Elastic similarity measures are a class of similarity measures specifically designed to work with time series data. When scoring the similarity between two time series, they allow points that do not correspond in timestamps to be aligned. This can compensate for misalignments in the time axis of time series data, and for similar processes that proceed at variable and differing paces. Elastic similarity measures are widely used in machine learning tasks such as classification, clustering and outlier detection when using time series data. There is a multitude of research on various univariate elastic similarity measures. However, except for multivariate versions of the well known Dynamic Time Warping (DTW) there is a lack of work to generalise other similarity measures for multivariate cases. This paper adapts two existing strategies used in multivariate DTW, namely, Independent and Dependent DTW, to several commonly used elastic similarity measures. Using 23 datasets from the University of East Anglia (UEA) multivariate archive, for nearest neighbour classification, we demonstrate that each measure outperforms all others on at least one dataset and that there are datasets for which either the dependent versions of all measures are more accurate than their independent counterparts or vice versa. This latter finding suggests that these differences arise from a fundamental property of the data. We also show that an ensemble of such nearest neighbour classifiers is highly competitive with other state-of-the-art multivariate time series classifiers.

This paper studies Time Series Extrinsic Regression (TSER): a regression task of which the aim is to learn the relationship between a time series and a continuous scalar variable; a task closely related to time series classification (TSC), which aims to learn the relationship between a time series and a categorical class label. This task generalizes time series forecasting (TSF), relaxing the requirement that the value predicted be a future value of the input series or primarily depend on more recent values. In this paper, we motivate and study this task, and benchmark existing solutions and adaptations of TSC algorithms on a novel archive of 19 TSER datasets which we have assembled. Our results show that the state-of-the-art TSC algorithm Rocket, when adapted for regression, achieves the highest overall accuracy compared to adaptations of other TSC algorithms and state-of-the-art machine learning (ML) algorithms such as XGBoost, Random Forest and Support Vector Regression. More importantly, we show that much research is needed in this field to improve the accuracy of ML models. We also find evidence that further research has excellent prospects of improving upon these straightforward baselines.

Time series research has gathered lots of interests in the last decade, especially for Time Series Classification (TSC) and Time Series Forecasting (TSF). Research in TSC has greatly benefited from the University of California Riverside and University of East Anglia (UCR/UEA) Time Series Archives. On the other hand, the advancement in Time Series Forecasting relies on time series forecasting competitions such as the Makridakis competitions, NN3 and NN5 Neural Network competitions, and a few Kaggle competitions. Each year, thousands of papers proposing new algorithms for TSC and TSF have utilized these benchmarking archives. These algorithms are designed for these specific problems, but may not be useful for tasks such as predicting the heart rate of a person using photoplethysmogram (PPG) and accelerometer data. We refer to this problem as Time Series Extrinsic Regression (TSER), where we are interested in a more general methodology of predicting a single continuous value, from univariate or multivariate time series. This prediction can be from the same time series or not directly related to the predictor time series and does not necessarily need to be a future value or depend heavily on recent values. To the best of our knowledge, research into TSER has received much less attention in the time series research community and there are no models developed for general time series extrinsic regression problems. Most models are developed for a specific problem. Therefore, we aim to motivate and support the research into TSER by introducing the first TSER benchmarking archive. This archive contains 19 datasets from different domains, with varying number of dimensions, unequal length dimensions, and missing values. In this paper, we introduce the datasets in this archive and did an initial benchmark on existing models.

Stream classification methods classify a continuous stream of data as new labelled samples arrive. They often also have to deal with concept drift. This paper focuses on seasonal drift in stream classification, which can be found in many real-world application data sources. Traditional approaches of stream classification consider seasonal drift by including seasonal dummy/indicator variables or building separate models for each season. But these approaches have strong limitations in high-dimensional classification problems, or with complex seasonal patterns. This paper explores how to best handle seasonal drift in the specific context of news article categorization (or classification/tagging), where seasonal drift is overwhelmingly the main type of drift present in the data, and for which the data are high-dimensional. We introduce a novel classifier named Seasonal Averaged One-Dependence Estimators (SAODE), which extends the AODE classifier to handle seasonal drift by including time as a super parent. We assess our SAODE model using two large real-world text mining related datasets each comprising approximately a million records, against nine state-of-the-art stream and concept drift classification models, with and without seasonal indicators and with separate models built for each season. Across five different evaluation techniques, we show that our model consistently outperforms other methods by a large margin where the results are statistically significant.

