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\begin{document}

\title[Supplementary Material]{Supplementary Material: A novel generalized additive scalar-on-function regression model for partially observed multidimensional functional data:  an application to air quality classification}

\author*[1]{\fnm{Pavel} \sur{Hernández-Amaro}}\email{pahernan@est-econ.uc3m.es}

\author[1]{\fnm{Maria} \sur{Durban}}\email{mdurban@est-econ.uc3m.es}

\author[2]{\fnm{M. Carmen} \sur{Aguilera-Morillo}}\email{mdagumor@eio.upv.es}

\affil*[1]{\orgdiv{Estadística y Econometría}, \orgname{Universidad Carlos III de Madrid}, \orgaddress{\street{Avda. Universidad, 31}, \city{Leganes}, \postcode{28911}, \state{Madrid}, \country{España}}}

\affil[2]{\orgdiv{Estadística e Investigación Operativa Aplicadas y Calidad}, \orgname{Universitat Politècnica de València}, \orgaddress{\street{Camino de Vera, s/n}, \city{Valencia}, \postcode{46022}, \state{Valencia}, \country{España}}}

\maketitle

\section*{Introduction}

This supplementary material provides comprehensive simulation results for the methods evaluated in the main manuscript. The document is organized as follows:

\begin{itemize}
    \item \textbf{Section 1:} Complete results for the one-dimensional (1D) simulation study, including violin plots for scenarios with Normal and Binomial responses across all levels of missingness and gap sizes.
    
    \item \textbf{Section 2:} Complete results for the two-dimensional (2D) simulation study, presenting violin plots for Normal and Binomial responses, including estimation accuracy, predictive performance, and computational efficiency metrics.
\end{itemize}

Throughout this document, \textbf{POFRM} denotes the proposed \emph{Partially Observed Functional Regression Model}, which directly estimates parameters from partially observed functional data. \textbf{POFRM-I} refers to the imputation-based approach, where missing segments are first reconstructed using the method explained in the main manuscript before model estimation.

Each violin plot displays the distribution of performance metrics across 100 simulation iterations, allowing for detailed assessment of both central tendency and variability. The figures are organized by the proportion of curves (or surfaces) containing gaps and by gap size, facilitating direct comparison between methods under varying levels of data incompleteness.


\section{One-Dimensional Simulation Study}

This section presents the complete set of violin plots for the 1D simulation study involving two functional covariates. Results are organized by response type (Normal and Binomial) and performance metric.

\subsection{Normal Response Results}

Figures \ref{fig:normal_beta1_page1}--\ref{fig:normal_nu_page2} display the distributions of the Integrated Mean Square Error (IMSE) for both functional coefficients ($\beta_1(t)$ and $\beta_2(t)$) and the Root Mean Square Error (RMSE) of the response variable across all scenarios with Normal responses.

\begin{figure}[H]
    \centering
    \includegraphics[width=0.85\textwidth]{Plots/normal_beta1_violin_page_1.png}
    \caption{IMSE for the functional coefficient $\beta_1(t)$ for the 1D simulation study with Normal response. The left column corresponds to scenarios with 20\% of curves containing gaps, and the right column to 40\%. Rows correspond (top to bottom) to gap sizes of 5\%, 10\%, and 15\%.}
    \label{fig:normal_beta1_page1}
\end{figure}

\begin{figure}[H]
    \centering
    \includegraphics[width=0.85\textwidth]{Plots/normal_beta1_violin_page_2.png}
    \caption{IMSE for the functional coefficient $\beta_1(t)$ for the 1D simulation study with Normal response. The left column corresponds to scenarios with 60\% of curves containing gaps, and the right column to 80\%. Rows correspond (top to bottom) to gap sizes of 5\%, 10\%, and 15\%.}
    \label{fig:normal_beta1_page2}
\end{figure}

\begin{figure}[H]
    \centering
    \includegraphics[width=0.85\textwidth]{Plots/normal_beta2_violin_page_1.png}
    \caption{IMSE for the functional coefficient $\beta_2(t)$ for the 1D simulation study with Normal response. The left column corresponds to scenarios with 20\% of curves containing gaps, and the right column to 40\%. Rows correspond (top to bottom) to gap sizes of 5\%, 10\%, and 15\%.}
    \label{fig:normal_beta2_page1}
\end{figure}

\begin{figure}[H]
    \centering
    \includegraphics[width=0.85\textwidth]{Plots/normal_beta2_violin_page_2.png}
    \caption{IMSE for the functional coefficient $\beta_2(t)$ for the 1D simulation study with Normal response. The left column corresponds to scenarios with 60\% of curves containing gaps, and the right column to 80\%. Rows correspond (top to bottom) to gap sizes of 5\%, 10\%, and 15\%.}
    \label{fig:normal_beta2_page2}
\end{figure}

