Time series alignment methods call for highly expressive, differentiable and invertible warping functions which preserve temporal topology, i.e diffeomorphisms. Diffeomorphic warping functions can be generated from the integration of velocity fields governed by an ordinary differential equation (ODE). Gradient-based optimization frameworks containing diffeomorphic transformations require to calculate derivatives to the differential equation's solution with respect to the model parameters, i.e. sensitivity analysis. Unfortunately, deep learning frameworks typically lack automatic-differentiation-compatible sensitivity analysis methods; and implicit functions, such as the solution of ODE, require particular care. Current solutions appeal to adjoint sensitivity methods, ad-hoc numerical solvers or ResNet's Eulerian discretization. In this work, we present a closed-form expression for the ODE solution and its gradient under continuous piecewise-affine (CPA) velocity functions. We present a highly optimized implementation of the results on CPU and GPU. Furthermore, we conduct extensive experiments on several datasets to validate the generalization ability of our model to unseen data for time-series joint alignment. Results show significant improvements both in terms of efficiency and accuracy.
Aggregate time-series data like traffic flow and site occupancy repeatedly sample statistics from a population across time. Such data can be profoundly useful for understanding trends within a given population, but also pose a significant privacy risk, potentially revealing e.g., who spends time where. Producing a private version of a time-series satisfying the standard definition of Differential Privacy (DP) is challenging due to the large influence a single participant can have on the sequence: if an individual can contribute to each time step, the amount of additive noise needed to satisfy privacy increases linearly with the number of time steps sampled. As such, if a signal spans a long duration or is oversampled, an excessive amount of noise must be added, drowning out underlying trends. However, in many applications an individual realistically cannot participate at every time step. When this is the case, we observe that the influence of a single participant (sensitivity) can be reduced by subsampling and/or filtering in time, while still meeting privacy requirements. Using a novel analysis, we show this significant reduction in sensitivity and propose a corresponding class of privacy mechanisms. We demonstrate the utility benefits of these techniques empirically with real-world and synthetic time-series data.
Shrinkage algorithms are of great importance in almost every area of statistics due to the increasing impact of big data. Especially time series analysis benefits from efficient and rapid estimation techniques such as the lasso. However, currently lasso type estimators for autoregressive time series models still focus on models with homoscedastic residuals. Therefore, an iteratively reweighted adaptive lasso algorithm for the estimation of time series models under conditional heteroscedasticity is presented in a high-dimensional setting. The asymptotic behaviour of the resulting estimator is analysed. It is found that the proposed estimation procedure performs substantially better than its homoscedastic counterpart. A special case of the algorithm is suitable to compute the estimated multivariate AR-ARCH type models efficiently. Extensions to the model like periodic AR-ARCH, threshold AR-ARCH or ARMA-GARCH are discussed. Finally, different simulation results and applications to electricity market data and returns of metal prices are shown.
The recent increase in the scale and complexity of software systems has introduced new challenges to the time series monitoring and anomaly detection process. A major drawback of existing anomaly detection methods is that they lack contextual information to help stakeholders identify the cause of anomalies. This problem, known as root cause detection, is particularly challenging to undertake in today's complex distributed software systems since the metrics under consideration generally have multiple internal and external dependencies. Significant manual analysis and strong domain expertise is required to isolate the correct cause of the problem. In this paper, we propose a method that isolates the root cause of an anomaly by analyzing the patterns in time series fluctuations. Our method considers the time series as observations from an underlying process passing through a sequence of discretized hidden states. The idea is to track the propagation of the effect when a given problem causes unaligned but homogeneous shifts of the underlying states. We evaluate our approach by finding the root cause of anomalies in Zillows clickstream data by identifying causal patterns among a set of observed fluctuations.
Ice accumulation in the blades of wind turbines can cause them to describe anomalous rotations or no rotations at all, thus affecting the generation of electricity and power output. In this work, we investigate the problem of ice accumulation in wind turbines by framing it as anomaly detection of multi-variate time series. Our approach focuses on two main parts: first, learning low-dimensional representations of time series using a Variational Recurrent Autoencoder (VRAE), and second, using unsupervised clustering algorithms to classify the learned representations as normal (no ice accumulated) or abnormal (ice accumulated). We have evaluated our approach on a custom wind turbine time series dataset, for the two-classes problem (one normal versus one abnormal class), we obtained a classification accuracy of up to 96$\%$ on test data. For the multiple-class problem (one normal versus multiple abnormal classes), we present a qualitative analysis of the low-dimensional learned latent space, providing insights into the capacities of our approach to tackle such problem. The code to reproduce this work can be found here https://github.com/agrija9/Wind-Turbines-VRAE-Paper.
