# PMD: Get a penalized matrix decomposition for a data matrix. In PMA: Penalized Multivariate Analysis

## Description

Performs a penalized matrix decomposition for a data matrix. Finds factors u and v that summarize the data matrix well. u and v will both be sparse, and v can optionally also be smooth.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20``` ```PMD( x, type = c("standard", "ordered"), sumabs = 0.4, sumabsu = 5, sumabsv = NULL, lambda = NULL, niter = 20, K = 1, v = NULL, trace = TRUE, center = TRUE, chrom = NULL, rnames = NULL, cnames = NULL, upos = FALSE, uneg = FALSE, vpos = FALSE, vneg = FALSE ) ```

## Arguments

 `x` Data matrix of dimension \$n x p\$, which can contain NA for missing values. `type` "standard" or "ordered": Do we want v to simply be sparse, or should it also be smooth? If the columns of x are ordered (e.g. CGH spots along a chromosome) then choose "ordered". Default is "standard". If "standard", then the PMD function will make use of sumabs OR sumabsu&sumabsv. If "ordered", then the function will make use of sumabsu and lambda. `sumabs` Used only if type is "standard". A measure of sparsity for u and v vectors, between 0 and 1. When sumabs is specified, and sumabsu and sumabsv are NULL, then sumabsu is set to \$sqrt(n)*sumabs\$ and sumabsv is set to \$sqrt(p)*sumabs\$. If sumabs is specified, then sumabsu and sumabsv should be NULL. Or if sumabsu and sumabsv are specified, then sumabs should be NULL. `sumabsu` Used for types "ordered" AND "standard". How sparse do you want u to be? This is the sum of absolute values of elements of u. It must be between 1 and the square root of the number of rows in data matrix. The smaller it is, the sparser u will be. `sumabsv` Used only if type is "standard". How sparse do you want v to be? This is the sum of absolute values of elements of v. It must be between 1 and square root of number of columns of data. The smaller it is, the sparser v will be. `lambda` Used only if type is "ordered". This is the tuning parameter for the fused lasso penalty on v, which takes the form \$lambda ||v||1 + lambda |v_j - v(j-1)|\$. \$lambda\$ must be non-negative. If NULL, then it is chosen adaptively from the data. `niter` How many iterations should be performed. It is best to run at least 20 of so. Default is 20. `K` The number of factors in the PMD to be returned; default is 1. `v` The first right singular vector(s) of the data. (If missing data is present, then the missing values are imputed before the singular vectors are calculated.) v is used as the initial value for the iterative PMD algorithm. If x is large, then this step can be time-consuming; therefore, if PMD is to be run multiple times, then v should be computed once and saved. `trace` Print out progress as iterations are performed? Default is TRUE. `center` Subtract out mean of x? Default is TRUE. `chrom` If type is "ordered", then this gives the option to specify that some columns of x (corresponding to CGH spots) are on different chromosomes. Then v will be sparse, and smooth within each chromosome but not between chromosomes. Length of chrom should equal number of columns of x, and each entry in chrom should be a number corresponding to which chromosome the CGH spot is on. `rnames` An optional vector containing a name for each row of x. `cnames` An optional vector containing a name for each column of x. `upos` Constrain the elements of u to be positive? TRUE or FALSE. `uneg` Constrain the elements of u to be negative? TRUE or FALSE. `vpos` Constrain the elements of v to be positive? TRUE or FALSE. Cannot be used if type is "ordered". `vneg` Constrain the elements of v to be negative? TRUE or FALSE. Cannot be used if type is "ordered."

## Details

The criterion for the PMD is as follows: we seek vectors \$u\$ and \$v\$ such that \$u'Xv\$ is large, subject to \$||u||_2=1, ||v||_2=1\$ and additional penalties on \$u\$ and \$v\$. These additional penalties are as follows: If type is "standard", then lasso (\$L_1\$) penalties (promoting sparsity) are placed on u and v. If type is "ordered", then lasso penalty is placed on u and a fused lasso penalty (promoting sparsity and smoothness) is placed on v.

If type is "standard", then arguments sumabs OR sumabsu&sumabsv are used. If type is "ordered", then sumabsu AND lambda are used. Sumabsu is the bound of absolute value of elements of u. Sumabsv is bound of absolute value of elements of v. If sumabs is given, then sumabsu is set to sqrt(nrow(x))*sumabs and sumabsv is set to sqrt(ncol(x))*sumabs. \$lambda\$ is the parameter for the fused lasso penalty on v when type is "ordered": \$lambda(||v||1 + sum_j |v_j - v(j-1))\$.

