scholarly journals General Fitting Methods Based on Lq Norms and their Optimization

Stats ◽  
2020 ◽  
Vol 3 (1) ◽  
pp. 16-31 ◽  
Author(s):  
George Livadiotis
Keyword(s):  
Metric Spaces ◽  
Least Square Method ◽  
Least Square ◽  
Method Of Least Squares ◽  
Optimal Method ◽  
Least Absolute Deviations ◽  
Fitting Method ◽  
The One ◽  
The Given ◽  
Fitting Parameters

The widely used fitting method of least squares is neither unique nor does it provide the most accurate results. Other fitting methods exist which differ on the metric norm can be used for expressing the total deviations between the given data and the fitted statistical model. The least square method is based on the Euclidean norm L2, while the alternative least absolute deviations method is based on the Taxicab norm, L1. In general, there is an infinite number of fitting methods based on metric spaces induced by Lq norms. The most accurate, and thus optimal method, is the one with the (i) highest sensitivity, given by the curvature at the minimum of total deviations, (ii) the smallest errors of the fitting parameters, (iii) best goodness of fitting. The first two cases concern fitting methods where the given curve functions or datasets do not have any errors, while the third case deals with fitting methods where the given data are assigned with errors.

scholarly journals The Parameter Estimation of Double Exponential Pulse Based on a Least Infinity Norm Fitting Method

2021 ◽  
Vol 2083 (4) ◽  
pp. 042002
Author(s):  
Yuewu Shi ◽  
Wei Wang ◽  
Zhizhen Zhu ◽  
Xin Nie
Keyword(s):  
Estimation Error ◽  
Estimation Method ◽  
Maximum Amplitude ◽  
Least Square Method ◽  
Least Square ◽  
Physical Parameters ◽  
Infinity Norm ◽  
Fitting Method ◽  
Double Exponential ◽  
Exponential Pulse

Abstract This paper presents an estimation method of double exponential pulse (DEP) between the physical parameters rise time (t r), full width at half maximum amplitude (t FWHM) and the mathematical parameters α, β. A newly fitting method based on the least infinity norm criterion is proposed to deal with the estimation problem of DEP. The calculation process and equation of parameters of this method is proposed based on an m-th-order polynomial fitting model. This estimation method is compared with the least square method by the same data and fitting function. The results show that the maximum estimation error of parameters of double exponential pulse obtained by the least infinity norm method is 1.5 %.

Parameter sensitivity analysis and optimum model of the magnetorheological damper’s Bouc–Wen model

2020 ◽  
pp. 107754632095929
Author(s):  
Min Jiang ◽  
Xiaoting Rui ◽  
Wei Zhu ◽  
Fufeng Yang ◽  
Hongtao Zhu ◽  
...  
Keyword(s):  
Parameter Identification ◽  
Fitness Function ◽  
Least Square Method ◽  
Least Square ◽  
Identification Process ◽  
Long Time ◽  
Method Optimization ◽  
Optimum Model ◽  
The One ◽  
Sensitive Parameters

To overcome the shortcomings of the Bouc–Wen model, such as too many parameters, complex identification process, and long time consuming, the sensitivity of parameters was analyzed. A Bouc–Wen optimum model with sensitive parameters to guarantee calculating accuracy was established. First, according to the results of the magnetorheological damper’s mechanical property test, the sensitivity of Bouc–Wen model’s parameters was analyzed by the one-at-a-time method. Optimization of the Bouc–Wen model was completed. Second, the parameters of the Bouc–Wen optimum model were identified under three harmonic excitations. Compared with the original Bouc–Wen model, the differences of calculation accuracy were 0.0055, 0.0007, and 0.0070 respectively. And the convergence rate of the fitness function for parameter identification increased by 67.89%, 49.94%, and 67.24%, respectively. And the iteration time of 1000 iterations was shortened by 36.52%, 25.95%, and 64.11%, respectively. It indicates that the Bouc–Wen optimum model had higher efficiency and certain accuracy in parameter identification process. Then, the calculation accuracy of Bouc–Wen optimum model with independent and coupled mean parameters were analyzed respectively. Finally, the parameters of the Bouc–Wen optimum model and current were fitted by the least square method. The results showed that the Bouc–Wen optimum model can accurately and efficiently simulate the dynamic characteristics of magnetorheological dampers.

