Theory Of Optimal Experimental Design Research Articles

Active Data Selection and Information Seeking

Bayesian inference typically focuses upon two issues. The first is estimating the parameters of some model from data, and the second is quantifying the evidence for alternative hypotheses—formulated as alternative models. This paper focuses upon a third issue. Our interest is in the selection of data—either through sampling subsets of data from a large dataset or through optimising experimental design—based upon the models we have of how those data are generated. Optimising data-selection ensures we can achieve good inference with fewer data, saving on computational and experimental costs. This paper aims to unpack the principles of active sampling of data by drawing from neurobiological research on animal exploration and from the theory of optimal experimental design. We offer an overview of the salient points from these fields and illustrate their application in simple toy examples, ranging from function approximation with basis sets to inference about processes that evolve over time. Finally, we consider how this approach to data selection could be applied to the design of (Bayes-adaptive) clinical trials.

Active SLAM: A Review on Last Decade.

This article presents a comprehensive review of the Active Simultaneous Localization and Mapping (A-SLAM) research conducted over the past decade. It explores the formulation, applications, and methodologies employed in A-SLAM, particularly in trajectory generation and control-action selection, drawing on concepts from Information Theory (IT) and the Theory of Optimal Experimental Design (TOED). This review includes both qualitative and quantitative analyses of various approaches, deployment scenarios, configurations, path-planning methods, and utility functions within A-SLAM research. Furthermore, this article introduces a novel analysis of Active Collaborative SLAM (AC-SLAM), focusing on collaborative aspects within SLAM systems. It includes a thorough examination of collaborative parameters and approaches, supported by both qualitative and statistical assessments. This study also identifies limitations in the existing literature and suggests potential avenues for future research. This survey serves as a valuable resource for researchers seeking insights into A-SLAM methods and techniques, offering a current overview of A-SLAM formulation.

Toward more numerically stable computations of optimal experimental designs

The optimal experimental design theory is based on working with the so-called information matrix which measures the amount of information provided by a design. As a consequence, the crucial expressions that are evaluated in algorithms for finding optimal designs generally involve the information matrix. However, we show that these expressions can be based on a matrix of a simpler form, which we denote as half-information matrix. In particular, we show that by using suitable matrix decompositions, many algorithms can be formulated just in the terms of with no need to compute the information matrix. Crucially, working with the half-information matrix is more stable: its condition number is the square root of that of the information matrix. As a consequence, stabilized versions of algorithms (i.e. those based on working with ) can be applied to much more ill-conditioned models. We show on examples for both exact and approximate designs that the stabilized algorithms work on models that are unfeasible for the standard versions. We also show that the use of suitable rank-one updates of the utilized matrix decompositions can lead to a significant speed-up of the exchange-type algorithms.

Practical implementation of optimal experimental design using the fractional-order Fricke–Morse bioimpedance model

This paper provides, for the first time, the application of the Optimal Experimental Design (OED) theory. Two algorithms for computing exact and approximate optimal designs have been adapted for the fractional-order Fricke–Morse circuit model (which is widely used to describe experimental bioimpedance data). Frequencies at which the impedance is measured are optimized, while reducing the measurement acquisition time and maximizing the information about the fractional-order electrical behaviour of the biological tissue. As a practical implementation of this methodology, for a sample of apple tissue, D-optimal approximate and exact designs are computed to obtain the best estimates of the parameters values according to a criteria. These designs were compared with the classical design commonly used by practitioners showing the efficiencies of the optimal designs. The application of OED theory to this type of problems opens up many possibilities for future research.

