Nonlinear Least-squares Method Research Articles

Improved Multi-Echo Gradient-Echo-Based Myelin Water Fraction Mapping Using Dimensionality Reduction.

Multi-echo gradient-echo (mGRE)-based myelin water fraction (MWF) mapping is a promising myelin water imaging (MWI) modality but is vulnerable to noise and artifact corruption. The linear dimensionality reduction (LDR) method has recently shown improvements with regard to these challenges. However, the magnitude value based low rank operators have been shown to misestimate the MWF for regions with [Formula: see text] anisotropy. This paper presents a nonlinear dimensionality reduction (NLDR) method to estimate the MWF map better by encouraging nonlinear low dimensionality of mGRE signal sources. Specifically, we implemented a fully connected deep autoencoder to extract the low-dimensional features of complex-valued signals and incorporated a sparse regularization to separate the anomaly sources that do not reside in the low-dimensional manifold. Simulations and in vivo experiments were performed to evaluate the accuracy of the MWF map under various situations. The proposed NLDR-based MWF improves the accuracy of the MWF map over the conventional nonlinear least-squares method and the LDR-based MWF and maintains robustness against noise and artifact corruption.

A new limited memory method for unconstrained nonlinear least squares

This paper suggests a new limited memory trust region algorithm for large unconstrained black box least squares problems, called LMLS. Main features of LMLS are a new non-monotone technique, a new adaptive radius strategy, a new Broyden-like algorithm based on the previous good points, and a heuristic estimation for the Jacobian matrix in a subspace with random basis indices. Our numerical results show that LMLS is robust and efficient, especially in comparison with solvers using traditional limited memory and standard quasi-Newton approximations.

Multi-surface phase-shifting algorithm using the window function fitted by the nonlinear least squares method

A multi-surface phase demodulation algorithm is proposed to measure surfaces of the transparent plate simultaneously. A non-linear least-squares method based on the trust region is utilized to fit the window function derived from the characteristic polynomials. The fitted window function requires fewer parameters and has a flexible form, which means the window length can be stretched freely and used flexibly. Then, a multi-surface phase demodulation technique combining the discrete Fourier principle and the fitted window function is investigated, allowing simultaneous reconstruction of the measured front surface, rear surface, and thickness variation. During the implementation of our algorithm, the processing time for the interferograms with 850×850 pixels is around 5.1s. Measurement errors caused by several factors are analyzed, including wavelength-tuning error, frequency order error, cavity length error, and average thickness error. The experimental result on a rectangular transparent plate further verifies the effectiveness of our algorithm.

Linear and non-linear regression analysis for the biosorption kinetics of methylene blue

The nonviable biomass of Rhizopus sp. R-18, Penicillium candidum and Penicillium chrysogenum was studied for biosorption of methylene blue (MB). The sorption of MB was studied be performing batch kinetic experiments. Kinetic measurements showed that sorption of MB reached equilibrium in 4 hours. The batch sorption models, based on a pseudo-first, pseudo-second and pseudo-nth order were applied to predict the rate constant of sorption and the equilibrium capacity. The linear and nonlinear least-square methods were used to obtain the kinetic parameters. The best-fit model was identified using statistic analysis. The results showed that both linear and nonlinear form of pseudo- second order expression could be used to fit the experimental data but nonlinear method may be a better way to obtain the desired parameters. As well the pseudo n-th order kinetic model was successfully applied to the kinetic data. The order (n) of adsorption reaction was found for all employed biosorbents: for Rhizopus sp. R-18 it had value 3.1, P. candidum 3.0 and P. chrysogenum 3.8.

A Fast Algorithm for Exact Maximum Likelihood Estimation of Multiple Input Transfer Function Models with Auto-Correlated Errors

This paper proposes a fast algorithm for the exact maximum likelihood estimation of parameters of multiple inputs transfer function models. This algorithm is a generalization of that proposed by Mélard (1984) which is a combination of an improved version of an algorithm of Pearlman (1980) which uses an algorithm of Morf, Sidhu et Kailath (1974) and consists to replace the (matrix) Riccati-type difference equation used in the Kalman filter by a (vector) Chandrasekhar-type difference equation with the quick recursion switching suggested by Gardner, Harvey and Phillips (1980) and an algorithm of Wilson (1979). Simulations and practical examples are used to illustrate the algorithm by comparing it with the method of Poskitt (1989), the generalized least squares method suggested by Sabiti (1993), and the nonlinear least-squares method of Box and Jenkins (1976).

