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Register a new userTrainable Time Warping: Aligning Time-Series in the Continuous-Time Domain
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Added by
Soheil Khorram
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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DTW calculates the similarity or alignment between two signals, subject to temporal warping. However, its computational complexity grows exponentially with the number of time-series. Although there have been algorithms developed that are linear in the number of time-series, they are generally quadratic in time-series length. The exception is generalized time warping (GTW), which has linear computational cost. Yet, it can only identify simple time warping functions. There is a need for a new fast, high-quality multisequence alignment algorithm. We introduce trainable time warping (TTW), whose complexity is linear in both the number and the length of time-series. TTW performs alignment in the continuous-time domain using a sinc convolutional kernel and a gradient-based optimization technique. We compare TTW and GTW on 85 UCR datasets in time-series averaging and classification. TTW outperforms GTW on 67.1% of the datasets for the averaging tasks, and 61.2% of the datasets for the classification tasks.
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Time series data analytics has been a problem of substantial interests for decades, and Dynamic Time Warping (DTW) has been the most widely adopted technique to measure dissimilarity between time series. A number of global-alignment kernels have since been proposed in the spirit of DTW to extend its use to kernel-based estimation method such as support vector machine. However, those kernels suffer from diagonal dominance of the Gram matrix and a quadratic complexity w.r.t. the sample size. In this work, we study a family of alignment-aware positive definite (p.d.) kernels, with its feature embedding given by a distribution of emph{Random Warping Series (RWS)}. The proposed kernel does not suffer from the issue of diagonal dominance while naturally enjoys a emph{Random Features} (RF) approximation, which reduces the computational complexity of existing DTW-based techniques from quadratic to linear in terms of both the number and the length of time-series. We also study the convergence of the RF approximation for the domain of time series of unbounded length. Our extensive experiments on 16 benchmark datasets demonstrate that RWS outperforms or matches state-of-the-art classification and clustering methods in both accuracy and computational time. Our code and data is available at { url{https://github.com/IBM/RandomWarpingSeries}}.
During the last decades there is a continuing international endeavor in developing realistic space weather prediction tools aiming to forecast the conditions on the Sun and in the interplanetary environment. These efforts have led to the need of developing appropriate metrics in order to assess the performance of those tools. Metrics are necessary for validating models, comparing different models and monitoring adjustments or improvements of a certain model over time. In this work, we introduce the Dynamic Time Warping (DTW) as an alternative way to validate models and, in particular, to quantify differences between observed and synthetic (modeled) time series for space weather purposes. We present the advantages and drawbacks of this method as well as applications on WIND observations and EUHFORIA modeled output at L1. We show that DTW is a useful tool that permits the evaluation of both the fast and slow solar wind. Its distinctive characteristic is that it warps sequences in time, aiming to align them with the minimum cost by using dynamic programming. It can be applied in two different ways for the evaluation of modeled solar wind time series. The first way calculates the so-called sequence similarity factor (SSF), a number that provides a quantification of how good the forecast is, compared to a best and a worst case prediction scenarios. The second way quantifies the time and amplitude differences between the points that are best matched between the two sequences. As a result, it can serve as a hybrid metric between continuous measurements (such as, e.g., the correlation coefficient) and point-by-point comparisons. We conclude that DTW is a promising technique for the assessment of solar wind profiles offering functions that other metrics do not, so that it can give at once the most complete evaluation profile of a model.
Time-series representation learning is a fundamental task for time-series analysis. While significant progress has been made to achieve accurate representations for downstream applications, the learned representations often lack interpretability and do not expose semantic meanings. Different from previous efforts on the entangled feature space, we aim to extract the semantic-rich temporal correlations in the latent interpretable factorized representation of the data. Motivated by the success of disentangled representation learning in computer vision, we study the possibility of learning semantic-rich time-series representations, which remains unexplored due to three main challenges: 1) sequential data structure introduces complex temporal correlations and makes the latent representations hard to interpret, 2) sequential models suffer from KL vanishing problem, and 3) interpretable semantic concepts for time-series often rely on multiple factors instead of individuals. To bridge the gap, we propose Disentangle Time Series (DTS), a novel disentanglement enhancement framework for sequential data. Specifically, to generate hierarchical semantic concepts as the interpretable and disentangled representation of time-series, DTS introduces multi-level disentanglement strategies by covering both individual latent factors and group semantic segments. We further theoretically show how to alleviate the KL vanishing problem: DTS introduces a mutual information maximization term, while preserving a heavier penalty on the total correlation and the dimension-wise KL to keep the disentanglement property. Experimental results on various real-world benchmark datasets demonstrate that the representations learned by DTS achieve superior performance in downstream applications, with high interpretability of semantic concepts.
We consider a continuous-time multi-arm bandit problem (CTMAB), where the learner can sample arms any number of times in a given interval and obtain a random reward from each sample, however, increasing the frequency of sampling incurs an additive penalty/cost. Thus, there is a tradeoff between obtaining large reward and incurring sampling cost as a function of the sampling frequency. The goal is to design a learning algorithm that minimizes regret, that is defined as the difference of the payoff of the oracle policy and that of the learning algorithm. CTMAB is fundamentally different than the usual multi-arm bandit problem (MAB), e.g., even the single-arm case is non-trivial in CTMAB, since the optimal sampling frequency depends on the mean of the arm, which needs to be estimated. We first establish lower bounds on the regret achievable with any algorithm and then propose algorithms that achieve the lower bound up to logarithmic factors. For the single-arm case, we show that the lower bound on the regret is $Omega((log T)^2/mu)$, where $mu$ is the mean of the arm, and $T$ is the time horizon. For the multiple arms case, we show that the lower bound on the regret is $Omega((log T)^2 mu/Delta^2)$, where $mu$ now represents the mean of the best arm, and $Delta$ is the difference of the mean of the best and the second-best arm. We then propose an algorithm that achieves the bound up to constant terms.
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.
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