Lieven Tack

Budget constrained run orders in optimum design

When experiments are to be performed in a time sequence, the observed responses may be affected by an unknown time trend. Run orders that are optimally balanced for time trends usually involve huge costs due to the large number of factor level changes. Therefore, trend-free run orders can be of low practical value in view of economical considerations. Tack and Vandebroek (J. Statist. Plann. Inference 98 (2001) 293) recently presented a design algorithm to construct trend-resistant run orders that give maximal information per unit cost. In this paper, the latter approach is extended to the construction of trendresistant run orders for which the total cost has to be lower than a specified budget. A new design algorithm is proposed that offers the experimenter a general method for solving a wide range of practical design problems.

Trend-Resistant and Cost-Efficient Block Designs with Fixed or Random Block Effects

When performing an experiment in a time sequence, the experimenter may have reason to believe that the observed responses will be influenced by a temporal trend over the course of the experiment. This paper presents an algorithm for the construction of blocked designs that are optimally balanced for time trends. Cost considerations are allowed for in the construction process. Cost-efficient and trend-resistant run orders are constructed for regression designs in the presence of either fixed or random block effects. The proposed algorithm is intended to provide the experimenter with a general method for solving a wide range of important problems. Examples clarify practical utility, and show that the computed trend-resistant run orders have outstanding performance in terms of both cost and 𝓓-optimality.

(Dt,C)-optimal run orders

Cost considerations have rarely been taken into account in optimum design theory. A few authors consider measurement costs, i.e. the cost associated with a particular factor level combination. A second cost approach results from the fact that it is often expensive to change factor levels from one observation to another. We refer to these costs as transition costs. In view of cost minimization, one should minimize the number of factor level changes. However, there is a substantial likelihood that there is some time-order dependence in the results. Consequently, when considering both time-order dependence and transition costs, an optimal ordering is not easy to find. There is precious little in the literature on how to select good time-order sequences for arbitrary design problems and up to now, no thorough analysis of both costs is found in the literature. Our proposed algorithm incorporates both costs in optimum design construction and enables one to compute cost-efficient and trend-free run orders for arbitrary design problems. The results show that cost considerations in the construction of trend-resistant run orders entail considerable reductions in the total cost of an experiment and imply a large increase in the amount of information per unit cost.

optimal run orders

Cost considerations have rarely been taken into account in optimum design theory. A few authors consider measurement costs, i.e. the cost associated with a particular factor level combination. A second cost approach results from the fact that it is often expensive to change factor levels from one observation to another. We refer to these costs as transition costs. In view of cost minimization, one should minimize the number of factor level changes. However, there is a substantial likelihood that there is some time-order dependence in the results. Consequently, when considering both time-order dependence and transition costs, an optimal ordering is not easy to find. There is precious little in the literature on how to select good time-order sequences for arbitrary design problems and up to now, no thorough analysis of both costs is found in the literature. Our proposed algorithm incorporates both costs in optimum design construction and enables one to compute cost-efficient and trend-free run orders for arbitrary design problems. The results show that cost considerations in the construction of trend-resistant run orders entail considerable reductions in the total cost of an experiment and imply a large increase in the amount of information per unit cost.

Efficient Bayesian designs under heteroscedasticity

In optimum design theory designs are constructed that maximize the information on the unknown parameters of the response function. The major part deals with designs optimal for response function estimation under the assumption of homoscedasticity. In this paper, optimal designs are derived in case of multiplicative heteroscedasticity for either response function estimation or response and variance function estimation by using a Bayesian approach. The efficiencies of the Bayesian designs derived with various priors are compared to those of the classic designs with respect to various variance functions. The results show that any prior knowledge about the sign of the variance function parameters leads to designs that are considerably more efficient than the classic ones based on homoscedastic assumptions.

An adjustment algorithm for optimal run orders in design of experiments

Using a systematic run order can be the proper way to conduct an experiment when a temporal trend is present. The construction of run orders that are optimally balanced for time trend effects is based on maximization of the information on the parameters of interest whereas the parameters of the postulated time trend are treated as nuisance parameters. An adjustment algorithm to improve the efficiency of the run orders obtained from a search over a predefined set of candidate points is presented. This is done by repeatedly moving the design points or the time points of the candidate list a small amount along their axes as long as an improvement in the efficiency is obtained. It is illustrated that the adjustment algorithm involves substantial increases in the efficiency of the run orders. The use of the adjustment algorithm in addition to a search over a coarse grid of candidate points is especially recommended in situations where the computation time has to be kept within reasonable limits.

