On the geostatistical approach to the inverse problem

Investigation on drought characteristics such as severity, duration, and frequency is crucial for water resources planning and management in a river basin. While the methodology for multivariate drought frequency analysis is well established by applying the copulas, the estimation on the associated parameters by various parameter estimation methods and the effects on the obtained results have not yet been investigated. This research aims at conducting a comparative analysis between the maximum likelihood parametric and non-parametric method of the Kendall \(\tau \) estimation method for copulas parameter estimation. The methods were employed to study joint severity–duration probability and recurrence intervals in Karkheh River basin (southwest Iran) which is facing severe water-deficit problems. Daily streamflow data at three hydrological gauging stations (Tang Sazbon, Huleilan and Polchehr) near the Karkheh dam were used to draw flow duration curves (FDC) of these three stations. The \(Q_{75}\) index extracted from the FDC were set as threshold level to abstract drought characteristics such as drought duration and severity on the basis of the run theory. Drought duration and severity were separately modeled using the univariate probabilistic distributions and gamma–GEV, LN2–exponential, and LN2–gamma were selected as the best paired drought severity–duration inputs for copulas according to the Akaike Information Criteria (AIC), Kolmogorov–Smirnov and chi-square tests. Archimedean Clayton, Frank, and extreme value Gumbel copulas were employed to construct joint cumulative distribution functions (JCDF) of droughts for each station. Frank copula at Tang Sazbon and Gumbel at Huleilan and Polchehr stations were identified as the best copulas based on the performance evaluation criteria including AIC, BIC, log-likelihood and root mean square error (RMSE) values. Based on the RMSE values, nonparametric Kendall-\(\tau \) is preferred to the parametric maximum likelihood estimation method. The results showed greater drought return periods by the parametric ML method in comparison to the nonparametric Kendall \(\tau \) estimation method. The results also showed that stations located in tributaries (Huleilan and Polchehr) have close return periods, while the station along the main river (Tang Sazbon) has the smaller return periods for the drought events with identical drought duration and severity.

The stochastic geostatistical inversion approach is widely used in subsurface inverse problems to estimate unknown parameter fields and corresponding uncertainty from noisy observations. However, the approach requires a large number of forward model runs to determine the Jacobian or sensitivity matrix, thus the computational and storage costs become prohibitive when the number of unknowns, m, and the number of observations, n increase. To overcome this challenge in large‐scale geostatistical inversion, the Principal Component Geostatistical Approach (PCGA) has recently been developed as a “matrix‐free” geostatistical inversion strategy that avoids the direct evaluation of the Jacobian matrix through the principal components (low‐rank approximation) of the prior covariance and the drift matrix with a finite difference approximation. As a result, the proposed method requires about K runs of the forward problem in each iteration independently of m and n, where K is the number of principal components and can be much less than m and n for large‐scale inverse problems. Furthermore, the PCGA is easily adaptable to different forward simulation models and various data types for which the adjoint‐state method may not be implemented suitably. In this paper, we apply the PCGA to representative subsurface inverse problems to illustrate its efficiency and scalability. The low‐rank approximation of the large‐dimensional dense prior covariance matrix is computed through a randomized eigen decomposition. A hydraulic tomography problem in which the number of observations is typically large is investigated first to validate the accuracy of the PCGA compared with the conventional geostatistical approach. Then the method is applied to a large‐scale hydraulic tomography with 3 million unknowns and it is shown that underlying subsurface structures are characterized successfully through an inversion that involves an affordable number of forward simulation runs. Lastly, we present a joint inversion of head and tracer test data using MODFLOW and MT3DMS as coupled black‐box forward simulation solvers. These applications demonstrate the advantages of the PCGA, i.e., the scalability to high‐dimensional inverse problems and the ability to utilize multiple forward models as black boxes.

On the estimation of parameters of log-unsaturated conductivity covariance from solute transport data

Pulsars are the most stable natural frequency standards. They can be applied to a number of principal problems of modern astronomy and time-keeping metrology. The full exploration of pulsar properties requires obtaining unbiased estimates of the spin and orbital parameters. These estimates depend essentially on the random noise component being revealed in the residuals of time of arrivals (TOA). In the present paper, the influence of low-frequency ("red") timing noise with spectral indices from 1 to 6 on TOA residuals, variances, and covariances of estimates of measured parameters of single and binary pulsars are studied. In order to determine their functional dependence on time, an analytic technique of processing of observational data in time domain is developed which takes into account both stationary and non-stationary components of noise. Our analysis includes a simplified timing model of a binary pulsar in a circular orbit and procedure of estimation of pulsar parameters and residuals under the influence of red noise. We reconfirm that uncorrelated white noise of errors of measurements of TOA brings on gradually decreasing residuals, variances and covariances of all parameters. On the other hand, we show that any red noise causes the residuals, variances, and covariances of certain parameters to increase with time. Hence, the low frequency noise corrupts our observations and reduces experimental possibilities for better tests of General Relativity Theory. We also treat in detail the influence of a polynomial drift of noise on the residuals and fitting parameters. Results of the analitic analysis are used for discussion of a statistic describing stabilities of kinematic and dynamic pulsar time scales.

