refund (version 0.1-23)

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.

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

# NOT RUN {
# }
# 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)
# }
# NOT RUN {
# }