Kai-Tai Fang

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.

Uniform design and its applications in chemistry and chemical engineering

Abstract Experimental designs are very important in chemometrics, since chemistry, until now, is still essentially a field of science strongly dependent on chemical experiments. So, for a long time, chemists have been trying to use the techniques of experimental design developed in statistics to improve experimental works. Three major methods of experimental design, such as factorial design including fractional factorial design and orthogonal design (OD), D-optimal design and uniform design (UD), and their applications in chemistry and chemical engineering are reviewed and compared. The features of uniform design are especially addressed. Uniformity of space filling is the most important and essential feature of the uniform design. Based on this feature, its cost-efficiency, robustness and flexibility make it very useful in the fields of chemistry and chemical engineering. Examples of successful applications of uniform design on improving technologies of various fields such as textile industry, pharmaceuticals, fermentation industry and others have been consistently reported in China since the 1980s. More recently, several various cases applying uniform design to chemical researches showed that the uniform design is indeed a very promising and powerful experimental design method.

The effective dimension and quasi-Monte Carlo integration

Quasi-Monte Carlo (QMC) methods are successfully used for high-dimensional integrals arising in many applications. To understand this success, the notion of effective dimension has been introduced. In this paper, we analyse certain function classes commonly used in QMC methods for empirical and theoretical investigations and show that the problem of determining their effective dimension is analytically tractable. For arbitrary square integrable functions, we propose a numerical algorithm to compute their truncation dimension. We also consider some realistic problems from finance: the pricing of options. We study the special structure of the corresponding integrands by determining their effective dimension and show how large the effective dimension can be reduced and how much the accuracy of QMC estimates can be improved by using the Brownian bridge and the principal component analysis techniques. A critical discussion of the influence of these techniques on the QMC error is presented. The connection between the effective dimension and the performance of QMC methods is demonstrated by examples.

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

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.

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

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.

Maximum-likelihood estimates and likelihood-ratio criteria for multivariate elliptically contoured distributions

For a class of multivariate elliptically contoured distributions the maximum-likelihood estimators of the mean vector and covariance matrix are found under certain conditions. Likelihood-ratio criteria are obtained for a class of null hypotheses. These have the same form as in the normal case.

On the construction of multi-level supersaturated designs

New criteria of comparing multi-level supersaturated designs are proposed and their properties are studied. A new class of multi-level supersaturated designs are obtained by collapsing a U-type uniform design to an orthogonal array. A global optimization algorithm, the threshold accepting algorithm, is then applied to search for the best supersaturated designs under any prespecified criterion. Examples show that these newly constructed supersaturated designs have good modeling properties.

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

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.

Uniform design applied to nonlinear multivariate calibration by ANN

Abstract In this paper, a new experimental design called uniform design (UD) has been introduced as a promising candidate for experimental design in nonlinear multivariate calibration by ANN. In its application to ANN, the UD has the following merits: 1. it is capable of producing samples with high representativeness; 2. it imposes no strong assumption on the model; and 3. it allows many levels for each factor. A comparative study has been made among UD, orthogonal array design (OAD) and central composite design (CCD). The results by UD in simulated and real-fluorescence systems are satisfactory.

Ch. 4. Uniform experimental designs and their applications in industry

The uniform experimental design is one kind of space filling designs that can be used for computer experiments and also for industrial experiments when the underlying model is unknown. The uniform design seeks its design points to be uniformly scattered on the experimental domain. In this chapter we shall introduce the theory and method of the uniform design and related data analysis and modelling methods. Applications of the uniform design to industry and other areas are discussed.