Gauss radial basis functions(GRBF)are frequently employed in data fitting and machine learning.Their linear independence property can theoretically guarantee the avoidance of data redundancy.In this paper,one of the m...Gauss radial basis functions(GRBF)are frequently employed in data fitting and machine learning.Their linear independence property can theoretically guarantee the avoidance of data redundancy.In this paper,one of the main contributions is proving this property using linear algebra instead of profound knowledge.This makes it easy to read and understand this fundamental fact.The proof of linear independence of a set of Gauss functions relies on the constructing method for one-dimensional space and on the deducing method for higher dimensions.Additionally,under the condition of preserving the same moments between the original function and interpolating function,both the interpolating existence and uniqueness are proven for GRBF in one-dimensional space.The final work demonstrates the application of the GRBF method to locate lunar olivine.By combining preprocessed data using GRBF with the removing envelope curve method,a program is created to find the position of lunar olivine based on spectrum data,and the numerical experiment shows that it is an effective scheme.展开更多
Based on the output-voltage error function, a novel time discrete modulation technique is proposed for matrix converters （MCs） and time-discrete difference equations of a MC circuit are derived. Switch states of MC ...Based on the output-voltage error function, a novel time discrete modulation technique is proposed for matrix converters （MCs） and time-discrete difference equations of a MC circuit are derived. Switch states of MC are obtained when the output voltage error function is minimized, thus the optimum combination of switch states is derived for the closed-loop control of MC. Meanwhile, advantages of the least calculation workload, the simple process, and the convenient for implementation are brought while switch states are described as space vectors in the α-β coordination system. Simulation and experimental results demonstrate the validity of the time-discrete modulation technique and the feasibility of the control approach.展开更多
Forecasting crop yields based on remote sensing data is one of the most important tasks in agriculture.Soybean is the main crop in the Russian Far East.It is desirable to forecast soybean yield as early as possible wh...Forecasting crop yields based on remote sensing data is one of the most important tasks in agriculture.Soybean is the main crop in the Russian Far East.It is desirable to forecast soybean yield as early as possible while maintaining high accuracy.This study aimed to investigate seasonal time series of the normalized difference vegetation index(NDVI) to achieve early forecasting of soybean yield.This research used data from the Moderate Resolution Image Spectroradiometer(MODIS),an arable-land mask obtained from the VEGA-Science web service,and soybean yield data for 2008-2017 for the Jewish Autonomous Region(JAR) districts.Four approximating functions were fitted to model the NDVI time series:Gaussian,double logistic(DL),and quadratic and cubic polynomials.In the period from calendar weeks 22-42(end of May to mid-October),averaged over two districts,the model using the DL function showed the highest accuracy(mean absolute percentage error-4.0%,root mean square error(RMSE)-0.029,P <0.01).The yield forecast accuracy of prediction in the period of weeks 25-30 in JAR municipalities using the parameters of the Gaussian function was higher(P <0.05) than that using the other functions.The mean forecast error for the Gaussian function was 14.9% in week 25(RMSE was0.21 t ha) and 5.1%-12.9% in weeks 26-30(RMSE varied from 0.06 to 0.15 t ha) according to the2013-2017 data.In weeks 31-32,the error was 5.0%-5.4%(RMSE was 0.07 t ha) using the Gaussian parameters and 7.4%-7.7%(RMSE was 0.09-0.11 t ha) for the DL function.When the method was applied to municipal districts of other soy-producing regions of the Russian Far East.RMSE was0.14-0.32 t hain weeks 25-26 and did not exceed 0.20 t hain subsequent weeks.展开更多
Traditionally, it is widely accepted that measurement error usually obeys the normal distribution. However, in this paper a new idea is proposed that the error in digitized data which is a major derived data source in...Traditionally, it is widely accepted that measurement error usually obeys the normal distribution. However, in this paper a new idea is proposed that the error in digitized data which is a major derived data source in GIS does not obey the normal distribution but the p-norm distribution with a determinate parameter. Assuming that the error is random and has the same statistical properties, the probability density function of the normal distribution, Laplace distribution and p-norm distribution are derived based on the arithmetic mean axiom, median axiom and p-median axiom, which means that the normal distribution is only one of these distributions but not the least one. Based on this ideal distribution fitness tests such as Skewness and Kurtosis coefficient test, Pearson chi-square chi(2) test and Kolmogorov test for digitized data are conducted. The results show that the error in map digitization obeys the p-norm distribution whose parameter is close to 1.60. A least p-norm estimation and the least square estimation of digitized data are further analyzed, showing that the least p-norm adjustment is better than the least square adjustment for digitized data processing in GIS.展开更多
