Parameters Of Nonlinear Function Research Articles

Effect of Smoothing in Generalized Linear Mixed Models on the Estimation of Covariance Parameters for Longitudinal Data.

Besides being mainly used for analyzing clustered or longitudinal data, generalized linear mixed models can also be used for smoothing via restricting changes in the fit at the knots in regression splines. The resulting models are usually called semiparametric mixed models (SPMMs). We investigate the effect of smoothing using SPMMs on the correlation and variance parameter estimates for serially correlated longitudinal normal, Poisson and binary data. Through simulations, we compare the performance of SPMMs to other simpler methods for estimating the nonlinear association such as fractional polynomials, and using a parametric nonlinear function. Simulation results suggest that, in general, the SPMMs recover the true curves very well and yield reasonable estimates of the correlation and variance parameters. However, for binary outcomes, SPMMs produce biased estimates of the variance parameters for high serially correlated data. We apply these methods to a dataset investigating the association between CD4 cell count and time since seroconversion for HIV infected men enrolled in the Multicenter AIDS Cohort Study.

Application of Adaptive DP-optimality to Design a Pilot Study for a Clotting Time Test for Enoxaparin.

Dosing of enoxaparin, like other anticoagulants, may result in bleeding following excessive doses and clot formation if the dose is too low. We recently showed that a factor Xa based clotting time test could potentially assess the effect of enoxaparin on the clotting system. However, the test did not perform well in subsequent individuals and effectiveness of an exogenous phospholipid, Actin FS, in reducing the variability in the clotting time was assessed. The aim of this work was to conduct an adaptive pilot study to determine the range of concentrations of Xa and Actin FS to take forward into a proof-of-concept study. A nonlinear parametric function was developed to describe the response surface over the factors of interest. An adaptive method was used to estimate the parameters using a D-optimal design criterion. In order to provide a reasonable probability of observing a success of the clotting time test, a P-optimal design criterion was incorporated using a loss function to describe the hybrid DP-optimality. The use of adaptive DP-optimality method resulted in an efficient estimation of model parameters using data from only 6 healthy volunteers. The use of response surface modelling identified a range of sets of Xa and Actin FS concentrations, any of which could be used for the proof-of-concept study. This study shows that parsimonious adaptive DP-optimal designs may provide both precise parameter estimates for response surface modelling as well as clinical confidence in the potential benefits of the study.

Confidence bounds for nonlinear dose-response relationships.

An important aim of drug trials is to characterize the dose-response relationship of a new compound. Such a relationship can often be described by a parametric (nonlinear) function that is monotone in dose. If such a model is fitted, it is useful to know the uncertainty of the fitted curve. It is well known that Wald confidence intervals are based on linear approximations and are often unsatisfactory in nonlinear models. Apart from incorrect coverage rates, they can be unreasonable in the sense that the lower confidence limit of the difference to placebo can be negative, even when an overall test shows a significant positive effect. Bootstrap confidence intervals solve many of the problems of the Wald confidence intervals but are computationally intensive and prone to undercoverage for small sample sizes. In this work, we propose a profile likelihood approach to compute confidence intervals for the dose-response curve. These confidence bounds have better coverage than Wald intervals and are more precise and generally faster than bootstrap methods. Moreover, if monotonicity is assumed, the profile likelihood approach takes this automatically into account. The approach is illustrated using a public dataset and simulations based on the Emax and sigmoid Emax models.

General and mechanistic optimal relationships for tensile strength of doubly convex tablets under diametrical compression

We propose a general framework for determining optimal relationships for tensile strength of doubly convex tablets under diametrical compression. This approach is based on the observation that tensile strength is directly proportional to the breaking force and inversely proportional to a non-linear function of geometric parameters and materials properties. This generalization reduces to the analytical expression commonly used for flat faced tablets, i.e., Hertz solution, and to the empirical relationship currently used in the pharmaceutical industry for convex-faced tablets, i.e., Pitt's equation. Under proper parametrization, optimal tensile strength relationship can be determined from experimental results by minimizing a figure of merit of choice. This optimization is performed under the first-order approximation that a flat faced tablet and a doubly curved tablet have the same tensile strength if they have the same relative density and are made of the same powder, under equivalent manufacturing conditions. Furthermore, we provide a set of recommendations and best practices for assessing the performance of optimal tensile strength relationships in general. Based on these guidelines, we identify two new models, namely the general and mechanistic models, which are effective and predictive alternatives to the tensile strength relationship currently used in the pharmaceutical industry.

