Type:	Package
Title:	Scalar-on-Function Regression with Measurement Error Correction
Version:	0.2.1
Maintainer:	Heyang Ji <jihx1015@outlook.com>
Description:	Solve scalar-on-function linear models, including generalized linear mixed effect model and quantile linear regression model, and bias correction estimation methods due to measurement error. Details about the measurement error bias correction methods, see Luan et al. (2023) <doi:10.48550/arXiv.2305.12624>, Tekwe et al. (2022) <doi:10.1093/biostatistics/kxac017>, Zhang et al. (2023) <doi:10.5705/ss.202021.0246>, Tekwe et al. (2019) <doi:10.1002/sim.8179>.
License:	GPL-3
Encoding:	UTF-8
LazyData:	true
RoxygenNote:	7.3.2
Depends:	R (≥ 2.10),
Imports:	stats, utils, lme4, quantreg, splines, dplyr, MASS, Matrix, gss, corpcor, fda, magrittr, methods, nlme, glme, mgcv, refund, pracma
Suggests:	knitr, rmarkdown
VignetteBuilder:	knitr
NeedsCompilation:	no
Packaged:	2025-06-30 07:14:05 UTC; jihx1
Author:	Heyang Ji [aut, cre, ctb, dtc], Ufuk Beyaztas [aut, ctb, rev], Nicolas Escobar-Velasquez [com], Yuanyuan Luan [aut, ctb], Xiwei Chen [aut, ctb], Mengli Zhang [aut, ctb], Roger Zoh [aut, ths], Lan Xue [aut, ths], Carmen Tekwe [aut, ths]
Repository:	CRAN
Date/Publication:	2025-06-30 07:30:02 UTC

Pipe operator

Description

See magrittr::%>% for details.

Usage

lhs %>% rhs

Arguments

Value

The result of calling 'rhs(lhs)'.

Functional principal component basis expansion for functional variable data

Description

For a function f(t), t\in\Omega, and a basis function sequence \{\rho_k\}_{k\in\kappa}, basis expansion is to compute \int_\Omega f(t)\rho_k(t) dt. Here we do basis expansion for all f_i(t), t\in\Omega = [t_0,t_0+T] in functional variable data, i=1,\dots,n. We compute a matrix (b_{ik})_{n\times p}, where b_{ik} = \int_\Omega f(t)\rho_k(t) dt. The basis we use here is the functional principal component (FPC) basis induced by the covariance function of the functional variable. Suppose K(s,t)\in L^2(\Omega\times \Omega), f(t)\in L^2(\Omega). Then K induces an linear operator \mathcal{K} by

(\mathcal{K}f)(x) = \int_{\Omega} K(t,x)f(t)dt

If \xi(\cdot)\in L^2(\Omega) such that

\mathcal{K}\xi = \lambda \xi

where \lambda\in {C}, we call \xi an eigenfunction/eigenvector of \mathcal{K}, and \lambda an eigenvalue associated with \xi.
For a stochastic process \{X(t),t\in\Omega\} we call the orthogonal basis \{\xi_k\}_{k=1}^\infty corresponding to eigenvalues \{\lambda_k\}_{k=1}^\infty (\lambda_1\geq\lambda_2\geq\dots), induced by

K(s,t)=\text{Cov}(X(t),X(s))

a functional principal component (FPC) basis for L^2(\Omega).

Usage

FPC_basis_expansion(object, npc)

## S4 method for signature 'functional_variable,integer'
FPC_basis_expansion(object, npc)

Arguments

Value

Returns a numeric matrix, (b_{ik})_{n\times p}, with an extra attribute numeric_basis, which represents the FPC basis. The attribute numeric_basis is a numeric_basis object. See numeric_basis. The slot basis_function is also a numeric matrix, denoted as (\zeta_{jk})_{m\times p}

b_{ik} = \int_\Omega f(t)\xi_k(t) dt

\zeta_{jk} = \xi_k(t_j)

Author(s)

Heyang Ji

Examples

n<-50; ti<-seq(0,1,length.out=101)
X<-t(sin(2*pi*ti)%*%t(rnorm(n,0,1)))
object = functional_variable(X = X, t_0 = 0, period = 1, t_points = ti)
a = FPC_basis_expansion(object,3L)
dim(a)

Compute the value of the Fourier summation series

Description

Compute the value of the Fourier summation series

f(x) = \frac{a_0}{2} + \sum_{k=1}^{p_a} a_k \cos{(\frac{2\pi}{T}k(x-t_0))} + \sum_{k=1}^{p_b} b_k \sin{(\frac{2\pi}{T}k(x-t_0))}, \qquad x\in[t_0,t_0+T]

at some certain point(s).

