Additive Error Model Research Articles

A test of homogeneity of distributions when observations are subject to measurement errors.

When the observed data are contaminated with errors, the standard two-sample testing approaches that ignore measurement errors may produce misleading results, including a higher type-I error rate than the nominal level. To tackle this inconsistency, a nonparametric test is proposed for testing equality of two distributions when the observed contaminated data follow the classical additive measurement error model. The proposed test takes into account the presence of errors in the observed data, and the test statistic is defined in terms of the (deconvoluted) characteristic functions of the latent variables. Proposed method is applicable to a wide range of scenarios as no parametric restrictions are imposed either on the distribution of the underlying latent variables or on the distribution of the measurement errors. Asymptotic null distribution of the test statistic is derived, which is given by an integral of a squared Gaussian process with a complicated covariance structure. For data-based calibration of the test, a new nonparametric Bootstrap method is developed under the two-sample measurement error framework and its validity is established. Finite sample performance of the proposed test is investigated through simulation studies, and the results show superior performance of the proposed method than the standard tests that exhibit inconsistent behavior. Finally, the proposed method was applied to real data sets from the National Health and Nutrition Examination Survey. An R package MEtest is available throughCRAN.

An Iterative Ensemble Kalman Smoother in Presence of Additive Model Error

Ensemble variational methods are being increasingly used in the field of geophysical data assimilation. Their efficiency comes from the combined use of ensembles, which provide statistics estimates, and a variational analysis, which handles nonlinear operators through iterative optimization techniques. Taking model error into account in four-dimensional ensemble variational algorithms is challenging because the state trajectory over the data assimilation window (DAW) is no longer determined by its sole initial condition. In particular, the control variable dimension scales with the DAW length, which yields a high numerical complexity. This is unfortunate since accuracy improvement is expected with longer DAWs. Building upon the work of [P. Sakov and M. Bocquet, Tellus A, 70 (2018), 1414545], this paper discusses how to algorithmically construct and numerically test an iterative ensemble Kalman smoother with additive model error (IEnKS-Q) which is thought to be the natural weak constraint generalization of the IEnKS [M. Bocquet and P. Sakov, Quart. J. Roy. Meteorol. Soc., 140 (2014), pp. 1521--1535], as well as the generalization of IEnKF-Q [P. Sakov, J. Haussaire, and M. Bocquet, Quart. J. Roy. Meteorol. Soc., 144 (2018), pp. 1297--1309] to general DAWs. The number of model evaluations per cycle of the IEnKS-Q is also examined. Solutions based on perturbation decomposition are proposed to dissociate those numerically costly evaluations from the control variable dimension.

Analysis of Parameters for the Accurate and Fast Estimation of Tree Diameter at Breast Height Based on Simulated Point Cloud

Terrestrial laser scanning (TLS) is a high-potential technology in forest surveys. Estimating the diameters at breast height (DBH) accurately and quickly has been considered a key step in estimating forest structural parameters by using TLS technology. However, the accuracy and speed of DBH estimation are affected by many factors, which are classified into three groups in this study. We adopt an additive error model and propose a simple and common simulation method to evaluate the impacts of three groups of parameters, which include the range error, angular errors in the vertical and horizontal directions, angular step width, trunk distance, slice thickness, and real DBH. The parameters were evaluated statistically by using many simulated point cloud datasets that were under strict control. Two typical circle fitting methods were used to estimate DBH, and their accuracy and speed were compared. The results showed that the range error and the angular error in horizontal direction played major roles in the accuracy of DBH estimation, the angular step widths had a slight effect in the case of high range accuracy, the distance showed no relationship with the accuracy of the DBH estimation, increasing the scanning angular width was relatively beneficial to the DBH estimation, and the algebraic circle fitting method was relatively fast while performing DBH estimation, as is the geometrical method, in the case of high range accuracy. Possible methods that could help to obtain accurate and fast DBH estimation results were proposed and discussed to optimize the design of forest inventory experiments.