In order to</span></span><span><span><span style="font-family:""><span style="font-family:Verdana;"> reduce the influence of nonlinear </span><span...In order to</span></span><span><span><span style="font-family:""><span style="font-family:Verdana;"> reduce the influence of nonlinear </span><span style="font-family:Verdana;">characteristic</span><span style="font-family:Verdana;"> and temperature on the measuring accuracy of </span><span style="font-family:Verdana;">inclinometer</span><span style="font-family:Verdana;">, the application of </span><span style="font-family:Verdana;">polynomial</span><span style="font-family:Verdana;"> fitting principle to compensate </span><span style="font-family:Verdana;">the</span><span style="font-family:Verdana;"> measuring error of </span><span style="font-family:Verdana;">inclinometer</span><span style="font-family:Verdana;"> is studied. According to the analysis of the experimental data of inclinometer, a polynomial model of the nonlinear error and the measured value is constructed, and then the relation between the coefficient of the polynomial model and the temperature is obtained by fitting, and </span><span style="font-family:Verdana;">finally</span><span style="font-family:Verdana;"> the function of the measurement error of inclinometer on the measured inclination and temperature is obtained. The results show that this method is feasible and effective, which can not only reduce the influence of </span><span style="font-family:Verdana;">temperature,</span><span style="font-family:Verdana;"> but also correct its nonlinear error.展开更多
Mapping function errors are usually not taken into consideration, when space geodetic data observed by VLBI, GNSS and some other techniques are utilized to estimate troposphere delay, which could, however, probably br...Mapping function errors are usually not taken into consideration, when space geodetic data observed by VLBI, GNSS and some other techniques are utilized to estimate troposphere delay, which could, however, probably bring non-ignorable errors to solutions. After analyzing the variation of mapping function errors with elevation angles based on several-year meteorological data, this paper constructed a model of this error and then proposed a two-step estimation method of troposphere delay with consideration of mapping function errors. The experimental results indicate that the method put forward by this paper could reduce the slant path delay residuals efficiently and improve the estimation accuracy of wet tropospheric delay to some extent.展开更多
Solving large radial basis function （RBF） interpolation problem with non-customized methods is computationally expensive and the matrices that occur are typically badly conditioned. In order to avoid these difficult...Solving large radial basis function （RBF） interpolation problem with non-customized methods is computationally expensive and the matrices that occur are typically badly conditioned. In order to avoid these difficulties, we present a fitting based on radial basis functions satisfying side conditions by least squares, although compared with interpolation the method loses some accuracy, it reduces the computational cost largely. Since the fitting accuracy and the non-singularity of coefficient matrix in normal equation are relevant to the uniformity of chosen centers of the fitted RBE we present a choice method of uniform centers. Numerical results confirm the fitting efficiency.展开更多
Based on the existing continuous borehole strain observation,the multiquadric function fitting method was used to deal with time series data. The impact of difference kernel function parameters was discussed to obtain...Based on the existing continuous borehole strain observation,the multiquadric function fitting method was used to deal with time series data. The impact of difference kernel function parameters was discussed to obtain a valuable fitting result,from which the physical connotation of the original data and its possible applications were analyzed.Meanwhile,a brief comparison was made between the results of multiquadric function fitting and polynomial fitting.展开更多
We consider some classes of generalized gap functions for two kinds of generalized variational inequality problems. We obtain error bounds for the underlying variational inequalities using the generalized gap function...We consider some classes of generalized gap functions for two kinds of generalized variational inequality problems. We obtain error bounds for the underlying variational inequalities using the generalized gap functions under the condition that the involved mapping F is g-strongly monotone with respect to the solution, but not necessarily continuous differentiable, even not locally Lipschitz.展开更多
In this paper, we consider the global error bound for the generalized complementarity problem (GCP) with analytic functions. Based on the new technique, we establish computable global error bound under milder conditio...In this paper, we consider the global error bound for the generalized complementarity problem (GCP) with analytic functions. Based on the new technique, we establish computable global error bound under milder conditions, which refines the previously known results.展开更多
