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Mathematics
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Optimal Experimental Design and Statistical Modeling
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Optimal Experimental Design and Statistical Modeling

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Probability and Statistics".

Deadline for manuscript submissions: closed (20 December 2023) | Viewed by 13282

Special Issue Editors

Prof. Dr. Raul Martin Martin

E-Mail Website
Guest Editor
Institute of Applied Mathematics for Science and Engineering (IMACI), University of Castilla-La Mancha, 45071 Toledo, Spain
Interests: optimal experimental design; survival analysis; algorithms; model robust design; mixture experiments
Prof. Dr. Weng Kee Wong

E-Mail Website
Guest Editor
Department of Biostatistics, UCLA School of Public Health, University of California Los Angeles, Los Angeles, CA 90095-1772, USA
Interests: public health; statistics; data analytics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

It is well known that the design and analysis of experiments have been used to improve estimates and predictions in empirical processes. It is remarkable to note the utility of measuring the efficiency of the implemented designs in practice and of providing more competitive designs for the optimal use of available resources. Optimal Experimental Design (OED) is a discipline that addresses this problem from different points of view. An important feature of optimal design is that it estimates statistical models with fewer experimental runs and thereby provides valid and precise statistical inference at a minimal cost given the constraints of the study. In the last few years, researchers have applied the ideas of their methodological developments of OED to different multidisciplinary areas such as engineering, biomedical and pharmaceutical research, spatial sampling, epidemiological studies, social studies, and drug development. The Big Data boom has made an impact in this field in what is referred to as active learning. Thus, there is a challenge to develop a basis for suggesting designs in nonstandard situations.

This Special Issue will focus on contributions to the statistical theory and practice of design. The main objective is to present the solutions that modern experimental design brings to the major challenges that arise in the different disciplines. Topics include but are not limited to:

  • Algorithms for the design of experiments
  • Discrimination
  • Robustness of experimental designs
  • Reliability/Survival experimental designs
  • Machine learning methodologies
  • Experiments with mixtures
  • Designs for nonlinear models
  • Split-plot designs
  • Computer experiments
  • Multi-objective optimal design
  • Bayesian experimental designs, etc.

Prof. Dr. Raul Martin Martin
Prof. Dr. Wengkee Wong
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • optimal design
  • constrained optimality
  • algorithms
  • Bayesian design
  • robustness
  • mixture design
  • machine learning
  • model discrimination
  • survival analysis
  • reliability analysis

Published Papers (6 papers)