Dual adaptive extremum control of Hammerstein systems

A novel dual adaptive controller for extremum control of stochastic and uncertain nonlinear Hammerstein systems is proposed. The design is based on the innovations dual control cost function originally developed for conventional adaptive control of linear systems. However, the design process is extensively modified and developed so as to cater for the extremum control scenario. This is a more challenging problem because the reference input is itself a nonlinear function of the unknown system parameters, rather than an independent and predefined external reference signal. As in all dual adaptive schemes, it leads to a control law that balances out the need for caution, due to parameter uncertainty, with the conflicting requirement of probing that acts to quickly reduce parameter uncertainty. The proposed controller's performance is analysed through extensive Monte Carlo simulation trials and compared with several other non-dual adaptive extremum controllers. It is shown that the novel extremum innovations dual controller is superior to other types of adaptive control systems that are based on a certainty equivalence assumption. In addition, a novel statistical measure is introduced that yields a more objective evaluation of the Monte Carlo simulation results.

Linear parameter-varying modeling and identification for condition-based monitoring of systems

A linear parameter-varying (LPV) model and its new identification scheme are proposed for monitoring the status of a system. As the subsystem parameters are generally inaccessible, during the offline identification stage, emulators, which are transfer function blocks, are included at the measurement outputs to simulate different operating scenarios, including the nominal and abnormal ones. That is, instead of relying on the past or historical data, which may be unavailable, various operating scenarios are obtained from real-time simulation using emulators. The feature vector (coefficients of the system transfer function) is, in general, a nonlinear function of the emulator parameters and this nonlinear relationship is completely characterized by influence vectors which are the partial derivatives of the feature vector with respect to the emulator parameters. Influence vectors are identified off-line using linear least-squares fit to the emulated input–output data. The LPV model is comprised of (a) scheduling variables, and (b) emulator parameters to track respectively the variations in the operating points and to generate operational data. A Kalman filter is designed using the identified model of the system driven by the residual and is adapted to all the trajectories of the scheduling variables. The Kalman filter residual forms the backbone of the condition-monitoring and fault-diagnosis tasks. The proposed scheme is successfully evaluated on simulated systems as well as on a physical position control system.

A semiparametric approach for joint modeling of median and skewness

We motivate this paper by showing through Monte Carlo simulation that ignoring the skewness of the response variable distribution in non-linear regression models may introduce biases on the parameter estimates and/or on the estimation of the associated variability measures. Then, we propose a semiparametric regression model suitable for data set analysis in which the distribution of the response is strictly positive and asymmetric. In this setup, both median and skewness of the response variable distribution are explicitly modeled, the median using a parametric non-linear function and the skewness using a semiparametric function. The proposed model allows for the description of the response using the log-symmetric distribution, which is a generalization of the log-normal distribution and is flexible enough to consider bimodal distributions in special cases as well as distributions having heavier or lighter tails than those of the log-normal one. An iterative estimation process as well as some diagnostic methods are derived. Two data sets previously analyzed under parametric models are reanalyzed using the proposed methodology.

Saturated RISE Feedback Control for a Class of Second-Order Nonlinear Systems

A saturated controller is developed for a class of uncertain, second-order, nonlinear systems which includes time-varying and nonlinearly parameterized functions with bounded disturbances using a continuous control law with smooth saturation functions. Based on the robust integral of the sign of the error (RISE) control methodology, the developed controller is able to utilize the benefits of high gain control strategies while guaranteeing saturation limits are not surpassed. The bounds on the control are known a priori and can be adjusted by changing the feedback gains. The saturated controller yields asymptotic tracking despite model uncertainty and added disturbances in the dynamics. Experimental results using a two-link robot manipulator demonstrate the performance of the developed controller.

