# 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  
Here, we demonstrate the scalability regarding both the size of the measurements and the dimensionality of our framework. For this, we apply it to the ARGO dataset, which consists of ocean temperature measurements from more than 4000 locations, at multiple depths and time points [37]. ARGO is a nearly global observing system for the ocean temperature, salinity, and other key variables via autonomous profiling floats. As of 2019, ARGO has generated over 338 gigabytes of data from 15,231 floats [37]. We focused on _high-quality_ (\"research\" mode option in the database API) data from 1998 to 2024 for depths between 0 and 200 m in the open-access snapshot of Argo GDAC of 9 November 2024 [43]. The number of measurement points per year varies widely--from 38,931 to over 11 million, with 127 million in total. Figure 8 illustrates a global map of the sea surface temperature measurements from February 2021, highlighting the dataset's extensive spatial coverage. Figure 6: Outcomes from the proposed method applied to MACS CD4 datasets. (**a**) Estimated curves for a random selection of nine sampled functions and the mean function. (**b**) Estimated eigenfunctions (eigenvalues). In our modeling, each year's data are treated as a single underlying function of four variables: latitude, longitude (on the spherical Earth), depth, and intra-annual time (modeled as a periodic variable). Note that the spatial data lie on a sphere and the time is a circle, assuming the periodicity of the time of the year. Our approach models these measurements holistically--without resorting to moving windows or submodeling--thereby preserving the continuous nature of the data and enabling the extraction of meaningful global, seasonal, and depth-dependent trends. Furthermore, the unique geospatial and temporal structure of the ARGO data, with spatial coordinates on a sphere and time exhibiting periodicity, necessitates specialized modeling techniques. Given that our model is 4D, the 4D kernel is defined as a product of the following kernels, following the design strategy for climatological data in [10]. The geospatial kernel on the sphere is a radial basis function (RBF) on geodesic distances. To ensure periodicity, the temporal kernel is an Exp-Sine-Squared \(k(x,x^{\prime})=\exp\left(-\frac{2\sin^{2}\left(\pi|x-x^{\prime}|\right)}{l_{ \mathrm{s}}}\right)\), where \(l_{\mathrm{s}}\) is the length scale. For depth, we use a Gaussian kernel. The numeric data (excluding metadata) as input to the model were approximately 4 GB. For length scale selection, we used Gaussian process regression on a small subset of  
Figure 8: Temperature measurements in February 2021 near the sea surface in the ARGO dataset. Figure 7: Outcomes from the proposed method applied to a wind speed dataset. (**a**) Estimated curves for a random selection of 9 sampled functions (denoted by different colors) and the mean function. (**b**) Estimated eigenfunctions (eigenvalues) denoted as EF. (**c**) Estimated covariance. 2000 randomly selected data in 2016 (medium size of measurements) for a cross-validated RMSE, which we optimized with a grid search. The specific length scales were set as follows: geodesic length scale of \(2\times 10^{3}\) km, depth length scale of 70 m, time length scale of 3, and periodicity of 1. For evaluation, we held out 10% of the depth profiles (a single round trip of a buoy from the surface to a depth at the same coordinate) from each year as testing data, following [44]. The total training set contained roughly 114 million points. Because the sample spacing was typically small relative to the selected length scales, we applied agglomerative clustering to 10,000 randomly chosen index points, reducing them to 2000 candidate basis functions. These candidate basis functions--precomputed for efficiency--took roughly 1.7 TB of memory. Computations were performed with 24 threads on a server equipped with 192 Intel(r) Xeon(r) Platinum 8360H CPUs @ 3.00 GHz and 3 TB RAM.