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We address two shortcomings in online travel time estimation methods for congested urban traffic. The first shortcoming is related to the determination of the number of mixture modes, which can change dynamically, within day and from day to day. The second shortcoming is the wide-spread use of Gaussian probability densities as mixture components. Gaussian densities fail to capture the positive skew in travel time distributions and, consequently, large numbers of mixture components are needed for reasonable fitting accuracy when applied as mixture components. They also assign positive probabilities to negative travel times. To address these issues, this paper derives a mixture distribution with Gamma component densities, which are asymmetric and supported on the positive numbers. We use sparse estimation techniques to ensure parsimonious models and propose a generalization of Gamma mixture densities using Mittag-Leffler functions, which provides enhanced fitting flexibility and improved parsimony. In order to accommodate within-day variability and allow for online implementation of the proposed methodology (i.e., fast computations on streaming travel time data), we introduce a recursive algorithm which efficiently updates the fitted distribution whenever new data become available. Experimental results using real-world travel time data illustrate the efficacy of the proposed methods.
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This note considers softmax parameter estimation when little/no labeled training data is available, but a priori information about the relative geometry of class label log-odds boundaries is available. It is shown that `data-free softmax model synthesis corresponds to solving a linear system of parameter equations, wherein desired dominant class log-odds boundaries are encoded via convex polytopes that decompose the input feature space. When solvable, the linear equations yield closed-form softmax parameter solution families using class boundary polytope specifications only. This allows softmax parameter learning to be implemented without expensive brute force data sampling and numerical optimization. The linear equations can also be adapted to constrained maximum likelihood estimation in data-sparse settings. Since solutions may also fail to exist for the linear parameter equations derived from certain polytope specifications, it is thus also shown that there exist probabilistic classification problems over m convexly separable classes for which the log-odds boundaries cannot be learned using an m-class softmax model.
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Estimating the travel time of any route is of great importance for trip planners, traffic operators, online taxi dispatching and ride-sharing platforms, and navigation provider systems. With the advance of technology, many traveling cars, including online taxi dispatch systems vehicles are equipped with Global Positioning System (GPS) devices that can report the location of the vehicle every few seconds. This paper uses GPS data and the Matrix Factorization techniques to estimate the travel times on all road segments and time intervals simultaneously. We aggregate GPS data into a matrix, where each cell of the original matrix contains the average vehicle speed for a segment and a specific time interval. One of the problems with this matrix is its high sparsity. We use Alternating Least Squares (ALS) method along with a regularization term to factorize the matrix. Since this approach can solve the sparsity problem that arises from the absence of cars in many road segments in a specific time interval, matrix factorization is suitable for estimating the travel time. Our comprehensive evaluation results using real data provided by one of the largest online taxi dispatching systems in Iran, shows the strength of our proposed method.
Continuous-time Bayesian Networks (CTBNs) represent a compact yet powerful framework for understanding multivariate time-series data. Given complete data, parameters and structure can be estimated efficiently in closed-form. However, if data is incomplete, the latent states of the CTBN have to be estimated by laboriously simulating the intractable dynamics of the assumed CTBN. This is a problem, especially for structure learning tasks, where this has to be done for each element of a super-exponentially growing set of possible structures. In order to circumvent this notorious bottleneck, we develop a novel gradient-based approach to structure learning. Instead of sampling and scoring all possible structures individually, we assume the generator of the CTBN to be composed as a mixture of generators stemming from different structures. In this framework, structure learning can be performed via a gradient-based optimization of mixture weights. We combine this approach with a new variational method that allows for a closed-form calculation of this mixture marginal likelihood. We show the scalability of our method by learning structures of previously inaccessible sizes from synthetic and real-world data.
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