Residual Error Model Research Articles

A Bayesian method for missing rainfall estimation using a conceptual rainfall–runoff model

ABSTRACTThe estimation of missing rainfall data is an important problem for data analysis and modelling studies in hydrology. This paper develops a Bayesian method to address missing rainfall estimation from runoff measurements based on a pre-calibrated conceptual rainfall–runoff model. The Bayesian method assigns posterior probability of rainfall estimates proportional to the likelihood function of measured runoff flows and prior rainfall information, which is presented by uniform distributions in the absence of rainfall data. The likelihood function of measured runoff can be determined via the test of different residual error models in the calibration phase. The application of this method to a French urban catchment indicates that the proposed Bayesian method is able to assess missing rainfall and its uncertainty based only on runoff measurements, which provides an alternative to the reverse model for missing rainfall estimates.

Effect of heteroscedasticity treatment in residual error models on model calibration and prediction uncertainty estimation

The heteroscedasticity treatment in residual error models directly impacts the model calibration and prediction uncertainty estimation. This study compares three methods to deal with the heteroscedasticity, including the explicit linear modeling (LM) method and nonlinear modeling (NL) method using hyperbolic tangent function, as well as the implicit Box-Cox transformation (BC). Then a combined approach (CA) combining the advantages of both LM and BC methods has been proposed. In conjunction with the first order autoregressive model and the skew exponential power (SEP) distribution, four residual error models are generated, namely LM-SEP, NL-SEP, BC-SEP and CA-SEP, and their corresponding likelihood functions are applied to the Variable Infiltration Capacity (VIC) hydrologic model over the Huaihe River basin, China. Results show that the LM-SEP yields the poorest streamflow predictions with the widest uncertainty band and unrealistic negative flows. The NL and BC methods can better deal with the heteroscedasticity and hence their corresponding predictive performances are improved, yet the negative flows cannot be avoided. The CA-SEP produces the most accurate predictions with the highest reliability and effectively avoids the negative flows, because the CA approach is capable of addressing the complicated heteroscedasticity over the study basin.

Challenges in longitudinal exposure-response modeling of data from complex study designs: a case study of modeling CDAI score for ustekinumab in patients with Crohn's disease.

Informative exposure-response modeling of clinical endpoints is important in drug development to identify optimum dose and dosing regimens. Despite much recent progress in mechanism-based longitudinal modeling of clinical data, challenges remain in clinical trials of diseases such as Crohn's disease, where a commonly used composite endpoint Crohn's Disease Activity Index (CDAI) has considerable variation in its administration and scoring between different assessors and complex study designs typically include maintenance phases with randomized withdrawal re-randomizations and other response driven dose adjustments. This manuscript illustrates the complexities of exposure-response modeling of such composite endpoint data through a latent-variable based Indirect Response model framework for CDAI scores using data from three phase III trials of ustekinumab in patients with moderate-to-severe Crohn's Disease. Visual predictive check was used to evaluate model performance. Potential impacts of the study design on model development and evaluation of the E-R relationship in the induction and maintenance phases of treatment are discussed. Certain biases appeared difficult to overcome, and an autocorrelated residual error model was found to provide improvement.

Combined proportional and additive residual error models in population pharmacokinetic modelling

IntroductionIn pharmacokinetic modelling, a combined proportional and additive residual error model is often preferred over a proportional or additive residual error model. Different approaches have been proposed, but a comparison between approaches is still lacking. MethodsThe theoretical background of the methods is described. Method VAR assumes that the variance of the residual error is the sum of the statistically independent proportional and additive components; this method can be coded in three ways. Method SD assumes that the standard deviation of the residual error is the sum of the proportional and additive components. Using datasets from literature and simulations based on these datasets, the methods are compared using NONMEM. ResultsThe different coding of methods VAR yield identical results. Using method SD, the values of the parameters describing residual error are lower than for method VAR, but the values of the structural parameters and their inter-individual variability are hardly affected by the choice of the method. ConclusionBoth methods are valid approaches in combined proportional and additive residual error modelling, and selection may be based on OFV. When the result of an analysis is used for simulation purposes, it is essential that the simulation tool uses the same method as used during analysis.

DTI Image Registration under Probabilistic Fiber Bundles Tractography Learning.

