fabiap: Factor Analysis for Bicluster Acquisition: Post-Projection (FABIAP)

Description

fabiap: C implementation of fabiap.

Usage

fabiap(X,p=13,alpha=0.01,cyc=500,spl=0,spz=0.5,sL=0.6,sZ=0.6,non_negative=0,random=1.0,center=2,norm=1,scale=0.0,lap=1.0,nL=0,lL=0,bL=0)

Arguments

the data matrix.

number of hidden factors = number of biclusters; default = 13.

alpha

sparseness loadings (0-1.0); default = 0.01.

cyc

number of iterations; default = 500.

spl

sparseness prior loadings (0 - 2.0); default = 0 (Laplace).

spz

sparseness factors (0.5 - 2.0); default = 0.5 (Laplace).

final sparseness loadings; default = 0.6.

final sparseness factors; default = 0.6.

non_negative

Non-negative factors and loadings if non_negative > 0; default = 0.

random

<=0: by="" svd,="">0: random initialization of loadings in [-random,random]; default = 1.0.

center

data centering: 1 (mean), 2 (median), > 2 (mode), 0 (no); default = 2.

norm

data normalization: 1 (0.75-0.25 quantile), >1 (var=1), 0 (no); default = 1.

scale

loading vectors are scaled in each iteration to the given variance. 0.0 indicates non scaling; default = 0.0.

lap

minimal value of the variational parameter; default = 1.0.

maximal number of biclusters at which a row element can participate; default = 0 (no limit)

maximal number of row elements per bicluster; default = 0 (no limit)

cycle at which the nL or lL maximum starts; default = 0 (start at the beginning)

Value

object of the class Factorization. Containing LZ (estimated noise free data $L Z$), L (loadings $L$), Z (factors $Z$), U (noise $X-LZ$), center (centering vector), scaleData (scaling vector), X (centered and scaled data $X$), Psi (noise variance $\sigma$), lapla (variational parameter), avini (the information which the factor $z_{ij}$ contains about $x_j$ averaged over $j$) xavini (the information which the factor $z_{j}$ contains about $x_j$) ini (for each $j$ the information which the factor $z_{ij}$ contains about $x_j$).

concept

biclustering
factor analysis
sparse coding
Laplace distribution
EM algorithm
non-negative matrix factorization
multivariate analysis
latent variables

Details

Biclusters are found by sparse factor analysis where both the factors and the loadings are sparse. Post-processing by projecting the final results to a given sparseness criterion.

Essentially the model is the sum of outer products of vectors: $$X = \sum_{i=1}^{p} \lambda_i z_i^T + U$$ where the number of summands $p$ is the number of biclusters. The matrix factorization is $$X = L Z + U$$

Here $\lambda_i$ are from $R^n$, $z_i$ from $R^l$, $L$ from $R^{n \times p}$, $Z$ from $R^{p \times l}$, and $X$, $U$ from $R^{n \times l}$.

If the nonzero components of the sparse vectors are grouped together then the outer product results in a matrix with a nonzero block and zeros elsewhere.

The model selection is performed by a variational approach according to Girolami 2001 and Palmer et al. 2006.

We included a prior on the parameters and minimize a lower bound on the posterior of the parameters given the data. The update of the loadings includes an additive term which pushes the loadings toward zero (Gaussian prior leads to an multiplicative factor).

Post-processing: The final results of the loadings and the factors are projected to a sparse vector according to Hoyer, 2004: given an $l_1$-norm and an $l_2$-norm minimize the Euclidean distance to the original vector (currently the $l_2$-norm is fixed to 1). The projection is a convex quadratic problem which is solved iteratively where at each iteration at least one component is set to zero. Instead of the $l_1$-norm a sparseness measurement is used which relates the $l_1$-norm to the $l_2$-norm:

The code is implemented in C and the projection in R.

References

S. Hochreiter et al., FABIA: Factor Analysis for Bicluster Acquisition, Bioinformatics 26(12):1520-1527, 2010. http://bioinformatics.oxfordjournals.org/cgi/content/abstract/btq227 Mark Girolami, A Variational Method for Learning Sparse and Overcomplete Representations, Neural Computation 13(11): 2517-2532, 2001. J. Palmer, D. Wipf, K. Kreutz-Delgado, B. Rao, Variational EM algorithms for non-Gaussian latent variable models, Advances in Neural Information Processing Systems 18, pp. 1059-1066, 2006.

Patrik O. Hoyer, Non-negative Matrix Factorization with Sparseness Constraints, Journal of Machine Learning Research 5:1457-1469, 2004.