A Note on Testing Serial Correlation in Partially Linear Additive Measurement Error Models

The paper considers the problem of testing error serial correlation of partially linear additive measurement error model. We propose a test statistic and show that it converges to the standard chi-square distribution under the null hypothesis. Finally, a simulation study is conducted to illustrate the performance of the test approach.

Exploring a copula-based alternative to additive error models—for non-negative and autocorrelated time series in hydrology

Inaccurate description of uncertainty in the error models can cause biases in parameter estimation. When the parameters of the deterministic model and the error model are inferred jointly from the observations, the posterior converges to regions that reflect the processes in both high and low flows. If the nature of errors in low and high flows is different to the extent that the same error description cannot be used for both, biases in inference are introduced. In such cases, the parameter posterior will adjust to the region of the hydrograph with longer proportionate presence in the calibration time series. In this paper we demonstrate that the autoregressive order 1 (AR1) description of errors can lead to sub-optimally performing predictive models if the calibration period has substantial sections of inadequately modelled flows. Inference is performed within the Bayesian framework. We show this for a synthetic example as well as a case study. We also see that the predictive uncertainty bands that we get using the AR1 description can be overconfident and also admit negative values. To mitigate this, we analyze an alternative to additive error models. We use a distribution with a non-negative support, gamma in this study, reflecting the uncertainty in the system response at every time step. The gamma distribution is conditioned on the deterministic model output, which determines its mode and standard deviation. We capture autocorrelation in time using copulas. Given that copulas can capture dependence between different marginals, we use different specifications of the marginal distribution for high and low flows. The results show that 1) biases in parameter estimation can be reduced if a representative error description is attained using the flexibility of a copula-based likelihood. 2) The non-negative support allows to make more realistic uncertainty intervals for low flows. 3) However, the autocorrelation parameter in copulas severely interacts with the model and heteroscedasticity parameters. 4) While the formulation, in principle, should be of added value for parameter inference, in case of less informative priors, the flexibility of this description can produce non-robust inference.

Accounting for model errors in iterative ensemble smoothers

In the strong-constraint formulation of the history-matching problem, we assume that all the model errors relate to a selection of uncertain model input parameters. One does not account for additional model errors that could result from, e.g., excluded uncertain parameters, neglected physics in the model formulation, the use of an approximate model forcing, or discretization errors resulting from numerical approximations. If parameters with significant uncertainties are unaccounted for, there is a risk for an unphysical update, of some uncertain parameters, that compensates for errors in the omitted parameters. This paper gives the theoretical foundation for introducing model errors in ensemble methods for history matching. In particular, we explain procedures for practically including model errors in iterative ensemble smoothers like ESMDA and IES, and we demonstrate the impact of adding (or neglecting) model errors in the parameter-estimation problem. Also, we present a new result regarding the consistency of using the sample covariance of the predicted nonlinear measurements in the update schemes.

Oracally efficient spline-backfitted kernel smoothing of additive partial linear measurement error model

We consider statistical inference for additive partial linear models when the linear covariate is measured with error. A bias-corrected spline-backfitted kernel smoothing method is proposed. Under mild assumptions, the proposed component function and parameter estimator are oracally efficient and fast to compute. The nonparametric function estimator’s pointwise distribution is asymptotically equivalent to an function estimator in partial linear model. Finite-sample performance of the proposed estimators is assessed by simulation experiments. The proposed methods are applied to Boston house data set.

Copula-based stochastic uncertainty analysis of satellite precipitation products

Satellite products, like all datasets, are subject to errors and uncertainties. Due to inherent biases embedded in satellite precipitation estimates, we present an error-adjustment approach based on the statistical differences between satellite precipitation products and in-situ observations (observed errors) employing two widely-used error models, namely that additive and the multiplicative error models, in an attempt to assess their suitability for the error correction of satellite-based daily precipitation estimates over northeast Austria. An error-adjustment technique based on the concept of the copula is adopted and applied to correct the supplied precipitation fields. It was found that IMERG precipitation estimates improved after error adjustment when compared to original satellite precipitation estimate (OSPE). The additive error model resulted in a better improvement by fitting the entire range of data when compared with the multiplicative error model. Moreover, the additive error model extracted the error with more accuracy and produced a better estimation of their characteristics, while the method based on the multiplicative error was less robust. However, the overall spatial dependence of the observed errors is reasonably preserved as that of the generated errors by copula. In addition, the validation results implied that the simulated realizations error-adjusted band, encompassed the observed data reasonably. Moreover, the copula-based simulations associated with the additive error model performed much better in comparison to the multiplicative error model. Overall, by using the t-copula model with an emphasis on the additive error model and imposing the simulated error fields on the OSPE, one may generate multiple realizations of precipitation fields.

