Path signatures are powerful nonparametric tools for time series analysis, shown to form a universal and characteristic feature map for Euclidean valued time series data. We lift the theory of path signatures to the setting of Lie group valued time series, adapting these tools for time series with underlying geometric constraints. We prove that this generalized path signature is universal and characteristic. To demonstrate universality, we analyze the human action recognition problem in computer vision, using $SO(3)$ representations for the time series, providing comparable performance to other shallow learning approaches, while offering an easily interpretable feature set. We also provide a two-sample hypothesis test for Lie group-valued random walks to illustrate its characteristic property. Finally we provide algorithms and a Julia implementation of these methods.
Time series has wide applications in the real world and is known to be difficult to forecast. Since its statistical properties change over time, its distribution also changes temporally, which will cause severe distribution shift problem to existing methods. However, it remains unexplored to model the time series in the distribution perspective. In this paper, we term this as Temporal Covariate Shift (TCS). This paper proposes Adaptive RNNs (AdaRNN) to tackle the TCS problem by building an adaptive model that generalizes well on the unseen test data. AdaRNN is sequentially composed of two novel algorithms. First, we propose Temporal Distribution Characterization to better characterize the distribution information in the TS. Second, we propose Temporal Distribution Matching to reduce the distribution mismatch in TS to learn the adaptive TS model. AdaRNN is a general framework with flexible distribution distances integrated. Experiments on human activity recognition, air quality prediction, and financial analysis show that AdaRNN outperforms the latest methods by a classification accuracy of 2.6% and significantly reduces the RMSE by 9.0%. We also show that the temporal distribution matching algorithm can be extended in Transformer structure to boost its performance.
Inferring dynamics from time series is an important objective in data analysis. In particular, it is challenging to infer stochastic dynamics given incomplete data. We propose an expectation maximization (EM) algorithm that iterates between alternating two steps: E-step restores missing data points, while M-step infers an underlying network model of restored data. Using synthetic data generated by a kinetic Ising model, we confirm that the algorithm works for restoring missing data points as well as inferring the underlying model. At the initial iteration of the EM algorithm, the model inference shows better model-data consistency with observed data points than with missing data points. As we keep iterating, however, missing data points show better model-data consistency. We find that demanding equal consistency of observed and missing data points provides an effective stopping criterion for the iteration to prevent overshooting the most accurate model inference. Armed with this EM algorithm with this stopping criterion, we infer missing data points and an underlying network from a time-series data of real neuronal activities. Our method recovers collective properties of neuronal activities, such as time correlations and firing statistics, which have previously never been optimized to fit.
Time series forecasting has gained lots of attention recently; this is because many real-world phenomena can be modeled as time series. The massive volume of data and recent advancements in the processing power of the computers enable researchers to develop more sophisticated machine learning algorithms such as neural networks to forecast the time series data. In this paper, we propose various neural network architectures to forecast the time series data using the dynamic measurements; moreover, we introduce various architectures on how to combine static and dynamic measurements for forecasting. We also investigate the importance of performing techniques such as anomaly detection and clustering on forecasting accuracy. Our results indicate that clustering can improve the overall prediction time as well as improve the forecasting performance of the neural network. Furthermore, we show that feature-based clustering can outperform the distance-based clustering in terms of speed and efficiency. Finally, our results indicate that adding more predictors to forecast the target variable will not necessarily improve the forecasting accuracy.
In time series analysis, when fitting an autoregressive model, one must solve a Toeplitz ordinary least squares problem numerous times to find an appropriate model, which can severely affect computational times with large data sets. Two recent algorithms (LSAR and Repeated Halving) have applied randomized numerical linear algebra (RandNLA) techniques to fitting an autoregressive model to big time-series data. We investigate and compare the quality of these two approximation algorithms on large-scale synthetic and real-world data. While both algorithms display comparable results for synthetic datasets, the LSAR algorithm appears to be more robust when applied to real-world time series data. We conclude that RandNLA is effective in the context of big-data time series.
