PrincipalComponents

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`PrincipalComponents`

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PrincipalComponents[matrix]

transforms elements of matrix into unscaled principal components.

Details and Options

PrincipalComponents gives the principal component transform of matrix.
The principal components of matrix are linear transformations of the original columns into uncorrelated columns arranged in order of decreasing variance.
PrincipalComponents supports a Method option. The following explicit settings can be specified:
"Covariance" uses covariance method (default)

"Correlation" uses correlation method
If principal components of scaled columns (standardized principal components) are required, the option Method"Correlation" should be used.
The dimensions of PrincipalComponents[matrix] are the same as the dimensions of matrix.
If matrix consists of exact numbers or symbols, the result is also exact or symbolic, respectively.

Examples

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Basic Examples (1)Summary of the most common use cases

Principal components of two datasets, treated as the columns of a matrix:

In[1]:=1

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https://wolfram.com/xid/01yyrkv68q7-c268kf

Out[1]=1

Scope (3)Survey of the scope of standard use cases

Principal components computed with arbitrary-precision numbers:

In[1]:=1

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https://wolfram.com/xid/01yyrkv68q7-ijpqbf

Out[1]=1

Principal components of exact numbers:

In[1]:=1

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https://wolfram.com/xid/01yyrkv68q7-le51c4

Out[1]=1

Principal components computation involving symbolic expressions:

In[1]:=1

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https://wolfram.com/xid/01yyrkv68q7-6vtlpi

Out[1]=1

Options (1)Common values & functionality for each option

Method (1)

Principal components using correlation scaling:

In[10]:=10

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https://wolfram.com/xid/01yyrkv68q7-2mj2os

Out[10]=10

Properties & Relations (2)Properties of the function, and connections to other functions

The principal component columns are ordered by decreasing variance:

In[11]:=11

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https://wolfram.com/xid/01yyrkv68q7-y07o2m

In[12]:=12

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https://wolfram.com/xid/01yyrkv68q7-mv5rmp

Out[12]=12

The mean of each principal component column is zero:

In[13]:=13

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https://wolfram.com/xid/01yyrkv68q7-lsjdfc

Out[13]=13

The principal component columns are not correlated:

In[14]:=14

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https://wolfram.com/xid/01yyrkv68q7-to1afi

Out[14]=14

The setting Method->"Correlation" yields the same results as standardizing the input matrix:

In[1]:=1

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https://wolfram.com/xid/01yyrkv68q7-t2at3d

Out[1]=1

Possible Issues (1)Common pitfalls and unexpected behavior

For certain symbolic matrices the result may be very large:

In[16]:=16

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https://wolfram.com/xid/01yyrkv68q7-mxktt7

Out[16]=16

Neat Examples (1)Surprising or curious use cases

Align the principal axis of a two-dimensional shape with the horizontal axis:

In[1]:=1

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https://wolfram.com/xid/01yyrkv68q7-u79ozx

Out[2]=2

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