EVOLUTIONARY HIERARCHICAL CREDIBILITY

2017 ◽  
Vol 48 (1) ◽  
pp. 339-374
Author(s):  
Greg Taylor
Keyword(s):  
Parameter Estimation ◽  
Kalman Filter ◽  
Random Walk ◽  
Specific Form ◽  
Tree Structure ◽  
Parameter Estimates ◽  
Model Parameters ◽  
Estimation Of Parameters ◽  
The Matrix ◽  
Over Time

AbstractThe hierarchical credibility model was introduced, and extended, in the 70s and early 80s. It deals with the estimation of parameters that characterize the nodes of a tree structure. That model is limited, however, by the fact that its parameters are assumed fixed over time. This causes the model's parameter estimates to track the parameters poorly when the latter are subject to variation over time. This paper seeks to remove this limitation by assuming the parameters in question to follow a process akin to a random walk over time, producing an evolutionary hierarchical model. The specific form of the model is compatible with the use of the Kalman filter for parameter estimation and forecasting. The application of the Kalman filter is conceptually straightforward, but the tree structure of the model parameters can be extensive, and some effort is required to retain organization of the updating algorithm. This is achieved by suitable manipulation of the graph associated with the tree. The graph matrix then appears in the matrix calculations inherent in the Kalman filter. A numerical example is included to illustrate the application of the filter to the model.

scholarly journals Sequential Parameter Estimation for Mammalian Cell Model Based on In Silico Design of Experiments

Processes ◽  
2018 ◽  
Vol 6 (8) ◽  
pp. 100 ◽  
Author(s):  
Zhenyu Wang ◽  
Hana Sheikh ◽  
Kyongbum Lee ◽  
Christos Georgakis
Keyword(s):  
Parameter Estimation ◽  
Mammalian Cells ◽  
Cell Model ◽  
Cell Metabolism ◽  
Model Performance ◽  
Sequential Estimation ◽  
Parameter Estimates ◽  
Model Parameters ◽  
Algebraic Equations ◽  
Estimation Of Parameters

Due to the complicated metabolism of mammalian cells, the corresponding dynamic mathematical models usually consist of large sets of differential and algebraic equations with a large number of parameters to be estimated. On the other hand, the measured data for estimating the model parameters are limited. Consequently, the parameter estimates may converge to a local minimum far from the optimal ones, especially when the initial guesses of the parameter values are poor. The methodology presented in this paper provides a systematic way for estimating parameters sequentially that generates better initial guesses for parameter estimation and improves the accuracy of the obtained metabolic model. The model parameters are first classified into four subsets of decreasing importance, based on the sensitivity of the model’s predictions on the parameters’ assumed values. The parameters in the most sensitive subset, typically a small fraction of the total, are estimated first. When estimating the remaining parameters with next most sensitive subset, the subsets of parameters with higher sensitivities are estimated again using their previously obtained optimal values as the initial guesses. The power of this sequential estimation approach is illustrated through a case study on the estimation of parameters in a dynamic model of CHO cell metabolism in fed-batch culture. We show that the sequential parameter estimation approach improves model accuracy and that using limited data to estimate low-sensitivity parameters can worsen model performance.

Multi-day flow forecasting using the Kalman filter

1991 ◽  
Vol 18 (2) ◽  
pp. 320-327 ◽  
Author(s):  
Murray A. Fitch ◽  
Edward A. McBean
Keyword(s):  
Kalman Filter ◽  
Prediction Model ◽  
Linear Prediction ◽  
Measured Data ◽  
System Model ◽  
Parameter Estimates ◽  
Model Parameters ◽  
Forecasting Performance ◽  
Optimal Forecasting ◽  
Linear Prediction Model

A model is developed for the prediction of river flows resulting from combined snowmelt and precipitation. The model employs a Kalman filter to reflect uncertainty both in the measured data and in the system model parameters. The forecasting algorithm is used to develop multi-day forecasts for the Sturgeon River, Ontario. The algorithm is shown to develop good 1-day and 2-day ahead forecasts, but the linear prediction model is found inadequate for longer-term forecasts. Good initial parameter estimates are shown to be essential for optimal forecasting performance. Key words: Kalman filter, streamflow forecast, multi-day, streamflow, Sturgeon River, MISP algorithm.

