Parameter identification and uncertainty quantification in stochastic state space models and its application to texture analysis

The main goal of this Focus Point volume is to explore new advances on theoretical and computational techniques for uncertainty quantification (UQ) of modeling and simulation in relevant problems appearing in physics sciences. The papers submitted to this volume were received by invitation. As a result we have collected a number of high-quality contributions dealing with a rich and update picture of UQ analysis that includes its interplay with laws in physics problems, mainly, formulated via differential equations. The following text highlights some of the primary aspects of the topics covered as well as the contributions of the published papers. The consideration of randomness in the analysis of mathematical models for analyzing relevant problems related to health is currently a main research topic. The paper "The role of noise in the tumor dynamics under chemotherapy treatment" proposes a two-dimensional stochastic nonlinear differential system, taking into account the impact of chemotherapy on both the tumor and the healthy cells. The randomness is driven by two uncorrelated Gaussian noises with different levels of intensity. Authors analyze this interesting problem combining a number of techniques that include bifurcation diagrams, phase portraits, time series and probability density plots. The contribution is aimed at offering a trustworthy explanation of tumor growth under medical treatment. The Focus Point volume continues with the study of a second model also belonging to Medicine, specifically, to Epidemiology. In "Random resampling numerical simulations applied to a SEIR compartmental model," authors revisit the SEIR model addressed to study the dynamics of COVID-19 in Spain during its first pandemic-wave in 2020. Authors concentrate on estimating model parameters taking into account data uncertainties by means of bootstrap sampling. The validation of the stochastic analysis is performed by different goodness-of-fit measures. They show that the point estimates obtained by the bootstrap samples improve the ones of the original data. The use of fractional and stochastic calculus has emerged, mainly, during the last decades with important applications in Physics. In "Hölder continuity of mild solutions of space-time fractional stochastic heat equation driven by colored noise," authors combine the Caputo fractional derivative with the fractional Laplace operator to analyze the heat equation with noise driven by a colored noise. The paper establishes Hölder continuity of mild random field solutions of the aforementioned PDE by, first, taking advantage of key properties associated to the space-time fractional Green function. In "Asymptotic controllability of nonlinear Fokker-Planck equations," the author studies a problem related with stabilization of the control for a Fokker-Plank PDE. This equation often appears in kinetic theory of statistical mechanics and, contains, as a particular case, the classical Einstein-Smoluchowski equation, which describes the time evolution of the probability density of Brownian particles in a field of forms with a given potential. In the paper, one identifies a class of final states (reachable states) for which the problem has solution. The results can have interesting applications in control theory and physics. The stochastic of multiphysics systems is often a challenging problem because different physical scales and physical effects must be properly combined to accurately conduct the corresponding analysis. When the study is done in the practical setting, it usually involves the rigorous treatment of large sets of data. In the paper "Random stick-slip oscillations in a multiphysics system" (Fig. 12 ), one combines stick-slip oscillations, electromechanical coupling and uncertainties to characterize and quantify the stochastic response of a multiphysics system in terms of two physical quantities, duration and position, which are dependent random variables that must be jointly studied.

Model and analysis of single event transient sensitivity based on uncertainty quantification

Highly efficient Bayesian updating using metamodels: An adaptive Kriging-based approach

Cone penetration test (CPT) is widely used in geotechnical site investigation. Several methods have been developed to automatically identify soil stratigraphy based on a single CPT sounding, which is an essential step in interpreting CPT data. Among existing methods, Bayesian methods allow probabilistic reasoning of soil stratigraphy from a given CPT sounding with explicit uncertainty quantification. However, Bayesian soil stratification methods based on CPT data tend to be computationally demanding or even prohibitive, particularly at sites with excessive soil strata due to the high dimensionality of soil stratification models. To address this issue, this study proposes a Bayesian method for fast soil stratification based on a single CPT sounding using a trans-dimensional sampling technique known as reversible jump Markov chain Monte Carlo (RJMCMC) simulation. The RJMCMC simulation is specifically tailored for Bayesian soil stratification by developing proposal distributions for three different Markov chain moves. The proposed method is illustrated and validated using benchmark examples and two real-life CPT soundings. It accomplishes CPT-based probabilistic soil stratification within 1–2 min. The computational time is generally unaffected by CPT sounding depth or the number of soil layers, making it feasible to stratify soil profiles with an excessive number of soil layers.

