rp.flm.test: Goodness-of fit test for the functional linear model using...

In egarpor/rp.flm.test: Goodness-of-fit Test for the Functional Linear Model with Scalar Response Based on Random Projections

Description

Tests the composite null hypothesis of a Functional Linear Model with scalar response (FLM),

H_0: Y = <X, β> + ε vs H_1: Y != <X, β> + ε.

If β=β_0 is provided, then the simple hypothesis H_0: Y = <X, β_0> + ε is tested. The way of testing the null hypothesis is via a norm (Cramer-von Mises or Kolmogorov-Smirnov) in the empirical process indexed by the projections.

Usage

rp.flm.test(X.fdata, Y, beta0.fdata = NULL, B = 1000, n.proj = 3,
  est.method = "pc", p = NULL, p.criterion = "SICc", pmax = 10,
  type.basis = "bspline", projs = 0.95, verbose = TRUE, ...)

Arguments

X.fdata	functional observations in the class fdata.
Y	scalar responses 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. The argvals and rangeval arguments of beta0.fdata must be the same of X.fdata. 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.
n.proj	vector with the number of projections to consider. The maximum number must be smaller than 100 (which is more than reasonable for the use of the test).
est.method	estimation method for β, only used in the composite case. There are three methods: "pc" if p is given, then β is estimated by fregre.pc. Otherwise, p is chosen using fregre.pc.cv and the p.criterion criterion. "pls" if p is given, β is estimated by fregre.pls. Otherwise, p is chosen using fregre.pls.cv and the p.criterion criterion. "basis" if p is given, β is estimated by fregre.basis. Otherwise, p is chosen using fregre.basis.cv and the p.criterion criterion. Both in fregre.basis and fregre.basis.cv, the same basis for basis.x and basis.b is considered.
p	number of elements for the basis representation of beta0.fdata and X.fdata with the est.method (only composite hypothesis). If not supplied, it is estimated from the data.
p.criterion	for est.method equal to "pc" or "pls", either "SIC", "SICc" or one of the criteria described in fregre.pc.cv. For "basis" a value for type.CV in fregre.basis.cv such as GCV.S.
pmax	maximum size of the basis expansion to consider in when using p.criterion.
type.basis	type of basis if est.method = "basis".
projs	a fdata object containing the random directions employed to project X.fdata. If numeric, the convenient value for ncomp in rdir.pc.
verbose	whether to show or not information about the testing progress.
...	further arguments passed to create.basis (not rangeval that is taken as the rangeval of X.fdata).

Details

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

Value

An object with class "htest" whose underlying structure is a list containing the following components:

p.values.fdr

a matrix of size c(n.proj, 2), containing in each row the FDR p-values of the CvM and KS tests up to that projection.

proj.statistics

a matrix of size c(max(n.proj), 2) with the value of the test statistic on each projection.

boot.proj.statistics

an array of size c(max(n.proj), 2, B) with the values of the bootstrap test statistics for each projection.

proj.p.values

a matrix of size c(max(n.proj), 2)

method

information about the test performed and the kind of estimation performed.

B

number of bootstrap replicates used.

n.proj

number of projections specified

projs

random directions employed to project X.fdata.

type.basis

type of basis for est.method = "basis".

beta.est

estimated functional parameter \hat β in the composite hypothesis. For the simple hypothesis, beta0.fdata.

p

number of basis elements considered for estimation of β.

p.criterion

criterion employed for selecting p.

data.name

the character string "Y = <X, b> + e"

Author(s)

Eduardo García-Portugués ([e-mail]) and Manuel Febrero-Bande ([e-mail]).

References

Cuesta-Albertos, J.A., García-Portugués, E., Febrero-Bande, M. and González-Manteiga, W. (2019). Goodness-of-fit tests for the functional linear model based on randomly projected empirical processes. Annals of Statistics, 47(1):439-467. https://doi.org/10.1214/18-AOS1693

García-Portugués, E., González-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

Examples

# Simulated example

set.seed(345678)
t <- seq(0, 1, l = 101)
n <- 100
X <- r.ou(n = n, t = t, alpha = 2, sigma = 0.5)
beta0 <- fdata(mdata = cos(2 * pi * t) - (t - 0.5)^2, argvals = t,
               rangeval = c(0,1))
Y <- inprod.fdata(X, beta0) + rnorm(n, sd = 0.1)

## Not run: 
# Test all cases
rp.flm.test(X.fdata = X, Y = Y, est.method = "pc")
rp.flm.test(X.fdata = X, Y = Y, est.method = "pls")
rp.flm.test(X.fdata = X, Y = Y, est.method = "basis",
            p.criterion = fda.usc::GCV.S)
rp.flm.test(X.fdata = X, Y = Y, est.method = "pc", p = 5)
rp.flm.test(X.fdata = X, Y = Y, est.method = "pls", p = 5)
rp.flm.test(X.fdata = X, Y = Y, est.method = "basis", p = 5)
rp.flm.test(X.fdata = X, Y = Y, beta0.fdata = beta0)

