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Spectral Laplacian methods, widely used in computer graphics and manifold learning, have been recently proposed for the Statistical Process Control (SPC) of a sequence of manufactured parts, whose 3-dimensional metrology is acquired with non-contact sensors. These techniques provide an {\em intrinsic} solution to the SPC problem, that is, a solution exclusively based on measurements on the scanned surfaces or 2-manifolds without making reference to their ambient space. These methods, therefore, avoid the computationally expensive, non-convex registration step needed to align the parts, as required by previous methods for SPC based on 3-dimensional measurements. Once a SPC mechanism triggers and out-of-control alarm, however, an additional problem remains: that of locating where on the surface of the part that triggered the SPC alarm there is a significant shape difference with respect to either an in-control part or its nominal (CAD) design. In the past, only registration-based solutions existed for this problem. In this paper, we present a new registration-free solution to the part localization problem. Our approach uses a functional map between the manifolds to be compared, that is, a map between functions defined on each manifold based on intrinsic differential operators, in particular, the Laplace-Beltrami operator, in order to construct a point to point mapping between the two manifolds and be able to locate defects on the suspected part. A recursive partitioning algorithm is presented to define a region of interest on the surface of the part where defects are likely to occur, which results in considerable computational advantages. The functional map method involves a very large number of point-to-point comparisons based on noisy measurements, and a statistical thresholding method is presented to filter the false positives in the underlying massive multiple comparisons problem., Comment: 32 pages, 12 figures

Recent work in on-line Statistical Process Control (SPC) of manufactured 3-dimensional (3-D) objects has been proposed based on the estimation of the spectrum of the Laplace-Beltrami (LB) operator, a differential operator that encodes the geometrical features of a manifold and is widely used in Machine Learning (i.e., Manifold Learning). The resulting spectra are an intrinsic geometrical feature of each part, and thus can be compared between parts avoiding the part to part registration (or "part localization") pre-processing or the need for equal size meshes, characteristics which are required in previous approaches for SPC of 3D parts. The recent spectral SPC methods, however, are limited to monitoring surface data from objects such that the scanned meshes have no boundaries, holes, or missing portions. In this paper, we extend spectral methods by first considering a more accurate and general estimator of the LB spectrum that is obtained by application of Finite Element Methods (FEM) to the solution of Helmholtz's equation with boundaries. It is shown how the new spectral FEM approach, while it retains the advantages of not requiring part localization/registration or equal size datasets scanned from each part, it provides more accurate spectrum estimates, which results in faster detection of out of control conditions than earlier methods, can be applied to both mesh or volumetric (solid) scans, and furthermore, it is shown how it can be applied to partial scans that result in open meshes (surface or volumetric) with boundaries, increasing the practical applicability of the methods. The present work brings SPC methods closer to contemporary research in Computer Graphics and Manifold Learning. MATLAB code that reproduces the examples of this paper is provided in the supplementary materials.

A bivariate extension to Box and Jenkins (1963) feedback adjustment problem is presented in this paper. The model balances the fixed cost of making an adjustment, which is assumed independent of the magnitude of the adjustments, with the cost of running the process off-target, which is assumed quadratic. It is also assumed that two controllable factors are available to compensate for the deviations from target of two responses in the presence of a bivariate IMA(1,1) disturbance. The optimal policy has the form of a "dead band", in which adjustments are justified only when the predicted process responses exceed some boundary in $\mathbb{R}^2$. This boundary indicates when the responses are predicted to be far enough from their targets that an additional adjustment or intervention in the process is justified. Although originally developed to control a machine tool, dead band control policies have application in other areas. For example, they could be used to control a disease through the application of a drug to a patient depending on the level of a substance in the body (e.g., diabetes control). This paper presents analytical formulae for the computation of the loss function that combines off-target and adjustment costs per time unit. Expressions are derived for the average adjustment interval and for the scaled mean square deviations from target. The minimization of the loss function and the practical use of the resulting dead band adjustment strategy is illustrated with an application to a semiconductor manufacturing process.

The theory of optimal design of experiments has been traditionally developed on an Euclidean space. In this paper, new theoretical results and an algorithm for finding the optimal design of an experiment located on a Riemannian manifold are provided. It is shown that analogously to the results in Euclidean spaces, D-optimal and G-optimal designs are equivalent on manifolds, and we provide a lower bound for the maximum prediction variance of the response evaluated over the manifold. In addition, a converging algorithm that finds the optimal experimental design on manifold data is proposed. Numerical experiments demonstrate the importance of considering the manifold structure in a designed experiment when present, and the superiority of the proposed algorithm.

