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peer_old

Functional Models with Structured Penalties

Description

Implements functional model with structured penalties (Randolph et al., 2012) with scalar outcome and single functional predictor through mixed model equivalence.

Usage

peer_old(
  Y,
  funcs,
  argvals = NULL,
  pentype = "Ridge",
  L.user = NULL,
  Q = NULL,
  phia = 10^3,
  se = FALSE,
  ...
)

Arguments

`Y`	vector of all outcomes
`funcs`	matrix containing observed functional predictors as rows. Rows with `NA` and `Inf` values will be deleted.
`argvals`	matrix (or vector) of indices of evaluations of X_i(t); i.e. a matrix with ith row (t_{i1},.,t_{iJ})
`pentype`	type of penalty. It can be either decomposition based penalty (`DECOMP`) or ridge (`RIDGE`) or second-order difference penalty (`D2`) or any user defined penalty (`USER`). For decomposition based penalty user need to specify Q matrix in Q argument (see details). For user defined penalty user need to specify L matrix in L argument (see details). For Ridge and second-order difference penalty, specification for arguments L and Q will be ignored. Default is `RIDGE`.
`L.user`	penalty matrix. Need to be specified with `pentype='USER'`. Number of columns need to be equal with number of columns of matrix specified to `funcs`. Each row represents a constraint on functional predictor. This argument will be ignored when value of `pentype` is other than `USER`.
`Q`	Q matrix to derive decomposition based penalty. Need to be specified with `pentype='DECOMP'`. Number of columns need to be equal with number of columns of matrix specified to `funcs`. Each row represents a basis function where functional predictor is expected lie according to prior belief. This argument will be ignored when value of `pentype` is other than `DECOMP`.
`phia`	Scalar value of a in decomposition based penalty. Need to be specified with `pentype='DECOMP'`.
`se`	logical; calculate standard error when `TRUE`.
`...`	additional arguments passed to the `lme` function.

Details

If there are any missing or infinite values in Y, and funcs, the corresponding row (or observation) will be dropped. Neither Q nor L may contain missing or infinite values.

peer_old() fits the following model:

y_i=\int {W_i(s)γ(s) ds} + ε_i

where ε_i ~ N(0,σ^2). For all the observations, predictor function W_i(s) is evaluated at K sampling points. Here, γ (s) denotes the regression function.

Values of y_i and W_i(s)are passed through argument Y and funcs, respectively. Number of elements or rows in Y and funcs need to be equal.

The estimate of regression functions γ(s) is obtained as penalized estimated. Following 3 types of penalties can be used:

i. Ridge: I_K

ii. Second-order difference: [d_{i,j}] with d_{i,i} = d_{i,i+2} = 1, d_{i,i+1} = -2, otherwise d_{i,i} =0

iii. Decomposition based penalty: bP_Q+a(I-P_Q) where P_Q= Q^T(QQ^T)^{-1}Q

For Decomposition based penalty user need to specify pentype='DECOMP' and associated Q matrix need to be passed through Q argument.

Alternatively, user can pass directly penalty matrix through argument L. For this user need to specify pentype='USER' and associated L matrix need to be passed through L argument.

Default penalty is Ridge penalty and user needs to specify RIDGE. For second-order difference penalty, user needs to specify D2.

Value

a list containing:

`fit`	result of the call to `lme`
`fitted.vals`	predicted outcomes
`Gamma`	estimates with standard error for regression function
`GammaHat`	estimates of regression function
`se.Gamma`	standard error associated with `GammaHat`
`AIC`	AIC value of fit (smaller is better)
`BIC`	BIC value of fit (smaller is better)
`logLik`	(restricted) log-likelihood at convergence
`lambda`	estimates of smoothing parameter
`N`	number of subjects
`K`	number of Sampling points in functional predictor
`sigma`	estimated within-group error standard deviation.

Author(s)

Madan Gopal Kundu mgkundu@iupui.edu

References

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

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.

Examples

## Not run: 
#------------------------------------------------------------------------
# Example 1: Estimation with D2 penalty
#------------------------------------------------------------------------

## Load Data
data(DTI)

## Extract values for arguments for peer() from given data
cca = DTI$cca[which(DTI$case == 1),]
DTI = DTI[which(DTI$case == 1),]

##1.1 Fit the model
fit.cca.peer1 = peer(Y=DTI$pasat, funcs = cca, pentype='D2', se=TRUE)
plot(fit.cca.peer1)

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

## Load Data
data(PEER.Sim)

## Extract values for arguments for peer() from given data
PEER.Sim1<- subset(PEER.Sim, t==0)
W<- PEER.Sim1$W
Y<- PEER.Sim1$Y

##Load Q matrix containing structural information
data(Q)

##2.1 Fit the model
Fit1<- peer(Y=Y, funcs=W, pentype='Decomp', Q=Q, se=TRUE)
plot(Fit1)

## End(Not run)

refund

Regression with Functional Data

v0.1-23

GPL (>= 2)

Authors

Jeff Goldsmith [aut], Fabian Scheipl [aut], Lei Huang [aut], Julia Wrobel [aut, cre], Chongzhi Di [aut], Jonathan Gellar [aut], Jaroslaw Harezlak [aut], Mathew W. McLean [aut], Bruce Swihart [aut], Luo Xiao [aut], Ciprian Crainiceanu [aut], Philip T. Reiss [aut], Yakuan Chen [ctb], Sonja Greven [ctb], Lan Huo [ctb], Madan Gopal Kundu [ctb], So Young Park [ctb], David L. Miller [ctb], Ana-Maria Staicu [ctb]

Initial release

2020-12-03