Estimation of physical parameters under location uncertainty using an ensemble2–expectation–maximization algorithm

Estimating the parameters of geophysical dynamic models is an important task in the data assimilation (DA) techniques used to forecast initialization and reanalysis. In the past, most parameter estimation strategies were derived by state augmentation, yielding algorithms that are easy to implement but may exhibit convergence difficulties. The expectation–maximization (EM) algorithm is considered advantageous because it employs two iterative steps to estimate the model state and the model parameter separately. In this work, we propose a novel ensemble formulation of the maximization step in EM that allows a direct optimal estimation of the physical parameters using iterative methods for linear systems. This departs from current EM formulations that are only capable of dealing with additive model error structures. This contribution shows how the EM technique can be used for dynamics identification problems with the model error parameterized as an arbitrary complex form. The proposed technique is used here for the identification of stochastic subgrid terms that account for processes unresolved by a geophysical fluid model. This method, together with the augmented state technique, is evaluated to estimate such subgrid terms through high‐resolution data. As compared to the augmented state technique, our method is shown to yield considerably more accurate parameters. In addition, in terms of prediction capacity, it leads to a smaller generalization error as a result of the overfitting of the trained model on the presented data and eventually better forecasts.

Modeling Euro STOXX 50 volatility with common and market-specific components

Similar volatility patterns are observables in the Euro area across national indices, suggesting the possibility of an underlying common component as a consequence of financial and monetary integration. This peculiar interdependence across market volatilities is captured by an additive component vector Multiplicative Error Model (vMEM) where the volatility dynamics is split between a common and a vector of market-specific components. When extracted from five major market indices and used as additional regressors in a HAR specification for the Euro STOXX 50 (a Euro area wide index) volatility, these components replace the terms that mimic long memory in the HAR, providing an interesting interpretation for volatility dynamics.

Proportional-Integral-Observer-Based Backstepping Approach for Position Control of a Hydraulic Differential Cylinder System With Model Uncertainties and Disturbances

This paper focuses on the design of an observer-based backstepping controller (BC) for a nonlinear hydraulic differential cylinder system. The system is affected by some uncertainties including modeling errors, external disturbances, and measurement noise. An observer-based control approach is proposed to assure suitable tracking performance and to increase robustness against unknown inputs. The task to estimate system states as well as unknown inputs is performed by a linear proportional-integral-observer (PIO). Input–output linearization is used to linearize the nonlinear system model to be used for the PIO structure. On the other hand, BC is utilized based on nonlinear system model to construct the Lyapunov function and to design the control input simultaneously. Stability or negativeness of the derivative of every-step Lyapunov function is fulfilled. Structural improvement regarding the combination of BC and PIO is the main aim of this contribution. This is supported by a novel stability proof and new conditions for the whole control loop with integrated PIO. Furthermore, parameter selection of BC is elaborately considered by defining a performance/energy criterion. A complete robustness evaluation considering different levels of additional measurement noise, modeling errors, and external disturbances is presented for the first time in this contribution. Experimental results validate the advantages of proposed observer-based approach compared to PIO-based sliding mode control (PIO-SMC) and industrial standard P-controller.