Gasoline Engine Fuel Pump Pressure Nonlinear Function Fuzzy Control

In order to improve the accuracy of electronic fuel injection (EFI) engine fuel pump pressure, according to the characteristics of pump directly driven by motor, a nonlinear mathematical model is built for the fuel supply system of EFI engine pressure control. And then, the nonlinear-function-optimized flow signal is introduced to adjust the parameters of nonlinear function. With the PI control algorithm, the precise tracking control of injection pressure is achieved. The simulation and experiment results show that, it can indirectly realize from the speed control to pressure control. DOI : http://dx.doi.org/10.11591/telkomnika.v12i4.4220

Probabilistic optimization of a continuum mechanics model to predict differential stress-induced damage in claystone

Phenomenological modeling of anisotropic damage in rock raises many fundamental thermodynamic and mechanical issues. In this paper, the maximum likelihood method is used to analyze the performance of the Differential Stress Induced Damage (DSID) model recently formulated by Xu and Arson [1]. The stress/strain relationship is a nonlinear function of parameters including unknown constants (i.e., damage constitutive parameters) and known variables (e.g., elastic parameters and controlled stress state). Logarithmic transformation, normalization and forward deletion are employed, in order to find the optimum number of constitutive parameters, as a trade off between accuracy and simplicity. For Eastern France claystone subject to deviatoric stress loading (e.g., triaxial and proportional compression loading), it is found that (1) only one damage parameter (a2) is needed in the expression of the free energy to predict stress/strain curves; (2) a2 controls the deviation of the current principal directions of stress to the principal directions of damage; (3) model parameters involved in the damage criterion can be related to a2. As a result, a2 is the only parameter needed to model differential-stress induced damage in Eastern France claystone. It is also shown that within the set of assumptions made in this study, the DSID model is not sensitive to the initial damage threshold C0, except for C0>106Pa, a range of values in which only one constitutive parameter becomes insufficient to predict the stress/strain curves of damaged claystone. Coupling probabilistic calibration and optimization methods to numerical codes promises to allow adapting the complexity of anisotropic damage models to different rocks and stress paths.

Interpolation of temperature in a mountainous region using nonlinear profiles and non‐Euclidean distances

ABSTRACTIn mountain regions, the distribution of surface air temperature happens to show marked horizontal gradients and nonlinear variations with topographic height. These pose a major challenge for the construction of area‐wide temperature datasets on a regular grid. This study introduces a new deterministic method's for the spatial interpolation of daily temperature from station measurements. Building on a scale‐separation concept, the methods main features are (1) a nonlinear parametric function to model nonlinearities in the vertical thermal profile at the scale of major basins, and (2) a distance weighting scheme with a non‐Euclidean metric that accounts for terrain effects on the spatial representativity of measurements. The method is configured for the territory of Switzerland (European Alps) and is applied for the construction of a km‐scale grid dataset from 70‐100 stations over all days from 1961 to 2010. Several illustrative cases attest the method's potential, also under challenging conditions. Temperature patterns from basin‐scale inversions and surface heated/cooled boundary layers are realistically reproduced. In situations with valley‐scale cold‐air pools and foehn, the method is less prone to artificial upslope/downslope extrapolation often observed with other techniques. With a network of 100 stations in Switzerland, typical interpolation errors (mean absolute error MAE, cross‐validation) range from 0.5 °C over flat and hilly terrain in summer to 1.5 °C in the Alps in winter. Larger and partly systematic errors (MAE ≥ 3 °C) must be expected in un‐sampled valleys in winter due to the missing out of local‐scale cold pools. Interpolation accuracy was found to vary with the change in station density over time, demonstrating improvements in the overall representativity of the measurement network. But this also compromises the long‐term homogeneity of the grid dataset, despite it being based on homogeneous records. The presented method may be applicable in other mountain regions after some configuration.

Noise properties of linear molecular communication networks

Molecular communication networks consist of transmitters and receivers distributed in a fluid medium. The communication in these networks is realised by the transmitters emitting signalling molecules, which are diffused in the medium to reach the receivers. This paper investigates the properties of noise, or the variance of the receiver output, in molecular communication networks. The noise in these networks come from multiple sources: stochastic emission of signalling molecules by the transmitters, diffusion in the fluid medium and stochastic reaction kinetics at the receivers. We model these stochastic fluctuations by using an extension of the master equation. We show that, under certain conditions, the receiver outputs of linear molecular communication networks are Poisson distributed. The derivation also shows that noise in these networks is a nonlinear function of the network parameters and is non-additive. Numerical examples are provided to illustrate the properties of this type of Poisson channels.

