D-optimal designs for complex Ornstein–Uhlenbeck processes

Complex Ornstein–Uhlenbeck (OU) processes have various applications in statistical modelling. They play role e.g. in the description of the motion of a charged test particle in a constant magnetic field or in the study of rotating waves in time-dependent reaction diffusion systems, whereas Kolmogorov used such a process to model the so-called Chandler wobble, small deviation in the Earth’s axis of rotation. In these applications parameter estimation and model fitting is based on discrete observations of the underlying stochastic process, however, the accuracy of the estimation strongly depend on the observation points. This paper studies the properties of D-optimal designs for estimating the parameters of a complex OU process with a trend. In special situations we show that in contrast with the case of the classical real OU process, a D-optimal design exists not only for the trend parameter, but also for joint estimation of the covariance parameters, moreover, these optimal designs are equidistant

MINIMIZING THE CONDITION NUMBER TO CONSTRUCT DESIGN POINTS FOR POLYNOMIAL REGRESSION MODELS ∗

Abstract. In this paper we study a new optimality criterion, the K-optimality criterion, for constructing optimal experimental designs for polynomial regression models. We focus on the pth order polynomial regression model with symmetric design space [−1, 1]. For this model, we show that there is always a symmetric K-optimal design with exactly p + 1 support points including the boundary points −1 and 1. It is well known that the condition number for a positive definite matrix as the ratio of the maximum eigenvalue to the minimum eigenvalue is usually nonsmooth. We show that for our model, the condition number of the information matrix is continuously differentiable. Theoretical K-optimal designs are derived for p = 1 and 2. Numerical results are presented for 3 ≤ p ≤ 10

Machining accuracy enhancement using machine tool error compensation and metrology

This dissertation aims to enhance machining accuracy by machine tool error reduction and workpiece metrology. The error characteristics are studied by building a quasi-static error model. Perturbed forward kinematic model is used for modeling a 5-axis Computer Numerical Control (CNC) machine with one redundant linear axis. It is found that the 1st order volumetric error model of the 5-axis machine is attributed to 32 error parameter groups. To identify the model by estimating these parameter groups using the least-squares fitting, errors at 290 quasi-randomly generated measurement points over the machine’s workspace are measured using a laser tracker. The identified error model explains 90% of the mean error of the training data sets. However, the measurements using the laser tracker take about 90 minutes, which may cause the identified error parameters to be inaccurate due to the slow varying and transient natures of thermal errors. To shorten the measurement time, an experimental design approach, which suggests the optimal observation locations such that the corresponding robustness of identification is maximized, is applied to design the optimal error observers. Since the observers must be uniformly distributed over the workspace for gaining redundancy, the constrained K-optimal designs are used to select 80 K-optimal observers for the 5-axis machine. Six measurement cycles using 80 observers are done at machine’s different thermal states within a 400-minute experiment. Six error models are trained with consistent performances and are found to be comparable to the one trained by 290 quasi-random observations. This shows the feasibility of using smaller but more strategical-chosen point-set in data-driven error models. More importantly, the growth on mean nominal (119.1 to 181.9 microns) and modeled error (26.3 to 33.9 microns) suggest the necessity of thermal error tracking for enhancing the machining accuracy. A point-set based metrology is also developed to compensate the inaccuracies introduced by workpiece and fixtures and enhance machining accuracy. The machinability of all planar features is examined by virtually comparing the scanned data with the nominal machining planes, which are also known as virtual gages. The virtual gaging problem is modeled as a constrained linear program. The optimal solution to the problem can compensate the displacement introduced by workpiece and fixtures and hence guarantee a conforming finished part. To transfer point-set data into mathematical constraints, algorithms that align, segment, downsize and filter the point-set data are exploited. The concept of virtual gage analysis is demonstrated using experimental data for a simple raw casting. However, for the case where the casting is defective, and some virtual gages are not feasible, the corresponding linear program was found to have no solution. By introducing slack variables to the original linear programming problem, the extended problem has been solved. The extended model is validated for the data obtained for another casting. Further, the analysis predicts the machining allowances on all functional features. Cylindrical surface and its tolerance verification play important role in machining process. Although there exist many approaches that can fit the maximum, minimum and minimum zone cylinders, the cylinder fitting problems can be even simplified. The proposed methodology seeks to reduce the number of parameters used in cylinder fitting model by using the projection model that considers the degenerated tolerance specifications of the projected 2-D point-set. Also, to avoid the problem of local optimum by introducing the optimal direction of projection such that the 2-D point projected onto this direction has optimal tolerance specifications (maximum, minimum and minimum zone circles), global optimum solver such as Particle Swarm Optimization (PSO) is used. The proposed simplified method shows consistent results compared with the results from literature