Uncertainty In Parameter Estimates Research Articles

A Coupled Sampling Design for Parameter Estimation in Microalgae Growth Experiment: Maximizing the Benefits of Uniform and Non-Uniform Sampling

As an important primary producer in aquatic ecosystems, the various parameters within the mathematical models are used to describe the growth of microalgae and need to be estimated by carefully designed experiments. Non-uniform sampling has proved to generate a deliberately optimized sampling temporal schedule that can benefit parameter estimation. However, the current non-uniform sampling method depends on prior knowledge of the nominal values of the model parameters. It also largely ignores the uncertainty associated with the nominal values, thus inducing unacceptable parameter estimates. This study focuses on the uncertainty problem and describes a new sampling design that couples the traditional uniform and non-uniform sampling schedules to benefit from the merits of both methods. Based on D-optimal design, we first derive the non-uniform optimal sampling points by maximizing the determinant of the Fisher information matrix. Then the confidence interval around the non-uniform sampling points is determined by Monte Carlo simulations based on the prior knowledge of parameter distribution. Finally, we wrap the non-uniform sampling points with the uniform sampling points within the confidence interval to obtain the ultimate optimal experimental design. Scenedesmus obliquus, whose growth curve follows a four-parameter model, was used as a case study. Compared with the traditional sampling design, the simulation results show that our proposed coupled sampling schedule can partly eliminate the uncertainty in parameter estimates caused by fixed systematic errors in observations. Our coupled sampling can also retain some advantages belonging to non-uniform sampling, in exploiting information maximization and managing the cost of sampling.

Bayesian analysis of quasar light curves with a running optimal average: new time delay measurements of COSMOGRAIL gravitationally lensed quasars

ABSTRACT We present a new method of modelling time-series data based on the running optimal average (ROA). By identifying the effective number of parameters for the ROA model, in terms of the shape and width of its window function and the times and accuracies of the data, we enable a Bayesian analysis, optimizing the ROA width, along with other model parameters, by minimizing the Bayesian information criterion (BIC) and sampling joint posterior parameter distributions using MCMC methods. For analysis of quasar light curves, our implementation of ROA modelling can measure time delays among light curves at different wavelengths or from different images of a lensed quasar and, in future work, be used to inter-calibrate light-curve data from different telescopes and estimate the shape and thus the power-density spectrum of the light curve. Our noise model implements a robust treatment of outliers and error-bar adjustments to account for additional variance or poorly quantified uncertainties. Tests with simulated data validate the parameter uncertainty estimates. We compare ROA delay measurements with results from cross-correlation and from javelin, which models light curves with a prior on the power-density spectrum. We analyse published COSMOGRAIL light curves of multilensed quasar light curves and present the resulting measurements of the inter-image time delays and detection of microlensing effects.

Bayesian estimation of copula parameters for wind speed models of dependence

Modelling the uncertainty of wind speed is essential in power flow analysis. Having abundant knowledge of the wind speed in an area is critical. A low volume of data can increase uncertainty in wind speed analysis. Spatial dependencies are often modelled before running probabilistic power flow and load flow analysis. Copulas are a popular way of capturing spatial dependence between multiple wind farms. Using NREL data from seven Northeastern United States wind farm sites, Bayesian inference will be used to determine the copula parameter uncertainty between weekly, daily, and hourly wind speed observations. This approach will be used on elliptical and single parameter Archimedean copulas. For each possible wind farm pair, an uninformative prior will be placed on the copula parameter. The resulting posterior will contain a distribution of copula parameter values based on the prior and the observed wind speed data. The posterior's credible interval is reviewed to determine the uncertainty in parameter estimation. The results show that using a data volume considerably more petite than 8760 hourly data points will result in more uncertainty in parameter estimation and inaccuracies in wind speed forecasting if using non-Bayesian methods for copula parameter estimation.

