The time series classification literature has expanded rapidly over the last decade, with many new classification approaches published each year. Prior research has mostly focused on improving the accuracy and efficiency of classifiers, with interpretability being somewhat neglected. This aspect of classifiers has become critical for many application domains and the introduction of the EU GDPR legislation in 2018 is likely to further emphasize the importance of interpretable learning algorithms. Currently, state-of-the-art classification accuracy is achieved with very complex models based on large ensembles (COTE) or deep neural networks (FCN). These approaches are not efficient with regard to either time or space, are difficult to interpret and cannot be applied to variable-length time series, requiring pre-processing of the original series to a set fixedlength. In this paper we propose new time series classification algorithms to address these gaps. Our approach is based on symbolic representations of time series, efficient sequence mining algorithms and linear classification models. Our linear models are as accurate as deep learning models but are more efficient regarding running time and memory, can work with variable-length time series and can be interpreted by highlighting the discriminative symbolic features on the original time series. We advance the state-of-the-art in time series classification by proposing new algorithms built using the following three key ideas: (1) Multiple resolutions of symbolic representations: we combine symbolic representations obtained using different parameters, rather than one fixed representation (e.g., multiple SAX representations); (2) Multiple domain representations: we combine symbolic representations in time (e.g., SAX) and frequency (e.g., SFA) domains, to be more robust across problem types; (3) Efficient navigation in a huge symbolic-words space: we extend a symbolic sequence classifier (SEQL) to work with multiple symbolic representations and use its greedy feature selection strategy to effectively filter the best features for each representation. We show that our multi-resolution multi-domain linear classifier (mtSS-SEQL+LR) achieves a similar accuracy to the state-of-the-art COTE ensemble, and to recent deep learning methods (FCN, ResNet), but uses a fraction of the time and memory required by either COTE or deep models. To further analyse the interpretability of our classifier, we present a case study on a human motion dataset collected by the authors. We discuss the accuracy, efficiency and interpretability of our proposed algorithms and release all the results, source code and data to encourage reproducibility.

Abstract-Existing approaches to time series classification can be grouped into shape-based (numeric) and structure-based (symbolic). Shape-based techniques use the raw numeric time series with Euclidean or Dynamic Time Warping distance and a 1-Nearest Neighbor classifier. They are accurate, but computationally intensive. Structure-based methods discretize the raw data into symbolic representations, then extract features for classifiers. Recent symbolic methods have outperformed numeric ones regarding both accuracy and efficiency. Most approaches employ a bag-of-symbolic-words representation, but typically the word-length is fixed across all time series, an issue identified as a major weakness in the literature. Also, there are no prior attempts to use efficient sequence learning techniques to go beyond single words, to features based on variable-length sequences of words or symbols. We study an efficient linear classification approach, SEQL, originally designed for classification of symbolic sequences. SEQL learns discriminative subsequences from training data by exploiting the all-subsequence space using greedy gradient descent. We explore different discretization approaches, from none at all to increasing smoothing of the original data, and study the effect of these transformations on the accuracy of SEQL classifiers. We propose two adaptations of SEQL for time series data, SAX-VSEQL, can deal with X-axis offsets by learning variable-length symbolic words, and SAX-VFSEQL, can deal with X-axis and Y-axis offsets, by learning fuzzy variable-length symbolic words. Our models are linear classifiers in rich feature spaces. Their predictions are based on the most discriminative subsequences learned during training, and can be investigated for interpreting the classification decision.

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Symbolic representations of time series have proven to be effective for time series classification, with many recent approaches including SAX-VSM, BOSS, WEASEL, and MrSEQL. The key idea is to transform numerical time series to symbolic representations in the time or frequency domain, i.e., sequences of symbols, and then extract features from these sequences. While achieving high accuracy, existing symbolic classifiers are computationally expensive. In this paper we present MrSQM, a new time series classifier which uses multiple symbolic representations and efficient sequence mining, to extract important time series features. We study four feature selection approaches on symbolic sequences, ranging from fully supervised, to unsupervised and hybrids. We propose a new approach for optimal supervised symbolic feature selection in all-subsequence space, by adapting a Chi-squared bound developed for discriminative pattern mining, to time series. Our extensive experiments on 112 datasets of the UEA/UCR benchmark demonstrate that MrSQM can quickly extract useful features and learn accurate classifiers with the classic logistic regression algorithm. Interestingly, we find that a very simple and fast feature selection strategy can be highly effective as compared with more sophisticated and expensive methods. MrSQM advances the state-of-the-art for symbolic time series classifiers and it is an effective method to achieve high accuracy, with fast runtime.