Usage

FourierSeries2fun(object, x)

## S4 method for signature 'Fourier_series,numeric'
FourierSeries2fun(object, x)

Arguments

Value

A numeric atomic vector

Author(s)

Heyang Ji

Examples

fsc = Fourier_series(
           double_constant = 0.5,
           cos = c(0,0.3),
           sin = c(1,0.7),
           k_cos = 1:2,
           )
          FourierSeries2fun(fsc,1:5)

s4 class of Fourier summation series

Description

A s4 class that represents the linear combination of Fourier basis functions below:

\frac{a_0}{2} + \sum_{k=1}^{p_a} a_k \cos{(\frac{2\pi}{T}k(x-t_0))} + \sum_{k=1}^{p_b} b_k \sin{(\frac{2\pi}{T}k(x-t_0))}, \qquad x\in[t_0,t_0+T]

Details

If not assigned, t_0 = 0, T = 2\pi. If not assigned, k_cos and k_sin equals 1, 2, 3, ...

Slots

double_constant

value of a_0.

cos

values of coefficients of \cos waves, a_k.

sin

values of coefficients of \sin waves, b_k.

k_cos

values of k corresponding to the coefficients of \cos waves

k_sin

values of k corresponding to the coefficients of \sin waves

t_0

left end of the domain interval, t_0

period

length of the domain interval, T.

Author(s)

Heyang Ji

Examples

fsc = Fourier_series(
           double_constant = 0.5,
           cos = c(0,0.3),
           sin = c(1,0.7),
           k_cos = 1:2,
           )

Bias correction method of applying linear regression to one functional covariate with measurement error using instrumental variable.

Description

See detailed model in reference

Usage

ME.fcLR_IV(
  data.Y,
  data.W,
  data.M,
  t_interval = c(0, 1),
  t_points = NULL,
  CI.bootstrap = FALSE
)

Arguments

Value

Returns a ME.fcLR_IV class object. It is a list that contains the following elements.

References

Tekwe, Carmen D., et al. "Instrumental variable approach to estimating the scalar‐on‐function regression model w ith measurement error with application to energy expenditure assessment in childhood obesity." Statistics in medicine 38.20 (2019): 3764-3781.

Examples

data(MECfda.data.sim.0.3)
res = ME.fcLR_IV(data.Y = MECfda.data.sim.0.3$Y,
              data.W = MECfda.data.sim.0.3$W,
              data.M = MECfda.data.sim.0.3$M)

Bias correction method of applying quantile linear regression to dataset with one functional covariate with measurement error using corrected loss score method.

Description

Zhang et al. proposed a new corrected loss function for a partially functional linear quantile model with functional measurement error in this manuscript. They established a corrected quantile objective function of the observed measurement that is an unbiased estimator of the quantile objective function that would be obtained if the true measurements were available. The estimators of the regression parameters are obtained by optimizing the resulting corrected loss function. The resulting estimator of the regression parameters is shown to be consistent.

Usage

ME.fcQR_CLS(
  data.Y,
  data.W,
  data.Z,
  tau = 0.5,
  t_interval = c(0, 1),
  t_points = NULL,
  grid_k,
  grid_h,
  degree = 45,
  observed_X = NULL
)

Arguments

Value

Returns a ME.fcQR_CLS class object. It is a list that contains the following elements.

References

Zhang, Mengli, et al. "PARTIALLY FUNCTIONAL LINEAR QUANTILE REGRESSION WITH MEASUREMENT ERRORS." Statistica Sinica 33 (2023): 2257-2280.