Noname manuscript No. (will be inserted by the editor)Time series classification for varying length series Chang Wei Tan · Fran cois Petitjean· Eamonn Keogh · Geoffrey I. Webb the date of receipt and acceptance should be inserted later Abstract Research into time series classification has tended to focus on the case of series of uniform length. However, it is common for real-world time series data to have unequal lengths. Differing time series lengths may arise from a number of fundamentally different mechanisms. In this work, we identify and evaluate two classes of such mechanisms - variations in sampling rate relative to the relevant signal and variations between the start and end points of one time series relative to one another. We investigate how time series generated by each of these classes of mechanism are best addressed for time series classification. We perform extensive experiments and provide practical recommendations on how variations in length should be handled in time series classification. Keywords Time Series Classification, Proximity Forest, Dynamic Time Warping 1 Introduction Time series classification (TSC) is an important task in many modern world applications such as remote sensing (Pelletier et al., 2019; Petitjean et al., 2012), astronomy (Batista et al., 2011), speech recognition (Hamooni et al., 2016), and insect classification (Chen et al., 2014). The time series to be classified are the observed outputs generated by some process. The classification task often relates to identifying the class of the process that generated the series. Each class of process might be considered as a realization of one or more ideals (in the Platonic sense) or prototypes. The resulting time series can then beChang Wei Tan · Fran cois Petitjean· Geoffrey I. Webb Faculty of Information Technology 25 Exhibition Walk Monash University, Melbourne VIC 3800, Australia Email: chang.tan@monash.edu,francois.petitjean@monash.edu,geoff.webb@monash.edu An observed time series might differ from the ideal in many ways. Much of the research on time series distance measures in the last decade can be seen as the introduction of techniques to mitigate these differences, either as a preprocessing step or directly in a distance measure. For example, variations in amplitude and offset are typically addressed in time series classification by normalization of the series (Rakthanmanon et al., 2012). Some observed values may be erroneous and might be addressed by outlier detection (Basu and Meckesheimer, 2007) and subsequent reinterpolation (Pelletier et al., 2019).

Time Series Classification (TSC) has seen enormous progress over the last two decades. HIVE-COTE (Hierarchical Vote Collective of Transformation-based Ensembles) is the current state of the art in terms of classification accuracy. HIVE-COTE recognizes that time series are a specific data type for which the traditional attribute-value representation, used predominantly in machine learning, fails to provide a relevant representation. HIVE-COTE combines multiple types of classifiers: each extracting information about a specific aspect of a time series, be it in the time domain, frequency domain or summarization of intervals within the series. However, HIVE-COTE (and its predecessor, FLAT-COTE) is often infeasible to run on even modest amounts of data. For instance, training HIVE-COTE on a dataset with only 1,500 time series can require 8 days of CPU time. It has polynomial runtime w.r.t training set size, so this problem compounds as data quantity increases. We propose a novel TSC algorithm, TS-CHIEF, which is highly competitive to HIVE-COTE in accuracy, but requires only a fraction of the runtime. TS-CHIEF constructs an ensemble classifier that integrates the most effective embeddings of time series that research has developed in the last decade. It uses tree-structured classifiers to do so efficiently. We assess TS-CHIEF on 85 datasets of the UCR archive, where it achieves state-of-the-art accuracy with scalability and efficiency. We demonstrate that TS-CHIEF can be trained on 130k time series in 2 days, a data quantity that is beyond the reach of any TSC algorithm with comparable accuracy.

