refund (version 0.1-1)

peer: Functional Models with Structured Penalties

Description

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

Usage

peer(Y, funcs, 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.

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 ne

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 ign

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 functi

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.

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.

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() fits the following model: $y_i=\int {W_i(s)\gamma(s) ds} + \epsilon_i$ where $\epsilon_i ~ N(0,\sigma^2)$. For all the observations, predictor function $W_i(s)$ is evaluated at K sampling points. Here, $\gamma (s)$ 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 $\gamma(s)$ 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.

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.

See Also

lpeer, plot.peer

Examples

#------------------------------------------------------------------------
# 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)
K<- 100
W<- PEER.Sim1[,c(3:(K+2))]
Y<- PEER.Sim1[,K+3]

##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)