Identification of linear models in view of robust control design requires the identification of a control‐relevant nominal model, and a quantification of model uncertainty. In this paper a procedure is presented to quantify the model uncertainty of any prespecified nominal model, from a sequence of measurement data of input and output signals from a plant. By employing a nonparametric empirical transfer function estimate (ETFE), we are able to split the model uncertainty into three parts: the inherent uncertainty in the data due to data imperfections, the unmodelled dynamics in the nominal model, and the uncertainty due to interpolation. A frequency‐dependent hard error bound is constructed, and results are given for tightening the bound through appropriate input design.

A new method is proposed based on the finite difference method (FDM), differential evolution algorithm and Markov Chain Monte Carlo (MCMC) simulation to identify water quality model parameters of an open channel in a long distance water transfer project. Firstly, this parameter identification problem is considered as a Bayesian estimation problem and the forward numerical model is solved by FDM, and the posterior probability density function of the parameters is deduced. Then these parameters are estimated using a sampling method with differential evolution algorithm and MCMC simulation. Finally this proposed method is compared with FDM-MCMC by a twin experiment. The results show that the proposed method can be used to identify water quality model parameters of an open channel in a long distance water transfer project under different scenarios better with fewer iterations, higher reliability and anti-noise capability compared with FDM-MCMC. Therefore, it provides a new idea and method to solve the traceability problem in sudden water pollution accidents.

In an age of both big data and increasing strain on water resources, sound management decisions often rely on numerical models. Numerical models provide a physics-based framework for assimilating and making sense of information that by itself only provides a limited description of the hydrologic system. Often, numerical models are the best option for quantifying even intuitively obvious connections between human activities and water resource impacts. However, despite many recent advances in model data assimilation and uncertainty quantification, the process of constructing numerical models remains laborious, expensive, and opaque, often precluding their use in decision making. Modflow-setup aims to provide rapid and consistent construction of MODFLOW groundwater models through robust and repeatable automation. Common model construction tasks are distilled in an open-source, online code base that is tested and extensible through collaborative version control. Input to Modflow-setup consists of a single configuration file that summarizes the workflow for building a model, including source data, construction options, and output packages. Source data providing model structure and parameter information including shapefiles, rasters, NetCDF files, tables, and other (geolocated) sources to MODFLOW models are read in and mapped to the model discretization, using Flopy and other general open-source scientific Python libraries. In a few minutes, an external array-based MODFLOW model amenable to parameter estimation and uncertainty quantification is produced. This paper describes the core functionality of Modflow-setup, including a worked example of a MODFLOW 6 model for evaluating pumping impacts to a lake in central Wisconsin, United States.

This paper proposes a data-driven approximate Bayesian computation framework for parameter estimation and uncertainty quantification of epidemic models, which incorporates two novelties: (i) the identification of the initial conditions by using plausible dynamic states that are compatible with observational data; (ii) learning of an informative prior distribution for the model parameters via the cross-entropy method. The new methodology's effectiveness is illustrated with the aid of actual data from the COVID-19 epidemic in Rio de Janeiro city in Brazil, employing an ordinary differential equation-based model with a generalized SEIR mechanistic structure that includes time-dependent transmission rate, asymptomatics, and hospitalizations. A minimization problem with two cost terms (number of hospitalizations and deaths) is formulated, and twelve parameters are identified. The calibrated model provides a consistent description of the available data, able to extrapolate forecasts over a few weeks, making the proposed methodology very appealing for real-time epidemic modeling.

This paper is one in a sequence of presentations that consider the problem of uncertainty quantification (UQ) in model verification and validation. The particular sub-topic considered here is uncertainty quantification in modeling. The objectives of the paper are to enumerate some of the various sources of uncertainty in models, as-built systems and experimental data, to define uncertainty and errors and differences between them, to introduce different types of comparative difference metrics that can be used for uncertainty quantification, and to present examples illustrating their uses. Particular emphasis is placed on the ability to quantify modeling uncertainty that includes model form uncertainty, i.e. an epistemic uncertainty that results from an incomplete understanding of the physics of the phenomenon being modeled, and/or undetected modeling error.

