Type:	Package
Title:	Robust Functional Linear Regression
Version:	1.2
Date:	2024-01-24
Depends:	R (≥ 3.5.0), fda, MASS, robustbase
Imports:	expm, fda.usc, goffda, mvtnorm, pcaPP, fields
LazyLoad:	yes
ByteCompile:	TRUE
Maintainer:	Ufuk Beyaztas <ufukbeyaztas@gmail.com>
Description:	Functions for implementing robust methods for functional linear regression. In the functional linear regression, we consider scalar-on-function linear regression and function-on-function linear regression. More details, see Beyaztas, U., and Shang, H. L. (2021) <doi:10.48550/arXiv.2111.01238> and Beyaztas, U., and Shang, H. L. (2022) <doi:10.48550/arXiv.2203.05065>.
License:	GPL-3
NeedsCompilation:	no
Packaged:	2024-01-23 21:05:46 UTC; hanlinshang
Author:	Ufuk Beyaztas [aut, cre, cph], Han Lin Shang [aut]
Repository:	CRAN
Date/Publication:	2024-01-23 21:30:03 UTC

Robust function-on-function regression

Description

This package presents robust methods for analyzing functional linear regression.

Author(s)

Ufuk Beyaztas and Han Lin Shang

Maintainer: Ufuk Beyaztas <ufukbeyaztas@gmail.com>

References

B. Akturk, U. Beyaztas, H. L. Shang, A. Mandal (2024) Robust functional logistic regression, Advances in Data Analysis and Classification, in press.

U. Beyaztas, H. L. Shang and A. Mandal (2024) Robust function-on-function interaction regression, Statistical Modelling: An International Journal, in press.

U. Beyaztas, M. Tez and H. L. Shang (2024) Robust scalar-on-function partial quantile regression, Journal of Applied Statistics, in press.

U. Beyaztas and H. L. Shang (2023) Robust functional linear regression models, The R Journal, 15(1), 212-233.

M. Mutis, U. Beyaztas, G. G. Simsek and H. L. Shang (2023) A robust scalar-on-function logistic regression for classification, Communications in Statistics - Theory and Methods, 52(23), 8538-8554.

S. Saricam, U. Beyaztas, B. Asikgil and H. L. Shang (2022) On partial least-squares estimation in scalar-on-function regression models, Journal of Chemometrics, 36(12), e3452.

U. Beyaztas and H. L. Shang (2022) A comparison of parameter estimation in function-on-function regression, Communications in Statistics - Simulation and Computation, 51(8), 4607-4637.

U. Beyaztas and H. L. Shang (2022) A robust functional partial least squares for scalar-on-multiple-function regression, Journal of Chemometrics, 36(4), e3394.

U. Beyaztas, H. L. Shang and A. Alin (2022) Function-on-function partial quantile regression, Journal of Agricultural, Biological, and Environmental Statistics, 27(1), 149-174.

U. Beyaztas and H. L. Shang (2021) A partial least squares approach for function-on-function interaction regression, Computational Statistics, 36(2), 911-939.

U. Beyaztas and H. L. Shang (2021) A robust partial least squares approach for function-on-function regression, Brazilian Journal of Probability and Statistics, 36(2), 199-219.

U. Beyaztas and H. L. Shang (2021) Function-on-function linear quantile regression, Mathematical Modelling and Analysis, 27(2), 322-341.

U. Beyaztas and H. L. Shang (2020) On function-on-function regression: partial least squares approach, Environmental and Ecological Statistics, 27(1), 95-114.

U. Beyaztas and H. L. Shang (2019) Forecasting functional time series using weighted likelihood methodology, 89(16), 3046-3060.

Hourly River Flow Measurements in the Mery River

Description

Hourly river flow measurements obtained from January 2009 to December 2014 (6 years in total) in the Mery River, Australia.