Diffusion Tensor Imaging (DTI) image registration is an essential step for diffusion tensor image analysis. Most of the fiber bundle based registration algorithms use deterministic fiber tracking technique to get the white matter fiber bundles, which will be affected by the noise and volume. In order to overcome the above problem, we proposed a Diffusion Tensor Imaging image registration method under probabilistic fiber bundles tractography learning. Probabilistic tractography technique can more reasonably trace to the structure of the nerve fibers. The residual error estimation step in active sample selection learning is improved by modifying the residual error model using finite sample set. The calculated deformation field is then registered on the DTI images. The results of our proposed registration method are compared with 6 state-of-the-art DTI image registration methods under visualization and 3 quantitative evaluation standards. The experimental results show that our proposed method has a good comprehensive performance.

A strategy for residual error modeling incorporating scedasticity of variance and distribution shape.

Nonlinear mixed effects models parameters are commonly estimated using maximum likelihood. The properties of these estimators depend on the assumption that residual errors are independent and normally distributed with mean zero and correctly defined variance. Violations of this assumption can cause bias in parameter estimates, invalidate the likelihood ratio test and preclude simulation of real-life like data. The choice of error model is mostly done on a case-by-case basis from a limited set of commonly used models. In this work, two strategies are proposed to extend and unify residual error modeling: a dynamic transform-both-sides approach combined with a power error model (dTBS) capable of handling skewed and/or heteroscedastic residuals, and a t-distributed residual error model allowing for symmetric heavy tails. Ten published pharmacokinetic and pharmacodynamic models as well as stochastic simulation and estimation were used to evaluate the two approaches. dTBS always led to significant improvements in objective function value, with most examples displaying some degree of right-skewness and variances proportional to predictions raised to powers between 0 and 1. The t-distribution led to significant improvement for 5 out of 10 models with degrees of freedom between 3 and 9. Six models were most improved by the t-distribution while four models benefited more from dTBS. Changes in other model parameter estimates were observed. In conclusion, the use of dTBS and/or t-distribution models provides a flexible and easy-to-use framework capable of characterizing all commonly encountered residual error distributions.Electronic supplementary materialThe online version of this article (doi:10.1007/s10928-015-9460-y) contains supplementary material, which is available to authorized users.

A simulation study with a new residual ionospheric error model for GPS radio occultation climatologies

Abstract. In this study, a new model was explored which corrects for higher order ionospheric residuals in Global Positioning System (GPS) radio occultation (RO) data. Recently, the theoretical basis of this new "residual ionospheric error model" has been outlined (Healy and Culverwell, 2015). The method was tested in simulations with a one-dimensional model ionosphere. The proposed new model for computing the residual ionospheric error is the product of two factors, one of which expresses its variation from profile to profile and from time to time in terms of measurable quantities (the L1 and L2 bending angles), while the other describes the weak variation with altitude. A simple integral expression for the residual error (Vorob’ev and Krasil’nikova, 1994) has been shown to be in excellent numerical agreement with the exact value, for a simple Chapman layer ionosphere. In this case, the "altitudinal" element of the residual error varies (decreases) by no more than about 25 % between ~10 and ~100 km for physically reasonable Chapman layer parameters. For other simple model ionospheres the integral can be evaluated exactly, and results are in reasonable agreement with those of an equivalent Chapman layer. In this follow-up study the overall objective was to explore the validity of the new residual ionospheric error model for more detailed simulations, based on modeling through a complex three-dimensional ionosphere. The simulation study was set up, simulating day and night GPS RO profiles for the period of a solar cycle with and without an ionosphere. The residual ionospheric error was studied, the new error model was tested, and temporal and spatial variations of the model were investigated. The model performed well in the simulation study, capturing the temporal variability of the ionospheric residual. Although it was not possible, due to high noise of the simulated bending-angle profiles at mid- to high latitudes, to perform a thorough latitudinal investigation of the performance of the model, first positive and encouraging results were found at low latitudes. Furthermore, first application tests of the model on the data showed a reduction in temperature level of the ionospheric residual at 40 km from about −2.2 to −0.2 K.

