The process of collecting and organizing sets of observations represents a common theme throughout the history of science. However, despite the ubiquity of scientists measuring, recording, and analyzing the dynamics of different processes, an extensive organization of scientific time-series data and analysis methods has never been performed. Addressing this, annotated collections of over 35 000 real-world and model-generated time series and over 9000 time-series analysis algorithms are analyzed in this work. We introduce reduced representations of both time series, in terms of their properties measured by diverse scientific methods, and of time-series analysis methods, in terms of their behaviour on empirical time series, and use them to organize these interdisciplinary resources. This new approach to comparing across diverse scientific data and methods allows us to organize time-series datasets automatically according to their properties, retrieve alternatives to particular analysis methods developed in other scientific disciplines, and automate the selection of useful methods for time-series classification and regression tasks. The broad scientific utility of these tools is demonstrated on datasets of electroencephalograms, self-affine time series, heart beat intervals, speech signals, and others, in each case contributing novel analysis techniques to the existing literature. Highly comparative techniques that compare across an interdisciplinary literature can thus be used to guide more focused research in time-series analysis for applications across the scientific disciplines.
This work presents an introduction to feature-based time-series analysis. The time series as a data type is first described, along with an overview of the interdisciplinary time-series analysis literature. I then summarize the range of feature-based representations for time series that have been developed to aid interpretable insights into time-series structure. Particular emphasis is given to emerging research that facilitates wide comparison of feature-based representations that allow us to understand the properties of a time-series dataset that make it suited to a particular feature-based representation or analysis algorithm. The future of time-series analysis is likely to embrace approaches that exploit machine learning methods to partially automate human learning to aid understanding of the complex dynamical patterns in the time series we measure from the world.
Recent years have witnessed the unprecedented rising of time series from almost all kindes of academic and industrial fields. Various types of deep neural network models have been introduced to time series analysis, but the important frequency information is yet lack of effective modeling. In light of this, in this paper we propose a wavelet-based neural network structure called multilevel Wavelet Decomposition Network (mWDN) for building frequency-aware deep learning models for time series analysis. mWDN preserves the advantage of multilevel discrete wavelet decomposition in frequency learning while enables the fine-tuning of all parameters under a deep neural network framework. Based on mWDN, we further propose two deep learning models called Residual Classification Flow (RCF) and multi-frequecy Long Short-Term Memory (mLSTM) for time series classification and forecasting, respectively. The two models take all or partial mWDN decomposed sub-series in different frequencies as input, and resort to the back propagation algorithm to learn all the parameters globally, which enables seamless embedding of wavelet-based frequency analysis into deep learning frameworks. Extensive experiments on 40 UCR datasets and a real-world user volume dataset demonstrate the excellent performance of our time series models based on mWDN. In particular, we propose an importance analysis method to mWDN based models, which successfully identifies those time-series elements and mWDN layers that are crucially important to time series analysis. This indeed indicates the interpretability advantage of mWDN, and can be viewed as an indepth exploration to interpretable deep learning.
Koopman spectral analysis has attracted attention for nonlinear dynamical systems since we can analyze nonlinear dynamics with a linear regime by embedding data into a Koopman space by a nonlinear function. For the analysis, we need to find appropriate embedding functions. Although several neural network-based methods have been proposed for learning embedding functions, existing methods require long time-series for training neural networks. This limitation prohibits performing Koopman spectral analysis in applications where only short time-series are available. In this paper, we propose a meta-learning method for estimating embedding functions from unseen short time-series by exploiting knowledge learned from related but different time-series. With the proposed method, a representation of a given short time-series is obtained by a bidirectional LSTM for extracting its properties. The embedding function of the short time-series is modeled by a neural network that depends on the time-series representation. By sharing the LSTM and neural networks across multiple time-series, we can learn common knowledge from different time-series while modeling time-series-specific embedding functions with the time-series representation. Our model is trained such that the expected test prediction error is minimized with the episodic training framework. We experimentally demonstrate that the proposed method achieves better performance in terms of eigenvalue estimation and future prediction than existing methods.