Research on Zero Bias Characteristic of FBW Gyro

2013 ◽  
Vol 562-565 ◽  
pp. 182-187
Author(s):  
Lun Chao Zhong ◽  
Jian Zhong Wang ◽  
Ying Xian Duo
Keyword(s):  
Random Error ◽  
Least Square Method ◽  
Temperature Characteristic ◽  
Least Square ◽  
Measuring Error ◽  
Zero Bias ◽  
Temperature Bias ◽  
Bias Stability ◽  
Gyro Bias ◽  
The Given

Zero bias is an important performance index of FBW gyroscope. Based on the given gyroscope experiments,we studied zero bias of random error properties,temperature characteristic, large overload environment variation, and then calculate FBW gyro bias, bias stability, zero bias repeatability.We use Kalman filter into FBW gyroscope for random error compensation.Through least-square method,we fit temperature scale factor and temperature bias of FBW gyro.The experimental results show that the compensation method has the advantages of simple operation,effectively compensation for measuring error of FBW gyroscope which is induced by temperature, and has strong engineering using value.

scholarly journals Calculation and realization of new method grey residual error correction model

PLoS ONE ◽  
2021 ◽  
Vol 16 (7) ◽  
pp. e0254154
Author(s):  
Lifang Xiao ◽  
Xiangyang Chen ◽  
Hao Wang
Keyword(s):  
Prediction Model ◽  
Least Square Method ◽  
Residual Error ◽  
Least Square ◽  
Grey Model ◽  
Grey Prediction ◽  
Grey Prediction Model ◽  
True Value ◽  
Prediction Function ◽  
Fitting Method

Aiming at the problem of prediction accuracy of stochastic volatility series, this paper proposes a method to optimize the grey model(GM(1,1)) from the perspective of residual error. In this study, a new fitting method is firstly used, which combines the wavelet function basis and the least square method to fit the residual data of the true value and the predicted value of the grey model(GM(1,1)). The residual prediction function is constructed by using the fitting method. Then, the prediction function of the grey model(GM(1,1)) is modified by the residual prediction function. Finally, an example of the wavelet residual-corrected grey prediction model (WGM) is obtained. The test results show that the fitting accuracy of the wavelet residual-corrected grey prediction model has irreplaceable advantages.

scholarly journals Optimal attitude determination method in presence of noise and bias on different star sensors

2014 ◽  
Vol 3 (2) ◽  
pp. 155 ◽  
Author(s):  
Amir Moghtadaei Rad
Keyword(s):  
Attitude Determination ◽  
Least Square Method ◽  
Least Square ◽  
Optimal Method ◽  
Input Noise ◽  
Advantages And Disadvantages ◽  
Celestial Navigation ◽  
Stochastic Characteristic ◽  
Determination Methods ◽  
N Vector

There are different attitude determination methods which have been used in satellite and spacecraft by star tracker. Each of these methods have its own advantages and disadvantages depending on their application, stochastic characteristic of noise on sensors (bias or noise), and weight of noise falling on different sensors. The present study has thus explored the major methods from two perspectives: the effect of input noise or bias on each star sensor and the corresponding weight of each noise or bias falling on each sensor. These aspects are compared in each method and the optimal method according to each condition is introduced. N Vector, Triad, Quest, Q method and least square method are the methods studied and simulated in this article. Finally, a comparison is made between the methods and the optimal method is introduced theoretically and practically. Keywords: Attitude Determination, Celestial Navigation, Triad, Quest, Least Square, Satellite.

Reliability and Usefulness of the Fitting of a Modified Gamma Function to Time-Activity Curves by Means of a Non-Linear Iterative Least-Square Method

1981 ◽  
Vol 20 (04) ◽  
pp. 195-197
Author(s):  
J. M. Fránquiz
Keyword(s):  
Gamma Function ◽  
Least Square Method ◽  
Least Square ◽  
Base Line ◽  
Time Activity ◽  
Appearance Time ◽  
Least Square Fitting ◽  
Fitting Method ◽  
Non Linear ◽  
Least Square Fitting Method

A non-linear iterative least-square fitting method is presented for calculating the parameters of a modified gamma function. The method permits the correction of the appearance time (AT) and the curve parameters in those situations in which AT cannot be estimated with accuracy. The reliability and accuracy of the method is studied in experimental and simulated curves by means of a computer, comparing the results with those obtained by the method of Starmer and Clark for different initial selections of AT and noise at the base line. The usefulness of the method is shown in situations where the curves are distorted in their initial part.

scholarly journals Geometric Interpretation of Errors in Multi-Parametrical Fitting Methods Based on Non-Euclidean Norms