We propose a new computational approach for tracking and detecting statistically significant linguistic shifts in the meaning and usage of words. Such linguistic shifts are especially prevalent on the Internet, where the rapid exchange of ideas can quickly change a word's meaning. Our meta-analysis approach constructs property time series of word usage, and then uses statistically sound change point detection algorithms to identify significant linguistic shifts. We consider and analyze three approaches of increasing complexity to generate such linguistic property time series, the culmination of which uses distributional characteristics inferred from word co-occurrences. Using recently proposed deep neural language models, we first train vector representations of words for each time period. Second, we warp the vector spaces into one unified coordinate system. Finally, we construct a distance-based distributional time series for each word to track it's linguistic displacement over time. We demonstrate that our approach is scalable by tracking linguistic change across years of micro-blogging using Twitter, a decade of product reviews using a corpus of movie reviews from Amazon, and a century of written books using the Google Book-ngrams. Our analysis reveals interesting patterns of language usage change commensurate with each medium.
An advanced conceptual validation framework for multimodal multivariate time series defines a multi-level contextual anomaly detection ranging from an univariate context definition, to a multimodal abstract context representation learnt by an Autoencoder from heterogeneous data (images, time series, sounds, etc.) associated to an industrial process. Each level of the framework is either applicable to historical data and/or live data. The ultimate level is based on causal discovery to identify causal relations in observational data in order to exclude biased data to train machine learning models and provide means to the domain expert to discover unknown causal relations in the underlying process represented by the data sample. A Long Short-Term Memory Autoencoder is successfully evaluated on multivariate time series to validate the learnt representation of abstract contexts associated to multiple assets of a blast furnace. A research roadmap is identified to combine causal discovery and representation learning as an enabler for unsupervised Root Cause Analysis applied to the process industry.
This paper proposes a new method for determining similarity and anomalies between time series, most practically effective in large collections of (likely related) time series, with a particular focus on measuring distances between structural breaks within such a collection. We consolidate and generalise a class of semi-metric distance measures, which we term MJ distances. Experiments on simulated data demonstrate that our proposed family of distances uncover similarity within collections of time series more effectively than measures such as the Hausdorff and Wasserstein metrics. Although our class of distances do not necessarily satisfy the triangle inequality requirement of a metric, we analyse the transitivity properties of respective distance matrices in various contextual scenarios. There, we demonstrate a trade-off between robust performance in the presence of outliers, and the triangle inequality property. We show in experiments using real data that the contrived scenarios that severely violate the transitivity property rarely exhibit themselves in real data; instead, our family of measures satisfies all the properties of a metric most of the time. We illustrate three ways of analysing the distance and similarity matrices, via eigenvalue analysis, hierarchical clustering, and spectral clustering. The results from our hierarchical and spectral clustering experiments on simulated data demonstrate that the Hausdorff and Wasserstein metrics may lead to erroneous inference as to which time series are most similar with respect to their structural breaks, while our semi-metrics provide an improvement.
Accurate measurement of blast-induced rock fragmentation is of great importance for many mining operations. The post-blast rock size distribution can significantly influence the efficiency of all the downstream mining and comminution processes. Image analysis methods are one of the most common methods used to measure rock fragment size distribution in mines regardless of criticism for lack of accuracy to measure fine particles and other perceived deficiencies. The current practice of collecting rock fragmentation data for image analysis is highly manual and provides data with low temporal and spatial resolution. Using UAVs for collecting images of rock fragments can not only improve the quality of the image data but also automate the data collection process. Ultimately, real-time acquisition of high temporal- and spatial-resolution data based on UAV technology will provide a broad range of opportunities for both improving blast design without interrupting the production process and reducing the cost of the human operator. This paper presents the results of a series of laboratory-scale rock fragment measurements using a quadrotor UAV equipped with a camera. The goal of this work is to highlight the benefits of aerial fragmentation analysis in terms of both prediction accuracy and time effort. A pile of rock fragments with different fragment sizes was placed in a lab that is equipped with a motion capture camera system for precise UAV localization and control. Such an environment presents optimal conditions for UAV flight and thus, is well-suited for conducting proof-of-concept experiments before testing them in large-scale field experiments. The pile was photographed by a camera attached to the UAV, and the particle size distribution curves were generated in almost real-time. The pile was also manually photographed and the results of the manual method were compared to the UAV method.
Seasonality is a distinctive characteristic which is often observed in many practical time series. Artificial Neural Networks (ANNs) are a class of promising models for efficiently recognizing and forecasting seasonal patterns. In this paper, the Particle Swarm Optimization (PSO) approach is used to enhance the forecasting strengths of feedforward ANN (FANN) as well as Elman ANN (EANN) models for seasonal data. Three widely popular versions of the basic PSO algorithm, viz. Trelea-I, Trelea-II and Clerc-Type1 are considered here. The empirical analysis is conducted on three real-world seasonal time series. Results clearly show that each version of the PSO algorithm achieves notably better forecasting accuracies than the standard Backpropagation (BP) training method for both FANN and EANN models. The neural network forecasting results are also compared with those from the three traditional statistical models, viz. Seasonal Autoregressive Integrated Moving Average (SARIMA), Holt-Winters (HW) and Support Vector Machine (SVM). The comparison demonstrates that both PSO and BP based neural networks outperform SARIMA, HW and SVM models for all three time series datasets. The forecasting performances of ANNs are further improved through combining the outputs from the three PSO based models.