fda.usc: flm.Ftest – R documentation

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flm.Ftest

F-test for the Functional Linear Model with scalar response

Description

The function flm.Ftest tests the null hypothesis of no interaction between a functional covariate and a scalar response inside the Functional Linear Model (FLM): Y=<X,β>+ε. The null hypothesis is H_0: β=0 and the alternative is H_1: β\neq 0. The null hypothesis is tested by a functional extension of the classical F-test (see Details).

Usage

Ftest.statistic(X.fdata, Y)

flm.Ftest(X.fdata, Y, B = 5000, verbose = TRUE)

Arguments

`X.fdata`	Functional covariate for the FLM. The object must be in the class `fdata`.
`Y`	Scalar response for the FLM. Must be a vector with the same number of elements as functions are in `X.fdata`.
`B`	Number of bootstrap replicates to calibrate the distribution of the test statistic. `B=5000` replicates are the recommended for carry out the test, although for exploratory analysis (not inferential), an acceptable less time-consuming option is `B=500`.
`verbose`	Either to show or not information about computing progress.

Details

The Functional Linear Model with scalar response (FLM), is defined as Y=<X,β>+ε, for a functional process X such that E[X(t)]=0, E[X(t)ε]=0 for all t and for a scalar variable Y such that E[Y]=0. The functional F-test is defined as

||\frac{1}{n} ∑_{i=1}^n (X_i-\bar X)(Y_i-\bar Y)||,

where \bar X is the functional mean of X, \bar Y is the ordinary mean of Y and ||.|| is the L^2 functional norm. The statistic is computed with the function Ftest.statistic. The distribution of the test statistic is approximated by a wild bootstrap resampling on the residuals, using the golden section bootstrap.

Value

The value for Ftest.statistic is simply the F-test statistic. The value for flm.Ftest is an object with class "htest" whose underlying structure is a list containing the following components:

statistic The value of the F-test statistic.
boot.statistics A vector of length B with the values of the bootstrap F-test statistics.
p.value The p-value of the test.
method The character string "Functional Linear Model F-test".
B The number of bootstrap replicates used.
data.name The character string "Y=<X,0>+e"

Note

No NA's are allowed neither in the functional covariate nor in the scalar response.

Author(s)

Eduardo Garcia-Portugues. Please, report bugs and suggestions to egarcia@math.ku.dk

References

Garcia-Portugues, E., Gonzalez-Manteiga, W. and Febrero-Bande, M. (2014). A goodness–of–fit test for the functional linear model with scalar response. Journal of Computational and Graphical Statistics, 23(3), 761-778. http://dx.doi.org/10.1080/10618600.2013.812519

Gonzalez-Manteiga, W., Gonzalez-Rodriguez, G., Martinez-Calvo, A. and Garcia-Portugues, E. Bootstrap independence test for functional linear models. arXiv:1210.1072. http://arxiv.org/abs/1210.1072

Examples

## Not run: 
## Simulated example ##
X=rproc2fdata(n=50,t=seq(0,1,l=101),sigma="OU")
beta0=fdata(mdata=rep(0,length=101)+rnorm(101,sd=0.05),
argvals=seq(0,1,l=101),rangeval=c(0,1))
beta1=fdata(mdata=cos(2*pi*seq(0,1,l=101))-(seq(0,1,l=101)-0.5)^2+
rnorm(101,sd=0.05),argvals=seq(0,1,l=101),rangeval=c(0,1))

# Null hypothesis holds
Y0=drop(inprod.fdata(X,beta0)+rnorm(50,sd=0.1))
# Null hypothesis does not hold
Y1=drop(inprod.fdata(X,beta1)+rnorm(50,sd=0.1))

# Do not reject H0
flm.Ftest(X,Y0,B=100)
flm.Ftest(X,Y0,B=5000)

# Reject H0
flm.Ftest(X,Y1,B=100)
flm.Ftest(X,Y1,B=5000)

## End(Not run)

fda.usc

Functional Data Analysis and Utilities for Statistical Computing

v2.0.2

GPL-2

Authors

Manuel Febrero Bande [aut], Manuel Oviedo de la Fuente [aut, cre], Pedro Galeano [ctb], Alicia Nieto [ctb], Eduardo Garcia-Portugues [ctb]

Initial release

2020-02-17