We present a new method for statistical process control (SPC) of a discrete part manufacturing system based on intrinsic geometrical properties of the parts, estimated from three-dimensional sensor data. An intrinsic method has the computational advantage of avoiding the difficult part registration problem, necessary in previous SPC approaches of three-dimensional geometrical data, but inadequate if noncontact sensors are used. The approach estimates the spectrum of the Laplace-Beltrami (LB) operator of the scanned parts and uses a multivariate nonparametric control chart for online process control. Our proposal brings SPC closer to computer vision and computer graphics methods aimed to detect large differences in shape (but not in size). However, the SPC problem differs in that small changes in either shape or size of the parts need to be detected, keeping a controllable false alarm rate and without completely filtering noise. An online or "Phase II" method and a scheme for starting up in the absence of prior data ("Phase I") are presented. Comparison with earlier approaches that require registration shows the LB spectrum method to be more sensitive to rapidly detect small changes in shape and size, including the practical case when the sequence of part datasets is in the form of large, unequal size meshes. A post-alarm diagnostic method to investigate the location of defects on the surface of a part is also presented. While we focus in this article on surface (triangulation) data, the methods can also be applied to point cloud and voxel metrology data.

We study the U.S. Operations Research/Industrial-Systems Engineering (ORIE) faculty hiring network, consisting of 1,179 faculty origin and destination data together with attribute data from 83 ORIE departments. A social network analysis of faculty hires can reveal important patterns in an academic field, such as the existence of a hierarchy or sociological aspects such as the presence of communities of departments. We first statistically test for the existence of a linear hierarchy in the network and for its steepness. We find a near linear hierarchical order of the departments, proposing a new index for hiring networks, which we contrast with other indicators of hierarchy, including published rankings. A single index is not capable to capture the full structure of a complex network, however, so we next fit a latent exponential random graph model (ERGM) to the network, which is able to reproduce its main observed characteristics: high incidence of self-hiring, skewed out-degree distribution, low density and clustering. Finally, we use the latent variables in the ERGM to simplify the network to one where faculty hires take place among three groups of departments. We contrast our findings with those reported for other related disciplines, Computer Science and Business.

We present a new method for estimating multivariate, second-order stationary Gaussian Random Field (GRF) models based on the Sparse Precision matrix Selection (SPS) algorithm, proposed by Davanloo et al. (2015) for estimating scalar GRF models. Theoretical convergence rates for the estimated between-response covariance matrix and for the estimated parameters of the underlying spatial correlation function are established. Numerical tests using simulated and real datasets validate our theoretical findings. Data segmentation is used to handle large data sets.

Multihead weighing machines (MWMs) are ubiquitous in industry for fast and accurate packaging of a wide variety of foods and vegetables, small hardware items and office supplies. A MWM consists of a system of multiple hoppers that are filled with product which when discharged through a funnel fills a package to a desired weight. Operating this machine requires first to specify the product weight targets or setpoints that each hopper should contain on average in each cycle, which do not need to be identical. The selection of these setpoints has a major impact on the performance of a MWM. Each cycle, the machine fills a package running a built-in knapsack algorithm that opens --or leaves shut-- different combinations of hoppers releasing their content such that the total weight of each package is near to its target, minimizing the amount of product ``given away". In this paper, we address the practical open problem for industry of how to determine the setpoint weights for each of the hoppers given a desired total package weight for a widely used type of MWM. An order statistic formulation based on a characterization of near-optimal solutions is presented. This is shown to be computationally intractable, and a faster heuristic that utilizes a lower bound approximation of the expected smallest order statistic is proposed instead. The setup solutions obtained with the proposed methods can result in substantial savings for MWM users. Alternatively, the analysis presented could be used by management to justify the acquisition of new MWM machines., Comment: 21 pages, 5 figures

Iterative methods for fitting a Gaussian Random Field (GRF) model via maximum likelihood (ML) estimation requires solving a nonconvex optimization problem. The problem is aggravated for anisotropic GRFs where the number of covariance function parameters increases with the dimension. Even evaluation of the likelihood function requires $O(n^3)$ floating point operations, where $n$ denotes the number of data locations. In this paper, we propose a new two-stage procedure to estimate the parameters of second-order stationary GRFs. First, a convex likelihood problem regularized with a weighted $\ell_1$-norm, utilizing the available distance information between observation locations, is solved to fit a sparse precision (inverse covariance) matrix to the observed data. Second, the parameters of the covariance function are estimated by solving a least squares problem. Theoretical error bounds for the solutions of stage I and II problems are provided, and their tightness are investigated., Comment: Two new sections 4.2.1, 5.1 and four new figures 5-8 are added. The title, abstract, and concluding remarks are revised