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Shape analysis for time series
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Proposes TS-LDDMM, an unsupervised shape-based representation of irregularly sampled multivariate time series that encodes each series as a diffeomorphic deformation of a reference graph, with a representation theorem restricting deformations to time-series graphs.
Problem
Representing time series for inter-individual variability analysis, where existing URL methods and shape representations do not respect the graph structure of time series or handle irregular sampling and variable lengths well.
Approach
Adapts LDDMM to time series graphs by choosing a velocity-field RKHS kernel that factorizes time and space so that deformations preserve the graph structure (time reparametrization composed with space transformation), and compares graphs with an oriented varifold loss; the momentum parameters of the geodesic serve as features.
Key results
- Under 0%, 30%, and 50% missing rates, TS-LDDMM achieves the best average macro F1 and rank among compared methods; at 70% missing rate Neural LNSDE is best.
- On synthetic data with well-specified kernel, the mean average relative error of parameter recovery is 0.10 (max 0.17) across settings.
- On mice ventilation, TS-LDDMM PCs correspond to interpretable respiratory cycle deformations (cycle duration, inspiration/expiration trade-off) with varifold relative error 0.06 vs 0.11 for LDDMM.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use TS-LDDMM representations instead of the compared methods | Classification on 15 shape-based datasets · Regular sampling, 30% dropped, and 50% dropped | macro F1-score and rank | Helped |
| Use Neural LNSDE instead of TS-LDDMM representations | Classification on 15 shape-based datasets · 70% dropped | macro F1-score and rank | Helped |
| Use a well-specified velocity-field kernel to estimate the synthetic deformation parameters (ablation) | TS-LDDMM · Synthetic time-series graph deformation | average relative error: 0.10, 0.03, 0.17 | Helped |
| Set σT,0=1 rather than σT,0=100 during estimation while generating with σT,0=100 (ablation) | TS-LDDMM · Synthetic time-series graph reconstruction | varifold loss: 4e-6 -> 4e-2 | Hurt |
| Increase additive Gaussian noise when learning the reference graph (ablation) | TS-LDDMM · Synthetic data, noise sensitivity experiment | reference graph shape preservation | Mixed |
Surprising findings
- Neural LNSDE becomes the best performer at 70% missing rate, outperforming TS-LDDMM, which the authors attribute to the varifold loss poorly approximating shapes under heavy sampling loss.
Limitations
- Requires tuning several kernel and optimization hyperparameters
- Kernel computations may not scale to very large datasets or high-dimensional time series such as video
- Designed for homogeneous datasets; performance on heterogeneous data relies on a future hierarchical clustering extension
Experimental setup
Datasets: UCR/UEA shape-based datasets (15: 7 univariate, 8 multivariate) mice respiratory cycle dataset (Ventilation recordings, 14 mice) synthetic sinusoid graphs
Benchmarks: UCR/UEA shape-based classification (15 datasets) synthetic identifiability experiment mice ventilation analysis
Metrics: macro f1-score rank average relative error relative varifold loss error
Baselines: LDDMM RNN LSTM GRU MTAN MIAM ODE-LSTM Neural SDE Neural LNSDE Shape-FPCA
Builds on: LDDMM Neural ODE Neural CDE Shape-FPCA oriented varifold distance
Closest papers by meaning
- Abstracted Shapes as Tokens - A Generalizable and Interpretable Model for Time-series Classification (NeurIPS 2024 · similarity 0.85)
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- Integrating Sequence and Image Modeling in Irregular Medical Time Series Through Self-Supervised Learning (AAAI 2025 · similarity 0.85)
- Learning Soft Sparse Shapes for Efficient Time-Series Classification (ICML 2025 · similarity 0.84)
- Diffusion Language-Shapelets for Semi-supervised Time-Series Classification (AAAI 2024 · similarity 0.84)
- T-Rep: Representation Learning for Time Series using Time-Embeddings (ICLR 2024 · similarity 0.84)
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