# Integrated Model Selection and Scalability in Functional Data Analysis Through Bayesian Learning
## Appendix F Experiments  
### Benchmark Simulation  
Figure 14 presents the application of the proposed BSFDA to the simulation benchmark (Scenario 1) outlined in [21]. Even though prior analyses have utilized this benchmark, the current experimental configuration is specifically adapted to highlight the method's capacity for uncertainty quantification. The experimental design consists of 20 functional observations, each sampled at either three points (with a 20% probability) or 10 points (with an 80% probability), determined via random assignment. The number of sampled functions is decreased from 200 to 20 to underscore the effect and estimation of uncertainties. The actual white noise standard deviation is 0.4472, whereas the estimated standard deviation is 0.4839. The component number is also correctly estimated as 3. The figure depicts the true underlying function, the discrete observational data, and the corresponding functional estimates, accompanied by their respective 95% truncated uncertainty intervals. Notably, the uncertainty associated with sparsely sampled functions exhibits substantial inflation in regions devoid of observations. In contrast, in sampled regions, the uncertainty aligns closely with that of densely sampled functions, approximating twice the standard deviation of the white noise. Additionally, the uncertainty bounds for the estimated mean function are presented, demonstrating reduced variability relative to individual function estimates. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \(N_{i}\) & \(\hat{p}\) & **AIC\({}_{\text{PACE}}\)** & **AIC** & **BIC** & **PC\({}_{\text{PI}}\)** & **IC\({}_{\text{PI}}\)** & **AIC\({}_{\text{PACE}}^{\text{2022}}\)** & **BIC\({}_{\text{PACE}}^{\text{2022}}\)** & **fpca** & **BSFDA** & **BSFDA\({}^{\text{Fast}}\)** \\ \hline
5 & \(\leq\)1 & 0.000 & 0.000 & 0.315 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 \\  & =2 & 0.015 & 0.020 & 0.180 & 0.160 & 0.015 & 0.000 & 0.000 & 0.000 & 0.000 \\  & =3 & 0.015 & 0.710 & 0.410 & 0.560 & 0.515 & 0.575 & 0.370 & **1.000** & 0.975 \\  & =4 & 0.145 & 0.185 & 0.070 & 0.095 & 0.260 & 0.450 & 0.390 & 0.515 & 0.000 & 0.025 \\  & \(\geq\)5 & 0.825 & 0.085 & 0.025 & 0.105 & 0.165 & 0.035 & 0.035 & 0.115 & 0.000 & 0.000 \\ \hline
10 & \(\leq\)1 & 0.000 & 0.000 & 0.010 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 \\  & =2 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 \\  & =3 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 \\  & \(\geq\)5 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.015 & 0.000 \\  & =3 & 0.000 & 0.000 & 0.020 & 0.035 & 0.015 & 0.000 & 0.000 & 0.027 & 0.000 & 0.000 \\ \hline
50 & \(\leq\)1 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 \\  & =2 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 \\  & =3 & 0.000 & 0.030 & 0.075 & 0.920 & 0.900 & 0.750 & 0.760 & 0.350 & **0.995** & 0.990 \\  & =4 & 0.000 & 0.150 & 0.190 & 0.045 & 0.085 & 0.250 & 0.240 & 0.380 & 0.005 & 0.010 \\  & \(\geq\)5 & 1.000 & 0.020 & 0.020 & 0.035 & 0.015 & 0.000 & 0.000 & 0.270 & 0.000 & 0.000 \\ \hline
50 & \(\leq\)1 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 \\  & =2 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 \\  & =4 & 0.000 & 0.000 & 0.025 & 0.000 & 0.000 & 0.000 & 0.000 & 0.110 & 0.000 & 0.010 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Distributions of the estimated component number \(\hat{r}\) for Scenario 4 (r = 3). Appendix F.1.1. Performance of LFRM  
To compare the latent factor regression model (LFRM) [18] as a dimension reduction model to ours, i.e., Bayesian scalable functional data analysis (BSFDA), we set the covariates in LFRM to zero, thus assigning standard Gaussian priors to the latent variables, analogously to our approach.