Aspects of size effect on discrete elementmodeling of normal strength concrete

Present paper focuses on the modeling of size effect on the compressive strength of normal concrete with the application of Discrete Element Method (DEM). Test specimens with different size and shape were cast and uniaxial compressive strength test was performed on each sample. Five different concrete mixes were used, all belonging to a different normal strength concrete class (C20/25, C30/37, C35/45, C45/55, and C50/60). The numerical simulations were carried out by using the PFC 5 software, which applies rigid spheres and contacts between them to model the material. DEM modeling of size effect could be advantageous because the development of micro-cracks in the material can be observed and the failure mode can be visualized. The series of experiments were repeated with the model after calibration. The relationship of the parallel bond strength of the contacts and the laboratory compressive strength test was analyzed by aiming to determine a relation between the compressive strength and the bond strength of different sized models. An equation was derived based on Bazant's size effect law to estimate the parallel bond strength of differently sized specimens. The parameters of the equation were optimized based on measurement data using nonlinear least-squares method with SSE (sum of squared errors) objective function. The laboratory test results showed a good agreement with the literature data (compressive strength is decreasing with the increase of the size of the specimen regardless of the shape). The derived estimation models showed strong correlation with the measurement data. The results indicated that the size effect is stronger on concretes with lower strength class due to the higher level of inhomogeneity of the material. It was observed that size effect is more significant on cube specimens than on cylinder samples, which can be caused by the side ratios of the specimens and the size of the purely compressed zone. A limit value for the minimum size of DE model for cubes and cylinder was determined, above which the size effect on compressive strength can be neglected within the investigated size range. The relationship of model size (particle number) and computational time was analyzed and a method to decrease the computational time (number of iterations) of material genesis is proposed.

Design of Aerodynamic Ball Levitation Laboratory Plant

This paper presents the development of a new Aerodynamic Ball Levitation Laboratory Plant at the Center of Modern Control Techniques and Industrial Informatics (CMCT&II). The entire design process of the plant is described, including the component selection process, the physical construction of the plant, the design of a printed circuit board (PCB) powered by a microcontroller, and the implementation of its firmware. A parametric mathematical model of the laboratory plant is created, whose parameters are then estimated using a nonlinear least-squares method based on acquired experimental data. The Kalman filter and the optimal state-space feedback control are designed based on the obtained mathematical model. The designed controller is then validated using the physical plant.

Using particle swarm optimization to improve visibility-aerosol optical depth retrieval method

In view of the lack of long-term AOD (Aerosol Optical Depth) data, PSO (Particle Swarm Optimization) algorithm is introduced and joint used with NLSM (the nonlinear least square method) to improve visibility-AOD retrieval method, which is referred to as the PSO-M-Elterman model and significantly increases data available rate by 8% and correlation by about 20% with the true value in the experimental group. The mean absolute error, the proportion of the smaller absolute error and the root mean square error in the PSO-M-Elterman model experimental group are 0.0314 and 91.23%, 0.0509 respectively, which significantly outperforms other groups. The main increase of AOD was found in the eastern region (South China, East China, Central China) and Taklimakan with the trend coefficients of 2.67, 2.46, 2.13, and 1.45 (×10−3 yr−1) in recent 55 years, which may not be interpreted by the influence of relative humidity. Long-term change of AOD in east China is mainly caused by human activity, and the AOD is higher in cities with a larger population and more human activity. The PSO-M-Elterman model can maximize the advantage of visibility sequence length to obtain long-term AOD inversion results.