Optimal designs for detecting and characterizing hormesis in toxicological tests

Toxicological tests are experiments that show the effects of a toxic on organisms, ecosystems, etc. This study focuses on tests in the aquatic environment, in which the test involving Ceriodaphnia Dubia organism stands out. The literature indicates that in two out of every three experiments carried out with this organism, there is hormesis. This study applies optimal experimental design theory to a linear quadratic model with a Poisson distribution for the response, in order to obtain designs that allow efficient detection and characterization of hormesis. To this end, a variety of utility functions are used, including the dose for the zero equivalent point, the area under the curve, the dose at which maximum response is reached or the dose at which there is a given relative inhibition with respect to the control or the maximum. A study of cross efficiencies of the calculated designs shows the importance of correctly defining the goal of the experiment, in order to obtain the most appropriate design.

Optimal Array Geometric Structures for Direct Position Determination Systems

Millimeter-wave (mmWave) fifth-generation (5G) and beyond 5G localization enables the provisioning of extremely accurate positioning information, a feature that has attracted substantial research efforts. In this paper, we contribute to this effort by exploring optimal array geometric structures of direct position determination (DPD) systems, inspired by sensor placement problems that predominantly focus on two-step localization and have not yet been extended to DPD. Specifically, we research an optimal array placement and orientation strategy for two-dimensional (2-D) DPD systems that use sensors equipped with uniform linear arrays (ULA) to localize an agent. The A-optimality criterion in Bayesian optimal (experimental) design theory is invoked to formulate this problem. We use an optimization subproblem that optimizes array orientations when array locations are arbitrary but fixed to tackle this high-dimensional optimization problem. Then the optimization problem is converted into a typical optimal angular separation problem in two-step localization. Experimental results show that judiciously designed array geometric structures can lead to significant performance improvements.

Stream-based active learning with linear models

The proliferation of automated data collection schemes and the advances in sensorics are increasing the amount of data we are able to monitor in real-time. However, given the high annotation costs and the time required by quality inspections, data is often available in an unlabeled form. This is fostering the use of active learning for the development of soft sensors and predictive models. In production, instead of performing random inspections to obtain product information, labels are collected by evaluating the information content of the unlabeled data. Several query strategy frameworks for regression have been proposed in the literature but most of the focus has been dedicated to the static pool-based scenario. In this work, we propose a new strategy for the stream-based scenario, where instances are sequentially offered to the learner, which must instantaneously decide whether to perform the quality check to obtain the label or discard the instance. The approach is inspired by the optimal experimental design theory and the iterative aspect of the decision-making process is tackled by setting a threshold on the informativeness of the unlabeled data points. The proposed approach is evaluated using numerical simulations and the Tennessee Eastman Process simulator. The results confirm that selecting the examples suggested by the proposed algorithm allows for a faster reduction in the prediction error.

Strategies for robust designs in toxicological tests

Toxicological tests are widely used to study toxicity in aquatic environments. Reproduction is a possible endpoint of this type of experiment, whose response variable is given by counts. There is a literature on the most suitable probability distribution to be used for analyzing the data. In the theory of optimal experimental design, the assumption of this probability distribution is essential, and when this assumption is not appropriate, there may be a loss of efficiency in the design obtained. The main objective of this study is to propose robust designs when there is uncertainty about the probability distribution of the response variable. Three different strategies for attaining this goal are introduced and compared, and they are then applied to toxicological tests based on Ceriodaphnia dubia and Lemna minor. In addition, a simulation study is performed to test the estimation properties of the robust designs obtained.

Optimal Experimental Design for Parametric Identification of the Electrical Behaviour of Bioelectrodes and Biological Tissues

The electrical behaviour of a system, such as an electrode–tissue interface (ETI) or a biological tissue, can be used for its characterization. One way of accomplishing this goal consists of measuring the electrical impedance, that is, the opposition that a system exhibits to an alternating current flow as a function of frequency. Subsequently, experimental impedance data are fitted to an electrical equivalent circuit (EEC model) whose parameters can be correlated with the electrode processes occurring in the ETI or with the physiological state of a tissue. The EEC used in this paper is a reasonable approach for simple bio-electrodes or cell membranes, assuming ideal capacitances. We use the theory of optimal experimental design to identify the frequencies in which the impedance is measured, as well as the number of measurement repetitions, in such a way that the EEC parameters can be optimally estimated. Specifically, we calculate approximate and exact D-optimal designs by optimizing the determinant of the information matrix by adapting two of the most algorithms that are routinely used nowadays (REX random exchange algorithm and KL exchange algorithm). The D-efficiency of the optimal designs provided by the algorithms was compared with the design commonly used by experimenters and it is shown that the precision of the parameter estimates can be increased.

