cv.sof.spike: Cross-validation for linear scalar-on-function regression for...

In FRegSigCom: Functional Regression using Signal Compression Approach

Description

This function is used to perform cross-validation and build the final model for highly densely observed spiky data using the signal compression approach for the following linear scalar-on-function regression model:

Y= μ+\int X_1(s)β_1(s)ds+...+\int X_p(s)β_p(s)ds+ε,

where μ is the intercept. The {X_i(s),1≤ i≤ p} are p functional predictors and {β_i(s),1≤ i≤ p} are their corresponding coefficient functions, where p is a positive integer. The ε is the random noise.

We require that all the sample curves of each functional predictor are observed in a common dense grid of time points, but the grid can be different for different predictors. All the sample curves of the functional response are observed in a common dense grid.

Usage

cv.sof.spike(X, Y, t.x, K.cv = 10, upper.level = 10)

Arguments

X	a list of length p, the number of functional predictors. Its i-th element is the n*m_i data matrix for the i-th functional predictor X_i(s), where n is the sample size and m_i is the number of observation time points for X_i(s).
Y	an n dimensional vector of the observed values for the response, where n is the sample size.
t.x	a list of length p. Its i-th element is the vector of obesrvation time points of the i-th functional predictor X_i(s), 1≤ i≤ p.
K.cv	the number of CV folds. Default is 10.
upper.level	the upper bound of the maximum resolution level. The optimal maximum resolution level is chosen between 1 and "upper.level", together with other tuning parameters, by cross-validation.

Details

This method uses wavelet basis to expand X_i(s) and β_i(s), (1 ≤ i ≤ p), and estimates the expansion coefficients of β_i(s)'s by penalized least squares method with penalty

λ∑_{i=1}^p \{∑_{j=0}^{J_1}\{2^{-2α e^{-(j-τ)/α}}2^{2α j}||b_{ij}||^2+ κ ||b_i||^2\}\},

where b_{ij} denotes the vector of wavelet coefficient for β_i(s) at the jth level, and b_{i} is the vector concatenating all b_{ij}, (0≤ j ≤ J_1).

Value

An object of the “cv.sof.spike” class, which is used in the function pred.sof.spike for prediction.

mu	the estimated intercept.
coef	a list of p vectors, where the i-th vector contains the estimated values of the slope coefficient function beta_i(s) at t.x.
...	optimal tuning parameters

Author(s)

Xin Qi and Ruiyan Luo,

References

Xin Qi and Ruiyan Luo, (manuscript) Functional regression for highly densely observed functional data with novel regularity.

Examples

##########################################################################
# Example: scalar-on-function for highly-densely observed curves
##########################################################################


ptm <- proc.time()
library(FRegSigCom)
data(Pork)
X=Pork$X
Y=Pork$Y
ntrain=40 # in paper, we use 80 observations as training data
xtrange=c(0,1) # the range of t in x(t).
t.x.list=list(seq(0,1,length.out=dim(X)[2]))
train.index=sample(1:dim(X)[1], ntrain)
X.train <- X.test <- list()

X.train[[1]]=X[train.index,]
X.test[[1]]=X[-(train.index),]
Y.train <- Y[train.index]
Y.test <- Y[-(train.index)]

fit.cv=cv.sof.spike(X.train, Y.train, t.x.list)
Y.pred=pred.sof.spike(fit.cv, X.test)
pred.error=mean((Y.pred-Y.test)^2)

print(c("pred.error=",pred.error))

print(proc.time()-ptm)

FRegSigCom documentation built on May 1, 2019, 9:45 p.m.