Assessment of influence in non-linear measurement error models

Recognition that measurement error in the independent (predictor) variable of data used for regression modeling necessitates special methods of model fitting dates back to the late 1800s. In the ensuing 100 years attention was devoted almost entirely to measurement error in linear regression models, at which time advances in computing opened the door to the nonlinear statistical modeling generally, and to the study of measurement error in nonlinear regression modeling in particular. This article contains an introduction to the problem of measurement error in regression using linear regression measurement error models to illustrate key ideas. It includes a discussion of the types of measurement error commonly studied (classical and Berkson, differential and nondifferential) and how those types arise in practice. A summary of the consequences of ignoring measurement error when analyzing data measured with error is presented, with detailed results given for simple and multiple linear measurement error models. Pointers to the literature on both linear and nonlinear measurement error models are provided.

Analyses of variables measured with error are often flawed when measurement error is ignored. An extensive literature on linear measurement error models is available by [1], and more recently, for non-linear measurement error models by [2] and [3]. We investigate problems encountered in the analysis of longitudinal change, where the difference in some outcome variable is modeled on a set of regressor variables in a linear regression model. This type of analysis has become quite popular in biomedical research, and the results are often biased when the outcome variable is measured with error and its observed baseline value is included as a covariate in the fitted model. [4] demonstrated the effects of measurement error in the analysis of change in wall thickness of the common carotid artery. They showed the naive analysis led to erroneous findings due to the measurement error bias, even when the regressor variables were assumed to be measured precisely. In this paper, we present a method-of-moments correction for measurement error bias, provided the measurement error variance is known or can be estimated. This work extends the work of [4] to include regressor variables that are measured with error.

Much attention in robust identification and control has been focused on linear low order models approximating high order linear systems. We consider the more realistic situation with a linear model approximating a nonlinear system. We describe how a nonlinear model error model can be developed, that allows a complete linear design process that results in a closed loop system with performance robustness guarantees (in terms of gain from disturbance to output) against the nonlinear error. Clearly the design can be successful only if the linear model is a reasonably good approximation of the system. A particular aspect of the design process is to define a workable definition of ‘practical stability’ for robust control design, with possible non‐linear model errors. We use affine norms for that purpose. Copyright © 2004 John Wiley & Sons, Ltd.

Over the last years miRNA microarray platforms have provided insights in the biological mechanisms underlying onset and development of several diseases and have thus become a very popular instrument for profiling thousands of miRNA simultaneously. However, because of large variety of microarray platforms available, an assessment of their performance in terms of both within-platform reliability and between-platform agreement is needful. In particular, assessment of platform concordance has been a very relevant issue in the past decade. To date, only a few studies have evaluated this problem in the field of miRNA microarray, and mostly by using improper statistical methods such as the Pearson and Spearman correlation coefficients. In this work we suggest to use a recently proposed modified version of the classical Bland-Altman approach for comparing clinical measurement methods. This modified version is useful in that allows not only to evaluate agreement between different miRNA microarray platforms, but also to assess which are the potential sources of disagreement/bias between them.Two samples were profiled using Affymetrix, Agilent and Illumina miRNA platform using three technical replicates each, and pairwise agreement between platforms was evaluated within each sample. Our results suggest that, after bias correction, Illumina and Agilent show the best patterns of agreement for both samples involved in the experiment, whereas Affymetrix is the one which seem to ”disagree” most, suggesting that a linear relationship as that hypothesized by the measurement error model used is not able to capture the complexity of the phenomenon.In the future it will be interesting to apply this method also to the comparison of microarray and NGS platform, a topic which is becoming more and more relevant, also by adopting non-linear measurement error models to depict relationships between platforms.

In this paper, a uniform form for the nonlinear error model is proposed for the inertial navigation system (INS). A modified quaternion-based nonlinear error model is derived using the uniform coordinate frame for the INS. Based on the modified quaternion-based nonlinear error model, a uniform form of the nonlinear error model is proposed based on the Gibbs parameter. The uniform form can present the nonlinear error model based on low-order Rodrigues parameters or Rodrigues-like parameters for the INS. Field tests are carried out to assess the performance of the existing typical nonlinear error models.

A novel nonlinear attitude error model based on square root cubature information filter (SR-CIF) is proposed aiming to speed up the convergence rate of DVL-aided strapdown inertial navigation system (SINS) in-motion initial alignment for autonomous underwater vehicle (AUV) when both large initial heading error and large initial level attitude errors exist The new nonlinear attitude error model and measurement model are applicable for initial alignment after a short-time coarse alignment with large attitude errors exist. The SR-CIF embeds the whole iterative process into the data structure framework of the information filtering, replacing the covariance matrix in the square root cubature Kalman filter (SR-CKF) with an information matrix which simplifies the filter initialization. The square root technology ensures the symmetry and positive definiteness of the information matrix with enhanced stability of the filter. The simulation and experimental results indicate that the proposed DVL-aided alignment filter is effective with large initial attitude errors. The rate of convergence and estimation accuracy of the SR-CIF is higher than that of the conventional SR-CKF with large attitude misalignments.

