Diagonalizations.jl

Documentation

Diagonalizations.jl is a Julia signal processing package implementing several closed form and iterative diagonalization procedures for both real and complex data input:

Acronym	Full Name	Datasets ( m )	Observations ( k )
PCA	Principal Component Analysis	1	1
Whitening	Whitening (Sphering)	1	1
MCA	Maximum Covariance Analysis	2	1
CCA	Canonical Correlation Analysis	2	1
gMCA	generalized MCA	>1	1
gCCA	generalized CCA	>1	1
CSP	Common Spatial Pattern	1	2
CSTP	Common Spatio-Temporal Pattern	1	>1
AJD	Approximate Joint Diagonalization	1	>1
mAJD	multiple AJD	>1	>1

For example the MCA diagonalizes a cross-covariance matrix, like in this figure:

As compared to MultivariateStats.jl this package supports :

• the dims keyword like in the StatsBase.jl package
• shrinkage covariance matrix estimations throught package CovarianceEstimation
• average covariance estimations using metrics for the manifold of positive definite matrices using the PosDefManifold package
• facilities to set the subspace dimension upon construction
• diagonalization procedures for the case m≥2 and k≥2.

This package implements state-of-the-art approximate joint diagonalization algorithms. For some benchmarking see here.

Installation

To install the package execute the following command in Julia's REPL:

]add CovarianceEstimation PosDefManifold Diagonalizations

Examples

using Diagonalizations, PosDefManifold, Test

n, t=10, 100

# generate an nxt data matrix
X=genDataMatrix(n, t)

# principal component analysis
pX=pca(X)

# the following is an equivalent constructor taking the covariance matrix as input
pC=pca(Symmetric((X*X')/t))

@test pX==pC # the output of the two constructors above is equivalent

@test C≈pC.F*pC.D*pC.F'  

# get only the first p eigenvectors, where p is the smallest integer
# explaining at least 75% of the variance
pX=pca(X; eVar=0.75)

Y=genDataMatrix(n, t)

# maximum covariance analysis
mXY=mca(X, Y)

# canonical correlation analysis
cXY=cca(X, Y)

# approximate joint diagonalization
Xset=randP(5, 20)
aXset=ajd(Xset; algorithm=:JADE)
aXset=ajd(Xset; algorithm=:LogLike)

# etc., etc.

About the Authors

Marco Congedo, is a Research Director of CNRS (Centre National de la Recherche Scientifique), working at UGA (University of Grenoble Alpes). contact: marco dot congedo at gmail dot com

Documentation