Effect of Probability Distribution of the Response Variable in Optimal Experimental Design with Applications in Medicine

In optimal experimental design theory it is usually assumed that the response variable follows a normal distribution with constant variance. However, some works assume other probability distributions based on additional information or practitioner’s prior experience. The main goal of this paper is to study the effect, in terms of efficiency, when misspecification in the probability distribution of the response variable occurs. The elemental information matrix, which includes information on the probability distribution of the response variable, provides a generalized Fisher information matrix. This study is performed from a practical perspective, comparing a normal distribution with the Poisson or gamma distribution. First, analytical results are obtained, including results for the linear quadratic model, and these are applied to some real illustrative examples. The nonlinear 4-parameter Hill model is next considered to study the influence of misspecification in a dose-response model. This analysis shows the behavior of the efficiency of the designs obtained in the presence of misspecification, by assuming heteroscedastic normal distributions with respect to the D-optimal designs for the gamma, or Poisson, distribution, as the true one.

A Deep Reinforcement Learning Approach for Active SLAM

In this paper, we formulate the active SLAM paradigm in terms of model-free Deep Reinforcement Learning, embedding the traditional utility functions based on the Theory of Optimal Experimental Design in rewards, and therefore relaxing the intensive computations of classical approaches. We validate such formulation in a complex simulation environment, using a state-of-the-art deep Q-learning architecture with laser measurements as network inputs. Trained agents become capable not only to learn a policy to navigate and explore in the absence of an environment model but also to transfer their knowledge to previously unseen maps, which is a key requirement in robotic exploration.

An R-shiny application to calculate optimal designs for single substance and interaction trials in dose response experiments

Background and objectiveOptimal experimental design theory proposes choosing specific settings in experimental trials in order to maximize the precision of the resulting parameter estimates. In dose response experiments, this corresponds to choosing the optimal dose levels for every available observation, and can be applied both to singular dose-response relationships and to interaction experiments where two substances are given simultaneously at several different mixture ratios (“ray designs”). While the theory of experimental design for this situation is well developed, the mathematical complexity prevents widespread use in practical applications. A simple to use application making the theory accessible to practitioners is thus very desirable. MethodsResults from established optimal experimental design theory are applied to dose response applications, focusing on log-logistic and Weibull class dose response functions. Suitable optimal design algorithms to solve these problems are implemented into an R-shiny based online application. ResultsThe application provides an interface to easily calculate D-optimal designs not only for singular dose experiments, but also for interaction trials with several combination rays of substances. Furthermore, the app also allows evaluating the efficiency of existing candidate designs, and finally allows construction of designs which perform robustly under different assumptions in regard to the true parameters.

Optimal dose calibration in radiotherapy

In this paper, the tools provided by the theory of Optimal Experimental Design are applied to a nonlinear calibration model. This is motivated by the need of estimating radiation doses using radiochromic films for radiotherapy purposes. The calibration model is in this case nonlinear and the explanatory variable cannot be worked out explicitly from the model. In this case an experimental design has to be found on the dependent variable. For that, the inverse function theorem will be used to obtain an information matrix to be optimized. Optimal designs on the response variable are computed from two different perspectives, first for fitting the model and estimating each of the parameters and then for predicting the proper dose. While the first is a common point of view in a general context of the Optimal Experimental Design, the latter is actually the main objective of the calibration problem for the practitioners and algorithms for computing these optimal designs are also provided.The optimal designs obtained have just three different points in their support, but practitioners usually demand for more support points. Thus, a methodology for computing space-filling designs is also provided when the support points are forced to follow some mathematical rule, such as arithmetic or geometric sequences. Cross efficiencies of all these designs are computed in order to show their ability for different goals.