“Go Wild for a While!”: A New Test for Forecast Evaluation in Nested Models

In this paper, we present a new asymptotically normal test for out-of-sample evaluation in nested models. Our approach is a simple modification of a traditional encompassing test that is commonly known as Clark and West test (CW). The key point of our strategy is to introduce an independent random variable that prevents the traditional CW test from becoming degenerate under the null hypothesis of equal predictive ability. Using the approach developed by West (1996), we show that in our test, the impact of parameter estimation uncertainty vanishes asymptotically. Using a variety of Monte Carlo simulations in iterated multi-step-ahead forecasts, we evaluated our test and CW in terms of size and power. These simulations reveal that our approach is reasonably well-sized, even at long horizons when CW may present severe size distortions. In terms of power, results were mixed but CW has an edge over our approach. Finally, we illustrate the use of our test with an empirical application in the context of the commodity currencies literature.

Geological framework for assessing variability in subsurface piping parameters underneath dikes in the Rhine-Meuse delta, the Netherlands

Dike failure due to piping – concentrated flow of seepage water underneath the dike during periods of high flood water levels – has been recognized as a major component of flood risk. Simulation models to predict piping in risk assessments require detailed information on subsurface characteristics such as sediment grain size and thickness of overburden layers. Quantitative local determination of these characteristics poses a major challenge in natural environments with a heterogeneous substrate. Geological knowledge on the natural genesis and resulting structure of the subsurface may provide useful information on the spatial variability of substrate characteristics. When correctly implemented, available subsurface geological information can provide for a-priori identification of dike sections of which the subsurface is susceptible to piping (strategic to new data collection), and a-posteriori screening of new collected field data (identifying unexpected values). In these two ways it has potential to reduce the uncertainty in parameter estimates for calculations that determine the potential occurrence of piping at a specific site. Here we describe a framework for using geological subsurface information for these assessments. Based on existing digital geological mapping products and knowledge of the geological development of the Rhine-Meuse delta, we first compiled a map that distinguishes primary hydrologically relevant units: upper main aquifer sands (pre-deltaic subunits), topped by the deltaic wedge aquitard that is dissected by channel belt sand bodies (deltaic subunits). We then used these spatial divisions in an analysis of digital borehole data (>130.000 locations, UU-LLG dataset), to provide quantitative information on grain sizes of the very top of sand bodies and non-sand overburden thicknesses, split per subunit and summarized for sub regions (lower delta, central delta, upper delta, delta rim sectors). This framework enables us to demonstrate to what extent the median grain size of the top of sand deposits varies within the delta. We quantitatively determined (1) a delta-scale longitudinal downstream fining trend due to drops in specific stream power through changes in gradient and substrate erodibility, (2) pulsed variations associated with local uptake and reworking of pre-deltaic sediments within channel belts. Although tailored for the Rhine-Meuse delta, utilizing geological knowledge as a substantiation for grouping of subsurface data underneath local dike sections can be transposed for such applications in delta regions elsewhere. This regionalization of the delta plain can greatly help streamline data acquisition, anticipating growing availability of medium to high density subsurface datasets in increasingly urbanized deltas with managed rivers around the world.

Bayesian inference for generalized linear model with linear inequality constraints

Bayesian statistical inference for Generalized Linear Models (GLMs) with parameters lying on a constrained space is of general interest (e.g., in monotonic or convex regression), but often constructing valid prior distributions supported on a subspace spanned by a set of linear inequality constraints can be challenging, especially when some of the constraints might be binding leading to a lower dimensional subspace. For the general case with canonical link, it is shown that a generalized truncated multivariate normal supported on a desired subspace can be used. Moreover, it is shown that such prior distribution facilitates the construction of a general purpose product slice sampling method to obtain (approximate) samples from corresponding posterior distribution, making the inferential method computationally efficient for a wide class of GLMs with an arbitrary set of linear inequality constraints. The proposed product slice sampler is shown to be uniformly ergodic, having a geometric convergence rate under a set of mild regularity conditions satisfied by many popular GLMs (e.g., logistic and Poisson regressions with constrained coefficients). One of the primary advantages of the proposed Bayesian estimation method over classical methods is that uncertainty of parameter estimates is easily quantified by using the samples simulated from the path of the Markov Chain of the slice sampler. Numerical illustrations using simulated data sets are presented to illustrate the superiority of the proposed methods compared to some existing methods in terms of sampling bias and variances. In addition, real case studies are presented using data sets for fertilizer-crop production and estimating the SCRAM rate in nuclear power plants.

