# Toeplitz Inverse Covariance-Based Clustering of Multivariate Time Series Data

arXiv: https://arxiv.org/abs/1706.03161 · HTML: https://mlsanity.com/paper/1706.03161/

## Key facts

- Cited by 47 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Jun 2017.

## Built on by (most cited first, 30 of 47)

- [AdaRNN: Adaptive Learning and Forecasting of Time Series](https://mlsanity.com/paper/2108.04443/) (2021)
- [Learning Mixtures of Linear Dynamical Systems](https://mlsanity.com/paper/2201.11211/) (2022)
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- [Multivariate Time-Series Anomaly Detection based on Enhancing Graph Attention Networks with Topological Analysis](https://mlsanity.com/paper/2408.13082/) (2024)
- [Tensor Decompositions Meet Control Theory: Learning General Mixtures of Linear Dynamical Systems](https://mlsanity.com/paper/2307.06538/) (2023)
- [An Attention-based ConvLSTM Autoencoder with Dynamic Thresholding for Unsupervised Anomaly Detection in Multivariate Time Series](https://mlsanity.com/paper/2201.09172/) (2022)
- [Hybrid Variational Autoencoder for Time Series Forecasting](https://mlsanity.com/paper/2303.07048/) (2023)
- [Dynamic Multi-Network Mining of Tensor Time Series](https://mlsanity.com/paper/2402.11773/) (2024)
- [Fast and Multi-aspect Mining of Complex Time-stamped Event Streams](https://mlsanity.com/paper/2303.03789/) (2023)
- [E2USD: Efficient-yet-effective Unsupervised State Detection for Multivariate Time Series](https://mlsanity.com/paper/2402.14041/) (2024)
- [Review of Clustering Methods for Functional Data](https://mlsanity.com/paper/2210.00847/) (2022)
- [Long-Term EEG Partitioning for Seizure Onset Detection](https://mlsanity.com/paper/2412.15598/) (2024)
- [Modeling Time-evolving Causality over Data Streams](https://mlsanity.com/paper/2502.08963/) (2025)
- [HOPE: Human-Centric Off-Policy Evaluation for E-Learning and Healthcare](https://mlsanity.com/paper/2302.09212/) (2023)
- [Mining of Switching Sparse Networks for Missing Value Imputation in Multivariate Time Series](https://mlsanity.com/paper/2409.09930/) (2024)
- [Exploring Timeline Control for Facial Motion Generation](https://mlsanity.com/paper/2505.20861/) (2025)
- [STICC: A multivariate spatial clustering method for repeated geographic pattern discovery with consideration of spatial contiguity](https://mlsanity.com/paper/2203.09611/) (2022)
- [CLaP -- State Detection from Time Series](https://mlsanity.com/paper/2504.01783/) (2025)
- [Unsupervised Driving Event Discovery Based on Vehicle CAN-data](https://mlsanity.com/paper/2301.04988/) (2023)
- [MC-GTA: Metric-Constrained Model-Based Clustering using Goodness-of-fit Tests with Autocorrelations](https://mlsanity.com/paper/2405.18395/) (2024)
- [PromptTSS: A Prompting-Based Approach for Interactive Multi-Granularity Time Series Segmentation](https://mlsanity.com/paper/2506.11170/) (2025)
- [A Generalized Apprenticeship Learning Framework for Capturing Evolving Student Pedagogical Strategies](https://mlsanity.com/paper/2602.20527/) (2026)
- [Distributed Estimation of Sparse Inverse Covariances](https://mlsanity.com/paper/2109.12020/) (2021)
- [An Offline Time-aware Apprenticeship Learning Framework for Evolving Reward Functions](https://mlsanity.com/paper/2305.09070/) (2023)
- [Towards Dynamic Trend Filtering through Trend Point Detection with Reinforcement Learning](https://mlsanity.com/paper/2406.03665/) (2024)
- [Wormhole: Concept-Aware Deep Representation Learning for Co-Evolving Sequences](https://mlsanity.com/paper/2409.13857/) (2024)
- [CSTS: A Benchmark for the Discovery of Correlation Structures in Time Series Clustering](https://mlsanity.com/paper/2505.14596/) (2025)
- [Toward Interpretable Evaluation Measures for Time Series Segmentation](https://mlsanity.com/paper/2510.23261/) (2025)
- [Fast Mining and Dynamic Time-to-Event Prediction over Multi-sensor Data Streams](https://mlsanity.com/paper/2601.04741/) (2026)
- [GroupSegment-SHAP: Shapley Value Explanations with Group-Segment Players for Multivariate Time Series](https://mlsanity.com/paper/2601.06114/) (2026)

## Abstract (arXiv)

Subsequence clustering of multivariate time series is a useful tool for discovering repeated patterns in temporal data. Once these patterns have been discovered, seemingly complicated datasets can be interpreted as a temporal sequence of only a small number of states, or clusters. For example, raw sensor data from a fitness-tracking application can be expressed as a timeline of a select few actions (i.e., walking, sitting, running). However, discovering these patterns is challenging because it requires simultaneous segmentation and clustering of the time series. Furthermore, interpreting the resulting clusters is difficult, especially when the data is high-dimensional. Here we propose a new method of model-based clustering, which we call Toeplitz Inverse Covariance-based Clustering (TICC). Each cluster in the TICC method is defined by a correlation network, or Markov random field (MRF), characterizing the interdependencies between different observations in a typical subsequence of that cluster. Based on this graphical representation, TICC simultaneously segments and clusters the time series data. We solve the TICC problem through alternating minimization, using a variation of the expectation maximization (EM) algorithm. We derive closed-form solutions to efficiently solve the two resulting subproblems in a scalable way, through dynamic programming and the alternating direction method of multipliers (ADMM), respectively. We validate our approach by comparing TICC to several state-of-the-art baselines in a series of synthetic experiments, and we then demonstrate on an automobile sensor dataset how TICC can be used to learn interpretable clusters in real-world scenarios.