A Modified Algorithm for the Least Squares Identification

1983 ◽  
Vol 105 (1) ◽  
pp. 50-52
Author(s):  
C. Batur
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Mechanical Systems ◽  
Model Structure ◽  
Input Output ◽  
Unknown Parameters ◽  
Estimation Errors ◽  
Usual Practice ◽  
Unknown Disturbances ◽  
Modified Algorithm

To identify the dynamics of mechanical systems, the usual practice is to assume a certain model structure and try to estimate the unknown parameters of this model on the basis of input output observations. For mechanical systems operating under noisy industrial conditions, the number of unknowns of the problem exceeds the number of equations available. It is then inevitable that certain assumptions must be made on the unknown disturbances. This paper assumes that the only reliable feature of the disturbance is its independence of input. This yields a set of assumptions in excess of the minimal requirements and an endeavor has been made to exploit this excess to minimize the parameter estimation errors. Th resulting algorithm is similar to that of the Two Stage Least Squares method [1].

scholarly journals GM(1,1;λ) with Constrained Linear Least Squares

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 278
Author(s):  
Ming-Feng Yeh ◽  
Ming-Hung Chang
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Model Fitting ◽  
Ordinary Least Squares ◽  
Linear Least Squares ◽  
Unknown Parameters ◽  
Constrained Problems ◽  
Linear Least Squares Problems ◽  
Ordinary Least Squares Method ◽  
Better Than

The only parameters of the original GM(1,1) that are generally estimated by the ordinary least squares method are the development coefficient a and the grey input b. However, the weight of the background value, denoted as λ, cannot be obtained simultaneously by such a method. This study, therefore, proposes two simple transformation formulations such that the unknown parameters, and can be simultaneously estimated by the least squares method. Therefore, such a grey model is termed the GM(1,1;λ). On the other hand, because the permission zone of the development coefficient is bounded, the parameter estimation of the GM(1,1) could be regarded as a bound-constrained least squares problem. Since constrained linear least squares problems generally can be solved by an iterative approach, this study applies the Matlab function lsqlin to solve such constrained problems. Numerical results show that the proposed GM(1,1;λ) performs better than the GM(1,1) in terms of its model fitting accuracy and its forecasting precision.

scholarly journals Generalized Nonlinear Least Squares Method for the Calibration of Complex Computer Code Using a Gaussian Process Surrogate

Entropy ◽  
2020 ◽  
Vol 22 (9) ◽  
pp. 985
Author(s):  
Youngsaeng Lee ◽  
Jeong-Soo Park
Keyword(s):  
Gaussian Process ◽  
Least Squares ◽  
Covariance Matrix ◽  
Process Model ◽  
Least Squares Method ◽  
Nonlinear Least Squares ◽  
Computer Code ◽  
Test Functions ◽  
Unknown Parameters ◽  
Complex Computer

The approximated nonlinear least squares (ALS) method has been used for the estimation of unknown parameters in the complex computer code which is very time-consuming to execute. The ALS calibrates or tunes the computer code by minimizing the squared difference between real observations and computer output using a surrogate such as a Gaussian process model. When the differences (residuals) are correlated or heteroscedastic, the ALS may result in a distorted code tuning with a large variance of estimation. Another potential drawback of the ALS is that it does not take into account the uncertainty in the approximation of the computer model by a surrogate. To address these problems, we propose a generalized ALS (GALS) by constructing the covariance matrix of residuals. The inverse of the covariance matrix is multiplied to the residuals, and it is minimized with respect to the tuning parameters. In addition, we consider an iterative version for the GALS, which is called as the max-minG algorithm. In this algorithm, the parameters are re-estimated and updated by the maximum likelihood estimation and the GALS, by using both computer and experimental data repeatedly until convergence. Moreover, the iteratively re-weighted ALS method (IRWALS) was considered for a comparison purpose. Five test functions in different conditions are examined for a comparative analysis of the four methods. Based on the test function study, we find that both the bias and variance of estimates obtained from the proposed methods (the GALS and the max-minG) are smaller than those from the ALS and the IRWALS methods. Especially, the max-minG works better than others including the GALS for the relatively complex test functions. Lastly, an application to a nuclear fusion simulator is illustrated and it is shown that the abnormal pattern of residuals in the ALS can be resolved by the proposed methods.

