This paper uses sample data to study the problem of comparing populations on finite-dimensional parallelizable Riemannian manifolds and more general trivial vector bundles. Utilizing triviality, our framework represents populations as mixtures of Gaussians on vector bundles and estimates the population parameters using a mode-based clustering algorithm. We derive a Wasserstein-type metric between Gaussian mixtures, adapted to the manifold geometry, in order to compare estimated distributions. Our contributions include an identifiability result for Gaussian mixtures on manifold domains and a convenient characterization of optimal couplings of Gaussian mixtures under the derived metric. We demonstrate these tools on some example domains, including the preshape space of planar closed curves, with applications to the shape space of triangles and populations of nanoparticles. In the nanoparticle application, we consider a sequence of populations of particle shapes arising from a manufacturing process and utilize the Wasserstein-type distance to perform change-point detection.

Many semiconductor fabrication plants (fabs) prefer simulation-based decision making for vehicle dwelling policies because it can capture a fab’s scalability and complexity. Vehicle dwelling policies assign idle vehicles to intra-bay and outer loops in automated material handling systems (AMHSs) to respond quickly to transportation demands. Fabs are motivated to control vehicle dwelling policies when fabs experience significant fluctuations, i.e., changes in product mix. Fab operators evaluate manually designed candidate solutions because it is time-intensive to run a large-scale simulation with numerous potential solutions. To determine a vehicle dwelling policy, we propose a simulation optimization approach based on Bayesian optimization (BO) with class-based clustering. BO adaptively traces efficient vehicle dwelling policies based on a surrogate model and an acquisition function. Class-based clustering alleviates the high dimensionality of the design space by grouping bays into a small number of classes. By striking a balance between the complexity of the design space and the quality of the solutions, our proposed policy significantly reduces the number of simulation runs required to determine efficient vehicle dwelling policies. We conclude that BO with class-based clustering is more advantageous than using a genetic algorithm (GA) and using heuristics.Note to Practitioners— This study is motivated by the difficulties in simulation-based decision making for optimal vehicle dwelling policies in a semiconductor wafer fab’s AMHS. While existing research has demonstrated the effectiveness of simulation analysis on operational planning in a fab, simulation optimization for a vehicle dwelling policy is still problematical due to the heavy computation burden for simulation runs and its large design space with the increased number of control variables in a fab. Therefore, we develop a simulation optimization approach using BO with class-based clustering. The proposed approach significantly reduces the number of simulation runs to obtain efficient vehicle dwelling policies, resulting in decreased delivery times and vehicle utilization rates.

This paper presents a Gaussian process (GP) model for estimating piecewise continuous regression functions. In many scientific and engineering applications of regression analysis, the underlying regression functions are often piecewise continuous in that data follow different continuous regression models for different input regions with discontinuities across regions. However, many conventional GP regression approaches are not designed for piecewise regression analysis. There are piecewise GP models to use explicit domain partitioning and pose independent GP models over partitioned regions. They are not flexible enough to model real datasets where data domains are divided by complex and curvy jump boundaries. We propose a new GP modeling approach to estimate an unknown piecewise continuous regression function. The new GP model seeks a local GP estimate of an unknown regression function at each test location, using local data neighboring the test location. Considering the possibilities of the local data being from different regions, the proposed approach partitions the local data into pieces by a local data partitioning function. It uses only the local data likely from the same region as the test location for the regression estimate. Since we do not know which local data points come from the relevant region, we propose a data-driven approach to split and subset local data by a local partitioning function. We discuss several modeling choices of the local data partitioning function, including a locally linear function and a locally polynomial function. We also investigate an optimization problem to jointly optimize the partitioning function and other covariance parameters using a likelihood maximization criterion. Several advantages of using the proposed approach over the conventional GP and piecewise GP modeling approaches are shown by various simulated experiments and real data studies.

