flm_test: Goodness-of-fit test for functional linear models
In goffda: Goodness-of-Fit Tests for Functional Data
flm_test R Documentation
Goodness-of-fit test for functional linear models
Description
Goodness-of-fit test of a functional linear model with
functional response Y \in L^2([c, d]) and functional predictor
X \in L^2([a, b]), where L^2([a, b]) is the Hilbert space of
square-integrable functions in [a, b].
The goodness-of-fit test checks the linearity of the regression model
m:L^2([a, b])\rightarrow L^2([c, d]) that relates Y and X
by
Y(t) = m(X) + \varepsilon(t),
where \varepsilon is a random variable in
L^2([c, d]) and t \in [c, d]. The check is formalized as the
test of the composite hypothesis
H_0: m \in \{m_\beta : \beta \in L^2([a, b]) \otimes L^2([c, d])\},
where
m_\beta(X(s))(t) = \int_a^b \beta(s, t) X(s)\,\mathrm{d}s
is the linear, Hilbert–Schmidt, integral operator parametrized by
the bivariate kernel \beta. Its estimation is done by the
truncated expansion of \beta in the tensor product of the
data-driven bases of Functional Principal Components (FPC) of
X and Y. The FPC basis for X is truncated in p
components, while the FPC basis for Y is truncated in q
components.
The particular cases in which either X or Y are
constant functions give either a scalar predictor or response.
The simple linear model arises if both X and Y are scalar,
for which \beta is a constant.
Usage
flm_test(X, Y, beta0 = NULL, B = 500, est_method = "fpcr", p = NULL,
q = NULL, thre_p = 0.99, thre_q = 0.99, lambda = NULL,
boot_scores = TRUE, verbose = TRUE, plot_dens = TRUE,
plot_proc = TRUE, plot_max_procs = 100, plot_max_p = 2,
plot_max_q = 2, save_fit_flm = TRUE, save_boot_stats = TRUE,
int_rule = "trapezoid", refit_lambda = FALSE, ...)
Arguments
X, Y
samples of functional/scalar predictors and functional/scalar
response. Either fdata objects (for functional
variables) or vectors of length n (for scalar variables).
beta0
if provided (defaults to NULL), the simple null
hypothesis H_0: m = m_{\beta_0} is tested. beta0 must be a
matrix of size
c(length(X$argvals), length(Y$argvals)). If X
or Y are scalar, beta0 can be also an
fdata object, with the same argvals as
X or Y. Can also be a constant (understood as a shorthand for
a matrix with all its entries equal to the constant).
B
number of bootstrap replicates. Defaults to 500.
est_method
either "fpcr" (Functional Principal Components
Regression; FPCR), "fpcr_l2" (FPCR with ridge penalty),
"fpcr_l1" (FPCR with lasso penalty) or "fpcr_l1s"
(FPCR with lasso-selected FPC). If X is scalar, flm_est
only considers "fpcr" as estimation method. See details below.
Defaults to "fpcr_l1s".
p, q
either index vectors indicating the specific FPC to be
considered for the truncated bases expansions of X and Y,
respectively. If a single number for p is provided, then
p <- 1:max(p) internally (analogously for q) and the first
max(p) FPC are considered. If NULL (default), then a
data-driven selection of p and q is done. See details below.
thre_p, thre_q
thresholds for the proportion of variance
that is explained, at least, by the first p and q FPC
of X and Y, respectively. These thresholds are employed
for an (initial) automatic selection of p and q.
Default to 0.99. thre_p (thre_q) is ignored if
p (q) is provided.
lambda
regularization parameter \lambda for the estimation
methods "fpcr_l2", "fpcr_l1", and "fpcr_l1s". If
NULL (default), it is chosen with cv_glmnet.
boot_scores
flag to indicate if the bootstrap shall be applied to the
scores of the residuals, rather than to the functional residuals. This
improves the computational expediency notably. Defaults to TRUE.
verbose
flag to show information about the testing progress. Defaults
to TRUE.
plot_dens
flag to indicate if a kernel density estimation of the
bootstrap statistics shall be plotted. Defaults to TRUE.
plot_proc
whether to display a graphical tool to identify the
degree of departure from the null hypothesis. If TRUE (default),
the residual marked empirical process, projected in several FPC directions
of X and Y, is shown, together with bootstrap analogues.
