Compute the 'Amari' distance between two matrices

Description

The Amari distance is a measure between two nonsingular matrices. Useful for checking for convergence in ICA algorithms, and for comparing solutions.

Usage

amari(V, W, orth = FALSE)

Arguments

Details

Formula is given in second reference below, page 570.

Value

a numeric distance metween 0 and 1

Author(s)

Trevor Hastie

References

Bach, F. and Jordan, M. (2002). Kernel independent component analysis, Journal of Machine Learning Research 3: 1-48
Hastie, T., Tibshirani, R. and Friedman, J. (2009) Elements of Statistical Learning (2nd edition), Springer.
http://www-stat.stanford.edu/~hastie/Papers/ESLII.pdf

See Also

ProDenICA

Examples

dist="n" 
N=1024
p=2
A0<-mixmat(p)
s<-scale(cbind(rjordan(dist,N),rjordan(dist,N)))
x <- s %*% A0
###Whiten the data
x <- scale(x, TRUE, FALSE)
sx <- svd(x)	### orthogonalization function
x <- sqrt(N) * sx$u
target <- solve(A0)
target <- diag(sx$d) %*% t(sx$v) %*% target/sqrt(N)
W0 <- matrix(rnorm(2*2), 2, 2)
W0 <- ICAorthW(W0)
W1 <- ProDenICA(x, W0=W0,trace=TRUE,Gfunc=G1)$W
fit=ProDenICA(x, W0=W0,Gfunc=GPois,trace=TRUE, density=TRUE)
W2 <- fit$W
#distance of FastICA from target
amari(W1,target)
#distance of ProDenICA from target
amari(W2,target)

Results

R version 3.3.1 (2016-06-21) -- "Bug in Your Hair"
Copyright (C) 2016 The R Foundation for Statistical Computing
Platform: x86_64-pc-linux-gnu (64-bit)

R is free software and comes with ABSOLUTELY NO WARRANTY.
You are welcome to redistribute it under certain conditions.
Type 'license()' or 'licence()' for distribution details.

R is a collaborative project with many contributors.
Type 'contributors()' for more information and
'citation()' on how to cite R or R packages in publications.

Type 'demo()' for some demos, 'help()' for on-line help, or
'help.start()' for an HTML browser interface to help.
Type 'q()' to quit R.

> library(ProDenICA)
Loading required package: gam
Loading required package: splines
Loading required package: foreach
Loaded gam 1.12

> png(filename="/home/ddbj/snapshot/RGM3/R_CC/result/ProDenICA/amari.Rd_%03d_medium.png", width=480, height=480)
> ### Name: amari
> ### Title: Compute the 'Amari' distance between two matrices
> ### Aliases: amari
> ### Keywords: multivariate
> 
> ### ** Examples
> 
> dist="n" 
> N=1024
> p=2
> A0<-mixmat(p)
> s<-scale(cbind(rjordan(dist,N),rjordan(dist,N)))
> x <- s %*% A0
> ###Whiten the data
> x <- scale(x, TRUE, FALSE)
> sx <- svd(x)	### orthogonalization function
> x <- sqrt(N) * sx$u
> target <- solve(A0)
> target <- diag(sx$d) %*% t(sx$v) %*% target/sqrt(N)
> W0 <- matrix(rnorm(2*2), 2, 2)
> W0 <- ICAorthW(W0)
> W1 <- ProDenICA(x, W0=W0,trace=TRUE,Gfunc=G1)$W
Iter 1 G 0.3762347 crit 0.1537012 
Iter 2 G 0.3761222 crit 0.08628485 
Iter 3 G 0.3761717 crit 0.1100507 
Iter 4 G 0.376113 crit 0.1001546 
Iter 5 G 0.3761644 crit 0.1047464 
Iter 6 G 0.3761114 crit 0.1029682 
Iter 7 G 0.3761631 crit 0.1038262 
Iter 8 G 0.3761111 crit 0.103499 
Iter 9 G 0.3761629 crit 0.103658 
Iter 10 G 0.376111 crit 0.1035976 
Iter 11 G 0.3761629 crit 0.103627 
Iter 12 G 0.376111 crit 0.1036158 
Iter 13 G 0.3761628 crit 0.1036212 
Iter 14 G 0.376111 crit 0.1036192 
Iter 15 G 0.3761628 crit 0.1036202 
Iter 16 G 0.376111 crit 0.1036198 
Iter 17 G 0.3761628 crit 0.10362 
Iter 18 G 0.376111 crit 0.1036199 
Iter 19 G 0.3761628 crit 0.1036199 
Iter 20 G 0.376111 crit 0.1036199 
Iter 21 G 0.3761628 crit 0.1036199 
> fit=ProDenICA(x, W0=W0,Gfunc=GPois,trace=TRUE, density=TRUE)
Iter 1 G 0.02227652 crit 0.04740158 
Iter 2 G 0.0231778 crit 0.009062331 
Iter 3 G 0.02326334 crit 0.001627743 
Iter 4 G 0.02324134 crit 0.0004278 
Iter 5 G 0.02323232 crit 1.955837e-05 
Iter 6 G 0.02323071 crit 9.771971e-06 
Iter 7 G 0.02322848 crit 3.236821e-05 
Iter 8 G 0.02323007 crit 2.425095e-06 
Iter 9 G 0.02323008 crit 1.21645e-07 
Iter 10 G 0.02323008 crit 5.772103e-09 
> W2 <- fit$W
> #distance of FastICA from target
> amari(W1,target)
[1] 0.2493955
> #distance of ProDenICA from target
> amari(W2,target)
[1] 0.006857558
> 
> 
> 
> 
> 
> dev.off()
null device 
          1 
>