Victor Solo

Xinhui Rong, 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.

Xi Wang, Victor Solo

In the paper, we propose a higher-order geometry-preserving numerical method for stochastic differential equations (SDEs) evolving on the Lie groups SO(n) and SE(n). Most existing Lie group integrators rely on Magnus expansion of the exponential map, which makes the construction of higher-order stochastic schemes difficult. To overcome this limitation, we develop a tangent-space parameterization corrected Milstein method (TaSP-CM), extending the tangent space parameterization (TaSP) framework from Lie-group ODEs to the stochastic setting. Although TaSP is a well-established method for Lie ODEs, the extension to SDEs is non-trivial and requires new stochastic corrections that ensure both geometric consistency and higher-order accuracy. We prove that the proposed scheme achieves strong convergence of order 1 under both commutative and non-commutative noise. Numerical experiments illustrate the theoretical results and demonstrate the efficiency and robustness of the proposed method.

Zuogong Yue, Victor Solo

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

Zuogong Yue, Xinyi Wang, Victor Solo

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.

Victor Solo

The Pearson distance between a pair of random variables $X,Y$ with correlation $ρ_{xy}$, namely, 1-$ρ_{xy}$, has gained widespread use, particularly for clustering, in areas such as gene expression analysis, brain imaging and cyber security. In all these applications it is implicitly assumed/required that the distance measures be metrics, thus satisfying the triangle inequality. We show however, that Pearson distance is not a metric. We go on to show that this can be repaired by recalling the result, (well known in other literature) that $\sqrt{1-ρ_{xy}}$ is a metric. We similarly show that a related measure of interest, $1-|ρ_{xy}|$, which is invariant to the sign of $ρ_{xy}$, is not a metric but that $\sqrt{1-ρ_{xy}^2}$ is. We also give generalizations of these results.