# Integrated Model Selection and Scalability in Functional Data Analysis Through Bayesian Learning
## 5 Results
### Simulation Results
#### 5.1.2 Multidimensional Functional Data Simulation  
A simulation experiment with a 4D index set reveals the proposed method's advantages for high-dimensional data, where the gridding strategies of previous methods are impractical. The settings are as follows, with a length scale \(l_{\text{s}}=0.33\):  
\[Z_{i} \sim\mathcal{N}(0,I)\in R^{1\times 3} \tag{40}\] \[\phi_{0}(x) =(\pi l_{\text{s}}^{2})^{-2}\exp\left(-\frac{1}{2}\bigg{\|}\frac {x-[0.5,0.5,0.5,0.5]}{l_{\text{s}}}\bigg{\|}_{2}^{2}\right)\] (41) \[\phi_{1}(x) =(\pi l_{\text{s}}^{2})^{-2}\exp\left(-\frac{1}{2}\bigg{\|}\frac {x-[0.4,0.4,0.4,0.4]}{l_{\text{s}}}\bigg{\|}_{2}^{2}\right)\] (42) \[\phi_{2}(x) =(\pi l_{\text{s}}^{2})^{-2}\exp\left(-\frac{1}{2}\bigg{\|}\frac {x-[0.6,0.6,0.6,0.6]}{l_{\text{s}}}\bigg{\|}_{2}^{2}\right)\] (43) \[y_{i}(x) =Z_{i0}*\sqrt{0.6}*(\phi_{0}(x)-\phi_{1}(x))+Z_{i1}*\sqrt{0.3}* \phi_{1}(x)+Z_{i2}*\sqrt{0.4}*\phi_{2}(x) \tag{44}\] \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \(N_{i}\) & \(\mathbf{AIC_{PACE}}\) & \(\mathbf{AIC}\) & \(\mathbf{BIC}\) & \(\mathbf{PC_{\mathbf{p1}}}\) & \(\mathbf{IC_{\mathbf{p1}}}\) & \(\mathbf{AIC_{PACE}^{2022}}\) & \(\mathbf{BIC_{PACE}^{2022}}\) & \(\mathbf{fpca}\) & \(\mathbf{BSFDA}\) & \(\mathbf{BSFDA^{Fast}}\) \\ \hline
5 & 0.015 & 0.710 & 0.410 & 0.640 & 0.560 & 0.515 & 0.575 & 0.370 & \(\mathbf{1.000}\) & 0.975 \\ \hline
10 & 0.000 & 0.830 & 0.775 & 0.920 & 0.900 & 0.750 & 0.760 & 0.350 & \(\mathbf{0.995}\) & 0.990 \\ \hline
50 & 0.000 & 0.945 & 0.835 & \(\mathbf{1.000}\) & \(\mathbf{1.000}\) & \(\mathbf{1.000}\) & \(\mathbf{1.000}\) & 0.730 & 0.950 & 0.935 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Proportion of accurate estimations for Scenario 4 (r = 3). \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \(N_{i}\) & \(\mathbf{AIC_{PACE}}\) & \(\mathbf{AIC}\) & \(\mathbf{BIC}\) & \(\mathbf{PC_{\mathbf{p1}}}\) & \(\mathbf{IC_{\mathbf{p1}}}\) & \(\mathbf{AIC_{PACE}^{2022}}\) & \(\mathbf{BIC_{PACE}^{-2022}}\) & \(\mathbf{fpca}\) & \(\mathbf{BSFDA}\) & \(\mathbf{BSFDA^{Fast}}\) \\ \hline
5 & 0.705 & 0.470 & 0.090 & 0.070 & 0.545 & 0.425 & 0.410 & 0.855 & \(\mathbf{0.925}\) & 0.160 \\ \hline
10 & 0.065 & 0.570 & 0.525 & 0.775 & 0.705 & 0.575 & 0.575 & 0.500 & \(\mathbf{1.000}\) & 0.930 \\ \hline
50 & 0.000 The observations include additive noise with a sigma of 4.472 \(\times\) 10\({}^{-1}\). The cross-validation selects a length scale of 0.405. The estimated noise sigma is 4.637 \(\times\) 10\({}^{-1}\). The proposed method correctly estimates the number of principal components as three and selects 31 basis functions. As shown in Figure 5, the eigenfunctions are correctly estimated. In addition, the estimated mean function is zero, which is accurate. Next, we present a convergence comparison between BSFDA and BSFDAFast under four schedules for the coefficient noise \(\xi_{k}\). Specifically, we compare the default diminishing schedule from 10\({}^{-2}\) to 10\({}^{-5}\) with three fixed settings: 10\({}^{-2}\), 10\({}^{-3}\), and 10\({}^{-5}\). We evaluate the covariance error and the discrepancy between the estimated/true dimensionality in one replicate of each sample density in Scenario 5 and the 4D simulation. For the 4D case, we adopt a default initial \(\xi_{k}\) of 10\({}^{-3}\). As illustrated in Figure 4, BSFDAFast achieves comparable accuracy to BSFDA while converging significantly faster than BSFDA in terms of both covariance errors and component estimation for medium and densely sampled data. In the 4D case, BSFDAFast converges in covariance estimation after approximately 10,000 s and in dimensionality after around 4000 s, compared to roughly 100,000 s and 13,000 s, respectively, for BSFDA. However, for sparse data, BSFDAFast exhibits reduced estimation accuracy and underestimates the number of components by one. A similar decline in accuracy is observed in the 4D simulation when data sparsity is high.