# SPC: Perform sparse principal component analysis In PMA: Penalized Multivariate Analysis

## Description

Performs sparse principal components analysis by applying PMD to a data matrix with lasso (\$L_1\$) penalty on the columns and no penalty on the rows.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14``` ```SPC( x, sumabsv = 4, niter = 20, K = 1, orth = FALSE, trace = TRUE, v = NULL, center = TRUE, cnames = NULL, vpos = FALSE, vneg = FALSE, compute.pve = TRUE ) ```

## Arguments

 `x` Data matrix of dimension \$n x p\$, which can contain NA for missing values. We are interested in finding sparse principal components of dimension \$p\$. `sumabsv` 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. `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. `orth` If TRUE, then use method of Section 3.2 of Witten, Tibshirani and Hastie (2008) to obtain multiple sparse principal components. Default is FALSE. `trace` Print out progress as iterations are performed? Default is TRUE. `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(\$L_1\$, \$L_1\$) 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. `center` Subtract out mean of x? Default is TRUE `cnames` An optional vector containing a name for each column. `vpos` Constrain the elements of v to be positive? TRUE or FALSE. `vneg` Constrain the elements of v to be negative? TRUE or FALSE. `compute.pve` Compute percent variance explained? Default TRUE. If not needed, then choose FALSE to save time.

## Details

PMD(x,sumabsu=sqrt(nrow(x)), sumabsv=3, K=1) and SPC(x,sumabsv=3, K=1) give the same result, since the SPC method is simply PMD with an L1 penalty on the columns and no penalty on the rows.

In Witten, Tibshirani, and Hastie (2008), two methods are presented for obtaining multiple factors for SPC. The methods are as follows:

(1) If one has already obtained factors \$k-1\$ factors then oen can compute residuals by subtracting out these factors. Then \$u_k\$ and \$v_k\$ can be obtained by applying the SPC/PMD algorithm to the residuals.

(2) One can require that \$u_k\$ be orthogonal to \$u_i\$'s with \$i<k\$; the method is slightly more complicated, and is explained in WT&H(2008).

Method 1 is performed by running SPC with option orth=FALSE (the default) and Method 2 is performed using option orth=TRUE. Note that Methods 1 and 2 always give identical results for the first component, and often given quite similar results for later components.

## 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. These are the sparse principal components. 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; it is the diagonal of the matrix \$D\$ in the penalized matrix decomposition. In the case of the rank-1 decomposition, it is given in the formulation \$||X-duv'||_F^2\$ subject to \$||u||_1 <= sumabsu\$, \$||v||_1 <= sumabsv\$. 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. `prop.var.explained` A vector containing the proportion of variance explained by the first 1, 2, ..., K sparse principal components obtaineds. Formula for proportion of variance explained is on page 20 of Shen & Huang (2008), Journal of Multivariate Analysis 99: 1015-1034. `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 SPC 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``` ```# A simple simulated example #NOT RUN #set.seed(1) #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) ## Perform Sparse PCA - that is, decompose a matrix w/o penalty on rows ## and w/ L1 penalty on columns ## First, we perform sparse PCA and get 4 components, but we do not ## require subsequent components to be orthogonal to previous components #out <- SPC(x,sumabsv=3, K=4) #print(out,verbose=TRUE) ## We could have selected sumabsv by cross-validation, using function SPC.cv ## Now, we do sparse PCA using method in Section 3.2 of WT&H(2008) for getting ## multiple components - that is, we require components to be orthogonal #out.orth <- SPC(x,sumabsv=3, K=4, orth=TRUE) #print(out.orth,verbose=TRUE) #par(mfrow=c(1,1)) #plot(out\$u[,1], out.orth\$u[,1], xlab="", ylab="") ## Note that the first components w/ and w/o orth option are identical, ## since the orth option only affects the way that subsequent components ## are found #print(round(t(out\$u)%*%out\$u,4)) # not orthogonal #print(round(t(out.orth\$u)%*%out.orth\$u,4)) # orthogonal # ## Use SPC.cv to choose tuning parameters: #cv.out <- SPC.cv(x) #print(cv.out) #plot(cv.out) #out <- SPC(x, sumabsv=cv.out\$bestsumabsv) #print(out) ## or we could do #out <- SPC(x, sumabsv=cv.out\$bestsumabsv1se) #print(out) # # ```