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KarhunenLoeveDecomposition
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KarhunenLoeveDecomposition

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gives the Karhunen–Loeve transform {{b1,b2,…},m} of the numerical arrays {a1,a2,…}, where m.aibi.

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

Details and Options

  • Karhunen–Loeve 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 Karhunen–Loeve 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 StandardizedTrue, datasets ai are shifted so that their means are zero.

Examples

open allclose all

Basic Examples  (2)Summary of the most common use cases

Karhunen–Loeve decomposition of two datasets:

In[3]:=3
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https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-knb8up
Out[3]=3

Principal component decomposition of RGB color channels:

In[1]:=1
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https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-cistzr
Out[1]=1

Scope  (5)Survey of the scope of standard use cases

Principal components of two grayscale images:

In[1]:=1
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https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-j1ll2v
Out[1]=1

Karhunen–Loeve decomposition of three matrix-valued datasets:

In[1]:=1
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https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-pyroz
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-32j1v7
Out[2]=2

Principal components of a list of color images:

In[1]:=1
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https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-ouqstd
Out[1]=1

Specify the transformation matrix:

In[1]:=1
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https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-zsnmgd
Out[1]=1

Use a transformation matrix and lesser datasets:

In[1]:=1
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https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-prlgd3
Out[1]=1

Options  (1)Common values & functionality for each option

Standardized  (1)

Karhunen–Loeve decomposition with datasets shifted to mean zero:

In[1]:=1
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https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-9aifu2
Out[1]=1

Applications  (3)Sample problems that can be solved with this function

Enhance the color contrast of an RGB image:

In[1]:=1
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https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-c1h72f
Out[2]=2

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

In[1]:=1
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https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-iqpvla
Out[1]=1

Transform a list of pictorial faces:

In[1]:=1
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https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-exvhvo
Out[1]=1

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

In[2]:=2
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https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-9lk0x
Out[2]=2

Properties & Relations  (7)Properties of the function, and connections to other functions

The Karhunen–Loeve decomposition preserves the total variance:

In[1]:=1
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https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-xfttmi
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-m3ehi9
Out[2]=2

The Karhunen–Loeve decomposition yields uncorrelated sets:

In[1]:=1
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https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-f0wukw
In[2]:=2
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https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-f89dvf
Out[2]=2

The Karhunen–Loeve decomposition yields an orthogonal transformation matrix:

In[1]:=1
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https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-ng385z
In[2]:=2
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https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-t4u4e8
Out[2]=2

Relation to PrincipalComponents:

In[1]:=1
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https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-dnw4l2
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-bav5vr
Out[2]=2

A setting Standardized->True is equivalent to subtracting the mean from the input data:

In[1]:=1
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https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-7ff1rn
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-nvp2ii
Out[2]=2

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

In[1]:=1
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https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-gucqhn
Out[1]=1

KarhunenLoeveDecomposition normally returns images of a real type:

In[1]:=1
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https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-buunxk
Out[1]=1
Wolfram Research (2010), KarhunenLoeveDecomposition, Wolfram Language function, https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html (updated 2015).
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Wolfram Research (2010), KarhunenLoeveDecomposition, Wolfram Language function, https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html (updated 2015).

Text

Wolfram Research (2010), KarhunenLoeveDecomposition, Wolfram Language function, https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html (updated 2015).

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Wolfram Research (2010), KarhunenLoeveDecomposition, Wolfram Language function, https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html (updated 2015).

CMS

Wolfram Language. 2010. "KarhunenLoeveDecomposition." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2015. https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html.

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Wolfram Language. 2010. "KarhunenLoeveDecomposition." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2015. https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html.

APA

Wolfram Language. (2010). KarhunenLoeveDecomposition. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html

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Wolfram Language. (2010). KarhunenLoeveDecomposition. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html

BibTeX

@misc{reference.wolfram_2025_karhunenloevedecomposition, author="Wolfram Research", title="{KarhunenLoeveDecomposition}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html}", note=[Accessed: 13-May-2025 ]}

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@misc{reference.wolfram_2025_karhunenloevedecomposition, author="Wolfram Research", title="{KarhunenLoeveDecomposition}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html}", note=[Accessed: 13-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_karhunenloevedecomposition, organization={Wolfram Research}, title={KarhunenLoeveDecomposition}, year={2015}, url={https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html}, note=[Accessed: 13-May-2025 ]}

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@online{reference.wolfram_2025_karhunenloevedecomposition, organization={Wolfram Research}, title={KarhunenLoeveDecomposition}, year={2015}, url={https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html}, note=[Accessed: 13-May-2025 ]}