Nonlinear Least-squares Method Research Articles

Artificial neural network for multi-echo gradient echo-based myelin water fraction estimation.

To demonstrate robust myelin water fraction (MWF) mapping using an artificial neural network (ANN) with multi-echo gradient-echo (GRE) signal. Multi-echo gradient-echo signals simulated with a three-pool exponential model were used to generate the training data set for the ANN, which was designed to yield the MWF. We investigated the performance of our proposed ANN for various conditions using both numerical simulations and in vivo data. Simulations were conducted with various SNRs to investigate the performance of the ANN. In vivo data with high spatial resolutions were applied in the analyses, and results were compared with MWFs derived by the nonlinear least-squares algorithm using a complex three-pool exponential model. The network results for the simulations show high accuracies against noise compared with nonlinear least-squares MWFs: RMS-error value of 5.46 for the nonlinear least-squares MWF and 3.56 for the ANN MWF at an SNR of 150 (relative gain=34.80%). These effects were also found in the in vivo data, with reduced SDs in the region-of-interest analyses. These effects of the ANN demonstrate the feasibility of acquiring high-resolution myelin water images. The simulation results and in vivo data suggest that the ANN facilitates more robust MWF mapping in multi-echo gradient-echo sequences compared with the conventional nonlinear least-squares method.

Toward a Common Framework and Database of Materials for Soft Robotics.

To advance the field of soft robotics, a unified database of material constitutive models and experimental characterizations is of paramount importance. This will facilitate the use of finite element analysis to simulate their behavior and optimize the design of soft-bodied robots. Samples from seventeen elastomers, namely Body Double™ SILK, Dragon Skin™ 10 MEDIUM, Dragon Skin 20, Dragon Skin 30, Dragon Skin FX-Pro, Dragon Skin FX-Pro + Slacker, Ecoflex™ 00-10, Ecoflex 00-30, Ecoflex 00-50, Rebound™ 25, Mold Star™ 16 FAST, Mold Star 20T, SORTA-Clear™ 40, RTV615, PlatSil® Gel-10, Psycho Paint®, and SOLOPLAST 150318, were subjected to uniaxial tensile tests according to the ASTM D412 standard. Sample preparation and tensile test parameters are described in detail. The tensile test data are used to derive parameters for hyperelastic material models using nonlinear least-squares methods, which are provided to the reader. This article presents the mechanical characterization and the resulting material properties for a wide set of commercially available hyperelastic materials, many of which are recognized and commonly applied in the field of soft robotics, together with some that have never been characterized. The experimental raw data and the algorithms used to determine material parameters are shared on the Soft Robotics Materials Database GitHub repository to enable accessibility, as well as future contributions from the soft robotics community. The presented database is aimed at aiding soft roboticists in designing and modeling soft robots while providing a starting point for future material characterizations related to soft robotics research.

Parameter Identification and Nonparametric Calibration of the Tri-Pyramid Robot

The Tri-pyramid Robot is a 3-degree-of-freedom overconstrained parallel robot designed for the rapid flexible forming of three-dimensional thin sheets without geometry-specific dies used in conventional forming processes. In this article, a combined parametric and nonparametric calibration method for the geometric and nongeometric errors of the Tri-pyramid Robot is presented. The geometry-based inverse and forward kinematic equations are derived. With the actuator values and the relative end-effector positions measured through experiments, the real structural parameters are identified using the nonlinear least-squares method. A neural network is trained to further calibrate the position- and direction-dependent nongeometric errors, such as backlash and link deformations. Combining the end-effector position calculated from the kinematic model and the nongeometric errors predicted with the trained neural network, the end-effector positions can be predicted. The validation experiments show that the accuracy of the robot can be improved by 60% with the proposed method.

Investigation on linearisation of data‐driven transport research: two representative case studies

Transportation engineering, as a practical engineering discipline, relies heavily on the accurate calibration of importation parameters from field data. In the real world, most transport relations possess inherent non-linearity. Two prevailing methods for handling non-linear regression are the non-linear least-squares method (LSM) with an iterative solution, and linearisation for the non-linear regression function. The second method applies a linear regression method to solve the non-linear regression problem but requires a data transformation of the observations from variant coordinates, and the objective function is suspected to be changed accordingly. This work describes the authors' investigation into the problem of non-linear regression through two illustrative examples, the calibration of three non-linear (either exponential or logarithmic) single-regime models for fundamental diagram and the regression of non-linear (power) bunker-consumption model, by applying the weighted LSM (WLSM) and the ordinary LSM to calibrate. It is found that linearising the regression model leads to deviations, and the data transformation can create even more concern with the WLSM because the weights can be redistributed after the data transformation. A further investigation into the linear regression and the non-linear regression gives more suggestions on the choice of regression method.