\begin{figure}[H]
    \centering
    \includegraphics[width=0.85\textwidth]{Plots/normal_nu_violin_page_1.png}
    \caption{RMSE of the response variable for the 1D simulation study with Normal response. The left column corresponds to scenarios with 20\% of curves containing gaps, and the right column to 40\%. Rows correspond (top to bottom) to gap sizes of 5\%, 10\%, and 15\%.}
    \label{fig:normal_nu_page1}
\end{figure}

\begin{figure}[H]
    \centering
    \includegraphics[width=0.85\textwidth]{Plots/normal_nu_violin_page_2.png}
    \caption{RMSE of the response variable for the 1D simulation study with Normal response. The left column corresponds to scenarios with 60\% of curves containing gaps, and the right column to 80\%. Rows correspond (top to bottom) to gap sizes of 5\%, 10\%, and 15\%.}
    \label{fig:normal_nu_page2}
\end{figure}

\clearpage

\subsection{Binomial Response Results}

Figures \ref{fig:binomial_beta1_page1}--\ref{fig:binomial_auc_page2} present the distributions for the Binomial response scenarios, including IMSE for both functional coefficients, misclassification error, and Area Under the Curve (AUC).

\begin{figure}[H]
    \centering
    \includegraphics[width=0.85\textwidth]{Plots/binomial_beta1_violin_page_1.png}
    \caption{IMSE for the functional coefficient $\beta_1(t)$ for the 1D simulation study with Binomial response. The left column corresponds to scenarios with 20\% of curves containing gaps, and the right column to 40\%. Rows correspond (top to bottom) to gap sizes of 5\%, 10\%, and 15\%.}
    \label{fig:binomial_beta1_page1}
\end{figure}

\begin{figure}[H]
    \centering
    \includegraphics[width=0.85\textwidth]{Plots/binomial_beta1_violin_page_2.png}
    \caption{IMSE for the functional coefficient $\beta_1(t)$ for the 1D simulation study with Binomial response. The left column corresponds to scenarios with 60\% of curves containing gaps, and the right column to 80\%. Rows correspond (top to bottom) to gap sizes of 5\%, 10\%, and 15\%.}
    \label{fig:binomial_beta1_page2}
\end{figure}

\begin{figure}[H]
    \centering
    \includegraphics[width=0.85\textwidth]{Plots/binomial_beta2_violin_page_1.png}
    \caption{IMSE for the functional coefficient $\beta_2(t)$ for the 1D simulation study with Binomial response. The left column corresponds to scenarios with 20\% of curves containing gaps, and the right column to 40\%. Rows correspond (top to bottom) to gap sizes of 5\%, 10\%, and 15\%.}
    \label{fig:binomial_beta2_page1}
\end{figure}

\begin{figure}[H]
    \centering
    \includegraphics[width=0.85\textwidth]{Plots/binomial_beta2_violin_page_2.png}
    \caption{IMSE for the functional coefficient $\beta_2(t)$ for the 1D simulation study with Binomial response. The left column corresponds to scenarios with 60\% of curves containing gaps, and the right column to 80\%. Rows correspond (top to bottom) to gap sizes of 5\%, 10\%, and 15\%.}
    \label{fig:binomial_beta2_page2}
\end{figure}

\begin{figure}[H]
    \centering
    \includegraphics[width=0.85\textwidth]{Plots/binomial_error_violin_page_1.png}
    \caption{Misclassification error for the 1D simulation study with Binomial response. The left column corresponds to scenarios with 20\% of curves containing gaps, and the right column to 40\%. Rows correspond (top to bottom) to gap sizes of 5\%, 10\%, and 15\%.}
    \label{fig:binomial_error_page1}
\end{figure}

\begin{figure}[H]
    \centering
    \includegraphics[width=0.85\textwidth]{Plots/binomial_error_violin_page_2.png}
    \caption{Misclassification error for the 1D simulation study with Binomial response. The left column corresponds to scenarios with 60\% of curves containing gaps, and the right column to 80\%. Rows correspond (top to bottom) to gap sizes of 5\%, 10\%, and 15\%.}
    \label{fig:binomial_error_page2}
\end{figure}

\begin{figure}[H]
    \centering
    \includegraphics[width=0.85\textwidth]{Plots/binomial_AUC_violin_page_1.png}
    \caption{Area Under the Curve (AUC) for the 1D simulation study with Binomial response. The left column corresponds to scenarios with 20\% of curves containing gaps, and the right column to 40\%. Rows correspond (top to bottom) to gap sizes of 5\%, 10\%, and 15\%.}
    \label{fig:binomial_auc_page1}
\end{figure}

\begin{figure}[H]
    \centering
    \includegraphics[width=0.85\textwidth]{Plots/binomial_AUC_violin_page_2.png}
    \caption{Area Under the Curve (AUC) for the 1D simulation study with Binomial response. The left column corresponds to scenarios with 60\% of curves containing gaps, and the right column to 80\%. Rows correspond (top to bottom) to gap sizes of 5\%, 10\%, and 15\%.}
    \label{fig:binomial_auc_page2}
\end{figure}

\clearpage

\section{Two-Dimensional Simulation Study}

This section presents the complete set of violin plots for the 2D simulation study involving a single two-dimensional functional covariate. Results include estimation accuracy, predictive performance, and computational efficiency metrics for both Normal and Binomial responses.

