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 realvalued matrices or vectors of equal length.

Examples

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Basic Examples  (3)

Compute Pillai's trace for two matrices:

Pillai's trace for two vectors:

Pillai's trace for a matrix and a vector:

Scope  (3)

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

Values tend to be near 1 for dependent matrices:

The value is much smaller for independent matrices:

Pillai's trace for machine-precision reals:

Use arbitrary precision:

Properties & Relations  (4)

Pillai's trace measures linear dependence:

Pillai's trace cannot detect nonlinear dependency:

HoeffdingD can be used to detect some nonlinear dependence structures:

Pillai's trace is asymptotically equivalent to WilksW:

Statistical significance can be tested using PillaiTraceTest:

Alternatively, use IndependenceTest to automatically choose a test:

Introduced in 2012
 (9.0)