WOLFRAM

Wolfram Language & System Documentation Center
Wolfram Language Home Page »

PillaiTrace
✖
PillaiTrace

✖
PillaiTrace[m1,m2]

gives Pillai's trace for the matrices m1 and m2.

Details

  • PillaiTrace[m1,m2] gives Pillai's trace between m1 and m2.
  • Pillai's trace is a measure of linear dependence based on partitions of the pooled covariance matrix.
  • Pillai's trace is computed as Tr[TemplateBox[{{(, {Sigma, _, {(, 11, )}}, )}}, Inverse].Sigma_(12).TemplateBox[{{(, {Sigma, _, {(, 22, )}}, )}}, Inverse].Sigma_(21)] where is the covariance matrix of the pooled sample, which can be partitioned into where and correspond to the covariance matrices of the individual datasets.
  • The arguments m1 and m2 can be any real‐valued matrices or vectors of equal length.

Examples

open allclose all

Basic Examples  (3)Summary of the most common use cases

Compute Pillai's trace for two matrices:

In[1]:=1
✖
https://wolfram.com/xid/0btno843o486-bv14aj
In[2]:=2
✖
https://wolfram.com/xid/0btno843o486-ce2wv
In[3]:=3
✖
https://wolfram.com/xid/0btno843o486-i51zwj
Out[3]=3

Pillai's trace for two vectors:

In[1]:=1
✖
https://wolfram.com/xid/0btno843o486-gugb6d
In[2]:=2
✖
https://wolfram.com/xid/0btno843o486-fulot0
Out[2]=2

Pillai's trace for a matrix and a vector:

In[1]:=1
✖
https://wolfram.com/xid/0btno843o486-cuzz7n
In[2]:=2
✖
https://wolfram.com/xid/0btno843o486-c4pua3
In[3]:=3
✖
https://wolfram.com/xid/0btno843o486-oz384
Out[3]=3

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

Pillai's trace is typically used to detect linear dependence between random matrices:

In[1]:=1
✖
https://wolfram.com/xid/0btno843o486-htfobe
In[3]:=3
✖
https://wolfram.com/xid/0btno843o486-b1crnn
In[4]:=4
✖
https://wolfram.com/xid/0btno843o486-bwvkw2

Values tend to be near 1 for dependent matrices:

In[5]:=5
✖
https://wolfram.com/xid/0btno843o486-bqlb9
Out[5]=5

The value is much smaller for independent matrices:

In[6]:=6
✖
https://wolfram.com/xid/0btno843o486-bozysc
Out[6]=6

Pillai's trace for machine-precision reals:

In[1]:=1
✖
https://wolfram.com/xid/0btno843o486-f29p
Out[1]=1

Use arbitrary precision:

In[1]:=1
✖
https://wolfram.com/xid/0btno843o486-ng52h
Out[1]=1

Properties & Relations  (4)Properties of the function, and connections to other functions

Pillai's trace measures linear dependence:

In[1]:=1
✖
https://wolfram.com/xid/0btno843o486-gzq8gw
In[2]:=2
✖
https://wolfram.com/xid/0btno843o486-gcdotf
Out[2]=2

Pillai's trace cannot detect nonlinear dependency:

In[1]:=1
✖
https://wolfram.com/xid/0btno843o486-7nvfr
In[2]:=2
✖
https://wolfram.com/xid/0btno843o486-nazb6e
In[3]:=3
✖
https://wolfram.com/xid/0btno843o486-j9qi3
In[4]:=4
✖
https://wolfram.com/xid/0btno843o486-gtfa0z
Out[4]=4

HoeffdingD can be used to detect some nonlinear dependence structures:

In[5]:=5
✖
https://wolfram.com/xid/0btno843o486-jor85u
Out[5]=5

Pillai's trace is asymptotically equivalent to WilksW:

In[1]:=1
✖
https://wolfram.com/xid/0btno843o486-bt6r2k
Out[1]=1

Statistical significance can be tested using PillaiTraceTest:

In[1]:=1
✖
https://wolfram.com/xid/0btno843o486-cpexjt
In[2]:=2
✖
https://wolfram.com/xid/0btno843o486-g80c7y
In[3]:=3
✖
https://wolfram.com/xid/0btno843o486-bkwj60
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0btno843o486-gj193z
Out[4]=4

Alternatively, use IndependenceTest to automatically choose a test:

In[5]:=5
✖
https://wolfram.com/xid/0btno843o486-dow0ew
Out[5]=5
Wolfram Research (2012), PillaiTrace, Wolfram Language function, https://reference.wolfram.com/language/ref/PillaiTrace.html.
✖
Wolfram Research (2012), PillaiTrace, Wolfram Language function, https://reference.wolfram.com/language/ref/PillaiTrace.html.

Text

Wolfram Research (2012), PillaiTrace, Wolfram Language function, https://reference.wolfram.com/language/ref/PillaiTrace.html.

✖
Wolfram Research (2012), PillaiTrace, Wolfram Language function, https://reference.wolfram.com/language/ref/PillaiTrace.html.

CMS

Wolfram Language. 2012. "PillaiTrace." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/PillaiTrace.html.

✖
Wolfram Language. 2012. "PillaiTrace." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/PillaiTrace.html.

APA

Wolfram Language. (2012). PillaiTrace. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PillaiTrace.html

✖
Wolfram Language. (2012). PillaiTrace. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PillaiTrace.html

BibTeX

@misc{reference.wolfram_2025_pillaitrace, author="Wolfram Research", title="{PillaiTrace}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/PillaiTrace.html}", note=[Accessed: 19-May-2025 ]}

✖
@misc{reference.wolfram_2025_pillaitrace, author="Wolfram Research", title="{PillaiTrace}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/PillaiTrace.html}", note=[Accessed: 19-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_pillaitrace, organization={Wolfram Research}, title={PillaiTrace}, year={2012}, url={https://reference.wolfram.com/language/ref/PillaiTrace.html}, note=[Accessed: 19-May-2025 ]}

✖
@online{reference.wolfram_2025_pillaitrace, organization={Wolfram Research}, title={PillaiTrace}, year={2012}, url={https://reference.wolfram.com/language/ref/PillaiTrace.html}, note=[Accessed: 19-May-2025 ]}