Temporal-Aware Density Peak Clustering

• Temporal-aware density peak clustering is a method that extends traditional DPC by integrating temporal similarity measures like DTW to accurately cluster time series data.
• The algorithm employs an admissible pruning strategy with upper and lower DTW bounds, reducing computational cost by up to 94% without sacrificing clustering quality.
• TADPole uses anytime optimization and multidimensional aggregation to achieve scalable, robust, and interpretable clustering across diverse real-world domains.

A temporal-aware density peak clustering algorithm extends the classical density peak clustering (DPC) paradigm to time series and temporally structured data, leveraging temporal relationships, distance bounds, and computational optimizations. Its core principle is to efficiently and accurately cluster high-dimensional temporal data by adapting the density peak mechanism—where cluster centers reside in regions of high local density and large separation from denser points—using distance metrics and algorithms that directly address temporal alignment and complexity.

1. Foundational Principles of Density Peak Clustering for Time Series

Density peak clustering treats clusters as density maxima in data space, assigning points to clusters centered on high-density, well-separated exemplars. Temporal-aware adaptations use time series similarity metrics, such as Dynamic Time Warping (DTW), instead of simple Euclidean distances, to capture sequence alignment invariance. In the temporal setting, the local density $Q_i$ of a time series $i$ counts the number of series within the cutoff distance $d_c$; the separation distance $d_i$ is the shortest distance from $i$ to any object with higher $Q$. Cluster centers are selected for their combined high density and separation, and non-center objects inherit the label of their nearest, denser neighbor.

For multidimensional time series, distances and bounds are aggregated across each dimension, maintaining temporal sensitivity and invariance.

2. Admissible Pruning Strategy for Efficient Temporal Clustering

Computational bottlenecks in time series clustering arise from the cost of DTW evaluations. The admissible pruning strategy addresses this by leveraging both upper and lower bounds on pairwise DTW distances:

• Case A – Identical Objects: For duplicates, the exact DTW value is known; full computation is skipped.
• Case B – Definite Inclusion: If the upper bound $UB(i,j) < d_c$, then $D(i,j) < d_c$; the pair is confirmed in the cutoff.
• Case C – Definite Exclusion: If $LB(i,j) > d_c$, $D(i,j) > d_c$; computation is skipped.
• Case D – Uncertain: If $LB(i, j) < d_c < UB(i, j)$, the DTW distance is computed.

The pruning rule is summarized:

$$\begin{array}{ll} \text{If } & \text{UB}(i,j) < d_c, \quad \text{skip } D(i,j) \ \text{If } & \text{LB}(i,j) > d_c, \quad \text{skip } D(i,j) \ \text{Else} & \text{compute } D(i,j) \end{array}$$

Empirical evaluation demonstrates up to $\sim94\%$ pruning of DTW calculations (StarLightCurves dataset), providing order-of-magnitude speedups with no loss in clustering accuracy. For cases requiring full computation, an anytime ordering heuristic ensures that the most influential distances are prioritized: cluster assignment converges quickly, and intermediate results are meaningful even before all distances are calculated.

3. Algorithmic Framework: The TADPole Approach

“Time-series Anytime DP” (TADPole) is the temporal-aware instantiation of density peak clustering. It integrates pruning, DTW-based distance computation, and anytime optimization:

• Local Density Calculation: For each series, count of neighbors within $d_c$, with pruning applied at each pairwise step.
• Cluster Center Selection: Points with high $Q_i \cdot d_i$.
• Label Propagation: Non-centers inherit cluster from nearest denser neighbor.
• Anytime Optimization: Order unpruned computations so the iterative solution is always the best available given completed calculations.
• Multidimensional Extension: Aggregate bounds across dimensions, preserving admissibility.

Parameters such as $d_c$ and DTW warping window are heuristically set via pseudo-labeled data, automating unsupervised setup.

4. Empirical Applications and Domain-Specific Case Studies

TADPole is evaluated on diverse temporal domains:

• Astronomy (StarLightCurves): Achieves up to 94% pruning, reducing runtime from 9 hours to 9 minutes while matching brute-force clustering accuracy.
• Speech Physiology (EMA Articulograph 3D traces): Attains interactive performance with 94% avoidance of DTW computation.
• Medicine (PPG, Pulsus Paradoxus detection): Accurately separates severe from non-severe conditions; clusters retain semantic interpretability.
• Entomology (insect time series): Maintains clustering quality, robust to outliers.
• Sequence Clustering (protein data): Extends to discrete Edit Distance; pruning rates depend on bound tightness (biological example: 28% pruning).

Across these domains, clustering quality is validated by scores such as Rand Index, and the method’s pruning maintains exactness with significant gains in efficiency.

5. Comparative Analysis: Performance, Limitations, and Robustness

Relative to baseline approaches:

• Efficiency: TADPole dramatically reduces costly DTW calculations, often pruning >90%; exhibits much lower runtime than DP with brute-force DTW.
• Quality: Outperforms k-means, DBSCAN, k-Shape, DP with Euclidean distance, and is robust to outliers. Rand Index is consistently higher.
• Robustness: Maintains exact clustering assignments due to admissibility of pruning.
• Limitations: Pruning effectiveness varies with bound tightness; in some challenging biological cases, pruning is less substantial. Improved bounding techniques provide a direction for more efficiency.

6. Generalization and Future Research Directions

The admissible pruning framework is applicable to any pairwise distance measure with computable bounds, not limited to DTW. Potential avenues include:

• Incorporating new distance functions (e.g., RNA/DNA alignment scores, Graph Edit Distance, Earth Mover's Distance).
• Developing more adaptive parameter selection mechanisms for clustering thresholds and warping windows, especially in unsupervised contexts.
• Exploring online and incremental clustering for streaming temporal data, leveraging the anytime property for evolving datasets.
• Tightening upper and lower bounds through domain-specific heuristics and scalable approximations.

A plausible implication is that these extensions may enable clustering of extremely large, continuously collected temporal datasets in interactive computational environments.

7. Integration with Biological Data and External Methods

TADPole's generality is exemplified by its integration into biologically relevant toolkits such as DMRIntTk, enabling high-confidence aggregation of differentially methylated region (DMR) sets. By weighting genomic bins according to both methylation difference and reliability across methods, adapted DPC algorithms (as in DMRIntTk) trim low-difference noise and enrich the proportion of biologically significant DMRs, enhancing downstream pathway analysis and biomarker discovery. This suggests broad applicability outside canonical time series analysis, including multi-modal and multi-source biological data integration (Zhang et al., 14 Jul 2024).

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The temporal-aware density peak clustering algorithm (TADPole) systematically combines density-based cluster identification with temporal distance metrics and admissible pruning, yielding an exact, efficient, and generalizable clustering framework for time series data. Its design addresses computational challenges inherent in dynamic data analysis, supporting robust, scalable, and interpretable clustering across scientific and biomedical domains (Begum et al., 2016).