scICA {coloredICA} R Documentation

## spatial colored Independent Component Analysis

### Description

This function implements the spatial colored Independent Component Analysis (scICA) algorithm, where sources are treated as spatial stochastic processes on a lattice.

### Usage

scICA(Xin, M = dim(Xin)[1], Win = diag(M), tol = 1e-04, maxit = 20, nmaxit = 1,
unmixing.estimate = "eigenvector", n1, n2, nx01 = n1, nx02 = n2, h)


### Arguments

 Xin Data matrix with p rows (representing variables) and n columns (representing observations). M Number of components to be extracted. Win Initial guess for the unmixing matrix W. Dimensions need to be M x M. tol Tolerance used to establish the convergence of the algorithm. maxit Maximum number of iterations. nmaxit If the algorithm does not converge, it is run again with a new initial guess for the unmixing matrix W. This operation is done nmaxit times. unmixing.estimate The method used in the unmixing matrix estimation step. The two allowed choices are "eigenvector" and "newton" (see Details). n1 Number of rows of the lattice. n2 Number of columns of the lattice. nx01 Number of rows of the lattice where the spectral density is evaluate. Default value is n1. nx02 Number of columns of the lattice where the spectral density is evaluate. Default value is n2. h Kernel bandwidth used for the nonparametric estimation of the sources spectral densities.

### Details

In the Independent Component Analysis approach, the data matrix X is considered to be a linear combination of independent components, i.e. X = AS, where rows of S contain the unobserved realizations of the independent components and A is a linear mixing matrix. According to classical ICA procedures data matrix X is centered and, then, whitened by projecting the data onto its principal component directions, i.e. X \rightarrow KX = \widetilde{X} where K is a M x p pre-whitening matrix. The scICA algorithm then estimates the unmixing matrix W, with W\widetilde{X} = S, according to the procedure described below. Then, defining \widetilde{W}=WK, the mixing matrix A is recovered through A=\widetilde{W}^T(\widetilde{W}\widetilde{W}^T)^{-1}.

Spatial colored Independent Component Analysis assumes that the independent sources are spatial stochastic processes on a lattice. To perform ICA, the Whittle log-likelihood is exploited. In particular the log-likelihood is written in function of the unmixing matrix W and the spectral densities f_{S_j} of the spatial autocorrelated sources as follows:

l(W,\boldsymbol{f}_{\boldsymbol{S}};\widetilde{X})=\sum_{j=1}^p\sum_{k=1}^n\left(\frac{\boldsymbol{e}_j^T W \widetilde{\boldsymbol{f}}(r_k,\widetilde{X})W^T \boldsymbol{e}_j}{f_{S_j}(r_k)}+\ln f_{S_j}(r_k)\right) + n\ln|\det(W)|.

Due to whitening, W is orthogonal and the last term of the objective function can be dropped. The orthogonality of the unmixing matrix W can be imposed in two different ways, setting the argument unmixing.estimate. In this way the estimate of the unmixing matrix W can be found according two different procedures:

• as described in Shen et al. (2014). A penalty term is added to the objective function. In particular \tau\bold{w}'_{j}C_{j}\bold{w}_{j}, where \bold{w}'_{j} is the jth column of W, C_j=\sum_{k\neq j}\bold{w}_k\bold{w}'_k and \tau is a tuning parameter. The matrix C_j provides an orthogonality constraint in the sense that \bold{w}'_{j}C_{j}\bold{w}_{j}=\sum_{k\neq j}. In this way the objective function assumes a symmetric and positive-definite form and the argmin correspond to the lower eigenvalue. This choice is obtained setting unmixing.matrix = "eigenvector".

• as described in Lee et al. (2011). The orthogonality constraint is considered performing the minimization of the objective function according a Newton-Raphson method with Lagrange multiplier. This choice is obtained setting unmixing.matrix = "newton".