Jointly Calibrating Hydrologic Model Parameters and State Adjustments

AbstractA method is presented to address model state uncertainty in hydrologic model simulation. This is achieved by introducing tuneable parameters that allow adjustments to the model states. Excessive dimensionality is avoided by introducing only a limited number of parameters that control the index (timing) and size of the state adjustments. The method is designed to compensate for issues with hydrologic model structures, particularly those relevant to the soil moisture state in a rainfall‐runoff model. In the context of water resource planning and management, errors in the model states have often been overlooked as an important source of uncertainty and have the potential to significantly degrade model simulations. A synthetic study shows that a classical parameter estimation approach will produce biased distributions when state errors exist, and that the proposed state and parameter uncertainty estimation (SPUE) can remove the bias in parameter estimates for improved model simulations. In a real case study, SPUE and the classical approach are implemented in 46 sites around Australia. The results show that hydrologic parameter distributions for a selected conceptual model can be significantly different when accounting for state uncertainty. This has large implications for scenario modeling since it puts into dispute how to determine appropriate parameters for such studies. SPUE outperforms the classical approach in a range of calibration and validation metrics, particularly for sites that contain zero flows. Future work involves testing SPUE with different hydrologic models and likelihood formulations, and enhancing rigor by explicitly accounting for observational data uncertainty.

Predicting the dynamics of establishing tree populations: A framework for statistical inference and lessons for data collection

Abstract Under global change, there is an urgent need to forecast the dynamics of establishing tree populations. However, tree population dynamics are slow and historical data on these dynamics are rare. This raises the question whether tree population dynamics can be reconstructed from data collected at a single time point. Doing so poses challenges for modelling, data collection and model‐data integration. We present a Bayesian framework that uses multiple data types to parametrize an individual‐based model (IBM) for the growth of establishing tree populations. The framework combines likelihood‐based Bayesian inference and approximate Bayesian computation (ABC). Using this framework, we assess the information content of three data types (recruitment data, dendrochronological data describing individual growth and molecular markers characterizing within‐population pedigrees) by comparing the bias and uncertainty of parameter estimates and model forecasts obtained under different simulated scenarios of data availability. The combination of all data types leads to accurate forecasts of the future state of tree populations, despite large uncertainties in some parameter estimates. Dendrochronological data were the most informative of the examined data types. Combining data types improved forecasts of population state. Nevertheless, for a given parameter related to a given process, combining data types did not improve estimates compared to using only the data type most closely related to the process. The presented Bayesian framework allows to infer the dynamics of establishing tree populations from data collected at a single time point. It helps to optimally allocate limited resources for data collection in order to rapidly improve the understanding and forecasting of tree population dynamics.

A residual-based test for multivariate GARCH models using transformed quadratic residuals

This paper provides a residual-based approach to examine the adequacy of multivariate GARCH models. We employ the transformed quadratic residuals to construct the residual-based statistic and derive its correct asymptotic distribution by taking into account the impact of parameter estimation uncertainty. The simulation results indicate that the residual-based test achieves reasonable sizes and comparable powers. An empirical application further shows the usefulness of the proposed test.

Determining the maximum information gain and optimizing experimental design in neutron reflectometry using the Fisher information.

An approach based on the Fisher information (FI) is developed to quantify the maximum information gain and optimal experimental design in neutron reflectometry experiments. In these experiments, the FI can be calculated analytically and used to provide sub-second predictions of parameter uncertainties. This approach can be used to influence real-time decisions about measurement angle, measurement time, contrast choice and other experimental conditions based on parameters of interest. The FI provides a lower bound on parameter estimation uncertainties, and these are shown to decrease with the square root of the measurement time, providing useful information for the planning and scheduling of experimental work. As the FI is computationally inexpensive to calculate, it can be computed repeatedly during the course of an experiment, saving costly beam time by signalling that sufficient data have been obtained or saving experimental data sets by signalling that an experiment needs to continue. The approach's predictions are validated through the introduction of an experiment simulation framework that incorporates instrument-specific incident flux profiles, and through the investigation of measuring the structural properties of a phospholipid bilayer.

