Package 'MECfda'

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

object	An object that represents a functional basis.
x	point(s) to take value.

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

---

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

object	a functional_variable class object.
n_splines	the number of splines, equal to $k+p+1$. See df in bs.
bs_degree	the degree of the piecewise polynomial of the b-splines. See degree in bs.

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-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 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
)

---

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

object	an object of bspline_series class.
x	Value of $x$.

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)

---

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

x	a functional_variable object.

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

object

an object of Fourier_series class.

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

object	An object that represents a functional covariates linear model.
...	More arguments.
FC	An integer, represent the ordinal number of the functional covariate. Default is 1, which is take the first functional covariate.
t_points	Sequence of the measurement (time) points.

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

Y	Response variable, can be an atomic vector, a one-column matrix or data frame, recommended form is a one-column data frame with column name
FC	Functional covariate(s), can be a "functional_variable" object or a matrix or a data frame or a list of these object(s)
Z	Scalar covariate(s), can be NULL or not input (when there's no scalar covariate), an atomic vector (when only one scalar covariate), a matrix or data frame, recommended form is a data frame with column name(s)
formula.Z	A formula without the response variable, contains only scalar covariate(s). If not assigned, include all scalar covariates and intercept term.
tau	Quantile $\tau\in(0,1)$, default is 0.5. See rq.
basis.type	Type of funtion basis. Can only be assigned as one type even if there is more than one functional covariates. Available options: 'Fourier' or 'Bspline' or 'FPC', represent Fourier basis, b-spline basis, and functional principal component (FPC) basis respectively. For the detailed form for Fourier, b-splines, and FPC basis, see fourier_basis_expansion, bspline_basis_expansion, and FPC_basis_expansion.
basis.order	Indicate number of the function basis. When using Fourier basis $\frac{1}{2},\sin k t, \cos k t, k = 1,\dots,K$, basis.order is the number $K$. When using b-splines basis $\{B_{i,p}(x)\}_{i=-p}^{k}$, basis.order is the number of splines, equal to $k+p+1$. When using FPC basis, basis.order is the number of functional principal components. (same as arguement df in bs.) May set a individual number for each functional covariate. When the element of this argument is less than the number of functional covariates, it will be used recursively.
bs_degree	Degree of the piecewise polynomials if use b-splines basis, default is 3. See degree in bs.

Value

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

regression_result	Result of the regression.
FC.BasisCoefficient	A list of Fourier_series or bspline_series object(s), represents the functional linear coefficient(s) of the functional covariates.
function.basis.type	Type of funtion basis used.
basis.order	Same as in the arguemnets.
data	Original data.
bs_degree	Degree of the splines, returned only if b-splines basis is used.

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

Y	Response variable, can be an atomic vector, a one-column matrix or data frame, recommended form is a one-column data frame with column name.
FC	Functional covariate(s), can be a "functional_variable" object or a matrix or a data frame or a list of these object(s).
Z	Scalar covariate(s), can be NULL or not input (when there's no scalar covariate), an atomic vector (when only one scalar covariate), a matrix or data frame, recommended form is a data frame with column name(s).
formula.Z	A formula without the response variable, contains only scalar covariate(s) (or intercept), use the format of lme4 package if random effects exist. e.g. ~ Z_1 + (1|Z_2). (See lmer and glmer) If not assigned, include all scalar covariates and intercept term as fixed effects.
family	A description of the error distribution and link function to be used in the model, see family.
basis.type	Type of funtion basis. Can only be assigned as one type even if there is more than one functional covariates. Available options: 'Fourier' or 'Bspline' or 'FPC', represent Fourier basis, b-spline basis, and functional principal component (FPC) basis respectively. For the detailed form for Fourier, b-splines, and FPC basis, see fourier_basis_expansion, bspline_basis_expansion, and FPC_basis_expansion.
basis.order	Indicate number of the function basis. When using Fourier basis $\frac{1}{2},\sin k t, \cos k t, k = 1,\dots,p_f$, basis.order is the number $p_f$. When using b-splines basis $\{B_{i,p}(x)\}_{i=-p}^{k}$, basis.order is the number of splines, equal to $k+p+1$. (same as arguement df in bs.) When using FPC basis, basis.order is the number of functional principal components. May set a individual number for each functional covariate. When the element of this argument is less than the number of functional covariates, it will be used recursively.
bs_degree	Degree of the piecewise polynomials if use b-splines basis, default is 3. See degree in bs.

