MultiCCA.permute {PMA}

Select tuning parameters for sparse multiple canonical correlation analysis using the penalized matrix decomposition.

Description

This function can be used to automatically select tuning parameters for sparse multiple CCA. This is the analog of sparse CCA, when >2 data sets are available. Each data set may have features of type="standard" or type="ordered" (e.g. CGH data). Assume that there are K data sets, called $X1,...,XK$.

Usage

MultiCCA.permute(
  xlist,
  penalties = NULL,
  ws = NULL,
  type = "standard",
  nperms = 10,
  niter = 3,
  trace = TRUE,
  standardize = TRUE
)

Arguments

xlist

A list of length K, where K is the number of data sets on which to perform sparse multiple CCA. Data set k should be a matrix of dimension $n x p_k$ where $p_k$ is the number of features in data set k.

penalties

The penalty terms to be considered in the cross-validation. If the same penalty term is desired for each data set, then this should be a vector of length equal to the number of penalty terms to be considered. If different penalty terms are desired for each data set, then this should be a matrix with rows equal to the number of data sets, and columns equal to the number of penalty terms to be considered. For a given data set Xk, if type is "standard" then the penalty term should be a number between 1 and $sqrt(p_k)$ (the number of features in data set k); it is a L1 bound on wk. If type is "ordered", on the other hand, the penalty term is of the form lambda in the fused lasso penalty. Therefore, the interpretation of the argument depends on whether type is "ordered" or "standard" for this data set.

ws

A list of length K; the kth element contanis the first ncomponents columns of the v matrix of the SVD of Xk. If NULL, then the SVD of Xk will be computed inside this function. However, if you plan to run this function multiple times, then save a copy of this argument so that it does not need to be re-computed.

type

A K-vector containing elements "standard" or "ordered" - or a single value. If a single value, then it is assumed that all elements are the same (either "standard" or "ordered"). If columns of v are ordered (e.g. CGH spots ordered along the chromosome) then "ordered", otherwise use "standard". "standard" will result in a lasso ($L_1$) penalty on v, which will result in smoothness. "ordered" will result in a fused lasso penalty on v, yielding both sparsity and smoothness.

nperms

How many times should the data be permuted? Default is 25. A large value of nperms is very important here, since the formula for computing the z-statistics requires a standard deviation estimate for the correlations obtained via permutation, which will not be accurate if nperms is very small.

niter

How many iterations should be performed each time CCA is called? Default is 3, since an approximate estimate of u and v is acceptable in this case, and otherwise this function can be quite time-consuming.

trace

Print out progress?

standardize

Should the columns of X and Z be centered (to have mean zero) and scaled (to have standard deviation 1)? Default is TRUE.

Details

The tuning parameters are selected using a permutation scheme. For each candidate tuning parameter value, the following is performed: (1) Repeat the following n times, for n large: (a) The samples in $(X1,...,XK)$ are randomly permuted to obtain data sets $(X1*,...,XK*)$. (b) Sparse multiple CCA is run on the permuted data sets $(X1*,...,XK*)$ to get canonical variates $(w1*,...,wK*)$. (c) Record $t* = sum_(i<j) Cor(Xi* wi*, Xj* wj*)$. (2) Sparse CCA is run on the original data $(X1,...,XK)$ to obtain canonical variates $(w1,...,wK)$. (3) Record $t = sum_(i<j) Cor(Xi wi, Xj wj)$. (4) The resulting p-value is given by $mean(t* > t)$; that is, the fraction of permuted totals that exceed the total on the real data. Then, choose the tuning parameter value that gives the smallest value in Step 4.

This function only selets tuning parameters for the FIRST sparse multiple CCA factors.

Note that $x1,...,xK$ must have same number of rows. This function performs just a one-dimensional search in tuning parameter space.

Value

zstat

The vector of z-statistics, one per element of penalties.

pvals

The vector of p-values, one per element of penalties.

bestpenalties

The best set of penalties (the one with the highest zstat).

cors

The value of $sum_(j<k) cor(Xk wk, Xj wj)$ obtained for each value of penalties.

corperms

The nperms values of $sum_(j<k) cor(Xk* wk*, Xj* wj*)$ obtained for each value of penalties, where Xk* indicates the Xk matrix with permuted rows, and wk* is the canonical variate corresponding to the permuted data.

ws.init

Initial values used for ws in sparse multiple CCA algorithm.

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

MultiCCA, CCA.permute, CCA

Examples

# See examples in MultiCCA function