Hugo Maruri-Aguilar

Nonlinear Matroid Optimization and Experimental Design

We study the problem of optimizing nonlinear objective functions over matroids presented by oracles or explicitly. Such functions can be interpreted as the balancing of multicriteria optimization. We provide a combinatorial polynomial time algorithm for arbitrary oracle-presented matroids, that makes repeated use of matroid intersection and an algebraic algorithm for vectorial matroids. Our work is partly motivated by applications to minimum-aberration model-fitting in experimental design in statistics, which we discuss and demonstrate in detail.

An Efficient Screening Method for Computer Experiments

Computer simulators of real-world processes are often computationally expensive and require many inputs. The problem of the computational expense can be handled using emulation technology; however, highly multidimensional input spaces may require more simulator runs to train and validate the emulator. We aim to reduce the dimensionality of the problem by screening the simulator’s inputs for nonlinear effects on the output rather than distinguishing between negligible and active effects. Our proposed method is built upon the elementary effects (EE) method for screening and uses a threshold value to separate the inputs with linear and nonlinear effects. The technique is simple to implement and acts in a sequential way to keep the number of simulator runs down to a minimum, while identifying the inputs that have nonlinear effects. The algorithm is applied on a set of simulated examples and a rabies disease simulator where we observe run savings ranging between 28% and 63% compared with the batch EE method. Supplementary materials for this article are available online.

Smooth supersaturated models

In areas such as kernel smoothing and non-parametric regression, there is emphasis on smooth interpolation. We concentrate on pure interpolation and build smooth polynomial interpolators by first extending the monomial (polynomial) basis and then minimizing a measure of roughness with respect to the extra parameters in the extended basis. Algebraic methods can help in choosing the extended basis. We get arbitrarily close to optimal smoothing for any dimension over an arbitrary region, giving simple models close to splines. We show in examples that smooth interpolators perform much better than straight polynomial fits and for small sample size, better than kriging-type methods, used, for example in computer experiments.

Betti numbers of polynomial hierarchical models for experimental designs

Polynomial models, in statistics, interpolation and other fields, relate an output η to a set of input variables (factors), x = (x1,..., xd), via a polynomial η(x1,...,xd). The monomials terms in η(x) are sometimes referred to as “main effect” terms such as x1, x2, ..., or “interactions” such as x1x2, x1x3, ... Two theories are related in this paper. First, when the models are hierarchical, in a well-defined sense, there is an associated monomial ideal generated by monomials not in the model. Second, the so-called “algebraic method in experimental design” generates hierarchical models which are identifiable when observations are interpolated with η(x) based at a finite set of points: the design. We study conditions under which ideals associated with hierarchical polynomial models have maximal Betti numbers in the sense of Bigatti (Commun Algebra 21(7):2317–2334, 1993). This can be achieved for certain models which also have minimal average degree in the design theory, namely “corner cut models”.

Optimal design of measurements on queueing systems

We examine the optimal design of measurements on queues with particular reference to the M/M/1 queue. Using the statistical theory of design of experiments, we calculate numerically the Fisher information matrix for an estimator of the arrival rate and the service rate to find optimal times to measure the queue when the number of measurements is limited for both interfering and non-interfering measurements. We prove that in the non-interfering case, the optimal design is equally spaced. For the interfering case, optimal designs are not necessarily equally spaced. We compute optimal designs for a variety of queuing situations and give results obtained under the

A Model Selection Algorithm for Mixture Experiments Including Process Variables

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Algebraic method in experimental design

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