Usage

data(MaryRiverFlow)

Author(s)

Ufuk Beyaztas and Han Lin Shang

Examples

data(MaryRiverFlow)
# Plot
library(fda.usc)
fflow <- fdata(MaryRiverFlow, argvals = 1:24)
plot(fflow, lty = 1, ylab = "", xlab = "Hour",
main = "", mgp = c(2, 0.5, 0), ylim = range(fflow))

Generate functional data for the function-on-function regression model

Description

This function provides a unified simulation structure for the function-on-function regression model

Y(t) = \sum_{m=1}^M \int X_m(s) \beta_m(s,t) ds + \epsilon(t),

where Y(t) denotes the functional response, X_m(s) denotes the m-th functional predictor, \beta_m(s,t) denotes the m-th bivariate regression coefficient function, and \epsilon(t) is the error function.

Usage

generate.ff.data(n.pred, n.curve, n.gp, out.p = 0)

Arguments

Details

In the data generation process, first, the functional predictors are simulated based on the following process:

X_m(s) = \sum_{j=1}^5 \kappa_j v_j(s),

where \kappa_j is a vector generated from a Normal distribution with mean one and variance \sqrt{a} j^{-1/2}, a is a uniformly generated random number between 1 and 4, and

v_j(s) = \sin(j \pi s) - \cos(j \pi s).

The bivariate regression coefficient functions are generated from a coefficient space that includes ten different functions such as:

b \sin(2 \pi s) \sin(\pi t)

and

b e^{-3 (s - 0.5)^2} e^{-4 (t - 1)^2},

where b is generated from a uniform distribution between 1 and 3. The error function \epsilon(t), on the other hand, is generated from the Ornstein-Uhlenbeck process:

\epsilon(t) = l + [\epsilon_0(t) - l] e^{-\theta t} + \sigma \int_0^t e^{-\theta (t-u)} d W_u,

where l, \theta > 0, \sigma > 0 are constants, \epsilon_0(t) is the initial value of \epsilon(t) taken from W_u, and W_u is the Wiener process. If outliers are allowed in the generated data, i.e., out.p > 0, then, the randomly selected n.curve \times out.p of the data are generated in a different way from the aforementioned process. In more detail, if out.p > 0, the bivariate regression coefficient functions (possibly different from the previously generated coefficient functions) generated from the coefficient space with b^* (instead of b), where b^* is generated from a uniform distribution between 1 and 2, are used to generate the outlying observations. In addition, in this case, the following process is used to generate functional predictors:

X_m^*(s) = \sum_{j=1}^5 \kappa_j^* v_j^*(s),

where \kappa_j^* is a vector generated from a Normal distribution with mean one and variance \sqrt{a} j^{-3/2} and

v_j^*(s) = 2 \sin(j \pi s) - \cos(j \pi s).

All the functions are generated equally spaced point in the interval [0, 1].

Value

A list object with the following components:

Author(s)

Ufuk Beyaztas and Han Lin Shang

References

E. Garcia-Portugues and J. Alvarez-Liebana J and G. Alvarez-Perez G and W. 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.

Examples

library(fda)
library(fda.usc)
set.seed(2022)
sim.data <- generate.ff.data(n.pred = 5, n.curve = 200, n.gp = 101, out.p = 0.1)
Y <- sim.data$Y
X <- sim.data$X
coeffs <- sim.data$f.coef
out.indx <- sim.data$out.indx
fY <- fdata(Y, argvals = seq(0, 1, length.out = 101))
plot(fY[-out.indx,], lty = 1, ylab = "", xlab = "Grid point", 
     main = "Response", mgp = c(2, 0.5, 0), ylim = range(fY))
lines(fY[out.indx,], lty = 1, col = "black") # Outlying functions

Generate functional data for the scalar-on-function regression model

Description

This function is used to simulate data for the scalar-on-function regression model

Y = \sum_{m=1}^M \int X_m(s) \beta_m(s) ds + \epsilon,

where Y denotes the scalar response, X_m(s) denotes the m-th functional predictor, \beta_m(s) denotes the m-th regression coefficient function, and \epsilon is the error process.

Usage

generate.sf.data(n, n.pred, n.gp, out.p = 0)

Arguments

Details

In the data generation process, first, the functional predictors are simulated based on the following process:

X_m(s) = \sum_{j=1}^5 \kappa_j v_j(s),

where \kappa_j is a vector generated from a Normal distribution with mean one and variance \sqrt{a} j^{-3/2}, a is a uniformly generated random number between 1 and 4, and

v_j(s) = \sin(j \pi s) - \cos(j \pi s).

