# Integrated Model Selection and Scalability in Functional Data Analysis Through Bayesian Learning
## 5 Results
### Results on Public Datasets
#### 5.2.3 Modeling Large-Scale, Dynamic, Geospatial Data
Initialization was conducted using the modified RVM for 200 iterations for initial basis functions, using stochastic optimization with a 1000 batch size per year. Then, BSFDAFast was executed for 10,000 iterations, where the heuristic to include new bases also used a 1000 batch size per year. With these computational strategies and heuristics, the entire modeling process was completed in 15. The proposed approach selected 163 effective basis functions and condensed them into 16 principle components. The final model occupied merely 50 MB of storage. The interpolation yielded a root mean square error (RMSE) of 1.95 and an \(R^{2}\) of 94.2% on the testing data, reflecting a reasonable balance between global dimension reduction and fidelity. The estimated white noise level was also 1.95, indicating that the training data adequately covered the underlying variability in the ARGO observations, and the final model was reasonably generalizable. Figure 9 presents 2D visualizations of geospatial interpolations at three depths (in decibars, roughly meters) and a specific time (29 May 2021) around 1\({}^{\circ}\) S and 30\({}^{\circ}\) W, each with three views. We have chosen one measurement as the central point, denoted by the red circle, and selected a narrow window (\(\pm\)1 decibar, \(\pm\)1 day) around this center. The cyan and fuchsia circles represent training and testing data, respectively, within this window. Their sizes indicate the distance along the unplotted dimensions (depth and time here), reflecting variations in these dimensions. The visualizations show that the temperatures are warmer near the equator and decrease with depth. The match between the interpolated values and actual measurements demonstrates consistency in capturing broad spatial and vertical variations. Figure 10 complements this by illustrating interpolation in the depth-time slices while holding the geospatial coordinates fixed, focusing on mixed layer characteristics. The \"mixed layer\" refers to a region of nearly uniform temperature, which is crucial in understanding thermodynamic potential and nutrient cycling [45]. Here, the plot uses a window of 50 km to include actual measurements, and the circle sizes denote the geodesic distance from the chosen center. We plot every fifth measurement vertically to reduce overlap and improve the clarity. Figure 10a uses the same center point, 1\({}^{\circ}\) S and 30\({}^{\circ}\) W, as in Figure 9, exhibiting a shallow mixed layer with pronounced vertical gradients. In contrast, Figure 10b adopts a center at a higher latitude, 49\({}^{\circ}\) N and 29\({}^{\circ}\) W, where the model reveals a deeper mixed layer. The temperature here remains relatively stable below the surface. The dominant variations are cyclic seasonal changes, which are warmer near the surface around September. As is shown, the vertical sequence of the center and the nearby testing sequence match the interpolation closely. These results confirm that the mid-latitudes exhibit a stronger seasonal cycle [45] and that BSFDAFast accurately approximates the actual measurements. To our knowledge, this is the first time that the ARGO dataset has been modeled in a full 4D principal component model, with the correct domain topology. We incorporate the entire period of 27 years, rather than shorter spans (e.g., 2004-2008 or 2007-2016) [44; 46; 47]. Instead of segmenting the dataset into localized spatiotemporal windows, we process the entire 4D domain (latitude, longitude, depth, and intra-annual time) in a single holistic framework. Previous studies were typically tailored to ARGO datasets and handled each depth, month, or spatial region separately, restricting the correlation estimates to limited windows (e.g., 1000 km and three months) while excluding data with large offsets [44; 47]. In addition, they required repeated on-demand model fitting, which can hinder scalability. Our kernel-based framework, by contrast, is broadly applicable to general functional data, only requiring kernel definitions for the domain.