Examples

data(MECfda.data.sim.0.1)

res = ME.fcQR_CLS(data.Y = MECfda.data.sim.0.1$Y,
                data.W = MECfda.data.sim.0.1$W,
               data.Z = MECfda.data.sim.0.1$Z,
               tau = 0.5,
               grid_k = 4:7,
               grid_h = 1:2)

Bias correction method of applying quantile linear regression to dataset with one functional covariate with measurement error using instrumental variable.

Description

Perform a two-stage strategy to correct the measurement error of a function-valued covariate and then fit a linear quantile regression model. In the first stage, an instrumental variable is used to estimate the covariance matrix associated with the measurement error. In the second stage, simulation extrapolation (SIMEX) is used to correct for measurement error in the function-valued covariate.
See detailed model in the reference.

Usage

ME.fcQR_IV.SIMEX(
  data.Y,
  data.W,
  data.Z,
  data.M,
  tau = 0.5,
  t_interval = c(0, 1),
  t_points = NULL,
  formula.Z,
  basis.type = c("Fourier", "Bspline"),
  basis.order = NULL,
  bs_degree = 3
)

Arguments

Value

Returns a ME.fcQR_IV.SIMEX class object. It is a list that contains the following elements.

References

Tekwe, Carmen D., et al. "Estimation of sparse functional quantile regression with measurement error: a SIMEX approach." Biostatistics 23.4 (2022): 1218-1241.

Examples

data(MECfda.data.sim.0.2)

res = ME.fcQR_IV.SIMEX(data.Y = MECfda.data.sim.0.2$Y,
                       data.W = MECfda.data.sim.0.2$W,
                       data.Z = MECfda.data.sim.0.2$Z,
                       data.M = MECfda.data.sim.0.2$M,
                       tau = 0.5,
                       basis.type = 'Bspline')

Use UP_MEM or MP_MEM substitution to apply (generalized) linear regression with one functional covariate with measurement error.

Description

The Mixed-effect model (MEM) approach is a two-stage-based method that employs functional mixed-effects models. It allows us to delve into the nonlinear measurement error model, where the relationship between the true and observed measurements is not constrained to be linear, and the distribution assumption on the observed measurement is relaxed to encompass the exponential family rather than being limited to the Gaussian distribution. The MEM approach employs point-wise (UP_MEM) and multi-point-wise (MP_MEM) estimation procedures to avoid potential computational complexities caused by analyses of multi-level functional data and computations of potentially intractable and complex integrals.

Usage

ME.fcRegression_MEM(
  data.Y,
  data.W,
  data.Z,
  method = c("UP_MEM", "MP_MEM", "average"),
  t_interval = c(0, 1),
  t_points = NULL,
  d = 3,
  family.W = c("gaussian", "poisson"),
  family.Y = "gaussian",
  formula.Z,
  basis.type = c("Fourier", "Bspline"),
  basis.order = NULL,
  bs_degree = 3,
  smooth = FALSE,
  silent = TRUE
)

Arguments

Value

Returns a fcRegression object. See fcRegression.

References

Luan, Yuanyuan, et al. "Scalable regression calibration approaches to correcting measurement error in multi-level generalized functional linear regression models with heteroscedastic measurement errors." arXiv preprint arXiv:2305.12624 (2023).

Examples

data(MECfda.data.sim.0.1)
res = ME.fcRegression_MEM(data.Y = MECfda.data.sim.0.1$Y,
                          data.W = MECfda.data.sim.0.1$W,
                          data.Z = MECfda.data.sim.0.1$Z,
                          method = 'UP_MEM',
                          family.W = "gaussian",
                          basis.type = 'Bspline')

Simulated data

Description

Simulated data

Simulated data

Description

Simulated data

Simulated data

Description

Simulated data

Simulated data

Description

Simulated data

Simulated data

Description

Simulated data

Simulated data

Description

Simulated data

Simulated data

Description

Simulated data

Simulated data

Description

Simulated data

Simulation Data Generation: Measurement Error Bias Correction of Scalar-on-function Regression

Description

Generate data set for measurement error bias correction methods for scalar-on-function regression in package MECfda

Usage

MECfda_simDataGen_ME(
  N = 100,
  J_W = 7,
  forwhich = c("MEM", "IV", "CLS", "IV.SIMEX"),
  t_interval = c(0, 1),
  n_t = 24,
  seed = 0
)

Arguments

Value

return a list that possibly contains following elements.

