vows (version 0.3.1)

qplsc.mp: Quadratically penalized least squares with constraints

Description

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.

Usage

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

Arguments

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

lsp

vector of candidate tuning parameters ($\log(\lambda)$).

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.

Value

• 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.

References

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.

Examples

## see semipar.mp