Bayesian A-Optimal Design of Experiment with Quantitative and Qualitative Responses

Summary When Shannon entropy is used as a criterion in the optimal design of experiments, advantage can be taken of the classical identity representing the joint entropy of parameters and observations as the sum of the marginal entropy of the observations and the preposterior conditional entropy of the parameters. Following previous work in which this idea was used in spatial sampling, the method is applied to standard parameterized Bayesian optimal experimental design. Under suitable conditions, which include non-linear as well as linear regression models, it is shown in a few steps that maximizing the marginal entropy of the sample is equivalent to minimizing the preposterior entropy, the usual Bayesian criterion, thus avoiding the use of conditional distributions. It is shown using this marginal formulation that under normality assumptions every standard model which has a two-point prior distribution on the parameters gives an optimal design supported on a single point. Other results include a new asymptotic formula which applies as the error variance is large and bounds on support size.

The demand for 3D models that represent real-world objects such as structures and buildings has increased in recent years. It is becoming increasingly important that the reconstructions are not only visually convincing but also feature high geometric accuracy. This includes, for example, the fields of civil engineering, terrestrial surveying and archeology, where precise measurements are made in the models for documentation and analysis purposes. There are different approaches to create such a reconstruction. The photogrammetric method Structure from Motion and laser scanning are among the most widely used methods here, as they do not require a complicated setup and can be used for scenarios at small to large scale. Recent developments are enabling unmanned robotic systems, especially sensor mounted UAVs, to assist in the recording of areas which are otherwise difficult to observe. The demand for a high geometric accuracy, however, comes at the expense of high computational complexity of up to several days. Hence, especially real-time reconstructions are unfeasible, such that recording and reconstruction procedure must be executed consecutively. The resulting model quality, i.e. completeness and accuracy, is only assessable afterwards. Since it is often difficult or even impossible to improve these models with additional measurements afterwards, methods that ensure a reliable acquisition of sufficient data is required.
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\nIn this thesis we develop new methods and theory that address this problem for the mentioned sensor types. For both, a probabilistic description of the expected surface reconstruction error is maintained cost-efficiently as an estimate for the model quality during the recording procedure. For image sensors this is realized by incrementally constructing confidence ellipsoids that describe the information obtained from all views. With depth sensors the surface quality is described by the variance of a Gaussian process implicit surface regression fit to point cloud data using polyharmonic kernel functions. Sensor poses are then assessed by the information they add to the subsequent reconstruction up to a desired geometric accuracy using a formulation that is motivated from Optimal Experimental Design. This quantity is further used in an iterative next-best-view selection framework as a subproblem of a coverage path planning problem.
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\nThe general formulations presented in this thesis enables a wide range of applications, such as offline and online view planning or various autonomous robot systems under consideration of dynamic and geometric constraints. We present the first multi-view coverage path planning approach, specifically targeted at autonomous Structure from Motion data acquisition. Its correctness is validated in simulation using the physics simulator Gazebo. Furthermore, we lay a foundation for similar applications with depth sensors. All presented algorithms were developed with scalability in mind and show promising results regarding real-time usability.

Optimal experimental design is an important methodology for most efficiently allocating resources in an experiment to best achieve some goal. Bayesian experimental design considers the potential impact that various choices of the controllable variables has on the posterior distribution of the unknowns. Optimal Bayesian design involves optimising an expected utility function, which is an analytically intractable integral over the prior predictive distribution. These integrals are typically estimated via standard Monte Carlo methods. In this paper, we demonstrate that the use of randomised quasi-Monte Carlo can bring significant reductions to the variance of the estimated expected utility. This variance reduction will then lead to a more efficient optimisation algorithm for maximising the expected utility.

The sequential optimal experimental design formulated as an information-theoretic sensitivity analysis is applied to the ignition delay problem using real experimental. The optimal design is obtained by maximizing the statistical dependence between the model parameters and observables, which is quantified in this study using mutual information. This is naturally posed in the Bayesian framework. The study shows that by monitoring the information gain after each measurement update, one can design a stopping criteria for the experimental process which gives a minimal set of experiments to efficiently learn the Arrhenius parameters.

The Bayesian optimal experimental design (OED) problem seeks to identify data, sensor configurations, or experiments which can optimally reduce uncertainty. The goal of OED is to find an experiment that maximizes the expected information gain (EIG) about quantities of interest given prior knowledge about expected data. Therefore, within the context of seismic monitoring, we can use Bayesian OED to configure sensor networks by choosing sensor locations, types, and fidelity in order to improve our ability to identify and locate seismic sources. In this work, we develop the framework necessary to use Bayesian OED to optimize the ability to locate seismic events from arrival time data of detected seismic phases. In order to do utilize Bayesian OED we must develop four elements:1. A likelihood function that describes the uncertainty of detection and travel times; 2. A Bayesian solver that takes a prior and likelihood to identify the posterior; 3. An algorithm to compute EIG; and, 4. An optimizer that finds a sensor network which maximizes EIG. Once we have developed this framework, we can explore many relevant questions to monitoring such as: how and what multiphenomenology data can be used to optimally reduce uncertainty, how to trade off sensor fidelity and earth model uncertainty, and how sensor types, number, and locations influence uncertainty