Logistic growth modelling of COVID-19 proliferation in China and its international implications

ObjectiveAs the coronavirus disease 2019 (COVID-19) pandemic continues to proliferate globally, this paper shares the findings of modelling the outbreak in China at both provincial and national levels. This paper examines the applicability of the logistic growth model, with implications for the study of the COVID-19 pandemic and other infectious diseases. MethodsAn NLS (Non-Linear Least Squares) method was employed to estimate the parameters of a differentiated logistic growth function using new daily COVID-19 cases in multiple regions in China and in other selected countries. The estimation was based upon training data from January 20, 2020 to March 13, 2020. A restriction test was subsequently implemented to examine whether a designated parameter was identical among regions or countries, and the diagnosis of residuals was also conducted. The model's goodness of fit was checked using testing data from March 14, 2020 to April 18, 2020. ResultsThe model presented in this paper fitted time-series data exceedingly well for the whole of China, its eleven selected provinces and municipalities, and two other countries - South Korea and Iran - and provided estimates of key parameters. This study rejected the null hypothesis that the growth rates of outbreaks were the same among ten selected non-Hubei provinces in China, as well as between South Korea and Iran. The study found that the model did not provide reliable estimates for countries that were in the early stages of outbreaks. Furthermore, this study concured that the R2 values might vary and mislead when compared between different portions of the same non-linear curve. In addition, the study identified the existence of heteroskedasticity and positive serial correlation within residuals in some provinces and countries. ConclusionsThe findings suggest that there is potential for this model to contribute to better public health policy in combatting COVID-19. The model does so by providing a simple logistic framework for retrospectively analyzing outbreaks in regions that have already experienced a maximal proliferation in cases. Based upon statistical findings, this study also outlines certain challenges in modelling and their implications for the results.

The Impact of Outliers on Parameter Estimation Method for Bilinear Model

Nonlinear least squares (NLS) method along with Newton-Raphson (NR) iterative procedure is the best method to estimate parameters for bilinear model. However, the existence of outliers will affect the estimated value of the parameter and its validity can be doubtful. This statement was proven by conducting simulation analysis for the bilinear model, especially on bilinear (1,0,1,1) model without and with the existence of additive outlier (AO), innovational outlier (IO), temporary change (TC) and level change (LC) in the data. The performance of the NLS method is measured in terms of bias. Numerical results show that, in general, the NLS method performs better in estimating the parameters without the existence of AO, IO, TC or LC in the data. Keywords: bilinear model; nonlinear least squares; Newton-Raphson; additive outlier; innovational outlier; temporary change; level change

Blind back-propagation method for fiber nonlinearity compensation with low computational complexity and high performance.

In this paper, a blind back-propagation (BP) method for fiber nonlinearity compensation with low computational complexity and high performance is proposed. The BP method compensates the fiber chromatic dispersion step by step. Between two linear steps, the proposed method compensates the fiber nonlinearity with the nonlinear tap coefficients optimized by the nonlinear least square method (NLSM). Unlike the traditional BP method, the proposed method takes into account the SPM, the intra-channel XPM and the intra-channel FWM effects while it is purely blind and requires no prior information of the transmission link except the total accumulated chromatic dispersion, e.g., the BP step in the proposed algorithm can be set as an arbitrary value which has no connection to the physical span length. The computational complexity of the proposed method is much lower (less than 50%) than the conventional BP method with one step per span, because of the reduction of the total number of steps. Meanwhile, the method improves the nonlinearity compensation performance in comparison to the standard BP method with one step per span at the optimal input power while maintaining the same computational complexity.

Separable Nonlinear Least-Squares Parameter Estimation for Complex Dynamic Systems.

Nonlinear dynamic models are widely used for characterizing processes that govern complex biological pathway systems. Over the past decade, validation and further development of these models became possible due to data collected via high-throughput experiments using methods from molecular biology. While these data are very beneficial, they are typically incomplete and noisy, which renders the inference of parameter values for complex dynamic models challenging. Fortunately, many biological systems have embedded linear mathematical features, which may be exploited, thereby improving fits and leading to better convergence of optimization algorithms. In this paper, we explore options of inference for dynamic models using a novel method of separable nonlinear least-squares optimization and compare its performance to the traditional nonlinear least-squares method. The numerical results from extensive simulations suggest that the proposed approach is at least as accurate as the traditional nonlinear least-squares, but usually superior, while also enjoying a substantial reduction in computational time.