An iterative ensemble Kalman filter in the presence of additive model error

The iterative ensemble Kalman filter (IEnKF) in a deterministic framework was introduced in Sakov et al. [Mon. Wea. Rev. 140: 1988–2004 ()] to extend the ensemble Kalman filter (EnKF) and improve its performance in mildly up to strongly nonlinear cases. However, the IEnKF assumes that the model is perfect. This assumption simplified the update of the system at a time different from the observation time, which made it natural to apply the IEnKF for smoothing. In this study, we generalize the IEnKF to the case of an imperfect model with additive model error.The new method called IEnKF‐Q conducts a Gauss–Newton minimization in ensemble space. It combines the propagated analysed ensemble anomalies from the previous cycle and model noise ensemble anomalies into a single ensemble of anomalies, and by doing so takes an algebraic form similar to that of the IEnKF. The performance of the IEnKF‐Q is tested in a number of experiments with the Lorenz'96 model, which show that the method consistently outperforms both the EnKF and the IEnKF naively modified to accommodate additive model noise.

Asymptotic Forecast Uncertainty and the Unstable Subspace in the Presence of Additive Model Error

It is well understood that dynamic instability is among the primary drivers of forecast uncertainty in chaotic, physical systems. Data assimilation techniques have been designed to exploit this phenomenon, reducing the effective dimension of the data assimilation problem to the directions of rapidly growing errors. Recent mathematical work has, moreover, provided formal proofs of the central hypothesis of the assimilation in the unstable subspace methodology of Anna Trevisan and her collaborators: for filters and smoothers in perfect, linear, Gaussian models, the distribution of forecast errors asymptotically conforms to the unstable-neutral subspace. Specifically, the column span of the forecast and posterior error covariances asymptotically align with the span of backward Lyapunov vectors with nonnegative exponents. Earlier mathematical studies have focused on perfect models, and this current work now explores the relationship between dynamical instability, the precision of observations, and the evolution of forecast error in linear models with additive model error. We prove bounds for the asymptotic uncertainty, explicitly relating the rate of dynamical expansion, model precision, and observational accuracy. Formalizing this relationship, we provide a novel, necessary criterion for the boundedness of forecast errors. Furthermore, we numerically explore the relationship between observational design, dynamical instability, and filter boundedness. Additionally, we include a detailed introduction to the multiplicative ergodic theorem and to the theory and construction of Lyapunov vectors. While forecast error in the stable subspace may not generically vanish, we show that even without filtering, uncertainty remains uniformly bounded due its dynamical dissipation. However, the continuous reinjection of uncertainty from model errors may be excited by transient instabilities in the stable modes of high variance, rendering forecast uncertainty impractically large. In the context of ensemble data assimilation, this requires rectifying the rank of the ensemble-based gain to account for the growth of uncertainty beyond the unstable and neutral subspace, additionally correcting stable modes with frequent occurrences of positive local Lyapunov exponents that excite model errors.

Comparison of various estimation methods for the parameters of Michaelis-Menten equation based on in vitro elimination kinetic simulation data.

The Michaelis-Menten equation is one of the best-known models describing the enzyme kinetics of in vitro drug elimination experiments, and takes a form of equation relating reaction rate (V) to the substrate concentration ([S]) via the maximum reaction rate (Vmax) and the Michaelis constant (Km). The current study was conducted to compare the accuracy and precision of the parameter estimates in the Michaelis-Menten equation from various estimation methods using simulated data. One thousand replicates of simulated [S] over serial time data were generated using the results of a previous study, incorporating additive or combined error models as a source of random variables in the Monte-Carlo simulation using R. From each replicate of simulated data, Vmax and Km were estimated by five different methods, including traditional linearization methods and nonlinear ones without linearization using NONMEM. The relative accuracy and precision of the estimated parameters were compared by the median values and their 90% confidence intervals. Overall, Vmax and Km estimation by nonlinear methods (NM) provided the most accurate and precise results from the tested 5 estimation methods. The superiority of parameter estimation by NM was even more evident in the simulated data incorporating the combined error model. The current simulation study suggests that NMs using a program such as NONMEM provide more reliable and accurate parameter estimates of the Michaelis-Menten equation than traditional linearization methods in in vitro drug elimination kinetic experiments.

