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Specifying and detecting temporal patterns with shape expressions

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  • Special Issue: RV 2019
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Abstract

Modern cyber-physical systems (CPS) and the Internet of things (IoT) are data factories generating, measuring and recording huge amounts of time series. The useful information in time series is usually present in the form of sequential patterns. We propose shape expressions as a declarative language for specification and extraction of rich temporal patterns from possibly noisy data. Shape expressions are regular expressions with arbitrary (linear, exponential, sinusoidal, etc.) shapes with parameters as atomic predicates and additional constraints on these parameters. We associate with shape expressions novel noisy semantics that combines regular expression matching semantics with statistical regression. We study essential properties of the language and propose an efficient heuristic for approximate matching of shape expressions. We demonstrate the applicability of this technique on two case studies from the health and the avionics domains.

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Notes

  1. The signal with the empty time domain is equivalent to the empty word in the classical language theory

  2. We use \(\underline{l}\) instead of \(\underline{l}_{\sigma ,x}\) whenever its association to \(\sigma _{x}\) is clear from the context and omit \(\underline{l}_{\sigma ,x}\) altogether when not interested in the duration of the shape.

  3. We omit the duration variable \(\underline{l}\) whenever we are not interested in the duration of a shape—for instance, we then use the notation \(\textsf {sin}(a,b,c,d)\).

  4. We abuse the notation and replace a parameter variable by a constant, for instance, \(\textsf {lin}_x(0,b)\), as a shortcut for \(\textsf {lin}_x(a_1,b)~:~a_1 = 0\).

  5. We also assume that the SMA \(\hat{\mathcal {A}}\), the signal w, the noise tolerance threshold \(\nu \) and the minimum match length \(\uplambda \) are given as global parameters to the main procedure \(\textsf {policy\_scheduler}\) and are implicitly propagated to all the other methods

  6. Recall that we require atomic matches of minimum length \(\uplambda \).

  7. The figure is under copyright by A. Rad.

  8. We recall that \(\nu = 0\) denotes zero noise tolerance and \(\nu = 1\) allows arbitrary level of noise.

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Acknowledgements

This research was supported by the Austrian Science Fund (FWF) under grants S11402-N23 (RiSE/SHiNE) and Z211-N23 (Wittgenstein Award), by the Productive 4.0 project (ECSEL 737459) and by the National Science Foundation under the FMitF grant CCF-1837131.

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Authors and Affiliations

  1. AIT Austrian Institute of Technology, Vienna, Austria

    Dejan Ničković & Cristinel Mateis

  2. University of Southern California, Los Angeles, US

    Xin Qin & Jyotirmoy Deshmukh

  3. Imagination Technologies, Dacorum, UK

    Thomas Ferrère

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  1. Dejan Ničković
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  2. Xin Qin
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  5. Jyotirmoy Deshmukh
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Correspondence to Dejan Ničković.

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Ničković, D., Qin, X., Ferrère, T. et al. Specifying and detecting temporal patterns with shape expressions. Int J Softw Tools Technol Transfer 23, 565–577 (2021). https://doi.org/10.1007/s10009-021-00627-x

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