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 scientist of CNRS (Centre National de la Recherche Scientifique), working at UGA (University of Grenoble Alpes). contact: marco dot congedo at gmail dot com

Documentation

oschulz / diagonalizations.jl Goto Github PK