The regression coefficient functions are generated from a coefficient space that includes ten different functions such as:

b \sin(2 \pi s)

and

b \cos(2 \pi s),

where b is generated from a uniform distribution between 1 and 3. The error process is generated from the standard normal distribution. If outliers are allowed in the generated data, i.e., out.p > 0, then, the randomply selected n \times out.p of the data are generated in a different way from the aforementioned process. In more detail, if out.p > 0, the regression coefficient functions (possibly different from the previously generated coefficient functions) generated from the coefficient space with b^* (instead of b), where b^* is generated from a uniform distribution between 3 and 5, are used to generate the outlying observations. In addition, in this case, the following process is used to generate functional predictors:

X_m^*(s) = \sum_{j=1}^5 \kappa_j^* v_j^*(s),

where \kappa_j^* is a vector generated from a Normal distribution with mean one and variance \sqrt{a} j^{-1/2} and

v_j^*(s) = 2 \sin(j \pi s) - \cos(j \pi s).

Moreover, the error process is generated from a normal distribution with mean 1 and variance 1. All the functional predictors are generated equally spaced point in the interval [0, 1].

Value

A list object with the following components:

Author(s)

Ufuk Beyaztas and Han Lin Shang

Examples

library(fda.usc)
library(fda)
set.seed(2022)
sim.data <- generate.sf.data(n = 400, n.pred = 5, n.gp = 101, out.p = 0.1)
Y <- sim.data$Y
X <- sim.data$X
coeffs <- sim.data$f.coef
out.indx <- sim.data$out.indx
plot(Y[-out.indx,], type = "p", pch = 16, xlab = "Index", ylab = "",
main = "Response", ylim = range(Y))
points(out.indx, Y[out.indx,], type = "p", pch = 16, col = "blue") # Outliers
fX1 <- fdata(X[[1]], argvals = seq(0, 1, length.out = 101))
plot(fX1[-out.indx,], lty = 1, ylab = "", xlab = "Grid point",
     main = expression(X[1](s)), mgp = c(2, 0.5, 0), ylim = range(fX1))
lines(fX1[out.indx,], lty = 1, col = "black") # Leverage points

Get the estimated bivariate regression coefficient functions for function-on-function regression model

Description

This function is used to obtain the estimated bivariate regression coefficient functions \beta_m(s,t) for function-on-function regression model (see the description in rob.ff.reg based on output object obtained from rob.ff.reg).

Usage

get.ff.coeffs(object)

Arguments

Details

In the estimation of bivariate regression coefficient functions, the estimated functional principal components of response \hat{\Phi}(t) and predictor \hat{\Psi}_m(s) variables and the estimated regression parameter function obtained from the regression model between the principal component scores of response and predictor variables \hat{B} are used, i.e., \hat{\beta}_m(s,t) = \hat{\Psi}_m^\top(s) \hat{B} \hat{\Phi}(t).

Value

A list object with the following components:

Author(s)

Ufuk Beyaztas and Han Lin Shang

Examples

sim.data <- generate.ff.data(n.pred = 5, n.curve = 200, n.gp = 101)
Y <- sim.data$Y
X <- sim.data$X
gpY = seq(0, 1, length.out = 101) # grid points of Y
gpX <- rep(list(seq(0, 1, length.out = 101)), 5) # grid points of Xs
model.fit <- rob.ff.reg(Y, X, model = "full", emodel = "classical", 
                        gpY = gpY, gpX = gpX)
coefs <- get.ff.coeffs(model.fit)

Get the estimated regression coefficient functions for scalar-on-function regression model

Description

This function is used to obtain the estimated regression coefficient functions \beta_m(s) and the estimated regression coefficients \gamma_r (if X.scl \neq NULL) for scalar-on-function regression model (see the description in rob.sf.reg based on output object obtained from rob.sf.reg).

