The modeling of samples of distributions is a major challenge since distributions do not form a vector space. While various approaches exist for univariate distributions, including transformations to a Hilbert space, far less is known about the multivariate case. We utilize a transformation approach to map multivariate distributions to a Hilbert space via a Wasserstein slicing method that is invertible. This approach combines functional data analysis tools, such as functional principal component analysis and modes of variation, with the facility to map back to interpretable distributions. We also provide convergence guarantees for the Hilbert space representations under a broad class of such transforms. The method is illustrated using joint systolic and diastolic blood pressure data.

Analysis of generalization properties of the proposed framework is given. An iterative Operator based Minimum Residual (OpMINRES) algorithm is proposed for solving the loss stabilization problem.

Classification of functional data where observations are curves or trajectories poses unique challenges, particularly under severe class imbalance. Traditional Random Forest algorithms, while robust for tabular data, often fail to capture the intrinsic structure of functional observations and struggle with minority class detection. This paper introduces Functional Random Forest with Adaptive Cost-Sensitive Splitting (FRF-ACS), a novel ensemble framework designed for imbalanced functional data classification. The proposed method leverages basis expansions and Functional Principal Component Analysis (FPCA) to represent curves efficiently, enabling trees to operate on low dimensional functional features. To address imbalance, we incorporate a dynamic cost sensitive splitting criterion that adjusts class weights locally at each node, combined with a hybrid sampling strategy integrating functional SMOTE and weighted bootstrapping. Additionally, curve specific similarity metrics replace traditional Euclidean measures to preserve functional characteristics during leaf assignment. Extensive experiments on synthetic and real world datasets including biomedical signals and sensor trajectories demonstrate that FRF-ACS significantly improves minority class recall and overall predictive performance compared to existing functional classifiers and imbalance handling techniques. This work provides a scalable, interpretable solution for high dimensional functional data analysis in domains where minority class detection is critical.

Artificial Intelligence (AI) models deployed in production frequently face challenges in maintaining their performance in non-stationary environments. This issue is particularly noticeable in medical settings, where temporal dataset shifts often occur. These shifts arise when the distributions of training data differ from those of the data encountered during deployment over time. Further, new labeled data to continuously retrain AI is not typically available in a timely manner due to data access limitations. To address these challenges, we propose a proactive self-adaptive AI approach, or pro-adaptive, where we model the temporal trajectory of AI parameters, allowing us to short-term forecast parameter values. To this end, we use polynomial spline bases, within an extensible Functional Data Analysis framework. We validate our methodology with a logistic regression model addressing prior probability shift, covariate shift, and concept shift. This validation is conducted on both a controlled simulated dataset and a publicly available real-world COVID-19 dataset from Mexico, with various shifts occurring between 2020 and 2024. Our results indicate that this approach enhances the performance of AI against shifts compared to baseline stable models trained at different time distances from the present, without requiring updated training data. This work lays the foundation for pro-adaptive AI research against dynamic, non-stationary environments, being compatible with data protection, in resilient AI production environments for health.

This study presents a novel representation learning model tailored for dynamic networks, which describes the continuously evolving relationships among individuals within a population. The problem is encapsulated in the dimension reduction topic of functional data analysis. With dynamic networks represented as matrix-valued functions, our objective is to map this functional data into a set of vector-valued functions in a lower-dimensional learning space. This space, defined as a metric functional space, allows for the calculation of norms and inner products. By constructing this learning space, we address (i) attribute learning, (ii) community detection, and (iii) link prediction and recovery of individual nodes in the dynamic network. Our model also accommodates asymmetric low-dimensional representations, enabling the separate study of nodes' regulatory and receiving roles. Crucially, the learning method accounts for the time-dependency of networks, ensuring that representations are continuous over time. The functional learning space we define naturally spans the time frame of the dynamic networks, facilitating both the inference of network links at specific time points and the reconstruction of the entire network structure without direct observation. We validated our approach through simulation studies and real-world applications. In simulations, we compared our methods link prediction performance to existing approaches under various data corruption scenarios. For real-world applications, we examined a dynamic social network replicated across six ant populations, demonstrating that our low-dimensional learning space effectively captures interactions, roles of individual ants, and the social evolution of the network. Our findings align with existing knowledge of ant colony behavior.

This paper introduces a novel supervised classification strategy that integrates functional data analysis (FDA) with tree-based methods, addressing the challenges of high-dimensional data and enhancing the classification performance of existing functional classifiers. Specifically, we propose augmented versions of functional classification trees and functional random forests, incorporating a new tool for assessing the importance of functional principal components. This tool provides an ad-hoc method for determining unbiased permutation feature importance in functional data, particularly when dealing with correlated features derived from successive derivatives. Our study demonstrates that these additional features can significantly enhance the predictive power of functional classifiers. Experimental evaluations on both real-world and simulated datasets showcase the effectiveness of the proposed methodology, yielding promising results compared to existing methods.

We introduce FDApy, an open-source Python package for the analysis of functional data. The package provides tools for the representation of (multivariate) functional data defined on different dimensional domains and for functional data that is irregularly sampled. Additionally, dimension reduction techniques are implemented for multivariate and/or multidimensional functional data that are regularly or irregularly sampled. A toolbox for generating functional datasets is also provided. The documentation includes installation and usage instructions, examples on simulated and real datasets and a complete description of the API. FDApy is released under the MIT license. The code and documentation are available at https://github.com/StevenGolovkine/FDApy.

This paper introduces a Random Survival Forest (RSF) method for functional data. The focus is specifically on defining a new functional data structure, the Censored Functional Data (CFD), for dealing with temporal observations that are censored due to study limitations or incomplete data collection. This approach allows for precise modelling of functional survival trajectories, leading to improved interpretation and prediction of survival dynamics across different groups. A medical survival study on the benchmark SOFA data set is presented. Results show good performance of the proposed approach, particularly in ranking the importance of predicting variables, as captured through dynamic changes in SOFA scores and patient mortality rates.

Most of the recent results in polynomial functional regression have been focused on an in-depth exploration of single-parameter regularization schemes. In contrast, in this study we go beyond that framework by introducing an algorithm for multiple parameter regularization and presenting a theoretically grounded method for dealing with the associated parameters. This method facilitates the aggregation of models with varying regularization parameters. The efficacy of the proposed approach is assessed through evaluations on both synthetic and some real-world medical data, revealing promising results.

Functional Data Analysis (FDA) is a statistical domain developed to handle functional data characterized by high dimensionality and complex data structures. Sequential Neural Networks (SNNs) are specialized neural networks capable of processing sequence data, a fundamental aspect of functional data. Despite their great flexibility in modeling functional data, SNNs have been inadequately employed in the FDA community. One notable advantage of SNNs is the ease of implementation, making them accessible to a broad audience beyond academia. Conversely, FDA-based methodologies present challenges, particularly for practitioners outside the field, due to their intricate complexity. In light of this, we propose utilizing SNNs in FDA applications and demonstrate their effectiveness through comparative analyses against popular FDA regression models based on numerical experiments and real-world data analysis. SNN architectures allow us to surpass the limitations of traditional FDA methods, offering scalability, flexibility, and improved analytical performance. Our findings highlight the potential of SNN-based methodologies as powerful tools for data applications involving functional data.