Based on the strain formulation of the quasi-conforming finite element, displacement functions are constructed which have definite physical meaning, and a conclusion can be obtained that the coefficients of the consta...Based on the strain formulation of the quasi-conforming finite element, displacement functions are constructed which have definite physical meaning, and a conclusion can be obtained that the coefficients of the constant and the linear strain are uniquely determined, and the quasi-conforming finite element method is convergent to constant strain. There are different methods for constructing the rigid displacement items, and different methods correspond to different order node errors, and this is different from ordinary displacement method finite element.展开更多
Residual based on a posteriori error estimates for conforming finite element solutions of incompressible Navier-Stokes equations with stream function form which were computed with seven recently proposed two-level met...Residual based on a posteriori error estimates for conforming finite element solutions of incompressible Navier-Stokes equations with stream function form which were computed with seven recently proposed two-level method were derived. The posteriori error estimates contained additional terms in comparison to the error estimates for the solution obtained by the standard finite element method. The importance of these additional terms in the error estimates was investigated by studying their asymptotic behavior. For optimal scaled meshes, these bounds are not of higher order than of convergence of discrete solution.展开更多
In refractive surgery, the cubic spline fit for the transition zone breaks down for myopia and myopic meridians in mixed astigmatism as in many cases the cubic spline function runs into negative values. In this paper,...In refractive surgery, the cubic spline fit for the transition zone breaks down for myopia and myopic meridians in mixed astigmatism as in many cases the cubic spline function runs into negative values. In this paper, the complementary error function is proposed instead of the cubic spline function as the transition zone function, due to the availability of analytical expression of its derivatives and the nonnegativity fact. It is shown that with the use of the complementary error function, transition zones for all refractive types work correctly.展开更多
Motivated by the effort to understand the mathematical structure underlying the Teukolsky equations in a Kerr metric background, a homogeneous integral equation related to the prolate spheroidal function is studied. F...Motivated by the effort to understand the mathematical structure underlying the Teukolsky equations in a Kerr metric background, a homogeneous integral equation related to the prolate spheroidal function is studied. From the consideration of the Fredholm determinant of the integral equation, a family of generalized error function is defined, with which the Fredholm determinant of the sinc kernel is also evaluated. An analytic solution of a special ease of the fifth Painlev~ transcendent is then worked out explicitly.展开更多
One of the classical approaches in the analysis of a variational inequality problem is to transform it into an equivalent optimization problem via the notion of gap function. The gap functions are useful tools in deri...One of the classical approaches in the analysis of a variational inequality problem is to transform it into an equivalent optimization problem via the notion of gap function. The gap functions are useful tools in deriving the error bounds which provide an estimated distance between a specific point and the exact solution of variational inequality problem. In this paper, we follow a similar approach for set-valued vector quasi variational inequality problems and define the gap functions based on scalarization scheme as well as the one with no scalar parameter. The error bounds results are obtained under fixed point symmetric and locally α-Holder assumptions on the set-valued map describing the domain of solution space of a set-valued vector quasi variational inequality problem.展开更多
The error-sum function of alternating Sylvester series is introduced. Some elementary properties of this function are studied. Also, the hausdorff dimension of the graph of such function is determined.
In this paper, the estimation of the parameters in partial functional linear models with ARCH(p) errors is discussed. With employing the functional principle component, a hybrid estimating method is suggested. The asy...In this paper, the estimation of the parameters in partial functional linear models with ARCH(p) errors is discussed. With employing the functional principle component, a hybrid estimating method is suggested. The asymptotic normality of the proposed estimators for both the linear parameter in the mean model and the parameter in the ARCH error model is obtained, and the convergence rate of the slope function estimate is established. Besides, some simulations and a real data analysis are conducted for illustration, and it is shown that the proposed method performs well with a finite sample.展开更多
A generalized form of the error function, Gp(x)=pΓ(1/p)∫0xe−tpdt, which is directly associated with the gamma function, is evaluated for arbitrary real values of p>1and 0x≤+∞by employing a fast-converging power...A generalized form of the error function, Gp(x)=pΓ(1/p)∫0xe−tpdt, which is directly associated with the gamma function, is evaluated for arbitrary real values of p>1and 0x≤+∞by employing a fast-converging power series expansion developed in resolving the so-called Grandi’s paradox. Comparisons with accurate tabulated values for well-known cases such as the error function are presented using the expansions truncated at various orders.展开更多
In this paper,the kernel of the cubic spline interpolation is given.An optimal error bound for the cu- bic spline interpolation of lower smooth functions is obtained.