Convergence of Levenberg-Marquardt Estimates of Nonlinear Parameters in EIS Analysis of Coated Aluminum Panels

Equivalent circuit models are frequently employed to analyze electrochemical impedance spectroscopy (EIS) data. However, aside from some simple cases in which the equivalent circuit element parameters can be estimated directly from a plot of the data, the parameters for each of the equivalent circuit elements must often be estimated through the implementation of a nonlinear least-squares regression method. One frequently-used approach is the powerful Levenberg-Marquardt algorithm. This algorithm functions as a damped least-squares method that performs an interpolation between the steepest-descent and the Gauss-Newton methods for estimating nonlinear parameters. However, there are several challenges to using a Levenberg-Marquardt algorithm to model EIS data. The nonlinear nature of the impedance equations along with the difference in the magnitude of the parameter values can result in the algorithm failing to converge to a local minimum in the parameter space. This convergence challenge can be seen in determining equivalent circuit values for undamaged coating systems that have a high pore resistance, a small capacitance, as well as contributions from the underlying metal substrate. A Levenberg-Marquardt algorithm, along with a heuristic approach to use the raw data to improve the initial parameter estimates, is implemented that provides an improved approach for nonlinear regression fits of equivalent circuit models. A comparison of three equivalent circuit models fits, with residuals, for a two-layer coating system on an aluminum substrate is shown in the attached figure.Figure caption: Plot of EIS data (+ symbols) from a cured coating system prior to any environmental exposures compared with 3 equivalent circuit model fits (solid lines). The model fits were obtained using an NRL-developed Levenberg-Marquardt nonlinear regression fitting routine.The Department of Defense Strategic Environmental Research and Development Program (SERDP) sponsored this project under work unit 5399. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Office of Naval Research, the U.S. Navy, the Department of Defense, or the U.S. government. Figure 1

Age-Related Change in Hepatic Clearance Inferred from Multiple Population Pharmacokinetic Studies: Comparison with Renal Clearance and Their Associations with Organ Weight and Blood Flow.

This study aimed to examine the magnitude of age-related change in hepatic clearance by integrating the data of multiple drugs and to compare this with renal clearance, considering associations with age-related changes in organ weight and blood flow. The results of multiple population pharmacokinetic analyses that detected age-related clearance changes in hepatically eliminated drugs were collected. The relationship between hepatic clearance of the unbound drug and age was then analyzed using the nonlinear least-squares method, adjusting for interdrug differences. The obtained change in hepatic clearance was compared with age-related changes in liver weight and hepatic blood flow in Japanese and Westerners. For comparison, the changes in renal clearance were analyzed similarly. In total, 18 drugs were analyzed. The hepatic unbound clearance decreased by 32% at age 80years and by 40% at age 90years, compared with age 40years, suggesting that it decreased by 0.80% per year with aging. The rate of the decrease was consistent with decreases in hepatic weight per person or blood flow per person, regardless of ethnicity and sex. Since age-related change in body weight varied somewhat by sex or ethnicity, hepatic weight per body weight was less consistent to account for age-related change in hepatic clearance. As for an index of renal clearance, the changes in inulin clearance with age were similar to those in renal blood flow, with a decrease of 0.97% per year from the age of 40 years. Hepatic clearance consistently decreased by 0.80% per year from the age of 40years, with aging for multiple drugs analyzed in this study. Changes in organ weight and blood flow are considered to be primarily responsible for the age-related changes in hepatic and renal clearance.

Mathematical modeling and projections of a vector-borne disease with optimal control strategies: A case study of the Chikungunya in Chad

In this work, we extend existing models of vector-borne diseases by including density-dependent rates and some existing control mechanisms to decrease the disease burden in the human population. We begin by analyzing the vector model dynamics and by determining the offspring reproductive number denoted by N as well as the trivial and nontrivial equilibria. Using theory of cooperative systems and the general theory of Lyapunov, we prove that, although there is a possibility that the trivial equilibrium coexists with a positive equilibrium, it remains globally asymptotically stable whenever N ≤ 1 . The fact that the non-trivial equilibrium is globally asymptotically stable permits us to reduce the study of the full model to the study of a reduced model whenever N > 1 . Thus, we analyze the reduced model by computing the basic reproduction number R 0 , equilibrium points as well as asymptotic stability of each equilibrium point. We also explore the nature of the bifurcation for the disease-free equilibrium from R 0 = 1 . By the application of the centre manifold theory, we prove that the backward bifurcation phenomenon can occur in our model, which means that the necessary condition R 0 < 1 is not sufficient to guarantee the final extinction of the disease in human populations. To calibrate our model, we estimate model parameters on clinical data from the last Chikungunya epidemic which occurred in Chad, using the non-linear least-square method. We find out that R 0 = 1.8519 , which means that we are in an endemic state since R 0 > 1 . To determine model parameters that are responsible for disease spread in the human community, we perform sensitivity analysis (SA) using a global method. It follows that the density-dependent death rate of mosquitoes and the average number of mosquito bites are key parameters in the disease dynamics. Following this, we thus formulate an optimal control model by including in the autonomous model, four time-dependent control functions to fight the disease spread. Pontryagin’s maximum principle is used to characterize our optimal controls. Numerical simulations, using parameter values of Chikungunya transmission dynamics, and efficiency analysis, are conducted to determine the better control strategy which guaranteed the final extinction of the disease in human populations.