Prediction of Partial Molar Enthalpies and Mixing Enthalpies for Sn-Based Binary Alloys by the Wilson Equation

Expressions of partial molar enthalpies and enthalpies of mixing of binary systems are derived using the Wilson equation, according to basic thermodynamic principles. The partial molar enthalpies and mixing enthalpies of Sn-based binary alloys are calculated based on the Wilson equation’s interaction parameters, which were obtained by the method of nonlinear least-squares. Calculated values are statistically in agreement with experimental data reported in the literature. The results indicate that the method proposed in this work is reliable for predicting thermodynamic properties for Sn-based binary alloys.

Simulation of forest ecosystems dynamic processes message

Dynamic characteristics of changes in the diameter of larch forests in the Lower Angara region are being analyzed. Larch stands of the herbal group of fox types is the object of the present research. This research is based on the data of field inventory 858 forest areas with larch predominance. The structure of the forest average diameter is approximated by the exponential function, statistical analysis of which has been carried out using a non-linear least-squares method and simulation methods by conducting 50000 statistical tests. Regularities of error distribution, parameters and calculation accuracy of approximating function have been defined. Estimations of parameter correlations approximating function have been suggested. Holetsky’s method helped to receive two dimensional parameter generator. The defined regularities of the dynamics of average diameters allow to design economic activities in forest stands of larch herbal groups types in the region under study.

Specimen Size and Shape Effect on the Compressive Strength of Normal Strength Concrete

Present study deals with the effect of size and shape on the compressive strength of concrete. Mixes of five different strengths were designed, all between C20/25 and C50/60 strength class (normal strength concretes). Different size specimens were casted from all mixes in cube (edge length: 50 to 200 mm) and in cylinder (diameter: 60 to 150 mm) shapes as well. An equation was derived to estimate the compressive strength of the differently sized specimen. The parameters of the equation were optimized based on measurement data using nonlinear least-squares method with SSE (sum of squared errors) cost function. The parameter optimization was performed in different models (estimation based on the standard strength of cube or cylinder to approximate the different size specimens' cylinder or cube strength with the edge length, the surface area or the volume as dimension data). The test results showed good agreement between the laboratory measurements and the literature data (compressive strength is decreasing with the increase of the size of the specimen). The derived estimation models showed good correlation with the measurement data and with the literature estimation models, in some cases even with lower errors. The results indicated that 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 size of both cube and cylinder specimens was determined, above which the size effect on compressive strength can be neglected.

Instability of low-moisture carrageenans as affected by water vapor sorption at moderate storage temperatures

Isothermal microcalorimetry was used to study the exothermic heat flow caused by the absorption of water into carrageenans and a furcellaran, leading to their thermodynamic instability under moderate storage conditions (35 °C, τ = 12 h), where τ is the reaction time (h). The net heats evolved by furcellaran, ι-carrageenan, κ-carrageenan, and a mixture of κ/λ-carrageenan were 43.5 J/g, 45.9 J/g, 31.6 J/g, and 28.1 J/g, respectively. The pronounced exothermic behavior of furcellaran was attributed to the enthalpic association of water molecules with the thermally degraded carbohydrate matrix. The responses of a heat treatment of carrageenans at 55 °C, 85 °C, and 105 °C for 15 min on exothermic heat Q (J/g) at different water activities (aw = 0.26, 0.51, 0.76, and 1.0) were measured. The dependence of the net recorded heat Q on water activity can be satisfactorily approximated by the equation Q = a[1 − (1 − aw)b], where a and b are the coefficients found by the nonlinear least-squares method. It was concluded that carrageenans affected by excessive heat treatment should be preferably stored with limited water access.

Government Size, the Composition of Public Spending and Economic Growth in Netherland

The performance of the fiscal policy is influenced significantly by the relationships and associations among the governmental size, the composition public and private spending and the economic growth of a country. The primary goal of this research study is to evaluate the impact of these factors and to evaluate the significance of these factors in the economic growth of the Netherlands. The economy of Netherlands is characterized as the 17th largest in the world and it presents stable and sustainable growth. In this altercation the researcher aimed to evaluate the significance of the governmental spending and size. For this purpose, the researcher used the data from 2004-2014 from the 12 provinces of Netherlands. The data was subjected to a unit root analysis so that the stationarity properties of the panel data can be evaluated. The unit root test results showed that the variables were stationary at I(0). In order to abstain from the endogeneity issues that can be present in such types of datasets the researcher used level and per capita variables as a robustness evaluation. The empirical framework was based upon the Cobb-Douglas production function and used the modern CES substitution elasticities to compute as the inputs of private capital and government spending in the production function. The nonlinear least-squares regression estimation method was used to evaluate the impact of the variables upon one another. The results indicate that the public investments and current governmental disbursements are conjoined in order to account for the inflexibility of the communal budget. Moreover, the governmental spending was found to be greater than 85 percent indicating that the provincial sectors are focused upon the stimulation of the economy.

