Data Whitening¶
A whitening transformation is a decorrelation transformation that transforms a set of random variables into a set of new random variables with identity covariance (uncorrelated with unit variances).
In particular, suppose a random vector has covariance , then a whitening transform is one that satisfy:
Note that is generally not unique. In particular, if is a whitening transform, so is any of its rotation with .
Whitening¶
The package uses Whitening
defined below to represent a whitening transform:
immutable Whitening{T<:FloatingPoint}
mean::Vector{T} # mean vector (can be empty to indicate zero mean), of length d
W::Matrix{T} # the transform coefficient matrix, of size (d, d)
end
An instance of Whitening
can be constructed by Whitening(mean, W)
.
There are several functions to access the properties of a whitening transform f
:

indim
(f)¶ Get the input dimension, i.e
d
.

outdim
(f)¶ Get the out dimension, i.e
d
.

mean
(f)¶ Get the mean vector.
Note: if
f.mean
is empty, this function returns a zero vector of lengthd
.

transform
(f, x)¶ Apply the whitening transform to a vector or a matrix with samples in columns, as .
Data Analysis¶
Given a dataset, one can use the fit
method to estimate a whitening transform.

fit
(Whitening, X; ...)¶ Estimate a whitening transform from the data given in
X
. Here,X
should be a matrix, whose columns give the samples.This function returns an instance of
Whitening
.Keyword Arguments:
name description default regcoef The regularization coefficient. The covariance will be regularized as follows when
regcoef
is positive:C + (eigmax(C) * regcoef) * eye(d)
zero(T)
mean The mean vector, which can be either of:
0
: the input data has already been centralizednothing
: this function will compute the mean a precomputed mean vector
nothing
Note: This function internally relies on
cov_whiten
to derive the transformationW
. The functioncov_whiten
itself is also a useful function.

cov_whitening
(C)¶ Derive the whitening transform coefficient matrix
W
given the covariance matrixC
. Here,C
can be either a square matrix, or an instance ofCholesky
.Internally, this function solves the whitening transform using Cholesky factorization. The rationale is as follows: let and , then .
Note: The return matrix
W
is an upper triangular matrix.

cov_whitening
(C, regcoef) Derive a whitening transform based on a regularized covariance, as
C + (eigmax(C) * regcoef) * eye(d)
.
In addition, the package also provides cov_whiten!
, in which the input matrix C
will be overwritten during computation. This can be more efficient when C
is no longer used.

invsqrtm
(C)¶ Compute
inv(sqrtm(C))
through symmetric eigenvalue decomposition.