Robust and Efficient Noisy-Label Time-Series Classification via Dynamic Time Warping Based Granular Ball Computing

Time series classification involves learning a mapping from a continuous, temporally ordered sequence of real-valued observations to a discrete response variable, like class labels. This task is fundamental in domains, including health monitoring, where the temporal structure of data is critical for accurate prediction. Dynamic Time Warping (DTW) is a standard technique for measuring similarity between sequences varying in time or speed. However, DTW is restricted to discrete point matching. To move beyond pairwise alignment, we propose a theoretical framework that learns mappings between real-valued functions. These mappings approximate the flow associated with the characteristic curves of a linear transport equation with a space-dependent velocity field, providing a diffeomorphic transformation between two time series. Using the method of characteristics, we transform this partial differential equation into ordinary differential equations (ODEs) modeling system dynamics. The objective function used to learn these ODEs derives from the fundamental theorem of calculus. To enable flexible, expressive representations of the velocity field, we utilize reproducing kernel Hilbert spaces and optimal control methods. Our method, Diffeomorphic Time Warping (DiffTW), provides a theoretically grounded dissimilarity measure. Using a 1-nearest neighbor classifier, DiffTW outperforms DTW on 60 of 86 datasets.

Time series segmentation assigns a label to each part of a sequence. We formulate dense univariate segmentation as node classification on a graph whose nodes are the original time points. A local window provides node features without setting output granularity. We evaluate the approach on a TSSB-derived inductive benchmark built from disjoint UCR training and test instances. Under one fixed Graph Attention Network (GAT), visibility-based transformations achieve the highest mean ranks among eleven graph constructions. WDPVG, directed NVG, and weighted NVG form a statistically indistinguishable top group after Holm correction. On the 59-dataset Time Series Segmentation Benchmark, WDPVG+GAT reaches a weighted F1 of $0.916$, below seq2point at $0.951$ and statistically indistinguishable from same-feature MLP, random-forest, and 1-NN controls, because at this downsampled resolution each segment is short and the fixed $81$-sample window already spans most of it. At native resolution, where each segment is longer than that window, WDPVG+GAT is less sensitive to feature-window width and remains above the same-feature MLP at every tested window. The graph's advantage over these point-wise classifiers comes from context beyond the local window, which the visibility graph's long-range edges can supply, rather than from better features within it. In a separate resolution sweep, it is statistically tied with a window-searched seq2point while using about $70\times$ fewer parameters and $46\times$ less measured peak memory, although seq2point moves ahead after downsampling. This identifies a practical operating regime for finely sampled series when model size and repeated window tuning matter.

Existing granular-ball classification methods are often driven by handcrafted quality measures, neighborhood rules, or heuristic splitting and stopping criteria, which may reduce the transparency of local construction decisions and hinder explicit modeling of boundary-sensitive regions. To address this issue, this paper proposes a Minimum Description Length based Granular-Ball Classifier (MDL-GBC), a boundary-aware non-parametric and interpretable granular-ball classifier. MDL-GBC formulates class-conditional granular-ball construction as a local model selection problem under the Minimum Description Length principle. For each class, samples from the target class provide positive class evidence, while samples from the remaining classes provide negative boundary evidence. For each current granular ball, three candidate explanations are compared under a unified description-length criterion: a single-ball model, a two-ball model, and a core-boundary model. The selected model determines whether the ball is retained, geometrically split, or refined into core and boundary-sensitive child balls, thereby making local construction decisions consistent with the MDL-based classification mechanism. During prediction, a class-level mixture coding rule aggregates stable granular balls of the same class and assigns the test sample by comparing class-wise coding costs. Experiments on 18 benchmark datasets show that MDL-GBC achieves competitive classification performance against classical classifiers and representative granular-ball-based methods, obtaining the best average Accuracy, Macro-F1, and average rank. These results indicate that MDL-GBC provides an effective and interpretable alternative to conventional heuristic granular-ball classification strategies.