A two-phase substrate model for enzymatic hydrolysis of lignocellulose: application to batch and continuous reactors

BackgroundEnzymatic hydrolysis continues to have a significant projected production cost for the biological conversion of biomass to fuels and chemicals, motivating research into improved enzyme and reactor technologies in order to reduce enzyme usage and equipment costs. However, technology development is stymied by a lack of accurate and computationally accessible enzymatic-hydrolysis reaction models. Enzymatic deconstruction of cellulosic materials is an exceedingly complex physico-chemical process. Models which elucidate specific mechanisms of deconstruction are often too computationally intensive to be accessible in process or multi-physics simulations, and empirical models are often too inflexible to be effectively applied outside of their batch contexts. In this paper, we employ a phenomenological modeling approach to represent rate slowdown due to substrate structure (implemented as two substrate phases) and feedback inhibition, and apply the model to a continuous reactor system.ResultsA phenomenological model was developed in order to predict glucose and solids concentrations in batch and continuous enzymatic-hydrolysis reactors from which liquor is continuously removed by ultrafiltration. A series of batch experiments were performed, varying initial conditions (solids, enzyme, and sugar concentrations), and best-fit model parameters were determined using constrained nonlinear least-squares methods. The model achieved a good fit for overall sugar yield and insoluble solids concentration, as well as for the reduced rate of sugar production over time. Additionally, without refitting model coefficients, good quantitative agreement was observed between results from continuous enzymatic-hydrolysis experiments and model predictions. Finally, the sensitivity of the model to its parameters is explored and discussed.ConclusionsAlthough the phenomena represented by the model correspond to behaviors that emerge from clusters of mechanisms, and hence a set of model coefficients are unique to the substrate and the enzyme system, the model is efficient to solve and may be applied to novel reactor schema and implemented in computational fluid dynamics (CFD) simulations. Hence, this modeling approach finds the right balance between model complexity and computational efficiency. These capabilities have broad application to reactor design, scale-up, and process optimization.

The effect of different inorganic anions on mineralization of acrylic acid in wet air oxidation.

Wet Air Oxidation (WAO) process is being developed as a very promising technique for efficient removal of high-concentration organic pollutants. However, because of the technical constraints, many chemical wastewater contain inorganic anions (e.g., Cl-, SO42-, PO43- and NO3-). These inorganic salts can cause equipment corrosion and also affect WAO reactivity. But very limited studies of the effect of inorganic anions on WAO in chemical wastewater have been performed. Here we for the first time intensively investigate the effect of different inorganic anions on kinetics and pathways of WAO reaction, acrylic acid was selected as substrate. We used a nonlinear least-squares curve fitting method using Matlab 2014® to obtain the kinetic model. Importantly, it was demonstrated that this proposed kinetic model represented the experimental data well, and acetic acid was the only residual short-chain carboxylic acid. Moreover, the higher concentration anions the higher selectivity of WAO of CO2 by acrylic acid. And the selectivity order of WAO to acetic acid is PO43->NO3->Cl->SO42- at low concentration (100mmolL-1). Finally, the theory calculation disclosed the feasibility of the reactions between these anions and acrylic acid, calculation results revealed that atoms 1#, 7# and 8# have the strongest chargeability and are more vulnerable to oxidant attack because of their high charge density. And the Total Organic Carbon (TOC) removal was positively correlated with the electronegativity of the central atom of oxidizing acid.

Parameter estimation, sensitivity and control strategies analysis in the spread of influenza in Mexico

In this paper we address a parameter estimation, sensitivity and control strategies analyses for influenza disease using a model the flows of people between four states: susceptible, exposed, infectious, recovered. We solved a curve-fitting mathematical model to Mexican influenza data using a nonlinear least-square method and the Landweber iteration. An optimal control problem is formulated and analyzed based on models between four states: susceptible, exposed, infectious, recovered; model considering educational campaign, vaccination and medical treatment as strategies for disease control. The sensitivity analysis is performed to determine which model parameters are the most important to disease transmission and prevalence. The numeric results suggest that an adequate implementation of these strategies during the outbreak of an epidemic could significantly mitigate the propagation of the disease.

SAD phasing of XFEL data depends critically on the error model.

