We study sparse principal component analysis for high dimensional vector autoregressive time series under a doubly asymptotic framework, which allows the dimension $d$ to scale with the series length $T$. We treat the transition matrix of time series as a nuisance parameter and directly apply sparse principal component analysis on multivariate time series as if the data are independent. We provide explicit non-asymptotic rates of convergence for leading eigenvector estimation and extend this result to principal subspace estimation. Our analysis illustrates that the spectral norm of the transition matrix plays an essential role in determining the final rates. We also characterize sufficient conditions under which sparse principal component analysis attains the optimal parametric rate. Our theoretical results are backed up by thorough numerical studies.
Despite the rapid progress on research in adversarial robustness of deep neural networks (DNNs), there is little principled work for the time-series domain. Since time-series data arises in diverse applications including mobile health, finance, and smart grid, it is important to verify and improve the robustness of DNNs for the time-series domain. In this paper, we propose a novel framework for the time-series domain referred as {\em Dynamic Time Warping for Adversarial Robustness (DTW-AR)} using the dynamic time warping measure. Theoretical and empirical evidence is provided to demonstrate the effectiveness of DTW over the standard Euclidean distance metric employed in prior methods for the image domain. We develop a principled algorithm justified by theoretical analysis to efficiently create diverse adversarial examples using random alignment paths. Experiments on diverse real-world benchmarks show the effectiveness of DTW-AR to fool DNNs for time-series data and to improve their robustness using adversarial training. The source code of DTW-AR algorithms is available at https://github.com/tahabelkhouja/DTW-AR
Neural networks are becoming more and more popular for the analysis of physiological time-series. The most successful deep learning systems in this domain combine convolutional and recurrent layers to extract useful features to model temporal relations. Unfortunately, these recurrent models are difficult to tune and optimize. In our experience, they often require task-specific modifications, which makes them challenging to use for non-experts. We propose U-Time, a fully feed-forward deep learning approach to physiological time series segmentation developed for the analysis of sleep data. U-Time is a temporal fully convolutional network based on the U-Net architecture that was originally proposed for image segmentation. U-Time maps sequential inputs of arbitrary length to sequences of class labels on a freely chosen temporal scale. This is done by implicitly classifying every individual time-point of the input signal and aggregating these classifications over fixed intervals to form the final predictions. We evaluated U-Time for sleep stage classification on a large collection of sleep electroencephalography (EEG) datasets. In all cases, we found that U-Time reaches or outperforms current state-of-the-art deep learning models while being much more robust in the training process and without requiring architecture or hyperparameter adaptation across tasks.
Time series motifs play an important role in the time series analysis. The motif-based time series clustering is used for the discovery of higher-order patterns or structures in time series data. Inspired by the convolutional neural network (CNN) classifier based on the image representations of time series, motif difference field (MDF) is proposed. Compared to other image representations of time series, MDF is simple and easy to construct. With the Fully Convolution Network (FCN) as the classifier, MDF demonstrates the superior performance on the UCR time series dataset in benchmark with other time series classification methods. It is interesting to find that the triadic time series motifs give the best result in the test. Due to the motif clustering reflected in MDF, the significant motifs are detected with the help of the Gradient-weighted Class Activation Mapping (Grad-CAM). The areas in MDF with high weight in Grad-CAM have a high contribution from the significant motifs with the desired ordinal patterns associated with the signature patterns in time series. However, the signature patterns cannot be identified with the neural network classifiers directly based on the time series.
A highly comparative, feature-based approach to time series classification is introduced that uses an extensive database of algorithms to extract thousands of interpretable features from time series. These features are derived from across the scientific time-series analysis literature, and include summaries of time series in terms of their correlation structure, distribution, entropy, stationarity, scaling properties, and fits to a range of time-series models. After computing thousands of features for each time series in a training set, those that are most informative of the class structure are selected using greedy forward feature selection with a linear classifier. The resulting feature-based classifiers automatically learn the differences between classes using a reduced number of time-series properties, and circumvent the need to calculate distances between time series. Representing time series in this way results in orders of magnitude of dimensionality reduction, allowing the method to perform well on very large datasets containing long time series or time series of different lengths. For many of the datasets studied, classification performance exceeded that of conventional instance-based classifiers, including one nearest neighbor classifiers using Euclidean distances and dynamic time warping and, most importantly, the features selected provide an understanding of the properties of the dataset, insight that can guide further scientific investigation.
