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PETSC: pattern-based embedding for time series classification

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Abstract

Efficient and interpretable classification of time series is an essential data mining task with many real-world applications. Recently several dictionary- and shapelet-based time series classification methods have been proposed that employ contiguous subsequences of fixed length. We extend pattern mining to efficiently enumerate long variable-length sequential patterns with gaps. Additionally, we discover patterns at multiple resolutions thereby combining cohesive sequential patterns that vary in length, duration and resolution. For time series classification we construct an embedding based on sequential pattern occurrences and learn a linear model. The discovered patterns form the basis for interpretable insight into each class of time series. The pattern-based embedding for time series classification (PETSC) supports both univariate and multivariate time series datasets of varying length subject to noise or missing data. We experimentally validate that MR-PETSC performs significantly better than baseline interpretable methods such as DTW, BOP and SAX-VSM on univariate and multivariate time series. On univariate time series, our method performs comparably to many recent methods, including BOSS, cBOSS, S-BOSS, ProximityForest and ResNET, and is only narrowly outperformed by state-of-the-art methods such as HIVE-COTE, ROCKET, TS-CHIEF and InceptionTime. Moreover, on multivariate datasets PETSC performs comparably to the current state-of-the-art such as HIVE-COTE, ROCKET, CIF and ResNET, none of which are interpretable. PETSC scales to large datasets and the total time for training and making predictions on all 85 ‘bake off’ datasets in the UCR archive is under 3 h making it one of the fastest methods available. PETSC is particularly useful as it learns a linear model where each feature represents a sequential pattern in the time domain, which supports human oversight to ensure predictions are trustworthy and fair which is essential in financial, medical or bioinformatics applications.

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Notes

  1. We remark that alternatives to sliding-window based frequency for sequential patterns have been investigated that do not require choosing \(\varDelta t\) (Cule et al. 2019). However, this is not compatible with window-based normalisation performed by SAX.

  2. This observation has led to adaptations for numerosity reduction in time series classification (Lin et al. 2012) or non-overlapping minimal windows in frequent pattern (or episode) mining (Zhu et al. 2010; Cule et al. 2019).

  3. We use the ordinal values for SAX symbols when computing Euclidean distance, that is \(b-a\) is 1 and \(c-a\) is 2.

  4. Note that pattern mining has a worst-case time complexity which is exponential in the size of the pattern and the alphabet size. That is, with a pattern size (or word size) of w and \(\alpha \) different symbols, there are \(\alpha ^w\) possible sequential patterns of length w. However, we assume parameters such as w, \(\alpha \), k and \( rdur \) are constants. That is, we argue that in the context of time series classification, and not pattern mining, it is less relevant to perform a detailed analysis of the efficiency of our method for large values of k or rdur, since we do not observe an increase in time series classification accuracy for large values of both k and rdur.

  5. Source code of PETSC: https://bitbucket.org/len_feremans/petsc.

  6. We remark that there are differences in our creation of BeetleFly dataset compared to the UCR version due to small changes in the pre-processing of the original MPEG-7 source images.

  7. Full experimental results: https://bitbucket.org/len_feremans/petsc/src/master/Results.html.

  8. We use the implementations available in the sktime library (Löning et al. 2019).

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Correspondence to Len Feremans.

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Feremans, L., Cule, B. & Goethals, B. PETSC: pattern-based embedding for time series classification. Data Min Knowl Disc 36, 1015–1061 (2022). https://doi.org/10.1007/s10618-022-00822-7

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