# Integrated Model Selection and Scalability in Functional Data Analysis Through Bayesian Learning
## 6 Discussion and Conclusions  
This paper proposes BSFDA, a novel framework for functional data analysis with irregular sampling, integrating model selection and scalability in one unique, coherent, and effective algorithm. Our extensive empirical studies, including both simulations and real-world applications, show that BSFDA offers superior covariance estimation accuracy with remarkable efficiency. In terms of accuracy, our method excels in model selection, consistently achieving top-tier performance. The accuracy of the covariance operator estimation also rivals that of the best existing methodologies in the field. This shows that our approach can not only handle large and complex datasets but also ensure high accuracy and precision in the results that it produces. Our method's superiority compared to existing techniques is expected owing to the inherent iterative nature of data smoothing and covariance estimation in our approach. In terms of scalability, our method demonstrates linear growth in time complexity with the size of the dataset, and, impressively, the computations are executed in a small, \(K^{(a)}\)-dimensional subspace. This property ensures that, as the datasets grow larger and more complex, the performance of our model remains robust and efficient. Additionally, we introduce a faster variant, BSFDAFast, which performs similarly to BSFDA on medium and dense datasets with a significantly reduced computational cost. This leap in efficiency enabled the full 4D functional modeling, for the first time, of a large-scale oceanic temperature dataset across 27 years (ARGO) [37]. Although BSFDAFast can underestimate the signal strength under very sparse sampling, the vanilla BSFDA effectively complements and alleviates this issue. Although the proposed framework proves effective in various real-world scenarios, it relies on the specific scheduling of coefficient noise levels. This schedule transitions from a faster, more biased model to a slower, less biased one, balancing the convergence speed against the estimation accuracy. Whereas empirical tests validate its advantages, there remain exploratory directions for further enhancement. For instance, the proper incorporation of structured variational inference or injection of artificial, compensatory noise in the observations could enable fast inference with reduced bias at the same time. However, it is necessary to address the increased complexity in optimization and variance in the estimation. Additionally, variational inference based on mean field approximations may underestimate the posterior variance, which is acceptable in many tasks [38] but leaves an open question when the independence assumptions are severely violated. Moreover, variational inference prioritizes computational efficiency over strict theoretical optimal  
Figure 10: Depth–time interpolation from BSFDAFast vs. actual ARGO global oceanic measurements at two sites focusing on mixed layer behavior. Measurements are represented by circles (green for training and pink for testing data), with the filling color indicating the temperature. Circle sizes show distance in geodesic space from the central point (denoted as red circles). (**a**) Shallow mixed layers at \(1^{\circ}\) S and \(30^{\circ}\) W. (**b**) Deep mixed layers at \(49^{\circ}\) N and \(29^{\circ}\) W.  
ity [23; 38], and the inherent infinite-dimensionality of fPCA further complicates any formal asymptotic analysis, even though our empirical studies demonstrate decreasing errors as the sampling density increases. Looking ahead, it would be interesting to explore how extensions of regular PCA, such as simplified PCA and robust PCA [25], can be integrated within our proposed framework. Domains such as finance, in particular, include large-time-series datasets that often contain more outliers. These extensions will enhance the flexibility and robustness of our method, further improving its adaptability to various data conditions.