ON CONFIDENCE REGIONS OF SEMIPARAMETRIC NONLINEAR REGRESSION MODELS(A GEOMETRIC APPROACH)

A geometric framework is proposed for semiparametric nonlinear regression models based on the concept of least favorable curve, introduced by Severini and Wong (1992). The authors use this framework to drive three kinds of improved approximate confidence regions for the parameter and parameter subset in terms of curvatures. The results obtained by Hamilton et al. (1982), Hamilton (1986) and Wei (1994) are extended to semiparametric nonlinear regression models.

CONSERVATIVE ESTIMATING FUNCTION IN THE NONLINEAR REGRESSION MODEL WITH AGGREGATED DATA

The purpose of this paper is to study the theory of conservative estimating functions in nonlinear regression model with aggregated data. In this model, a quasi-score function with aggregated data is defined. When this function happens to be conservative, it is projection of the true score function onto a class of estimation functions. By constructing, the potential function for the projected score with aggregated data is obtained, which have some properties of log-likelihood function.

CBPS-Based Inference in Nonlinear Regression Models with Missing Data

In this article, to improve the doubly robust estimator, the nonlinear regression models with missing responses are studied. Based on the covariate balancing propensity score (CBPS), estimators for the regression coefficients and the population mean are obtained. It is proved that the proposed estimators are asymptotically normal. In simulation studies, the proposed estimators show improved performance relative to usual augmented inverse probability weighted estimators.

TESTING OF CORRELATION AND HETEROSCEDASTICITY IN NONLINEAR REGRESSION MODELS WITH DBL(p,q,1) RANDOM ERRORS

Chaos theory has taught us that a system which has both nonlinearity and random input will most likely produce irregular data. If random errors are irregular data, then random error process will raise nonlinearity （Kantz and Schreiber （1997））. Tsai （1986） introduced a composite test for autocorrelation and heteroscedasticity in linear models with AR（1） errors. Liu （2003） introduced a composite test for correlation and heteroscedasticity in nonlinear models with DBL（p, 0, 1） errors. Therefore, the important problems in regression model axe detections of bilinearity, correlation and heteroscedasticity. In this article, the authors discuss more general case of nonlinear models with DBL（p, q, 1） random errors by score test. Several statistics for the test of bilinearity, correlation, and heteroscedasticity are obtained, and expressed in simple matrix formulas. The results of regression models with linear errors are extended to those with bilinear errors. The simulation study is carried out to investigate the powers of the test statistics. All results of this article extend and develop results of Tsai （1986）, Wei, et al （1995）, and Liu, et al （2003）.

CONFIDENCE REGIONS IN TERMS OF STATISTICAL CURVATURE FOR AR(q) NONLINEAR REGRESSION MODELS

This paper constructs a set of confidence regions of parameters in terms of statistical curvatures for AR(q) nonlinear regression models. The geometric frameworks are proposed for the model. Then several confidence regions for parameters and parameter subsets in terms of statistical curvatures are given based on the likelihood ratio statistics and score statistics. Several previous results, such as [1] and [2] are extended to AR(q) nonlinear regression models.

ON A LARGE SAMPLE PROBLEM IN NONLINEAR REGRESSION

Assume that in the nonlinear regression model, independent variable sequence {xi, i ≥ 1} is a known constant-vector sequence. This article proposes a condition on {xi}, which can be tested and verified easily. The condition is essential for proving the consistency and asymptotic normality of the estimator.

Attenuation Parameters of Blasting Vibration by Fuzzy Nonlinear Regression Analysis

In order to reduce the influence of outliers on the parameter estimate of the attenuation formula for the blasting vibration velocity,a fuzzy nonlinear regression method of Sadov’s vibration formula was proposed on the basis of the fuzziness of blasting engineering,and the algorithm was described in details as well.In accordance with an engineering case,the vibration attenuation formula was regressed by the fuzzy nonlinear regression method and the nonlinear least square method,respectively.The calculation results showed that the fuzzy nonlinear regression method is more suitable to the field test data.It differs from the nonlinear least square method because the weight of residual square in the objective function can be adjusted according to the membership of each data.And the deviation calculation of least square estimate of parameters in the nonlinear regression model verified the rationality of using the membership to assign the weight of residual square.The fuzzy nonlinear regression method provides a calculation basis for estimating Sadov’s vibration formula’s parameters more accurately.

Local Curvature and Centering Effects in Nonlinear Regression Models

The effects of centering response and explanatory variables as a way of simplifying fitted linear models in the presence of correlation are reviewed and extended to include nonlinear models, common in many biological and economic applications. In a nonlinear model, the use of a local approximation can modify the effect of centering. Even in the presence of uncorrelated explanatory variables, centering may affect linear approximations and related test statistics. An approach to assessing this effect in relation to intrinsic curvature is developed and applied. Mis-specification bias of linear versus nonlinear models also reflects this centering effect.

