fabiap {fabia} | R Documentation

## 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

`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).
`sL` | final sparseness loadings; default = 0.6.
`sZ` | 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.
`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. Post-processing by projecting the final results to a given sparseness criterion.

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).

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.

### 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).

### Author(s)

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.

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

### See Also

`fabia`, `fabias`, `fabiap`, `spfabia`, `fabi`, `fabiasp`, `mfsc`, `nmfdiv`, `nmfeu`, `nmfsc`, `extractPlot`, `extractBic`, `plotBicluster`, `Factorization`, `projFuncPos`, `projFunc`, `estimateMode`, `makeFabiaData`, `makeFabiaDataBlocks`, `makeFabiaDataPos`, `makeFabiaDataBlocksPos`, `matrixImagePlot`, `fabiaDemo`, `fabiaVersion`

### Examples

    #---------------
    # 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)

    ## Not run:

    #-----------------
    # 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)

    }

    ## End(Not run)

[Package fabia version 2.33.3 Index]