Toward Interpretable Time Series Modeling: A Kernel Representation Perspective

1 Introduction

Time series analysis is crucial in fields like finance, healthcare, and meteorology. The complexities of time series data - such as nonlinearity, noise, and concept drift - make accurate modeling and prediction challenging [Chen et al., 2022a; Chen et al., 2022b; ?]. Recent deep learning advances, particularly Transformer-based architectures, have improved forecasting by capturing long-range dependencies. However, most models focus on prediction accuracy at the expense of interpretability. This lack of interpretability is a concern in many fields. In finance, for instance, shifts in market regimes and nonlinear relationships are as important as prediction accuracy. Detecting and understanding these shifts is vital for adapting to changing market conditions. Similarly, in healthcare, understanding a model's reasoning is crucial, as clinical decisions can significantly impact patient outcomes.

An additional challenge arises with co-evolving time series [Xu et al., 2024c], where multiple time series exhibit interdependent behavior over time. In such cases, understanding the interactions and dynamics among these series becomes crucial for both prediction and interpretation. A more holistic approach is required - one that not only forecasts future values but also explains the underlying behaviors and relationships that govern the evolution of these time series [Xu,

2025]. Accordingly, we explore the following research question:

RQ - Can we model co-evolving or large-scale time series as an ecosystem globe and track its evolution to balance predictive power with interpretability?

2 Our Contributions

Rethinking time series modeling from a kernel perspective, we introduce an innovative approach to capture complex, nonlinear relationships within data. Our goal is to construct a robust representation that not only enhances predictive accuracy but also offers deeper interpretability. In the subsections that follow, we present our key contributions:

2.1 KRL for Time Series Modeling

Kernel-induced Representation. We introduce CORAL [Xu et al., 2025a], a method designed to address the dynamic complexities of co-evolving time series. By leveraging kernel-induced self-representation, CORAL captures intricate interdependencies and adapts to concept drift through a representation matrix transformation of the data. This paradigm offers several advantages: (1) a novel approach for modeling complex interdependencies that integrates easily with deep learning frameworks, (2) adaptive detection and response to concept/distribution drift without prior assumptions, and (3) a transformation of time series concept/distribution drift into a matrix optimization problem that enhances interpretability.

Domain-agnostic Twin Learning Framework. We propose a unified twin learning framework [Xu et al., 2024a] for domain-agnostic time series analysis that simultaneously identifies behavioral patterns and optimal segmentation boundaries. Our KRL-based approach determines pattern durations and captures dynamic switches in non-linear time series data. Leveraging the learned kernel representation, we introduce a heuristic segmentation method based on a segmentscore function that quantifies the behavior-consistency of the time series data. Optimizing this score enables our method to adaptively identify segmentation boundaries that preserve and elucidate the core behaviors in the time series.

Adaptive Kernel Learning. We propose a data-driven kernel subspace learning method [Xu et al., 2022a; Xu et al., 2025b] that adaptively learns kernel mappings from selfrepresentation, eliminating the need for predefined kernel functions. The approach preserves the intrinsic manifold structure of the time series while promoting a block-diagonal structure in the representation matrix. Under certain conditions, the representation behaves as adaptive-weighted kernel learning and satisfies the triangular inequality, providing a theoretical lower bound and enhancing learning efficiency.

2.2 Theoretical Connection

We explore how kernel representation learning (KRL) offers insights into modern deep learning models, improving both interpretability and theoretical understanding.

Transformer-based models. The self-attention mechanism in Transformers can be viewed as a kernel operation, where the Query-Key dot product acts as a kernel mapping inputs into a high-dimensional space. This perspective clarifies attention dynamics and links Transformers to kernel methods.

Kolmogorov-Arnold Networks. We further show that KAN's learnable activations and nonlinear mappings align with kernel-induced self-representation under certain settings [Xu et al., 2024e]. Building on this, our WormKAN model [Xu et al., 2024d] leverages KAN to detect concept drift in time series, drawing inspiration from physics (e.g., wormholes) to model dynamic transitions. This bridges kernel theory and interpretable deep learning.

2.3 Real-world Applications

Financial Market. We tackle the task of discovering and forecasting regime switches in financial markets, which reflect changing market behaviors over time. To this end, we propose RHINE-a kernel-based model that captures nonlinear interactions and dynamic shifts [Xu et al., 2024b]. It also employs eigengap thresholding to automatically infer the number of regimes, enhancing adaptability.

DNA/RNA Sequences. We propose a scalable multi-view kernel learning framework for categorical sequences (e.g., DNA/RNA) [Xu et al., 2022b]. Each view is encoded as a kernel matrix, and instance weights are jointly learned with data partitioning. The model adaptively constructs kernels without relying on predefined functions and employs a kernel approximation to reduce the O(N 2 ) space complexity.

3 Ongoing and FutureWorks

We will continue refining the theoretical connections between KRL and deep models. We are also developing a kernelbased diffusion model to generate a public synthetic time series dataset-preserving privacy while adapting to distribution shifts. To support the community, we will launch an open-access website hosting this dataset to facilitate research and collaboration in time series representation learning.