The FPC directions are ones selected at the estimation stage.
plot_max_procs
maximum number of bootstrapped processes to plot in
the graphical tool. Set as the minimum of plot_max_procs and B.
Defaults to 100.
plot_max_p, plot_max_q
maximum number of FPC directions to be
considered in the graphical tool. They limit the resulting plot to be at
most of size c(plot_max_p, plot_max_q). Default to 2.
save_fit_flm, save_boot_stats
flag to return fit_flm and
boot_*. If FALSE, these memory-expensive objects
are set to NA. Default to TRUE.
int_rule
quadrature rule for approximating the definite
unidimensional integral: trapezoidal rule (int_rule = "trapezoid")
and extended Simpson rule (int_rule = "Simpson") are available.
Defaults to "trapezoid".
refit_lambda
flag to reselect lambda in each bootstrap
replicate, incorporating its variability in the bootstrap calibration.
Much more time consumig. Defaults to FALSE.
...
further parameters to be passed to cv_glmnet
(and then to cv.glmnet) such as cv_1se,
cv_nlambda or cv_parallel, among others.
Details
The function implements the bootstrap-based goodness-of-fit test for
the functional linear model with functional/scalar response and
functional/scalar predictor, as described in Algorithm 1 in
García-Portugués et al. (2021). The specifics are detailed there.
By default cv_1se = TRUE for cv_glmnet is
considered, unless it is changed via .... This is the recommended
choice for conducting the goodness-of-fit test based on regularized
estimators, as the oversmoothed estimate of the regression model under the
null hypothesis notably facilitates the calibration of the test (see
García-Portugués et al., 2021).
The graphical tool obtained with plot_proc = TRUE is based on
an extension of the tool described in García-Portugués et al. (2014).
Repeated observations on X are internally removed, as otherwise they
would cause NaNs in Adot. Missing values on X and
Y are also automatically removed.
Value
An object of the htest class with the following elements:
statistic
test statistic.
p.value
p-value of the test.
boot_statistics
the bootstrapped test statistics, a vector
of length B.
method
information on the type of test performed.
parameter
a vector with the dimensions p and q
considered in the test statistic. These are the lengths of the outputs
p and q.
p
the index of the FPC considered for X.
q
the index of the FPC considered for Y.
fit_flm
the output resulted from calling flm_est.
boot_lambda
bootstrapped lambda.
boot_p
a list with the bootstrapped indexes of the FPC considered
for X.
data.name
name of the value of data.
Author(s)
Eduardo García-Portugués.
References
García-Portugués, E., Álvarez-Liébana, J., Álvarez-Pérez, G. and
Gonzalez-Manteiga, W. (2021). A goodness-of-fit test for the functional
linear model with functional response. Scandinavian Journal of
Statistics, 48(2):502–528. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1111/sjos.12486")}
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.
\Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/10618600.2013.812519")}
Examples
## Quick example for functional response and predictor
# Generate data under H0
n <- 100
set.seed(987654321)
X_fdata <- r_ou(n = n, t = seq(0, 1, l = 101), sigma = 2)
epsilon <- r_ou(n = n, t = seq(0, 1, l = 101), sigma = 0.5)
Y_fdata <- epsilon
# Test the FLMFR
flm_test(X = X_fdata, Y = Y_fdata)
# Simple hypothesis
flm_test(X = X_fdata, Y = Y_fdata, beta0 = 0)
# Generate data under H1
n <- 100
set.seed(987654321)
sample_frm_fr <- r_frm_fr(n = n, scenario = 3, s = seq(0, 1, l = 101),
t = seq(0, 1, l = 101), nonlinear = "quadratic")
X_fdata <- sample_frm_fr[["X_fdata"]]
Y_fdata <- sample_frm_fr[["Y_fdata"]]
# Test the FLMFR
flm_test(X = X_fdata, Y = Y_fdata)
## Functional response and predictor
# Generate data under H0
n <- 50
B <- 100
set.seed(987654321)
t <- seq(0, 1, l = 201)
X_fdata <- r_ou(n = n, t = t, sigma = 2)
epsilon <- r_ou(n = n, t = t, sigma = 0.5)
Y_fdata <- epsilon
# With boot_scores = TRUE
flm_test(X = X_fdata, Y = Y_fdata, est_method = "fpcr", B = B)
flm_test(X = X_fdata, Y = Y_fdata, est_method = "fpcr_l2", B = B)
flm_test(X = X_fdata, Y = Y_fdata, est_method = "fpcr_l1s", B = B)
# With boot_scores = FALSE
flm_test(X = X_fdata, Y = Y_fdata, est_method = "fpcr",
boot_scores = FALSE, B = B)
flm_test(X = X_fdata, Y = Y_fdata, est_method = "fpcr_l2",
boot_scores = FALSE, B = B)
flm_test(X = X_fdata, Y = Y_fdata, est_method = "fpcr_l1",
boot_scores = FALSE, B = B)
flm_test(X = X_fdata, Y = Y_fdata, est_method = "fpcr_l1s",
boot_scores = FALSE, B = B)
# Simple hypothesis
flm_test(X = X_fdata, Y = Y_fdata, beta0 = 2, est_method = "fpcr", B = B)
flm_test(X = X_fdata, Y = Y_fdata, beta0 = 0, est_method = "fpcr", B = B)
flm_test(X = X_fdata, Y = Y_fdata, beta0 = 0, est_method = "fpcr_l1s", B = B)
# Generate data under H1
n <- 50
B <- 100
set.seed(987654321)
sample_frm_fr <- r_frm_fr(n = n, scenario = 3, s = t, t = t,
nonlinear = "quadratic")
X_fdata <- sample_frm_fr$X_fdata
Y_fdata <- sample_frm_fr$Y_fdata
# With boot_scores = TRUE
flm_test(X = X_fdata, Y = Y_fdata, est_method = "fpcr", B = B)
flm_test(X = X_fdata, Y = Y_fdata, est_method = "fpcr_l2", B = B)
flm_test(X = X_fdata, Y = Y_fdata, est_method = "fpcr_l1s", B = B)
# With boot_scores = FALSE
flm_test(X = X_fdata, Y = Y_fdata, est_method = "fpcr",
boot_scores = FALSE, B = B)
flm_test(X = X_fdata, Y = Y_fdata, est_method = "fpcr_l2",
boot_scores = FALSE, B = B)
flm_test(X = X_fdata, Y = Y_fdata, est_method = "fpcr_l1",
boot_scores = FALSE, B = B)
flm_test(X = X_fdata, Y = Y_fdata, est_method = "fpcr_l1s",
boot_scores = FALSE, B = B)
## Scalar response and functional predictor
# Generate data under H0
n <- 50
B <- 100
set.seed(987654321)
t <- seq(0, 1, l = 201)
X_fdata <- r_ou(n = n, t = t, sigma = 2)
beta <- r_ou(n = 1, t = t, sigma = 0.5, x0 = 2)
epsilon <- rnorm(n = n)
Y <- drop(inprod_fdata(X_fdata1 = X_fdata, X_fdata2 = beta) + epsilon)
# With boot_scores = TRUE
flm_test(X = X_fdata, Y = Y, est_method = "fpcr", B = B)
flm_test(X = X_fdata, Y = Y, est_method = "fpcr_l2", B = B)
flm_test(X = X_fdata, Y = Y, est_method = "fpcr_l1s", B = B)
# With boot_scores = FALSE
flm_test(X = X_fdata, Y = Y, est_method = "fpcr",
boot_scores = FALSE, B = B)
flm_test(X = X_fdata, Y = Y, est_method = "fpcr_l2",
boot_scores = FALSE, B = B)
flm_test(X = X_fdata, Y = Y, est_method = "fpcr_l1",
boot_scores = FALSE, B = B)