Linear causal analysis is central to a wide range of important application spanning finance, the physical sciences, and engineering. Much of the existing literature in linear causal analysis operates in the time domain. Unfortunately, the direct application of time domain linear causal analysis to many real-world time series presents three critical challenges: irregular temporal sampling, long range dependencies, and scale. Moreover, real-world data is often collected at irregular time intervals across vast arrays of decentralized sensors and with long range dependencies which make naive time domain correlation estimators spurious. In this paper we present a frequency domain based estimation framework which naturally handles irregularly sampled data and long range dependencies while enabled memory and communication efficient distributed processing of time series data. By operating in the frequency domain we eliminate the need to interpolate and help mitigate the effects of long range dependencies. We implement and evaluate our new work-flow in the distributed setting using Apache Spark and demonstrate on both Monte Carlo simulations and high-frequency financial trading that we can accurately recover causal structure at scale.
Most machine learning methods are used as a black box for modelling. We may try to extract some knowledge from physics-based training methods, such as neural ODE (ordinary differential equation). Neural ODE has advantages like a possibly higher class of represented functions, the extended interpretability compared to black-box machine learning models, ability to describe both trend and local behaviour. Such advantages are especially critical for time series with complicated trends. However, the known drawback is the high training time compared to the autoregressive models and long-short term memory (LSTM) networks widely used for data-driven time series modelling. Therefore, we should be able to balance interpretability and training time to apply neural ODE in practice. The paper shows that modern neural ODE cannot be reduced to simpler models for time-series modelling applications. The complexity of neural ODE is compared to or exceeds the conventional time-series modelling tools. The only interpretation that could be extracted is the eigenspace of the operator, which is an ill-posed problem for a large system. Spectra could be extracted using different classical analysis methods that do not have the drawback of extended time. Consequently, we reduce the neural ODE to a simpler linear form and propose a new view on time-series modelling using combined neural networks and an ODE system approach.
Similarity-based approaches represent a promising direction for time series analysis. However, many such methods rely on parameter tuning, and some have shortcomings if the time series are multivariate (MTS), due to dependencies between attributes, or the time series contain missing data. In this paper, we address these challenges within the powerful context of kernel methods by proposing the robust \emph{time series cluster kernel} (TCK). The approach taken leverages the missing data handling properties of Gaussian mixture models (GMM) augmented with informative prior distributions. An ensemble learning approach is exploited to ensure robustness to parameters by combining the clustering results of many GMM to form the final kernel. We evaluate the TCK on synthetic and real data and compare to other state-of-the-art techniques. The experimental results demonstrate that the TCK is robust to parameter choices, provides competitive results for MTS without missing data and outstanding results for missing data.
This paper describes the R package crqa to perform cross-recurrence quantification analysis of two time series of either a categorical or continuous nature. Streams of behavioral information, from eye movements to linguistic elements, unfold over time. When two people interact, such as in conversation, they often adapt to each other, leading these behavioral levels to exhibit recurrent states. In dialogue, for example, interlocutors adapt to each other by exchanging interactive cues: smiles, nods, gestures, choice of words, and so on. In order for us to capture closely the goings-on of dynamic interaction, and uncover the extent of coupling between two individuals, we need to quantify how much recurrence is taking place at these levels. Methods available in crqa would allow researchers in cognitive science to pose such questions as how much are two people recurrent at some level of analysis, what is the characteristic lag time for one person to maximally match another, or whether one person is leading another. First, we set the theoretical ground to understand the difference between 'correlation' and 'co-visitation' when comparing two time series, using an aggregative or cross-recurrence approach. Then, we describe more formally the principles of cross-recurrence, and show with the current package how to carry out analyses applying them. We end the paper by comparing computational efficiency, and results' consistency, of crqa R package, with the benchmark MATLAB toolbox crptoolbox. We show perfect comparability between the two libraries on both levels.
Many businesses and industries nowadays rely on large quantities of time series data making time series forecasting an important research area. Global forecasting models that are trained across sets of time series have shown a huge potential in providing accurate forecasts compared with the traditional univariate forecasting models that work on isolated series. However, there are currently no comprehensive time series archives for forecasting that contain datasets of time series from similar sources available for the research community to evaluate the performance of new global forecasting algorithms over a wide variety of datasets. In this paper, we present such a comprehensive time series forecasting archive containing 20 publicly available time series datasets from varied domains, with different characteristics in terms of frequency, series lengths, and inclusion of missing values. We also characterise the datasets, and identify similarities and differences among them, by conducting a feature analysis. Furthermore, we present the performance of a set of standard baseline forecasting methods over all datasets across eight error metrics, for the benefit of researchers using the archive to benchmark their forecasting algorithms.
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