LUCIDGames: Online Unscented Inverse Dynamic Games for Adaptive Trajectory Prediction and Planning

Existing game-theoretic planning methods assume that the robot knows the objective functions of the other agents a priori while, in practical scenarios, this is rarely the case. This letter introduces LUCIDGames, an inverse optimal control algorithm that is able to estimate the other agents' objective functions in real time, and incorporate those estimates online into a receding-horizon game-theoretic planner. LUCIDGames solves the inverse optimal control problem by recasting it in a recursive parameter-estimation framework. LUCIDGames uses an unscented Kalman filter (UKF) to iteratively update a Bayesian estimate of the other agents' cost function parameters, improving that estimate online as more data is gathered from the other agents' observed trajectories. The planner then takes account of the uncertainty in the Bayesian parameter estimates of other agents by planning a trajectory for the robot subject to uncertainty ellipse constraints. The algorithm assumes no explicit communication or coordination between the robot and the other agents in the environment. An MPC implementation of LUCIDGames demonstrates real-time performance on complex autonomous driving scenarios with an update frequency of 40 Hz. Empirical results demonstrate that LUCIDGames improves the robot's performance over existing game-theoretic and traditional MPC planning approaches. Our implementation of LUCIDGames is available at https://github.com/RoboticExplorationLab/LUCIDGames.jl.

Data Needs for Economic Evaluations of Screening in Pediatric Primary Care: A Research Framework.

Data Needs for Economic Evaluations of Screening in Pediatric Primary Care: A Research Framework.

Stochastic Local Interaction Model: An Alternative to Kriging for Massive Datasets

Classical geostatistical methods face serious computational challenges if they are confronted with large sets of spatially distributed data. We present a simplified stochastic local interaction (SLI) model for computationally efficient spatial prediction that can handle large data. The SLI method constructs a spatial interaction matrix (precision matrix) that accounts for the data values, their locations, and the sampling density variations without user input. We show that this precision matrix is strictly positive definite. The SLI approach does not require matrix inversion for parameter estimation, spatial prediction, and uncertainty estimation, leading to computational procedures that are significantly less intensive computationally than kriging. The precision matrix involves compact kernel functions (spherical, quadratic, etc.) which enable the application of sparse matrix methods, thus improving computational time and memory requirements. We investigate the proposed SLI method with a data set that includes approximately 11500 drill-hole data of coal thickness from Campbell County (Wyoming, USA). We also compare SLI with ordinary kriging (OK) in terms of estimation performance, using cross validation analysis, and computational time. According to the validation measures used, SLI performs slightly better in estimating seam thickness than OK in terms of cross-validation measures. In terms of computation time, SLI prediction is 3 to 25 times (depending on the size of the kriging neighborhood) faster than OK for the same grid size.

A spatially explicit N-mixture model for the estimation of disease prevalence

This article develops an approximate N-mixture model for infectious disease counts that accounts for under-reporting as well as spatial dependence induced by person-to-person spread of disease. We employ the model to estimate actual case counts in Oregon of chlamydia, an easily-treated but usually asymptomatic sexually transmitted disease. We describe a combined parametric bootstrap to account for uncertainty in parameter estimates as well as sampling variability in actual case counts. A simulation study illustrates that our method performs well in many scenarios when the model is correctly specified, and also gives reasonable results when the model is misspecified, and no spatial dependence exists.

Uncertainty Estimation in Hydrogeological Forecasting with Neural Networks: Impact of Spatial Distribution of Rainfalls and Random Initialization of the Model

Neural networks are used to forecast hydrogeological risks, such as droughts and floods. However, uncertainties generated by these models are difficult to assess, possibly leading to a low use of these solutions by water managers. These uncertainties are the result of three sources: input data, model architecture and parameters and their initialization. The aim of the study is, first, to calibrate a model to predict Champagne chalk groundwater level at Vailly (Grand-Est, France), and, second, to estimate related uncertainties, linked both to the spatial distribution of rainfalls and to the parameter initialization. The parameter uncertainties are assessed following a previous methodology, using nine mixed probability density functions (pdf), thus creating models of correctness. Spatial distribution of rainfall uncertainty is generated by swapping three rainfall inputs and then observing dispersion of 27 model outputs. This uncertainty is incorporated into models of correctness. We show that, in this case study, an ensemble model of 40 different initializations is sufficient to estimate parameter uncertainty while preserving quality. Logistic, Gumbel and Raised Cosine laws fit the distribution of increasing and decreasing groundwater levels well, which then allows the establishment of models of correctness. These models of correctness provide a confidence interval associated with the forecasts, with an arbitrary degree of confidence chosen by the user. These methodologies have proved to have significant advantages: the rigorous design of the neural network model has allowed the realisation of models able to generalize outside of the range of the data used for training. Furthermore, it is possible to flexibly choose the confidence index according to the hydrological configuration (e.g., recession or rising water table).