Calibration of microscopic traffic-flow models using multiple data sources

2010 ◽  
Vol 368 (1928) ◽  
pp. 4497-4517 ◽  
Author(s):  
Serge Hoogendoorn ◽  
Raymond Hoogendoorn
Keyword(s):  
Driving Simulator ◽  
Likelihood Estimation ◽  
Joint Estimation ◽  
Data Sources ◽  
Parameter Estimates ◽  
Model Parameters ◽  
Ratio Test ◽  
Estimation Of Parameters ◽  
Flow Models ◽  
Car Following

Parameter identification of microscopic driving models is a difficult task. This is caused by the fact that parameters—such as reaction time, sensitivity to stimuli, etc.—are generally not directly observable from common traffic data, but also due to the lack of reliable statistical estimation techniques. This contribution puts forward a new approach to identifying parameters of car-following models. One of the main contributions of this article is that the proposed approach allows for joint estimation of parameters using different data sources, including prior information on parameter values (or the valid range of values). This is achieved by generalizing the maximum-likelihood estimation approach proposed by the authors in previous work. The approach allows for statistical analysis of the parameter estimates, including the standard error of the parameter estimates and the correlation of the estimates. Using the likelihood-ratio test, models of different complexity (defined by the number of model parameters) can be cross-compared. A nice property of this test is that it takes into account the number of parameters of a model as well as the performance. To illustrate the workings, the approach is applied to two car-following models using vehicle trajectories of a Dutch freeway collected from a helicopter, in combination with data collected with a driving simulator.

Integrating Parameter Estimation Solutions From the Time and Time-Scale Domains

2009 ◽  
Author(s):  
James R. McCusker ◽  
Kourosh Danai
Keyword(s):  
Parameter Estimation ◽  
Least Squares ◽  
Time Scale ◽  
Prediction Error ◽  
Nonlinear Least Squares ◽  
Integrated Approach ◽  
Parameter Estimates ◽  
Model Parameters ◽  
Single Model ◽  
Iterative Estimation

A method of parameter estimation was recently introduced that separately estimates each parameter of the dynamic model [1]. In this method, regions coined as parameter signatures, are identified in the time-scale domain wherein the prediction error can be attributed to the error of a single model parameter. Based on these single-parameter associations, individual model parameters can then be estimated for iterative estimation. Relative to nonlinear least squares, the proposed Parameter Signature Isolation Method (PARSIM) has two distinct attributes. One attribute of PARSIM is to leave the estimation of a parameter dormant when a parameter signature cannot be extracted for it. Another attribute is independence from the contour of the prediction error. The first attribute could cause erroneous parameter estimates, when the parameters are not adapted continually. The second attribute, on the other hand, can provide a safeguard against local minima entrapments. These attributes motivate integrating PARSIM with a method, like nonlinear least-squares, that is less prone to dormancy of parameter estimates. The paper demonstrates the merit of the proposed integrated approach in application to a difficult estimation problem.

scholarly journals Application of ensemble transform data assimilation methods for parameter estimation in reservoir modeling

2018 ◽  
Vol 25 (4) ◽  
pp. 731-746 ◽  
Author(s):  
Sangeetika Ruchi ◽  
Svetlana Dubinkina
Keyword(s):  
Parameter Estimation ◽  
Kalman Filter ◽  
Data Assimilation ◽  
Particle Filter ◽  
Reservoir Modeling ◽  
Model Parameters ◽  
Uncertain Parameters ◽  
Initial Ensemble ◽  
Leading Mode ◽  
Ensemble Transform

Abstract. Over the years data assimilation methods have been developed to obtain estimations of uncertain model parameters by taking into account a few observations of a model state. The most reliable Markov chain Monte Carlo (MCMC) methods are computationally expensive. Sequential ensemble methods such as ensemble Kalman filters and particle filters provide a favorable alternative. However, ensemble Kalman filter has an assumption of Gaussianity. Ensemble transform particle filter does not have this assumption and has proven to be highly beneficial for an initial condition estimation and a small number of parameter estimations in chaotic dynamical systems with non-Gaussian distributions. In this paper we employ ensemble transform particle filter (ETPF) and ensemble transform Kalman filter (ETKF) for parameter estimation in nonlinear problems with 1, 5, and 2500 uncertain parameters and compare them to importance sampling (IS). The large number of uncertain parameters is of particular interest for subsurface reservoir modeling as it allows us to parameterize permeability on the grid. We prove that the updated parameters obtained by ETPF lie within the range of an initial ensemble, which is not the case for ETKF. We examine the performance of ETPF and ETKF in a twin experiment setup, where observations of pressure are synthetically created based on the known values of parameters. For a small number of uncertain parameters (one and five) ETPF performs comparably to ETKF in terms of the mean estimation. For a large number of uncertain parameters (2500) ETKF is robust with respect to the initial ensemble, while ETPF is sensitive due to sampling error. Moreover, for the high-dimensional test problem ETPF gives an increase in the root mean square error after data assimilation is performed. This is resolved by applying distance-based localization, which however deteriorates a posterior estimation of the leading mode by largely increasing the variance due to a combination of less varying localized weights, not keeping the imposed bounds on the modes via the Karhunen–Loeve expansion, and the main variability explained by the leading mode. A possible remedy is instead of applying localization to use only leading modes that are well estimated by ETPF, which demands knowledge of which mode to truncate.

