peer: Construct a PEER regression term in a `pfr` formula

Description

Defines a term $\int_{T} β (t) X_{i} (t) d t$ for inclusion in a pfr formula, where $β (t)$ is estimated with structured penalties (Randloph et al., 2012).

Usage

peer(X, argvals = NULL, pentype = "RIDGE", Q = NULL, phia = 10^3,
  L = NULL, ...)

Arguments

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. $(t_{i 1}, ., t_{i J})$ 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.)

pentype

the type of penalty to apply, one of "RIDGE", "D", "DECOMP", or "USER"; see Details.

matrix $Q$ used for pentype="DECOMP"; see Details.

phia

scalar $a$ used for pentype="DECOMP"; see Details.

user-supplied penalty matrix for pentype="USER"; see Details.

...

additional arguments to be passed to lf (and then possibly s). Arguments processed by lf include, for example, integration for specifying the method of numerical integration. Arguments processed by s include information related to basis and penalization, such as m for specifying the order of the difference penalty; See Details. xt-argument is not allowed for peer-terms and will cause an error.

Details

peer is a wrapper for lf, which defines linear functional predictors for any type of basis. It simply calls lf with the appropriate options for the peer basis and penalty construction. The type of penalty is determined by the pentype argument. There are four types of penalties available:

pentype=="RIDGE" for a ridge penalty, the default
pentype=="D" for a difference penalty. The order of the difference penalty may be specified by supplying an m argument (default is 2).
pentype=="DECOMP" for a decomposition-based penalty, $b P_{Q} + a (I - P_{Q})$ , where $P_{Q} = Q^{t} (Q Q^{t})^{- 1} Q$ . The $Q$ matrix must be specified by Q, and the scalar $a$ by phia. The number of columns of Q must be equal to the length of the data. Each row represents a basis function where the functional predictor is expected to lie, according to prior belief.
pentype=="USER" for a user-specified penalty matrix, supplied by the L argument.

The original stand-alone implementation by Madan Gopal Kundu is available in peer_old.

References

Randolph, T. W., Harezlak, J, and Feng, Z. (2012). Structured penalties for functional linear models - partially empirical eigenvectors for regression. Electronic Journal of Statistics, 6, 323-353.

Kundu, M. G., Harezlak, J., and Randolph, T. W. (2012). Longitudinal functional models with structured penalties (arXiv:1211.4763 [stat.AP]).

Examples

Run this code

# NOT RUN {
#------------------------------------------------------------------------
# Example 1: Estimation with D2 penalty
#------------------------------------------------------------------------

data(DTI)
DTI = DTI[which(DTI$case == 1),]
fit.D2 = pfr(pasat ~ peer(cca, pentype="D"), data=DTI)
plot(fit.D2)

#------------------------------------------------------------------------
# Example 2: Estimation with structured penalty (need structural
#            information about regression function or predictor function)
#------------------------------------------------------------------------

data(PEER.Sim)
data(Q)
PEER.Sim1<- subset(PEER.Sim, t==0)

# Setting k to max possible value
fit.decomp <- pfr(Y ~ peer(W, pentype="Decomp", Q=Q, k=99), data=PEER.Sim1)
plot(fit.decomp)
# }

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