We propose a framework, named Aggregated Wasserstein, for computing a dissimilarity measure or distance between two Hidden Markov Models with state conditional distributions being Gaussian. For such HMMs, the marginal distribution at any time position follows a Gaussian mixture distribution, a fact exploited to softly match, aka register, the states in two HMMs. We refer to such HMMs as Gaussian mixture model-HMM (GMM-HMM). The registration of states is inspired by the intrinsic relationship of optimal transport and the Wasserstein metric between distributions. Specifically, the components of the marginal GMMs are matched by solving an optimal transport problem where the cost between components is the Wasserstein metric for Gaussian distributions. The solution of the optimization problem is a fast approximation to the Wasserstein metric between two GMMs. The new Aggregated Wasserstein distance is a semi-metric and can be computed without generating Monte Carlo samples. It is invariant to relabeling or permutation of states. The distance is defined meaningfully even for two HMMs that are estimated from data of different dimensionality, a situation that can arise due to missing variables. This distance quantifies the dissimilarity of GMM-HMMs by measuring both the difference between the two marginal GMMs and that between the two transition matrices. Our new distance is tested on tasks of retrieval, classification, and t-SNE visualization of time series. Experiments on both synthetic and real data have demonstrated its advantages in terms of accuracy as well as efficiency in comparison with existing distances based on the Kullback-Leibler divergence. Index Terms-HiddenMarkov Model, Gaussian Mixture Model, Wasserstein Distance, Optimal Transport✦
We propose a framework, named Aggregated Wasserstein, for computing a dissimilarity measure or distance between two Hidden Markov Models with state conditional distributions being Gaussian. For such HMMs, the marginal distribution at any time spot follows a Gaussian mixture distribution, a fact exploited to softly match, aka register, the states in two HMMs. We refer to such HMMs as Gaussian mixture model-HMM (GMM-HMM). The registration of states is inspired by the intrinsic relationship of optimal transport and the Wasserstein metric between distributions. Specifically, the components of the marginal GMMs are matched by solving an optimal transport problem where the cost between components is the Wasserstein metric for Gaussian distributions. The solution of the optimization problem is a fast approximation to the Wasserstein metric between two GMMs. The new Aggregated Wasserstein distance is a semi-metric and can be computed without generating Monte Carlo samples. It is invariant to relabeling or permutation of the states. This distance quantifies the dissimilarity of GMM-HMMs by measuring both the di↵erence between the two marginal GMMs and the difference between the two transition matrices. Our new distance is tested on the tasks of retrieval and classification of time series. Experiments on both synthetic data and real data have demonstrated its advantages in terms of accuracy as well as e ciency in comparison with existing distances based on the Kullback-Leibler divergence.
Meteorologists use shapes and movements of clouds in satellite images as indicators of several major types of severe storms. Yet, because satellite image data are in increasingly higher resolution, both spatially and temporally, meteorologists cannot fully leverage the data in their forecasts. Automatic satellite image analysis methods that can find storm-related cloud patterns are thus in demand. We propose a machine-learning and pattern-recognition-based approach to detect "comma-shaped" clouds in satellite images, which are specific cloud distribution patterns strongly associated with cyclone formulation. In order to detect regions with the targeted movement patterns, we use manually annotated cloud examples represented by both shape and motion-sensitive features to train the computer to analyze satellite images. Sliding windows in different scales ensure the capture of dense clouds, and we implement effective selection rules to shrink the region of interest among these sliding windows. Finally, we evaluate the method on a hold-out annotated commashaped cloud dataset and cross-match the results with recorded storm events in the severe weather database. The validated utility and accuracy of our method suggest a high potential for assisting meteorologists in weather forecasting.
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