Yifei Wang

K\mathcal{Y}=\{y_{k}\}_{k=1}^{K}=\{(t_{\texttt{on}}^{(k)},t_{\texttt{off}}^{(k)},e^{(k)})\}_{k=1}^{K} (1) Objective. Minimizing the discrepancy between the predicted set 𝒴\mathcal{Y} and the ground truth 𝒴^\hat{\mathcal{Y}}, which requires: (1) Detection Accuracy: For any ground-truth y^j∈𝒴^\hat{y}_{j}\in\hat{\mathcal{Y}}, a matched prediction yi∈𝒴y_{i}\in\mathcal{Y} (e(i)=e^(j)e^{(i)}=\hat{e}^{(j)}) should reach a certain alignment threshold (e.g. temporal IoU) with y^j\hat{y}_{j}. (2) Localization Precision: For each matched event instance pair (yi,y^j)(y_{i},\hat{y}_{j}) under the objective (1), the deviation between the predicted temporal boundaries [ton(i),toff(i)][t_{\texttt{on}}^{(i)},t_{\texttt{off}}^{(i)}] and the ground truth [t^on(j),t^off(j)][\hat{t}_{\texttt{on}}^{(j)},\hat{t}_{\texttt{off}}^{(j)}] should be minimized. 3 Related Work 3.1 Time Series Symbolic Representation To achieve the goal of accurate event detection with explainability, the internal structures of events need to be explicitly modeled. Based on the example introduced in Section 1, we set the following desiderata for designing the logic structure: D1. Hierarchical Representation: the ability to capture hierarchical structures of events: the way atomic patterns recursively form sub-patterns, and finally the global pattern. D2. Semantic Quantification: the ability to quantify the coherence between the signal morphology and the semantics. D3. Topological Elasticity: the ability to define events based on the internal temporal-logic structure, agnostic to actual temporal durations: the representation must support “time warping” (elastic stretching or compressing of time) while maintaining logical validity. We analyze existing symbolic time series representation frameworks based on our desiderata (summarized in Table 1). (1) SAX [Lin et al., 2003; Malinowski et al., 2013] and ABBA (Elsworth and Güttel, 2020) map time series into string sequences using unsupervised learning. Each character represents a specific shape or amplitude. However, the unsupervised learning is performed on the dataset level, making the symbolic representation dataset-dependent. Moreover, the repeated sub-strings carry semantics but do not quantify coherence (partially satisfying D2), and do not allow hierarchical structure (failing D1) or complex topological relationships (failing D3). (2) Logical-Shapelets (Mueen et al., 2011) represents entire time series with a series of frequently appearing sub-sequences with fixed length, i.e., shapelets (failing D3). Z-Time (Lee et al., 2024) improves the representation by making the length variable (satisfying D3). Both the shapelets and discrete representations of Z-Time carry no semantic abstraction (failing D2). Regarding structural modeling, Logical-Shapelets only considers boolean relations, and while Z-Time uses Allen’s algebra (Allen, 1983), only limited hierarchy structures can be represented by stacking temporal relation pairs. Therefore, both approaches partially satisfy D1. (3) Chronicle System (Dousson and Le Maigat, 2007) applies graph structure to represent temporal events, where each sub-event is considered as a node in the graph, and STL (Maler and Nickovic, 2004) represents events as recursive logical formulas over signals intervals, both satisfying D1. However, both systems model semantic coherence as binary values (true/false) (partially satisfying D2), and their reliance on actual interval durations does not meet D3. D1: Hierarchical