Peter Winker

Uniform Design: Theory and Application

A uniform design (UD) seeks design points that are uniformly scattered on the domain. It has been popular since 1980. A survey of UD is given in the first portion: The fundamental idea and construction method are presented and discussed and examples are given for illustration. It is shown that UDs have many desirable properties for a wide variety of applications. Furthermore, we use the global optimization algorithm, threshold accepting, to generate UDs with low discrepancy. The relationship between uniformity and orthogonality is investigated. It turns out that most UDs obtained here are indeed orthogonal.

A global optimization heuristic for estimating agent based models

A continuous global optimization heuristic for a stochastic approximation of an objective function, which itself is not globally convex, is introduced. The objective function arises from the simulation based indirect estimation of the parameters of agent based models of financial markets. The function is continuous in the variables but non-differentiable. Due to Monte Carlo variance, only a stochastic approximation of the objective function is available. The algorithm combines features of the Nelder-Mead simplex algorithm with those of a local search heuristic called threshold accepting. The Monte Carlo variance of the simulation procedure is also explicitly taken into account. We present details of the algorithm and some results of the estimation of the parameters for a specific agent based model of the DM/US-

Centered L 2 -discrepancy of random sampling and Latin hypercube design, and construction of uniform designs

foreign exchange market.

Application of Threshold-Accepting to the Evaluation of the Discrepancy of a Set of Points

In this paper properties and construction of designs under a centered version of the L2-discrepancy are analyzed. The theoretic expectation and variance of this discrepancy are derived for random designs and Latin hypercube designs. The expectation and variance of Latin hypercube designs are significantly lower than that of random designs. While in dimension one the unique uniform design is also a set of equidistant points, low-discrepancy designs in higher dimension have to be generated by explicit optimization. Optimization is performed using the threshold accepting heuristic which produces low discrepancy designs compared to theoretic expectation and variance.

Lower bounds for centered and wrap-around L 2 -discrepancies and construction of uniform designs by threshold accepting

Efficient routines for multidimensional numerical integration are provided by quasi--Monte Carlo methods. These methods are based on evaluating the integrand at a set of representative points of the integration area. A set may be called representative if it shows a low discrepancy. However, in dimensions higher than two and for a large number of points the evaluation of discrepancy becomes infeasible. The use of the efficient multiple-purpose heuristic threshold-accepting offers the possibility to obtain at least good approximations to the discrepancy of a given set of points. This paper presents an implementation of the threshold-accepting heuristic, an assessment of its performance for some small examples, and results for larger sets of points with unknown discrepancy.

Applications of optimization heuristics to estimation and modelling problems

We study the uniformity of two- and three-level U-type designs based on the centered and wrap-around L2-discrepancies. By analyzing the known formulae, we find it possible to reexpress them as functions of column balance, and also as functions of Hamming distances of the rows. These new representations allow to obtain two kinds of lower bounds, which can be used as bench marks in searching uniform U-type designs. An efficient updating procedure for the local search heuristic threshold accepting is developed based on these novel formulations of the centered and wrap-around L2-discrepancies. Our implementation of this heuristic for the two- and three-level case efficiently generates low discrepancy U-type designs. Their quality is assessed using the available lower bounds.

Improving the computation of censored quantile regressions

Abstract Estimation and modelling problems as they arise in many fields often turn out to be intractable by standard numerical methods. One way to deal with such a situation consists in simplifying models and procedures. However, the solutions to these simplified problems might not be satisfying. A different approach consists in applying optimization heuristics such as evolutionary algorithms (simulated annealing, threshold accepting), neural networks, genetic algorithms, tabu search, hybrid methods, etc., which have been developed over the last two decades. Although the use of these methods became more standard in several fields of sciences, their use in estimation and modelling in statistics appears to be still limited. A brief introduction to the computational complexity of problems encountered in the fields of statistical modelling and econometrics as well as an overview and classification of the optimization heuristics used is provided. Given the applications presented and the growing availability of optimization heuristics, it is expected that their use will become more frequent in statistics in the near future.

Optimal U—Type Designs

Censored quantile regressions (CQR) are a valuable tool in economics and engineering. The computation of estimators is highly complex and the performance of standard methods is not satisfactory, in particular if a high degree of censoring is present. Due to an interpolation property the computation of CQR estimates corresponds to the solution of a large scale discrete optimization problem. This feature motivates the use of the global optimization heuristic threshold accepting (TA) in comparison to other algorithms. Simulation results presented in this paper indicate that it can improve finding the exact CQR estimator considerably though it uses more computing time.

Lower bounds and stochastic optimization algorithms for uniform designs with three or four levels

Designs with low discrepancy are of interest in many areas of statistical work. U—type designs are among the most widely studied design classes. In this paper a heuristic global optimization algorithm, Threshold Accepting, is used to find optimal U—type designs (uniform designs) or at least good approximations to uniform designs. As the evaluation of the discrepancy of a given point set is performed by an exact algorithm, the application presented here is restricted to small numbers of experiments in low dimensional spaces. The comparison with known optimal results for the two—factor uniform design and good designs for three to five factors shows a good performance of the algorithm.

Identification of multivariate AR-models by threshold accepting

New lower bounds for three- and four-level designs under the centered L 2 -discrepancy are provided. We describe necessary conditions for the existence of a uniform design meeting these lower bounds. We consider several modifications of two stochastic optimization algorithms for the problem of finding uniform or close to uniform designs under the centered L 2 -discrepancy. Besides the threshold accepting algorithm, we introduce an algorithm named balance-pursuit heuristic. This algorithm uses some combinatorial properties of inner structures required for a uniform design. Using the best specifications of these algorithms we obtain many designs whose discrepancy is lower than those obtained in previous works, as well as many new low-discrepancy designs with fairly large scale. Moreover, some of these designs meet the lower bound, i.e., are uniform designs.