Improved Decision-Making Confidence Using Item-Based Pharmacometric Model: Illustration with a Phase II Placebo-Controlled Trial

This study aimed to illustrate how a new methodology to assess clinical trial outcome measures using a longitudinal item response theory–based model (IRM) could serve as an alternative to mixed model repeated measures (MMRM). Data from the EXACT (Exacerbation of chronic pulmonary disease tool) which is used to capture frequency, severity, and duration of exacerbations in COPD were analyzed using an IRM. The IRM included a graded response model characterizing item parameters and functions describing symptom-time course. Total scores were simulated (month 12) using uncertainty in parameter estimates. The 50th (2.5th, 97.5th) percentiles of the resulting simulated differences in average total score (drug minus placebo) represented the estimated drug effect (95%CI), which was compared with published MMRM results. Furthermore, differences in sample size, sensitivity, specificity, and type I and II errors between approaches were explored. Patients received either oral danirixin 75 mg twice daily (n = 45) or placebo (n = 48) on top of standard of care over 52 weeks. A step function best described the COPD symptoms-time course in both trial arms. The IRM improved precision of the estimated drug effect compared to MMRM, resulting in a sample size of 2.5 times larger for the MMRM analysis to achieve the IRM precision. The IRM showed a higher probability of a positive predictive value (34%) than MMRM (22%). An item model–based analysis data gave more precise estimates of drug effect than MMRM analysis for the same endpoint in this one case study.

PDG10 Cost-Effectiveness Analysis of Gnrh Analogue ALONE Versus Combined Growth Hormone for Girls with Central Precocious Puberty

Central precocious puberty (CPP) can lead to short final height in adult-hood and social problems during childhood. Cost of treatment is high and need to be added to the cost and stress associated with the follow-up clinic visits and periodic injections. We aimed to evaluate the cost-effectiveness of GnRHa alone versus combined growth hormone (GH) treatment for girls with CPP in a Chinese societal setting. A decision model was constructed to evaluate costs and effectiveness (quality-adjusted life years, QALYs) of two clinical treatment from the third-party payer's perspective. Estimates for probabilistic parameters and QALYs were derived from published studies. Screening and treatment costs were taken from Chinese tertiary hospitals. Incremental cost-effectiveness ratios (ICER) were calculated to compare the treatments. All costs and treatment effects were discounted at 5%. Sensitivity analyses were performed to evaluate the impact of uncertainty in parameter estimation. The total cost per patient was CNY¥122,919.96 (US$18,978.84) for GnRHa plus GH and CNY¥34,277.12 (US$5292.39) for GnRHa alone. The ICER for GnRHa plus GH was CNY¥9,915,381.32(US$1,530,934.88) /QALY versus GnRHa. The most sensitive parameters in the model were the effective of height gain (final adult height - predictive adult height). Our study indicated that GnRH combined GH was not cost-effective versus GnRH in the Chinese population.

Estimation of groundwater recharge from groundwater levels using nonlinear transfer function noise models and comparison to lysimeter data