Examples

for (i in 1:4) {MECfda_simDataGen_ME(forwhich = c('MEM','IV','CLS','IV.SIMEX')[i])}

Simulation Data Generation: Scalar-on-function Regression

Description

Generate data set for scalar-on-function regression

Usage

MECfda_simDataGen_fcReg(
  N = 100,
  distribution = c("Gaussian", "Bernoulli"),
  t_interval,
  t_points,
  n_t = 100,
  seed = 0
)

Arguments

Value

return a list with following elements.

Examples

dat_sim = MECfda_simDataGen_fcReg(100,"Bernoulli")
res = fcRegression(FC = dat_sim$FC, Y=dat_sim$Y, Z=dat_sim$Z,
                   basis.order = 3, basis.type = c('Fourier'),
                   family = binomial(link = "logit"))

Get MEM substitution for (generalized) linear regression with one functional covariate with measurement error.

Description

The function to get the data of \hat X_i(t) using the mixed model based measurement error bias correction method proposed by Luan et al. See ME.fcRegression_MEM

Usage

MEM_X_hat(
  data.W,
  method = c("UP_MEM", "MP_MEM", "average"),
  d = 3,
  family.W = c("gaussian", "poisson"),
  smooth = FALSE
)

Arguments

Value

A numeric value matrix of \hat X_i(t).

References

Luan, Yuanyuan, et al. "Scalable regression calibration approaches to correcting measurement error in multi-level generalized functional linear regression models with heteroscedastic measurement errors." arXiv preprint arXiv:2305.12624 (2023).

Examples

data(MECfda.data.sim.0.1)
X_hat = MEM_X_hat(data.W = MECfda.data.sim.0.1$W,
                  method = 'UP_MEM',
                  family.W = "gaussian")

From the summation series of a functional basis to function value

Description

Generic function to compute function value from summation series of a functional basis.

Usage

basis2fun(object, x)

## S4 method for signature 'bspline_series,numeric'
basis2fun(object, x)

## S4 method for signature 'Fourier_series,numeric'
basis2fun(object, x)

## S4 method for signature 'numericBasis_series,numeric'
basis2fun(object, x)

Arguments

Details

When applied to bspline_series object, equivalent to bsplineSeries2fun.
When applied to Fourier_series object, equivalent to FourierSeries2fun.
When applied to numericBasis_series object, equivalent to numericBasisSeries2fun.

Value

A numeric atomic vector. See bsplineSeries2fun and FourierSeries2fun.

Author(s)

Heyang Ji

Compute the value of the b-splines summation series at certain points.

Description

Compute the function f(x) = \sum_{i=0}^{k}b_i B_{i,p}(x)

Usage

bsplineSeries2fun(object, x)

## S4 method for signature 'bspline_series,numeric'
bsplineSeries2fun(object, x)

Arguments

Value

A numeric atomic vector

Author(s)

Heyang Ji

Examples

bsb = bspline_basis(
            Boundary.knots = c(0,24),
            intercept      = TRUE,
            df             = NULL,
            degree         = 3
)
bss = bspline_series(
          coef = c(2,1,1.5,0.5),
          bspline_basis = bsb
)
bsplineSeries2fun(bss,(1:239)/10)

b-spline basis

Description

A s4 class that represents a b-spline basis \{B_{i,p}(x)\}_{i=-p}^{k} on the interval [t_0,t_{k+1}], where B_{i,p}(x) is defined as

B_{i,0}(x) = \left\{ \begin{aligned} &I_{(t_i,t_{i+1}]}(x), & i = 0,1,\dots,k\\ &0, &i<0\ or\ i>k \end{aligned} \right.

