qplsc.mp: Quadratically penalized least squares with constraints
In vows: Voxelwise Semiparametrics

Description Usage Arguments Value Author(s) References Examples

Fits a possibly very large number of models, with common design matrix, by quadratically penalized least squares, with identifiability constraints imposed. This function serves as the fitting engine for semipar.mp.

1 2	qplsc.mp(Y, modmat, penmat, constr.list = NULL, lsp, nulldim = NULL, store.reml = FALSE, store.fitted = FALSE)

`Y`	an n \times V response matrix (V refers to number of models fitted in parallel, e.g., voxels in neuroimaging applications).
`modmat`	model matrix, e.g., a matrix of B-spline basis functions.
`penmat`	penalty matrix.
`constr.list`	a list of length equal to number of constraints to be imposed, containing information for reparametization to an unconstrained optimization. Attribute `'C'` is the constraint matrix, and `'start'` and `'end'` refer to the corresponding column positions of the model matrix.
`lsp`	vector of candidate tuning parameters (\log(λ)).
`nulldim`	null space dimension, ordinarily equal to the order of the derivative penalty.
`store.reml`	logical: should the pointwise REML criterion at each grid point be included in the output? `FALSE` by default, as this output can be very large.
`store.fitted`	logical: should the fitted values be included in the output? `FALSE` by default.

An object of class "qplsc.mp", which is a list with elements:

`fitted`	fitted value matrix, if `store.fitted = TRUE`.
`edf`	matrix giving the effective degrees of freedom per parameter, as in Wood (2004), for each model.
`pwdf`	vector of point-wise degrees of freedom, equal to the column sums of `edf`.
`pwlsp`	vector of point-wise log smoothing parameters.
`coef`	matrix of coefficients.
`reml`	matrix giving the point-wise REML criterion at each grid point, if `store.reml = TRUE`.
`modmat`	model matrix.
`penmat`	penalty matrix.
`RinvU`	R^{-1}U, as in Reiss et al. (2014); this and `tau` are used for plotting.
`tau`	singular values of R^{-T}PR^{-1}, as in Reiss et al. (2014).
`sigma2`	vector of variance estimates.
`ttu`	matrix for transformation to an unconstrained problem.

Lei Huang huangracer@gmail.com, Yin-Hsiu Chen enjoychen0701@gmail.com, and Philip Reiss phil.reiss@nyumc.org

Reiss, P. T., Huang, L., Chen, Y.-H., Huo, L., Tarpey, T., and Mennes, M. (2014). Massively parallel nonparametric regression, with an application to developmental brain mapping. Journal of Computational and Graphical Statistics, Journal of Computational and Graphical Statistics, 23(1), 232–248.

Wood, S. N. (2004). Stable and efficient multiple smoothing parameter estimation for generalized additive models. Journal of the American Statistical Association, 99, 673–686.