\subsection{Normal Response Results}

Figures \ref{fig:gaussian_rmse_all}--\ref{fig:gaussian_timing_all} display the distributions of RMSE for the response variable, IMSE for the functional coefficient $\beta(t_1,t_2)$, and computation time across all scenarios with Normal responses.

\begin{figure}[H]
    \centering
    \includegraphics[width=0.85\textwidth]{Plots/gaussian_functional_effect_all_scenarios.png}
    \caption{RMSE of the response variable for the 2D simulation study with Normal response. Columns correspond (left to right) to gap sizes of 5\%, 10\%, and 15\%, while rows correspond (top to bottom) to proportions of surfaces with gaps of 20\%, 40\%, and 60\%.}
    \label{fig:gaussian_rmse_all}
\end{figure}

\begin{figure}[H]
    \centering
    \includegraphics[width=0.85\textwidth]{Plots/gaussian_beta_all_scenarios.png}
    \caption{IMSE for the functional coefficient $\beta(t_1,t_2)$ for the 2D simulation study with Normal response. Columns correspond (left to right) to gap sizes of 5\%, 10\%, and 15\%, while rows correspond (top to bottom) to proportions of surfaces with gaps of 20\%, 40\%, and 60\%.}
    \label{fig:gaussian_beta_all}
\end{figure}

\begin{figure}[H]
    \centering
    \includegraphics[width=0.85\textwidth]{Plots/gaussian_timing_all_scenarios.png}
    \caption{Computation time (in seconds) for the Normal-response scenarios in the 2D simulation study. Columns correspond (left to right) to gap sizes of 5\%, 10\%, and 15\%, while rows correspond (top to bottom) to proportions of surfaces with gaps of 20\%, 40\%, and 60\%.}
    \label{fig:gaussian_timing_all}
\end{figure}

\clearpage

\subsection{Binomial Response Results}

Figures \ref{fig:binomial_error_all}--\ref{fig:binomial_timing_all} present the distributions for the Binomial response scenarios in the 2D study, including misclassification error, AUC, and computation time.

\begin{figure}[H]
    \centering
    \includegraphics[width=0.85\textwidth]{Plots/binomial_misclassification_all_scenarios.png}
    \caption{Misclassification error for the 2D simulation study with Binomial response. Columns correspond (left to right) to gap sizes of 5\%, 10\%, and 15\%, while rows correspond (top to bottom) to proportions of surfaces with gaps of 20\%, 40\%, and 60\%.}
    \label{fig:binomial_error_all}
\end{figure}

\begin{figure}[H]
    \centering
    \includegraphics[width=0.85\textwidth]{Plots/binomial_AUC_all_scenarios.png}
    \caption{Area Under the Curve (AUC) for the 2D simulation study with Binomial response. Columns correspond (left to right) to gap sizes of 5\%, 10\%, and 15\%, while rows correspond (top to bottom) to proportions of surfaces with gaps of 20\%, 40\%, and 60\%.}
    \label{fig:binomial_auc_all}
\end{figure}

\begin{figure}[H]
    \centering
    \includegraphics[width=0.85\textwidth]{Plots/binomial_timing_all_scenarios.png}
    \caption{Computation time (in seconds) for the Binomial-response scenarios in the 2D simulation study. Columns correspond (left to right) to gap sizes of 5\%, 10\%, and 15\%, while rows correspond (top to bottom) to proportions of surfaces with gaps of 20\%, 40\%, and 60\%.}
    \label{fig:binomial_timing_all}
\end{figure}

\section*{Summary}

The comprehensive simulation results presented in this supplementary material demonstrate the consistent performance advantages of the proposed POFRM method across a wide range of scenarios. Key findings include:

\begin{itemize}
    \item \textbf{1D Study:} POFRM consistently outperforms POFRM-I in terms of estimation accuracy (lower IMSE) and predictive performance (lower RMSE and misclassification error), particularly as the proportion of incomplete curves and gap size increase.
    
    \item \textbf{2D Study:} Both methods achieve comparable predictive accuracy, but POFRM demonstrates substantially greater computational efficiency (approximately three times faster) while maintaining or improving estimation quality.
    
    \item \textbf{Robustness:} POFRM exhibits stable performance across all levels of missingness, whereas POFRM-I shows increasing variability and degraded performance as data incompleteness intensifies.
\end{itemize}

These results support the main conclusions of the manuscript and highlight the practical advantages of the proposed methodology for handling partially observed functional data in both one- and two-dimensional settings.

\end{document}