Nonlinear Regression Fuzzy Analysis of Real Variable and Fuzzy Parameter

As far as the nonlinear regression method is concerned, the condition when both independent and dependent variable take the Fuzzy value, while the parameter, θ∈ΘR m the real value, have been discussed in . But for most of actual conditions, the independent variable generally takes the real value, while both parameter and dependent variable take the Fuzzy value. This paper propounded a method for the latter and its relevant Fuzzy regreession model. In addition the Fuzzy observation, matrix distribution and the rational estimation of modeling parameter have also been discussed. Furthermore, the Max min estimation of modeling parameter and its corresponding calculating sequence have also been offered to and the calculating example shows the method is feasible.

Constitutive equations of 1060 pure aluminum based on modified double multiple nonlinear regression model

In order to study the work-ability and establish the optimum hot formation processing parameters for industrial 1060 pure aluminum, the compressive deformation behavior of pure aluminum was investigated at temperatures of 523?823 K and strain rates of 0.005?10 s?1 on a Gleeble?1500 thermo-simulation machine. The influence rule of processing parameters (strain, strain rate and temperature) on flow stress of pure aluminum was investigated. Nine analysis factors consisting of material parameters and according weights were optimized. Then, the constitutive equations of multilevel series rules, multilevel parallel rules and multilevel series &parallel rules were established. The correlation coefficients (R) are 0.992, 0.988 and 0.990, respectively, and the average absolute relative errors (AAREs) are 6.77%, 8.70% and 7.63%, respectively, which proves that the constitutive equations of multilevel series rules can predict the flow stress of pure aluminum with good correlation and precision.

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.

Geometric Properties of AR(q) Nonlinear Regression Models

This paper is devoted to a study of geometric properties of AR(q) nonlinear regression models. We present geometric frameworks for regression parameter space and autoregression parameter space respectively based on the weighted inner product by fisher information matrix. Several geometric properties related to statistical curvatures are given for the models. The results of this paper extended the work of Bates & Watts(1980,1988)[1.2] and Seber & Wild (1989)[3].

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.

Responses of River Runoff to Climate Change Based on Nonlinear Mixed Regression Model in Chaohe River Basin of Hebei Province, China

Taking the nonlinear nature of runoff system into account,and combining auto-regression method and multi-regression method,a Nonlinear Mixed Regression Model (NMR) was established to analyze the impact of temperature and precipitation changes on annual river runoff process. The model was calibrated and verified by using BP neural network with observed meteorological and runoff data from Daiying Hydrological Station in the Chaohe River of Hebei Province in 1956–2000. Compared with auto-regression model,linear multi-regression model and linear mixed regression model,NMR can improve forecasting precision remarkably. Therefore,the simulation of climate change scenarios was carried out by NMR. The results show that the nonlinear mixed regression model can simulate annual river runoff well.

Unconfined compressive strength and failure behaviour of completely weathered granite from a fault zone

Understanding the strength characteristics and deformation behaviour of the tunnel surrounding rock in a fault zone is significant for tunnel stability evaluation.In this study,a series of unconfined compression tests were conducted to investigate the mechanical characteristics and failure behaviour of completely weathered granite(CWG)from a fault zone,considering with height-diameter(h/d)ratio,dry densities(ρd)and moisture contents(ω).Based on the experimental results,a regression mathematical model of unconfined compressive strength(UCS)for CWG was developed using the Multiple Nonlinear Regression method(MNLR).The research results indicated that the UCS of the specimen with a h/d ratio of 0.6 decreased with the increase ofω.When the h/d ratio increased to 1.0,the UCS increasedωwith up to 10.5%and then decreased.Increasingρd is conducive to the improvement of the UCS at anyω.The deformation and rupture process as well as final failure modes of the specimen are controlled by h/d ratio,ρd andω,and the h/d ratio is the dominant factor affecting the final failure mode,followed byωandρd.The specimens with different h/d ratio exhibited completely different fracture mode,i.e.,typical splitting failure(h/d=0.6)and shear failure(h/d=1.0).By comparing the experimental results,this regression model for predicting UCS is accurate and reliable,and the h/d ratio is the dominant factor affecting the UCS of CWG,followed byρd and thenω.These findings provide important references for maintenance of the tunnel crossing other fault fractured zones,especially at low confining pressure or unconfined condition.

Diagnostics in generalized nonlinear models based on maximum L_q-likelihood estimation

In order to detect whether the data conforms to the given model, it is necessary to diagnose the data in the statistical way. The diagnostic problem in generalized nonlinear models based on the maximum Lq-likelihood estimation is considered. Three diagnostic statistics are used to detect whether the outliers exist in the data set. Simulation results show that when the sample size is small, the values of diagnostic statistics based on the maximum Lq-likelihood estimation are greater than the values based on the maximum likelihood estimation. As the sample size increases, the difference between the values of the diagnostic statistics based on two estimation methods diminishes gradually. It means that the outliers can be distinguished easier through the maximum Lq-likelihood method than those through the maximum likelihood estimation method.

Height-diameter models for King Boris fir(Abies borisii regis Mattf.) and Scots pine(Pinus sylvestris L.) in Olympus and Pieria Mountains, Greece

In forest science and practice, the total tree height is one of the basic morphometric attributes at the tree level and it has been closely linked with important stand attributes. In the current research, sixteen nonlinear functions for height prediction were tested in terms of their fitting ability against samples of Abies borisii regis and Pinus sylvestris trees from mountainous forests in central Greece. The fitting procedure was based on generalized nonlinear weighted regression. At the final stage, a five-quantile nonlinear height-diameter model was developed for both species through a quantile regression approach, to estimate the entire conditional distribution of tree height, enabling the evaluation of the diameter impact at various quantiles and providing a comprehensive understanding of the proposed relationship across the distribution. The results clearly showed that employing the diameter as the sole independent variable, the 3-parameter Hossfeld function and the 2-parameter N?slund function managed to explain approximately 84.0% and 81.7% of the total height variance in the case of King Boris fir and Scots pine species, respectively. Furthermore, the models exhibited low levels of error in both cases(2.310m for the fir and 3.004m for the pine), yielding unbiased predictions for both fir(-0.002m) and pine(-0.004m). Notably, all the required assumptions for homogeneity and normality of the associated residuals were achieved through the weighting procedure, while the quantile regression approach provided additional insights into the height-diameter allometry of the specific species. The proposed models can turn into valuable tools for operational forest management planning, particularly for wood production and conservation of mountainous forest ecosystems.

An Adaptive Design for Six Sigma(ADFSS): a Simulated Annealing and Regression Analysis Embedded Approach

Design for six sigma （DFSS） is a powerful approach of designing products, processes, and services with the objective of meeting the needs of customers in a cost-effective maimer. DFSS activities are classified into four major phases viz. identify, design, optimize, and validate （IDOV）. And an adaptive design for six sigma （ADFSS） incorporating the traits of artifidai intelligence and statistical techniques is presented. In the identify phase of the ADFSS, fuzzy relation measures between customer attributes （CAs） and engineering characteristics （ECs） as well as fuzzy correlation measures among ECs are determined with the aid of two fuzzy logic controllers （FLCs）. These two measures are then used to establish the cumulative impact factor for ECs. In the next phase （ i. e. design phase）, a transfer function is developed with the aid of robust multiple nonlinear regression analysis. Furthermore, 1this transfer function is optimized with the simulated annealing （ SA ） algorithm in the optimize phase. In the validate phase, t-test is conducted for the validation of the design resulted in earlier phase. Finally, a case study of a hypothetical writing instrument is simulated to test the efficacy of the proposed ADFSS.