It becomes increasingly popular to perform mediation analysis for complex data from sophisticated experimental studies. In this paper, we present Granger Mediation Analysis (GMA), a new framework for causal mediation analysis of multiple time series. This framework is motivated by a functional magnetic resonance imaging (fMRI) experiment where we are interested in estimating the mediation effects between a randomized stimulus time series and brain activity time series from two brain regions. The stable unit treatment assumption for causal mediation analysis is thus unrealistic for this type of time series data. To address this challenge, our framework integrates two types of models: causal mediation analysis across the variables and vector autoregressive models across the temporal observations. We further extend this framework to handle multilevel data to address individual variability and correlated errors between the mediator and the outcome variables. These models not only provide valid causal mediation for time series data but also model the causal dynamics across time. We show that the modeling parameters in our models are identifiable, and we develop computationally efficient methods to maximize the likelihood-based optimization criteria. Simulation studies show that our method reduces the estimation bias and improve statistical power, compared to existing approaches. On a real fMRI data set, our approach not only infers the causal effects of brain pathways but accurately captures the feedback effect of the outcome region on the mediator region.
We propose an algorithm to impute and forecast a time series by transforming the observed time series into a matrix, utilizing matrix estimation to recover missing values and de-noise observed entries, and performing linear regression to make predictions. At the core of our analysis is a representation result, which states that for a large model class, the transformed matrix obtained from the time series via our algorithm is (approximately) low-rank. This, in effect, generalizes the widely used Singular Spectrum Analysis (SSA) in literature, and allows us to establish a rigorous link between time series analysis and matrix estimation. The key is to construct a matrix with non-overlapping entries rather than with the Hankel matrix as done in the literature, including in SSA. We provide finite sample analysis for imputation and prediction leading to the asymptotic consistency of our method. A salient feature of our algorithm is that it is model agnostic both with respect to the underlying time dynamics as well as the noise model in the observations. Being noise agnostic makes our algorithm applicable to the setting where the state is hidden and we only have access to its noisy observations a la a Hidden Markov Model, e.g., observing a Poisson process with a time-varying parameter without knowing that the process is Poisson, but still recovering the time-varying parameter accurately. As part of the forecasting algorithm, an important task is to perform regression with noisy observations of the features a la an error- in-variable regression. In essence, our approach suggests a matrix estimation based method for such a setting, which could be of interest in its own right. Through synthetic and real-world datasets, we demonstrate that our algorithm outperforms standard software packages (including R libraries) in the presence of missing data as well as high levels of noise.
With developing of computation tools in the last years, data analysis methods to find insightful information are becoming more common among industries and researchers. This paper is the first part of the times series analysis of New England electricity price and demand to find anomaly in the data. In this paper time-series stationary criteria to prepare data for further times-series related analysis is investigated. Three main analysis are conducted in this paper, including moving average, moving standard deviation and augmented Dickey-Fuller test. The data used in this paper is New England big data from 9 different operational zones. For each zone, 4 different variables including day-ahead (DA) electricity demand, price and real-time (RT) electricity demand price are considered.
Multivariate time series analysis is an important problem in data mining because of its widespread applications. With the increase of time series data available for training, implementing deep neural networks in the field of time series analysis is becoming common. Res2Net, a recently proposed backbone, can further improve the state-of-the-art networks as it improves the multi-scale representation ability through connecting different groups of filters. However, Res2Net ignores the correlations of the feature maps and lacks the control on the information interaction process. To address that problem, in this paper, we propose a backbone convolutional neural network based on the thought of gated mechanism and Res2Net, namely Gated Res2Net (GRes2Net), for multivariate time series analysis. The hierarchical residual-like connections are influenced by gates whose values are calculated based on the original feature maps, the previous output feature maps and the next input feature maps thus considering the correlations between the feature maps more effectively. Through the utilization of gated mechanism, the network can control the process of information sending hence can better capture and utilize the both the temporal information and the correlations between the feature maps. We evaluate the GRes2Net on four multivariate time series datasets including two classification datasets and two forecasting datasets. The results demonstrate that GRes2Net have better performances over the state-of-the-art methods thus indicating the superiority
In many real-world application, e.g., speech recognition or sleep stage classification, data are captured over the course of time, constituting a Time-Series. Time-Series often contain temporal dependencies that cause two otherwise identical points of time to belong to different classes or predict different behavior. This characteristic generally increases the difficulty of analysing them. Existing techniques often depended on hand-crafted features that were expensive to create and required expert knowledge of the field. With the advent of Deep Learning new models of unsupervised learning of features for Time-series analysis and forecast have been developed. Such new developments are the topic of this paper: a review of the main Deep Learning techniques is presented, and some applications on Time-Series analysis are summaried. The results make it clear that Deep Learning has a lot to contribute to the field.