This paper focuses on resolving the identification problem of a neuro-fuzzy model(NFM) applied in batch processes. A hybrid learning algorithm is introduced to identify the proposed NFM with the idea of auxiliary erro...This paper focuses on resolving the identification problem of a neuro-fuzzy model(NFM) applied in batch processes. A hybrid learning algorithm is introduced to identify the proposed NFM with the idea of auxiliary error model and the identification principle based on the probability density function(PDF). The main contribution is that the NFM parameter updating approach is transformed into the shape control for the PDF of modeling error. More specifically, a virtual adaptive control system is constructed with the aid of the auxiliary error model and then the PDF shape control idea is used to tune NFM parameters so that the PDF of modeling error is controlled to follow a targeted PDF, which is in Gaussian or uniform distribution. Examples are used to validate the applicability of the proposed method and comparisons are made with the minimum mean square error based approaches.展开更多
基金Supported by the National Basic Research Program of China(2012CB025904)Zhengzhou Shengda University of Economics,Business and Management(SD-YB2025085)。
文摘Gauss radial basis functions(GRBF)are frequently employed in data fitting and machine learning.Their linear independence property can theoretically guarantee the avoidance of data redundancy.In this paper,one of the main contributions is proving this property using linear algebra instead of profound knowledge.This makes it easy to read and understand this fundamental fact.The proof of linear independence of a set of Gauss functions relies on the constructing method for one-dimensional space and on the deducing method for higher dimensions.Additionally,under the condition of preserving the same moments between the original function and interpolating function,both the interpolating existence and uniqueness are proven for GRBF in one-dimensional space.The final work demonstrates the application of the GRBF method to locate lunar olivine.By combining preprocessed data using GRBF with the removing envelope curve method,a program is created to find the position of lunar olivine based on spectrum data,and the numerical experiment shows that it is an effective scheme.
文摘Based on the output-voltage error function, a novel time discrete modulation technique is proposed for matrix converters （MCs） and time-discrete difference equations of a MC circuit are derived. Switch states of MC are obtained when the output voltage error function is minimized, thus the optimum combination of switch states is derived for the closed-loop control of MC. Meanwhile, advantages of the least calculation workload, the simple process, and the convenient for implementation are brought while switch states are described as space vectors in the α-β coordination system. Simulation and experimental results demonstrate the validity of the time-discrete modulation technique and the feasibility of the control approach.
文摘Forecasting crop yields based on remote sensing data is one of the most important tasks in agriculture.Soybean is the main crop in the Russian Far East.It is desirable to forecast soybean yield as early as possible while maintaining high accuracy.This study aimed to investigate seasonal time series of the normalized difference vegetation index(NDVI) to achieve early forecasting of soybean yield.This research used data from the Moderate Resolution Image Spectroradiometer(MODIS),an arable-land mask obtained from the VEGA-Science web service,and soybean yield data for 2008-2017 for the Jewish Autonomous Region(JAR) districts.Four approximating functions were fitted to model the NDVI time series:Gaussian,double logistic(DL),and quadratic and cubic polynomials.In the period from calendar weeks 22-42(end of May to mid-October),averaged over two districts,the model using the DL function showed the highest accuracy(mean absolute percentage error-4.0%,root mean square error(RMSE)-0.029,P <0.01).The yield forecast accuracy of prediction in the period of weeks 25-30 in JAR municipalities using the parameters of the Gaussian function was higher(P <0.05) than that using the other functions.The mean forecast error for the Gaussian function was 14.9% in week 25(RMSE was0.21 t ha) and 5.1%-12.9% in weeks 26-30(RMSE varied from 0.06 to 0.15 t ha) according to the2013-2017 data.In weeks 31-32,the error was 5.0%-5.4%(RMSE was 0.07 t ha) using the Gaussian parameters and 7.4%-7.7%(RMSE was 0.09-0.11 t ha) for the DL function.When the method was applied to municipal districts of other soy-producing regions of the Russian Far East.RMSE was0.14-0.32 t hain weeks 25-26 and did not exceed 0.20 t hain subsequent weeks.