Genotyping costs still preclude analysis of a comprehensive SNP map in thousands of individual subjects in the search for disease susceptibility loci. Allele frequency estimation in DNA pools from cases and controls offers a partial solution, but variance in these estimates will result in some loss of statistical power. However, there has been no systematic attempt to quantify the several sources of error in previous studies. We report an analysis of the magnitude of variance components of each experimental stage in DNA pooling studies, and find that a design based on the formation of numerous small pools of approximately 50 individuals is superior to the formation of fewer, larger pools and the replication of any of the experimental stages. We conclude that this approach may retain an effective sample size greater than 68% of the true sample size, whilst offering a 60-fold reduction in DNA usage and a greater than 30-fold saving in cost, compared to individual genotyping. The possibility of combining pooling with informed selection of haplotype tag SNPs is also considered. In this way further savings in efficiency may be possible by using pooled allele frequency estimates to infer haplotype frequencies and hence, allele frequencies at untyped markers.

The shortage of current method which is used during the optimal design of the uniform magnetic field is that it only takes how to obtain the optimum dimensions of coil system into account without taking much consideration to the practicability of the optimum system(stability of coil system). Considering the current method's shortage, a kind of method with the purpose to enhance the uniformity of the magnetic field and stability is proposed in this paper. Therefore, a multi-objective optimization model of the coils dimensions optimal design is built firstly with the coils' endpoint co-ordinates as the optimized parameters and uniformity of coils system and stability of the system as optimized target. Then the Pareto solution set of the optimization problem is obtained by a kind of multi-objective particle swarm optimization (MOPSO). Last, the method's effectiveness is verified by a certain example and the results are analyzed. The result show that the method can enhance both the uniform magnetic field and stability of magnetic field at the same time, and it can provide a set of different dimensions for the designer's choices.

Free AccessAboutSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack CitationsPermissionsReprints ShareShare onFacebookTwitterLinked InEmail Go to SectionFree Access HomeOperations ResearchVol. 15, No. 5 Letter to the Editor—Some Comments on the Paper by B. V. Dean and E. S. Marks, “Optimal Design of Optimization Experiments”Regina C. Elandt-JohnsonRegina C. Elandt-JohnsonPublished Online:1 Oct 1967https://doi.org/10.1287/opre.15.5.958"Letter to the Editor—Some Comments on the Paper by B. V. Dean and E. S. Marks, “Optimal Design of Optimization Experiments”." Operations Research, 15(5), pp. 958–959 Previous Back to Top Next FiguresReferencesRelatedInformation Volume 15, Issue 5September-October 1967Pages 779-983 Article Information Metrics Information Published Online:October 01, 1967 © 1967 INFORMSCite asRegina C. Elandt-Johnson, (1967) Letter to the Editor—Some Comments on the Paper by B. V. Dean and E. S. Marks, “Optimal Design of Optimization Experiments”. Operations Research 15(5):958-959. https://doi.org/10.1287/opre.15.5.958 PDF download

Intravenous glucose tolerance test (IVGTT) provocations are informative, but complex and laborious, for studying the glucose-insulin system. The objective of this study was to evaluate, through optimal design methodology, the possibilities of more informative and/or less laborious study design of the insulin modified IVGTT in type 2 diabetic patients. A previously developed model for glucose and insulin regulation was implemented in the optimal design software PopED 2.0. The following aspects of the study design of the insulin modified IVGTT were evaluated; (1) glucose dose, (2) insulin infusion, (3) combination of (1) and (2), (4) sampling times, (5) exclusion of labeled glucose. Constraints were incorporated to avoid prolonged hyper- and/or hypoglycemia and a reduced design was used to decrease run times. Design efficiency was calculated as a measure of the improvement with an optimal design compared to the basic design. The results showed that the design of the insulin modified IVGTT could be substantially improved by the use of an optimized design compared to the standard design and that it was possible to use a reduced number of samples. Optimization of sample times gave the largest improvement followed by insulin dose. The results further showed that it was possible to reduce the total sample time with only a minor loss in efficiency. Simulations confirmed the predictions from PopED. The predicted uncertainty of parameter estimates (CV) was low in all tested cases, despite the reduction in the number of samples/subject. The best design had a predicted average CV of parameter estimates of 19.5%. We conclude that improvement can be made to the design of the insulin modified IVGTT and that the most important design factor was the placement of sample times followed by the use of an optimal insulin dose. This paper illustrates how complex provocation experiments can be improved by sequential modeling and optimal design.

