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$k$-Graph: A Graph Embedding for Interpretable Time Series Clustering

Paul Boniol, Donato Tiano, Angela Bonifati, Themis Palpanas

TL;DR

Graph outperforms current state-of-the-art time series clustering algorithms in accuracy, while providing users with meaningful explanations and interpretations of the clustering outcomes.

Abstract

Time series clustering poses a significant challenge with diverse applications across domains. A prominent drawback of existing solutions lies in their limited interpretability, often confined to presenting users with centroids. In addressing this gap, our work presents $k$-Graph, an unsupervised method explicitly crafted to augment interpretability in time series clustering. Leveraging a graph representation of time series subsequences, $k$-Graph constructs multiple graph representations based on different subsequence lengths. This feature accommodates variable-length time series without requiring users to predetermine subsequence lengths. Our experimental results reveal that $k$-Graph outperforms current state-of-the-art time series clustering algorithms in accuracy, while providing users with meaningful explanations and interpretations of the clustering outcomes.

$k$-Graph: A Graph Embedding for Interpretable Time Series Clustering

TL;DR

Graph outperforms current state-of-the-art time series clustering algorithms in accuracy, while providing users with meaningful explanations and interpretations of the clustering outcomes.

Abstract

Time series clustering poses a significant challenge with diverse applications across domains. A prominent drawback of existing solutions lies in their limited interpretability, often confined to presenting users with centroids. In addressing this gap, our work presents -Graph, an unsupervised method explicitly crafted to augment interpretability in time series clustering. Leveraging a graph representation of time series subsequences, -Graph constructs multiple graph representations based on different subsequence lengths. This feature accommodates variable-length time series without requiring users to predetermine subsequence lengths. Our experimental results reveal that -Graph outperforms current state-of-the-art time series clustering algorithms in accuracy, while providing users with meaningful explanations and interpretations of the clustering outcomes.

Paper Structure

This paper contains 24 sections, 2 theorems, 10 equations, 12 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background and Related Work
  3. Time Series and Graph Notation
  4. Time Series Clustering
  5. Raw-Based Approaches
  6. Feature-Based Approaches
  7. Deep Learning Approaches
  8. Graph Embedding of Time Series
  9. Problem Formulation
  10. Proposed Approach
  11. Graph Embedding
  12. Graph Clustering
  13. Consensus Clustering
  14. Interpretability and Explainability
  15. Complexity Analysis
  16. ...and 9 more sections

Key Result

Lemma 1

For a given clustering partition $C = \{C_1,C_2,...,C_k\}$, if $\lambda \leq k$, then $\bigcup_{C_i \in C} \mathcal{G}^{\lambda}_{C_i} = \mathcal{G}$. if $\lambda > 0.5$, then $\bigcap_{C_i \in C} \mathcal{G}^{\lambda}_{C_i} = \emptyset$.

Figures (12)

  • Figure 1: $k$-Graph resulting graph $\mathcal{G}$ when applied on the Trace dataset Dau2018TheUT
  • Figure 2: $\lambda$-Graphoids and $\gamma$-Graphoids for different $\lambda$ and $\gamma$.
  • Figure 3: $k$-Graph pipeline.
  • Figure 4: $W_c$ and $W_e$ for $k$-Graph (with $k=4$ and $M=10$) applied on the Trace dataset of the UCR-Archive.
  • Figure 5: Experimental comparison of $k$-Graph versus the baselines on the UCR archive. In (a), the mean values are represented as a white square. The horizontal red dotted line represents the best mean.
  • ...and 7 more figures

Theorems & Definitions (8)

  • Definition 1: Graph Embedding
  • Definition 2: $Graphoid$
  • Definition 3: Node representativity
  • Definition 4: Node Exclusivity
  • Definition 5: $\lambda$-$Graphoid$
  • Definition 6: $\gamma$-$Graphoid$
  • Lemma 1
  • Lemma 2