This paper proposes a probabilistic neural network developed on the basis of time-series discriminant component analysis (TSDCA) that can be used to classify high-dimensional time-series patterns. TSDCA involves the compression of high-dimensional time series into a lower-dimensional space using a set of orthogonal transformations and the calculation of posterior probabilities based on a continuous-density hidden Markov model with a Gaussian mixture model expressed in the reduced-dimensional space. The analysis can be incorporated into a neural network, which is named a time-series discriminant component network (TSDCN), so that parameters of dimensionality reduction and classification can be obtained simultaneously as network coefficients according to a backpropagation through time-based learning algorithm with the Lagrange multiplier method. The TSDCN is considered to enable high-accuracy classification of high-dimensional time-series patterns and to reduce the computation time taken for network training. The validity of the TSDCN is demonstrated for high-dimensional artificial data and EEG signals in the experiments conducted during the study.
A fundamental problem in statistical data analysis is testing whether two phenomena are related. When the phenomena in question are time series, many challenges emerge. The first is defining a dependence measure between time series at the population level, as well as a sample level test statistic. The second is computing or estimating the distribution of this test statistic under the null, as the permutation test procedure is invalid for most time series structures. This work aims to address these challenges by combining distance correlation and multiscale graph correlation (MGC) from independence testing literature and block permutation testing from time series analysis. Two hypothesis tests for testing the independence of time series are proposed. These procedures also characterize whether the dependence relationship between the series is linear or nonlinear, and the time lag at which this dependence is maximized. For strictly stationary auto-regressive moving average (ARMA) processes, the proposed independence tests are proven valid and consistent. Finally, neural connectivity in the brain is analyzed using fMRI data, revealing linear dependence of signals within the visual network and default mode network, and nonlinear relationships in other regions. This work opens up new theoretical and practical directions for many modern time series analysis problems.
Data presented in the form of time series as its analysis and applications recently have become increasingly important in different areas and domains. Prediction and classification of time-series data play a vital role in multiple fields. In this paper, the time series analysis related to power consumption at 12 o'clock every day in the period of 2012 to 2014 has been compared for two distribution networks of Sistan and one of the four networks of Tehran. By analyzing the power consumption of these two networks, a comparison can be made between these two regions in terms of development and climate difference and the impact of social, industrial and environmental phenomena. The reason for choosing these two networks was to compare a deprived area with an area in the capital. CRP tool software and toolkits have been used to analyze and compare time series, and various tools have been used to compare two time series.
A time series consists of a series of values or events obtained over repeated measurements in time. Analysis of time series represents and important tool in many application areas, such as stock market analysis, process and quality control, observation of natural phenomena, medical treatments, etc. A vital component in many types of time-series analysis is the choice of an appropriate distance/similarity measure. Numerous measures have been proposed to date, with the most successful ones based on dynamic programming. Being of quadratic time complexity, however, global constraints are often employed to limit the search space in the matrix during the dynamic programming procedure, in order to speed up computation. Furthermore, it has been reported that such constrained measures can also achieve better accuracy. In this paper, we investigate two representative time-series distance/similarity measures based on dynamic programming, Dynamic Time Warping (DTW) and Longest Common Subsequence (LCS), and the effects of global constraints on them. Through extensive experiments on a large number of time-series data sets, we demonstrate how global constrains can significantly reduce the computation time of DTW and LCS. We also show that, if the constraint parameter is tight enough (less than 10-15% of time-series length), the constrained measure becomes significantly different from its unconstrained counterpart, in the sense of producing qualitatively different 1-nearest neighbor graphs. This observation explains the potential for accuracy gains when using constrained measures, highlighting the need for careful tuning of constraint parameters in order to achieve a good trade-off between speed and accuracy.
The analysis of nonstationary time series is of great importance in many scientific fields such as physics and neuroscience. In recent years, Gaussian process regression has attracted substantial attention as a robust and powerful method for analyzing time series. In this paper, we introduce a new framework for analyzing nonstationary time series using locally stationary Gaussian process analysis with parameters that are coupled through a hidden Markov model. The main advantage of this framework is that arbitrary complex nonstationary covariance functions can be obtained by combining simpler stationary building blocks whose hidden parameters can be estimated in closed-form. We demonstrate the flexibility of the method by analyzing two examples of synthetic nonstationary signals: oscillations with time varying frequency and time series with two dynamical states. Finally, we report an example application on real magnetoencephalographic measurements of brain activity.
Over the last few years, traffic data has been exploding and the transportation discipline has entered the era of big data. It brings out new opportunities for doing data-driven analysis, but it also challenges traditional analytic methods. This paper proposes a new Divide and Combine based approach to do K means clustering on activity-travel behavior time series using features that are derived using tools in Time Series Analysis and Topological Data Analysis. Clustering data from five waves of the National Household Travel Survey ranging from 1990 to 2017 suggests that activity-travel patterns of individuals over the last three decades can be grouped into three clusters. Results also provide evidence in support of recent claims about differences in activity-travel patterns of different survey cohorts. The proposed method is generally applicable and is not limited only to activity-travel behavior analysis in transportation studies. Driving behavior, travel mode choice, household vehicle ownership, when being characterized as categorical time series, can all be analyzed using the proposed method.