A nonlinear least-squares method for refining a parametric expression describing the estimated errors of reflection intensities in serial crystallographic (SX) data is presented. This approach, which is similar to that used in the rotation method of crystallographic data collection at synchrotrons, propagates error estimates from photon-counting statistics to the merged data. Here, it is demonstrated that the application of this approach to SX data provides better SAD phasing ability, enabling the autobuilding of a protein structure that had previously failed to be built. Estimating the error in the merged reflection intensities requires the understanding and propagation of all of the sources of error arising from the measurements. One type of error, which is well understood, is the counting error introduced when the detector counts X-ray photons. Thus, if other types of random errors (such as readout noise) as well as uncertainties in systematic corrections (such as from X-ray attenuation) are completely understood, they can be propagated along with the counting error, as appropriate. In practice, most software packages propagate as much error as they know how to model and then include error-adjustment terms that scale the error estimates until they explain the variance among the measurements. If this is performed carefully, then during SAD phasing likelihood-based approaches can make optimal use of these error estimates, increasing the chance of a successful structure solution. In serial crystallography, SAD phasing has remained challenging, with the few examples of de novo protein structure solution each requiring many thousands of diffraction patterns. Here, the effects of different methods of treating the error estimates are estimated and it is shown that using a parametric approach that includes terms proportional to the known experimental uncertainty, the reflection intensity and the squared reflection intensity to improve the error estimates can allow SAD phasing even from weak zinc anomalous signal.

A Spatial-Socioeconomic Urban Development Status Curve from NPP-VIIRS Nighttime Light Data

Urban development status is closely related to the urban economy, environment, ecology, and health. Spatial and socioeconomic processes are the two key aspects of urban development, so the absence of any of them will affect the assessment of urban development status. In this study, using both spatial and socioeconomic information from land cover data and nighttime light data, respectively, we proposed an exponential model, Spatial–Socioeconomic Urban Development Curve (SSUDC), to provide a quantitative expression of the relationship between the two key processes of urban development and analyze urban development status. The SSUDC was calculated from the artificial surface ratio at 1% intervals obtained from Globeland30 land cover data and the corresponding average NPP-VIIRS nighttime light radiance data, using a nonlinear least-squares method. We generated SSUDCs for 330 prefecture-level cities in Mainland China, 208 of which had coefficients of determination (R2) greater than 0.6. Taking Ordos and Guiyang as two typical examples, we analyzed the importance and advantages of SSUDC. The coefficients α and β of the exponential SSUDC were shown to indicate the base intensity socioeconomic activity and the concentration of socioeconomic activities, respectively, and can be used to reveal the urban socioeconomic development status and functional type of cities. At the internal urban level, the residuals of SSUDC can imply the demand for urban physical or economic construction in different areas of the city, and even the urban growth type, together with the distribution of the artificial surface ratio. In summary, the proposed SSUDC provides a simple way to combine the spatial and socioeconomic processes of urban development, which is beneficial to the analysis of urban development at different scales and a rewarding tool for urban planning.

Correcting the Bias in the Practitioner Black-Scholes Method

We address a number of technical problems with the popular Practitioner Black-Scholes (PBS) method for valuing options. The method amounts to a two-stage procedure in which fitted values of implied volatilities (IV) from a linear regression are plugged into the Black-Scholes formula to obtain predicted option prices. Firstly we ensure that the prediction from stage one is positive by using log-linear regression. Secondly, we correct the bias that results from the transformation applied to the fitted values (i.e., the Black-Scholes formula) being a highly non-linear function of implied volatility. We apply the smearing technique in order to correct this bias. An alternative means of implementing the PBS approach is to use the market option price as the dependent variable and estimate the parameters of the IV equation by the method of non-linear least squares (NLLS). A problem we identify with this method is one of model incoherency: the IV equation that is estimated does not correspond to the set of option prices used to estimate it. We use the Monte Carlo method to verify that (1) standard PBS gives biased option values, both in-sample and out-of-sample; (2) using standard (log-linear) PBS with smearing almost completely eliminates the bias; (3) NLLS gives biased option values, but the bias is less severe than with standard PBS. We are led to conclude that, of the range of possible approaches to implementing PBS, log-linear PBS with smearing is preferred on the basis that it is the only approach that results in valuations with negligible bias.

Analysis of vertical multijunction solar cells

ABSTRACTThe electrical parameters saturation current density; series resistance; shunt resistance; ideality factor and photocurrent density of multi-junction solar cells have been extracted from characteristics data J-V. We found that experimental results on photo-current voltage characteristics can be reliably reproduced using electrical extraction parameters using the non-linear least-squares method. We realized high voltage silicon solar modules with a voltage of up to Voc =25.491 V at 500 suns.