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Motion Code: Robust Time Series Classification and Forecasting via Sparse Variational Multi-Stochastic Processes Learning

Chandrajit Bajaj, Minh Nguyen

TL;DR

A novel framework that views each time series, regardless of length, as a realization of a continuous-time stochastic process, enabling simultaneous classification and forecasting of time series and demonstrates strong performance against established benchmarks for time series classification and forecasting.

Abstract

Despite extensive research, time series classification and forecasting on noisy data remain highly challenging. The main difficulties lie in finding suitable mathematical concepts to describe time series and effectively separate noise from the true signals. Unlike traditional methods treating time series as static vectors or fixed sequences, we propose a novel framework that views each time series, regardless of length, as a realization of a continuous-time stochastic process. This mathematical approach captures dependencies across timestamps and detects hidden, time-varying signals within the noise. However, real-world data often involves multiple distinct dynamics, making it insufficient to model the entire process with a single stochastic model. To address this, we assign each dynamic a unique signature vector and introduce the concept of "most informative timestamps" to infer a sparse approximation of the individual dynamics from these vectors. The resulting model, called Motion Code, includes parameters that fully capture diverse underlying dynamics in an integrated manner, enabling simultaneous classification and forecasting of time series. Extensive experiments on noisy datasets, including real-world Parkinson's disease sensor tracking, demonstrate Motion Code's strong performance against established benchmarks for time series classification and forecasting.

Motion Code: Robust Time Series Classification and Forecasting via Sparse Variational Multi-Stochastic Processes Learning

TL;DR

A novel framework that views each time series, regardless of length, as a realization of a continuous-time stochastic process, enabling simultaneous classification and forecasting of time series and demonstrates strong performance against established benchmarks for time series classification and forecasting.

Abstract

Despite extensive research, time series classification and forecasting on noisy data remain highly challenging. The main difficulties lie in finding suitable mathematical concepts to describe time series and effectively separate noise from the true signals. Unlike traditional methods treating time series as static vectors or fixed sequences, we propose a novel framework that views each time series, regardless of length, as a realization of a continuous-time stochastic process. This mathematical approach captures dependencies across timestamps and detects hidden, time-varying signals within the noise. However, real-world data often involves multiple distinct dynamics, making it insufficient to model the entire process with a single stochastic model. To address this, we assign each dynamic a unique signature vector and introduce the concept of "most informative timestamps" to infer a sparse approximation of the individual dynamics from these vectors. The resulting model, called Motion Code, includes parameters that fully capture diverse underlying dynamics in an integrated manner, enabling simultaneous classification and forecasting of time series. Extensive experiments on noisy datasets, including real-world Parkinson's disease sensor tracking, demonstrate Motion Code's strong performance against established benchmarks for time series classification and forecasting.
Paper Structure (27 sections, 1 theorem, 21 equations, 9 figures, 6 tables, 1 algorithm)

This paper contains 27 sections, 1 theorem, 21 equations, 9 figures, 6 tables, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. Related Works
  3. MOTION CODE: JOINT LEARNING ON COLLECTIONS OF TIME SERIES
  4. Stochastic Process Formulation and Data Assumption
  5. The Most Informative Timestamps
  6. Approximate Formula for $\mathcal{L}^{max}$
  7. Motion Code Learning
  8. Classification and Forecasting with Motion Code
  9. EXPERIMENTS
  10. Datasets
  11. Experimental Setups
  12. Evaluation on Time Series Classification
  13. Evaluation on Time Series Forecasting
  14. MOTION CODE'S BENEFITS
  15. Interpretable Features
  16. ...and 12 more sections

Key Result

Lemma 1

Let $G = \left\{ g(t) \right\}_{t \geq 0}$ be a stochastic process with the underlying signal $g$. Assume the data collection $\mathcal{C}$ for the process $G$ consists of $B$ noisy time series $\left\{ y^i \right\}_{i=1}^B$ with data points $(y^i)_{T_i}$, where $(y^i)_{T_i} \sim \mathcal{N}(g_{T_i} Then $\mathcal{L}^{max}$ defined in Section 2.2 has the approximate form: where $p_\mathcal{N}(X|\

Figures (9)

  • Figure 1: (a): Two Collections Of Time Series Representing Pronunciation Audio Data For The Words Absorptivity And Anything. (b) And (c): Most Informative Timestamps For The Pronunciation Of Absorptivity (Red) And Anything (Blue).
  • Figure 2: Forecasting With Uncertainty For Time Series In Chinatown (Pedestrian Count On Weekends Vs Weekdays) And MoteStrain (Humidity Vs Temperature Sensor Values). Motion Code Is Trained On $[0, 0.8]$ And Predicted On $[0.8, 1]$.
  • Figure 3: (a) And (b): Most Informative Timestamps For The Pronunciation Of Absorptivity And Anything, Highlighting Key Linguistic Components.
  • Figure 4: Interpretable Features Showing Tremor Patterns And Disease Stages For Parkinson Data: (a) Normal, (b) Light Tremor, And (c) Noticeable Tremor.
  • Figure 5: Forecasting with Uncertainty and Interpretable Features for ItalyPowerDemand.
  • ...and 4 more figures

Theorems & Definitions (5)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 1
  • proof