Usage

get.sf.coeffs(object)

Arguments

Details

In the estimation of regression coefficient functions, the estimated functional principal components of predictor \hat{\Psi}_m(s), 1\le m\le M variables and the estimated regression parameter function obtained from the regression model of scalar response on the principal component scores of the functional predictor variables \hat{B} are used, i.e., \hat{\beta}_m(s) = \hat{\Psi}_m^\top(s) \hat{B}.

Value

A list object with the following components:

Author(s)

Ufuk Beyaztas and Han Lin Shang

Examples

sim.data <- generate.sf.data(n = 400, n.pred = 5, n.gp = 101)
Y <- sim.data$Y
X <- sim.data$X
gp <- rep(list(seq(0, 1, length.out = 101)), 5) # grid points of Xs
model.fit <- rob.sf.reg(Y, X, emodel = "classical", gp = gp)
coefs <- get.sf.coeffs(model.fit)

Functional principal component analysis

Description

This function is used to perform functional principal component analysis.

Usage

getPCA(data, nbasis, ncomp, gp, emodel = c("classical", "robust"))

Arguments

Details

The functional principal decomposition of a functional data X(s) is given by

X(s) = \bar{X}(s) + \sum_{k=1}^K \xi_k \psi_k(s),

where \bar{X}(s) is the mean function, \psi_k(s) is the k-th weight function, and \xi_k is the corresponding principal component score which is given by

\xi_k = \int (X(s) - \bar{X}(s)) \psi_k(s) ds.

When computing the estimated functional principal components, first, the functional data is expressed by a set of B-spline basis expansion. Then, the functional principal components are equal to the principal components extracted from the matrix D \varphi^{1/2}, where D is the matrix of basis expansion coefficients and \varphi is the inner product matrix of the basis functions, i.e., \varphi = \int \varphi(s) \varphi^\top(s) ds. Finally, the k-th weight function is given by \psi_k(s) = \varphi^{-1/2} a_k, where a_k is the k-th eigenvector of the sample covariance matrix of D \varphi^{1/2}.

If emodel = "classical", then, the standard functional principal component decomposition is used as given by Ramsay and Dalzell (1991).

If emodel = "robust", then, the robust principal component algorithm of Hubert, Rousseeuw and Verboven (2002) is used.

Value

A list object with the following components:

Author(s)

Ufuk Beyaztas and Han Lin Shang

References

J. O. Ramsay and C. J. Dalzell (1991) "Some tools for functional data analysis (with discussion)", Journal of the Royal Statistical Society: Series B, 53(3), 539-572.

M. Hubert and P. J. Rousseeuw and S. Verboven (2002) "A fast robust method for principal components with applications to chemometrics", Chemometrics and Intelligent Laboratory Systems, 60(1-2), 101-111.

P. Filzmoser and H. Fritz and K Kalcher (2021) pcaPP: Robust PCA by Projection Pursuit, R package version 1.9-74, URL: https://cran.r-project.org/web/packages/pcaPP/index.html.

J. L. Bali and G. Boente and D. E. Tyler and J.-L. Wang (2011) "Robust functional principal components: A projection-pursuit approach", The Annals of Statistics, 39(6), 2852-2882.

Examples

sim.data <- generate.ff.data(n.pred = 5, n.curve = 200, n.gp = 101)
Y <- sim.data$Y
gpY <- seq(0, 1, length.out = 101) # grid points
rob.fpca <- getPCA(data = Y, nbasis = 20, ncomp = 4, gp = gpY, emodel = "robust")

Get the functional principal component scores for a given test sample

Description

This function is used to compute the functional principal component scores of a test sample based on outputs obtained from getPCA.

Usage

getPCA.test(object, data)

Arguments

Details

See getPCA for details.

Value

A matrix of principal component scores for the functional data.

Author(s)

Ufuk Beyaztas and Han Lin Shang

Examples

sim.data <- generate.ff.data(n.pred = 5, n.curve = 200, n.gp = 101)
Y <- sim.data$Y
Y.train <- Y[1:100,]
Y.test <- Y[101:200,]
gpY = seq(0, 1, length.out = 101) # grid points
rob.fpca <- getPCA(data = Y.train, nbasis = 20, ncomp = 4,
gp = gpY, emodel = "robust")
rob.fpca.test <- getPCA.test(object = rob.fpca, data = Y.test)

Image plot of bivariate regression coefficient functions of a function-on-function regression model

Description

This function is used to obtain image plots of bivariate regression coefficient functions of a function-on-function regression model based on output object obtained from get.ff.coeffs.