Distance-covariance-based tests for heteroscedasticity in nonlinear regressions

In this paper,we propose a new numerical scheme for the coupled Stokes-Darcy model with the Beavers-Joseph-Saffman interface condition.We use the weak Galerkin method to discretize the Stokes equation and the mixed finite element method to discretize the Darcy equation.A discrete inf-sup condition is proved and the optimal error estimates are also derived.Numerical experiments validate the theoretical analysis.

Interaction behavior and load sharing pattern of piled raft using nonlinear regression and LM algorithm-based artificial neural network

In the recent era,piled raft foundation(PRF)has been considered an emergent technology for offshore and onshore structures.In previous studies,there is a lack of illustration regarding the load sharing and interaction behavior which are considered the main intents in the present study.Finite element(FE)models are prepared with various design variables in a double-layer soil system,and the load sharing and interaction factors of piled rafts are estimated.The obtained results are then checked statistically with nonlinear multiple regression(NMR)and artificial neural network(ANN)modeling,and some prediction models are proposed.ANN models are prepared with Levenberg-Marquardt(LM)algorithm for load sharing and interaction factors through backpropagation technique.The factor of safety(FS)of PRF is also estimated using the proposed NMR and ANN models,which can be used for developing the design strategy of PRF.

A nonlinear regression model applied to kinetic multi-component analysis of esters with a SAW-impedance sensor

A kinetic nonlinear regression model for multi-component assay of esters was proposed based on their different alkaline-catalysed hydrolysis rate. The reaction rate was determined by monitoring the conductance change in solution with a liquid-purpose surface acoustic wave impedance sensor(SAW). The model was tested theoretically and experimentally with the mixture of methyl acetate and n-propyl acetate. The experimental detection limit of methyl acetate and n-propyl acetate (within 10 min) was 0.5 mu mol/L and 1.0 mu mol/L respectively and the recovery of the sensor system ranged from 93% to 106% (n=6).

Nonlinear regression without i.i.d.assumption

In this paper,we consider a class of nonlinear regression problems without the assumption of being independent and identically distributed.We propose a correspondent mini-max problem for nonlinear regression and give a numerical algorithm.Such an algorithm can be applied in regression and machine learning problems,and yields better results than traditional least squares and machine learning methods.

NONLINEAR WAVELET SMOOTHING OF ERROR DENSITY IN A SEMIPARAM ETRIC REGRESSION MODEL

In this paper,a sem iparam etric regression m odel in w hich errors are i.i.d random variables from an unknow n density f(·) is considered.Based on Hallet al.(1995),a nonlinear w avelet estim ation of f(·) withoutrestrictions ofcontinuity everyw here on f(·) is given,and the convergence rate ofthe estim ators in L2 is obtained.

Heteroscedasticity check in nonlinear semiparametric models based on nonparametric variance function

The assumption of homoscedasticity has received much attention in classical analysis of regression. Heteroscedasticity tests have been well studied in parametric and nonparametric regressions. The aim of this paper is to present a test of heteroscedasticity for nonlinear semiparametric regression models with nonparametric variance function. The validity of the proposed test is illustrated by two simulated examples and a real data example.

Mathematical modelling of nonlinear behaviour of seismicity

In this paper,the nonlinear behaviour of seismic activities has been studied by means of the threshold autoregressive model and the exponential autoregressive model. The contents are as follows: ① The theories and modelling methods of this two models have been studied.② One kind of explanation for the seismic cycle and order structure are given by means of the threshold autoregressive model.③ According to the exponential autoregressive model,an inherent structure of the magnitude series are discussed,the different relations between magnitude and frequency in active period and quiet period are also explained in this paper.

Characterizing prediction errors of a new tree height model for cut-to-length Pinus radiata stems through the Burr TypeⅫdistribution

Unlike height-diameter equations for standing trees commonly used in forest resources modelling,tree height models for cut-to-length(CTL)stems tend to produce prediction errors whose distributions are not conditionally normal but are rather leptokurtic and heavy-tailed.This feature was merely noticed in previous studies but never thoroughly investigated.This study characterized the prediction error distribution of a newly developed such tree height model for Pin us radiata(D.Don)through the three-parameter Burr TypeⅫ(BⅫ)distribution.The model’s prediction errors(ε)exhibited heteroskedasticity conditional mainly on the small end relative diameter of the top log and also on DBH to a minor extent.Structured serial correlations were also present in the data.A total of 14 candidate weighting functions were compared to select the best two for weightingεin order to reduce its conditional heteroskedasticity.The weighted prediction errors(εw)were shifted by a constant to the positive range supported by the BXII distribution.Then the distribution of weighted and shifted prediction errors(εw+)was characterized by the BⅫdistribution using maximum likelihood estimation through 1000 times of repeated random sampling,fitting and goodness-of-fit testing,each time by randomly taking only one observation from each tree to circumvent the potential adverse impact of serial correlation in the data on parameter estimation and inferences.The nonparametric two sample Kolmogorov-Smirnov(KS)goodness-of-fit test and its closely related Kuiper’s(KU)test showed the fitted BⅫdistributions provided a good fit to the highly leptokurtic and heavy-tailed distribution ofε.Random samples generated from the fitted BⅫdistributions ofεw+derived from using the best two weighting functions,when back-shifted and unweighted,exhibited distributions that were,in about97 and 95%of the 1000 cases respectively,not statistically different from the distribution ofε.Our results for cut-tolength P.radiata stems represented the first case of any tree species where a non-normal error distribution in tree height prediction was described by an underlying probability distribution.The fitted BXII prediction error distribution will help to unlock the full potential of the new tree height model in forest resources modelling of P.radiata plantations,particularly when uncertainty assessments,statistical inferences and error propagations are needed in research and practical applications through harvester data analytics.