## Value

 `u` u is output. If you asked for multiple factors then each column of u is a factor. u has dimension nxK if you asked for K factors. `v` v is output. If you asked for multiple factors then each column of v is a factor. v has dimension pxK if you asked for K factors. `d` d is output. Computationally, \$d=u'Xv\$ where \$u\$ and \$v\$ are the sparse factors output by the PMD function and \$X\$ is the data matrix input to the PMD function. When K=1, the residuals of the rank-1 PMD are given by \$X - duv'\$. `v.init` The first right singular vector(s) of the data; these are returned to save on computation time if PMD will be run again. `meanx` Mean of x that was subtracted out before PMD was performed.

## References

Witten D. M., Tibshirani R., and Hastie, T. (2009) A penalized matrix decomposition, with applications to sparse principal components and canonical correlation analysis, Biostatistics, Gol 10 (3), 515-534, Jul 2009

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70``` ```# Try PMD with L1 penalty on rows and columns: type="standard" # A simple simulated example set.seed(1) # Our data is a rank-one matrix, plus noise. The underlying components # contain 50 and 75 non-zero elements, respectively. u <- matrix(c(rnorm(50), rep(0,150)), ncol=1) v <- matrix(c(rnorm(75),rep(0,225)), ncol=1) x <- u%*%t(v)+ matrix(rnorm(200*300),ncol=300) # We can use cross-validation to try to find optimal value of sumabs cv.out <- PMD.cv(x, type="standard", sumabss=seq(0.1, 0.6, len=20)) print(cv.out) plot(cv.out) # The optimal value of sumabs is 0.4157, but we can get within one # standard error of that CV error using sumabs=0.337, which corresponds to # an average of 45.8 and 71.8 non-zero elements in each component - pretty # close to the true model. # We can fit the model corresponding to the lowest cross-validation error: out <- PMD(x, type="standard", sumabs=cv.out\$bestsumabs, K=1, v=cv.out\$v.init) print(out) par(mfrow=c(2,2)) par(mar=c(2,2,2,2)) plot(out\$u[,1], main="Est. u") plot(out\$v[,1], main="Est. v") plot(u, main="True u") plot(v, main="True v") # And if we want to control sumabsu and sumabsv separately, we can do # that too. Let's get 2 components while we're at it: out2 <- PMD(x, type="standard", K=2, sumabsu=6, sumabsv=8, v=out\$v.init, cnames=paste("v", sep=" ", 1:ncol(x)), rnames=paste("u", sep=" ", 1:nrow(x))) print(out2) # Now check out PMD with L1 penalty on rows and fused lasso penalty on # columns: type="ordered". We'll use the Chin et al (2006) Cancer Cell # data set; try "?breastdata" for more info. ## Not run: data(breastdata) attach(breastdata) # dna contains CGH data and chrom contains chromosome of each CGH spot; # nuc contains position of each CGH spot. dna <- t(dna) # Need samples on rows and CGH spots on columns # First, look for shared regions of gain/loss on chromosome 1. # Use cross-validation to choose tuning parameter value par(mar=c(2,2,2,2)) cv.out <- PMD.cv(dna[,chrom==1],type="ordered",chrom=chrom[chrom==1], nuc=nuc[chrom==1], sumabsus=seq(1, sqrt(nrow(dna)), len=15)) print(cv.out) plot(cv.out) out <- PMD(dna[,chrom==1],type="ordered", sumabsu=cv.out\$bestsumabsu,chrom=chrom[chrom==1],K=1,v=cv.out\$v.init, cnames=paste("Pos",sep="", nuc[chrom==1]), rnames=paste("Sample", sep=" ", 1:nrow(dna))) print(out, verbose=TRUE) # Which samples actually have that region of gain/loss? par(mfrow=c(3,1)) par(mar=c(2,2,2,2)) PlotCGH(dna[which.min(out\$u[,1]),chrom==1],chrom=chrom[chrom==1], main=paste(paste(paste("Sample ", sep="", which.min(out\$u[,1])), sep="; u=", round(min(out\$u[,1]),3))),nuc=nuc[chrom==1]) PlotCGH(dna[88,chrom==1], chrom=chrom[chrom==1], main=paste("Sample 88; u=", sep="", round(out\$u[88,1],3)), nuc=nuc[chrom==1]) PlotCGH(out\$v[,1],chrom=chrom[chrom==1], main="V",nuc=nuc[chrom==1]) detach(breastdata) ## End(Not run) ```