Usage

plot_ff_coeffs(object, b)

Arguments

Value

No return value, called for side effects.

Author(s)

Ufuk Beyaztas and Han Lin Shang

References

D. Nychka and R. Furrer and J. Paige and S. Sain (2021) fields: Tools for spatial data. R package version 14.1, URL: https://github.com/dnychka/fieldsRPackage.

Examples

sim.data <- generate.ff.data(n.pred = 5, n.curve = 200, n.gp = 101)
Y <- sim.data$Y
X <- sim.data$X
gpY = seq(0, 1, length.out = 101) # grid points of Y
gpX <- rep(list(seq(0, 1, length.out = 101)), 5) # grid points of Xs
model.fit <- rob.ff.reg(Y, X, model = "full", emodel = "classical", 
                        gpY = gpY, gpX = gpX)
coefs <- get.ff.coeffs(model.fit)
plot_ff_coeffs(object = coefs, b = 1)

Plot of regression coefficient functions of a scalar-on-function regression model

Description

This function is used to obtain the plots of regression coefficient functions of a scalar-on-function regression model based on output object obtained from get.sf.coeffs.

Usage

plot_sf_coeffs(object, b)

Arguments

Value

No return value, called for side effects.

Author(s)

Ufuk Beyaztas and Han Lin Shang

Examples

sim.data <- generate.sf.data(n = 400, n.pred = 5, n.gp = 101)
Y <- sim.data$Y
X <- sim.data$X
gp <- rep(list(seq(0, 1, length.out = 101)), 5) # grid points of Xs
model.fit <- rob.sf.reg(Y, X, emodel = "classical", gp = gp)
coefs <- get.sf.coeffs(model.fit)
plot_sf_coeffs(object = coefs, b = 1)

Prediction for a function-on-function regression model

Description

This function is used to make prediction for a new set of functional predictors based upon a fitted function-on-function regression model in the output of rob.ff.reg.

Usage

predict_ff_regression(object, Xnew)

Arguments

Value

An n_{test} \times p-dimensional matrix of predicted functions of the response variable for the given set of new functional predictors Xnew. Here, n_{test}, the number of rows of the matrix of predicted values, equals to the number of rows of Xnew, and p equals to the number of columns of Y, the input in the rob.ff.reg.

Author(s)

Ufuk Beyaztas and Han Lin Shang

Examples

set.seed(2022)
sim.data <- generate.ff.data(n.pred = 5, n.curve = 200, n.gp = 101, out.p = 0.1)
out.indx <- sim.data$out.indx
Y <- sim.data$Y
X <- sim.data$X
indx.test <- sample(c(1:200)[-out.indx], 60)
indx.train <- c(1:200)[-indx.test]
Y.train <- Y[indx.train,]
Y.test <- Y[indx.test,]
X.train <- X.test <- list()
for(i in 1:5){
  X.train[[i]] <- X[[i]][indx.train,]
  X.test[[i]] <- X[[i]][indx.test,]
}
gpY = seq(0, 1, length.out = 101) # grid points of Y
gpX <- rep(list(seq(0, 1, length.out = 101)), 5) # grid points of Xs

model.MM <- rob.ff.reg(Y = Y.train, X = X.train, model = "full", emodel = "robust",
                       fmodel = "MM", gpY = gpY, gpX = gpX)
pred.MM <- predict_ff_regression(object = model.MM, Xnew = X.test)
round(mean((Y.test - pred.MM)^2), 4)        # 0.5925 (MM method)

Prediction for a scalar-on-function regression model

Description

This function is used to make prediction for a new set of functional and scalar (if any) predictors based upon a fitted scalar-on-function regression model in the output of rob.sf.reg.