Abstract. The estimation of groundwater recharge is of paramount importance to assess the sustainability of groundwater use in aquifers around the world. Estimation of the recharge flux, however, remains notoriously difficult. In this study the application of nonlinear transfer function noise (TFN) models using impulse response functions is explored to simulate groundwater levels and estimate groundwater recharge. A nonlinear root zone model that simulates recharge is developed and implemented in a TFN model and is compared to a more commonly used linear recharge model. An additional novel aspect of this study is the use of an autoregressive–moving-average noise model so that the remaining noise fulfills the statistical conditions to reliably estimate parameter uncertainties and compute the confidence intervals of the recharge estimates. The models are calibrated on groundwater-level data observed at the Wagna hydrological research station in the southeastern part of Austria. The nonlinear model improves the simulation of groundwater levels compared to the linear model. The annual recharge rates estimated with the nonlinear model are comparable to the average seepage rates observed with two lysimeters. The recharges estimates from the nonlinear model are also in reasonably good agreement with the lysimeter data at the smaller timescale of recharge per 10 d. This is an improvement over previous studies that used comparable methods but only reported annual recharge rates. The presented framework requires limited input data (precipitation, potential evaporation, and groundwater levels) and can easily be extended to support applications in different hydrogeological settings than those presented here.

A Bayesian model to estimate land surface phenology parameters with harmonized Landsat 8 and Sentinel-2 images

We develop a Bayesian Land Surface Phenology (LSP) model and examine its performance using Enhanced Vegetation Index (EVI) observations derived from the Harmonized Landsat Sentinel-2 (HLS) dataset. Building on previous work, we propose a double logistic function that, once couched within a Bayesian model, yields posterior distributions for all LSP parameters. We assess the efficacy of the Normal, Truncated Normal, and Beta likelihoods to deliver robust LSP parameter estimates. Three case studies are presented and used to explore aspects of the proposed model. The first case study, conducted over forested pixels within an HLS tile, explores choice of likelihood and space-time varying HLS data availability for long-term average LSP parameter point and uncertainty estimation. The second case study, conducted over the same pixels using only 2018 data, compares annual LSP parameter estimates from our proposed models with those generated using methods described in Bolton et al. (2020). The third case study, conducted on a small area of interest within the HLS tile on an annual time-step (2014–2019), further examines the impact of sample size and choice of likelihood on annual LSP parameter estimates in addition to assessing potential for the proposed models to inform LSP change detection analysis. Results indicate that while the Truncated Normal and Beta likelihoods are theoretically preferable when the vegetation index is bounded, all three likelihoods performed similarly when the number of index observations is sufficiently large and values are not near the index bounds. The case studies demonstrate how pixel-level LSP parameter posterior distributions can be used to propagate uncertainty through subsequent analysis. As a companion to this article, we provide an open-source R package rsBayes and supplementary data and code used to reproduce the analysis results. The proposed model specification and software implementation delivers computationally efficient, statistically robust, and inferentially rich LSP parameter posterior distributions at the pixel-level across massive raster time series datasets. Modeling functions in the rsBayes package and supplementary code sections are threaded, allowing for the use of multiple processing cores to further speed up model fitting for massive datasets. Using a 64 CPU workstation, the first case study analysis took ~3 days to run using the Beta likelihood model. However, processing time decreases linearly as the number of CPU cores increases. We expect that run times for this LSP modeling approach will decrease substantially as the power of new computing systems increases over time.

A new elastic characterization method for anisotropic bilayer specimens via Bayesian resonant ultrasound spectroscopy

A novel nondestructive method for complete elastic characterization of substrate-coating bilayer specimens with distinct anisotropic layers via resonant ultrasound spectroscopy (RUS) and Bayesian inversion is developed here. Bayesian formulations of the RUS inversion problem—of quantifying elastic properties given a measured list of resonance frequencies recorded from a single, typically small, precisely fabricated, macroscopically homogeneous, linear-elastic specimen—are a recent development. Here we report the first Bayesian formulation of the bilayer problem, and through a series of practical examples, demonstrate novel parameter estimation capabilities of our open-source CmdStan-RUS code. Finding specimen geometry and the number of resonance modes used for inversion strongly govern the ability to retrieve individual elastic moduli. The concept of “invertability” is explored for a range of relevant geometries using virtual specimens that resemble experimental bilayers of plasma sprayed ceramic coatings on single crystal metallic substrates. A range of Bayesian posterior evaluation methods are addressed, particularly considering the large computational cost of the bilayer forward model. Laplace approximation methods are thus developed and implemented for bilayer geometry design space modeling and expedient estimates of parameter uncertainties. Ideal specimen design, different noise models, the influence of prior distributions, dual-likelihood fits incorporating measurements of the bare substrate, and how Bayesian RUS methods differ from traditional RUS optimization are discussed.