B_{i,r}(x) = \frac{x - t_{i}}{t_{i+r}-t_{i}} B_{i,r-1}(x) + \frac{t_{i+r+1} - x} {t_{i+r+1} - t_{i+1}}B_{i+1,r-1}(x)

For all the discontinuity points of B_{i,r} (r>0) in the interval (t_0,t_k), let the value equals its limit, which means

B_{i,r}(x) = \lim_{t\to x} B_{i,r}(t)

Slots

Boundary.knots

boundary of the domain of the splines (start and end), which is t_0 and t_{k+1}. Default is [0,1]. See Boundary.knots in bs.

knots

knots of the splines, which is (t_1,\dots,t_k), equally spaced sequence is chosen by the function automatically with equal space (t_j = t_0 + j\cdot\frac{t_{k+1}-t_0}{k+1}) when not assigned. See knots in bs.

intercept

Whether an intercept is included in the basis, default value is TRUE, and must be TRUE. See intercept bs.

df

degree of freedom of the basis, which is the number of the splines, equal to p+k+1. By default k = 0, and df = p+1. See df bs.

degree

degree of the splines, which is the degree of piecewise polynomials p, default value is 3. See degree in bs.

Author(s)

Heyang Ji

Examples

bsb = bspline_basis(
            Boundary.knots = c(0,24),
            intercept      = TRUE,
            df             = NULL,
            degree         = 3
)

B-splines basis expansion for functional variable data

Description

For a function f(t), t\in\Omega, and a basis function sequence \{\rho_k\}_{k\in\kappa}, basis expansion is to compute \int_\Omega f(t)\rho_k(t) dt. Here we do basis expansion for all f_i(t), t\in\Omega = [t_0,t_0+T] in functional variable data, i=1,\dots,n. We compute a matrix (b_{ik})_{n\times p}, where b_{ik} = \int_\Omega f(t)\rho_k(t) dt. The basis used here is the b-splines basis, \{B_{i,p}(x)\}_{i=-p}^{k}, x\in[t_0,t_{k+1}], where t_{k+1} = t_0+T and B_{i,p}(x) is defined as

B_{i,0}(x) = \left\{ \begin{aligned} &I_{(t_i,t_{i+1}]}(x), & i = 0,1,\dots,k\\ &0, &i<0\ or\ i>k \end{aligned} \right.

B_{i,r}(x) = \frac{x - t_{i}}{t_{i+r}-t_{i}} B_{i,r-1}(x) + \frac{t_{i+r+1} - x} {t_{i+r+1} - t_{i+1}}B_{i+1,r-1}(x)

Usage

bspline_basis_expansion(object, n_splines, bs_degree)

## S4 method for signature 'functional_variable,integer'
bspline_basis_expansion(object, n_splines, bs_degree)

Arguments

Value

Returns a numeric matrix, (b_{ik})_{n\times p}, where b_{ik} = \int_\Omega f(t)\rho_k(t) dt.

Author(s)

Heyang Ji

b-splines summation series.

Description

A s4 class that represents the summation \sum_{i=0}^{k}b_i B_{i,p}(x) by a bspline_basis object and coefficients b_i (i = 0,\dots,k).

Slots

coef

coefficients of the b-splines, b_i (i = 0,\dots,k).

bspline_basis

a bspline_basis object, represents the b-splines basis used, \{B_{i,p}(x)\}_{i=-p}^{k}.

Author(s)

Heyang Ji

Examples

bsb = bspline_basis(
            Boundary.knots = c(0,24),
            intercept      = TRUE,
            df             = NULL,
            degree         = 3
)
bss = bspline_series(
          coef = c(2,1,1.5,0.5),
          bspline_basis = bsb
)

Extract dimensionality of functional data.

Description

Extract the dimensionality of slot X of functional_variable object.

Usage

## S4 method for signature 'functional_variable'
dim(x)

Arguments

Value

Retruns a 2-element numeric vector.

Author(s)

Heyang Ji

Examples

fv = functional_variable(X=array(rnorm(12),dim = 4:3),period = 3)
dim(fv)

Method of class Fourier_series to extract Fourier coefficients

Description

Method of class Fourier_series to extract Fourier coefficients

Usage

extractCoef(object)

## S4 method for signature 'Fourier_series'
extractCoef(object)

Arguments

Value

A list that contains the coefficients.