Projections and fractional dynamics of the typhoid fever: A case study of Mbandjock in the Centre Region of Cameroon

In this work, we formulate a mathematical model with a non-integer order derivative to investigate typhoid fever transmission dynamics. To combat the spread of this disease in the human community, control measures like vaccination are included in the proposed model. We calculate the epidemiological threshold called the control reproduction number, Rc, and perform the asymptotic stability of the typhoid-free equilibrium point. We prove that the typhoid-free equilibrium for both integer and non-integer models is locally and globally asymptotically stable whenever Rc is less than one. We also prove that both models admit only one endemic equilibrium point which is globally asymptotically stable whenever Rc>1 and no endemic equilibrium point otherwise. This means that the backward bifurcation phenomenon does not occur. In absence of vaccination, Rc is equal to the basic reproduction number R0. We found out that Rc<R0 which means that vaccination can permit to decrease of the spread of typhoid fever in the human community. Using fixed point theory, we perform existence and uniqueness analysis of solutions of the fractional model. We use the Adams-Bashforth-Moulton method to construct a numerical scheme of the fractional model. We prove the stability of the proposed numerical scheme. To calibrate our model, we estimate model parameters on clinical data of Mbandjock District Hospital in the Centre Region of Cameroon, using the Non-linear Least-Square method. This permits us to find Rc=1.3722, which means that we are in an endemic state (since Rc>1), and then to predict new cases of typhoid fever per month at Mbandjock in the next new year. To determine model parameters that are responsible for disease spread in the human community, we perform sensitivity analysis (SA). This analysis shows that the vaccination rate, the human-bacteria contact rate, as well as the recovery rate, are the most important parameters in the disease spread. To validate our analytical results, and to see the impact of some control measures in the spread of typhoid fever in the human community, as well as the impact of the fractional-order on typhoid transmission dynamics, we perform several numerical simulations.

Novel study on the electron density distribution projection maps calculated via the discrete cosine transform

It is already known that a curve estimated via the discrete cosine transform (DCT) always passes through all the measured points without using imaginary numbers in the DCT coefficients, unlike the discrete Fourier transform. Moreover, owing to its character, the DCT can be used instead of the nonlinear least-squares method to express various theoretical curves. Because the DCT is a kind of Fourier transform, there is a possibility that the DCT could be employed to draw electron density distribution maps of crystals. If so, the probability that the DCT could be used to investigate the internal structure of materials by analysing the theoretical curves would increase. This article reports an attempt to draw the electron density distribution maps of the Mg3BN3 low-pressure phase [Mg3BN3(L)] by using the DCT in order to confirm the utility of the DCT for analysing the internal structure of materials. It is found that the DCT can provide mirror-symmetric electron density distribution projection maps and a modified DCT can be used to calculate whole standard electron density distribution projection maps for the surface plane of the unit cell. Moreover, a real crystal structure that has a centre of symmetry can be determined by the DCT by transforming a 1/4 part of the mirror-symmetric electron density distribution projection map.