This paper presents a new approach for Gaussian process (GP) regression for large datasets. The approach involves partitioning the regression input domain into multiple local regions with a different local GP model fitted in each region. Unlike existing local partitioned GP approaches, we introduce a technique for patching together the local GP models nearly seamlessly to ensure that the local GP models for two neighboring regions produce nearly the same response prediction and prediction error variance on the boundary between the two regions. This largely mitigates the well-known discontinuity problem that degrades the prediction accuracy of existing local partitioned GP methods over regional boundaries. Our main innovation is to represent the continuity conditions as additional pseudo-observations that the differences between neighboring GP responses are identically zero at an appropriately chosen set of boundary input locations. To predict the response at any input location, we simply augment the actual response observations with the pseudo-observations and apply standard GP prediction methods to the augmented data. In contrast to heuristic continuity adjustments, this has an advantage of working within a formal GP framework, so that the GP-based predictive uncertainty quantification remains valid. Our approach also inherits a sparse block-like structure for the sample covariance matrix, which results in computationally efficient closed-form expressions for the predictive mean and variance. In addition, we provide a new spatial partitioning scheme based on a recursive space partitioning along local principal component directions, which makes the proposed approach applicable for regression domains having more than two dimensions. Using three spatial datasets and three higher dimensional datasets, we investigate the numerical performance of the approach and compare it to several state-of-the-art approaches.

This paper develops an efficient computational method for solving a Gaussian process (GP) regression for large spatial data sets using a collection of suitably defined local GP regressions. The conventional local GP approach first partitions a domain into multiple non-overlapping local regions, and then fits an independent GP regression for each local region using the training data belonging to the region. Two key issues with the local GP are (1) the prediction around the boundary of a local region is not as accurate as the prediction at interior of the local region, and (2) two local GP regressions for two neighboring local regions produce different predictions at the boundary of the two regions, creating undesirable discontinuity in the prediction. We address these issues by constraining the predictions of local GP regressions sharing a common boundary to satisfy the same boundary constraints, which in turn are estimated by the data. The boundary constrained local GP regressions are solved by a finite element method. Our approach shows competitive performance when compared with several state- of-the-art methods using two synthetic data sets and three real data sets.

This paper presents a general formulation for a minimum cost data association problem which associates data features via one-to-one, m-to-one and one-to-n links with minimum total cost of the links. A motivating example is a problem of tracking multiple interacting nanoparticles imaged on video frames, where particles can aggregate into one particle or a particle can be split into multiple particles. Many existing multitarget tracking methods are capable of tracking non-interacting targets or tracking interacting targets of restricted degrees of interactions. The proposed formulation solves a multitarget tracking problem for general degrees of inter-object interactions. The formulation is in the form of a binary integer programming problem. We propose a polynomial time solution approach that can obtain a good relaxation solution of the binary integer programming, so the approach can be applied for multitarget tracking problems of a moderate size (for hundreds of targets over tens of time frames). The resulting solution is always integral and obtains a better duality gap than the simple linear relaxation solution of the corresponding problem. The proposed method was validated through applications to simulated multitarget tracking problems and a real multitarget tracking problem.

This paper presents a method that enables automated morphology analysis of partially overlapping nanoparticles in electron micrographs. In the undertaking of morphology analysis, three tasks appear necessary: separate individual particles from an agglomerate of overlapping nano-objects; infer the particle's missing contours; and ultimately, classify the particles by shape based on their complete contours. Our specific method adopts a two-stage approach: the first stage executes the task of particle separation, and the second stage conducts simultaneously the tasks of contour inference and shape classification. For the first stage, a modified ultimate erosion process is developed for decomposing a mixture of particles into markers, and then, an edge-to-marker association method is proposed to identify the set of evidences that eventually delineate individual objects. We also provided theoretical justification regarding the separation capability of the first stage. In the second stage, the set of evidences become inputs to a Gaussian mixture model on B-splines, the solution of which leads to the joint learning of the missing contour and the particle shape. Using twelve real electron micrographs of overlapping nanoparticles, we compare the proposed method with seven state-of-the-art methods. The results show the superiority of the proposed method in terms of particle recognition rate.