Victor Solo

Despite the widespread occurrence of classification problems and the increasing collection of point process data across many disciplines, study of error probability for point process classification only emerged very recently. Here, we consider classification of renewal processes. We obtain asymptotic expressions for the Bhattacharyya bound on misclassification error probabilities for heavy-tailed renewal processes.

In this paper, we study the numerical simulation of stochastic differential equations (SDEs) on the special orthogonal Lie group . We propose a geometry-preserving numerical scheme based on the stochastic tangent space parametrization (S-TaSP) method for state-dependent multiplicative SDEs on . The convergence analysis of the S-TaSP scheme establishes a strong convergence order of (δ1-2), which matches the convergence order of the previous stochastic Lie Euler-Maruyama scheme while avoiding the computational cost of the exponential map. Numerical simulations illustrate the theoretical results.

Clustering of time series based on their underlying dynamics is keeping attracting researchers due to its impacts on assisting complex system modelling. Most current time series clustering methods handle only scalar time series, treat them as white noise, or rely on domain knowledge for high-quality feature construction, where the autocorrelation pattern/feature is mostly ignored. Instead of relying on heuristic feature/metric construction, the system identification approach allows treating vector time series clustering by explicitly considering their underlying autoregressive dynamics. We first derive a clustering algorithm based on a mixture autoregressive model. Unfortunately it turns out to have significant computational problems. We then derive a `small-noise' limiting version of the algorithm, which we call k-LMVAR (Limiting Mixture Vector AutoRegression), that is computationally manageable. We develop an associated BIC criterion for choosing the number of clusters and model order. The algorithm performs very well in comparative simulations and also scales well computationally.

Point processes are finding growing applications in numerous fields, such as neuroscience, high frequency finance and social media. So classic problems of classification and clustering are of increasing interest. However, analytic study of misclassification error probability in multi-class classification has barely begun. In this paper, we tackle the multi-class likelihood classification problem for point processes and develop, for the first time, both asymptotic upper and lower bounds on the error rate in terms of pair-wise affinities. We apply these general results to classifying renewal processes. Under some technical conditions, we show that the bounds have exponential decay and give explicit associated constants. The results are illustrated with non-trivial simulations, where we demonstrate the practical usage of our results and show their computational efficiency.

In this paper, we develop a novel method for deriving a global optimal control strategy for stochastic attitude kinematics on the special orthogonal group SO(3). We first introduce a stochastic Lie-Hamilton-Jacobi-Bellman (SL-HJB) equation on SO(3), which theoretically provides an optimality condition for the global optimal control strategy of the stochastic attitude kinematics. Then we propose a novel numerical method, the Successive Wigner-Galerkin Approximation (SWGA) method, to solve the SL-HJB equation on SO(3). The SWGA method leverages the Wigner-D functions to represent the Galerkin solution of the SL-HJB equation in a policy iteration framework, providing a computationally efficient approach to derive a global optimal control strategy for systems on SO(3). We demonstrate the effectiveness of the SWGA method through numerical simulation on stochastic attitude stabilization.

We develop hard clustering based on likelihood rather than distance and prove convergence. We also provide simulations and real data examples.

Subspace tracking has a long and still growing list of applications with recent interest stimulated by the advent of ’streaming’ data. Despite the considerable literature, all of it motivated by the desire to track time varying subspaces, there seems to be almost no analysis of that time varying problem. Rather existing analyses assume the subspace is fixed. Here, for the first time we develop a true subspace tracking analysis in continuous time. It turns out to be a very hard problem and so we tackle the problem of a rank 1 subspace.