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Research

12 pages, 346 KiB  
Article
Nested Maximum Entropy Designs for Computer Experiments
by Weiyan Mu, Chengxin Liu and Shifeng Xiong
Mathematics 2023, 11(16), 3572; https://doi.org/10.3390/math11163572 - 18 Aug 2023
Viewed by 659
Abstract
Presently, computer experiments with multiple levels of accuracy are widely applied in science and engineering. This paper introduces a class of nested maximum entropy designs for such computer experiments. A multi-layer DETMAX algorithm is proposed to construct nested maximum entropy designs. Based on [...] Read more.
Presently, computer experiments with multiple levels of accuracy are widely applied in science and engineering. This paper introduces a class of nested maximum entropy designs for such computer experiments. A multi-layer DETMAX algorithm is proposed to construct nested maximum entropy designs. Based on nested maximum entropy designs, we also propose an integer-programming procedure to specify the sample sizes in multi-fidelity computer experiments. Simulated annealing techniques are used to tackle complex optimization problems in the proposed methods. Illustrative examples show that the proposed nested entropy designs can yield better prediction results than nested Latin hypercube designs in the literature and that the proposed sample-size determination method is effective. Full article
(This article belongs to the Special Issue Optimal Experimental Design and Statistical Modeling)
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16 pages, 374 KiB  
Article
Optimal Designs for Antoine’s Equation: Compound Criteria and Multi-Objective Designs via Genetic Algorithms
by Carlos de la Calle-Arroyo, Miguel A. González-Fernández and Licesio J. Rodríguez-Aragón
Mathematics 2023, 11(3), 693; https://doi.org/10.3390/math11030693 - 30 Jan 2023
Viewed by 1610
Abstract
Antoine’s Equation is commonly used to explain the relationship between vapour pressure and temperature for substances of industrial interest. This paper sets out a combined strategy to obtain optimal designs for the Antoine Equation for D- and I-optimisation criteria and different variance structures [...] Read more.
Antoine’s Equation is commonly used to explain the relationship between vapour pressure and temperature for substances of industrial interest. This paper sets out a combined strategy to obtain optimal designs for the Antoine Equation for D- and I-optimisation criteria and different variance structures for the response. Optimal designs strongly depend not only on the criterion but also on the response’s variance, and their efficiency can be strongly affected by a lack of foresight in this selection. Our approach determines compound and multi-objective designs for both criteria and variance structures using a genetic algorithm. This strategy provides a backup for the experimenter providing high efficiencies under both assumptions and for both criteria. One of the conclusions of this work is that the differences produced by using the compound design strategy versus the multi-objective one are very small. Full article
(This article belongs to the Special Issue Optimal Experimental Design and Statistical Modeling)
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17 pages, 1283 KiB  
Article
Improved G-Optimal Designs for Small Exact Response Surface Scenarios: Fast and Efficient Generation via Particle Swarm Optimization
by Stephen J. Walsh and John J. Borkowski
Mathematics 2022, 10(22), 4245; https://doi.org/10.3390/math10224245 - 13 Nov 2022
Cited by 5 | Viewed by 4299
Abstract
G-optimal designs are those which minimize the worst-case prediction variance. Thus, such designs are of interest if prediction is a primary component of the post-experiment analysis and decision making. G-optimal designs have not attained widespread use in practical applications, in part, [...] Read more.
G-optimal designs are those which minimize the worst-case prediction variance. Thus, such designs are of interest if prediction is a primary component of the post-experiment analysis and decision making. G-optimal designs have not attained widespread use in practical applications, in part, because they are difficult to compute. In this paper, we review the last two decades of algorithm development for generating exact G-optimal designs. To date, Particle Swarm Optimization (PSO) has not been applied to construct exact G-optimal designs for small response surface scenarios commonly encountered in industrial settings. We were able to produce improved G-optimal designs for the second-order model and several sample sizes under experiments with K=1,2,3,4, and 5 design factors using an adaptation of PSO. Thereby, we publish updated knowledge on the best-known exact G-optimal designs. We compare computing cost/time and algorithm efficacy to all previous published results including those generated by the current state-of-the-art (SOA) algorithm, the G(Iλ)-coordinate exchange. PSO is hereby demonstrated to produce better designs than the SOA at commensurate cost. In all, the results of this paper suggest PSO should be adopted by more practitioners as a tool for generating exact optimal designs. Full article
(This article belongs to the Special Issue Optimal Experimental Design and Statistical Modeling)
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15 pages, 2394 KiB  
Article
A D-Optimal Sequential Calibration Design for Computer Models
by Huaimin Diao, Yan Wang and Dianpeng Wang
Mathematics 2022, 10(9), 1375; https://doi.org/10.3390/math10091375 - 20 Apr 2022
Cited by 1 | Viewed by 1378
Abstract
The problem with computer model calibration by tuning the parameters associated with computer models is significant in many engineering and scientific applications. Although several methods have been established to estimate the calibration parameters, research focusing on the design of calibration parameters remains limited. [...] Read more.
The problem with computer model calibration by tuning the parameters associated with computer models is significant in many engineering and scientific applications. Although several methods have been established to estimate the calibration parameters, research focusing on the design of calibration parameters remains limited. Therefore, this paper proposes a sequential computer experiment design based on the D-optimal criterion, which can efficiently tune the calibration parameters while improving the prediction ability of the calibrated computer model. Numerical comparisons of the simulated and real data demonstrate the efficiency of the proposed technique. Full article
(This article belongs to the Special Issue Optimal Experimental Design and Statistical Modeling)
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13 pages, 569 KiB  
Article
Design Plan for an Evolution Study of Related Characteristics of a Population
by Juan M. Rodríguez-Díaz, Rosa E. Pruneda and Mercedes Rodríguez-Hernández
Mathematics 2022, 10(5), 792; https://doi.org/10.3390/math10050792 - 02 Mar 2022
Cited by 1 | Viewed by 1451
Abstract
The objective is to study the evolution of different characteristics of a population through time. These response variables may be related for each experimental unit, and in addition, the observations for each response may as well be correlated with time, producing a complex [...] Read more.
The objective is to study the evolution of different characteristics of a population through time. These response variables may be related for each experimental unit, and in addition, the observations for each response may as well be correlated with time, producing a complex correlation structure. The number of responses that can be observed is usually limited for budget, resources, or time reasons, and thus the selection of the most informative time points when data must be taken is quite convenient. This will be performed by using the optimal design of experiments techniques. Some analytical results will be shown, and the results will be applied to obtain the most convenient points when tests about two variables related with the capability of the resolution of mathematical problems in primary school students should be performed. Full article
(This article belongs to the Special Issue Optimal Experimental Design and Statistical Modeling)
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16 pages, 346 KiB  
Article
Recent Advances in Robust Design for Accelerated Failure Time Models with Type I Censoring
by María J. Rivas-López, Raúl Martín-Martín and Irene García-Camacha Gutiérrez
Mathematics 2022, 10(3), 379; https://doi.org/10.3390/math10030379 - 26 Jan 2022
Cited by 1 | Viewed by 1879
Abstract
Many fields including clinical and manufacturing areas usually perform life-testing experiments and accelerated failure time models (AFT) play an essential role in these investigations. In these models the covariate causes an accelerant effect on the course of the event through the term named [...] Read more.
Many fields including clinical and manufacturing areas usually perform life-testing experiments and accelerated failure time models (AFT) play an essential role in these investigations. In these models the covariate causes an accelerant effect on the course of the event through the term named acceleration factor (AF). Despite the influence of this factor on the model, recent studies state that the form of AF is weakly or partially known in most real applications. In these cases, the classical optimal design theory may produce low efficient designs since they are highly model dependent. This work explores planning and techniques that can provide the best robust designs for AFT models with type I censoring when the form of the AF is misspecified, which is an issue little explored in the literature. Main idea is focused on considering the AF to vary over a neighbourhood of perturbation functions and assuming the mean square error matrix as the basis for measuring the design quality. A key result of this research was obtaining the asymptotic MSE matrix for type I censoring under the assumption of known variance regardless the selected failure time distribution. In order to illustrate the applicability of previous result to a study case, analytical characterizations and numerical approaches were developed to construct optimal robust designs under different contaminating scenarios for a failure time following a log-logistic distribution. Full article
(This article belongs to the Special Issue Optimal Experimental Design and Statistical Modeling)
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