Feature-based time series representation has attracted substantial attention in a wide range of time series analysis methods. Recently, the use of time series features for forecast model selection and model averaging has been an emerging research focus in the forecasting community. Nonetheless, most of the existing approaches depend on the manual choice of an appropriate set of features. Exploiting machine learning methods to automatically extract features from time series becomes crucially important in the state-of-the-art time series analysis. In this paper, we introduce an automated approach to extract time series features based on images. Time series are first transformed into recurrence images, from which local features can be extracted using computer vision algorithms. The extracted features are used for forecast model selection and model averaging. Our experiments show that forecasting based on automatically extracted features, with less human intervention and a more comprehensive view of the raw time series data, yields comparable performances with the top best methods proposed in the largest forecasting competition M4.
This paper formulates the problem of learning discriminative features (\textit{i.e.,} segments) from networked time series data considering the linked information among time series. For example, social network users are considered to be social sensors that continuously generate social signals (tweets) represented as a time series. The discriminative segments are often referred to as \emph{shapelets} in a time series. Extracting shapelets for time series classification has been widely studied. However, existing works on shapelet selection assume that the time series are independent and identically distributed (i.i.d.). This assumption restricts their applications to social networked time series analysis, since a user's actions can be correlated to his/her social affiliations. In this paper we propose a new Network Regularized Least Squares (NetRLS) feature selection model that combines typical time series data and user network data for analysis. Experiments on real-world networked time series Twitter and DBLP data demonstrate the performance of the proposed method. NetRLS performs better than LTS, the state-of-the-art time series feature selection approach, on real-world data.
In this paper, we propose a technique for time series clustering using community detection in complex networks. Firstly, we present a method to transform a set of time series into a network using different distance functions, where each time series is represented by a vertex and the most similar ones are connected. Then, we apply community detection algorithms to identify groups of strongly connected vertices (called a community) and, consequently, identify time series clusters. Still in this paper, we make a comprehensive analysis on the influence of various combinations of time series distance functions, network generation methods and community detection techniques on clustering results. Experimental study shows that the proposed network-based approach achieves better results than various classic or up-to-date clustering techniques under consideration. Statistical tests confirm that the proposed method outperforms some classic clustering algorithms, such as $k$-medoids, diana, median-linkage and centroid-linkage in various data sets. Interestingly, the proposed method can effectively detect shape patterns presented in time series due to the topological structure of the underlying network constructed in the clustering process. At the same time, other techniques fail to identify such patterns. Moreover, the proposed method is robust enough to group time series presenting similar pattern but with time shifts and/or amplitude variations. In summary, the main point of the proposed method is the transformation of time series from time-space domain to topological domain. Therefore, we hope that our approach contributes not only for time series clustering, but also for general time series analysis tasks.
We analyze a variant of multivariate singular spectrum analysis (mSSA), a widely used multivariate time series method, which we find to perform competitively with respect to the state-of-art neural network time series methods (LSTM, DeepAR). Its restriction for single time series, singular spectrum analysis (SSA), has been analyzed recently. Despite its popularity, theoretical understanding of mSSA is absent. Towards this, we introduce a natural spatio-temporal factor model to analyze mSSA. We establish the in-sample prediction error for imputation and forecasting under mSSA scales as $1/\sqrt{NT}$, for $N$ time series with $T$ observations per time series. In contrast, for SSA the error scales as $1/\sqrt{T}$ and for matrix factorization based time series methods, the error scales as ${1}/{\min(N, T)}$. We utilize an online learning framework to analyze the one-step-ahead prediction error of mSSA and establish it has a regret of ${1}/{(\sqrt{N}T^{0.04})}$ with respect to in-sample forecasting error. By applying mSSA on the square of the time series observations, we furnish an algorithm to estimate the time-varying variance of a time series and establish it has in-sample imputation / forecasting error scaling as $1/\sqrt{NT}$. To establish our results, we make three technical contributions. First, we establish that the "stacked" Page Matrix time series representation, the core data structure in mSSA, has an approximate low-rank structure for a large class of time series models used in practice under the spatio-temporal factor model. Second, we extend the theory of online convex optimization to address the variant when the constraints are time-varying. Third, we extend the analysis prediction error analysis of Principle Component Regression beyond recent work to when the covariate matrix is approximately low-rank.
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