scholarly journals A Bayesian Approach to Estimating Reciprocal Effects with the Bivariate STARTS Model using Markov Chain Monte Carlo

2021 ◽  
Author(s):  
Oliver Lüdtke ◽  
Alexander Robitzsch ◽  
Esther Ulitzsch
Keyword(s):  
Bayesian Approach ◽  
Prior Distribution ◽  
Parameter Estimates ◽  
Model Parameters ◽  
Reciprocal Effects ◽  
Ml Estimation ◽  
Model Parameterization ◽  
Lagged Effects ◽  
The Bayesian Approach ◽  
Over Time

The bivariate Stable Trait, AutoRegressive Trait, and State (STARTS) model provides a general approach for estimating reciprocal effects between constructs over time. However, previous research has shown that this model is difficult to estimate using the maximum likelihood (ML) method (e.g., nonconvergence). In this article, we introduce a Bayesian approach for estimating the bivariate STARTS model and implement it in the software Stan. We discuss issues of model parameterization and show how appropriate prior distributions for model parameters can be selected. Specifically, we propose the four-parameter beta distribution as a flexible prior distribution for the autoregressive and cross-lagged effects. Using a simulation study, we show that the proposed Bayesian approach provides more accurate estimates than ML estimation in challenging data constellations. An example is presented to illustrate how the Bayesian approach can be used to stabilize the parameter estimates of the bivariate STARTS model.

scholarly journals Data assimilation in integrated hydrological modelling in the presence of observation bias

2015 ◽  
Vol 12 (8) ◽  
pp. 8131-8173 ◽  
Author(s):  
J. Rasmussen ◽  
H. Madsen ◽  
K. H. Jensen ◽  
J. C. Refsgaard
Keyword(s):  
Parameter Estimation ◽  
Kalman Filter ◽  
Stream Flow ◽  
Groundwater Modeling ◽  
Optimal Parameter ◽  
Estimation Methods ◽  
Model Parameters ◽  
Stream Discharge ◽  
Groundwater Head ◽  
Observation Bias

Abstract. The use of bias-aware Kalman filters for estimating and correcting observation bias in groundwater head observations is evaluated using both synthetic and real observations. In the synthetic test, groundwater head observations with a constant bias and unbiased stream discharge observations are assimilated in a catchment scale integrated hydrological model with the aim of updating stream discharge and groundwater head, as well as several model parameters relating to both stream flow and groundwater modeling. The Colored Noise Kalman filter (ColKF) and the Separate bias Kalman filter (SepKF) are tested and evaluated for correcting the observation biases. The study found that both methods were able to estimate most of the biases and that using any of the two bias estimation methods resulted in significant improvements over using a bias-unaware Kalman Filter. While the convergence of the ColKF was significantly faster than the convergence of the SepKF, a much larger ensemble size was required as the estimation of biases would otherwise fail. Real observations of groundwater head and stream discharge were also assimilated, resulting in improved stream flow modeling in terms of an increased Nash-Sutcliffe coefficient while no clear improvement in groundwater head modeling was observed. Both the ColKF and the SepKF tended to underestimate the biases, which resulted in drifting model behavior and sub-optimal parameter estimation, but both methods provided better state updating and parameter estimation than using a bias-unaware filter.