文摘Traditionally, it is widely accepted that measurement error usually obeys the normal distribution. However, in this paper a new idea is proposed that the error in digitized data which is a major derived data source in GIS does not obey the normal distribution but the p-norm distribution with a determinate parameter. Assuming that the error is random and has the same statistical properties, the probability density function of the normal distribution, Laplace distribution and p-norm distribution are derived based on the arithmetic mean axiom, median axiom and p-median axiom, which means that the normal distribution is only one of these distributions but not the least one. Based on this ideal distribution fitness tests such as Skewness and Kurtosis coefficient test, Pearson chi-square chi(2) test and Kolmogorov test for digitized data are conducted. The results show that the error in map digitization obeys the p-norm distribution whose parameter is close to 1.60. A least p-norm estimation and the least square estimation of digitized data are further analyzed, showing that the least p-norm adjustment is better than the least square adjustment for digitized data processing in GIS.
文摘In order to</span></span><span><span><span style="font-family:""><span style="font-family:Verdana;"> reduce the influence of nonlinear </span><span style="font-family:Verdana;">characteristic</span><span style="font-family:Verdana;"> and temperature on the measuring accuracy of </span><span style="font-family:Verdana;">inclinometer</span><span style="font-family:Verdana;">, the application of </span><span style="font-family:Verdana;">polynomial</span><span style="font-family:Verdana;"> fitting principle to compensate </span><span style="font-family:Verdana;">the</span><span style="font-family:Verdana;"> measuring error of </span><span style="font-family:Verdana;">inclinometer</span><span style="font-family:Verdana;"> is studied. According to the analysis of the experimental data of inclinometer, a polynomial model of the nonlinear error and the measured value is constructed, and then the relation between the coefficient of the polynomial model and the temperature is obtained by fitting, and </span><span style="font-family:Verdana;">finally</span><span style="font-family:Verdana;"> the function of the measurement error of inclinometer on the measured inclination and temperature is obtained. The results show that this method is feasible and effective, which can not only reduce the influence of </span><span style="font-family:Verdana;">temperature,</span><span style="font-family:Verdana;"> but also correct its nonlinear error.
基金National Natural Science Foundation of China(No.41674082)National Natural Science Foundation of China(No.41774018)。
文摘Mapping function errors are usually not taken into consideration, when space geodetic data observed by VLBI, GNSS and some other techniques are utilized to estimate troposphere delay, which could, however, probably bring non-ignorable errors to solutions. After analyzing the variation of mapping function errors with elevation angles based on several-year meteorological data, this paper constructed a model of this error and then proposed a two-step estimation method of troposphere delay with consideration of mapping function errors. The experimental results indicate that the method put forward by this paper could reduce the slant path delay residuals efficiently and improve the estimation accuracy of wet tropospheric delay to some extent.
基金Supported by National Natural Science Youth Foundation (10401021).
文摘Solving large radial basis function （RBF） interpolation problem with non-customized methods is computationally expensive and the matrices that occur are typically badly conditioned. In order to avoid these difficulties, we present a fitting based on radial basis functions satisfying side conditions by least squares, although compared with interpolation the method loses some accuracy, it reduces the computational cost largely. Since the fitting accuracy and the non-singularity of coefficient matrix in normal equation are relevant to the uniformity of chosen centers of the fitted RBE we present a choice method of uniform centers. Numerical results confirm the fitting efficiency.
基金sponsored by the Annual Earthquake Tracking Task,CEA(2017010214)
文摘Based on the existing continuous borehole strain observation,the multiquadric function fitting method was used to deal with time series data. The impact of difference kernel function parameters was discussed to obtain a valuable fitting result,from which the physical connotation of the original data and its possible applications were analyzed.Meanwhile,a brief comparison was made between the results of multiquadric function fitting and polynomial fitting.
基金supported by the National Natural Science Foundation of China (No. 10671050)the Natural Science Foundation of Heilongjiang Province of China (No. A200607)
文摘We consider some classes of generalized gap functions for two kinds of generalized variational inequality problems. We obtain error bounds for the underlying variational inequalities using the generalized gap functions under the condition that the involved mapping F is g-strongly monotone with respect to the solution, but not necessarily continuous differentiable, even not locally Lipschitz.
基金supported by National Natural Science Foundation of China (Nos. 11171180 and 11101303)Specialized Research Fund for the Doctoral Program of Chinese Higher Education (No. 20113705110002)Shandong Provincial Natural Science Foundation (Nos. ZR2010AL005 and ZR2011FL017)
文摘In this paper, we consider the global error bound for the generalized complementarity problem (GCP) with analytic functions. Based on the new technique, we establish computable global error bound under milder conditions, which refines the previously known results.
