Supplementary material for Functional Response Model with Functional Response (FRMFR)

Supplementary codes and data used in the paper Darbalaei et al. (2022).

Please cite this paper as:

@article{darbalaei2022functional,
  title={Functional Regression Models with Functional Response: New Approaches and a Comparative Study},
  author={Darbalaei, Mohammad and Amini, Morteza and Febrero-Bande, Manuel and Oviedo de-la Fuente, Manuel},
  journal={arXiv preprint arXiv:2207.04773},
  year={2022}
}

Installation

In order to use paper implementation and run all files (numerical and real examples), the following prerequisites are needed:

1. fda.usc package

To install fda.usc.devel package (devel version of fda.usc) from Github with (2023/03/29):

# install.packages("devtools")
require(devtools)
devtools::install_github("moviedo5/FRMFR/pkg/fda.usc.devel")

To compares our proposed methods (namely, FLMFR, FSAMFR, and FKAMFR, which are available in the fda.usc package (devel version) through the commands fregre.mlm.fr, fregre.sam.fr and fregre.kam.fr) with the four mentioned competitor methods (namely, PFR, FAMM, LSC, and DISC).

2. refund package

PFR and FAMM methods are available in the refund package through the command pffr, where the argument formula allows us to include linear ffpc, ff or nonlinear term sff.

To install refund package from CRAN or Github.

# install.packages("refund")
devtools::install_version("refund", version = "0.1-30", 
                          repos = "http://cran.us.r-project.org")
# latest patched version directly from Github
# devtools::install_github("refunders/refund") 

3. FRegSigCom package

The authors considered the latter as an experimental feature. LSC and DISC methods are available in the FRegSigCom package through the commands cv.sigcom and cv.nonlinear.

This package is not currently maintained and its latest version was published in November 2018 but, anyway, it can be downloaded and installed from the Packages/Archive section of CRAN.

devtools::install_github("moviedo5/FRMFR/pkg/FRegSigCom")
# url <- "https://cran.r-project.org/src/contrib/Archive/FRegSigCom/FRegSigCom_0.3.0.tar.gz"
# install.packages(url, repos=NULL, type="source")

Simulation (Numerical Studies)

• ./inst/script/Simulation.R: Code for main simulation. Scenarios 1–4.

  • Linear smooth (LS)
  • Linear non-smooth (LNS)
  • Nonlinear smooth (NLS)
  • Nonlinear non-smooth (NLNS)

library(fda.usc.devel)
# source("./inst/script/Simulation.R")

Real Data Applications

Consult a detailed documentation of the data examples and R code of used.

1. Air Quality Data

Our last example is the Air Quality dataset (AQI) available from the UCI machine learning repository Qi and Luo (2019). AQI is a popular dataset consisting of five metal oxide chemical sensors embedded into an air quality multisensor device.

The sensors are

labeled with:

• Carbon monoxide (CO),
• Non-methane hydrocarbons (NMHC),
• Total Nitrogen Oxides (NOx),
• Ozone (O3) because it is supposed that its measures are related with the respective pollutants.

data("AirQuality")

The corresponding plots are displayed in Figure 16 of paper.

The goal of this study is to predict the content of (log transformation of) Benzene (C6H6) obtained through an independent analyzer considered the Ground Truth. These sensors were collected as 24 hourly averaged concentration values each day jointly with the relative humidity (rH) as an external factor.

• /RealDataApplications/AirQuality.R: Code for AirQuality data example (high computational time)

source("/RealDataApplications/AirQuality.R")

2. Bike-sharing Data

To illustrate how our proposed function-on-function methods work, we use the Bike-sharing data (Fanaee-T and Gama 2014) as our first example. This dataset is collected by Capital Bikeshare System (CBS), Washington D.C., USA. The logaritmo of number of casual bike rentals (NCBR) is considered as our functional response variable (log(NCBR+1)) and four functional predictors:

• Temperature (T),
• Humidity (H),
• Wind Speed (WS) and
• Feeling Temperature (FT)

data("BikeSharing") included in fda.usc.devel packages:

data("BikeSharing")

The corresponding plots are displayed in Figure 5 of paper.

## [1] "df"        "logNBCR"   "temp"      "feeltemp"  "humidity"  "windspeed"

These variables are recorded each hour from January 1, 2011, to December 31, 2012. Similar to Kim et al. (2018), we only consider the data for Saturday trajectories, and NBCR is log–transformed to avoid its natural heteroskedasticity. Ignoring three curves with missing values, the dataset contains 102 trajectories, each with 24 data points (hourly) for all variables.

• /RealDataApplications/BikeSharing.R: Code for example Air Quality

source("/RealDataApplications/BikeSharing.R")

3. Electricity Demand and Price Data

Daily profiles of Electricity Price and Demand, both measured hourly, are obtained from two biannual periods separated by ten years: 2008-2009 and 2018-2019 (source:omie.es).

• /RealDataApplications/omie2008vs2018.R: Code for Electricity Demand and Price example.

data(omel2008_09)
names(omel2008_09)

## [1] "df" "Pr" "En"

The corresponding plots are displayed in Figure 6 of paper.

Profiles for Electricity Demand (first row) and Electricity Price (second row) for the periods 2008-09 (first column) and 2018-19 (second column). The black line corresponds to the functional mean of each dataset.

• /RealDataApplications/omie2008vs2018.R: Code for Electricity Demand and Price example.

source("/RealDataApplications/omie2008vs2018.R")

References

Darbalaei, Mohammad, Morteza Amini, Manuel Febrero-Bande, and Manuel Oviedo de-la Fuente. 2022. “Functional Regression Models with Functional Response: New Approaches and a Comparative Study.” arXiv Preprint arXiv:2207.04773. https://doi.org/10.48550/arXiv.2207.04773.

Fanaee-T, Hadi, and Joao Gama. 2014. “Event Labeling Combining Ensemble Detectors and Background Knowledge.” Progress in Artificial Intelligence 2 (2): 113–27.

Kim, Janet S, Ana-Maria Staicu, Arnab Maity, Raymond J Carroll, and David Ruppert. 2018. “Additive Function-on-Function Regression.” Journal of Computational and Graphical Statistics 27 (1): 234–44.

Qi, Xin, and Ruiyan Luo. 2019. “NONLINEAR FUNCTION-ON-FUNCTION ADDITIVE MODEL WITH MULTIPLE PREDICTOR CURVES.” Statistica Sinica 29 (2): 719–39. https://www.jstor.org/stable/26705485.