Clustered iterative stochastic ensemble method for multi-modal calibration of subsurface flow models

BackgroundParameter estimation is often the bottlenecking step in biological system modeling. For ordinary differential equation (ODE) models, the challenge in this estimation has been attributed to not only the lack of parameter identifiability, but also computational issues such as finding globally optimal parameter estimates over highly multidimensional search space. Recent methods using incremental estimation approach could alleviate the computational difficulty by performing the parameter estimation one-reaction-at-a-time. However, incremental estimation strategies usually require data smoothing and are known to produce biased parameter estimates.ResultsIn this article, we presented a new parameter estimation method called integrated flux parameter estimation (IFPE). We employed the integral form of the ODE such that we could compute the integral of reaction fluxes from time-series concentration data without data smoothing. Here, we formulated the parameter estimation as a nested optimization problem. In the outer optimization, we performed a minimization of model prediction errors over parameters associated with a subset of reactions labeled as independent. The dimension of the independent reaction subset was equal to the degrees of freedom in the calculation of integrated fluxes (IF) from concentration data. We selected the independent reactions such that given their IF values, the IFs of the remaining (dependent) reactions could be uniquely determined. Meanwhile, in the inner optimization, we estimated the model parameters associated with the dependent reactions, one-reaction-at-a-time, by minimizing the dependent IF prediction errors. We demonstrated the performance of the IFPE method using two case studies: a generalized mass action model of a branched pathway and a lin-log ODE model of Lactococcus lactis glycolytic pathway.ConclusionsThe IFPE significantly outperformed standard simultaneous parameter estimation in terms of computational efficiency and scaling. In comparison to incremental parameter estimation (IPE) method, the IFPE produced parameter estimates with significantly lower bias and did not require time-series data smoothing. The advantages of IFPE over the IPE however came at the cost of a small increase in the computational time.Electronic supplementary materialThe online version of this article (doi:10.1186/s12918-014-0127-x) contains supplementary material, which is available to authorized users.

In this work we develop some parameter and spectral estimation methods in Magnetic Resonance Spectroscopy Signals (MRS). We do this with a dual purpose: first, to enable the full estimation process for the constituent parameters of the time-domain msignal: Amplitude, frequency, damping constant and phase. Second, to generate the parameter estimation in the frequency domain in order to evaluate the effect of the application of time domain estimation methods [2].

A long-standing problem at the frontier of biomechanical studies is to develop fast methods capable of estimating material properties from clinical data. In this paper, we have studied three surrogate models based on machine learning (ML) methods for fast parameter estimation of left ventricular (LV) myocardium. We use three ML methods named K-nearest neighbour (KNN), XGBoost and multi-layer perceptron (MLP) to emulate the relationships between pressure and volume strains during the diastolic filling. Firstly, to train the surrogate models, a forward finite-element simulator of LV diastolic filling is used. Then the training data are projected in a low-dimensional parametrized space. Next, three ML models are trained to learn the relationships of pressure–volume and pressure–strain. Finally, an inverse parameter estimation problem is formulated by using those trained surrogate models. Our results show that the three ML models can learn the relationships of pressure–volume and pressure–strain very well, and the parameter inference using the surrogate models can be carried out in minutes. Estimated parameters from both the XGBoost and MLP models have much less uncertainties compared with the KNN model. Our results further suggest that the XGBoost model is better for predicting the LV diastolic dynamics and estimating passive parameters than other two surrogate models. Further studies are warranted to investigate how XGBoost can be used for emulating cardiac pump function in a multi-physics and multi-scale framework.

Parameter estimation of the three-parameter Weibull distribution is an important problem in reliability analysis and statistical modeling. Random right-censored data are widely encountered in engineering practice. Conventional least squares (LS) methods usually construct the empirical cumulative distribution function (CDF) based on rank statistics. However, this empirical assumption cannot adequately capture the nonlinear variation in failure probability with time in the Weibull distribution. To address this limitation, an iterative conditional probability based on conditional failure probability (ICP-CDF) is proposed. The method uses the parameter estimates obtained from the conventional LS approach as initial values, adjusts the ranks of failure data according to conditional failure probabilities, and updates the empirical CDF accordingly. Within a unified least squares estimation framework, an ICP-CDF-LS parameter estimation method is developed, in which both the CDF and distribution parameters are updated iteratively. Simulation studies and case analyses demonstrate that, compared with the LS and MLE methods, the proposed approach achieves superior overall performance in terms of estimation accuracy and stability, making it more suitable for practical engineering applications.

Many existing software reliability models are based on the assumption of statistical independence among successive software failures. In reality, this assumption could be easily violated. In recent years, efforts have been made to relax this unrealistic assumption and a software reliability modeling framework considering failure correlation was developed by Goseva-Popstojanova and Trivedi. However, some important issues that are crucial to the proposed modeling framework to be used in practice remain unstudied, such as the method of estimation of model parameters. In this paper, we study the parameter estimation problem for the software reliability modeling framework developed in. We propose a relationship function among model parameters which could be essential to the reduction of the number of parameters to be estimated as well as to the reliability prediction using the proposed modeling framework. Two parameter estimation methods are developed based on deferent types of data available, using Maximum Likelihood Estimation (MLE) method. Simulation results preliminarily show that the accuracy of both proposed estimation methods seem to be satisfactory.