flm_test(X = X_fdata, Y = Y, est_method = "fpcr_l1s",
boot_scores = FALSE, B = B)
# Simple hypothesis
flm_test(X = X_fdata, Y = Y, beta0 = beta, est_method = "fpcr", B = B)
flm_test(X = X_fdata, Y = Y, beta0 = 0, est_method = "fpcr", B = B)
flm_test(X = X_fdata, Y = Y, beta0 = 0, est_method = "fpcr_l1s", B = B)
# Generate data under H1
n <- 50
B <- 100
set.seed(987654321)
X_fdata <- r_ou(n = n, t = t, sigma = 2)
beta <- r_ou(n = 1, t = t, sigma = 0.5)
epsilon <- rnorm(n = n)
Y <- drop(exp(inprod_fdata(X_fdata1 = X_fdata^2, X_fdata2 = beta)) + epsilon)
# With boot_scores = TRUE
flm_test(X = X_fdata, Y = Y, est_method = "fpcr", B = B)
flm_test(X = X_fdata, Y = Y, est_method = "fpcr_l2", B = B)
flm_test(X = X_fdata, Y = Y, est_method = "fpcr_l1s", B = B)
# With boot_scores = FALSE
flm_test(X = X_fdata, Y = Y, est_method = "fpcr",
boot_scores = FALSE, B = B)
flm_test(X = X_fdata, Y = Y, est_method = "fpcr_l2",
boot_scores = FALSE, B = B)
flm_test(X = X_fdata, Y = Y, est_method = "fpcr_l1",
boot_scores = FALSE, B = B)
flm_test(X = X_fdata, Y = Y, est_method = "fpcr_l1s",
boot_scores = FALSE, B = B)
## Functional response and scalar predictor
# Generate data under H0
n <- 50
B <- 100
set.seed(987654321)
X <- rnorm(n)
t <- seq(0, 1, l = 201)
beta <- r_ou(n = 1, t = t, sigma = 0.5, x0 = 3)
beta$data <- matrix(beta$data, nrow = n, ncol = ncol(beta$data),
byrow = TRUE)
epsilon <- r_ou(n = n, t = t, sigma = 0.5)
Y_fdata <- X * beta + epsilon
# With boot_scores = TRUE
flm_test(X = X, Y = Y_fdata, est_method = "fpcr", B = B)
# With boot_scores = FALSE
flm_test(X = X, Y = Y_fdata, est_method = "fpcr", boot_scores = FALSE, B = B)
# Simple hypothesis
flm_test(X = X, Y = Y_fdata, beta0 = beta[1], est_method = "fpcr", B = B)
flm_test(X = X, Y = Y_fdata, beta0 = 0, est_method = "fpcr", B = B)
# Generate data under H1
n <- 50
B <- 100
set.seed(987654321)
X <- rexp(n)
beta <- r_ou(n = 1, t = t, sigma = 0.5, x0 = 3)
beta$data <- matrix(beta$data, nrow = n, ncol = ncol(beta$data),
byrow = TRUE)
epsilon <- r_ou(n = n, t = t, sigma = 0.5)
Y_fdata <- log(X * beta) + epsilon
# With boot_scores = TRUE
flm_test(X = X, Y = Y_fdata, est_method = "fpcr", B = B)
# With boot_scores = FALSE
flm_test(X = X, Y = Y_fdata, est_method = "fpcr", boot_scores = FALSE, B = B)
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| flm_test | R Documentation |
Goodness-of-fit test for functional linear models
Description
Goodness-of-fit test of a functional linear model with
functional response Y \in L^2([c, d]) and functional predictor
X \in L^2([a, b]), where L^2([a, b]) is the Hilbert space of
square-integrable functions in [a, b].
The goodness-of-fit test checks the linearity of the regression model
m:L^2([a, b])\rightarrow L^2([c, d]) that relates Y and X
by
Y(t) = m(X) + \varepsilon(t),
where \varepsilon is a random variable in
L^2([c, d]) and t \in [c, d]. The check is formalized as the
test of the composite hypothesis
H_0: m \in \{m_\beta : \beta \in L^2([a, b]) \otimes L^2([c, d])\},
where
m_\beta(X(s))(t) = \int_a^b \beta(s, t) X(s)\,\mathrm{d}s
is the linear, Hilbert–Schmidt, integral operator parametrized by
the bivariate kernel \beta. Its estimation is done by the
truncated expansion of \beta in the tensor product of the
data-driven bases of Functional Principal Components (FPC) of
X and Y. The FPC basis for X is truncated in p
components, while the FPC basis for Y is truncated in q
components.