Bayesian based lifetime prediction for high-power white LEDs

The introduction of high-power white LEDs has revolutionized the lighting industry in the past few decades due to the multiple benefits in terms of high reliability, environmental friendliness and versatile applications. However, challenges have arisen in assessing the reliability and lifetime prediction because it is difficult to record the failure data in a short period of time. Currently, the nonlinear least squares (NLS) regression-based method is used in industry for projecting the lumen maintenance lifetime from degradation data. The model parameters estimated using the NLS regression approach are deterministic and introduce high prediction errors. In this paper, a Bayesian method is proposed to estimate the remaining useful lifetimes (RULs) of both high-power white LED packages and lamps. The accelerated degradation tests conducted for gathering lumen degradation data are used to validate the proposed method. The exponential decay model is used as the degradation model and the parameters are estimated based on Markov Chain Monte Carlo (MCMC) sampling and using the Metropolis-Hasting (MH) algorithm. The lifetime prediction results showed that the Bayesian method has better prediction accuracy compared to the NLS method. Thus, the proposed Bayesian method is shown to be a promising approach to address the lifetime prediction issue for high-power white LEDs with improved prediction accuracy.

Robust Motion Prediction of a Free-Tumbling Satellite with On-Ground Experimental Validation

The task of approaching and capturing a free-tumbling satellite on-orbit presents open challenges for autonomous guidance and control strategies. One of these is to robustly predict the satellite’s tumbling motion in view of measurement errors and of unfavorable free-body dynamic effects. A comparative study of solutions proposed in the literature is presented, considering tumbling scenarios that might offer low observability of the related parameters. To this end, this paper extends and compares nonlinear and linear least-squares batch techniques with an extended Kalman filter recursive technique to identify the necessary state and inertial parameters of a satellite for the purpose of motion prediction. These estimation methods are fed with attitude measurements generated by a model-based image-processing algorithm, which is applied to images produced on ground with two dedicated experimental facilities. It is shown that the attitude measurements present a non-Gaussian error distribution. Through experimental validation, the nonlinear least-squares method is shown to be the most robust for five representative tumbling states of the target satellite. The output of a statistical identification procedure provides an estimate of the motion prediction dispersion for long prediction times, which is a key input in robust tracking control methods.

TMF: A GNSS Tropospheric Mapping Function for the Asymmetrical Neutral Atmosphere

Tropospheric mapping function plays a vital role in the high precision Global Navigation Satellites Systems (GNSS) data processing for positioning. However, most mapping functions are derived under the assumption that atmospheric refractivity is spherically symmetric. In this paper, the pressure, temperature, and humidity fields of ERA5 data with the highest spatio-temporal resolution available from the European Centre for Medium-range Weather Forecast (ECMWF) were utilized to compute ray-traced delays by the software WHURT. Results reveal the universal asymmetry of the hydrostatic and wet tropospheric delays. To accurately represent these highly variable delays, a new mapping function that depends on elevation and azimuth angles—Tilting Mapping Function (TMF)—was applied. The basic idea is to assume an angle between the tropospheric zenith direction and the geometric zenith direction. Ray-traced delays served as the reference values. TMF coefficients were fitted by Levenberg–Marquardt nonlinear least-squares method. Comparisons demonstrate that the TMF can improve the MF-derived slant delay’s accuracy by 73%, 54% and 29% at the 5° elevation angle, against mapping functions based on the VMF3 concept, without, with a total and separate estimation of gradients, respectively. If all coefficients of a symmetric mapping function are determined together with gradients by a least-square fit at sufficient elevation angles, the accuracy is only 6% lower than TMF. By adopting the b and c coefficients of VMF3, TMF can keep its high accuracy with less computational cost, which could be meaningful for large-scale computing.

Prognostic Analysis of High-Speed Cylindrical Roller Bearing Using Weibull Distribution and k-Nearest Neighbor