Usage

predict_sf_regression(object, Xnew, Xnew.scl = NULL)

Arguments

Value

An n_{test} \times 1-dimensional matrix of predicted values of the scalar response variable for the given set of new functional and scalar (if any) predictors Xnew and Xnew.scl, respectively. Here, n_{test}, the number of rows of the matrix of predicted values, equals to the number of rows of Xnew and and Xnew.scl (if any).

Author(s)

Ufuk Beyaztas and Han Lin Shang

Examples

set.seed(2022)
sim.data <- generate.sf.data(n = 400, n.pred = 5, n.gp = 101, out.p = 0.1)
out.indx <- sim.data$out.indx
indx.test <- sample(c(1:400)[-out.indx], 120)
indx.train <- c(1:400)[-indx.test]
Y <- sim.data$Y
X <- sim.data$X
Y.train <- Y[indx.train,]
Y.test <- Y[indx.test,]
X.train <- X.test <- list()
for(i in 1:5){
  X.train[[i]] <- X[[i]][indx.train,]
  X.test[[i]] <- X[[i]][indx.test,]
}
gp <- rep(list(seq(0, 1, length.out = 101)), 5) # grid points of Xs

model.tau <- rob.sf.reg(Y.train, X.train, emodel = "robust", fmodel = "tau", gp = gp)
pred.tau <- predict_sf_regression(object = model.tau, Xnew = X.test)
round(mean((Y.test - pred.tau)^2), 4)        # 1.868 (tau method)

Robust function-on-function regression

Description

This function is used to perform both classical and robust function-on-function regression model

Y(t) = \sum_{m=1}^M \int X_m(s) \beta_m(s,t) ds + \epsilon(t),

where Y(t) denotes the functional response, X_m(s) denotes the m-th functional predictor, \beta_m(s,t) denotes the m-th bivariate regression coefficient function, and \epsilon(t) is the error function.

Usage

rob.ff.reg(Y, X, model = c("full", "selected"), emodel = c("classical", "robust"),
 fmodel = c("MCD", "MLTS", "MM", "S", "tau"), nbasisY = NULL, nbasisX = NULL,
 gpY = NULL, gpX = NULL, ncompY = NULL, ncompX = NULL)

Arguments

Details

When performing a function-on-function regression model based on the functional principal component analysis, first, both the functional response Y(t) and functional predictors X_m(s), 1\le m\le M are decomposed by the functional principal component analysis method:

Y(t) = \bar{Y}(t) + \sum_{k=1}^K \nu_k \phi_k(t),

X_m(s) = \bar{X}_m(s) + \sum_{l=1}^{K_m} \xi_{ml} \psi_{ml}(s),

where \bar{Y}(t) and \bar{X}_m(s) are the mean functions, \phi_k(t) and \psi_{ml}(s) are the weight functions, and \nu_k = \int (Y(t) - \bar{Y}(t)) \phi_k(t) and \xi_{ml} = \int (X_m(s) - \bar{X}_m(s)) \psi_{ml}(s) are the principal component scores for the functional response and m-th functional predictor, respectively. Assume that the m-th bivariate regression coefficient function admits the expansion

\beta_m(s,t) = \sum_{k=1}^K \sum_{l=1}^{K_m} b_{mkl} \phi_k(t) \psi_{ml}(s),

where b_{mkl} = \int \int \beta_m(s,t) \phi_k(t) \psi_{ml}(s) dt ds. Then, the following multiple regression model is obtained for the functional response:

\hat{Y}(t) = \bar{Y}(s) + \sum_{k=1}^K ( \sum_{m=1}^M \sum_{l=1}^{K_m} b_{mkl} \xi_{ml} ) \phi_k(t).

If model = "full", then, all the functional predictor variables are used in the model.

If model = "selected", then, only the significant functional predictor variables determined by the forward variable selection procedure of Beyaztas and Shang (2021) are used in the model.

If emodel = "classical", then, the least-squares method is used to estimate the function-on-function regression model.

If emodel = "robust", then, the robust functional principal component analysis of Bali et al. (2011) along with the method specified in fmodel is used to estimate the function-on-function regression model.

If fmodel = "MCD", then, the minimum covariance determinant estimator of Rousseeuw et al. (2004) is used to estimate the function-on-function regression model.