The particular cases in which either X or Y are
constant functions give either a scalar predictor or response.
The simple linear model arises if both X and Y are scalar,
for which \beta is a constant.
Usage
flm_test(X, Y, beta0 = NULL, B = 500, est_method = "fpcr", p = NULL,
q = NULL, thre_p = 0.99, thre_q = 0.99, lambda = NULL,
boot_scores = TRUE, verbose = TRUE, plot_dens = TRUE,
plot_proc = TRUE, plot_max_procs = 100, plot_max_p = 2,
plot_max_q = 2, save_fit_flm = TRUE, save_boot_stats = TRUE,
int_rule = "trapezoid", refit_lambda = FALSE, ...)
Arguments
X, Y |
samples of functional/scalar predictors and functional/scalar
response. Either |
beta0 |
if provided (defaults to |
B |
number of bootstrap replicates. Defaults to |
est_method |
either |
p, q |
either index vectors indicating the specific FPC to be
considered for the truncated bases expansions of |
thre_p, thre_q |
thresholds for the proportion of variance
that is explained, at least, by the first |
lambda |
regularization parameter |
boot_scores |
flag to indicate if the bootstrap shall be applied to the
scores of the residuals, rather than to the functional residuals. This
improves the computational expediency notably. Defaults to |
verbose |
flag to show information about the testing progress. Defaults
to |
plot_dens |
flag to indicate if a kernel density estimation of the
bootstrap statistics shall be plotted. Defaults to |
plot_proc |
whether to display a graphical tool to identify the
degree of departure from the null hypothesis. If |
plot_max_procs |
maximum number of bootstrapped processes to plot in
the graphical tool. Set as the minimum of |
plot_max_p, plot_max_q |
maximum number of FPC directions to be
considered in the graphical tool. They limit the resulting plot to be at
most of size |
save_fit_flm, save_boot_stats |
flag to return |
int_rule |
quadrature rule for approximating the definite
unidimensional integral: trapezoidal rule ( |
refit_lambda |
flag to reselect |
... |
further parameters to be passed to |
Details
The function implements the bootstrap-based goodness-of-fit test for the functional linear model with functional/scalar response and functional/scalar predictor, as described in Algorithm 1 in García-Portugués et al. (2021). The specifics are detailed there.
By default cv_1se = TRUE for cv_glmnet is
considered, unless it is changed via .... This is the recommended
choice for conducting the goodness-of-fit test based on regularized
estimators, as the oversmoothed estimate of the regression model under the
null hypothesis notably facilitates the calibration of the test (see
García-Portugués et al., 2021).
The graphical tool obtained with plot_proc = TRUE is based on
an extension of the tool described in García-Portugués et al. (2014).
Repeated observations on X are internally removed, as otherwise they
would cause NaNs in Adot. Missing values on X and
Y are also automatically removed.
Value
An object of the htest class with the following elements:
statistic |
test statistic. |
p.value |
|
boot_statistics |
the bootstrapped test statistics, a vector
of length |
method |
information on the type of test performed. |
parameter |
a vector with the dimensions |
p |
the index of the FPC considered for |
q |
the index of the FPC considered for |
fit_flm |
the output resulted from calling |
boot_lambda |
bootstrapped |
boot_p |
a list with the bootstrapped indexes of the FPC considered
for |
data.name |
name of the value of |
Author(s)
Eduardo García-Portugués.