文摘Based on the strain formulation of the quasi-conforming finite element, displacement functions are constructed which have definite physical meaning, and a conclusion can be obtained that the coefficients of the constant and the linear strain are uniquely determined, and the quasi-conforming finite element method is convergent to constant strain. There are different methods for constructing the rigid displacement items, and different methods correspond to different order node errors, and this is different from ordinary displacement method finite element.
文摘Residual based on a posteriori error estimates for conforming finite element solutions of incompressible Navier-Stokes equations with stream function form which were computed with seven recently proposed two-level method were derived. The posteriori error estimates contained additional terms in comparison to the error estimates for the solution obtained by the standard finite element method. The importance of these additional terms in the error estimates was investigated by studying their asymptotic behavior. For optimal scaled meshes, these bounds are not of higher order than of convergence of discrete solution.
文摘In refractive surgery, the cubic spline fit for the transition zone breaks down for myopia and myopic meridians in mixed astigmatism as in many cases the cubic spline function runs into negative values. In this paper, the complementary error function is proposed instead of the cubic spline function as the transition zone function, due to the availability of analytical expression of its derivatives and the nonnegativity fact. It is shown that with the use of the complementary error function, transition zones for all refractive types work correctly.
基金Supported by the National Natural Science Foundation of China under Grant Nos 11171329,11203003 and 11373013
文摘Motivated by the effort to understand the mathematical structure underlying the Teukolsky equations in a Kerr metric background, a homogeneous integral equation related to the prolate spheroidal function is studied. From the consideration of the Fredholm determinant of the integral equation, a family of generalized error function is defined, with which the Fredholm determinant of the sinc kernel is also evaluated. An analytic solution of a special ease of the fifth Painlev~ transcendent is then worked out explicitly.
文摘One of the classical approaches in the analysis of a variational inequality problem is to transform it into an equivalent optimization problem via the notion of gap function. The gap functions are useful tools in deriving the error bounds which provide an estimated distance between a specific point and the exact solution of variational inequality problem. In this paper, we follow a similar approach for set-valued vector quasi variational inequality problems and define the gap functions based on scalarization scheme as well as the one with no scalar parameter. The error bounds results are obtained under fixed point symmetric and locally α-Holder assumptions on the set-valued map describing the domain of solution space of a set-valued vector quasi variational inequality problem.
文摘The error-sum function of alternating Sylvester series is introduced. Some elementary properties of this function are studied. Also, the hausdorff dimension of the graph of such function is determined.
文摘In this paper, the estimation of the parameters in partial functional linear models with ARCH(p) errors is discussed. With employing the functional principle component, a hybrid estimating method is suggested. The asymptotic normality of the proposed estimators for both the linear parameter in the mean model and the parameter in the ARCH error model is obtained, and the convergence rate of the slope function estimate is established. Besides, some simulations and a real data analysis are conducted for illustration, and it is shown that the proposed method performs well with a finite sample.
文摘A generalized form of the error function, Gp(x)=pΓ(1/p)∫0xe−tpdt, which is directly associated with the gamma function, is evaluated for arbitrary real values of p>1and 0x≤+∞by employing a fast-converging power series expansion developed in resolving the so-called Grandi’s paradox. Comparisons with accurate tabulated values for well-known cases such as the error function are presented using the expansions truncated at various orders.
文摘In this paper,the kernel of the cubic spline interpolation is given.An optimal error bound for the cu- bic spline interpolation of lower smooth functions is obtained.
基金Supported by the National Natural Science Foundation of China(61374044)Shanghai Science Technology Commission(12510709400)+1 种基金Shanghai Municipal Education Commission(14ZZ088)Shanghai Talent Development Plan
文摘This paper focuses on resolving the identification problem of a neuro-fuzzy model(NFM) applied in batch processes. A hybrid learning algorithm is introduced to identify the proposed NFM with the idea of auxiliary error model and the identification principle based on the probability density function(PDF). The main contribution is that the NFM parameter updating approach is transformed into the shape control for the PDF of modeling error. More specifically, a virtual adaptive control system is constructed with the aid of the auxiliary error model and then the PDF shape control idea is used to tune NFM parameters so that the PDF of modeling error is controlled to follow a targeted PDF, which is in Gaussian or uniform distribution. Examples are used to validate the applicability of the proposed method and comparisons are made with the minimum mean square error based approaches.
