Information contained in design points of experiments with correlated observations

summary:A random process (field) with given parametrized mean and covariance function is observed at a finite number of chosen design points. The information about its parameters is measured via the Fisher information matrix (for normally distributed observations) or using information functionals depending on that matrix. Conditions are stated, under which the contribution of one design point to this information is zero. Explicit expressions are obtained for the amount of information coming from a selected subset of a given design. Relations to some algorithms for optimum design of experiments in case of correlated observations are indicated

Criteria for optimal design of small-sample experiments with correlated observations

summary:We consider observations of a random process (or a random field), which is modeled by a nonlinear regression with a parametrized mean (or trend) and a parametrized covariance function. Optimality criteria for parameter estimation are to be based here on the mean square errors (MSE) of estimators. We mention briefly expressions obtained for very small samples via probability densities of estimators. Then we show that an approximation of MSE via Fisher information matrix is possible, even for small or moderate samples, when the errors of observations are normal and small. Finally, we summarize some properties of optimality criteria known for the noncorrelated case, which can be transferred to the correlated case, in particular a recently published concept of universal optimality

On Supporting Wide Range of Attribute Types for Top-K Search

Searching top-k objects for many users face the problem of different user preferences. The family of Threshold algorithms computes top-k objects using sorted access to ordered lists. Each list is ordered w.r.t. user preference to one of objects' attributes. In this paper the index based methods to simulate the sorted access for different user preferences in parallel are presented. The simulation for different domain types -- ordinal, nominal, metric and hierarchical -- is presented

Strict Monotonicity and Convergence Rate of Titterington's Algorithm for Computing D-optimal Designs

We study a class of multiplicative algorithms introduced by Silvey et al. (1978) for computing D-optimal designs. Strict monotonicity is established for a variant considered by Titterington (1978). A formula for the rate of convergence is also derived. This is used to explain why modifications considered by Titterington (1978) and Dette et al. (2008) usually converge faster

Optimal design for correlated processes with input-dependent noise

Optimal design for parameter estimation in Gaussian process regression models with input-dependent noise is examined. The motivation stems from the area of computer experiments, where computationally demanding simulators are approximated using Gaussian process emulators to act as statistical surrogates. In the case of stochastic simulators, which produce a random output for a given set of model inputs, repeated evaluations are useful, supporting the use of replicate observations in the experimental design. The findings are also applicable to the wider context of experimental design for Gaussian process regression and kriging. Designs are proposed with the aim of minimising the variance of the Gaussian process parameter estimates. A heteroscedastic Gaussian process model is presented which allows for an experimental design technique based on an extension of Fisher information to heteroscedastic models. It is empirically shown that the error of the approximation of the parameter variance by the inverse of the Fisher information is reduced as the number of replicated points is increased. Through a series of simulation experiments on both synthetic data and a systems biology stochastic simulator, optimal designs with replicate observations are shown to outperform space-filling designs both with and without replicate observations. Guidance is provided on best practice for optimal experimental design for stochastic response models

D-optimal designs via a cocktail algorithm

A fast new algorithm is proposed for numerical computation of (approximate) D-optimal designs. This "cocktail algorithm" extends the well-known vertex direction method (VDM; Fedorov 1972) and the multiplicative algorithm (Silvey, Titterington and Torsney, 1978), and shares their simplicity and monotonic convergence properties. Numerical examples show that the cocktail algorithm can lead to dramatically improved speed, sometimes by orders of magnitude, relative to either the multiplicative algorithm or the vertex exchange method (a variant of VDM). Key to the improved speed is a new nearest neighbor exchange strategy, which acts locally and complements the global effect of the multiplicative algorithm. Possible extensions to related problems such as nonparametric maximum likelihood estimation are mentioned.Comment: A number of changes after accounting for the referees' comments including new examples in Section 4 and more detailed explanations throughou