Designs For Polynomial Regression Models Research Articles

Enumeration and Evaluation of Small Orthogonal Latin Hypercube Designs for Polynomial Regression Models

In this paper, we construct and evaluate all nonisomorphic Latin hypercube designs with n≤16 runs, the use of which guarantee that the estimates of the first‐order effects are uncorrelated with each other and also uncorrelated with the estimates of the second‐order effects, in polynomial regression models. The produced designs are evaluated using well‐known and popular criteria, and optimal designs are presented in every case studied. An effort to construct nonisomorphic small Latin hypercubes in which only the estimates of the first‐order effects are required to be uncorrelated with each other has also been made, and new designs are presented. All the constructed designs, besides their stand‐alone properties, are useful for the construction of bigger orthogonal Latin hypercubes with desirable properties, using well‐known techniques proposed in the literature. Copyright © 2015 John Wiley & Sons, Ltd.

On Exact K-optimal Designs Minimizing the Condition Number

A new design criterion based on the condition number of an information matrix is proposed to construct optimal designs for linear models, and the resulting designs are called K-optimal designs. The relationship between exact and asymptotic K-optimal designs is derived. Since it is usually hard to find exact optimal designs analytically, we apply a simulated annealing algorithm to compute K-optimal design points on continuous design spaces. Specific issues are addressed to make the algorithm effective. Through exact designs, we can examine some properties of the K-optimal designs such as symmetry and the number of support points. Examples and results are given for polynomial regression models and linear models for fractional factorial experiments. In addition, K-optimal designs are compared with A-optimal and D-optimal designs for polynomial regression models, showing that K-optimal designs are quite similar to A-optimal designs.

Minimizing the Condition Number to Construct Design Points for Polynomial Regression Models

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 $p$th 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 $3 \le p \le 10$.

An algorithm for calculating D-optimal designs for polynomial regression through a fixed point

Optimal designs are required to make efficient statistical experiments. By using canonical moments, in 1980, Studden found D s -optimal designs for polynomial regression models. On the other hand, integrable systems are dynamical systems whose solutions can be written down concretely. In this paper, polynomial regression models through a fixed point are discussed. In order to calculate D-optimal designs for these models, a useful relationship between canonical moments and discrete integrable systems is introduced. By using canonical moments and discrete integrable systems, a new algorithm for calculating D-optimal designs for these models is proposed.

Some robust designs for polynomial regression models

AbstractThe author identifies static optimal designs for polynomial regression models with or without intercept. His optimality criterion is an average between the D‐optimality criterion for the estimation of low‐degree terms and the D8‐optimality criterion for testing the significance of higher degree terms. His work relies on classical results concerning canonical moments and the theory of continued fractions.

D-optimal designs for polynomial regression models through origin

In this article we consider D-optimal designs for polynomial regression models with low-degree terms being missed, by applying the theory of canonical moments. It turns out that the optimal design places equal weight on each of the zeros of some Jacobi polynomial when the number of unknown parameters in the model is even. The procedure and examples of finding the optimal supports and weights are given when the number of unknown parameters in the model is odd.

Maximin designs for exponential growth models and heteroscedastic polynomial models

This paper is concerned with nonsequential optimal designs for a class of nonlinear growth models, which includes the asymptotic regression model. This design problem is intimately related to the problem of finding optimal designs for polynomial regression models with only partially known heteroscedastic structure. In each case, a straightforward application of the usual D­optimality criterion would lead to designs which depend on the unknown underlying parameters. To overcome this undesirable dependence a maximin approach is adopted. The theorem of Perron and Frobenius on primitive matrices plays a crucial role in the analysis.

On a mixture of the D- and D1-optimality criterion in polynomial regression

We consider a mixture of the D1-optimality criterion (minimizing the variance of the estimate for the highest coefficient) and the D-optimality criterion (minimizing the volume of the ellipsoid of concentration for the unknown parameter vector) in the polynomial regression model of degree n ⊆ N. The mixture is defined as a weighted product of both optimality criteria and explicit solutions are given for the proposed criterion. The derived designs have excellent efficiencies compared to the G-, D-, and D1-optimal design. This is illustrated in some examples for polynomial regression of lower degree. The optimal designs are calculated using the theory of canonical moments. Further applications are given in the field of model robust designs for polynomial regression models where only an upper bound for the degree of the polynomial regression model is known by the experimenter before the experiments are carried out.