On construction of orthogonal and nested orthogonal Latin hypercube designs

Abstract

Latin hypercube designs is a popular choice of experimental design when computer simulation is used for studying a physical process as the design points of a Latin hypercube are equally spaced in the design region when projected onto univariate margins. In computer experiments, changing the levels of variables is only a matter of setting different numbers for the input, whereas in physical experiments, taking more levels of variables often requires an additional cost of making prototypes and a more elaborate and time-consuming implementation of the experiment. Therefore, the differences between computer experiments and traditional physical experiments call for different considerations in design and analysis methods for computer experiments. To handle such type of experimental situations Orthogonal Latin Hypercube design was introduced. Orthogonal Array (OA) designs are used extensively for planning experiments and their success is due to the uniformity properties but when a large number of factors are to be studied in an experiment and only a few of them are virtually effective, OA designs projected onto the subspace spanned by the effective factors can result in repetition of points on effective part only which is undesirable for physical experiments in which the bias of the proposed model is more serious than the variance. In this case LHD may be preferred. But the projection of such design points onto even bivariate margins cannot be guaranteed to be uniformly scattered. Thus to handle this situation Orthogonal arrays based Latin hypercube has been proposed which generally have better space filling properties than random Latin hypercube designs. In some experiments large and expensive computer code can be executed at various degrees of fidelity, and result in computer experiments with multiple levels of cost and accuracy. Efficient data collection from these experiments is critical. Nested designs are useful for designing such experiments. The main drawback of this approach to use LHD is that it requires imputation of some responses of the high-accuracy and low-accuracy experiments when the two sources are aligned together. To mitigate this difficulty, Nested orthogonal LHD has been defined. A general method of construction of Orthogonal Latin Hypercube Designs has been describe. The methods of construction deal with both first order and second order orthogonal Latin hypercube designs. Orthogonal and nearly orthogonal space filling Latin Hypercube Designs has been constructed by modifying the OLH designs. A construction methods of Nested Orthogonal Latin Hypercube (NOLH) Designs has been described. Two general methods of constructing nested orthogonal Latin hypercube designs have been developed. First method deals with 2 layers of NOLH and the second methods deals with three or more layers of NOLH. The methods give many new nested orthogonal Latin hypercube designs with fewer number of runs as compared to existing nested orthogonal Latin hypercube designs.