GPFDA (version 2.2)

gpfr: Gaussian Process in functional data.

Description

Use functional regression to be the mean structure and Gaussian Process to be the covariance structure.

Usage

gpfr(response,lReg=NULL,fReg=NULL,fyList=NULL,fbetaList_l=NULL,
      fxList=NULL,fbetaList=NULL,concurrent=TRUE,fbetaList_f=NULL,
      gpReg=NULL,hyper=NULL,Cov=c('pow.ex','linear'),gamma=1,
      time=NULL,NewHyper=NULL,accuracy=c('high','normal','low'),
      trace.iter=5,fitting=FALSE,rPreIdx=FALSE)

Arguments

response

The training response. can be an fd object or a matrix with nrow samples, ncol time points

lReg

The input variable for functional linear regression with scale covariates. Expected to be a matrix with nrow samples.

fReg

The input variable for functional linear regression with functional covariates. Expected to be a matrix with nrow samples, or an fd object, or a list of matrices or fd objects.

fyList

The list to control the smoothing of response. See details for more info.

fbetaList_l

The list to control the smoothing of beta for functional regression with scale covariates. See details for more info.

fxList

The list to control the smoothing of functional covariates for functional regression with functional covariates. See details for more info.

fbetaList_f

The list to control the smoothing of beta for functional regression with functional covariates. See details for more info.

fbetaList

The list to control the smoothing of functional covariates for functional regression with functional covariates and scale response. Not available for now.

concurrent

Logical. If True concurrent functional regression will be carried out, otherwise the full functional regression will be carried out.

gpReg

Data for training Gaussian Process. Expecting matrix, fd object, list of matrices or list of fd objects.

Cov

Kernel function or covariance function type(s).

hyper

Hyper parameter initial value. Default to be NULL.

NewHyper

Vector of the names of the new hyper parameters from customized kernel function.

gamma

Power parameter used in powered exponential.

time

Time used in globle setting for functional objects.

accuracy

Optimization accuracy. Default to be high.

trace.iter

Print the processing of iterations of optimization.

fitting

Is fitting required or not. Default to be F.

rPreIdx

Logical. If True, do random selected index for pre-optimization, otherwise use fixed index.

Value

A list of

hyper

Estimated hyper-parameters

I

A vector of estimated standard deviation of hyper-parameters

modellist

List of models fitted before Gaussian process

CovFun

Covariance function

gamma

gamma used in Gaussian process powered exponential kernel

init_resp

Initial response value

resid_resp

Residual after the fitted value from models has been taken out

fitted

Fitted value

fitted.sd

Standard deviation of the fitted value

ModelType

The model applied in the function.

lTrain

Training data for functional regression with scalper covariates

fTrain

Training data for functional regression with functional covariates

mfTrainfd

List of fd objects that from training data for functional regression with functional covariates

gpTrain

Training data for Gaussian Process

time

Time used in training in Gaussian Process

iuuL

Inverse of covariance matrix for lReg

iuuF

Inverse of covariance matrix for fReg

fittedFM

Fitted value from functional regression

fyList

fyList used in the function

Details

fyList is a list with items: `time': a sequence of time points default to be 100 points from 0 to 1; `nbasis': number of basis functions used in smoothing, default to be less or equal to 23; `norder': the order of the functional curves default to be 6, `bSpline': logical, if True, b-spline is used, otherwise use Fourier basis, default to be True; `Pen': default to be c(0,0), means that the penalty is on the second order derivative of the curve, since the weight for zero-th and first order derivatives of the curve are zero, `lambda':default to be 1e-4, the smoothing parameter for the penalty.

fxList is a list of lists which are similar to fyList. Because it may contain different information for more than one functional covariates.

fbetaList, fbetaList_l and fbetaList_f are similar to each other. They are also expected to be list of lists. The items in each sub-list are: `rtime': range of time, default to be 0 and 1; `nbasis': number of basis functions used in smoothing, default to be less or equal to 19; norder: the order of the functional curves default to be 6;`bSpline': logical, if True, b-spline is used, otherwise use Fourier basis, default to be True; `Pen': default to be c(0,0);`lambda':default to be 1e4;`bivar':logical, if True, the bivariate basis will be calculated, otherwise normal basis, default to be False; `lambdas':the smoothing parameter for the penalty of the additional basis, default to be 1e4.

Note that user only write the item they need to change in the list, all items have default settings. See example below.

References

Shi, J Q., and Choi, T. (2011), Gaussian Process Regression Analysis for Functional Data, Springer, New York.

Ramsay, James O., and Silverman, Bernard W. (2006), Functional Data Analysis, 2nd ed., Springer, New York.

See Also

gpr

Examples

# NOT RUN {
library(GPFDA)

traindata=vector('list',20)
for(i in 1:20) traindata[[i]]=i
n=50
traindata=lapply(traindata,function(i) {
  x=seq(-3,3,len=n)
  y=sin(x^2)-x+0.2*rnorm(n,runif(1,-3,3),runif(1,0.5,3))
  x1=0.5*x^3+exp(x)+rnorm(n,runif(1,-3,3),runif(1,0.5,5))
  x2=cos(x^3)+0.2*rnorm(n,runif(1,-3,3),runif(1,0.5,5))
  mat=cbind(x,x1,x2,y)
  colnames(mat)=c('time','x1','x2','y')
  scale=t(c(2*(mean(y)>0.25)-1,(var(y)>3.6)*2-1,(sd(y)-sd(x)>1.4)*2-1))
  i=list(mat,scale)
})

lx=do.call('rbind',lapply(traindata,function(i)i[[2]]))
fx1=do.call('rbind',lapply(traindata,function(i)i[[1]][,2]))
fx2=do.call('rbind',lapply(traindata,function(i)i[[1]][,3]))
fy1=do.call('rbind',lapply(traindata,function(i)i[[1]][,4]))
time_old=traindata[[1]][[1]][,1]

## comment out because running time is a bit long
# system.time(a1<-gpfr(response=(fy1),lReg=lx,fReg=NULL,gpReg=list(fx1,fx2)
#                ,fyList=list(nbasis=23,lambda=0.1),fbetaList_l=
#                list(list(lambda=.01,nbasi=17)),hyper=NULL,
#                Cov=c('pow.ex','linear'),fitting=TRUE,
#                time=seq(-3,3,len=50),rPreIdx=TRUE,concurrent=TRUE))

# }