Population pharmacokinetics of fluticasone propionate/salmeterol using two different dry powder inhalers

The combination of fluticasone propionate (FLP) and salmeterol (SAL) is often used in clinical practice for the treatment of pulmonary disorders. The purpose of this study was to explore the pharmacokinetics (PK) of inhaled FLP and SAL, after concomitant administration, in healthy male and female subjects using two dry powder inhalers. Plasma concentration (C)–time (t) data were obtained from a single dose, two-sequence, two-period, crossover (2×2) bioequivalence (BE) study. Activated charcoal was co-administered in order to prohibit absorption from the gastrointestinal tract. A number of 60 subjects were recruited, while 57 of them completed the study and were included in the PK analysis. Initially, PK parameters of FLP and SAL were estimated using the classic non-compartmental methods. Subsequently, BE assessment was applied to the estimated PK parameters of the two dry powder inhalers. Special focus was placed on the population PK analysis of the C–t data, which were pooled together. ‘Treatment’ (i.e., test or reference) and ‘period’ of the BE study were considered as covariates. A variety of structural and residual error models were tested to find the one which best described the plasma C–t data of FLP and SAL. Demographic data were also evaluated for their impact on the PK parameters. Several goodness-of-fit criteria were utilized. The non-compartmental PK estimates of this study were in agreement with previously reported values. The population PK analysis showed that FLP data were described by a two-compartment model with first-order absorption and elimination kinetics. Body weight was found to affect significantly absorption rate constant, inter-compartmental clearance, and volume of distribution of the peripheral compartment. As body weight increases, the values of these PK parameters also rise. For SAL, the best results were obtained when a two-compartment disposition model was used assuming very rapid absorption kinetics (like intravenous bolus) and first-order elimination kinetics. Gender was found to be a significant covariate on clearance, with men exhibiting higher clearance than women.

Modeling residual hydrologic errors with Bayesian inference

Hydrologic modelers are confronted with the challenge of producing estimates of the uncertainty associated with model predictions across an array of catchments and hydrologic flow regimes. Formal Bayesian approaches are commonly employed for parameter calibration and uncertainty analysis, but are often criticized for making strong assumptions about the nature of model residuals via the likelihood function that may not be well satisfied (or even checked). This technical note outlines a residual error model (likelihood function) specification framework that aims to provide guidance for the application of more appropriate residual error models through a nested approach that is both flexible and extendible. The framework synthesizes many previously employed residual error models and has been applied to four synthetic datasets (of differing error structure) and a real dataset from the Black River catchment in Queensland, Australia. Each residual error model was investigated and assessed under a top-down approach focused on its ability to properly characterize the errors. The results of these test applications indicate that a multifaceted assessment strategy is necessary to determine the adequacy of an individual likelihood function.

Bayesian inversions of a dynamic vegetation model at four European grassland sites

Abstract. Eddy covariance data from four European grassland sites are used to probabilistically invert the CARAIB (CARbon Assimilation In the Biosphere) dynamic vegetation model (DVM) with 10 unknown parameters, using the DREAM(ZS) (DiffeRential Evolution Adaptive Metropolis) Markov chain Monte Carlo (MCMC) sampler. We focus on comparing model inversions, considering both homoscedastic and heteroscedastic eddy covariance residual errors, with variances either fixed a priori or jointly inferred together with the model parameters. Agreements between measured and simulated data during calibration are comparable with previous studies, with root mean square errors (RMSEs) of simulated daily gross primary productivity (GPP), ecosystem respiration (RECO) and evapotranspiration (ET) ranging from 1.73 to 2.19, 1.04 to 1.56 g C m−2 day−1 and 0.50 to 1.28 mm day−1, respectively. For the calibration period, using a homoscedastic eddy covariance residual error model resulted in a better agreement between measured and modelled data than using a heteroscedastic residual error model. However, a model validation experiment showed that CARAIB models calibrated considering heteroscedastic residual errors perform better. Posterior parameter distributions derived from using a heteroscedastic model of the residuals thus appear to be more robust. This is the case even though the classical linear heteroscedastic error model assumed herein did not fully remove heteroscedasticity of the GPP residuals. Despite the fact that the calibrated model is generally capable of fitting the data within measurement errors, systematic bias in the model simulations are observed. These are likely due to model inadequacies such as shortcomings in the photosynthesis modelling. Besides the residual error treatment, differences between model parameter posterior distributions among the four grassland sites are also investigated. It is shown that the marginal distributions of the specific leaf area and characteristic mortality time parameters can be explained by site-specific ecophysiological characteristics.