References
García-Portugués, E., Álvarez-Liébana, J., Álvarez-Pérez, G. and Gonzalez-Manteiga, W. (2021). A goodness-of-fit test for the functional linear model with functional response. Scandinavian Journal of Statistics, 48(2):502–528. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1111/sjos.12486")}
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. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/10618600.2013.812519")}
Examples
## Quick example for functional response and predictor
# Generate data under H0
n <- 100
set.seed(987654321)
X_fdata <- r_ou(n = n, t = seq(0, 1, l = 101), sigma = 2)
epsilon <- r_ou(n = n, t = seq(0, 1, l = 101), sigma = 0.5)
Y_fdata <- epsilon
# Test the FLMFR
flm_test(X = X_fdata, Y = Y_fdata)
# Simple hypothesis
flm_test(X = X_fdata, Y = Y_fdata, beta0 = 0)
# Generate data under H1
n <- 100
set.seed(987654321)
sample_frm_fr <- r_frm_fr(n = n, scenario = 3, s = seq(0, 1, l = 101),
t = seq(0, 1, l = 101), nonlinear = "quadratic")
X_fdata <- sample_frm_fr[["X_fdata"]]
Y_fdata <- sample_frm_fr[["Y_fdata"]]
# Test the FLMFR
flm_test(X = X_fdata, Y = Y_fdata)
## Functional response and predictor
# Generate data under H0
n <- 50
B <- 100
set.seed(987654321)
t <- seq(0, 1, l = 201)
X_fdata <- r_ou(n = n, t = t, sigma = 2)
epsilon <- r_ou(n = n, t = t, sigma = 0.5)
Y_fdata <- epsilon
# With boot_scores = TRUE
flm_test(X = X_fdata, Y = Y_fdata, est_method = "fpcr", B = B)
flm_test(X = X_fdata, Y = Y_fdata, est_method = "fpcr_l2", B = B)
flm_test(X = X_fdata, Y = Y_fdata, est_method = "fpcr_l1s", B = B)
# With boot_scores = FALSE
flm_test(X = X_fdata, Y = Y_fdata, est_method = "fpcr",
boot_scores = FALSE, B = B)
flm_test(X = X_fdata, Y = Y_fdata, est_method = "fpcr_l2",
boot_scores = FALSE, B = B)
flm_test(X = X_fdata, Y = Y_fdata, est_method = "fpcr_l1",
boot_scores = FALSE, B = B)
flm_test(X = X_fdata, Y = Y_fdata, est_method = "fpcr_l1s",
boot_scores = FALSE, B = B)
# Simple hypothesis
flm_test(X = X_fdata, Y = Y_fdata, beta0 = 2, est_method = "fpcr", B = B)
flm_test(X = X_fdata, Y = Y_fdata, beta0 = 0, est_method = "fpcr", B = B)
flm_test(X = X_fdata, Y = Y_fdata, beta0 = 0, est_method = "fpcr_l1s", B = B)
# Generate data under H1
n <- 50
B <- 100
set.seed(987654321)
sample_frm_fr <- r_frm_fr(n = n, scenario = 3, s = t, t = t,
nonlinear = "quadratic")
X_fdata <- sample_frm_fr$X_fdata
Y_fdata <- sample_frm_fr$Y_fdata
# With boot_scores = TRUE
flm_test(X = X_fdata, Y = Y_fdata, est_method = "fpcr", B = B)
flm_test(X = X_fdata, Y = Y_fdata, est_method = "fpcr_l2", B = B)
flm_test(X = X_fdata, Y = Y_fdata, est_method = "fpcr_l1s", B = B)
# With boot_scores = FALSE
flm_test(X = X_fdata, Y = Y_fdata, est_method = "fpcr",
boot_scores = FALSE, B = B)
flm_test(X = X_fdata, Y = Y_fdata, est_method = "fpcr_l2",
boot_scores = FALSE, B = B)
flm_test(X = X_fdata, Y = Y_fdata, est_method = "fpcr_l1",
boot_scores = FALSE, B = B)
flm_test(X = X_fdata, Y = Y_fdata, est_method = "fpcr_l1s",
boot_scores = FALSE, B = B)
## Scalar response and functional predictor
# Generate data under H0
n <- 50