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.

regression_result	Result of the regression.
FC.BasisCoefficient	A list of Fourier_series or bspline_series or numeric_basis object(s), represents the functional linear coefficient(s) of the functional covariates.
function.basis.type	Type of funtion basis used.
basis.order	Same as in the arguemnets.
data	Original data.
bs_degree	Degree of the splines, returned only if b-splines basis is used.

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

object	a functional_variable class object.
order_fourier_basis	the order of Fourier basis, $p_f$.

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

---

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,
           )

---

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

object	an object of Fourier_series class.
x	Value of $x$.

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)

---

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

object	a functional_variable class object. The minimum and maximum of the slot t_points should be respectively equal to the slot t_0 and slot t_0 plus slot period.
npc	The number of functional principal components. See npc in fpca.sc.

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)

---

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)

---

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

data.Y	Response variable, can be an atomic vector, a one-column matrix or data frame, recommended form is a one-column data frame with column name.
data.W	A dataframe or matrix, represents $W$, the measurement of $X$. Each row represents a subject. Each column represent a measurement (time) point.
data.M	A dataframe or matrix, represents $M$, the instrumental variable. Each row represents a subject. Each column represent a measurement (time) point.
t_interval	A 2-element vector, represents an interval, means the domain of the functional covariate. Default is c(0,1), represent interval $[0,1]$.
t_points	Sequence of the measurement (time) points, default is NULL.
CI.bootstrap	Whether to return the confidence using bootstrap method. Default is FALSE.

Value

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

beta_tW	Parameter estimates.
CI	Confidence interval, returnd only when CI.bootstrap is TRUE.

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

data.Y	Response variable, can be an atomic vector, a one-column matrix or data frame, recommended form is a one-column data frame with column name.
data.W	A 3-dimensional array, represents $W$, the measurement of $X$. Each row represents a subject. Each column represent a measurement (time) point. Each layer represents an observation.
data.Z	Scalar covariate(s), can be not input or NULL (when there's no scalar covariate), an atomic vector (when only one scalar covariate), a matrix or data frame, recommended form is a data frame with column name(s).
tau	Quantile $\tau\in(0,1)$, default is 0.5.
t_interval	A 2-element vector, represents an interval, means the domain of the functional covariate. Default is c(0,1), represent interval $[0,1]$.
t_points	Sequence of the measurement (time) points, default is NULL
grid_k	An atomic vector, of which each element is candidate number of basis.
grid_h	A non-zero-value atomic vector, of which each element is candidate value of tunning parameter.
degree	Used in computation for derivative and integral, defult is 45, large enough for most scenario.
observed_X	For estimating parametric variance. Default is NULL.

Value

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

estimated_beta_hat	Estimated coefficients from corrected loss function (including functional part)
estimated_beta_t	Estimated functional curve
SE_est	Estimated parametric variance. Returned only if observed_X is not NULL.
estimated_Xbasis	The basis matrix we used
res_naive	results of naive method