A population model of early rheumatoid arthritis disease activity during treatment with methotrexate, sulfasalazine and hydroxychloroquine.

To develop a population model describing the disease activity (DAS28) time course in patients with early rheumatoid arthritis (RA) treated with triple disease-modifying anti-rheumatic drug (DMARD) therapy (methotrexate, sulfasalazine and hydroxychloroquine). DAS28 was obtained in 263 patients with early RA from initiation of therapy until 60 weeks. Using NONMEM(®), base models (DAS28 vs. time) and covariate influences were investigated for the population. The best model was an exponential model of DAS28 vs. time that was additive to baseline DAS28, with covariance between parameters, and a combined residual error model. Age and patient smoking history were covariates significantly affecting response to therapy. Population estimates were baseline DAS28 (5.7), extent of change in DAS28 (-2.8) and the half-life of disease activity (6.2 weeks; time to steady disease state achieved within approximately 30 weeks). Older individuals exhibited more severe baseline DAS28, described by a power function centred around 57 years (baseline DAS28 for 40- and 70-year-old patients were 5.4 vs. 5.8, respectively) and current smokers took longer to achieve a steady disease state (approximately 50 weeks). There was considerable within-patient random variability in DAS28 over time (empirical 90% CI for DAS28 in a population typical patient at 60 weeks: 1.8, 4.2 with median value of 2.8). This is the first report of a disease activity model for early RA treated with triple DMARD therapy. Smoking and age were identified as covariates.

Incorporation of concentration data below the limit of quantification in population pharmacokinetic analyses

Handling of data below the lower limit of quantification (LLOQ), below the limit of quantification (BLOQ) in population pharmacokinetic (PopPK) analyses is important for reducing bias and imprecision in parameter estimation. We aimed to evaluate whether using the concentration data below the LLOQ has superior performance over several established methods. The performance of this approach (“All data”) was evaluated and compared to other methods: “Discard,” “LLOQ/2,” and “LIKE” (likelihood-based). An analytical and residual error model was constructed on the basis of in-house analytical method validations and analyses from literature, with additional included variability to account for model misspecification. Simulation analyses were performed for various levels of BLOQ, several structural PopPK models, and additional influences. Performance was evaluated by relative root mean squared error (RMSE), and run success for the various BLOQ approaches. Performance was also evaluated for a real PopPK data set. For all PopPK models and levels of censoring, RMSE values were lowest using “All data.” Performance of the “LIKE” method was better than the “LLOQ/2” or “Discard” method. Differences between all methods were small at the lowest level of BLOQ censoring. “LIKE” method resulted in low successful minimization (<50%) and covariance step success (<30%), although estimates were obtained in most runs (∼90%). For the real PK data set (7.4% BLOQ), similar parameter estimates were obtained using all methods. Incorporation of BLOQ concentrations showed superior performance in terms of bias and precision over established BLOQ methods, and shown to be feasible in a real PopPK analysis.

Population pharmacokinetic analysis of fexofenadine in Japanese pediatric patients

A population pharmacokinetic analysis was conducted to characterize the pharmacokinetics of fexofenadine in Japanese pediatric patients (6months through 16years) with perennial allergic rhinitis or atopic dermatitis. The dataset was composed of 515 patients (including 109 adults), for a total of 1,080 concentration-time points. The analysis was performed with NONMEM using the SAEM method. Several structural models and residual error models were evaluated. The relationship between the individual estimates and the potential covariates was then investigated: demographic and pathophysiologic characteristics were tested as potential model covariates (forward selection method). The qualification of the model was performed using visual predictive check and bootstrap. A two-compartment disposition model with first-order absorption best fitted the data. The inter-individual variability was modeled through an exponential error model for all parameters (except for ka for which no inter-individual term could be estimated), while a proportional error model was used to model the residual variability. The final model included two covariates on elimination clearance and one on the intercompartmental clearance. CL/F was related to BSA and patient's age (expressed in months) Q/F was also related to BSA. Once the model was correctly qualified, exposure parameters such as Cmax and AUCτ were computed and compared between each age sub-group and between Japanese and Caucasians patients. These comparisons did not reveal any major difference (less than 50%) between subgroups.