Research into the classification of time series has made enormous progress in the last decade. The UCR time series archive has played a significant role in challenging and guiding the development of new learners for time series classification. The largest dataset in the UCR archive holds 10 thousand time series only; which may explain why the primary research focus has been in creating algorithms that have high accuracy on relatively small datasets. This paper introduces Proximity Forest, an algorithm that learns accurate models from datasets with millions of time series, and classifies a time series in milliseconds. The models are ensembles of highly randomized Proximity Trees. Whereas conventional decision trees branch on attribute values (and usually perform poorly on time series), Proximity Trees branch on the proximity of time series to one exemplar time series or another; allowing us to leverage the decades of work into developing relevant measures for time series. Proximity Forest gains both efficiency and accuracy by stochastic selection of both exemplars and similarity measures. Our work is motivated by recent time series applications that provide orders of magnitude more time series than the UCR benchmarks. Our experiments demonstrate that Proximity Forest is highly competitive on the UCR archive: it ranks among the most accurate classifiers while being significantly faster. We demonstrate on a 1M time series Earth observation dataset that Proximity Forest retains this accuracy on datasets that are many orders of magnitude greater than those in the UCR repository, while learning its models at least 100,000 times faster than current state of the art models Elastic Ensemble and COTE.

There has been renewed recent interest in developing effective lower bounds for Dynamic Time Warping (DTW) distance between time series. These have many applications in time series indexing, clustering, forecasting, regression and classification. One of the key time series classification algorithms, the nearest neighbor algorithm with DTW distance (NN-DTW) is very expensive to compute, due to the quadratic complexity of DTW. Lower bound search can speed up NN-DTW substantially. An effective and tight lower bound quickly prunes off unpromising nearest neighbor candidates from the search space and minimises the number of the costly DTW computations. The speed up provided by lower bound search becomes increasingly critical as training set size increases. Different lower bounds provide different trade-offs between computation time and tightness. Most existing lower bounds interact with DTW warping window sizes. They are very tight and effective at smaller warping window sizes, but become looser as the warping window increases, thus reducing the pruning effectiveness for NN-DTW. In this work, we present a new class of lower bounds that are tighter than the popular Keogh lower bound, while requiring similar computation time. Our new lower bounds take advantage of the DTW boundary condition, monotonicity and continuity constraints to create a tighter lower bound. Of particular significance, they remain relatively tight even for large windows. A single parameter to these new lower bounds controls the speed-tightness trade-off. We demonstrate that these new lower bounds provide an exceptional balance between computation time and tightness for the NN-DTW time series classification task, resulting in greatly improved efficiency for NN-DTW lower bound search.

This paper presents a framework for exact discovery of the top-k sequential patterns under Leverage. It combines (1) a novel definition of the expected support for a sequential pattern - a concept on which most interestingness measures directly rely - with (2) SkOPUS: a new branch-and-bound algorithm for the exact discovery of top-k sequential patterns under a given measure of interest. Our interestingness measure employs the partition approach. A pattern is interesting to the extent that it is more frequent than can be explained by assuming independence between any of the pairs of patterns from which it can be composed. The larger the support compared to the expectation under independence, the more interesting is the pattern. We build on these two elements to exactly extract the k sequential patterns with highest leverage, consistent with our definition of expected support. We conduct experiments on both synthetic data with known patterns and real-world datasets; both experiments confirm the consistency and relevance of our approach with regard to the state of the art. This article was published in Data Mining and Knowledge Discovery and is accessible at http://dx.doi.org/10.1007/s10618-016-0467-9.

This paper introduces a novel parameter estimation method for the probability tables of Bayesian network classifiers (BNCs), using hierarchical Dirichlet processes (HDPs). The main result of this paper is to show that improved parameter estimation allows BNCs to outperform leading learning methods such as Random Forest for both 0-1 loss and RMSE, albeit just on categorical datasets. As data assets become larger, entering the hyped world of "big", efficient accurate classification requires three main elements: (1) classifiers with low-bias that can capture the fine-detail of large datasets (2) out-of-core learners that can learn from data without having to hold it all in main memory and (3) models that can classify new data very efficiently. The latest Bayesian network classifiers (BNCs) satisfy these requirements. Their bias can be controlled easily by increasing the number of parents of the nodes in the graph. Their structure can be learned out of core with a limited number of passes over the data. However, as the bias is made lower to accurately model classification tasks, so is the accuracy of their parameters' estimates, as each parameter is estimated from ever decreasing quantities of data. In this paper, we introduce the use of Hierarchical Dirichlet Processes for accurate BNC parameter estimation. We conduct an extensive set of experiments on 68 standard datasets and demonstrate that our resulting classifiers perform very competitively with Random Forest in terms of prediction, while keeping the out-of-core capability and superior classification time.