Ensemble methods present a practical framework for parameter estimation, performance prediction, and uncertainty quantification in subsurface flow and transport modeling. In particular, the ensemble Kalman filter (EnKF) has received significant attention for its promising performance in calibrating heterogeneous subsurface flow models. Since an ensemble of model realizations is used to compute the statistical moments needed to perform the EnKF updates, large ensemble sizes are needed to provide accurate updates and uncertainty assessment. However, for realistic problems that involve large-scale models with computationally demanding flow simulation runs, the EnKF implementation is limited to small-sized ensembles. As a result, spurious numerical correlations can develop and lead to inaccurate EnKF updates, which tend to underestimate or even eliminate the ensemble spread. Ad hoc practical remedies, such as localization, local analysis, and covariance inflation schemes, have been developed and applied to reduce the effect of sampling errors due to small ensemble sizes. In this paper, a fast linear approximate forecast method is proposed as an alternative approach to enable the use of large ensemble sizes in operational settings to obtain more improved sample statistics and EnKF updates. The proposed method first clusters a large number of initial geologic model realizations into a small number of groups. A representative member from each group is used to run a full forward flow simulation. The flow predictions for the remaining realizations in each group are approximated by a linearization around the full simulation results of the representative model (centroid) of the respective cluster. The linearization can be performed using either adjoint-based or ensemble-based gradients. Results from several numerical experiments with two-phase and three-phase flow systems in this paper suggest that the proposed method can be applied to improve the EnKF performance in large-scale problems where the number of full simulation is constrained.

International Financial Reporting Standard 17 (IFRS 17) is a global accounting standard developed by the International Accounting Standards Board (IASB). It guides the accounting treatment of insurance contracts in financial statements. In IFRS 17, claim reserving is vital for measuring insurance contracts and determining the Contractual Service Margin (CSM). The claim reserving process involves estimating expected claim payouts over the contract’s duration. These estimates are crucial for calculating the liability for incurred claims and determining the CSM, representing unearned profit. Claim data is treated as ordered data by sorting the run-off triangle row-wise, resulting univariate time series with missing values. The state space model provides a framework for modeling time series, assuming system development over time. By taking the log of claim data so that the data is normally distributed, the lognormal state space model, along with Kalman recursion algorithms, can be applied for Incurred but Not Reported (IBNR) projections. The lognormal state space model’s IBNR reserve estimate is more precise and sufficient to compensate loss than the chain ladder method. Furthermore, the performance of the lognormal state space model outperforms the chain ladder method by means of model validation using RMSE, MAE, RSE, and R2.

We propose a Bayesian approach for parameter uncertainty quantification in macroscopic traffic flow models from cross-sectional data. We consider both a simple first order model consisting in the mass conservation equation and its second order version including a speed evolution equation. A bias term is introduced and modeled as a Gaussian process to account for the traffic flow models limitations. We validate the results comparing the error in the macroscopic variables (flow, speed, density) for both models, showing that second order models globally perform better in reconstructing traffic quantities of interest.

<bold>background</bold>Uncertainty quantification and nuclear data adjustment of reactor core physical parameters can improve the calculation accuracy, thus improving the safety of reactor core design parameters. In recent years, domestic and foreign research institutions have developed a series of programs to analyze the sensitivity and uncertainty of core physical parameters, and use various methods to adjust the nuclear data to reduce the uncertainty of the calculation results.<bold>Purpose</bold>This study aims to conduct uncertainty quantification and nuclear data adjustment of the core physical parameters of the small lead-bismuth cooled fast reactor (SVBR), which holds broad prospects in the fourth generation nuclear energy technology.<bold>Methods</bold>Firstly, the SVBR was simplified, and the sensitivity analysis of critical reactivity, control rod value and power distribution were carried out by using NECP-SARAX and UNICORN programs. Then,138 groups of critical experimental devices were selected from the International Criticality Safety Benchmark Evaluation Project (ICSBEP) for uncertainty quantification. A similarity analysis was conducted between these 138 critical experiments and the SVBR. Finally, nuclear data adjustment was performed using the critical experiment devices with high similarity coefficients, with the isotopes ²³⁵U and ²³⁸U involved in the adjustment. The adjusted database was then applied to the back-substitution calculations.<bold>Results</bold>The selected critical experiment has a certain similarity with the SVBR. By using the adjusted nuclear data for back substitution calculations, the calculation uncertainty of critical reactivity is reduced from 2.59% to 0.496%, which improves the calculation accuracy.<bold>Conclusions</bold>It is verified in this study by back-substitution calculation that the uncertainty of the physical parameter calculation results after nuclear data adjustment has been significantly reduced, while the adjustments to the nuclear cross-sections are all smaller than the measurement errors. This indicates that the nuclear data adjustment can provide more reliable calculation results for the design of the lead-bismuth fast reactor and offers valuable reference for its design.

