refund (version 0.1-1)

peer_old: 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_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

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

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 fun

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_old() 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)
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)