Author(s)

Heyang Ji

Examples

fsc = Fourier_series(
           double_constant = 0.5,
           cos = c(0,0.3),
           sin = c(1,0.7),
           k_cos = 1:2,
           )
 extractCoef(fsc)

Extract the value of coefficient parameter function

Description

Generic function to extract the value of coefficient parameter function of the covariates from linear model with functional covariates at some certain points.

Usage

fc.beta(object, ...)

## S4 method for signature 'fcRegression'
fc.beta(object, FC = 1, t_points = NULL)

## S4 method for signature 'fcQR'
fc.beta(object, FC = 1, t_points = NULL)

Arguments

Value

A numeric atomic vector

Author(s)

Heyang Ji

Solve quantile regression models with functional covariate(s).

Description

Fit a quantile regression models below

Q_{Y_i|X_i,Z_i}(\tau) = \sum_{l=1}^L\int_\Omega \beta_l(\tau,t) X_{li}(t) dt + (1,Z_i^T)\gamma

where Q_{Y_i}(\tau) = F_{Y_i|X_i,Z_i}^{-1}(\tau) is the \tau-th quantile of Y_i given X_i(t) and Z_i, \tau\in(0,1).
Model allows one or multiple functional covariate(s) as fixed effect(s), and zero, one, or multiple scalar-valued covariate(s).

Usage

fcQR(
  Y,
  FC,
  Z,
  formula.Z,
  tau = 0.5,
  basis.type = c("Fourier", "Bspline"),
  basis.order = 6L,
  bs_degree = 3
)

Arguments

Value

fcQR returns an object of class "fcQR". It is a list that contains the following elements.

Author(s)

Heyang Ji

Examples

data(MECfda.data.sim.0.0)
res = fcQR(FC = MECfda.data.sim.0.0$FC, Y=MECfda.data.sim.0.0$Y, Z=MECfda.data.sim.0.0$Z,
           basis.order = 5, basis.type = c('Bspline'))

Solve linear models with functional covariate(s)

Description

Function to fit (generalized) linear model with functional covariate(s). Model allows one or multiple functional covariate(s) as fixed effect(s), and zero, one, or multiple scalar-valued fixed or random effect(s).

Usage

fcRegression(
  Y,
  FC,
  Z,
  formula.Z,
  family = gaussian(link = "identity"),
  basis.type = c("Fourier", "Bspline", "FPC"),
  basis.order = 6L,
  bs_degree = 3
)

Arguments

Details

Solve linear models with functional covariates below

g(E(Y_i|X_i,Z_i)) = \sum_{l=1}^{L} \int_{\Omega_l} \beta_l(t) X_{li}(t) dt + (1,Z_i^T)\gamma

where the scalar-valued covariates can be fixed or random effect or doesn't exist (may do not contain scalar-valued covariates).

Value

fcRegression returns an object of class "fcRegression". It is a list that contains the following elements.

Author(s)

Heyang Ji

Examples

data(MECfda.data.sim.0.0)
res = fcRegression(FC = MECfda.data.sim.0.0$FC, Y=MECfda.data.sim.0.0$Y, Z=MECfda.data.sim.0.0$Z,
                   basis.order = 5, basis.type = c('Bspline'),
                   formula.Z = ~ Z_1 + (1|Z_2))

Fourier basis expansion for functional variable data

Description

For a function f(x), x\in\Omega, and a basis function sequence \{\rho_k\}_{k\in\kappa}, basis expansion is to compute \int_\Omega f(t)\rho_k(t) dt. Here we do basis expansion for all f_i(t), t\in\Omega = [t_0,t_0+T] in functional variable data, i=1,\dots,n. We compute a matrix (b_{ik})_{n\times p}, where b_{ik} = \int_\Omega f(t)\rho_k(t) dt. The basis used here is the Fourier basis,

\frac{1}{2},\ \cos(\frac{2\pi}{T}k[x-t_0]),\ \sin (\frac{2\pi}{T}k[x-t_0])

where x\in[t_0,t_0+T] and k = 1,\dots,p_f.