Asynchronous data assimilation with the EnKF in presence of additive model error

The term ‘asynchronous data assimilation’ (ADA) refers to modifications of sequential data assimilation methods that take into consideration the observation time. In Sakov et al. [Tellus A, 62, 24–29 (2010)], a simple rule has been formulated for the ADA with the ensemble Kalman filter (EnKF). To assimilate scattered in time observations, one needs to calculate ensemble forecast observations using the forecast ensemble at observation time. Using then these ensemble observations in the EnKF update matches the optimal analysis in the linear perfect model case. In this note, we generalise this rule for the case of additive model error.

Causal inference with measurement error in outcomes: Bias analysis and estimation methods.

Inverse probability weighting estimation has been popularly used to consistently estimate the average treatment effect. Its validity, however, is challenged by the presence of error-prone variables. In this paper, we explore the inverse probability weighting estimation with mismeasured outcome variables. We study the impact of measurement error for both continuous and discrete outcome variables and reveal interesting consequences of the naive analysis which ignores measurement error. When a continuous outcome variable is mismeasured under an additive measurement error model, the naive analysis may still yield a consistent estimator; when the outcome is binary, we derive the asymptotic bias in a closed-form. Furthermore, we develop consistent estimation procedures for practical scenarios where either validation data or replicates are available. With validation data, we propose an efficient method for estimation of average treatment effect; the efficiency gain is substantial relative to usual methods of using validation data. To provide protection against model misspecification, we further propose a doubly robust estimator which is consistent even when either the treatment model or the outcome model is misspecified. Simulation studies are reported to assess the performance of the proposed methods. An application to a smoking cessation dataset is presented.

Enoxaparin Population Pharmacokinetics in the First Year of Life.

Enoxaparin dosing requirements in the first year of life can be highly variable. Characterization of pharmacokinetics in this patient population can assist in dosing. Patients less than 1 year postnatal age who received enoxaparin and had an anti-factor Xa activity level drawn as inpatients were identified through the pharmacy database over a 5-year period. Patients on renal replacement therapy or with hyperbilirubinemia were excluded. Data collection included demographic variables, indication for enoxaparin, enoxaparin doses, anti-factor Xa activity levels, serum creatinine, hemoglobin, hematocrit, platelet count, and urine output over the previous 24 hours. Population pharmacokinetic analysis was performed with NONMEM. A total of 182 patients [male 50%, median 100 days postnatal age (range: 4-353 days)] met the study criteria. Patients received median 22 doses (range: 1-526) at a mean starting dose of 1.38 ± 0.43 mg/kg with median 5 (range: 1-56) anti-factor Xa activity levels measured. A 1-compartment proportional and additive error model best fits the data. Allometrically scaled weight significantly decreased the objective function value, as did serum creatinine on clearance, and postmenstrual age (PMA) on volume of distribution. When evaluated graphically, dosing based on PMA appeared to have less variability as compared to postnatal age-based dosing. Dosing of enoxaparin in infants younger than 1 year should incorporate PMA.

Pyrazinamide clearance is impaired among HIV/tuberculosis patients with high levels of systemic immune activation.

Pyrazinamide is the main driver of sterilizing effect in the standard regimen in adults and older children, and this effect is concentration-dependent. Tuberculosis patients co-infected with human immunodeficiency virus (HIV) have an increased risk for poor tuberculosis treatment outcomes and adverse drug events. We sought to determine whether measures of systemic immune activation were related to pyrazinamide pharmacokinetics among HIV/tuberculosis patients. We conducted a prospective cohort study of pyrazinamide pharmacokinetics in HIV/tuberculosis patients in Gaborone, Botswana. Patients underwent intensive pharmacokinetic sampling before and after the initiation of antiretroviral therapy, which can increase immune activation in HIV/tuberculosis. Compartmental pharmacokinetic modeling was performed to determine whether variability in systemic immune activation was related to variability in pyrazinamide pharmacokinetic parameters. Forty HIV/tuberculosis patients completed the first pharmacokinetic sampling visit, and 24 patients returned for a second visit following antiretroviral therapy initiation. The pyrazinamide plasma concentration-versus-time data were best explained by a one-compartment model with first-order elimination, and a combined additive and proportional residual error model. Pyrazinamide clearance was higher in men than women. Expression of CD38 and HLA- DR on CD8+T cells, a measure of HIV-associated immune activation, was inversely related to pyrazinamide clearance, with increasing immune activation associated with decreasing pyrazinamide clearance. Future studies should verify this finding in larger numbers of tuberculosis patients with and without HIV co-infection.

