af_old: Construct an FGAM regression term

Description

Defines a term $\int_{T}F(X_i(t),t)dt$ for inclusion in an mgcv::gam-formula (or bam or gamm or gamm4:::gamm) as constructed by fgam, where $F(x,t)$$ is an unknown smooth bivariate function and $X_i(t)$ is a functional predictor on the closed interval $T$. Defaults to a cubic tensor product B-spline with marginal second-order difference penalties for estimating $F(x,t)$. The functional predictor must be fully observed on a regular grid

Usage

af_old(
  X,
  argvals = seq(0, 1, l = ncol(X)),
  xind = NULL,
  basistype = c("te", "t2", "s"),
  integration = c("simpson", "trapezoidal", "riemann"),
  L = NULL,
  splinepars = list(bs = "ps", k = c(min(ceiling(nrow(X)/5), 20), min(ceiling(ncol(X)/5),
    20)), m = list(c(2, 2), c(2, 2))),
  presmooth = TRUE,
  Xrange = range(X),
  Qtransform = FALSE
)

Value

A list with the following entries:

call - a "call" to te (or s, t2) using the appropriately constructed covariate and weight matrices.
argvals - the argvals argument supplied to af
L the matrix of weights used for the integration xindname the name used for the functional predictor variable in the formula used by mgcv. tindname - the name used for argvals variable in the formula used by mgcv Lname - the name used for the L variable in the formula used by mgcv presmooth - the presmooth argument supplied to af Qtranform - the Qtransform argument supplied to af Xrange - the Xrange argument supplied to af ecdflist - a list containing one empirical cdf function from applying ecdf to each (possibly presmoothed) column of X. Only present if Qtransform=TRUE Xfd - an fd object from presmoothing the functional predictors using smooth.basisPar. Only present if presmooth=TRUE. See fd.

Arguments

X: an N by J=ncol(argvals) matrix of function evaluations $X_i(t_{i1}),., X_i(t_{iJ}); i=1,.,N.$
argvals: matrix (or vector) of indices of evaluations of $X_i(t)$; i.e. a matrix with ith row $(t_{i1},.,t_{iJ})$
xind: Same as argvals. It will discard this argument in the next version of refund.
basistype: defaults to "te", i.e. a tensor product spline to represent $F(x,t)$ Alternatively, use "s" for bivariate basis functions (see s) or "t2" for an alternative parameterization of tensor product splines (see t2)
integration: method used for numerical integration. Defaults to "simpson"'s rule for calculating entries in L. Alternatively and for non-equidistant grids, "trapezoidal" or "riemann". "riemann" integration is always used if L is specified
L: optional weight matrix for the linear functional
splinepars: optional arguments specifying options for representing and penalizing the function $F(x,t)$. Defaults to a cubic tensor product B-spline with marginal second-order difference penalties, i.e. list(bs="ps", m=list(c(2, 2), c(2, 2)), see te or s for details
presmooth: logical; if true, the functional predictor is pre-smoothed prior to fitting; see smooth.basisPar
Xrange: numeric; range to use when specifying the marginal basis for the x-axis. It may be desired to increase this slightly over the default of range(X) if concerned about predicting for future observed curves that take values outside of range(X)
Qtransform: logical; should the functional be transformed using the empirical cdf and applying a quantile transformation on each column of X prior to fitting? This ensures Xrange=c(0,1). If Qtransform=TRUE and presmooth=TRUE, presmoothing is done prior to transforming the functional predictor

Author

Mathew W. McLean mathew.w.mclean@gmail.com and Fabian Scheipl

References