Conserving the Birds of Uganda’s Banana-Coffee Arc: Land Sparing and Land Sharing Compared

Reconciling the aims of feeding an ever more demanding human population and conserving biodiversity is a difficult challenge. Here, we explore potential solutions by assessing whether land sparing (farming for high yield, potentially enabling the protection of non-farmland habitat), land sharing (lower yielding farming with more biodiversity within farmland) or a mixed strategy would result in better bird conservation outcomes for a specified level of agricultural production. We surveyed forest and farmland study areas in southern Uganda, measuring the population density of 256 bird species and agricultural yield: food energy and gross income. Parametric non-linear functions relating density to yield were fitted. Species were identified as “winners” (total population size always at least as great with agriculture present as without it) or “losers” (total population sometimes or always reduced with agriculture present) for a range of targets for total agricultural production. For each target we determined whether each species would be predicted to have a higher total population with land sparing, land sharing or with any intermediate level of sparing at an intermediate yield. We found that most species were expected to have their highest total populations with land sparing, particularly loser species and species with small global range sizes. Hence, more species would benefit from high-yield farming if used as part of a strategy to reduce forest loss than from low-yield farming and land sharing, as has been found in Ghana and India in a previous study. We caution against advocacy for high-yield farming alone as a means to deliver land sparing if it is done without strong protection for natural habitats, other ecosystem services and social welfare. Instead, we suggest that conservationists explore how conservation and agricultural policies can be better integrated to deliver land sparing by, for example, combining land-use planning and agronomic support for small farmers.

On estimability of parsimonious term structure models: an experiment with the Nelson–Siegel specification

This study addresses operational issues in estimation of parsimonious term structure models. When using price errors, objective function in term structure estimation is a nonlinear function of the model parameters. This necessarily entails using numerical optimization techniques for estimation, which brings to fore the issue of (sensitivity of final results to) the choice of initialization of the optimization routine. This study assesses the sensitivity of the final objective function value and the final parameter vector to the choice of the ‘initial guess’ during the estimation of the popular Nelson–Siegel model. It turns out that there exist regions in the shape of the objective function where a slight change in (seemingly reasonable) initial vector takes one far from optimum. Choice of the (range of) ‘best’ starting vector turns out to be an empirical matter. Grid search is recommended. One must first get to a subset of initial values that results in the objective function value near a minimum and then assess the sensitivity of the final parameter vector to those relevant (subset of) initial values. The study illustrates the process using a typical trading day's data.

On feedback capability for a class of semiparametric uncertain systems

The main purpose of this paper is to understand and characterize the maximum capability of the feedback mechanism for a basic class of scalar discrete-time semiparametric minimum-phase control systems, where both parametric and nonparametric uncertainties are included. We will demonstrate that the necessary and sufficient condition to stabilize the class of systems is L<32+2, where L is the Lipschitz constant describing the “size” of the uncertainty of the nonparametric part. This critical value is the same as that first obtained in Xie and Guo (2000b) for a class of purely nonparametric systems, which shows that the capability of feedback is not influenced by the parameterized uncertainty in the systems, as long as the corresponding parametric nonlinear function has a linear growth rate. While the necessity proof is directly obtainable from Xie and Guo (2000b), the main task of this paper is to prove the sufficiency part.

BSS algorithm by diffusion mixing non-parametric density estimator

Diffusion mixing estimator (DME)-based non-parametric blind source separation (BSS) algorithm is proposed under the framework of non-Bayesian framework and natural gradient optimisation method. In order to improve the performance of signal separation by BSS, the probability distribution of source signals must be described as accurately as possible. Compared to the non-parametric fixed-width kernel density estimator (FKDE) method, the DME with a new data-driven bandwidth selection method can improve the performance of FKDE, which is inspired via a Langevin diffusion process. Moreover, the direct estimation of the score functions can separate the hybrid mixtures of sources that contain both symmetric and asymmetric distribution source signals and do not need to assume the parametric non-linear functions as them. The effectiveness of the proposed algorithm has been confirmed by simulation experiments.

