PMD.cv {PMA}

Do tuning parameter selection for PMD via cross-validation

Description

Performs cross-validation to select tuning parameters for rank-1 PMD, the penalized matrix decomposition for a data matrix.

Usage

PMD.cv(
  x,
  type = c("standard", "ordered"),
  sumabss = seq(0.1, 0.7, len = 10),
  sumabsus = NULL,
  lambda = NULL,
  nfolds = 5,
  niter = 5,
  v = NULL,
  chrom = NULL,
  nuc = NULL,
  trace = TRUE,
  center = TRUE,
  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.

sumabss

Used only if type is "standard". A vector of sumabs values to be used. Sumabs is a measure of sparsity for u and v vectors, between 0 and

1. When sumabss is specified, and sumabsus and sumabsvs are NULL, then sumabsus is set to $sqrt(n)*sumabss$ and sumabsvs is set at $sqrt(p)*sumabss$. If sumabss is specified, then sumabsus and sumabsvs should be NULL. Or if sumabsus and sumabsvs are specified, then sumabss should be NULL.

sumabsus

Used only for type "ordered". A vector of sumabsu values to be used. Sumabsu measures sparseness of u - it is the sum of absolute values of elements of u. Must be between 1 and sqrt(n).

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.

nfolds

How many cross-validation folds should be performed? Default is 5.

niter

How many iterations should be performed. For speed, only 5 are performed by default.

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.

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.

nuc

If type is "ordered", can specify the nucleotide position of each CGH spot (column of x), to be used in plotting. If NULL, then it is assumed that CGH spots are equally spaced.

trace

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

center

Subtract out mean of x? Default is TRUE

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

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.

Cross-validation of the rank-1 PMD is performed over sumabss (if type is "standard") or over sumabsus (if type is "ordered"). If type is "ordered", then lambda is chosen from the data without cross-validation.

The cross-validation works as follows: Some percent of the elements of $x$ is removed at random from the data matrix. The PMD is performed for a range of tuning parameter values on this partially-missing data matrix; then, missing values are imputed using the decomposition obtained. The value of the tuning parameter that results in the lowest sum of squared errors of the missing values if "best".

To do cross-validation on the rank-2 PMD, first the rank-1 PMD should be computed, and then this function should be performed on the residuals, given by $x-udv'$.

Value

cv

Average sum of squared errors obtained over cross-validation folds.

cv.error

Standard error of average sum of squared errors obtained over cross-validation folds.

bestsumabs

If type="standard", then value of sumabss resulting in smallest CV error is returned.

bestsumabsu

If type="ordered", then value of sumabsus resulting in smallest CV error is returned.

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.

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

See Also

PMD, SPC

Examples

# See examples in PMD help file