Usage

fourier_basis_expansion(object, order_fourier_basis)

## S4 method for signature 'functional_variable,integer'
fourier_basis_expansion(object, order_fourier_basis)

Arguments

Value

Returns a numeric matrix, (b_{ik})_{n\times p}, where b_{ik} = \int_\Omega f(t)\rho_k(t) dt.

Author(s)

Heyang Ji

Function-valued variable data.

Description

A s4 class that represents data of a function-valued variable. The format is f_i(t),\ t\in\Omega=[t_0,t_0 + T] where i is the observation (subject) index, t represents the measurement (time) points.

Slots

X

a matrix (x_{ij})_{n\times m}, where x_{ij} = f_i(t_j), represents the value of f_i(t_j), each row represent an observation (subject), each column is corresponding to a measurement (time) point.

t_0

start of the domain (time period), t_0. Default is 0.

period

length of the domain (time period), T. Default is 1.

t_points

sequence of the measurement points, (t_1,\dots,t_m). Default is t_k = t_0 + \frac{(2k-1)T}{2(m+1)}.

Author(s)

Heyang Ji

Examples

X = array(rnorm(12),dim = 4:3)
functional_variable(X=X,period = 3)

Compute the value of the basis function summation series at certain points.

Description

Compute the function f(x) = \sum_{k=0}^{p}c_k \rho_{k}(x), x\in\Omega

Usage

numericBasisSeries2fun(object, x)

## S4 method for signature 'numericBasis_series,numeric'
numericBasisSeries2fun(object, x)

Arguments

Value

A numeric atomic vector

Author(s)

Heyang Ji

Examples

t_0 = 0
period = 1
t_points = seq(0.05,0.95,length.out = 19)
nb = numeric_basis(
  basis_function = cbind(1/2,cos(2*pi*t_points),sin(2*pi*t_points)),
  t_points       = t_points,
  t_0            = t_0,
  period         = period
)
ns = numericBasis_series(coef = c(0.8,1.2,1.6),numeric_basis = nb)
numericBasisSeries2fun(ns,seq(0,1,length.out = 51))

Linear combination of a sequence of basis functions represented numerically

Description

A linear combination of basis function \{\rho_k\}_{k=1}^p,

\sum_{k=1}^p c_k \rho_k(t).

Slots

coef

linear coefficient \{c_k\}_{k=1}^p.

numeric_basis

\{\rho_k\}_{k=1}^p represented by a numeric_basis object. See numeric_basis.

Author(s)

Heyang Ji

Examples

t_0 = 0
period = 1
t_points = seq(0.05,0.95,length.out = 19)
nb = numeric_basis(
  basis_function = cbind(1,cos(t_points),sin(t_points)),
  t_points       = t_points,
  t_0            = t_0,
  period         = period
)
ns = numericBasis_series(coef = c(0.8,1.2,1.6),numeric_basis = nb)

Numeric representation of a function basis

Description

A s4 class that numerically represents a basis of linear space of function.
\{\rho_k\}_{k=1}^\infty denotes a basis of function linear space. Some times the basis cannot be expressed analytically. But we can numerically store the space by the value of a finite subset of the basis functions at some certain points in the domain, \rho_k(t_j), k = 1,\dots,p, j = 1,\dots,m. The s4 class is to represent a finite sequence of functions by their values at a finite sequence of points within their domain, in which all the functions have the same domain and the domain is an interval.

Details

The units of a basis of a linear space should be linearly independent. But the program doesn't check the linear dependency of the basis function when a numeric_basis object is initialized.

Slots

basis_function

matrix of the value of the functions, (\zeta_{jk})_{m\times p}, where \zeta_{ik} = \rho_k(t_j), j = 1,\dots,m, k = 1,\dots,p. Each row of the matrix is corresponding to a point of t. Each column of the matrix is corresponding to a basis function.

t_points

a numeric atomic vector, represents the points in the domains of the function where the function values are taken. The jth element is corresponding to jth row of slot basis_function.

t_0

left end of the domain interval.

period

length of the domain interval.