B <- 100
set.seed(987654321)
t <- seq(0, 1, l = 201)
X_fdata <- r_ou(n = n, t = t, sigma = 2)
beta <- r_ou(n = 1, t = t, sigma = 0.5, x0 = 2)
epsilon <- rnorm(n = n)
Y <- drop(inprod_fdata(X_fdata1 = X_fdata, X_fdata2 = beta) + epsilon)
# With boot_scores = TRUE
flm_test(X = X_fdata, Y = Y, est_method = "fpcr", B = B)
flm_test(X = X_fdata, Y = Y, est_method = "fpcr_l2", B = B)
flm_test(X = X_fdata, Y = Y, est_method = "fpcr_l1s", B = B)
# With boot_scores = FALSE
flm_test(X = X_fdata, Y = Y, est_method = "fpcr",
boot_scores = FALSE, B = B)
flm_test(X = X_fdata, Y = Y, est_method = "fpcr_l2",
boot_scores = FALSE, B = B)
flm_test(X = X_fdata, Y = Y, est_method = "fpcr_l1",
boot_scores = FALSE, B = B)
flm_test(X = X_fdata, Y = Y, est_method = "fpcr_l1s",
boot_scores = FALSE, B = B)
# Simple hypothesis
flm_test(X = X_fdata, Y = Y, beta0 = beta, est_method = "fpcr", B = B)
flm_test(X = X_fdata, Y = Y, beta0 = 0, est_method = "fpcr", B = B)
flm_test(X = X_fdata, Y = Y, beta0 = 0, est_method = "fpcr_l1s", B = B)
# Generate data under H1
n <- 50
B <- 100
set.seed(987654321)
X_fdata <- r_ou(n = n, t = t, sigma = 2)
beta <- r_ou(n = 1, t = t, sigma = 0.5)
epsilon <- rnorm(n = n)
Y <- drop(exp(inprod_fdata(X_fdata1 = X_fdata^2, X_fdata2 = beta)) + epsilon)
# With boot_scores = TRUE
flm_test(X = X_fdata, Y = Y, est_method = "fpcr", B = B)
flm_test(X = X_fdata, Y = Y, est_method = "fpcr_l2", B = B)
flm_test(X = X_fdata, Y = Y, est_method = "fpcr_l1s", B = B)
# With boot_scores = FALSE
flm_test(X = X_fdata, Y = Y, est_method = "fpcr",
boot_scores = FALSE, B = B)
flm_test(X = X_fdata, Y = Y, est_method = "fpcr_l2",
boot_scores = FALSE, B = B)
flm_test(X = X_fdata, Y = Y, est_method = "fpcr_l1",
boot_scores = FALSE, B = B)
flm_test(X = X_fdata, Y = Y, est_method = "fpcr_l1s",
boot_scores = FALSE, B = B)
## Functional response and scalar predictor
# Generate data under H0
n <- 50
B <- 100
set.seed(987654321)
X <- rnorm(n)
t <- seq(0, 1, l = 201)
beta <- r_ou(n = 1, t = t, sigma = 0.5, x0 = 3)
beta$data <- matrix(beta$data, nrow = n, ncol = ncol(beta$data),
byrow = TRUE)
epsilon <- r_ou(n = n, t = t, sigma = 0.5)
Y_fdata <- X * beta + epsilon
# With boot_scores = TRUE
flm_test(X = X, Y = Y_fdata, est_method = "fpcr", B = B)
# With boot_scores = FALSE
flm_test(X = X, Y = Y_fdata, est_method = "fpcr", boot_scores = FALSE, B = B)
# Simple hypothesis
flm_test(X = X, Y = Y_fdata, beta0 = beta[1], est_method = "fpcr", B = B)
flm_test(X = X, Y = Y_fdata, beta0 = 0, est_method = "fpcr", B = B)
# Generate data under H1
n <- 50
B <- 100
set.seed(987654321)
X <- rexp(n)
beta <- r_ou(n = 1, t = t, sigma = 0.5, x0 = 3)
beta$data <- matrix(beta$data, nrow = n, ncol = ncol(beta$data),
byrow = TRUE)
epsilon <- r_ou(n = n, t = t, sigma = 0.5)
Y_fdata <- log(X * beta) + epsilon
# With boot_scores = TRUE
flm_test(X = X, Y = Y_fdata, est_method = "fpcr", B = B)
# With boot_scores = FALSE
flm_test(X = X, Y = Y_fdata, est_method = "fpcr", boot_scores = FALSE, B = B)
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