Uncertainty Identification of Damage Growth Parameters Using Nonlinear Regression

R ECENTLY, it has been shown that structural health monitoring (SHM) systems can be used for inspection to detect damage [1]. SHM-based maintenance is effective as only those airplanes that are in danger will be sent for maintenance (condition-based maintenance). Furthermore, Coppe et al. [2] showed that in addition to damage diagnosis SHM can predict the remaining useful life by identifying damage growth parameters. They used Bayesian inference [3] to reduce uncertainty in the damage growth parameters using measured damage size information. Bayesian inference is a powerful method of quantifying uncertainty in the model parameters. It can take into account prior knowledge on the unknown parameters and improve them using experimental observations. However, in the case of SHM, the advantage of the prior information can be overpowered by the amount of data available. In addition, when many parameters are updated simultaneously, Bayesian inference becomes computationally expensive due to multidimensional integration. On the other hand, the traditional linear regression method [4] can be used to identify deterministic parameters when the model is a linear function of the parameters. This method is particularly powerful when many data are available, which is the case for SHM. By assuming that the noise in the experimental data is Gaussian, it is possible to estimate the uncertainty in the identified parameters. When the physical model is a nonlinear function of model parameters, uncertainty quantification is not straightforward [5]. As will be shown in the numerical examples, the crack growth in aircraft structures nonlinearly depends on the parameters that need to be identified. In this Note, a linear perturbation concept is used to quantify uncertainty in the nonlinear regression result. First, the nonlinear optimization is used to find the model parameters that minimize error between themodel and SHMdata. Then the nonlinear model is linearized at the identified parameter values, fromwhich the uncertainty quantification in the linear regression method can be used. This approach can introduce two errors into the uncertainty estimation: 1) linearization error and 2) error associated with assumption ofGaussian noise. In addition, it is assumed that noises at different experiments are uncorrelated. The objective of theNote is to examine their effect on accuracy of the uncertainty estimation. II. Uncertainty Quantification in Nonlinear Regression

Non-linear mixed logit

We develop an extension of the familiar linear mixed logit model to allow for the direct estimation of parametric non-linear functions defined over structural parameters. Classic applications include the estimation of coefficients of utility functions to characterize risk attitudes and discounting functions to characterize impatience. There are several unexpected benefits of this extension, apart from the ability to directly estimate structural parameters of theoretical interest.

Globally Optimal Linear Precoders for Finite Alphabet Signals Over Complex Vector Gaussian Channels

We study the design optimization of linear precoders for maximizing the mutual information between finite alphabet input and the corresponding output over complex-valued vector channels. This mutual information is a nonlinear and non-concave function of the precoder parameters, posing a major obstacle to precoder design optimization. Our work presents three main contributions: First, we prove that the mutual information is a concave function of a matrix which itself is a quadratic function of the precoder matrix. Second, we propose a parameterized iterative algorithm for finding optimal linear precoders to achieve the global maximum of the mutual information. The proposed iterative algorithm is numerically robust, computationally efficient, and globally convergent. Third, we demonstrate that maximizing the mutual information between a discrete constellation input and the corresponding output of a vector channel not only provides the highest practically achievable rate but also serves as an excellent criterion for minimizing the coded bit error rate. Our numerical examples show that the proposed algorithm achieves mutual information very close to the channel capacity for channel coding rate under 0.75, and also exhibits a large gain over existing linear precoding and/or power allocation algorithms. Moreover, our examples show that certain existing methods are susceptible to being trapped at locally optimal precoders.

Parametric Estimation for Delayed Nonlinear Time-Varying Dynamical Systems

A constructive algorithm is proposed and illustrated for parametric estimation in delayed nonlinear time-varying dynamic system models from available data. The algorithm uses Chebyshev spectral collocation and optimization. The problems addressed are estimations with complete state vector and incomplete state vector availability. Using an equivalent algebraic description of dynamical systems by Chebyshev spectral collocation and data, a standard least-squares residual cost function is set up for complete and incomplete information cases. Minimization of this cost yields the unique solution for the unknown parameters for estimation with complete state availability, only owing to the fact that the cost function is quadratic and positive definite. Such arguments cannot be made for estimation with incomplete state availability as the cost function is positive definite albeit a nonlinear function of the unknown parameters and states. All the algorithms are presented stepwise and are illustrated using suitable examples.