Comparison of joint versus postprocessor approaches for hydrological uncertainty estimation accounting for error autocorrelation and heteroscedasticity

AbstractThe paper appraises two approaches for the treatment of heteroscedasticity and autocorrelation in residual errors of hydrological models. Both approaches use weighted least squares (WLS), with heteroscedasticity modeled as a linear function of predicted flows and autocorrelation represented using an AR(1) process. In the first approach, heteroscedasticity and autocorrelation parameters are inferred jointly with hydrological model parameters. The second approach is a two‐stage “postprocessor” scheme, where Stage 1 infers the hydrological parameters ignoring autocorrelation and Stage 2 conditionally infers the heteroscedasticity and autocorrelation parameters. These approaches are compared to a WLS scheme that ignores autocorrelation. Empirical analysis is carried out using daily data from 12 US catchments from the MOPEX set using two conceptual rainfall‐runoff models, GR4J, and HBV. Under synthetic conditions, the postprocessor and joint approaches provide similar predictive performance, though the postprocessor approach tends to underestimate parameter uncertainty. However, the MOPEX results indicate that the joint approach can be nonrobust. In particular, when applied to GR4J, it often produces poor predictions due to strong multiway interactions between a hydrological water balance parameter and the error model parameters. The postprocessor approach is more robust precisely because it ignores these interactions. Practical benefits of accounting for error autocorrelation are demonstrated by analyzing streamflow predictions aggregated to a monthly scale (where ignoring daily‐scale error autocorrelation leads to significantly underestimated predictive uncertainty), and by analyzing one‐day‐ahead predictions (where accounting for the error autocorrelation produces clearly higher precision and better tracking of observed data). Including autocorrelation into the residual error model also significantly affects calibrated parameter values and uncertainty estimates. The paper concludes with a summary of outstanding challenges in residual error modeling, particularly in ephemeral catchments.

항로표지 정보를 이용한 해상감시레이더의 시스템 오차 보정

Vessel traffic system uses multiple sea surveillance radars as a primary sensor to obtain maritime traffic information like as ship's position, speed, course. The systematic errors such as the range bias and the azimuth bias of the two-dimensional radar system can significantly degrade the accuracy of the radar image and target tracking information. Therefore, the systematic errors of the radar system should be corrected precisely in order to provide the accurate target information in the vessel traffic system. In this paper, it is proposed that the method compensates the range bias and the azimuth bias using AtoN information installed at VTS coverage. The radar measurement residual error model is derived from the standard error model of two-dimensional radar measurements and the position information of AtoN, and then the linear Kalman filter is designed for estimation of the systematic errors of the radar system. The proposed method is validated via Monte-Carlo runs. Also, the convergence characteristics of the designed filter and the accuracy of the systematic error estimates according to the number of AtoN information are analyzed.

Population pharmacokinetics of peginterferon alfa-2b in pediatric patients with chronic hepatitis C

The aim of this study was to characterize the population pharmacokinetics of peginterferon (PEG-IFN) alfa-2b in pediatric patients with chronic hepatitis C and to identify covariates influencing PEG-IFN alfa-2b disposition. Pharmacokinetic data from a multicenter open-label study of subcutaneously administered peginterferon alfa-2b (60 μg/m(2)/wk) plus oral ribavirin (15 mg/kg/day) in patients with chronic hepatitis C aged 3-17 years old was used to develop a population pharmacokinetic nonlinear mixed-effects model. The final population pharmacokinetic analysis was conducted with the pooled data from 107 pediatric patients. A one-compartment model with first-order absorption, first-order elimination, exponential inter-individual variability on clearance, and a combination additive and proportional residual error model adequately described the PEG-IFN alfa-2b pharmacokinetic profile. Age (apparent clearance and apparent volume of distribution) and sex (apparent clearance) were significant covariates. The mean body surface area normalized apparent clearance of PEG-IFN alfa-2b was 0.56 L/h/m(2), and was similar when evaluated across the pediatric age groups. The final population model suggests age-dependent increases in clearance and volume of distribution of PEG-IFN alfa-2b in pediatric patients with chronic hepatitis C. The apparent clearance normalized to body surface area was similar across pediatric age groups, supporting the use of body size-adjusted dosing in pediatric subjects.

