Computation of -optimal designs for models with correlated observations
Introduction
Let us assume the linear model where denotes the observations vector, is the design matrix, with and the are linearly independent on the experimental domain , is the -vector parameters and the error terms vector, that will be assumed normally-distributed with covariance matrix .
For non-independent observations, the information matrix for a design is given by The inverse of the information matrix is proportional to the covariance matrix of the estimators of the parameters of the model, thus an optimality criterion typically minimizes a function of .
There is a known set of techniques for obtaining optimal designs when the model is linear in the parameters (see for instance Fedorov and Hackl, 1997 or Atkinson et al., 2007), but in most cases assuming independent observations. For a non-linear model, the usual approach is to linearize it and use the standard toolbox for the linearized model. In this case initial values are needed for the non-linear parameters, and thus the obtained designs will be locally optimal.
For a nonzero -dimensional vector , the -optimality criterion tries to find the design that minimizes the variance of the best linear unbiased estimator of . When is taken to be each one of the Euclidean vectors , , -optimality will provide the best designs for the estimation of each parameter, which in particular are needed for the standardized criteria (Dette, 1997), that take into account the scale of the parameters. They can be as well used for checking how good a specific design is for the estimation of each one of the model parameters. Specifically, a design is -optimal if minimizes , where is a generalized inverse (more information on generalized inverses can be found for instance in Yanai et al., 2011 or Pukelsheim, 2006, where a study of generalized inverses related with -optimality is performed). -optimal designs are very often singular (that is, the information matrix of the corresponding design is singular), which increases the difficulty of obtaining them. Elfving (1952) provided a graphical method for finding -optimal designs when the observations are independent. This identifies a key point corresponding to the intersection of the line defined by with the boundary of the ‘Elfving Set’ (the convex hull of ). This optimal point will be a convex combination of, at most, points of the set (Fellman, 1974). The result can be stated as follows: Theorem 1.1 Assuming model (1) and independent observations, the approximate designwhere the are the support points and is the weight that point has in the design. () is -optimal if there exists a point belonging to the line defined by vector and to the boundary of the convex hull of that can be expressed as . That is, it has a maximal norm within the points of the convex hull that can be expressed as with a scalar number.
This procedure is specially suitable for two-dimensional models, for which the required convex hull is easy to obtain, and produces approximate designs. Pukelsheim and Torsney (1991) give a method for computing -optimal weights given the support points, López-Fidalgo and Rodríguez-Díaz (2004) generalize the Elfving’s method to the multi-dimensional case and Harman and Jurik (2008) improve the computation task by using linear programming. Pukelsheim (2006) contains as well an updated approach to Elfving’s theorem.
In the next section, a procedure for obtaining -optimal designs when the observations are correlated is presented. Roughly speaking, the idea is to use a change of variables such that it turns the information matrix (2) into the shape , and thus some ideas from Elfving’s method for independent observations can be applied, now to the new design matrix . Although the method is not suitable for every covariance structure it can be used to solve some cases, and several examples of these applications are shown in Section 3. Finally, Section 4 describes a summary of the results appearing in the paper, introducing as well some other cases where the technique could be applied. Recent papers related to the keys of the present work are Tommasi et al. (2014), which computes -optimal designs for log-linear models, and Dette et al. (in press), where optimal designs for comparing models under correlation are computed.
Section snippets
A derivation of Elfving’s method for correlated observations
A strictly positive definite covariance matrix will be assumed, that is, every eigenvalue of will be assumed to be greater than zero. Through the paper, different examples using strictly positive definite stationary covariance kernels will be shown (see Dette et al., 2013, Dette et al., 2015), giving rise to the covariance structure cov. For this type of correlation, different support points should be assumed in order to avoid singular covariance matrices, thus
Examples of application
In the following, some examples dealing with different stationary covariance kernels will be shown. In first place, the case of constant covariance will be studied.
Conclusions and discussion
A new procedure for the computation of -optimal designs in the correlated setup is introduced, relating the problem to that of independent observations and thus being able to use some ideas from Elfving (1952). Analytical results have been obtained for two-parameter models with an intercept for two-point designs, both assuming constant covariance or a strictly positive definite covariance kernel. A general procedure for a number of observations greater than the number of parameters has been
Acknowledgments
Research was supported by the Spanish Ministry of Economy and Competitiveness and Junta de Castilla y León (Grants ‘MTM 2013-47879-C2-2-P’ and ‘SA130U14’).
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