Bayesian optimal experimental design (OED) plays an important role in minimizing model uncertainty with limited experimental data in a Bayesian framework. In many applications, rather than minimizing the uncertainty in the inference of model parameters, one seeks to minimize the uncertainty of a model-dependent quantity of interest (QoI). This is known as goal-oriented OED (GOOED). Here, we consider GOOED for linear Bayesian inverse problems governed by large-scale models represented by partial differential equations (PDE) that are computationally expensive to solve. In particular, we consider optimal sensor placement by maximizing an expected information gain (EIG) for the QoI. We develop an efficient method to solve such problems by deriving a new formulation of the goal-oriented EIG. Based on this formulation we propose an offline-online decomposition scheme that achieves significant computational reduction by computing all of the PDE-dependent quantities in an offline stage just once, and optimizing the sensor locations in an online stage without solving any PDEs. Moreover, in the offline stage we need only to compute low-rank approximations of two Hessian-related operators. The computational cost of these low-rank approximations, measured by the number of PDE solves, does not depend on the parameter or data dimensions for a large class of elliptic, parabolic, and sufficiently dissipative hyperbolic inverse problem that exhibit dimension-independent rapid spectra decay. We carry out detailed error analysis for the approximate goal-oriented EIG due to the low-rank approximations of the two operators. Furthermore, in the online stage we extend a swapping greedy method to optimize the sensor locations developed in our recent work that is demonstrated to be more efficient than a standard greedy method. We conduct a numerical experiment for a contaminant transport inverse problem with an infinite-dimensional parameter field to demonstrate the efficiency, accuracy, and both data- and parameter-dimension independence of the proposed algorithm.

Model Predictive Control (MPC) is a solution towards more energy-efficient waste treatment without compromising on treatment quality. A key component is the process model describing how the inputs and outputs correlate. MPC uses this model to predict future outputs over a finite horizon to decide on step changes to make at the input. These step changes are made so that the output reaches and maintains at a user specified set point. For MPC to be effective, the process model needs to accurately describe the process behaviour. This is a difficult challenge in waste treatment processes due to a combination of slow response, process complexity, and large disturbances. This research project investigated two research avenues towards developing better modelling techniques. This would result in more accurate models or achieve a sufficiently accurate model with fewer experiments. The first avenue is Constrained Model Identification (CMI). Model identification is an optimisation problem to estimate the model parameters. In CMI, process knowledge from first principles and operator experience is translated into optimisation constraints to aid data-driven model identification. The second avenue is Sequential Optimal Experiment Design (SOED). This uses the concept of measuring a value representing information content of a dataset. Like MPC, SOED uses the model to make output predictions. The expected output response to a sequence of input steps form a dataset, and SOED is an optimisation problem to maximise the information content of that expected dataset, by changing the input step sequence. Once optimised, this step sequence is applied in the next experiment. The third part of this work focused on farm-fed anaerobic digestion. It is a renewable energy technology fuelled by agricultural waste. They rely on government incentives to be profitable, but these incentives have steadily been decreased. This project investigated methods to help farmers in the day-to day operation of the unit, including biogas production estimation, automated fault identification and partial diagnosis.

In this paper, a PSO algorithm for optimal design of wind farm is presented. In particular, we solve the problem of minimizing the cost per unit of power produced in a wind farm. This cost depends on the number of turbines placed in a wind farm and the total power produced. In order to solve this problem, the algorithm PSO is used and in particular the gbest model. In this problem we take into account the number of turbines which are placed in a wind farm, the power produced by each one of them and the velocity of the wind which affects their function. The gbest model of the PSO algorithm is used for a wind farm which consists of 10, 20 and 30 wind turbines.

Researchers designing experiments in which a sample of participants responds to a sample of stimuli are faced with difficult questions about optimal study design. The conventional procedures of statistical power analysis fail to provide appropriate answers to these questions because they are based on statistical models in which stimuli are not assumed to be a source of random variation in the data, models that are inappropriate for experiments involving crossed random factors of participants and stimuli. In this article, we present new methods of power analysis for designs with crossed random factors, and we give detailed, practical guidance to psychology researchers planning experiments in which a sample of participants responds to a sample of stimuli. We extensively examine 5 commonly used experimental designs, describe how to estimate statistical power in each, and provide power analysis results based on a reasonable set of default parameter values. We then develop general conclusions and formulate rules of thumb concerning the optimal design of experiments in which a sample of participants responds to a sample of stimuli. We show that in crossed designs, statistical power typically does not approach unity as the number of participants goes to infinity but instead approaches a maximum attainable power value that is possibly small, depending on the stimulus sample. We also consider the statistical merits of designs involving multiple stimulus blocks. Finally, we provide a simple and flexible Web-based power application to aid researchers in planning studies with samples of stimuli.

A class of optimal shape design problems is studied in this paper. The boundary shape of a domain is determined such that the solution of the underlying partial differential equation matches, as well as possible, a given desired state. In the original optimal shape design problem, the variable domain is parameterized by a class of functions in such a way that the optimal design problem is changed to an optimal control problem on a fixed domain. Then, the resulting distributed control problem is embedded in a measure theoretical form, in fact, an infinite-dimensional linear programming problem. The optimal measure representing the optimal shape is approximated by a solution of a finite-dimensional linear programming problem. The method is evaluated via a numerical example.

Optimal measurement design for parameter identification in mechanized tunneling