Author(s)

Heyang Ji

Examples

t_0 = 0
period = 1
t_points = seq(0.05,0.95,length.out = 19)
numeric_basis(
  basis_function = cbind(1/2,cos(t_points),sin(t_points)),
  t_points       = t_points,
  t_0            = t_0,
  period         = period
)

Numeric basis expansion for functional variable data

Description

For a function f(t), t\in\Omega, and a basis function sequence \{\rho_k\}_{k\in\kappa}, basis expansion is to compute \int_\Omega f(t)\rho_k(t) dt. Here we do basis expansion for all f_i(t), t\in\Omega = [t_0,t_0+T] in functional variable data, i=1,\dots,n. We compute a matrix (b_{ik})_{n\times p}, where b_{ik} = \int_\Omega f(t)\rho_k(t) dt. The basis we use here is numerically represented by the value of basis functions at some points in the domain.

Usage

numeric_basis_expansion(object, num_basis)

## S4 method for signature 'functional_variable,numeric_basis'
numeric_basis_expansion(object, num_basis)

Arguments

Value

Returns a numeric matrix, (b_{ik})_{n\times p}, with an extra attribute numeric_basis, which is the numeric_basis object input by the argument num_basis.

Author(s)

Heyang Ji

Plot Fourier basis summation series.

Description

Plot Fourier basis summation series.

Usage

## S4 method for signature 'Fourier_series'
plot(x, y, ...)

Arguments

Author(s)

Heyang Ji

Examples

fsc = Fourier_series(
double_constant = 0.5,
cos = c(0,0.3),
sin = c(1,0.7),
k_cos = 1:2,
)
plot(fsc)

Plot b-splines basis summation series.

Description

Plot b-splines basis summation series.

Usage

## S4 method for signature 'bspline_series'
plot(x, y, ...)

Arguments

Author(s)

Heyang Ji

Examples

bsb = bspline_basis(
Boundary.knots = c(0,24),
intercept      = TRUE,
df             = NULL,
degree         = 3
)
bss = bspline_series(
coef = c(2,1,1.5,3),
bspline_basis = bsb
)
plot(bss)

Plot numeric basis function summation series.

Description

Plot numeric basis function summation series.

Usage

## S4 method for signature 'numericBasis_series'
plot(x, y, ...)

Arguments

Author(s)

Heyang Ji

Examples

t_0 = 0
period = 1
t_points = seq(0.05,0.95,length.out = 19)
nb = numeric_basis(
  basis_function = cbind(1/2,cos(2*pi*t_points),sin(2*pi*t_points)),
  t_points       = t_points,
  t_0            = t_0,
  period         = period
)
ns = numericBasis_series(coef = c(0.8,1.2,1.6),numeric_basis = nb)
plot(ns)

Predicted values based on fcQR object

Description

Predicted values based on the Quantile linear model with functional covariates represented by a "fcQR" class object.

Usage

## S3 method for class 'fcQR'
predict(object, newData.FC, newData.Z = NULL, ...)

Arguments

Details

If no new data is input, will return the fitted value.

Value

See predict.rq.

Author(s)

Heyang Ji

Predicted values based on fcRegression object

Description

Predicted values based on the linear model with functional covariates represented by a "fcRegression" class object.

Usage

## S3 method for class 'fcRegression'
predict(object, newData.FC, newData.Z = NULL, ...)

Arguments

Details

If no new data is input, will return the fitted value.

Value

See predict.lm, predict.glm, predict.merMod.

Author(s)

Heyang Ji

Examples

data(MECfda.data.sim.0.0)
res = fcRegression(FC = MECfda.data.sim.0.0$FC, Y=MECfda.data.sim.0.0$Y, Z=MECfda.data.sim.0.0$Z,
                   basis.order = 5, basis.type = c('Bspline'),
                   formula.Z = ~ Z_1 + (1|Z_2))
data(MECfda.data.sim.1.0)
predict(object = res, newData.FC = MECfda.data.sim.1.0$FC,newData.Z = MECfda.data.sim.1.0$Z)