scholarly journals An Optimal Estimation Method for Nonlinear Models of Mechanical Systems

1994 ◽  
Vol 116 (4) ◽  
pp. 805-810 ◽  
Author(s):  
M. J. G. van de Molengraft ◽  
F. E. Veldpaus ◽  
J. J. Kok
Keyword(s):  
Nonlinear Models ◽  
A Priori ◽  
Mechanical Systems ◽  
Estimation Method ◽  
Optimal Estimation ◽  
Model Structure ◽  
Unknown Parameters ◽  
Starting Point ◽  
Nonlinear Mechanical ◽  
Nonlinear Mechanical Systems

This paper presents an optimal estimation method for nonlinear mechanical systems. The a priori knowledge of the system in the form of a nonlinear model structure is taken as a starting point. The method determines estimates of the parameters and estimates of the positions, velocities, accelerations, and inputs of the system. The optimal estimation method is applied to an experimental mechanical system. The unknown parameters in this system relate to inertia, friction and elastic deformation. It is shown that the optimal estimation method on the basis of a relatively simple model structure can lead to a useful description of the system.

Incorporation of Neural Network State Estimator for GPS Attitude Determination

2004 ◽  
Vol 57 (1) ◽  
pp. 117-134 ◽  
Author(s):  
Dah-Jing Jwo ◽  
Chun-Fan Pai
Keyword(s):  
Neural Network ◽  
Kalman Filter ◽  
Least Squares ◽  
Kalman Filtering ◽  
Attitude Determination ◽  
Least Squares Method ◽  
State Estimator ◽  
Estimation Errors ◽  
Network State ◽  
The Neural Network

The Global Positioning System (GPS) can be employed as a free attitude determination interferometer when carrier phase measurements are utilized. Conventional approaches for the baseline vectors are essentially based on the least-squares or Kalman filtering methods. The raw attitude solutions are inherently noisy if the solutions of baseline vectors are obtained based on the least-squares method. The Kalman filter attempts to minimize the error variance of the estimation errors and will provide the optimal result while it is required that the complete a priori knowledge of both the process noise and measurement noise covariance matrices are available. In this article, a neural network state estimator, which replaces the Kalman filter, will be incorporated into the attitude determination mechanism for estimating the attitude angles from the noisy raw attitude solutions. Employing the neural network estimator improves robustness compared to the Kalman filtering method when uncertainty in noise statistical knowledge exists. Simulation is conducted and a comparative evaluation based on the neural network estimator and Kalman filter is provided.

Recursive least squares identification for piecewise affine Hammerstein models

2018 ◽  
Vol 11 (2) ◽  
pp. 234-253
Author(s):  
Wang Jian Hong ◽  
Daobo Wang
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Nonlinear Function ◽  
Recursive Least Squares ◽  
Bounded Noise ◽  
Unknown Parameters ◽  
Content Type ◽  
Piecewise Affine ◽  
Probabilistic Description ◽  
Recursive Least Squares Method

Purpose The purpose of this paper is to probe the recursive identification of piecewise affine Hammerstein models directly by using input-output data. To explain the identification process of a parametric piecewise affine nonlinear function, the authors prove that the inverse function corresponding to the given piecewise affine nonlinear function is also an equivalent piecewise affine form. Based on this equivalent property, during the detailed identification process with respect to piecewise affine function and linear dynamical system, three recursive least squares methods are proposed to identify those unknown parameters under the probabilistic description or bounded property of noise. Design/methodology/approach First, the basic recursive least squares method is used to identify those unknown parameters under the probabilistic description of noise. Second, multi-innovation recursive least squares method is proposed to improve the efficiency lacked in basic recursive least squares method. Third, to relax the strict probabilistic description on noise, the authors provide a projection algorithm with a dead zone in the presence of bounded noise and analyze its two properties. Findings Based on complex mathematical derivation, the inverse function of a given piecewise affine nonlinear function is also an equivalent piecewise affine form. As the least squares method is suited under one condition that the considered noise may be a zero mean random signal, a projection algorithm with a dead zone in the presence of bounded noise can enhance the robustness in the parameter update equation. Originality/value To the best knowledge of the authors, this is the first attempt at identifying piecewise affine Hammerstein models, which combine a piecewise affine function and a linear dynamical system. In the presence of bounded noise, the modified recursive least squares methods are efficient in identifying two kinds of unknown parameters, so that the common set membership method can be replaced by the proposed methods.

