# CCA.permute: Select tuning parameters for sparse canonical correlation... In PMA: Penalized Multivariate Analysis

## Description

This function can be used to automatically select tuning parameters for sparse CCA using the penalized matrix decompostion. For each data set x and z, two types are possible: (1) type "standard", which does not assume any ordering of the columns of the data set, and (2) type "ordered", which assumes that columns of the data set are ordered and thus that corresponding canonical vector should be both sparse and smooth (e.g. CGH data).

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22``` ```CCA.permute( x, z, typex = c("standard", "ordered"), typez = c("standard", "ordered"), penaltyxs = NULL, penaltyzs = NULL, niter = 3, v = NULL, trace = TRUE, nperms = 25, standardize = TRUE, chromx = NULL, chromz = NULL, upos = FALSE, uneg = FALSE, vpos = FALSE, vneg = FALSE, outcome = NULL, y = NULL, cens = NULL ) ```

## Arguments

 `x` Data matrix; samples are rows and columns are features. `z` Data matrix; samples are rows and columns are features. Note that x and z must have the same number of rows, but may (and generally will) have different numbers of columns. `typex` Are the columns of x unordered (type="standard") or ordered (type="ordered")? If "standard", then a lasso penalty is applied to v, to enforce sparsity. If "ordered" (generally used for CGH data), then a fused lasso penalty is applied, to enforce both sparsity and smoothness. `typez` Are the columns of z unordered (type="standard") or ordered (type="ordered")? If "standard", then a lasso penalty is applied to v, to enforce sparsity. If "ordered" (generally used for CGH data), then a fused lasso penalty is applied, to enforce both sparsity and smoothness. `penaltyxs` The set of x penalties to be considered. If typex="standard", then the L1 bound on u is penaltyxs*sqrt(ncol(x)). If "ordered", then it's the lambda for the fused lasso penalty. The user can specify a single value or a vector of values. If penaltyxs is a vector and penaltyzs is a vector, then the vectors must have the same length. If NULL, then the software will automatically choose a single lambda value if type is "ordered", or a grid of (L1 bounds)/sqrt(ncol(x)) if type is "standard". `penaltyzs` The set of z penalties to be considered. If typez="standard", then the L1 bound on v is penaltyzs*sqrt(ncol(z)). If "ordered", then it's the lambda for the fused lasso penalty. The user can specify a single value or a vector of values. If penaltyzs is a vector and penaltyzs is a vector, then the vectors must have the same length. If NULL, then the software will automatically choose a single lambda value if type is "ordered", or a grid of (L1 bounds)/sqrt(ncol(z)) if type is "standard". `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. `v` The first K columns of the v matrix of the SVD of X'Z. If NULL, then the SVD of X'Z 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 (since that process can be time-consuming if X and Z both have high dimension). `trace` Print out progress? `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. `standardize` Should the columns of X and Z be centered (to have mean zero) and scaled (to have standard deviation 1)? Default is TRUE. `chromx` Used only if typex="ordered"; a vector of length ncol(x) that allows you to specify which chromosome each CGH spot is on. If NULL, then it is assumed that all CGH spots are on same chromosome. `chromz` Used only if typex="ordered"; a vector of length ncol(z) that allows you to specify which chromosome each CGH spot is on. If NULL, then it is assumed that all CGH spots are on same chromosome. `upos` If TRUE, then require all elements of u to be positive in sign. Default is FALSE. Can only be used if type is standard. `uneg` If TRUE, then require all elements of u to be negative in sign. Default is FALSE. Can only be used if type is standard. `vpos` If TRUE, then require all elements of v to be positive in sign. Default is FALSE. Can only be used if type is standard. `vneg` If TRUE, then require all elements of v to be negative in sign. Default is FALSE. Can only be used if type is standard. `outcome` If you would like to incorporate a phenotype into CCA analysis - that is, you wish to find features that are correlated across the two data sets and also correlated with a phenotype - then use one of "survival", "multiclass", or "quantitative" to indicate outcome type. Default is NULL. `y` If outcome is not NULL, then this is a vector of phenotypes - one for each row of x and z. If outcome is "survival" then these are survival times; must be non-negative. If outcome is "multiclass" then these are class labels. Default NULL. `cens` If outcome is "survival" then these are censoring statuses for each observation. 1 is complete, 0 is censored. Default NULL.

## Details

For X and Z, the samples are on the rows and the features are on the columns.

The tuning parameters are selected using a permutation scheme. For each candidate tuning parameter value, the following is performed: (1) The samples in X are randomly permuted nperms times, to obtain matrices \$X*_1,X*_2,...\$. (2) Sparse CCA is run on each permuted data set \$(X*_i,Z)\$ to obtain factors \$(u*_i, v*_i)\$. (3) Sparse CCA is run on the original data (X,Z) to obtain factors u and v. (4) Compute \$c*_i=cor(X*_i u*_i,Z v*_i)\$ and \$c=cor(Xu,Zv)\$. (5) Use Fisher's transformation to convert these correlations into random variables that are approximately normally distributed. Let Fisher(c) denote the Fisher transformation of c. (6) Compute a z-statistic for Fisher(c), using \$(Fisher(c)-mean(Fisher(c*)))/sd(Fisher(c*))\$. The larger the z-statistic, the "better" the corresponding tuning parameter value.

This function also gives the p-value for each pair of canonical variates (u,v) resulting from a given tuning parameter value. This p-value is computed as the fraction of \$c*_i\$'s that exceed c (using the notation of the previous paragraph).

Using this function, only the first left and right canonical variates are considered in selection of the tuning parameter.

Note that x and z must have same number of rows. This function performs just a one-dimensional search in tuning parameter space, even if penaltyxs and penaltyzs both are vectors: the pairs `(penaltyxs,penaltyzs)`, `(penaltyxs,penaltyzs)`,.... are considered.

## Value

 `zstat` The vector of z-statistics, one per element of sumabss. `pvals` The vector of p-values, one per element of sumabss. `bestpenaltyx` The x penalty that resulted in the highest z-statistic. `bestpenaltyz` The z penalty that resulted in the highest z-statistic. `cors` The value of cor(Xu,Zv) obtained for each value of sumabss. `corperms` The nperms values of cor(Xu,Zv*) obtained for each value of sumabss, where X* indicates the X matrix with permuted rows, and u* and v* are the output of CCA using data (X*,Z). `ft.cors` The result of applying Fisher transformation to cors. `ft.corperms` The result of applying Fisher transformation to corperms. `nnonzerous` Number of non-zero u's resulting from applying CCA to data (X,Z) for each value of sumabss. `nnonzerouv` Number of non-zero v's resulting from applying CCA to data (X,Z) for each value of sumabss. `v.init` The first factor of the v matrix of the SVD of x'z. This is saved in case this function (or the CCA function) will be re-run later.

## 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` ```# See examples in CCA function ```