Constrained Adaptive Sensing

Suppose that we wish to estimate a vector $\mathbf{x} \in \mathbb{C}^n$ from a small number of noisy linear measurements of the form $\mathbf{y} = \mathbf{A x} + \mathbf{z}$, where $\mathbf{z}$ represents measurement noise. When the vector $\mathbf{x}$ is sparse, meaning that it has only $s$ nonzeros with $s \ll n$, one can obtain a significantly more accurate estimate of $\mathbf{x}$ by adaptively selecting the rows of $\mathbf{A}$ based on the previous measurements provided that the signal-to-noise ratio (SNR) is sufficiently large. In this paper we consider the case where we wish to realize the potential of adaptivity but where the rows of $\mathbf{A}$ are subject to physical constraints. In particular, we examine the case where the rows of $\mathbf{A}$ are constrained to belong to a finite set of allowable measurement vectors. We demonstrate both the limitations and advantages of adaptive sensing in this constrained setting. We prove that for certain measurement ensembles, the benefits offered by adaptive designs fall far short of the improvements that are possible in the unconstrained adaptive setting. On the other hand, we also provide both theoretical and empirical evidence that in some scenarios adaptivity does still result in substantial improvements even in the constrained setting. To illustrate these potential gains, we propose practical algorithms for constrained adaptive sensing by exploiting connections to the theory of optimal experimental design and show that these algorithms exhibit promising performance in some representative applications.

Quality and shelf-life prediction for retail fresh hake (Merluccius merluccius)

Fish quality has a direct impact on market price and its accurate assessment and prediction are of main importance to set prices, increase competitiveness, resolve conflicts of interest and prevent food wastage due to conservative product shelf-life estimations. In this work we present a general methodology to derive predictive models of fish freshness under different storage conditions.The approach makes use of the theory of optimal experimental design, to maximize data information and in this way reduce the number of experiments. The resulting growth model for specific spoilage microorganisms in hake (Merluccius merluccius) is sufficiently informative to estimate quality sensory indexes under time-varying temperature profiles. In addition it incorporates quantitative information of the uncertainty induced by fish variability.The model has been employed to test the effect of factors such as fishing gear or evisceration, on fish spoilage and therefore fish quality. Results show no significant differences in terms of microbial growth between hake fished by long-line or bottom-set nets, within the implicit uncertainty of the model. Similar conclusions can be drawn for gutted and un-gutted hake along the experiment horizon.In addition, whenever there is the possibility to carry out the necessary experiments, this approach is sufficiently general to be used in other fish species and under different stress variables.

Incorporating covariance estimation uncertainty in spatial sampling design for prediction with trans-Gaussian random fields

Recently, Spock and Pilz [38], demonstrated that the spatial sampling design problem for the Bayesian linear kriging predictor can be transformed to an equivalent experimental design problem for a linear regression model with stochastic regression coefficients and uncorrelated errors. The stochastic regression coefficients derive from the polar spectral approximation of the residual process. Thus, standard optimal convex experimental design theory can be used to calculate optimal spatial sampling designs. The design functionals considered in Spock and Pilz [38] did not take into account the fact that kriging is actually a plug-in predictor which uses the estimated covariance function. The resulting optimal designs were close to space-filling configurations, because the design criterion did not consider the uncertainty of the covariance function. In this paper we also assume that the covariance function is estimated, e.g., by restricted maximum likelihood (REML). We then develop a design criterion that fully takes account of the covariance uncertainty. The resulting designs are less regular and space- filling compared to those ignoring covariance uncertainty. The new designs, however, also require some closely spaced samples in order to improve the estimate of the covariance function. We also relax the assumption of Gaussian observations and assume that the data is transformed to Gaussianity by means of the Box-Cox transformation. The resulting prediction method is known as trans-Gaussian kriging. We apply the Smith and Zhu [37] approach to this kriging method and show that resulting optimal designs also depend on the available data. We illustrate our results with a data set of monthly rainfall measurements from Upper Austria.