Accurate needle placement is essential for prostate biopsy. Recently, transperineal prostate biopsies are receiving renewed interest due to concern over infection from conventional transrectal biopsies. However, accurate needle placement is more challenging in the transperineal approach than in the transrectal approach due to the long insertion distance leading to a large targeting error and repeated insertion attempts. Improved procedure planning tools that can predict the deviation of the needle can potentially reduce the targeting error and number of insertion attempts. Prediction of deflection magnitude requires a model of biopsy needle deflection, which in turn requires information about tissue material properties. However, material properties of tissue in patients cannot be easily obtained. Accounting for this uncertainty in patient tissue properties requires a model capable of quantifying uncertainty in needle deflection as a function of a distribution of tissue properties. A Monte Carlo uncertainty quantification requires 1000s of samples, but it is not possible to obtain this many samples in a short enough time for intraoperative procedure planning using published needle deflection predictionmodels. This work seeks to develop a model of needle deflection fast enough for use in intraoperative procedure planning, validate this model against experimental results, and integrate it into a Monte Carlo uncertainty quantificationmodel. This work used a mechanics-based model of biopsy needle deflection to train a Fourier feature neural network (FFNN) model in order to make predictions with a low computational cost. Both models were validated against experimental data. The neural network model was used in a Monte Carlo uncertainty quantification model to quantify uncertainty in needle deflection arising from uncertain tissue mechanicalproperties. This work (1) implemented a mechanics-based model and a FFNN model. Both models were validated against previously published experiments carried out with tissue phantoms. Both models showed close agreement with the experimental data. (2) We showed that our FFNN model was more accurate than a baseline ordinary least squares model, introducing only about 0.3-mm tip deflection error compared to the mechanics-based model. We also showed that our FFNN model makes unbiased predictions with respect to the amount of deflection. (3) We demonstrated a Monte Carlo uncertainty quantification model of needle deflection with a low computational cost of about 20 CPU s. We used our uncertainty quantification model to show how the depth, stiffness, and magnitude of uncertainty in a layer of tissue affect needle deflection. In addition, we showed a simple clinical example of the use of our model. This work demonstrates a Monte Carlo uncertainty quantification model of needle deflection with a low computational cost. This method shows promise for future applications in procedure planning for prostate biopsies as well as other transperineal procedures conducted with flexible needles such as cryoablation andbrachytherapy.

The directional flame thermometer (DFT) is a robust device used to measure heat fluxes in harsh environments such as fire scenarios but is large when compared to other standard heat flux measurement devices. To better understand the uncertainties associated with heat flux measurements in these environments, a Bayesian framework is utilized to propagate uncertainties of both known and unknown parameters describing the thermal model of a modified, smaller DFT. Construction of the modified DFT is described along with a derivation of the thermal model used to predict the incident heat flux to its sensing surface. Parameters of the model are calibrated to data collected using a Schmidt–Boelter (SB) gauge with an accuracy of ±3% at incident heat fluxes of 5 kW/m2, 10 kW/m2, and 15 kW/m2. Markov Chain Monte Carlo simulations were used to obtain posterior distributions for the free parameters of the thermal model as well as the modeling uncertainty. The parameter calibration process produced values for the free parameters that were similar to those presented in the literature with relative uncertainties at 5 kW/m2, 10 kW/m2, and 15 kW/m2 of 17%, 9%, and 7%, respectively. The derived model produced root-mean-squared errors between the prediction and SB gauge output of 0.37, 0.77, and 1.13 kW/m2 for the 5, 10, and 15 kW/m2 cases, respectively, compared to 0.53, 1.12, and 1.66 kW/m2 for the energy storage method (ESM) described in ASTM E3057.