# lf: Construct an FLM regression term In refund: Regression with Functional Data

## Description

Defines a term \int_{T}β(t)X_i(t)dt for inclusion in an mgcv::gam-formula (or bam or gamm or gamm4:::gamm) as constructed by pfr, where β(t) is an unknown coefficient function and X_i(t) is a functional predictor on the closed interval T. See smooth.terms for a list of basis and penalty options; the default is thin-plate regression splines, as this is the default option for s.

## Usage

  1 2 3 4 5 6 7 8 9 10 lf( X, argvals = NULL, xind = NULL, integration = c("simpson", "trapezoidal", "riemann"), L = NULL, presmooth = NULL, presmooth.opts = NULL, ... ) 

## Arguments

 X functional predictors, typically expressed as an N by J matrix, where N is the number of columns and J is the number of evaluation points. May include missing/sparse functions, which are indicated by NA values. Alternatively, can be an object of class "fd"; see fd. argvals indices of evaluation of X, i.e. (t_{i1},.,t_{iJ}) for subject i. May be entered as either a length-J vector, or as an N by J matrix. Indices may be unequally spaced. Entering as a matrix allows for different observations times for each subject. If NULL, defaults to an equally-spaced grid between 0 or 1 (or within X$basis$rangeval if X is a fd object.) xind same as argvals. It will not be supported in the next version of refund. 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". L an optional N by ncol(argvals) matrix giving the weights for the numerical integration over t. If present, overrides integration. presmooth string indicating the method to be used for preprocessing functional predictor prior to fitting. Options are fpca.sc, fpca.face, fpca.ssvd, fpca.bspline, and fpca.interpolate. Defaults to NULL indicating no preprocessing. See create.prep.func. presmooth.opts list including options passed to preprocessing method create.prep.func. ... optional arguments for basis and penalization to be passed to mgcv::s. These could include, for example, "bs", "k", "m", etc. See s for details.

## 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 lf 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 LXname the name used for the L variable in the formula used by mgcv presmooth the presmooth argument supplied to lf prep.func a function that preprocesses data based on the preprocessing method specified in presmooth. See create.prep.func

## Author(s)

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

## References

Goldsmith, J., Bobb, J., Crainiceanu, C., Caffo, B., and Reich, D. (2011). Penalized functional regression. Journal of Computational and Graphical Statistics, 20(4), 830-851.

Goldsmith, J., Crainiceanu, C., Caffo, B., and Reich, D. (2012). Longitudinal penalized functional regression for cognitive outcomes on neuronal tract measurements. Journal of the Royal Statistical Society: Series C, 61(3), 453-469.

pfr, af, mgcv's smooth.terms and linear.functional.terms; pfr for additional examples
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 data(DTI) DTI1 <- DTI[DTI$visit==1 & complete.cases(DTI),] # We can apply various preprocessing options to the DTI data fit1 <- pfr(pasat ~ lf(cca, k=30), data=DTI1) fit2 <- pfr(pasat ~ lf(cca, k=30, presmooth="fpca.sc", presmooth.opts=list(nbasis=8, pve=.975)), data=DTI1) fit3 <- pfr(pasat ~ lf(cca, k=30, presmooth="fpca.face", presmooth.opts=list(m=3, npc=9)), data=DTI1) fit4 <- pfr(pasat ~ lf(cca, k=30, presmooth="fpca.ssvd"), data=DTI1) fit5 <- pfr(pasat ~ lf(cca, k=30, presmooth="bspline", presmooth.opts=list(nbasis=8)), data=DTI1) fit6 <- pfr(pasat ~ lf(cca, k=30, presmooth="interpolate"), data=DTI1) # All models should result in similar fits fits <- as.data.frame(lapply(1:6, function(i) get(paste0("fit",i))$fitted.values)) names(fits) <- c("none", "fpca.sc", "fpca.face", "fpca.ssvd", "bspline", "interpolate") pairs(fits)