Value Prediction of Design Parameters Based on Weighting Least Squares

2014 ◽  
Vol 909 ◽  
pp. 379-385 ◽  
Author(s):  
Sheng Li ◽  
Hong Sheng Jia
Keyword(s):  
Least Squares ◽  
Design Parameter ◽  
Least Squares Method ◽  
Weighted Least Squares ◽  
Design Parameters ◽  
Test Results ◽  
Original Design ◽  
Unknown Parameters ◽  
Value Prediction ◽  
Weighted Least Squares Method

Parametric equipments or standard parts usually have many different types of original design parameters. So when designing some new specifications, it requires a lot of estimation or trial and error to determine the value trends and intervals of other unknown design parameters. Based on a finite number of historical examples of design parameter groups, the paper gives an algorithm to fit value trend line using multivariate linear weighted least squares method, whose weights are designed by using distance-proximity coefficient and correlation coefficient. The algorithm uses a small amount of new design parameters, fits value trend lines of other unknown parameters, predicts all other design parameters, finally makes up a design parameter group for a new specification. Two test results of standard parts from home and abroad show that, the accuracy of value prediction is able to meet the requirements of engineering applications.

IDENTIFICATION OF NOISY INPUT-OUTPUT SYSTEM USING BIAS-COMPENSATED LEAST-SQUARES METHOD

2005 ◽  
Vol 38 (1) ◽  
pp. 803-808 ◽  
Author(s):  
Masato Ikenoue ◽  
Shunshoku Kanae ◽  
Zi-Jiang Yang ◽  
Kiyoshi Wada
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Input Output ◽  
Noisy Input ◽  
Output System

scholarly journals Georeferencing of Multi-Sheet Maps Based on Least Squares with Constraints—First Military Mapping Survey Maps in the Area of Czechia

2020 ◽  
Vol 11 (1) ◽  
pp. 299
Author(s):  
Tomáš Janata ◽  
Jiří Cajthaml
Keyword(s):  
Problem Solving ◽  
Least Squares ◽  
Statistical Method ◽  
Least Squares Method ◽  
Topographic Maps ◽  
Unknown Parameters ◽  
Habsburg Monarchy ◽  
Typical Representative ◽  
Robust Statistical Method ◽  
M Estimate

The article deals with the possibility of georeferencing old multi-sheet map works. Various approaches to problem solving and a workable method for using the least squares method with the conditions of the adjacency of map sheets are discussed. To increase reliability, the IRLS robust statistical method is used, which uses iterative weighting of individual measurements based on Huber’s M-estimate. The method is applied to the First Military Mapping of the Habsburg monarchy as a typical representative of old topographic maps, which are not easy to georeference due to unknown parameters of the used cartographic projection. A georeferenced layer of the above mentioned mapping is available on the Mapire.eu portal as well. A basic analysis of the comparison of georeferencing results using our method and the mentioned portal is performed.

The Application of Total Least Squares Method in Data Fitting of Speed Radar

2012 ◽  
Vol 203 ◽  
pp. 69-75 ◽  
Author(s):  
Cheng Chen ◽  
Chang Jin Liu
Keyword(s):  
Least Squares ◽  
Initial Velocity ◽  
High Speed ◽  
Least Squares Method ◽  
Data Fitting ◽  
Total Least Squares ◽  
Measure Data ◽  
Observation Data ◽  
Unknown Parameters ◽  
Observation Matrix

For acquiring the initial velocity of high-speed object, it needs data fitting to get the unknown parameters. Least squares method(LS) is usually uses to complete this work, but LS method takes no account of the errors in the observation matrix, not only may makes error in unknown parameters' fitting, but also do harm to the further analysis. Therefore, this paper lead total least squares method(TLS) into data fitting, it can at the same time in consideration of observation data and its error margin, and at last in actually measure data analysis to prove TLS compare to LS enjoy higher accuracy.

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