The fact that Pearson's correlation coefficient and effect size are perspective functions of covariance parameters demonstrates that how covariance is defined is one of the most important issues in data analysis. We suggest that covariance analysis for pairwise numeric, categorical, and mixed numeric-categorical data types are mathematically distinct problems. This is because of the disparate algebraic properties and systematic effects associated with numeric and categorical quantities. We examine the weighted least squares (WLS) formulation of linear regression and obtain definitions for heteroscedastic covariance and variance. Covariance and variance as functions of centered variable vectors are instrumental quantities. Then it is essential that the instrumental effects cancel when dividing covariance by the variance to estimate the slope in linear regression. The tensor product form of the covariance demonstrates that the composite properties of variable vectors are intrinsic to covariate data analysis. The solution of the inverse problem for linear regression takes the form of a relation between slope and covariance parameters, and requires the specification of an error model for the data; otherwise, the inverse problem is ill-posed. We propose that, in current practice, the term “effect size” is ambiguous because it does not distinguish between the different algebraic components of the inverse problem in a case-control data analysis. Then, it is necessary to identify the analogs of WLS covariance for case-control data and to distinguish between covariance and functional parameters in effect size analysis. The development of effect size methodology for studies of complex systems is complicated by the fact that the functional inverse problem is ill-posed.

The Pareto distribution has been introduced in Chapter 19. Also discussed there is a brief review of literature and methods of estimating its parameters. Further elaboration of the distribution is given in Chapter 20. Methods of parameter estimation were reviewed by Hosking and Wallis (1987). The methods of moments (MOM), maximum likelihood estimation (MLE) and probability weighted moments (PWM) were included in the review. Guo and Singh (1992) and Singh and Guo (1995) employed the principle of maximum entropy (POME) to develop a new competitive method of parameter estimation (Singh and Rajagopal, 1986) for the 3-parameter generalized Pareto (GP3) distribution and compared it with MOM, MLE and PWM using Monte Carlo simulated data The parameter estimates yielded by POME were either superior or comparable for high skewness.KeywordsRoot Mean Square ErrorMaximum Likelihood EstimationPareto DistributionGeneralize Pareto DistributionQuantile EstimationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Wind power curve (WPC) is an important index of wind turbines, and it plays an important role in wind power prediction and condition monitoring of wind turbines. Motivated by model parameter estimation of logistic function in WPC modelling, aimed at the problem of selecting initial value of model parameter estimation and local optimum result, based on the combination of genetic algorithm and least square estimation method, a genetic least square estimation (GLSE) method of parameter estimation is proposed, and the global optimum estimation result can be obtained. Six evaluation indices including the root mean square error, the coefficient of determination R2, the mean absolute error, the mean absolute percentage error, the improved Akaike information criterion and the Bayesian information criterion are used to select the optimal power curve model in the different candidate models, and avoid the model’s over-fitting. Finally, to predict the annual energy production and output power of wind turbines, a two-component Weibull mixture distribution wind speed model and five-parameter logistic function power curve model are applied in a wind farm of Jiangsu Province, China. The results show that the GLSE approach proposed in this paper is feasible and effective in WPC modelling and wind power prediction, which can improve the accuracy of model parameter estimation, and five-parameter logistic function can be preferred compared with high-order polynomial and four-parameter logistic function when the fitting accuracy is close.

A new method of parameter estimation has been developed to estimate a newly chosen group of eight parameters of the canine arterial system. These parameters, which are physically meaningful, include arterial radii, Young's modulus for the aortic wall, aortic length, and peripheral resistances. The method is model-based, and depends upon minimization of a criterion which combines weighted integrals of absolute values of pressure and flow with errors in certain average and peak values of waveforms. A minimization algorithm which is a modification of the Hooke and Jeeves pattern search method was used. The entire scheme was implemented on a hybrid computer. This method of parameter estimation was developed and checked by use of model-to-model studies. Then dog-to-model parameter estimation runs were made, using data recorded in a dog experiment. Estimated parameters were found to check direct measurements where these were possible, and variations in estimated parameters from data obtained after various drug interventions were found to fall into expected patterns. Future applications of the methods developed would appear to have promise for obtaining rapid estimates of human parameters from noninvasive measurements made in clinical situations.

This paper is concerned with the parameter estimation and adaptive control problem for a sort of strict‐feedback nonlinear systems with unknown parameters. In order to deal with the nonlinear functions, we develop a kind of least‐squares estimator in the parameter estimation part. With such an estimator, we design the state‐feedback control law and analyze the properties of the closed‐loop system. To be concrete, the parameter estimator can guarantee the boundedness of parameter estimate error. At the same time, the adaptive control law is proposed to make the system states converge to zero and the equilibrium stable globally under some common assumptions. Finally, simulations are given to demonstrate the theoretical results.