Kevin C. Cheng

Kevin C. Cheng, Shuchin Aeron, Michael C. Hughes et al.

We consider probabilistic models for sequential observations which exhibit gradual transitions among a finite number of states. We are particularly motivated by applications such as human activity analysis where observed accelerometer time series contains segments representing distinct activities, which we call pure states, as well as periods characterized by continuous transition among these pure states. To capture this transitory behavior, the dynamical Wasserstein barycenter (DWB) model of Cheng et al. in 2021 [1] associates with each pure state a data-generating distribution and models the continuous transitions among these states as a Wasserstein barycenter of these distributions with dynamically evolving weights. Focusing on the univariate case where Wasserstein distances and barycenters can be computed in closed form, we extend [1] specifically relaxing the parameterization of the pure states as Gaussian distributions. We highlight issues related to the uniqueness in identifying the model parameters as well as uncertainties induced when estimating a dynamically evolving distribution from a limited number of samples. To ameliorate non-uniqueness, we introduce regularization that imposes temporal smoothness on the dynamics of the barycentric weights. A quantile-based approximation of the pure state distributions yields a finite dimensional estimation problem which we numerically solve using cyclic descent alternating between updates to the pure-state quantile functions and the barycentric weights. We demonstrate the utility of the proposed algorithm in segmenting both simulated and real world human activity time series.

Kevin C. Cheng, Shuchin Aeron, Michael C. Hughes et al.

Many time series can be modeled as a sequence of segments representing high-level discrete states, such as running and walking in a human activity application. Flexible models should describe the system state and observations in stationary "pure-state" periods as well as transition periods between adjacent segments, such as a gradual slowdown between running and walking. However, most prior work assumes instantaneous transitions between pure discrete states. We propose a dynamical Wasserstein barycentric (DWB) model that estimates the system state over time as well as the data-generating distributions of pure states in an unsupervised manner. Our model assumes each pure state generates data from a multivariate normal distribution, and characterizes transitions between states via displacement-interpolation specified by the Wasserstein barycenter. The system state is represented by a barycentric weight vector which evolves over time via a random walk on the simplex. Parameter learning leverages the natural Riemannian geometry of Gaussian distributions under the Wasserstein distance, which leads to improved convergence speeds. Experiments on several human activity datasets show that our proposed DWB model accurately learns the generating distribution of pure states while improving state estimation for transition periods compared to the commonly used linear interpolation mixture models.

Kevin C. Cheng, Eric L. Miller, Michael C. Hughes et al.

Non-parametric and distribution-free two-sample tests have been the foundation of many change point detection algorithms. However, randomness in the test statistic as a function of time makes them susceptible to false positives and localization ambiguity. We address these issues by deriving and applying filters matched to the expected temporal signatures of a change for various sliding window, two-sample tests under IID assumptions on the data. These filters are derived asymptotically with respect to the window size for the Wasserstein quantile test, the Wasserstein-1 distance test, Maximum Mean Discrepancy squared (MMD^2), and the Kolmogorov-Smirnov (KS) test. The matched filters are shown to have two important properties. First, they are distribution-free, and thus can be applied without prior knowledge of the underlying data distributions. Second, they are peak-preserving, which allows the filtered signal produced by our methods to maintain expected statistical significance. Through experiments on synthetic data as well as activity recognition benchmarks, we demonstrate the utility of this approach for mitigating false positives and improving the test precision. Our method allows for the localization of change points without the use of ad-hoc post-processing to remove redundant detections common to current methods. We further highlight the performance of statistical tests based on the Quantile-Quantile (Q-Q) function and show how the invariance property of the Q-Q function to order-preserving transformations allows these tests to detect change points of different scales with a single threshold within the same dataset.

Kevin C. Cheng, Shuchin Aeron, Michael C. Hughes et al.

Two common problems in time series analysis are the decomposition of the data stream into disjoint segments that are each in some sense "homogeneous" - a problem known as Change Point Detection (CPD) - and the grouping of similar nonadjacent segments, a problem that we call Time Series Segment Clustering (TSSC). Building upon recent theoretical advances characterizing the limiting distribution-free behavior of the Wasserstein two-sample test (Ramdas et al. 2015), we propose a novel algorithm for unsupervised, distribution-free CPD which is amenable to both offline and online settings. We also introduce a method to mitigate false positives in CPD and address TSSC by using the Wasserstein distance between the detected segments to build an affinity matrix to which we apply spectral clustering. Results on both synthetic and real data sets show the benefits of the approach.