D-optimal Experimental Designs Research Articles

Fast-dissolving calcium carbonate particles as a high-performance lime for amelioration of agricultural acidic soils

Effective soil management is essential for maximizing crop yields, as soil fertility plays a pivotal role in this process. Unfortunately, liming and fertilization practices are frequently ineffective, leading to overdoses and unnecessary product waste. Liming is essential for correcting soil acidity, mitigating aluminum toxicity, and promoting nutrient bioavailability before fertilizer application. This study explores the potential of vaterite, a highly soluble CaCO3 polymorph, as a sustainable liming agent. Vaterite particle synthesis is challenging because of its poor stability and rapid phase transformation into calcite, therefore, we analyze the effects of precursor concentration, Ca2+/CO32- ratio, and stirring rate on vaterite morphology and content in a reactive precipitation process through screening and D-optimal experimental designs, resulting in a design space to tailor the conditions that promote vaterite particles with a high dissolution profile at the lowest possible cost. To assess the efficacy of fast-dissolving CaCO3 particles, a comparative incubation test was conducted in andisol-type soil with a high buffering capacity. The results demonstrated that vaterite particles were capable of quickly adjusting the soil pH in the first few days using 360% fewer conventional liming agents. This innovative strategy aligns with Precision Agriculture principles, enabling efficient crop production while minimizing the environmental impact.

Mathematical modelling and parametric optimization of biomethane production with response surface methodology: A case of cassava vinasse from a bioethanol distillery

The high rate of cassava vinasse (CV) generation from distilleries and its anti-environmental characteristics threaten the sustainable production of bioethanol. In this study, the production of biomethane via anaerobic digestion (AD) of CV was modelled and optimized with response surface methodology (RSM). The D-Optimal experimental designs produced 27 digestion experiments randomly executed to investigate the effects of CV (55–100%), poultry dropping, PD (0–45%), organic loading rates, OLR (2–10 gVS/l.d), inoculum to substrate percentage, ISP (10–30%) and hydraulic residence time, HRT (20–30 days) on methane yield (MY). Experimental data were fitted to a quadratic model and its significance was assessed using the analysis of variance (ANOVA), and lack of fit test. Model efficiency was experimentally validated in the laboratory. Results showed that the CV had a high carbon-to-nitrogen ratio (C/N) of 43.5:1 and a biochemical methane potential (BMP) of 247.1 Nml/gVS demonstrates the potential for biogas production. The optimum conditions established for biomethane production were 55% CV, 10% OLR, 30% ISP and 20 days HRT with a MY of 254.4 ml/g VS. For the developed model, a coefficient of determination (R2), standard deviation (SDev.) and p-value of 0.9925, 4.91, <0.0001 respectively indicate high predictive accuracy. The study provided vital information for the bioethanol industry and the environmental agencies, on alternative treatment and utilization options for CV.

D-optimal experimental designs for precise parameter estimation of adsorption equilibrium models: initial concentration and solvent volume to adsorbent mass ratio as independent variables

D-optimal experimental designs for precise parameter estimation of adsorption equilibrium models: initial concentration and solvent volume to adsorbent mass ratio as independent variables

An algorithm for the simultaneous construction of D-optimal experimental designs

Construction of optimal multiresponse designs when the responses are measured simultaneously for the same settings of the input variables have been studied. However, in some experiments, the response variables can be measured at different time points and also defined in the same experimental space. The response variables are assumed to be uncorrelated. In this case, simultaneous construction of more than one optimal design is required. We present an algorithm for the simultaneous construction of optimal designs for these kind of experiments. The algorithm operates on two parallel platforms. On one, the same support points are selected for the response functions. The starting point of the search, direction vector, and step length are computed. On the other, the choice of initial design and exchange of design points are carried out. Examples are given to illustrate the construction of optimal designs using the algorithm. The algorithm converges rapidly and is more economical than constructing optimal design for one response function at-a-time.

D-optimal experimental designs for linear multiple regression under heteroscedastic observations

The problem of construction of «continuous» (number of observations is not fixed) and «exact» (number of observations is fixed) D-optimal experimental designs for linear multiple regression in the case when variance of errors of observations depends on regressor value is studied in this paper. Families of functions that determine heteroscedastic observations are found for which it is possible to construct «continuous» and «exact» D-optimal experimental designs. «Continuous» D-optimal experimental designs under four different types of heteroscedasticity are constructed for linear multiple regression with three regressors.

On greedy heuristics for computing D-efficient saturated subsets

D-efficient saturated subsets are natural initial solutions of various algorithms applied in statistics and computational geometry. We propose two greedy heuristics for the construction of D-efficient saturated subsets: an improvement of the method suggested by Galil and Kiefer in the context of D-optimal experimental designs and a modification of the Kumar–Yildirim method for the initiation of the minimum-volume enclosing ellipsoid algorithms. We provide mathematical insights into the two methods and compare them to the commonly used random and regularized heuristics.

Estimators and D-optimal experimental designs for mixtures of binary responses

Models of mixture distributions are of great interest in many areas where several populations are mixed up. If the response is binary there is a mixture of Bernoulli distributions, which has practical applications for the classification of texts and images, biochemistry, genetics, robotics, computer science or pattern recognition. A finite mixture of probability distributions includes a set of parameters such as the proportion of the mixture and the parameters of each distribution. This simple case allows us to make some explicit computations of the estimators as well as working on the EM algorithm doing some comparisons. Some of these parameters may depend on one or more covariates through some specific model. Mixtures of exponential family distributions, except in very particular cases, are no longer within the exponential family and this means, among other things, that the expectation for computing the Information Matrix must, in most cases, be approximated. One of the main contributions of this article is its handling of this. Linear and logistic models are considered either for the proportion of one of the two populations (clusters), or for the parameters of the Bernoulli distributions. For each of these cases the analytic expression of the information matrix is calculated and optimal designs determined.

D-optimal experimental designs for precise parameter estimation of adsorption equilibrium models

D-optimal experimental designs were obtained for Langmuir, Freundlich, Jovanovich, Sips, Redlich-Peterson and BET isotherms using two different approaches. In the first one, it is assumed that liquid solution equilibrium concentration is the independent system variable and solid equilibrium concentration is the dependent variable, which is the usual procedure for parameter estimation of adsorption isotherm models. In the second approach, initial liquid solution concentration, liquid solution volume and adsorbent mass are the independent variables and liquid solution equilibrium concentration is the dependent variable, which is closer to what actually occurs in adsorption experiments. In both cases, one of the designed conditions is an experiment using the maximum concentration value in the experimental range, while other conditions are functions of one or more isotherm parameter. When using equilibrium concentration as the independent variable, the linear parameter of the equilibrium model did not affect the experimental design. In this case, analytical solutions for 2-parameter equilibrium isotherm were obtained, while only numerical solutions were obtained for 3-parameter equilibrium models. When considering the initial concentration as the independent variable, with a constant liquid solution volume to adsorbent mass ratio, it was possible to observe that the experimental conditions of the optimum design depend on all model parameters, for all isotherms evaluated. Additionally, as liquid solution volume to adsorbent mass ratio increases, the experimental design obtained using the initial concentration as the independent variable approaches the one obtained when equilibrium concentration is assumed to be independent variable; therefore, this approach is just a particular case of the first one.

Optimization of mass spectrometry acquisition parameters for determination of polycarbonate additives, degradation products, and colorants migrating from food contact materials to chocolate.

The interest towards "substances of emerging concerns" referred to objects intended to come into contact with food is recently growing. Such substances can be found in traces in simulants and in food products put in contact with plastic materials. In this context, it is important to set up analytical systems characterized by high sensitivity and to improve detection parameters to enhance signals. This work was aimed at optimizing a method based on UHPLC coupled to high resolution mass spectrometry to quantify the most common plastic additives, and able to detect the presence of polymers degradation products and coloring agents migrating from plastic re-usable containers. The optimization of mass spectrometric parameter settings for quantitative analysis of additives has been achieved by a chemometric approach, using a full factorial and d-optimal experimental designs, allowing to evaluate possible interactions between the investigated parameters. Results showed that the optimized method was characterized by improved features in terms of sensitivity respect to existing methods and was successfully applied to the analysis of a complex model food system such as chocolate put in contact with 14 polycarbonate tableware samples. A new procedure for sample pre-treatment was carried out and validated, showing high reliability. Results reported, for the first time, the presence of several molecules migrating to chocolate, in particular belonging to plastic additives, such Cyasorb UV5411, Tinuvin 234, Uvitex OB, and oligomers, whose amount was found to be correlated to age and degree of damage of the containers.

On Bayesian A- and D-Optimal Experimental Designs in Infinite Dimensions

We consider Bayesian linear inverse problems in infinite-dimensional separable Hilbert spaces, with a Gaussian prior measure and additive Gaussian noise model, and provide an extension of the concept of Bayesian D-optimality to the infinite-dimensional case. To this end, we derive the infinite-dimensional version of the expression for the Kullback–Leibler divergence from the posterior measure to the prior measure, which is subsequently used to derive the expression for the expected information gain. We also study the notion of Bayesian A-optimality in the infinite-dimensional setting, and extend the well known (in the finite-dimensional case) equivalence of the Bayes risk of the MAP estimator with the trace of the posterior covariance, for the Gaussian linear case, to the infinite-dimensional Hilbert space case.

D-optimal experimental designs for a growth model applied to a Holstein-Friesian dairy farm

The body mass growth of organisms is usually represented in terms of what is known as ontogenetic growth models, which represent the relation of dependence between the mass of the body and time. This paper discusses design issues of West’s ontogenetic growth model applied to a Holstein-Friesian dairy farm in the northwest of Spain. D-optimal experimental designs were computed to obtain an optimal fitting of the model. A correlation structure has been included in the statistical model due to the fact that observations on a particular animal are not independent. The choice of a robust correlation structure is an important contribution of this paper; it provides a methodology that can be used for any correlation structure. The experimental designs undertaken provide a tool to control the proper weight of heifers, which will help improve their productivity and, by extension, the competitiveness of the dairy farm.

Optimal experimental designs for dose-response studies with continuous endpoints.

In most areas of clinical and preclinical research, the required sample size determines the costs and effort for any project, and thus, optimizing sample size is of primary importance. An experimental design of dose–response studies is determined by the number and choice of dose levels as well as the allocation of sample size to each level. The experimental design of toxicological studies tends to be motivated by convention. Statistical optimal design theory, however, allows the setting of experimental conditions (dose levels, measurement times, etc.) in a way which minimizes the number of required measurements and subjects to obtain the desired precision of the results. While the general theory is well established, the mathematical complexity of the problem so far prevents widespread use of these techniques in practical studies. The paper explains the concepts of statistical optimal design theory with a minimum of mathematical terminology and uses these concepts to generate concrete usable D-optimal experimental designs for dose–response studies on the basis of three common dose–response functions in toxicology: log-logistic, log-normal and Weibull functions with four parameters each. The resulting designs usually require control plus only three dose levels and are quite intuitively plausible. The optimal designs are compared to traditional designs such as the typical setup of cytotoxicity studies for 96-well plates. As the optimal design depends on prior estimates of the dose–response function parameters, it is shown what loss of efficiency occurs if the parameters for design determination are misspecified, and how Bayes optimal designs can improve the situation.

Optimal design of large-scale screening experiments: a critical look at the coordinate-exchange algorithm

We focus on the D-optimal design of screening experiments involving main-effects regression models, especially with large numbers of factors and observations. We propose a new selection strategy for the coordinate-exchange algorithm based on an orthogonality measure of the design. Computational experiments show that this strategy finds better designs within an execution time that is 30 % shorter than other strategies. We also provide strong evidence that the use of the prediction variance as a selection strategy does not provide any added value in comparison to simpler selection strategies. Additionally, we propose a new iterated local search algorithm for the construction of D-optimal experimental designs. This new algorithm outperforms the original coordinate-exchange algorithm.

A novel calibration procedure for dynamically tuned gyroscope designed by D-optimal approach

Effective calibration and compensation of the deterministic errors of Dynamically Tuned Gyroscope (DTG) are crucial in improving the application precision and performance of DTG. However, conventional calibration procedures used empirical parameter-setting and provided no evidence on optimization in terms of accuracy and calibration time. In this paper, D-optimal experimental designs were constructed to propose a practical 12-position calibration procedure of DTG which can be implemented by the medium-precision turntable. The proposed procedure was tested by a tactical-grade DTG. Test results show that, the fit uncertainty is reduced about 62% for the ox axis and 39% for the oy axis respectively. Furthermore, the new calibration procedure achieves D-optimal accuracy with half of the calibration time of the widely adopted 24-position calibration procedure.

D-optimal experimental design coupled with parallel factor analysis 2 decomposition a useful tool in the determination of triazines in oranges by programmed temperature vaporization–gas chromatography–mass spectrometry when using dispersive-solid phase extraction

The determination of triazines in oranges using a GC–MS system coupled to a programmed temperature vaporizer (PTV) inlet in the context of legislation is performed. Both pretreatment (using a Quick Easy Cheap Effective Rugged and Safe (QuEChERS) procedure) and injection steps are optimized using D-optimal experimental designs for reducing the experimental effort. The relative dirty extracts obtained and the elution time shifts make it necessary to use a PARAFAC2 decomposition to solve these two usual problems in the chromatographic determinations. The “second-order advantage” of the PARAFAC2 decomposition allows unequivocal identification according to document SANCO/12495/2011 (taking into account the tolerances for relative retention time and the relative abundance for the diagnostic ions), avoiding false negatives even in the presence of unknown co-eluents. The detection limits (CCα) found, from 0.51 to 1.05μgkg−1, are far below the maximum residue levels (MRLs) established by the European Union for simazine, atrazine, terbuthylazine, ametryn, simetryn, prometryn and terbutryn in oranges. No MRL violations were found in the commercial oranges analyzed.

