given in order of decreasing variance.
• The rows of the transformation matrix m returned by KarhunenLoeveDecomposition are the eigenvectors of the covariance matrix formed from the arrays .
• KarhunenLoeveDecomposition[{a1, a2, ...}, m] effectively computes the inverse Karhunen-Loeve 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)

Karhunen-Loeve decomposition of two datasets:

 Out[1]=

Principal component decomposition of RGB color channels:

 Out[1]=

### Properties & Relations (5)Properties & Relations (5)

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