Independently from the choice of the technique to minimize the objective function, the scICA algorithm is based on an iterative procedure. While the Amari error is greater than tol and the number of iteration is less or equal than maxit, the two following steps are repeated:

• nonparametric estimation of the sources spectral density through a multidimensional local linear kernel estimator \widehat{m}_{LK} (see Shen et al. (2014) for further details).

• estimate the unmixing matrix W according the method selected in unmixing.estimate.

### Value

A list containing the following components:

 W Estimate of the M x M unmixing matrix in the whitened space. K pre-whitening matrix that projects data onto the first M principal components. Dimensions are M x p. A Estimate of the p x M mixing matrix. S Estimate of the M x n source matrix. X Original p x n data matrix. iter number of iterations. NInv number of times the algorithm is rerun after it does not achieve convergence. den Estimate of the spectral density of the sources. Dimensions are M x n.

### Note

Note that source matrix S and spectral density matrix den are n x M matrices. Every row, that should be in a n1 x n2 grid, has been vectorized in a n vector by column, with n = n1 x n2.

### Author(s)

Lee, S., Shen, H., Truong, Y. and Zanini, P.

### References

Shen, H., Truong, Y., Zanini, P. (2014). Independent Component Analysis for Spatial Processes on a Lattice. MOX report 38/2014, Department of Mathematics, Politecnico di Milano.

Lee, S., Shen, H., Truong, Y., Lewis, M., Huang, X. (2011). Independent Component Analysis Involving Autocorrelated Sources With an Application to Funcional Magnetic Resonance Imaging. Journal of the American Statistical Association, 106, 1009–1024.

cICA

### Examples

## Not run:

require(fastICA)

n1=20
n2=20
M=2

# Fist source

sigma1=2
S1=matrix(0,n1,n2)
for (i in 1:n1){
S1[i,]=rnorm(n2,i*2,0.2)
}
for (j in 1:n2){
S1[,j]=S1[,j]+rnorm(n1,j*2,0.2)
}
S1=S1+matrix(rnorm(n1*n2,0,sigma1),n1,n2)

image(1:n2,1:n1,t(S1[n1:1,]),xlab="",ylab="",main="Source 1")

# Second source

val1=1
val2=1.2
val3=1.5
val4=2
sigma2=0.1

S2=matrix(0,n1,n2)
S2[2:5,4:10]=val1
S2[3:4,5:9]=val3
S2[13:18,16:19]=val2
S2[14:17,17:18]=val4
S2=S2+matrix(rnorm(n1*n2,0,sigma2),n1,n2)

image(1:n2,1:n1,t(S2[n1:1,]),xlab="",ylab="",main="Source 2")

# Generating data matrix X

A = rerow(matrix(runif(M^2)-0.5,M,M))
W = solve(A)
S=rbind(as.vector(S1),as.vector(S2))
X = A %*% S

# Solving Blind Source Separation problem with three different methods

cica = cICA(X,tol=0.001)
## scica = scICA(X,n1=n1,n2=n2,h=(2*pi/10),tol=0.001)
fica = fastICA(t(X),2)

amari_distance(t(A),t(cica$A)) ## amari_distance(t(A),t(scica$A))
amari_distance(t(A),fica$A) Shat1=cica$S
## Shat2=scica$S Shat3=t(fica$S)

par(mfrow=c(2,2))
image(t(S1[n1:1,]),xlab="",ylab="")
image(t(S2[n1:1,]),xlab="",ylab="")
image(t(matrix(Shat1[1,],n1,n2)[n1:1,]),xlab="",ylab="")
image(t(matrix(Shat1[2,],n1,n2)[n1:1,]),xlab="",ylab="")

## par(mfrow=c(2,2))
## image(t(S1[n1:1,]),xlab="",ylab="")
## image(t(S2[n1:1,]),xlab="",ylab="")
## image(t(matrix(Shat2[1,],n1,n2)[n1:1,]),xlab="",ylab="")
## image(t(matrix(Shat2[2,],n1,n2)[n1:1,]),xlab="",ylab="")

par(mfrow=c(2,2))
image(t(S1[n1:1,]),xlab="",ylab="")