Stats ◽  
2019 ◽  
Vol 2 (4) ◽  
pp. 426-438 ◽  
Author(s):  
Livadiotis
Keyword(s):  
Optimal Parameter ◽  
Least Square Method ◽  
Geometric Interpretation ◽  
Least Square ◽  
Geometric Representation ◽  
Geometric Meaning ◽  
Normal Equations ◽  
Optimal Values ◽  
Parameter Values ◽  
Fitting Parameters

The paper completes the multi-parametrical fitting methods, which are based on metrics induced by the non-Euclidean Lq-norms, by deriving the errors of the optimal parameter values. This was achieved using the geometric representation of the residuals sum expanded near its minimum, and the geometric interpretation of the errors. Typical fitting methods are mostly developed based on Euclidean norms, leading to the traditional least–square method. On the other hand, the theory of general fitting methods based on non-Euclidean norms is still under development; the normal equations provide implicitly the optimal values of the fitting parameters, while this paper completes the puzzle by improving understanding the derivations and geometric meaning of the optimal errors.

scholarly journals The simple method of dynamic visco-elastic analysis of road structure. on rheological foundation

2009 ◽  
Vol 4 (1) ◽  
pp. 073-083
Author(s):  
Sławomir Karaś ◽  
Magdalena Sawecka
Keyword(s):  
Dynamic Analysis ◽  
Elastic Constants ◽  
Deformation Process ◽  
Least Square Method ◽  
Least Square ◽  
Elastic Analysis ◽  
Simple Method ◽  
One Dimensional ◽  
Road Pavement ◽  
The One

In contrast to computationally advanced methods of road pavement dynamic analysis, the one-dimensional, simple method is derived on the basis of visco-elastic simple beam lying on generalized Winkler visco-elastic foundation. By virtue of least square method the visco-elastic constants could be estimated with technically admissible accuracy. The introduced method is useful enough to predict any pavement deformation process in the range of linear visco-elasticity.

Genetically Optimized Multi-Layer Fuzzy Polynomial Neural Networks: Analysis and Design

2006 ◽  
Vol 10 (1) ◽  
pp. 35-49 ◽  
Author(s):  
Sung-Kwun Oh ◽  
◽  
Witold Pedrycz ◽  
Ho-Sung Park ◽  
◽  
...  
Keyword(s):  
Neural Networks ◽  
Design Procedure ◽  
Least Square Method ◽  
Least Square ◽  
Genetic Optimization ◽  
Analysis And Design ◽  
Networks Analysis ◽  
The One ◽  
Neurofuzzy Networks ◽  
Selection Of

In this study, we introduce a new category of neurofuzzy networks – Fuzzy Polynomial Neural Networks and develop a comprehensive design methodology involving mechanisms of genetic optimization, and genetic algorithms, in particular. The augmented genetically optimized FPNN (referred to as gFPNN) is a structurally optimized architecture which comes with a higher level of flexibility in comparison to the one we have encountered in the conventional FPNN. The GA-based design procedure being applied to each layer of FPNN leads to the selection of preferred nodes (or FPNs) available within the FPNN. In the sequel, two general optimization mechanisms are explored. First, the structural optimization is realized via GAs whereas for the ensuing detailed parametric optimization is carried out in the setting of a standard least square method-based learning. The performance of the gFPNN is quantified through experimentation where we use a number of modeling benchmarks – synthetic and experimental data already experimented with in fuzzy or neurofuzzy modeling.

Functional Availability Modelling of CNC Machine Tools

2014 ◽  
Vol 915-916 ◽  
pp. 395-399 ◽  
Author(s):  
Xiao Bing Li ◽  
Jun Gao ◽  
Zheng Zhang ◽  
Xiao Cui Zhu
Keyword(s):  
Least Square Method ◽  
Smoothing Method ◽  
Least Square ◽  
Cnc Machine Tools ◽  
Linear Regression Method ◽  
Cnc Machine ◽  
Basis System ◽  
Functional Linear Regression ◽  
Availability Model ◽  
Fitting Parameters

A new method for calculating the instantaneous availability was proposed based on Functional Data Analysis method. It introduced the Quadratic Bernstein Polynomial into the smoothing method firstly for the reliability which estimated by median rank method and estimated the fitting parameters by least square method. Then, under the assumption that the maintenance difficult of the CNCs was decreasing over the work time, chosen the appropriate smoothing basis function based on the trend after the time section adjustment for the estimated maintainability value. The Fourier basis system and the non-linear least squares were selected for the maintainability function smoothing method and the fitting parameters. Finally, the instantaneous availability model of CNCs was built based on the functional linear regression method, and a case example of 15 CNCs was given.

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