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

data.Y	Response variable, can be an atomic vector, a one-column matrix or data frame, recommended form is a one-column data frame with column name.
data.W	A dataframe or matrix, represents $W$, the measurement of $X$. Each row represents a subject. Each column represent a measurement (time) point.
data.Z	Scalar covariate(s), can be not input or NULL (when there's no scalar covariate), an atomic vector (when only one scalar covariate), a matrix or data frame, recommended form is a data frame with column name(s).
data.M	A dataframe or matrix, represents $M$, the instrumental variable. Each row represents a subject. Each column represent a measurement (time) point.
tau	Quantile $\tau\in(0,1)$, default is 0.5.
t_interval	A 2-element vector, represents an interval, means the domain of the functional covariate. Default is c(0,1), represent interval $[0,1]$.
t_points	Sequence of the measurement (time) points, default is NULL.
formula.Z	A formula without the response variable, contains only scalar covariate(s), no random effects. If not assigned, include all scalar covariates and intercept term.
basis.type	Type of funtion basis. Can only be assigned as one type even if there is more than one functional covariates. Available options: 'Fourier' or 'Bspline', represent Fourier basis and b-spline basis respectively. For the detailed form for Fourier and b-splines basis, see fourier_basis_expansion and bspline_basis_expansion.
basis.order	Indicate number of the function basis. When using Fourier basis $\frac{1}{2},\sin k t, \cos k t, k = 1,\dots,K$, basis.order is the number $K$. When using b-splines basis $\{B_{i,p}(x)\}_{i=-p}^{k}$, basis.order is the number of splines, equal to $k+p+1$. (same as arguement df in bs.) May set a individual number for each functional covariate. When the element of this argument is less than the number of functional covariates, it will be used recursively.
bs_degree	Degree of the piecewise polynomials if use b-splines basis, default is 3. See degree in bs.

Value

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

coef.X	A Fourier_series or bspline_series object, represents the functional coefficient parameter of the functional covariate.
coef.Z	The estimate of the linear coefficients of the scalar covariates.
coef.all	Original estimate of linear coefficients.
function.basis.type	Type of funtion basis used.
basis.order	Same as in the input arguements.
t_interval	A 2-element vector, represents an interval, means the domain of the functional covariate.
t_points	Sequence of the measurement (time) points.
formula	Regression model.
formula.Z	formula object contains only the scalar covariate(s).
zlevels	levels of the non-continuous scalar covariate(s).

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

data.Y	Response variable, can be an atomic vector, a one-column matrix or data frame, recommended form is a one-column data frame with column name.
data.W	A 3-dimensional array, represents $W$, the measurement of $X$. Each row represents a subject. Each column represent a measurement (time) point. Each layer represents an observation.
data.Z	Scalar covariate(s), can be not input or NULL (when there's no scalar covariate), an atomic vector (when only one scalar covariate), a matrix or data frame, recommended form is a data frame with column name(s).
method	The method to construct the substitution $X$. Available options: 'UP_MEM', 'MP_MEM', 'average'.
t_interval	A 2-element vector, represents an interval, means the domain of the functional covariate. Default is c(0,1), represent interval $[0,1]$.
t_points	Sequence of the measurement (time) points, default is NULL.
d	The number of time points involved for MP_MEM (default and miniumn is 3).
family.W	Distribution of $W$ given $X$, Available options: "gaussian","poisson".
family.Y	A description of the error distribution and link function to be used in the model, see family.
formula.Z	A formula without the response variable, contains only scalar covariate(s), use the format of lme4 package if random effects exist. e.g. ~ Z_1 + (1|Z_2). If not assigned, include all scalar covariates and intercept term as fixed effects.
basis.type	Type of function basis. Can only be assigned as one type even if there is more than one functional covariates. Available options: 'Fourier' or 'Bspline', represent Fourier basis and b-spline basis respectively. For the detailed form for Fourier and b-splines basis, see fourier_basis_expansion and bspline_basis_expansion.
basis.order	Indicate number of the function basis. When using Fourier basis $\frac{1}{2},\sin k t, \cos k t, k = 1,\dots,K$, basis.order is the number $K$. When using b-splines basis $\{B_{i,p}(x)\}_{i=-p}^{k}$, basis.order is the number of splines, equal to $k+p+1$. (same as arguement df in bs.) May set a individual number for each functional covariate. When the element of this argument is less than the number of functional covariates, it will be used recursively.
bs_degree	Degree of the piecewise polynomials if use b-splines basis, default is 3. See degree in bs.
smooth	Whether to smooth the substitution of $X$. Default is FALSE.
silent	Whether not to show the state of the running of the function. Default is TRUE.

