# 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:

Principal component decomposition of RGB color channels:

## Scope(5)

Principal components of two grayscale images:

KarhunenLoeve decomposition of three matrix-valued datasets:

Principal components of a list of color images:

Specify the transformation matrix:

Use a transformation matrix and lesser datasets:

## Options(1)

### Standardized(1)

KarhunenLoeve decomposition with datasets shifted to mean zero:

## Applications(3)

Enhance the color contrast of an RGB image:

Reconstruct a multichannel image from 1, 2, or 3 components:

Transform a list of pictorial faces:

Show the residual images when using only the first three components:

## Properties & Relations(7)

The KarhunenLoeve decomposition preserves the total variance:

The KarhunenLoeve decomposition yields uncorrelated sets:

The KarhunenLoeve decomposition yields an orthogonal transformation matrix:

Relation to PrincipalComponents:

A setting is equivalent to subtracting the mean from the input data:

Normalizing by the square root of the number of datasets better preserves the input dynamics:

KarhunenLoeveDecomposition normally returns images of a real type: