Experimental design for model discrimination of thermodynamic models

The selection of items from a calibrated item bank for fixed-form tests is an optimal test design problem; this problem has been handled in the literature by mathematical programming models. A similar problem, however, arises when items are selected for an adaptive test or for testlets. This paper focuses on the similarities of optimal design of fixed-form tests, adaptive tests, and testlets within the framework of the general theory of optimal designs. A sequential design procedure is proposed that uses these similarities. This procedure not only enables optimal design of fixed-form tests, adaptive tests, and testlets, but is also very flexible. The procedure is easy to apply, and consistent estimates for the trait level distribution are obtained. Index terms: adaptive tests, consistency, efficiency, optimal test design, sequential procedure, test design, testlets.

This paper reviews the role of uncertainty in the identification of mathematical models of water quality and in the application of these models to problems of prediction. More specifically, four problem areas are examined in detail: uncertainty about model structure, uncertainty in the estimated model parameter values, the propagation of prediction errors, and the design of experiments in order to reduce the critical uncertainties associated with a model. The review is rather lengthy, and it has therefore been prepared in effect as two papers. There is a shorter, largely nontechnical version, which gives a quick impression of the current and future issues in the analysis of uncertainty in water quality modeling. Enclosed by this shorter discussion is the main body of the review dealing in turn with (1) identifiability and experimental design, (2) the generation of preliminary model hypotheses under conditions of sparse, grossly uncertain field data, (3) the selection and evaluation of model structure, (4) parameter estimation (model calibration), (5) checks and balances on the identified model, i.e., model “verification” and model discrimination, and (6) prediction error propagation. Much time is spent in discussing the algorithms of system identification, in particular, the methods of recursive estimation, and in relating these algorithms and the subject of identification to the problems of prediction uncertainty and first‐order error analysis. There are two obvious omissions from the review. It is not concerned primarily with either the development and solution of stochastic differential equations or the issue of decision making under uncertainty, although clearly some reference must be made to these topics. In brief, the review concludes (not surprisingly) that much work has been done on the analysis of uncertainty in the development of mathematical models of water quality, and much remains to be done. A lack of model identifiability has been an outstanding difficulty in the interpretation and explanation of past observed system behavior, and there is ample evidence to show that the “larger,” more “comprehensive” models are easily capable of generating highly uncertain predictions of future behavior. For the future of the subject, it is speculated that there is the possibility of progress in the development of novel algorithms for model structure identification, a need for new questions to be posed in the problem of prediction, and a distinct challenge to the conventional views of this review in the new forms of knowledge representation and manipulation now emerging from the field of artificial intelligence.

Using classical test theory and item response theory, this study applied sequential procedures to a real operational item pool in a variable-length computerized adaptive testing (CAT) to detect items whose security may be compromised. Moreover, this study proposed a hybrid threshold approach to improve the detection power of the sequential procedure while controlling the Type I error rate. The hybrid threshold approach uses a local threshold for each item in an early stage of the CAT administration, and then it uses the global threshold in the decision-making stage. Applying various simulation factors, a series of simulation studies examined which factors contribute significantly to the power rate and lag time of the procedure. In addition to the simulation study, a case study investigated whether the procedures are applicable to the real item pool administered in CAT and can identify potentially compromised items in the pool. This research found that the increment of probability of a correct answer (p-increment) was the simulation factor most important to the sequential procedures' ability to detect compromised items. This study also found that the local threshold approach improved power rates and shortened lag times when the p-increment was small. The findings of this study could help practitioners implement the sequential procedures using the hybrid threshold approach in real-time CAT administration.

Coarse-grained modeling is an outcome of scientific endeavors to address the broad spectrum of time and length scales encountered in polymer systems. However, providing a faithful structural and dynamic characterization/description is challenging for several reasons, particularly in the selection of appropriate model parameters. By using a hybrid particle- and field-based approach with a generalized energy functional expressed in terms of density fields, we explore model parameter spaces over a broad range and map the relation between parameter values with experimentally measurable quantities, such as single-chain scaling exponent, chain density, and interfacial and surface tension. The obtained parameter map allows us to successfully reproduce experimentally observed polymer solution assembly over a wide range of concentrations and solvent qualities. The approach is further applied to simulate structure and shape evolution in emulsified block copolymer droplets where concentration and domain shape change continuously during the process.

The Magma Chamber Simulator (MCS) is a thermodynamic model that computes the phase, thermal, and compositional evolution of a multiphase–multicomponent system of a Fractionally Crystallizing resident body of magma (i.e., melt ± solids ± fluid), linked wallrock that may either be assimilated as Anatectic melts or wholesale as Stoped blocks, and multiple Recharge reservoirs (RnASnFC system, where n is the number of user-selected recharge events). MCS calculations occur in two stages; the first utilizes mass and energy balance to produce thermodynamically constrained major element and phase equilibria information for an RnASnFC system; this tool is informally called MCS-PhaseEQ, and is described in a companion paper (Bohrson et al. 2020). The second stage of modeling, called MCS-Traces, calculates the RASFC evolution of up to 48 trace elements and seven radiogenic and one stable isotopic system (Sr, Nd, Hf, 3xPb, Os, and O) for the resident melt. In addition, trace element concentrations are calculated for bulk residual wallrock and each solid (± fluid) phase in the cumulate reservoir and residual wallrock. Input consists of (1) initial trace element concentrations and isotope ratios for the parental melt, wallrock, and recharge magmas/stoped wallrock blocks and (2) solid-melt and solid–fluid partition coefficients (optional temperature-dependence) for stable phases in the resident magma and residual wallrock. Output can be easily read and processed from tabulated worksheets. We provide trace element and isotopic results for the same example cases (FC, R2FC, AFC, S2FC, and R2AFC) presented in the companion paper. These simulations show that recharge processes can be difficult to recognize based on trace element data alone unless there is an independent reference frame of successive recharge events or if serial recharge magmas are sufficiently distinct in composition relative to the parental magma or magmas on the fractionation trend. In contrast, assimilation of wallrock is likely to have a notable effect on incompatible trace element and isotopic compositions of the contaminated resident melt. The magnitude of these effects depends on several factors incorporated into both stages of MCS calculations (e.g., phase equilibria, trace element partitioning, style of assimilation, and geochemistry of the starting materials). Significantly, the effects of assimilation can be counterintuitive and very different from simple scenarios (e.g., bulk mixing of magma and wallrock) that do not take account phase equilibria. Considerable caution should be practiced in ruling out potential assimilation scenarios in natural systems based upon simple geochemical “rules of thumb”. The lack of simplistic responses to open-system processes underscores the need for thermodynamical RASFC models that take into account mass and energy conservation. MCS-Traces provides an unprecedented and detailed framework for utilizing thermodynamic constraints and element partitioning to document trace element and isotopic evolution of igneous systems. Continued development of the Magma Chamber Simulator will focus on easier accessibility and additional capabilities that will allow the tool to better reproduce the documented natural complexities of open-system magmatic processes.

Stoichio-kinetic model discrimination and parameter identification in continuous microreactors

A method of model discrimination and parameter estimation in enzyme kinetics is proposed. The experimental design and analysis of the model are carried out simultaneously and the stopping rule for experimentation is deduced by the experimenter when the probabilities a posteriori indicate that one model is clearly superior to the rest. A FORTRAN77 program specifically developed for joint designs is given. The method is very powerful, as indicated by its usefulness in the discrimination between models. For example, it has been successfully applied to three cases of enzyme kinetics (a single-substrate Michaelian reaction with product inhibition, a single-substrate complex reaction and a two-substrate reaction). By using this method the most probable model and the estimates of the parameters can be obtained in one experimental session. The FORTRAN77 program is deposited as Supplementary Publication SUP 50134 (19 pages) at the British Library (Lending Division), Boston Spa, Wetherby, West Yorkshire LS23 7BQ, U.K., from whom copies can be obtained on the terms indicated in Biochem. J. (1986) 233, 5.

Bayesian experimental design (BED) is a framework that uses statistical models and decision making under uncertainty to optimise the cost and performance of a scientific experiment. Sequential BED, as opposed to static BED, considers the scenario where we can sequentially update our beliefs about the model parameters through data gathered in the experiment. A class of models of particular interest for the natural and medical sciences are implicit models, where the data generating distribution is intractable, but sampling from it is possible. Even though there has been a lot of work on static BED for implicit models in the past few years, the notoriously difficult problem of sequential BED for implicit models has barely been touched upon. We address this gap in the literature by devising a novel sequential design framework for parameter estimation that uses the Mutual Information (MI) between model parameters and simulated data as a utility function to find optimal experimental designs, which has not been done before for implicit models. Our approach uses likelihood-free inference by ratio estimation to simultaneously estimate posterior distributions and the MI. During the sequential BED procedure we utilise Bayesian optimisation to help us optimise the MI utility. We find that our framework is efficient for the various implicit models tested, yielding accurate parameter estimates after only a few iterations.

Thermodynamic modeling and uncertainty quantification of CO2-loaded aqueous MEA solutions

Thermodynamic models and experimental data exhibit systematic and random errors. The severity of their errors depends on their use, such as for process calculations in a process simulator. Similarly, the value of better thermodynamic models and/or data should be measured with reference to such use. Strategies for quantification of such thermodynamics-induced process uncertainties via Monte Carlo simulation, regression analysis, and analogies to optimization are described, with simple examples. Such approaches can be used for safety-factor/risk analysis, guidelines for process simulator use, experimental design, and model comparisons.