If fmodel = "MLTS", then, the multivariate least trimmed squares estimator Agullo et al. (2008) is used to estimate the function-on-function regression model.

If fmodel = "MM", then, the MM estimator of Kudraszow and Maronna (2011) is used to estimate the function-on-function regression model.

If fmodel = "S", then, the S estimator of Bilodeau and Duchesne (2000) is used to estimate the function-on-function regression model.

If fmodel = "tau", then, the tau estimator of Ben et al. (2006) is used to estimate the function-on-function regression model.

Value

A list object with the following components:

Author(s)

Ufuk Beyaztas and Han Lin Shang

References

J. Agullo and C. Croux and S. V. Aelst (2008), "The multivariate least-trimmed squares estimator", Journal of Multivariate Analysis, 99(3), 311-338.

M. G. Ben and E. Martinez and V. J. Yohai (2006), "Robust estimation for the multivariate linear model based on a \tau scale", Journal of Multivariate Analysis, 97(7), 1600-1622.

U. Beyaztas and H. L. Shang (2021), "A partial least squares approach for function-on-function interaction regression", Computational Statistics, 36(2), 911-939.

J. L. Bali and G. Boente and D. E. Tyler and J. -L.Wang (2011), "Robust functional principal components: A projection-pursuit approach", The Annals of Statistics, 39(6), 2852-2882.

M. Bilodeau and P. Duchesne (2000), "Robust estimation of the SUR model", The Canadian Journal of Statistics, 28(2), 277-288.

N. L. Kudraszow and R. A. Moronna (2011), "Estimates of MM type for the multivariate linear model", Journal of Multivariate Analysis, 102(9), 1280-1292.

P. J. Rousseeuw and K. V. Driessen and S. V. Aelst and J. Agullo (2004), "Robust multivariate regression", Technometrics, 46(3), 293-305.

Examples

sim.data <- generate.ff.data(n.pred = 5, n.curve = 200, n.gp = 101)
Y <- sim.data$Y
X <- sim.data$X
gpY <- seq(0, 1, length.out = 101) # grid points of Y
gpX <- rep(list(seq(0, 1, length.out = 101)), 5) # grid points of Xs
model.MM <- rob.ff.reg(Y = Y, X = X, model = "full", emodel = "robust",
                       fmodel = "MM", gpY = gpY, gpX = gpX)

Outlier detection in the functional response

Description

This function is used to detect outliers in the functional response based on a fitted function-on-function regression model in the output of rob.ff.reg.

Usage

rob.out.detect(object, alpha = 0.01, B = 200, fplot = FALSE)

Arguments

Details

The functional depth-based outlier detection method of Febrero-Bande et al. (2008) together with the h-modal depth proposed by Cuaves et al. (2007) is applied to the estimated residual functions obtained from rob.ff.reg to determine the outliers in the response variable. This method makes it possible to determine both magnitude and shape outliers in the response variable Hullait et al., (2021).

Value

A vector containing the indices of outlying observations in the functional response.

Author(s)

Ufuk Beyaztas and Han Lin Shang

References

M. Febrero-Bande and P. Galeano and W. Gonzalez-Mantelga (2008), "Outlier detection in functional data by depth measures, with application to identify abnormal NOx levels", Environmetrics, 19(4), 331-345.

A. Cuaves and M. Febrero and R Fraiman (2007), "Robust estimation and classification for functional data via projection-based depth notions", Computational Statistics, 22(3), 481-496.

H. Hullait and D. S. Leslie and N. G. Pavlidis and S. King (2021), "Robust function-on-function regression", Technometrics, 63(3), 396-409.