Design of experiment for hysteresis loops measurement

Hysteresis loop measurements are frequently used to assess the magnetic quality of a nanomaterial under an external magnetic field. Based on the values of the hysteresis parameters, it is possible to decide whether the nanomaterial meets requirements of a given application. In this work, we present a new approach to the measurement of the hysteresis loop based on the theory of optimal experimental design. We show that the maximin efficient design leads to a reduction in the measurements costs when compared to the standard equispaced measurement design. Moreover, a significantly higher accuracy in the estimation of hysteresis parameters is reached within a broad range of plausible values. The functionality of the proposed approach is successfully tested considering real experimental data obtained from the hysteresis loop measurements of the ϵ-Fe2O3 phase. The measurement procedure can be easily adapted to any magnetic nanomaterial for which the values of its hysteresis parameters are to be determined.

Optimal nonlinear design of marine borehole seismic surveys

We explore [Formula: see text]-optimization, a recently introduced method in optimal experimental design theory, as a tool to minimize the forecasted postinversion model uncertainty in marine borehole seismic applications. The [Formula: see text]-criterion is a design objective function crafted specifically for nonlinear data-model relationships and, when used in conjunction with greedy, sequential optimization algorithms, is among the first of its kind to be computationally efficient enough for industrial-scale applications. The benefits and disadvantages of greedy algorithms are briefly discussed, and one of the most popular, the Construction algorithm, is confirmed to take (nearly) linear time, as predicted by theory. The [Formula: see text]-criterion is rederived here to show that restrictive assumptions in its original derivation were unnecessary. This new derivation is cast in terms of expected likelihood ratios, which quantify the discriminability of geomodels constrained by observed seismic data. A 3D vertical seismic profiling (VSP) survey and a walkaway VSP survey are [Formula: see text]-optimized for a site in the Gulf of Mexico, from which several potential applications are identified, including: (1) prescribing the maximum radius of 3D VSP spirals; (2) a novel annular-spiral 3D VSP geometry to reduce acquisition time and cost; (3) a novel azimuthal-walkaway geometry for presurvey acquisition; (4) identifying optimal data for rapid postacquisition quality control and quick-look inversion; (5) optimizing data decimation to facilitate the analysis of massive data sets which cannot be practically analyzed in toto; and (6) optimizing source vessel placement for offset/azimuth checkshots.

Approximation of a maximum-submodular-coverage problem involving spectral functions, with application to experimental designs

We study a family of combinatorial optimization problems defined by a parameter p∈[0,1], which involves spectral functions applied to positive semidefinite matrices, and has some application in the theory of optimal experimental design. This family of problems tends to a generalization of the classical maximum coverage problem as p goes to 0, and to a trivial instance of the knapsack problem as p goes to 1.In this article, we establish a matrix inequality which shows that the objective function is submodular for all p∈[0,1], from which it follows that the greedy approach, which has often been used for this problem, always gives a design within 1−1/e of the optimum. We next study the design found by rounding the solution of the continuous relaxed problem, an approach which has been applied by several authors. We prove an inequality which generalizes a classical result from the theory of optimal designs, and allows us to give a rounding procedure with an approximation factor which tends to 1 as p goes to 1.

Optimal Design Robust to a Misspecified Model

Usually, in the Theory of Optimal Experimental Design the model is assumed to be known at the design stage. In practice, however, more competing models may be plausible for the same data. There is a possibility to find an optimum design which takes into account both model discrimination (for a subsequent application of a hypothesis test) and parameter estimation. In order to avoid the problem of hypothesis testing, a different approach is proposed: to determine an optimum design which is useful for estimation purposes and is robust to a misspecified model. In other words, the optimum design is “good” for estimating the unknown parameters whether or not the assumed model is correct.