Goodness-of-fit test for the Functional Linear Model with scalar response

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

Usage

flm.test(
  X.fdata,
  Y,
  beta0.fdata = NULL,
  B = 5000,
  est.method = "pls",
  p = NULL,
  type.basis = "bspline",
  verbose = TRUE,
  plot.it = TRUE,
  B.plot = 100,
  G = 200,
  ...
)

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.fdata. A possibility to do this is to consider, for example for $\beta_0=0$ (the simple null hypothesis of no interaction), beta0.fdata=fdata(mdata=rep(0,length(X.fdata$argvals)),argvals=X.fdata$argvals,rangeval=X.fdata$rangeval). If beta0.fdata=NULL (default), the function will test for the composite null hypothesis.

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.

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 criteria (see Details section for discussion). Then, it must be one of the following methods:

"pc": If p, the number of basis elements, is given, then $\beta$ is estimated by fregre.pc. Otherwise, an optimum p is chosen using fregre.pc.cv and the "SICc" criteria.
"pls": If p is given, $\beta$ is estimated by fregre.pls. Otherwise, an optimum p is chosen using fregre.pls.cv and the "SICc" criteria. This is the default argument as it has been checked empirically that provides a good balance between the performance of the test and the estimation of $\beta$.
"basis": If p is given, $\beta$ is estimated by fregre.basis. Otherwise, an optimum p is chosen using fregre.basis.cv and the "GCV.S" criteria. In these functions, the same basis for the arguments basis.x and basis.b is considered. The type of basis used will be the given by the argument type.basis and must be one of the class of create.basis. Further arguments passed to create.basis (not rangeval that is taken as the rangeval of X.fdata), can be passed throughout ....

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": If p is given, the functional process is expressed in a basis of p B-splines. If not, an optimal p will be chosen by optim.basis, using the "GCV.S" criteria.
- "fourier": If p is given, the functional process is expressed in a basis of p Fourier functions. If not, an optimal p will be chosen by optim.basis, using the "GCV.S" criteria.
- "pc": p must be given. Expresses the functional process in a basis of p principal components.
- "pls": p must be given. Expresses the functional process in a basis of p partial least squares.
Although other basis types supported by create.basis are possible,
"bspline" and "fourier" are recommended. Other basis types may cause incompatibilities.
Composite hypothesis. This argument is only used when est.method="basis" and, in this case, it specifies the type of basis used in the basis estimation method of the functional parameter. Again, basis "bspline" and "fourier" are recommended, as other basis types may cause incompatibilities.

verbose

Either to show or not information about computing progress.

plot.it

Either to show or not a graph of the observed trajectory, and the bootstrap trajectories under the null composite hypothesis, of the process $R_n(\cdot)$ (see Details). Note that if plot.it=TRUE, the function takes more time to run.

B.plot

Number of bootstrap trajectories to show in the resulting plot of the test. As the trajectories shown are the first B.plot of B, B.plot must be lower or equal to B.

G

Number of projections used to compute the trajectories of the process $R_n(\cdot)$ by Monte Carlo.

...

Further arguments passed to create.basis.

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.
method: The method used.
B: The number of bootstrap replicates 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.
ord: The optimal order for PC and PLS given by fregre.pc.cv and fregre.pls.cv. For other methods, it is set to 1:p.
data.name: The character string "Y=<X,b>+e".

Details

The Functional Linear Model with scalar response (FLM), is defined as $Y=\big<X,\beta\big>+\epsilon$, 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 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 resampling 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.

Note

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

References

Escanciano, J. C. (2006). A consistent diagnostic test for regression models using projections. Econometric Theory, 22, 1030-1051. doi:10.1017/S0266466606060506

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. doi:10.1080/10618600.2013.812519

Author

Eduardo Garcia-Portugues. Please, report bugs and suggestions to edgarcia@est-econ.uc3m.es

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)

if (FALSE) { # \dontrun{
# Composite hypothesis: do not reject FLM
pcvm.sim=flm.test(X,Y,B=50,B.plot=50,G=100,plot.it=TRUE)
pcvm.sim
flm.test(X,Y,B=5000)
 
# Estimated beta
dev.new()
plot(pcvm.sim$beta.est)

# Simple hypothesis: do not reject beta=beta0
flm.test(X,Y,beta0.fdata=beta0,B=50,B.plot=50,G=100)
flm.test(X,Y,beta0.fdata=beta0,B=5000) 

# AEMET dataset #
data(aemet)
# Remove the 5\
dev.new()
res.FM=depth.FM(aemet$temp,draw=TRUE)
qu=quantile(res.FM$dep,prob=0.05)
l=which(res.FM$dep<=qu)
lines(aemet$temp[l],col=3)
aemet$df$name[l]

# Data without outliers 
wind.speed=apply(aemet$wind.speed$data,1,mean)[-l]
temp=aemet$temp[-l]
# Exploratory analysis: accept the FLM
pcvm.aemet=flm.test(temp,wind.speed,est.method="pls",B=100,B.plot=50,G=100)
pcvm.aemet

# Estimated beta
dev.new()
plot(pcvm.aemet$beta.est,lwd=2,col=2)
# B=5000 for more precision on calibration of the test: also accept the FLM
flm.test(temp,wind.speed,est.method="pls",B=5000) 

# Simple hypothesis: rejection of beta0=0? Limiting p-value...
dat=rep(0,length(temp$argvals))
flm.test(temp,wind.speed, beta0.fdata=fdata(mdata=dat,argvals=temp$argvals,
                                            rangeval=temp$rangeval),B=100)
flm.test(temp,wind.speed, beta0.fdata=fdata(mdata=dat,argvals=temp$argvals,
                                            rangeval=temp$rangeval),B=5000) 
                                            
# Tecator dataset #
data(tecator)
names(tecator)
absorp=tecator$absorp.fdata
ind=1:129 # or ind=1:215
x=absorp[ind,]
y=tecator$y$Fat[ind]
tt=absorp[["argvals"]]

# Exploratory analysis for composite hypothesis with automatic choose of p
pcvm.tecat=flm.test(x,y,B=100,B.plot=50,G=100)
pcvm.tecat

# B=5000 for more precision on calibration of the test: also reject the FLM
flm.test(x,y,B=5000) 

# Distribution of the PCvM statistic
plot(density(pcvm.tecat$boot.statistics),lwd=2,xlim=c(0,10),
              main="PCvM distribution", xlab="PCvM*",ylab="Density")
rug(pcvm.tecat$boot.statistics)
abline(v=pcvm.tecat$statistic,col=2,lwd=2)
legend("top",legend=c("PCvM observed"),lwd=2,col=2)

# Simple hypothesis: fixed p
dat=rep(0,length(x$argvals))
flm.test(x,y,beta0.fdata=fdata(mdata=dat,argvals=x$argvals,
                               rangeval=x$rangeval),B=100,p=11)
                               
# Simple hypothesis, automatic choose of p
flm.test(x,y,beta0.fdata=fdata(mdata=dat,argvals=x$argvals,
                               rangeval=x$rangeval),B=100)
flm.test(x,y,beta0.fdata=fdata(mdata=dat,argvals=x$argvals,
                               rangeval=x$rangeval),B=5000)
} # }