Segmented taper equations with crown ratio and stand density for Dahurian Larch (Larix gmelinii) in Northeastern China

Segmented taper equation was selected to model stem profile of Dahurian larch （Larix gmelinii Rupr.）. The data were based on stem analysis of 74 trees from Dailing Forest Bureau in Heilongjiang Province, Northeastern China. Two taper equations with crown ratio and stand basal area were derived from the Max and Burkhart’s （1976） taper equation. Three taper equations were evaluated： （1） the original equation, （2） the original equation with crown ratio, and （3） the original equation with basal area. SAS NLIN and SYSNLIN procedures were used to fit taper equations. Fit statistics and cross-validation were used to evaluate the accuracy and precision of these models. Parameter estimates showed that the original equation with inclusion of crown ratio and basal area variables provided significantly different parameter estimates with lower standard errors. Overall fit statistics indicated that the root mean square error （RMSE） for diameter outside and inside bark decreased respectively by 10% and 7% in the original model with crown ratio and by 12% and 7.2% in the original model with basal area. Cross-validation further confirmed that the original equation with inclusion of crown ratio and basal area variables provided more accurate predictions at the lower section （relative heights, 10%） and upper section （relative heights, 50%） for both outside and inside bark diameters.

Stem taper functions for Betula platyphylla in the Daxing’an Mountains,northeast China

Accurate prediction of stem diameter is an important prerequisite of forest management.In this study,an appropriate stem taper function was developed for upper stem diameter estimation of white birch(Betula platyphylla Sukaczev)in ten sub-regions of the Daxing’an Mountains,northeast China.Three commonly used taper functions were assessed using a diameter and height dataset comprising 1344 trees.A first-order continuous-time error structure accounted for the inherent autocorrelation.The segmented model of Max and Burkhart(For Sci 22:283–289,1976.https://doi.org/10.1093/fores tscie nce/22.3.283)and the variable exponent taper function of Kozak(For Chron 80:507–515,2004.https://doi.org/10.5558/tfc80507-4)described the data accurately.Owing to its lower multicollinearity,the Max and Burkhart(1976)model is recommended for diameter estimation at specific heights along the stem for the ten sub-regions.After comparison,the Max and Burkhart(1976)model was refitted using nonlinear mixed-effects techniques.Mixed-effects models would be used only when additional upper stem diameter measurements are available for calibration.Differences in region-specific taper functions were indicated by the method of the non-linear extra sum of squares.Therefore,the particular taper function should be adjusted accordingly for each sub-region in the Daxing’an Mountains.

ON THE MONTE CARLO SIMULATION OF NORMAL GRAIN GROWTH

The microstructures and their kinetics of normal grain growth are simulated using different Monte Carlo （MC） algorithms. Compared with the relative figures and the theoretical normal grain growth exponents of n =0.5, the effects of some factors of MC algorithm, i.e. the lattice types, the methods of selecting lattice sites, and the neighbors selection for energy calculations, on the simulation results of grain growth are studied. Two methods of regression were compared, and the three-parameter nonlinear regression is much more suitable for fitting the grain growth kinetics. A better model with appropriate factors included triangular lattice, the attempted site randomly selected, and the first and second nearest neighbors for energy calculations is obtained.

Investigation of the compact torus plasma motion in the KTX-CTI device based on circuit analyses

Compact torus(CT)injection is one of the most promising methods for the central fuelling of next-generation reactor-grade fusion devices due to its high density,high velocity,and selfcontained magnetised structure.A newly compact torus injector(CTI)device in Keda Torus e Xperiment(KTX),named KTX-CTI,was successfully developed and tested at the University of Science and Technology in China.In this study,first,we briefly introduce the basic principles and structure of KTX-CTI,and then,present an accurate circuit model that relies on nonlinear regression analysis(NRA)for studying the current waveform of the formation region.The current waveform,displacement,and velocity of CT plasma in the acceleration region are calculated using this NRA-based one-dimensional point model.The model results were in good agreement with the experiments.The next-step upgrading reference scheme of the KTX-CTI device is preliminarily investigated using this NRA-based point model.This research can provide insights for the development of experiments and future upgrades of the device.