Pitfalls and improvements in the joint inference of heteroscedasticity and autocorrelation in hydrological model calibration

Residual errors of hydrological models are usually both heteroscedastic and autocorrelated. However, only a few studies have attempted to explicitly include these two statistical properties into the residual error model and jointly infer them with the hydrological model parameters. This technical note shows that applying autoregressive error models to raw heteroscedastic residuals, as done in some recent studies, can lead to unstable error models with poor predictive performance. This instability can be avoided by applying the autoregressive process to standardized residuals. The theoretical analysis is supported by empirical findings in three hydrologically distinct catchments. The case studies also highlight strong interactions between the parameters of autoregressive residual error models and the water balance parameters of the hydrological model.

Prediction of Shrinkage of Individual Parameters Using the Bayesian Information Matrix in Non-Linear Mixed Effect Models with Evaluation in Pharmacokinetics

When information is sparse, individual parameters derived from a non-linear mixed effects model analysis can shrink to the mean. The objective of this work was to predict individual parameter shrinkage from the Bayesian information matrix (M BF ). We 1) Propose and evaluate an approximation of M BF by First-Order linearization (FO), 2) Explore by simulations the relationship between shrinkage and precision of estimates and 3) Evaluate prediction of shrinkage and individual parameter precision. We approximated M BF using FO. From the shrinkage formula in linear mixed effects models, we derived the predicted shrinkage from M BF . Shrinkage values were generated for parameters of two pharmacokinetic models by varying the structure and the magnitude of the random effect and residual error models as well as the design. We then evaluated the approximation of M BF FO and compared it to Monte-Carlo (MC) simulations. We finally compared expected and observed shrinkage as well as the predicted and estimated Standard Errors (SE) of individual parameters. M BF FO was similar to M BF MC. Predicted and observed shrinkages were close . Predicted and estimated SE were similar. M BF FO enables prediction of shrinkage and SE of individual parameters. It can be used for design optimization.

An improved SAEM algorithm for maximum likelihood estimation in mixtures of non linear mixed effects models

We propose a new methodology for maximum likelihood estimation in mixtures of non linear mixed effects models (NLMEM). Such mixtures of models include mixtures of distributions, mixtures of structural models and mixtures of residual error models. Since the individual parameters inside the NLMEM are not observed, we propose to combine the EM algorithm usually used for mixtures models when the mixture structure concerns an observed variable, with the Stochastic Approximation EM (SAEM) algorithm, which is known to be suitable for maximum likelihood estimation in NLMEM and also has nice theoretical properties. The main advantage of this hybrid procedure is to avoid a simulation step of unknown group labels required by a full version of SAEM. The resulting MSAEM (Mixture SAEM) algorithm is now implemented in the Monolix software. Several criteria for classification of subjects and estimation of individual parameters are also proposed. Numerical experiments on simulated data show that MSAEM performs well in a general framework of mixtures of NLMEM. Indeed, MSAEM provides an estimator close to the maximum likelihood estimator in very few iterations and is robust with regard to initialization. An application to pharmacokinetic (PK) data demonstrates the potential of the method for practical applications.

Precise attitude determination strategy for spacecraft based on information fusion of attitude sensors: Gyros/GPS/Star-sensor

The rigorous requirements of modern spacecraft missions necessitate a precise attitude determination strategy. This paper mainly researches that, based on three space-borne attitude sensors: 3-axis rate gyros, 3-antenna GPS receiver and starsensor. To obtain global attitude estimation after an information fusion process, a feedback-involved Federated Kalman Filter (FKF), consisting of two subsystem Kalman filters (Gyros/GPS and Gyros/Star-sensor), is established. In these filters, the state equation is implemented according to the spacecraft’s kinematic attitude model, while the residual error models of GPS and star-sensor observed attitude are utilized, to establish two observation equations, respectively. Taking the sensors’ different update rates into account, these two subsystem filters are conducted under a variable step size state prediction method. To improve the fault tolerant capacity of the attitude determination system, this paper designs malfunction warning factors, based on the principle of χ² residual verification. Mathematical simulation indicates that the information fusion strategy overwhelms the disadvantages of each sensor, acquiring global attitude estimation with precision at a 2-arcsecs level. Although a subsystem encounters malfunction, FKF still reaches precise and stable accuracy. In this process, malfunction warning factors advice malfunctions correctly and effectively.