# fabia: Factor Analysis for Bicluster Acquisition: Laplace Prior... In fabia: FABIA: Factor Analysis for Bicluster Acquisition

## Description

fabia: C implementation of fabia.

## Usage

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

## Arguments

 X the data matrix. p 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). 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 nL maximal number of biclusters at which a row element can participate; default = 0 (no limit) lL maximal number of row elements per bicluster; default = 0 (no limit) bL cycle at which the nL or lL maximum starts; default = 0 (start at the beginning)

## Details

Biclusters are found by sparse factor analysis where both the factors and the loadings are sparse.

Essentially the model is the sum of outer products of vectors:

X = ∑_{i=1}^{p} λ_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 λ_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).

The code is implemented in C.

## 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 σ), 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).

Sepp Hochreiter

## 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.

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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 #### FABIA resToy1 <- fabia(X,13,0.01,400) extractPlot(resToy1,ti="FABIA",Y=Y) raToy1 <- extractBic(resToy1) if ((raToy1$bic[[1]][1]>1) && (raToy1$bic[[1]][2])>1) { plotBicluster(raToy1,1) } if ((raToy1$bic[[2]][1]>1) && (raToy1$bic[[2]][2])>1) { plotBicluster(raToy1,2) } if ((raToy1$bic[[3]][1]>1) && (raToy1$bic[[3]][2])>1) { plotBicluster(raToy1,3) } if ((raToy1$bic[[4]][1]>1) && (raToy1$bic[[4]][2])>1) { plotBicluster(raToy1,4) } colnames(X(resToy1)) <- clab rownames(X(resToy1)) <- llab plot(resToy1,dim=c(1,2),label.tol=0.1,col.group = groups,lab.size=0.6) plot(resToy1,dim=c(1,3),label.tol=0.1,col.group = groups,lab.size=0.6) plot(resToy1,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) resBreast1 <- fabia(X,5,0.1,400) extractPlot(resBreast1,ti="FABIA Breast cancer(Veer)") raBreast1 <- extractBic(resBreast1) if ((raBreast1$bic[[1]][1]>1) && (raBreast1$bic[[1]][2])>1) { plotBicluster(raBreast1,1) } if ((raBreast1$bic[[2]][1]>1) && (raBreast1$bic[[2]][2])>1) { plotBicluster(raBreast1,2) } if ((raBreast1$bic[[3]][1]>1) && (raBreast1$bic[[3]][2])>1) { plotBicluster(raBreast1,3) } if ((raBreast1$bic[[4]][1]>1) && (raBreast1$bic[[4]][2])>1) { plotBicluster(raBreast1,4) } plot(resBreast1,dim=c(1,2),label.tol=0.03,col.group=CBreast,lab.size=0.6) plot(resBreast1,dim=c(1,3),label.tol=0.03,col.group=CBreast,lab.size=0.6) plot(resBreast1,dim=c(1,4),label.tol=0.03,col.group=CBreast,lab.size=0.6) plot(resBreast1,dim=c(1,5),label.tol=0.03,col.group=CBreast,lab.size=0.6) plot(resBreast1,dim=c(2,3),label.tol=0.03,col.group=CBreast,lab.size=0.6) plot(resBreast1,dim=c(2,4),label.tol=0.03,col.group=CBreast,lab.size=0.6) plot(resBreast1,dim=c(2,5),label.tol=0.03,col.group=CBreast,lab.size=0.6) plot(resBreast1,dim=c(3,4),label.tol=0.03,col.group=CBreast,lab.size=0.6) plot(resBreast1,dim=c(3,5),label.tol=0.03,col.group=CBreast,lab.size=0.6) plot(resBreast1,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) resMulti1 <- fabia(X,5,0.06,300,norm=2) extractPlot(resMulti1,ti="FABIA Multiple tissues(Su)") raMulti1 <- extractBic(resMulti1) if ((raMulti1$bic[[1]][1]>1) && (raMulti1$bic[[1]][2])>1) { plotBicluster(raMulti1,1) } if ((raMulti1$bic[[2]][1]>1) && (raMulti1$bic[[2]][2])>1) { plotBicluster(raMulti1,2) } if ((raMulti1$bic[[3]][1]>1) && (raMulti1$bic[[3]][2])>1) { plotBicluster(raMulti1,3) } if ((raMulti1$bic[[4]][1]>1) && (raMulti1$bic[[4]][2])>1) { plotBicluster(raMulti1,4) } plot(resMulti1,dim=c(1,2),label.tol=0.01,col.group=CMulti,lab.size=0.6) plot(resMulti1,dim=c(1,3),label.tol=0.01,col.group=CMulti,lab.size=0.6) plot(resMulti1,dim=c(1,4),label.tol=0.01,col.group=CMulti,lab.size=0.6) plot(resMulti1,dim=c(1,5),label.tol=0.01,col.group=CMulti,lab.size=0.6) plot(resMulti1,dim=c(2,3),label.tol=0.01,col.group=CMulti,lab.size=0.6) plot(resMulti1,dim=c(2,4),label.tol=0.01,col.group=CMulti,lab.size=0.6) plot(resMulti1,dim=c(2,5),label.tol=0.01,col.group=CMulti,lab.size=0.6) plot(resMulti1,dim=c(3,4),label.tol=0.01,col.group=CMulti,lab.size=0.6) plot(resMulti1,dim=c(3,5),label.tol=0.01,col.group=CMulti,lab.size=0.6) plot(resMulti1,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) resDLBCL1 <- fabia(X,5,0.1,400,norm=2) extractPlot(resDLBCL1,ti="FABIA Lymphoma(Rosenwald)") raDLBCL1 <- extractBic(resDLBCL1) if ((raDLBCL1$bic[[1]][1]>1) && (raDLBCL1$bic[[1]][2])>1) { plotBicluster(raDLBCL1,1) } if ((raDLBCL1$bic[[2]][1]>1) && (raDLBCL1$bic[[2]][2])>1) { plotBicluster(raDLBCL1,2) } if ((raDLBCL1$bic[[3]][1]>1) && (raDLBCL1$bic[[3]][2])>1) { plotBicluster(raDLBCL1,3) } if ((raDLBCL1$bic[[4]][1]>1) && (raDLBCL1$bic[[4]][2])>1) { plotBicluster(raDLBCL1,4) } plot(resDLBCL1,dim=c(1,2),label.tol=0.03,col.group=CDLBCL,lab.size=0.6) plot(resDLBCL1,dim=c(1,3),label.tol=0.03,col.group=CDLBCL,lab.size=0.6) plot(resDLBCL1,dim=c(1,4),label.tol=0.03,col.group=CDLBCL,lab.size=0.6) plot(resDLBCL1,dim=c(1,5),label.tol=0.03,col.group=CDLBCL,lab.size=0.6) plot(resDLBCL1,dim=c(2,3),label.tol=0.03,col.group=CDLBCL,lab.size=0.6) plot(resDLBCL1,dim=c(2,4),label.tol=0.03,col.group=CDLBCL,lab.size=0.6) plot(resDLBCL1,dim=c(2,5),label.tol=0.03,col.group=CDLBCL,lab.size=0.6) plot(resDLBCL1,dim=c(3,4),label.tol=0.03,col.group=CDLBCL,lab.size=0.6) plot(resDLBCL1,dim=c(3,5),label.tol=0.03,col.group=CDLBCL,lab.size=0.6) plot(resDLBCL1,dim=c(4,5),label.tol=0.03,col.group=CDLBCL,lab.size=0.6) } ## End(Not run)