fda.usc (version 1.2.3)

rp.flm.test: Goodness-of-fit test for the Functional Linear Model with scalar response using random projections

Description

The function rp.flm.test tests the composite null hypothesis of a Functional Linear Model with scalar response (FLM), $$H_0:\,Y=\big+\epsilon,$$ versus a general alternative. If $\beta=\beta_0$ is provided, then the simple hypothesis $H_0:\,Y=\big+\epsilon$ is tested. The way of testing the null hypothesis is via a Projected Cramer-von Mises test (see Details).

Usage

rp.flm.test (X.fdata, Y,beta0.fdata=NULL, est.method="pc",
             p=NULL, type.basis="bspline", B=5000, n.proj=50,
             verbose = TRUE, F.code=TRUE, sigma="vexponential",
             par.list=list(theta=diff(range(X.fdata$argvals))/20),
             same.rwild=FALSE,...)

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.

beta0.fdata

Functional parameter for the simple null hypothesis, in the fdata class. Recall that the argvals and rangeval arguments of beta0.fdata must be the same of X.fda

est.method

Estimation method for the unknown parameter $\beta$, only used in the composite case. Mainly, there are two options: specify the number of basis elements for the estimated $\beta$ by p or optimally select p by a data-driven crite

p

Number of elements of the basis considered. If it is not given, an optimal p will be chosen using a specific criteria (see est.method and type.basis arguments).

type.basis

Type of basis used to represent the functional process. Depending on the hypothesis it will have a different interpretation:

• Simple hypothesis. One of these options:
  • "bspline"Ifpis given, the functional process

Value

• An object with class "htest" whose underlying structure is a list containing the following components:
• statistic
  The value of the test statistic.
• boot.statistics
  A vector of length B with the values of the bootstrap test statistics.
• p.value
  The p-value of the test.
• proj.p.values
• method
  The method used.
• B
  The number of bootstrap replicates used.
• n.proj
  The number of projections used
• type.basis
  The type of basis used.
• beta.est
  The estimated functional parameter $\beta$ in the composite hypothesis. For the simple hypothesis, the given beta0.fdata.
• p
  The number of basis elements passed or automatically chosen.
• data.name
  The character string "Y=+e".

item

• B
• n.proj
• verbose
• F.code
• sigma
• par.list
• same.rwild
• ...

code

create.basis

bold

not inferential

Details

The Functional Linear Model with scalar response (FLM), is defined as $Y=\big+\epsilon$, for a functional process $X$ such that $E[X(t)]=0$, $E[X(t)\epsilon]=0$ for all $t$ and for a scalar variable $Y$ such that $E[Y]=0$. Then, the test assumes that Y and X.fdata are centred and will automatically center them. So, bear in mind that when you apply the test for Y and X.fdata, actually, you are applying it to Y-mean(Y) and fdata.cen(X.fdata)$Xcen. The test statistic corresponds to the Cramer-von Mises norm of the Residual Marked empirical Process based on Projections $R_n(u,\gamma)$ defined in Garcia-Portugues et al. (2014). The expression of this process in a $p$-truncated basis of the space $L^2[0,T]$ leads to the $p$-multivariate process $R_{n,p}\big(u,\gamma^{(p)}\big)$, whose Cramer-von Mises norm is easily computed. The choice of an appropriate $p$ to represent the functional process $X$, in case that is not provided, is done via the estimation of $\beta$ for the composite hypothesis. For the simple hypothesis, as no estimation of $\beta$ is done, the choice of $p$ depends only on the functional process $X$. As the result of the test may change for different $p$'s, we recommend to use an automatic criterion to select $p$ instead of provide a fixed one. The distribution of the test statistic is approximated by a wild bootstrap on the residuals, using the golden section bootstrap. Finally, the graph shown if plot.it=TRUE represents the observed trajectory, and the bootstrap trajectories under the null, of the process RMPP integrated on the projections: $$R_n(u)\approx\frac{1}{G}\sum_{g=1}^G R_n(u,\gamma_g),$$ where $\gamma_g$ are simulated as Gaussians processes. This gives a graphical idea of how distant is the observed trajectory from the null hypothesis.

See Also

flm.test, rwild, fregre.pc, fregre.pls,latex{ } fregre.basis, fregre.pc.cv, fregre.pls.cv, fregre.basis.cv, min.basis, latex{ } create.basis

Examples

# Simulated example #

X=rproc2fdata(n=100,t=seq(0,1,l=101),sigma="OU")
beta0=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))
Y=inprod.fdata(X,beta0)+rnorm(100,sd=0.1)

dev.new(width=21,height=7)
par(mfrow=c(1,3))
plot(X,main="X")
plot(beta0,main="beta0")
plot(density(Y),main="Density of Y",xlab="Y",ylab="Density")
rug(Y)

# Composite hypothesis: do not reject FLM
res.rp=rp.flm.test(X,Y,B=50,n.proj=100)
res.pcvm=flm.test(X,Y,B=50,G=100)
res.rp
respcvm