A D-optimal Multi-position Calibration Method for Dynamically Tuned Gyroscopes

This paper presents a novel experimental design to greatly improve the calibration accuracy of the acceleration-insensitive bias and the acceleration-sensitive bias of the dynamically tuned gyroscopes (DTGs). In order to reduce experimental cost, the D-optimal criteria with constraints are constructed. The turntable positions and the number of test points are chosen to build D-optimal experimental designs. The D-optimal experimental designs are tested by multi-position calibration experiment for tactical-grade DTGs. Test results show that, with the same cost, the fit uncertainty is reduced by about 50% by using the D-optimal 8-position experimental procedure, compared to using a defacto standard experimental procedure in ANSI/IEEE Std 813–1988. Furthermore, the new experimental procedure almost achieves optimal accuracy with only 12-position which is half the cost of the widely adopted 24-position experimental procedure for achieving optimal accuracy.

D-optimal design of DSC experiments for nth order kinetics

D-optimal experimental designs are developed for differential scanning calorimetry experiments conducted under non-isothermal, constant heating modes under the assumption that the kinetics can be described by an nth order model. Two separate cases are considered: single-scan experiments in which the entire rate of reaction versus temperature curve is available, albeit only at selected points, and multiple-scan experiments in which only the temperature at the maximum in the DSC curve is available but it is known at several heating rates. For the single-scan experiment, the D-optimal heating rate and critical extents of reaction where the data should be analyzed are determined. For the multiple-scan experiment the three D-optimal heating rates are determined. For both cases, a hypothetical example is given which illustrates the success and limitations of this approach.

D-optimal experimental designs for Freundlich and Langmuir adsorption isotherms

In this paper, D-optimal experimental designs are derived for three common adsorption isotherm models: the 2-parameter Freundlich, the 2-parameter Langmuir, and the 3-parameter Langmuir. For each of these liquid-solid adsorption models, the D-optimal design equations are derived from the information matrix and then solved numerically to determine the specific design values as a function of the model parameters. The variation of the design with the assumed values of the model parameters is discussed. For all three isotherms, it was determined that one support point was always at the maximum of the scaled experimental region. It was also determined that the D-optimal designs for all three isotherms were independent of the proportionality parameter in the model. For the three models and their associated D-optimal designs, the design optimality was verified using the general equivalence theorem.

A D-optimal design approach to constrained multiresponse robust design with prioritized mean and variance considerations

The traditional measures of ensuring quality, derived from viewing quality as simply conforming to specifications, does not adequately address today’s global markets and value-driven customers. Consequently, process improvement projects are often performed to improve operational performance in order to increase customer satisfaction. Therefore, process improvement methods, such as robust design, are important to industrial quality improvement initiatives. Yet, little of the work in the area of robust design has specifically addressed problems involving physical processing constraints that create an irregularly shape experimental region and the simultaneous consideration of multiple quality characteristics. To address these issues, we propose a new approach to robust design that utilizes D-optimal experimental designs in the context of multiresponse optimization problems in order to overcome the limitations of standard experimental approaches often used in robust design studies. Specifically, we formulate our optimization models as a preemptive nonlinear goal programming problem that focuses on consideration of the mean and variance. We also investigate the extension of optimization models traditionally used in robust design investigations to address multiple responses and compare the outcomes of our proposed approaches using a numerical example.

Simultaneous population optimal design for pharmacokinetic-pharmacodynamic experiments

Multiple outputs or measurement types are commonly gathered in biological experiments. Often, these experiments are expensive (such as clinical drug trials) or require careful design to achieve the desired information content. Optimal experimental design protocols could help alleviate the cost and increase the accuracy of these experiments. In general, optimal design techniques ignore between-individual variability, but even work that incorporates it (population optimal design) has treated simultaneous multiple output experiments separately by computing the optimal design sequentially, first finding the optimal design for one output (eg, a pharmacokinetic [PK] measurement) and then determining the design for the second output (eg, a pharmacodynamic [PD] measurement). Theoretically, this procedure can lead to biased and imprecise results when the second model parameters are also included in the first model (as in PK-PD models). We present methods and tools for simultaneous population D-optimal experimental designs, which simultaneously compute the design of multiple output experiments, allowing for correlation between model parameters. We then apply these methods to simulated PK-PD experiments. We compare the new simultaneous designs to sequential designs that first compute the PK design, fix the PK parameters, and then compute the PD design in an experiment. We find that both population designs yield similar results in designs for low sample number experiments, with simultaneous designs being possibly superior in situations in which the number of samples is unevenly distributed between outputs. Simultaneous population D-optimality is a potentially useful tool in the emerging field of experimental design.