During the fabrication of casting parts sensor data is typically automatically recorded and accumulated for process monitoring and defect diagnosis. As casting is a thermal process with many interacting process parameters, root cause analysis tends to be tedious and ineffective. We show how a decomposition based on non-negative matrix factorization (NMF), which is guided by a knowledge-based initialization strategy, is able to extract physical meaningful sources from temperature time series collected during a thermal manufacturing process. The approach assumes the time series to be generated by a superposition of several simultaneously acting component processes. NMF is able to reverse the superposition and to identify the hidden component processes. The latter can be linked to ongoing physical phenomena and process variables, which cannot be monitored directly. Our approach provides new insights into the underlying physics and offers a tool, which can assist in diagnosing defect causes. We demonstrate our method by applying it to real world data, collected in a foundry during the series production of casting parts for the automobile industry.
tl;dr: no, it cannot, at least not on average on the standard archive problems. We assess whether using six smoothing algorithms (moving average, exponential smoothing, Gaussian filter, Savitzky-Golay filter, Fourier approximation and a recursive median sieve) could be automatically applied to time series classification problems as a preprocessing step to improve the performance of three benchmark classifiers (1-Nearest Neighbour with Euclidean and Dynamic Time Warping distances, and Rotation Forest). We found no significant improvement over unsmoothed data even when we set the smoothing parameter through cross validation. We are not claiming smoothing has no worth. It has an important role in exploratory analysis and helps with specific classification problems where domain knowledge can be exploited. What we observe is that the automatic application does not help and that we cannot explain the improvement of other time series classification algorithms over the baseline classifiers simply as a function of the absence of smoothing.
Human behavior modeling deals with learning and understanding behavior patterns inherent in humans' daily routines. Existing pattern mining techniques either assume human dynamics is strictly periodic, or require the number of modes as input, or do not consider uncertainty in the sensor data. To handle these issues, in this paper, we propose a novel clustering approach for modeling human behavior (named, MTpattern) from time-series data. For mining frequent human behavior patterns effectively, we utilize a three-stage pipeline: (1) represent time series data into a sequence of regularly sampled equal-sized unit time intervals for better analysis, (2) a new distance measure scheme is proposed to cluster similar sequences which can handle temporal variation and uncertainty in the data, and (3) exploit an exemplar-based clustering mechanism and fine-tune its parameters to output minimum number of clusters with given permissible distance constraints and without knowing the number of modes present in the data. Then, the average of all sequences in a cluster is considered as a human behavior pattern. Empirical studies on two real-world datasets and a simulated dataset demonstrate the effectiveness of MTpattern with respect to internal and external measures of clustering.
Self-supervision may boost model performance in downstream tasks. However, there is no principled way of selecting the self-supervised objectives that yield the most adaptable models. Here, we study this problem on daily time-series data generated from wearable sensors used to detect onset of influenza-like illness (ILI). We first show that using self-supervised learning to predict next-day time-series values allows us to learn rich representations which can be adapted to perform accurate ILI prediction. Second, we perform an empirical analysis of three different self-supervised objectives to assess their adaptability to ILI prediction. Our results show that predicting the next day's resting heart rate or time-in-bed during sleep provides better representations for ILI prediction. These findings add to previous work demonstrating the practical application of self-supervised learning from activity data to improve health predictions.
A large fraction of the electronic health records consists of clinical measurements collected over time, such as blood tests, which provide important information about the health status of a patient. These sequences of clinical measurements are naturally represented as time series, characterized by multiple variables and the presence of missing data, which complicate analysis. In this work, we propose a surgical site infection detection framework for patients undergoing colorectal cancer surgery that is completely unsupervised, hence alleviating the problem of getting access to labelled training data. The framework is based on powerful kernels for multivariate time series that account for missing data when computing similarities. Our approach show superior performance compared to baselines that have to resort to imputation techniques and performs comparable to a supervised classification baseline.