This paper proposes a twice rapid transfer alignment algorithm based on dual models in order to solve the problems such as long convergence time, poor accuracy, and heavy computation burden resulting from the traditional nonlinear error models. The quaternion matching method based on quaternion error model along with the extended Kalman filter (EKF) is applied to deal with the large misalignment in the first phase. Then in the second transfer alignment phase, velocity plus attitude matching method as well as classical Kalman filter is adopted. The simulation and the results of vehicle tests demonstrate that this method combines the advantages of both nonlinear and linear error models with the guarantee of accuracy and fastness.

In this paper, a robust unscented Kalman filter (UKF) based on the generalized maximum likelihood estimation (M-estimation) is proposed to improve the robustness of the integrated navigation system of Global Navigation Satellite System and Inertial Measurement Unit. The UKF is a variation of Kalman filter by which the Jacobian matrix calculation in a nonlinear system state model is not necessary. The proposed robust M–M unscented Kalman filter (RMUKF) applies the M-estimation principle to both functional model errors and measurement errors. Hence, this robust filter attenuates the influences of disturbances in the dynamic model and of measurement outliers without linearizing the nonlinear state space model. In addition, an equivalent weight matrix, composed of the bi-factor shrink elements, is proposed in order to keep the original correlation coefficients of the predicted state unchanged. Furthermore, a nonlinear error model is used as the dynamic equation to verify the performance of the proposed RMUKF with a simulation and field test. Compared with the conventional UKF, the impacts of measurement outliers and system disturbances on the state estimation are both controlled by RMUKF.

In recent years, quantum computing has been applied in optimizing different challenging systems in the domains of science and engineering. This work reflects the identification of the input nonlinear output error (I-NOE) model by using the Quantum-based avian navigation optimizer algorithm (QANA). QANA is an evolutionary method inspired by the precision navigation of migratory birds. The assessment of QANA is achieved on various iterations, population, and noise levels. The complexity, statistical, and convergence investigation of QANA with arithmetic optimization algorithm (AOA), coati optimization algorithm (COA), grey wolf optimizer (GWO), particle swarm optimization (PSO), synergistic swarm optimization algorithm (SSOA), and velocity pausing particle swarm optimization (VPPSO) approves the robustness of QANA for I-NOE model identification.

Disturbance observer based reliable H∞ fuzzy attitude tracking control for Mars entry vehicles with actuator failures

This paper presents a new method for in-field calibration of accelerometers to address the problems of low efficiency and high cost associated with traditional calibration methods. A nonlinear mathematical model of the accelerometer is established, and the cost function is analysed and deduced the cost function. Then, an adaptive Northern Goshawk Optimisation (NGO) algorithm based on prior knowledge enhancement is introduced. A method of collecting multi-position data with a hand-held accelerometer is introduced, and the proposed algorithm is used to in-field calibrate nine parameters of the accelerometer’s nonlinear error model. In addition, simulation is used to compare the results of calibrating the accelerometer using the proposed algorithm and the original algorithm, demonstrating the superiority of the proposed algorithm. Finally, experimental results confirm that the proposed method can rapidly calibrate accelerometer error parameters without relying on complex equipment and with greater accuracy than traditional methods.

The large misalignment angle error model of strapdown inertial navigation system (SINS) is a kind of nonlinear Gauss error model. The Gauss Hermite method can approach Gauss integral with any precision, and achieve high precision filtering effect. The number of Gauss Hermite integral points is exponentially related to the dimension of nonlinear system. In the process of real-time filtering, a large number of integral points of multidimensional system may lead to dimension disaster. According to the initial alignment error model of SINS with large misalignment angle, this paper presents a strong tracking reduced dimension sparse grid Gauss Hermite filtering method, which reduces the nonlinear integration dimension through separation of the linear and nonlinear terms, further reduces the number of Gauss hermit filtering integration points by sparse grid method, and ensures the stability of the system by strong tracking method.

This Special Issue presents efficient formulations and implementations of sequential and variational data assimilation methods. The methods address three important issues in the context of operational data assimilation: efficient implementation of localization methods, sampling methods for approaching posterior ensembles under non-linear model errors, and adjoint-free formulations of four dimensional variational methods.

On diagnostics in symmetrical nonlinear models

In this paper we consider the problem of set-membership identification of nonlinear polynomial output error models, where output measurements are affected by bounded additive noise known to enjoy certain peculiar properties like whiteness and uncorrelation with the noiseless output sequence. More precisely, we propose an original approach to compute the so-called parameter uncertainty intervals by properly taking into account all the available a-priori information about the model structure and the properties of the noise. Parameters bounds are obtained by computing the solution to a set of polynomial optimization problems, where the decision variable are suitably selected and constrained in order to enforce the available a-priori information on the noise properties. Convex relaxation techniques are exploited to solve the formulated optimization problem. A simulation example is reported in order to show the effectiveness of the proposed approach.