Abstract
Cyber-physical systems of today are generating large volumes of time-series data. As manual inspection of such data is not tractable, the need for learning methods to help discover logical structure in the data has increased. We propose a logic-based framework that allows domain-specific knowledge to be embedded into formulas in a parametric logical specification over time-series data. The key idea is to then map a time series to a surface in the parameter space of the formula. Given this mapping, we identify the Hausdorff distance between surfaces as a natural distance metric between two time-series data under the lens of the parametric specification. This enables embedding non-trivial domain-specific knowledge into the distance metric and then using off-the-shelf machine learning tools to label the data. After labeling the data, we demonstrate how to extract a logical specification for each label. Finally, we showcase our technique on real world traffic data to learn classifiers/monitors for slow-downs and traffic jams.
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Notes
- 1.
We note that such data can be obtained from fixed mounted cameras on a freeway, which is then converted into time-series data for individual vehicles, such as in [4].
- 2.
Nevertheless, the material presented in the sequel easily generalizes to other objects.
- 3.
Colloquially, if it looks like a duck and quacks like a duck, it should have a small distance to a duck.
- 4.
The co-domain of (14) could be tightened to \({\Big ( 2^{n} - 2\Big )}^{i}\), but to avoid also parameterizing the discretization function, we do not strengthen the type signature.
- 5.
If the rectangle being subdivided is degenerate, i.e., lies entirely within the boundary of the validity domain and thus all point intersect the boundary, then the halfway point of the diagonal is taken to be the subdivision point.
- 6.
again associated with (9).
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Acknowledgments
Some of the key ideas in this paper were influenced by discussions with Oded Maler, especially those pertaining to computing the boundaries of monotonic specifications. The work of the authors on this paper was funded in part by the NSF VeHICaL project (#1545126), NSF project #1739816, the DARPA BRASS program under agreement number FA8750–16–C0043, the DARPA Assured Autonomy program, Berkeley Deep Drive, the Army Research Laboratory under Cooperative Agreement Number W911NF–17–2–0196, and by Toyota under the iCyPhy center.
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Vazquez-Chanlatte, M., Ghosh, S., Deshmukh, J.V., Sangiovanni-Vincentelli, A., Seshia, S.A. (2018). Time-Series Learning Using Monotonic Logical Properties. In: Colombo, C., Leucker, M. (eds) Runtime Verification. RV 2018. Lecture Notes in Computer Science(), vol 11237. Springer, Cham. https://doi.org/10.1007/978-3-030-03769-7_22
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