Parameter Estimation of Nonlinear Biomedical Systems Using Extended Kalman Filter Algorithm

2017 ◽  
pp. 76-99 ◽  
Author(s):  
Kamalanand Krishnamurthy
Keyword(s):  
Parameter Estimation ◽  
Kalman Filter ◽  
Extended Kalman Filter ◽  
Computational Models ◽  
Three Dimensional ◽  
Model Parameters ◽  
State Variables ◽  
Three Dimensional Model ◽  
Biomedical Systems ◽  
Estimation Of Model Parameters

Parameter estimation is a central issue in mathematical modelling of biomedical systems and for the development of patient specific models. The technique of estimating parameters helps in obtaining diagnostic information from computational models of biological systems. However, in most of the biomedical systems, the estimation of model parameters is a challenging task due to the nonlinearity of mathematical models. In this chapter, the method of estimation of nonlinear model parameters from measurements of state variables, using the extended Kalman filter, is extensively explained using an example of the three-dimensional model of the HIV/AIDS system.

Interpretation of Simulation Studies for Efficient Estimation of Population Pharmacokinetic Parameters

1993 ◽  
Vol 27 (9) ◽  
pp. 1034-1039 ◽  
Author(s):  
Ene I. Ette ◽  
Andrew W. Kelman ◽  
Catherine A. Howie ◽  
Brian Whiting
Keyword(s):  
Parameter Estimation ◽  
Efficient Estimation ◽  
Experimental Unit ◽  
Pharmacokinetic Parameters ◽  
Population Pharmacokinetic ◽  
Parameter Estimates ◽  
Model Parameters ◽  
Good Precision ◽  
Simulation Studies ◽  
Individual Parameter

OBJECTIVE: To develop new approaches for evaluating results obtained from simulation studies used to determine sampling strategies for efficient estimation of population pharmacokinetic parameters. METHODS: One-compartment kinetics with intravenous bolus injection was assumed and the simulated data (one observation made on each experimental unit [human subject or animal]), were analyzed using NONMEM. Several approaches were used to judge the efficiency of parameter estimation. These included: (1) individual and joint confidence intervals (CIs) coverage for parameter estimates that were computed in a manner that would reveal the influence of bias and standard error (SE) on interval estimates; (2) percent prediction error (%PE) approach; (3) the incidence of high pair-wise correlations; and (4) a design number approach. The design number (Φ) is a new statistic that provides a composite measure of accuracy and precision (using SE). RESULTS: The %PE approach is useful only in examining the efficiency of estimation of a parameter considered independently. The joint CI coverage approach permitted assessment of the accuracy and reliability of all model parameter estimates. The Φ approach is an efficient method of achieving an accurate estimate of parameter(s) with good precision. Both the Φ for individual parameter estimation and the overall Φ for the estimation of model parameters led to optimal experimental design. CONCLUSIONS: Application of these approaches to the analyses of the results of the study was found useful in determining the best sampling design (from a series of two sampling times designs within a study) for efficient estimation of population pharmacokinetic parameters.

scholarly journals Fault Detection in Wastewater Treatment Systems Using Multiparametric Programming

Processes ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 231 ◽  
Author(s):  
Ernie Che Mid ◽  
Vivek Dua
Keyword(s):  
Wastewater Treatment ◽  
Parameter Estimation ◽  
Fault Detection ◽  
Normal Operation ◽  
Parameter Estimates ◽  
Model Parameters ◽  
Algebraic Equations ◽  
Multiparametric Programming ◽  
Nonlinear Algebraic Equations ◽  
Wastewater Treatment Systems

In this work, a methodology for fault detection in wastewater treatment systems, based on parameter estimation, using multiparametric programming is presented. The main idea is to detect faults by estimating model parameters, and monitoring the changes in residuals of model parameters. In the proposed methodology, a nonlinear dynamic model of wastewater treatment was discretized to algebraic equations using Euler’s method. A parameter estimation problem was then formulated and transformed into a square system of parametric nonlinear algebraic equations by writing the optimality conditions. The parametric nonlinear algebraic equations were then solved symbolically to obtain the concentration of substrate in the inflow, , inhibition coefficient, , and specific growth rate, , as an explicit function of state variables (concentration of biomass, ; concentration of organic matter, ; concentration of dissolved oxygen, ; and volume, ). The estimated model parameter values were compared with values from the normal operation. If the residual of model parameters exceeds a certain threshold value, a fault is detected. The application demonstrates the viability of the approach, and highlights its ability to detect faults in wastewater treatment systems by providing quick and accurate parameter estimates using the evaluation of explicit parametric functions.

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