Examples

sim.data <- generate.ff.data(n.pred = 5, n.curve = 200, n.gp = 101, out.p = 0.1)
out.indx <- sim.data$out.indx
Y <- sim.data$Y
X <- sim.data$X
gpY = seq(0, 1, length.out = 101) # grid points of Y
gpX <- rep(list(seq(0, 1, length.out = 101)), 5) # grid points of Xs

model.MM <- rob.ff.reg(Y = Y, X = X, model = "full", emodel = "robust", fmodel = "MM", 
                       gpY = gpY, gpX = gpX)
rob.out.detect(object = model.MM, fplot = TRUE)
sort(out.indx)

Robust scalar-on-function regression

Description

This function is used to perform both classical and robust scalar-on-function regression model

Y = \sum_{m=1}^M \int X_m(s) \beta_m(s) ds + X.scl \gamma + \epsilon,

where Y denotes the scalar response, X_m(s) denotes the m-th functional predictor, \beta_m(s) denotes the m-th regression coefficient function, X.scl denotes the matrix of scalar predictors, \gamma denotes the vector of coefficients for the scalar predictors' matrix, and \epsilon is the error function, which is assumed to follow standard normal distribution.

Usage

rob.sf.reg(Y, X, X.scl = NULL, emodel = c("classical", "robust"),
fmodel = c("LTS", "MM", "S", "tau"), nbasis = NULL, gp = NULL, ncomp = NULL)

Arguments

Details

When performing a scalar-on-function regression model based on the functional principal component analysis, first, the functional predictors X_m(s), 1\le m\le M are decomposed by the functional principal component analysis method:

X_m(s) = \bar{X}_m(s) + \sum_{l=1}^{K_m} \xi_{ml} \psi_{ml}(s),

where \bar{X}_m(s) is the mean function, \psi_{ml}(s) is the weight function, and \xi_{ml} = \int (X_m(s) - \bar{X}_m(s)) \psi_{ml}(s) is the principal component score for the m-th functional predictor. Assume that the m-th regression coefficient function admits the expansion

\beta_m(s) = \sum_{l=1}^{K_m} b_{ml} \psi_{ml}(s),

where b_{ml} = \int \beta_m(s) \psi_{m}(s) ds. Then, the following multiple regression model is obtained for the scalar response:

\hat{Y} = \bar{Y} + \sum_{m=1}^M \sum_{l=1}^{K_m} b_{ml} \xi_{ml} + X.scl \gamma.

If emodel = "classical", then, the least-squares method is used to estimate the scalar-on-function regression model.

If emodel = "robust", then, the robust functional principal component analysis of Bali et al. (2011) along with the method specified in fmodel is used to estimate the scalar-on-function regression model.

If fmodel = "LTS", then, the least trimmed squares robust regression of Rousseeuw (1984) is used to estimate the scalar-on-function regression model.

If fmodel = "MM", then, the MM-type regression estimator described in Yohai (1987) and Koller and Stahel (2011) is used to estimate the scalar-on-function regression model.

If fmodel = "S", then, the S estimator is used to estimate the scalar-on-function regression model.

If fmodel = "tau", then, the tau estimator proposed by Salibian-Barrera et al. (2008) is used to estimate the scalar-on-function regression model.

Value

A list object with the following components:

Author(s)

Ufuk Beyaztas and Han Lin Shang

References

J. L. Bali and G. Boente and D. E. Tyler and J. -L.Wang (2011), "Robust functional principal components: A projection-pursuit approach", The Annals of Statistics, 39(6), 2852-2882.

P. J. Rousseeuw (1984), "Least median of squares regression", Journal of the American Statistical Association, 79(388), 871-881.

P. J. Rousseeuw and K. van Driessen (1999) "A fast algorithm for the minimum covariance determinant estimator", Technometrics, 41(3), 212-223.

V. J. Yohai (1987), "High breakdown-point and high efficiency estimates for regression", The Annals of Statistics, 15(2), 642-65.

M. Koller and W. A. Stahel (2011), "Sharpening Wald-type inference in robust regression for small samples", Computational Statistics & Data Analysis, 55(8), 2504-2515.

M. Salibian-Barrera and G. Willems and R. Zamar (2008), "The fast-tau estimator for regression", Journal of Computational and Graphical Statistics, 17(3), 659-682

Examples

sim.data <- generate.sf.data(n = 400, n.pred = 5, n.gp = 101)
Y <- sim.data$Y
X <- sim.data$X
gp <- rep(list(seq(0, 1, length.out = 101)), 5) # grid points of Xs
model.tau <- rob.sf.reg(Y, X, emodel = "robust", fmodel = "tau", gp = gp)