The soil conservation service curve number (SCS-CN) model is a widely utilized tool for estimating runoff and relies on two empirical parameters: the CN and the ratio of initial abstraction to maximum potential retention (λ). The determination of the parameters is via the empirical method or calculations based on actual data. However, few studies address the effect of rainfall on parameter selection, and collecting runoff data for model analysis is challenging. This study, taking the Nemor River Basin as the research region, investigates how the combination of CN and λ impacts the model in different rainfall conditions. Using runoff plots and reanalysis product data, the study reveals that: (1) the calculated methods outperformed the empirical method, increasing the Nash Sutcliffe efficiency coefficient from 0.34 to 0.65. (2) A higher λ value (0.2 compared to 0.02) reduces runoff and smoothes the runoff curve, which becomes less obvious with increasing CN. (3) The CN values exhibit a non-monotonic relationship with rainfall, initially decreasing before rising, highlighting the need to adjust the CN based on rainfall. Moreover, the SCS-CN model's performance with reanalysis data approximates that with actual data, confirming the viability of reanalysis datasets in this region.

The average transmitted power of a Ti:sapphire femtosecond (fs) laser beam through thin, oxidized silicon (Si) membranes in the 10–30 \(\, \upmu \text{m}\) thickness range at \(45{^\circ }\) external angle of incidence at first increases with the incident average laser power and then decreases considerably. This stationary state behavior is quantitatively reproduced by a recently derived model using an effective absorption coefficient \(\alpha _{\text {eff}} = \alpha _{1} + \alpha _{\text {FCA}}\), adding the coefficients of two independent linear absorption processes, \(\alpha _{1}\) of the indirect interband transition and \(\alpha _{\text {FCA}}\) of free charge carrier absorption (FCA). The relations between the empirical parameter \(\alpha _{\text {FCA}}\) and the model parameters \(n_{\text {eh}}\) (density of free charge carriers) and \(\sigma _{\text {eh}}\) (absorption cross section of free charge carriers) are found to depend on the Si membrane thickness and are briefly discussed. Using different sample thicknesses, experimental evidence has been found for a nonlinear intensity dependence of \(\alpha _{\text {FCA}}\)∼\(I_{0}^{n}\), where \(1\le n \le 3\) shows systematic variation. The FCA model in combination with its empirical fit parameters allowed to fully calculate the transmission behavior of the Si membranes within the investigated sample thickness and laser power regimes.

The applicability of model predictive accuracy and the selection of suitable model parameters for the membrane flux prediction and prediction of the filtration characteristics of dead-end filtration systems were assessed. In determining the model predictive accuracy using model parameters, the selection of suitable model parameters is of great significance. A series of experiments were conducted. The relative deviations, average deviations, and root mean square between the experimental and predicted values were calculated to evaluate predictive accuracy. The results showed that the model parameter has changed nonlinearly with operating conditions. Particularly, in the membrane flux prediction model, the time-dependent character of the model parameter can result in a considerable error. However, model prediction accuracy can be improved indisputably and the model will get higher predictive accuracy than that of other predictive models if the constant membrane fouling index in the flux prediction model is either replaced by the average membrane fouling index or it takes a composite parameter with free combination of the initial membrane fouling index, meantime, minimum and the maximum of the model parameter. These results give us a possibility to see the panorama of the variation of the model parameters in different situations and an idea to establish the foundation for building mathematical models.

ABSTRACTThe way of investigating a distribution knowing its interesting properties might be often inadequate when the shapes of two distributions are almost similar. In each of these circumstances, the accurate decision about the genesis of a random sample from any of the two parent distributions will be very much ambiguous even with the availability of the existing testing procedure of the circular data. A sequential discrimination procedure has been suggested which is also invariant to the sample size. The performance of the proposed discrimination procedure has been evaluated by checking its capability of detecting the genesis of the known samples from the two identically shaped wrapped distributions.

The phase equilibrium and thermodynamic information of the Nb-N system was reviewed and assessed by using thermodynamic models for the Gibbs energy of individual phases. Although there was a large amount of experimental information of the system, heat capacity data of the Nb2N and NbN were not available either in low or high temperatures. In the present study, low-temperature heat capacity and theoS298 values were estimated using estimated entropy Debye temperatures. Only the Nb2N (hcp) and NbN (fcc) nitrides were considered to be the true binary phases and were included in the present evaluation in addition to the N2 gas, liquid, andα-solid solution (bcc). Three thermodynamic models were used: a two-sublattice model for the solid solution phases, a substitutional model for the liquid phase, and an ideal-gas model for the N2 gas. The model parameters were evaluated by fitting to the selected data by means of a computer program. A consistent set of parameters was obtained which satisfactorily described most of the experimental and estimated data.