The impact of body weight on rivaroxaban pharmacokinetics

Unlabelled Box BackgroundThere is concern amongst clinicians that the fixed dosing strategy of rivaroxaban for the treatment of venous thromboembolism (VTE) might not be optimal in those patients under or overweight. ObjectiveTo develop a pharmacokinetic model for rivaroxaban, based on real‐world patients, specifically focusing on the impact of patients’ body weight on rivaroxaban pharmacokinetics. Patients/methodsOne hundred and one patients prescribed rivaroxaban prophylactic or treatment doses for the prevention or treatment of VTE were recruited at a London teaching hospital. Subjects had up to 3 rivaroxaban concentrations measured during a single dosing period (trough, 1 and 3 hours post dose). Population pharmacokinetic analyses was conducted to develop a rivaroxaban model, which was subsequently evaluated. ResultsA one‐compartment model with between‐subject variability on rivaroxaban clearance and volume of distribution, with a combined (additive and proportional) error model, best fitted the data. Following a full covariate analysis, creatinine clearance on rivaroxaban clearance was found to be the significant covariate impacting on the pharmacokinetic profile of rivaroxaban in the dataset. ConclusionsOur results suggest that the most important covariate impacting on rivaroxaban pharmacokinetics is creatinine clearance and the weight alone has little effect. These findings are in line with previous studies for rivaroxaban. Larger datasets, from real‐world patients who are followed longitudinally, should be conducted to provide front‐line clinicians with further reassurance when prescribing rivaroxaban for the acute management of VTE.

Simulation of electrical machines: a FEM-BEM coupling scheme

PurposeElectrical machines commonly consist of moving and stationary parts. The field simulation of such devices can be demanding if the underlying numerical scheme is solely based on a domain discretization, such as in the case of the finite element method (FEM). This paper aims to present a coupling scheme based on FEM together with boundary element methods (BEMs) that neither hinges on re-meshing techniques nor deals with a special treatment of sliding interfaces. While the numerics are certainly more involved, the reward is obvious: the modeling costs decrease and the application engineer is provided with an easy-to-use, versatile and accurate simulation tool.Design/methodology/approachThe authors present the implementation of a FEM-BEM coupling scheme in which the unbounded air region is handled by the BEM, while only the solid parts are discretized by the FEM. The BEM is a convenient tool to tackle unbounded exterior domains, as it is based on the discretization of boundary integral equations (BIEs) that are defined only on the surface of the computational domain. Hence, no meshing is required for the air region. Further, the BIEs fulfill the decay and radiation conditions of the electromagnetic fields such that no additional modeling errors occur.FindingsThis work presents an implementation of a FEM-BEM coupling scheme for electromagnetic field simulations. The coupling eliminates problems that are inherent to a pure FEM approach. In detail, the benefits of the FEM-BEM scheme are: the decay conditions are fulfilled exactly, no meshing of parts of the exterior air region is necessary and, most importantly, the handling of moving parts is incorporated in an intriguingly simple manner. The FEM-BEM formulation in conjunction with a state-of-the-art preconditioner demonstrates its potency. The numerical tests not only reveal an accurate convergence behavior but also prove the algorithm to be suitable for industrial applications.Originality/valueThe presented FEM-BEM scheme is a mathematically sound and robust implementation of a theoretical work presented a decade ago. For the application within an industrial context, the original work has been extended by higher-order schemes, periodic boundary conditions and an efficient treatment of moving parts. While not intended to be used under all circumstances, it represents a powerful tool in case that high accuracies together with simple mesh-handling facilities are required.