## End(Not run)

# Composite hypothesis: do not reject FLM
rp.test <- rp.flm.test(X.fdata = X, Y = Y, est.method = "pc")
rp.test$p.values.fdr
pcvm.test <- flm.test(X.fdata = X, Y = Y, est.method = "pc", B = 1e3,
                      plot.it = FALSE)
pcvm.test

# Estimation of beta
par(mfrow = c(1, 3))
plot(X, main = "X")
plot(beta0, main = "beta")
lines(rp.test$beta.est, col = 2)
lines(pcvm.test$beta.est, col = 3)
plot(density(Y), main = "Density of Y", xlab = "Y", ylab = "Density")
rug(Y)

## Not run: 
# Simple hypothesis: do not reject beta = beta0
rp.flm.test(X.fdata = X, Y = Y, beta0.fdata = beta0)$p.values.fdr
flm.test(X.fdata = X, Y = Y, beta0.fdata = beta0, B = 1e3, plot.it = FALSE)

# Simple hypothesis: reject beta = beta0^2
rp.flm.test(X.fdata = X, Y = Y, beta0.fdata = beta0^2)$p.values.fdr
flm.test(X.fdata = X, Y = Y, beta0.fdata = beta0^2, B = 1e3, plot.it = FALSE)

## End(Not run)

## Not run: 
# Increasing n.proj
rp.flm.test(X.fdata = X, Y = Y, n.proj = 3, verbose = FALSE)$p.values.fdr
rp.flm.test(X.fdata = X, Y = Y, n.proj = 5, verbose = FALSE)$p.values.fdr

# Increasing B
rp.flm.test(X.fdata = X, Y = Y, B = 1e3, verbose = FALSE)$p.values.fdr
rp.flm.test(X.fdata = X, Y = Y, B = 5e3, verbose = FALSE)$p.values.fdr

## End(Not run)

## Not run: 
# Tecator dataset

# Load data
data(tecator)
absorp <- tecator$absorp.fdata
ind <- 1:215 # sometimes ind <- 1:129
x <- absorp[ind, ]
y <- tecator$y$Fat[ind]

# Composite hypothesis
rp.tecat <- rp.flm.test(X.fdata = x, Y = y, verbose = FALSE, B = 1e4)
rp.tecat$p.values.fdr

# Compare with flm.test
pcvm.tecat <- flm.test(X.fdata = x, Y = y, verbose = FALSE, p = rp.tecat$p, 
                       B = 1e4, plot.it = FALSE)
pcvm.tecat

# Simple hypothesis
zero <- fdata(mdata = rep(0, length(x$argvals)), argvals = x$argvals,
              rangeval = x$rangeval)
rp.flm.test(X.fdata = x, Y = y, beta0.fdata = zero, verbose = FALSE, B = 1e4)
flm.test(X.fdata = x, Y = y, beta0.fdata = zero, B = 1e3)

# With derivatives
rp.tecat.1 <- rp.flm.test(X.fdata = fdata.deriv(x, 1), Y = y, verbose = FALSE,
                          B = 1e4)
rp.tecat.1$p.values.fdr
rp.tecat.2 <- rp.flm.test(X.fdata = fdata.deriv(x, 2), Y = y, verbose = FALSE,
                          B = 1e4)
rp.tecat.2$p.values.fdr

## End(Not run)

## Not run: 
# AEMET dataset

# Load data
data(aemet)
wind.speed <- apply(aemet$wind.speed$data, 1, mean)
temp <- aemet$temp

# Remove the 5% of the curves with less depth (i.e. 4 curves)
par(mfrow = c(1, 1))
res.FM <- depth.FM(temp, draw = TRUE)
qu <- quantile(res.FM$dep, prob = 0.05)
l <- which(res.FM$dep <= qu)
lines(aemet$temp[l], col = 3)

# Data without outliers
wind.speed <- wind.speed[-l]
temp <- temp[-l]

# Composite hypothesis
rp.aemet <- rp.flm.test(X.fdata = temp, Y = wind.speed, verbose = FALSE, 
                        B = 1e4)
rp.aemet$p.values.fdr

# Compare with flm.test
pcvm.aemet <- flm.test(X.fdata = temp, Y = wind.speed, B = 1e4,
                       p = rp.aemet$p, verbose = FALSE, plot.it = FALSE)
pcvm.aemet                        

# Simple hypothesis
zero <- fdata(mdata = rep(0, length(temp$argvals)), argvals = temp$argvals,
              rangeval = temp$rangeval)
rp.flm.test(X.fdata = temp, Y = wind.speed, beta0.fdata = zero, 
            verbose = FALSE, B = 1e4)
flm.test(X.fdata = temp, Y = wind.speed, beta0.fdata = zero, B = 1e4,
         plot.it = FALSE, verbose = FALSE)

## End(Not run)

egarpor/rp.flm.test documentation built on Oct. 20, 2021, 12:31 a.m.