Optimal designs for variance function estimation using sample variances

Using sample variances for estimating a variance function is intuitively more appealing than using residuals. The main advantage of sample variances over residuals is that they do not require specification of a mean function. Based on maximum likelihood and weighted least squares estimation, two alternative approaches for the construction of optimal designs for variance function estimation with sample variances are proposed. Both methods are compared to existing approaches. A generic exchange algorithm and computational results are presented. Irrespective of the link function between the variance and the linear predictor, the algorithm serves as a useful tool to construct tailor-made designs for variance function estimation by means of sample variances.

Trend-resistant and cost-efficient cross-over designs for mixed models

A mixed model approach is used to construct optimal cross-over designs. In a cross-over experiment the same subject is tested at different points in time. Consider as an example an experiment to investigate the influence of physical attributes of the work environment such as luminance, ambient temperature and relative humidity on human performance of acceptance inspection in quality assurance. In a mixed model context, the subject effects are assumed to be independent and normally distributed. Besides the induction of correlated observations within the same inspector, the mixed model approach also enables one to specify the covariance structure of the inspection data. Here, several covariance structures are considered either depending on the time variable or not. Unfortunately, a serious drawback of the inspection experiment is that the results may be influenced by an unknown time trend because of inspector fatigue due to monotony of the inspection task. In other circumstances, time trend effects can be caused by learning effects of the test subjects in behavioural and life sciences, heating or ageing of material in prototype experiments, etc. In addition, the costs for using the subjects and for altering the factor levels between consecutive observations are also taken into account. An algorithm is presented to construct cost-efficient cross-over designs that are optimally balanced for time trend effects. The robustness of the computed run orders against misspecification of the covariance pattern is also investigated and a number of examples illustrate utility of the outlined design methodology.

Semiparametric Exact Optimal Run Orders

When performing an experiment, the observed responses are often influenced by a temporal trend possibly due to aging of material, learning effects, equipment wear-out, or warm-up effects. The construction of run orders that are optimally balanced for time trend effects usually relies on the incorporation of a parametric representation of the time dependence. Using a parametric approach works very well as long as the unknown time dependence is properly specified or overspecified. However, for complicated temporal trends of unknown periodicity, or when the design size is small compared to the complexity of the response model, a parametric approach may lead to underspecification of the true time trend. Serious problems of bias can result. In this paper we show that, contrary to a fully parametric approach with an underfitted time trend, modeling the time trend nonparametrically is very attractive in terms of both bias and precision of the parameter estimators. An algorithm is presented for the construction of optimal run orders when kernel smoothing is used to model the temporal trend. An industrial example illustrates the practical utility of the proposed design methodology.

Les Pierrot Lunaire d'Albert Giraud, O.E. Hartleben et Arnold Schönberg : Pour une analyse des transferts interculturels et intersémiotiques

This article develops some of the major issues and questions raised by the interdisciplinary study of Arnold Schönbergs Pierrot Lunaire (in itself an occurrence of exchange between text and music), and its earlier poetic counterparts. Indeed, Schönbergs creation of Pierrot Lunaire in Berlin (1912), comes at the end of a very complex path, whose starting point is a collection of poems by the French-speaking Belgian poet Albert Giraud, published in Paris in 1884; the poems were translated in German by the poet and translator O. E. Hartleben, in Leipzig and Berlin (1887–1893). It is through the articulation of three research perspectives (the systemic position of authors and texts, the principles of text composition, and the intercultural and intersemiotic transpositions), that this complex path is to be reconstructed. Attention will be paid to the following aspects: the generic inscription of the texts, the significance and transformation of the Pierrot-figure, the transfer conditions, the periodization problems from Décadence to Modernism, the translational norms and intersemiotic relations (concerning Schoenberg as well as the pictural and theatrical references in Giraud). These aspects will receive further consideration in a long term research project launched at the University of Leuven.