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')

---

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

N	Sample size.
distribution	Conditional distribution of response varaible given the covariate ($Y_i|X_i(t),Z_i$). There are two options: 'Gaussian' and 'Bernoulli'.
t_interval	A 2-element vector, represents an interval, means the domain of the functional covariate. Default is c(0,1), represent interval $[0,1]$.
t_points	the measurement points of functional variables, should be numeric vector.
n_t	Number of measurement time points. Overwritten if argument t_points is assigned.
seed	Pseudo-random number generation seed.

Value

return a list with following elements.

Y	An atomic vector of response variable
Z	A dataframe with a binary and a continuous scalar-valued covariate.
FC	A list of two 'functional_variable' class object.
t_interval	Same as in the input argument.
t_points	Sequence of the measurement (time) points.

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"))

---

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

N	Sample size.
J_W	Number of repeated measurement (period), if applicable.
forwhich	For which method of measurement error bias correction method the data set is generated. There are two options: 'MEM','IV','CLS','IV.SIMEX'.
t_interval	A 2-element vector, represents an interval, means the domain of the functional covariate. Default is c(0,1), represent interval $[0,1]$.
n_t	Number of measurement time points.
seed	Pseudo-random number generation seed.

Value

return a list that possibly contains following elements.

Y	An atomic vector of response variable
Z	A dataframe with a binary and a continuous scalar-valued covariate.
W	Observed values of function-valued covariate.
M	Instrumental vairable.
t_interval	Same as in the input argument.
t_points	Sequence of the measurement (time) points.

Examples

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

---

Simulated data

Description

Simulated data

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Simulated data

Description

Simulated data

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Simulated data

Description

Simulated data

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Simulated data

Description

Simulated data

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Simulated data

Description

Simulated data

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Simulated data

Description

Simulated data

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Simulated data

Description

Simulated data

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Simulated data

Description

Simulated data

---

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

data.W	A 3-dimensional array, represents $W$, the measurement of $X$. Each row represents a subject. Each column represent a measurement (time) point. Each layer represents an observation.
method	The method to construct the substitution $X$. Available options: 'UP_MEM', 'MP_MEM', 'average'.
d	The number of time points involved for MP_MEM (default and miniumn is 3).
family.W	Distribution of $W$ given $X$, Available options: "gaussian", "poisson".
smooth	Whether to smooth the substitution of $X$. Default is FALSE.

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")

---

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

object	a functional_variable class object. The minimum and maximum of the slot t_points should be respectively equal to the slot t_0 and slot t_0 plus slot period.
num_basis	a numeric_basis class object, representing the function basis. See numeric_basis.

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

---

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 $j$th element is corresponding to $j$th 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
)

---

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)

---

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

object	an object of numericBasis_series class.
x	Value of $x$.

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))

---

Plot b-splines basis summation series.

Description

Plot b-splines basis summation series.

Usage

## S4 method for signature 'bspline_series'
plot(x)

Arguments

x	A bspline_series object.

Value

No return value. Generate a scatter plot.

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 Fourier basis summation series.

Description

Plot Fourier basis summation series.

Usage

## S4 method for signature 'Fourier_series'
plot(x)

Arguments

x	A Fourier_series object.

Value

No return value. Generate a scatter plot.

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 numeric basis function summation series.

Description

Plot numeric basis function summation series.

Usage

## S4 method for signature 'numericBasis_series'
plot(x)

Arguments

x	A numericBasis_series object.

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

object	A fcQR class object produced by fcQR.
newData.FC	A atomic vector or a matrix or a dataframe or a functional_variable class object or a list of objects above. See argument FC in fcRegression.
newData.Z	A dataframe or a matrix or a atomic vector. See argument Z in fcRegression.
...	Further arguments passed to or from other methods predict.rq.

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

object	A fcRegression class object produced by fcRegression.
newData.FC	A atomic vector or a matrix or a dataframe or a functional_variable class object or a list of objects above. See argument FC in fcRegression.
newData.Z	A dataframe or a matrix or a atomic vector. See arguement Z in fcRegression.
...	Further arguments passed to or from other methods, including predict.lm, predict.glm, predict.merMod.

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)

---