Abstract Bearing remnant operational life can be determined by implementing a data-driven prognostics method. In this work, the bearing run-to-failure data from experimentation on test rig is used to extract time-domain features. The sudden change in time-domain information signifies the fault inception which led to failure stage promptly. The monotonicity metric is utilized to select the optimal feature set that best represents bearing degradation. Principal component analysis (PCA) is used for dimension reduction and fusion, and a unidimensional health indicator (HI) is constructed. Fluctuations of HI are smoothed by fitting it with a Weibull failure rate function (WFRF) and the corresponding parameters are estimated using nonlinear least-squares method. By inverting the model, the predicted time values are calculated, and hence remnant operational life of bearing is evaluated and compared with the actual life from experimental data. The performance assessment metrics utilized are mean absolute percentage error (MAPE), mean-square error (MSE), root-mean-square error (RMSE), and bias. Besides this, an online degradation state classification method using the k-nearest neighbor (KNN) classifier is implemented. The KNN model performance is assessed by constructing receiver operating characteristics (ROC) curve, which indicates the value of area under the curve (AUC) equal to 0.94, representing high accuracy of the KNN. The remaining useful life (RUL) is predicted within 95% confidence limits, and the predicted RUL almost follows the actual one with some fluctuations. The model performance is found promising and can be implemented to evaluate the remaining useful life of bearing.

Predicting of the Coronavirus Disease 2019 (COVID-19) Epidemic Using Estimation of Parameters in the Logistic Growth Model.

The COVID-19 pandemic was impacting the health and economy around the world. All countries have taken measures to control the spread of the epidemic. Because it is not known when the epidemic will end in several countries, then the prediction of the COVID-19 pandemic is a very important challenge. This study has predicted the temporal evolution of the COVID-19 pandemic in several countries using the logistic growth model. This model has analyzed several countries to describe the epidemic situation of these countries. The time interval of the actual data used as a comparison with the prediction results of this model was starting in the firstly confirmed COVID-19 cases to December 2020. This study examined an approach to the complexity spread of the COVID-19 pandemic using the logistic growth model formed from an ordinary differential equation. This model described the time-dependent population growth rate characterized by the three parameters of the analytical solution. The non-linear least-squares method was used to estimate the three parameters. These parameters described the rate growth constant of infected cases and the total number of confirmed cases in the final phase of the epidemic. This model is applied to the spread of the COVID-19 pandemic in several countries. The prediction results show the spread dynamics of COVID-19 infected cases which are characterized by time-dependent dynamics. In this study, the proposed model provides estimates for the model parameters that are good for predicting the COVID-19 pandemic because they correspond to actual data for all analyzed countries. It is based on the coefficient of determination, R2, and the R2 value of more than 95% which is obtained from the non-linear curves for all analyzed countries. It shows that this model has the potential to contribute to better public health policy-making in the prevention of the COVID-19 pandemic.

Measurement Time Reduction by Means of Mathematical Modeling of Enzyme Mediated RedOx Reaction in Food Samples Biosensors.

The possibility of measuring in real time the different types of analytes present in food is becoming a requirement in food industry. In this context, biosensors are presented as an alternative to traditional analytical methodologies due to their specificity, high sensitivity and ability to work in real time. It has been observed that the behavior of the analysis curves of the biosensors follow a trend that is reproducible among all the measurements and that is specific to the reaction that occurs in the electrochemical cell and the analyte being analyzed. Kinetic reaction modeling is a widely used method to model processes that occur within the sensors, and this leads to the idea that a mathematical approximation can mimic the electrochemical reaction that takes place while the analysis of the sample is ongoing. For this purpose, a novel mathematical model is proposed to approximate the enzymatic reaction within the biosensor in real time, so the output of the measurement can be estimated in advance. The proposed model is based on adjusting an exponential decay model to the response of the biosensors using a nonlinear least-square method to minimize the error. The obtained results show that our proposed approach is capable of reducing about 40% the required measurement time in the sample analysis phase, while keeping the error rate low enough to meet the accuracy standards of the food industry.

Optimal Control Applied to a Fractional-Order Foot-and-Mouth Disease Model

In this paper, we propose a nonlinear fractional-order model in order to explain and understand the outbreaks of foot-and-mouth disease. The proposed model rely on the Caputo operator. We computed the basic reproduction number and demonstrated that it is an important metric for extinction and persistence of the disease. Utilizing reported foot-and-mouth disease data for Zimbabwe and the nonlinear least-squares curve fitting method we estimated the model parameters. Meanwhile, we performed an optimal control study on the use of animal vaccination and culling of infectious animals as disease control measures against foot-and-mouth disease. Our findings showed combinations of optimal vaccination and culling rates that could lead to the effective management of the disease.