Examples

Run this code

#---------------
# TEST
#---------------

dat <- makeFabiaDataBlocks(n = 100,l= 50,p = 3,f1 = 5,f2 = 5,
  of1 = 5,of2 = 10,sd_noise = 3.0,sd_z_noise = 0.2,mean_z = 2.0,
  sd_z = 1.0,sd_l_noise = 0.2,mean_l = 3.0,sd_l = 1.0)

X <- dat[[1]]
Y <- dat[[2]]

resEx <- fabiap(X,3,0.1,50)


#-----------------
# DEMO1: Toy Data
#-----------------

n = 1000
l= 100
p = 10

dat <- makeFabiaDataBlocks(n = n,l= l,p = p,f1 = 5,f2 = 5,
  of1 = 5,of2 = 10,sd_noise = 3.0,sd_z_noise = 0.2,mean_z = 2.0,
  sd_z = 1.0,sd_l_noise = 0.2,mean_l = 3.0,sd_l = 1.0)

X <- dat[[1]]
Y <- dat[[2]]
ZC <- dat[[3]]
LC <- dat[[4]]

gclab <- rep.int(0,l)
gllab <- rep.int(0,n)
clab <- as.character(1:l)
llab <- as.character(1:n)
for (i in 1:p){
 for (j in ZC[i]){
     clab[j] <- paste(as.character(i),"_",clab[j],sep="")
 }
 for (j in LC[i]){
     llab[j] <- paste(as.character(i),"_",llab[j],sep="")
 }
 gclab[unlist(ZC[i])] <- gclab[unlist(ZC[i])] + p^i
 gllab[unlist(LC[i])] <- gllab[unlist(LC[i])] + p^i
}


groups <- gclab

#### FABIAP

resToy3 <- fabiap(X,13,0.1,400)

extractPlot(resToy3,ti="FABIAP",Y=Y)

raToy3 <- extractBic(resToy3)

if ((raToy3$bic[[1]][1]>1) && (raToy3$bic[[1]][2])>1) {
    plotBicluster(raToy3,1)
}
if ((raToy3$bic[[2]][1]>1) && (raToy3$bic[[2]][2])>1) {
    plotBicluster(raToy3,2)
}
if ((raToy3$bic[[3]][1]>1) && (raToy3$bic[[3]][2])>1) {
    plotBicluster(raToy3,3)
}
if ((raToy3$bic[[4]][1]>1) && (raToy3$bic[[4]][2])>1) {
    plotBicluster(raToy3,4)
}

colnames(X(resToy3)) <- clab

rownames(X(resToy3)) <- llab


plot(resToy3,dim=c(1,2),label.tol=0.1,col.group = groups,lab.size=0.6)
plot(resToy3,dim=c(1,3),label.tol=0.1,col.group = groups,lab.size=0.6)
plot(resToy3,dim=c(2,3),label.tol=0.1,col.group = groups,lab.size=0.6)


#------------------------------------------
# DEMO2: Laura van't Veer's gene expression  
#        data set for breast cancer 
#------------------------------------------


avail <- require(fabiaData)

if (!avail) {
    message("")
    message("")
    message("#####################################################")
    message("Package 'fabiaData' is not available: please install.")
    message("#####################################################")
} else {

data(Breast_A)

X <- as.matrix(XBreast)


resBreast3 <- fabiap(X,5,0.1,400)

extractPlot(resBreast3,ti="FABIAP Breast cancer(Veer)")

raBreast3 <- extractBic(resBreast3)

if ((raBreast3$bic[[1]][1]>1) && (raBreast3$bic[[1]][2])>1) {
    plotBicluster(raBreast3,1)
}
if ((raBreast3$bic[[2]][1]>1) && (raBreast3$bic[[2]][2])>1) {
    plotBicluster(raBreast3,2)
}
if ((raBreast3$bic[[3]][1]>1) && (raBreast3$bic[[3]][2])>1) {
    plotBicluster(raBreast3,3)
}
if ((raBreast3$bic[[4]][1]>1) && (raBreast3$bic[[4]][2])>1) {
    plotBicluster(raBreast3,4)
}

plot(resBreast3,dim=c(1,2),label.tol=0.03,col.group=CBreast,lab.size=0.6)
plot(resBreast3,dim=c(1,3),label.tol=0.03,col.group=CBreast,lab.size=0.6)
plot(resBreast3,dim=c(1,4),label.tol=0.03,col.group=CBreast,lab.size=0.6)
plot(resBreast3,dim=c(1,5),label.tol=0.03,col.group=CBreast,lab.size=0.6)
plot(resBreast3,dim=c(2,3),label.tol=0.03,col.group=CBreast,lab.size=0.6)
plot(resBreast3,dim=c(2,4),label.tol=0.03,col.group=CBreast,lab.size=0.6)
plot(resBreast3,dim=c(2,5),label.tol=0.03,col.group=CBreast,lab.size=0.6)
plot(resBreast3,dim=c(3,4),label.tol=0.03,col.group=CBreast,lab.size=0.6)
plot(resBreast3,dim=c(3,5),label.tol=0.03,col.group=CBreast,lab.size=0.6)
plot(resBreast3,dim=c(4,5),label.tol=0.03,col.group=CBreast,lab.size=0.6)

