PMA (version 1.0.4)

SPC: Perform sparse principal component 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

SPC(x, sumabsv=4, niter=20, K=1, orth=FALSE, trace=TRUE, v=NULL,
center=TRUE, cnames=NULL, vpos=FALSE, vneg=FALSE)

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.

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

trace

Print out progress as iterations are performed? Default is TRUE.

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.

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.

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="" pmd="" function="" $x$="" is="" data="" matrix="" input="" to="" function.<="" description="">
• 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.

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

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.

References

Witten, DM and Tibshirani, R and T Hastie (2008) A penalized matrix decomposition, with applications to sparse principal components and canonical correlation analysis. Submitted.

See Also

SPC.cv, PMD, PMD.cv

Examples

# A simple simulated example
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)