PrivShape: Extracting Shapes in Time Series under User-Level Local Differential Privacy

TL;DR

The paper tackles extracting essential shapes from time series under user-level local differential privacy, where direct release of raw or perturbed sequences would overly distort shapes. It introduces PrivShape, a trie-based mechanism that combines Compressive SAX with trie-expansion pruning and a two-level refinement to efficiently generate and perturb shape candidates. The method achieves superior utility over PatternLDP in offline settings, enabling accurate frequent-shape extraction, clustering, and classification while preserving strong privacy. Empirical results on real and synthetic data show that PrivShape maintains shape fidelity across varied lengths and distance measures, indicating practical impact for privacy-preserving time-series analytics. The work lays groundwork for broader applications like shapelets discovery under local privacy frameworks.

Abstract

Time series have numerous applications in finance, healthcare, IoT, and smart city. In many of these applications, time series typically contain personal data, so privacy infringement may occur if they are released directly to the public. Recently, local differential privacy (LDP) has emerged as the state-of-the-art approach to protecting data privacy. However, existing works on LDP-based collections cannot preserve the shape of time series. A recent work, PatternLDP, attempts to address this problem, but it can only protect a finite group of elements in a time series due to ω-event level privacy guarantee. In this paper, we propose PrivShape, a trie-based mechanism under user-level LDP to protect all elements. PrivShape first transforms a time series to reduce its length, and then adopts trie-expansion and two-level refinement to improve utility. By extensive experiments on real-world datasets, we demonstrate that PrivShape outperforms PatternLDP when adapted for offline use, and can effectively extract frequent shapes.

Figures (19)

• Figure 1: (a) illustrates the shapes of frequency features corresponding to the pronunciations of "No" by two speakers. The two time series cannot be matched exactly, but they do have similar essential shapes (red points) shown in (b). (c) shows the essential shapes captured from the original time series in (a) by SAX.
• Figure 2: (a) illustrates that two time series exhibit scaling along value-axis despite having similar shapes. (b) elucidates that the lack of warping between the two time series is attributed to delays along the time-axis.
• Figure 3: A time series of length $m=128$ is compressed into a sequence "aaaccccccbbbbaaa" with the segment length $w=8$ and symbol size $t=3$.
• Figure 4: Extracted shape after Compressive SAX.
• Figure 6: The trie expansion in PrivShape from Level 2 to Level 3 with $c=3$ and $k=2$. The candidates in Level 2 will be first pruned before expansion to Level 3. Moreover, the expansion candidates (i.e., Level 2-Level 3 sub-shapes) are also pruned.

Theorems & Definitions (9)

• Definition 1: Local Differential Privacy
• Definition 2: User-Level Neighboring Time Series
• Definition 3: Frequent Shape
• Definition 4: Top-$k$ Frequent Shapes
• Theorem 1
• Lemma 1
• Theorem 2
• Theorem 3
• Theorem 4