}


#-----------------------------------
# DEMO3: Su's multiple tissue types
#        gene expression data set 
#-----------------------------------


avail <- require(fabiaData)

if (!avail) {
    message("")
    message("")
    message("#####################################################")
    message("Package 'fabiaData' is not available: please install.")
    message("#####################################################")
} else {

data(Multi_A)

X <- as.matrix(XMulti)

resMulti3 <- fabiap(X,5,0.1,300)

extractPlot(resMulti3,ti="FABIAP Multiple tissues(Su)")


raMulti3 <- extractBic(resMulti3)

if ((raMulti3$bic[[1]][1]>1) && (raMulti3$bic[[1]][2])>1) {
    plotBicluster(raMulti3,1)
}
if ((raMulti3$bic[[2]][1]>1) && (raMulti3$bic[[2]][2])>1) {
    plotBicluster(raMulti3,2)
}
if ((raMulti3$bic[[3]][1]>1) && (raMulti3$bic[[3]][2])>1) {
    plotBicluster(raMulti3,3)
}
if ((raMulti3$bic[[4]][1]>1) && (raMulti3$bic[[4]][2])>1) {
    plotBicluster(raMulti3,4)
}

plot(resMulti3,dim=c(1,2),label.tol=0.01,col.group=CMulti,lab.size=0.6)
plot(resMulti3,dim=c(1,3),label.tol=0.01,col.group=CMulti,lab.size=0.6)
plot(resMulti3,dim=c(1,4),label.tol=0.01,col.group=CMulti,lab.size=0.6)
plot(resMulti3,dim=c(1,5),label.tol=0.01,col.group=CMulti,lab.size=0.6)
plot(resMulti3,dim=c(2,3),label.tol=0.01,col.group=CMulti,lab.size=0.6)
plot(resMulti3,dim=c(2,4),label.tol=0.01,col.group=CMulti,lab.size=0.6)
plot(resMulti3,dim=c(2,5),label.tol=0.01,col.group=CMulti,lab.size=0.6)
plot(resMulti3,dim=c(3,4),label.tol=0.01,col.group=CMulti,lab.size=0.6)
plot(resMulti3,dim=c(3,5),label.tol=0.01,col.group=CMulti,lab.size=0.6)
plot(resMulti3,dim=c(4,5),label.tol=0.01,col.group=CMulti,lab.size=0.6)

}


#-----------------------------------------
# DEMO4: Rosenwald's diffuse large-B-cell
#        lymphoma gene expression data set 
#-----------------------------------------

avail <- require(fabiaData)

if (!avail) {
    message("")
    message("")
    message("#####################################################")
    message("Package 'fabiaData' is not available: please install.")
    message("#####################################################")
} else {


data(DLBCL_B)

X <- as.matrix(XDLBCL)


resDLBCL3 <- fabiap(X,5,0.1,400)

extractPlot(resDLBCL3,ti="FABIAP Lymphoma(Rosenwald)")
raDLBCL3 <- extractBic(resDLBCL3)

if ((raDLBCL3$bic[[1]][1]>1) && (raDLBCL3$bic[[1]][2])>1) {
    plotBicluster(raDLBCL3,1)
}
if ((raDLBCL3$bic[[2]][1]>1) && (raDLBCL3$bic[[2]][2])>1) {
    plotBicluster(raDLBCL3,2)
}
if ((raDLBCL3$bic[[3]][1]>1) && (raDLBCL3$bic[[3]][2])>1) {
    plotBicluster(raDLBCL3,3)
}
if ((raDLBCL3$bic[[4]][1]>1) && (raDLBCL3$bic[[4]][2])>1) {
    plotBicluster(raDLBCL3,4)
}

plot(resDLBCL3,dim=c(1,2),label.tol=0.03,col.group=CDLBCL,lab.size=0.6)
plot(resDLBCL3,dim=c(1,3),label.tol=0.03,col.group=CDLBCL,lab.size=0.6)
plot(resDLBCL3,dim=c(1,4),label.tol=0.03,col.group=CDLBCL,lab.size=0.6)
plot(resDLBCL3,dim=c(1,5),label.tol=0.03,col.group=CDLBCL,lab.size=0.6)
plot(resDLBCL3,dim=c(2,3),label.tol=0.03,col.group=CDLBCL,lab.size=0.6)
plot(resDLBCL3,dim=c(2,4),label.tol=0.03,col.group=CDLBCL,lab.size=0.6)
plot(resDLBCL3,dim=c(2,5),label.tol=0.03,col.group=CDLBCL,lab.size=0.6)
plot(resDLBCL3,dim=c(3,4),label.tol=0.03,col.group=CDLBCL,lab.size=0.6)
plot(resDLBCL3,dim=c(3,5),label.tol=0.03,col.group=CDLBCL,lab.size=0.6)
plot(resDLBCL3,dim=c(4,5),label.tol=0.03,col.group=CDLBCL,lab.size=0.6)


}

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