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 A Method for Reconstruction of Boiler Combustion Temperature Field Based on Acoustic Tomography

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Yuhui Wu ◽  
Xinzhi Zhou ◽  
Li Zhao ◽  
Chenlong Dong ◽  
Hailin Wang
Keyword(s):  
Temperature Field ◽  
Temperature Distribution ◽  
Optimal Parameter ◽  
Interpolation Method ◽  
Least Square Method ◽  
Reconstruction Algorithm ◽  
Least Square ◽  
Temperature Fields ◽  
Acoustic Tomography ◽  
Measured Temperature

Acoustic tomography (AT), as a noninvasive temperature measurement method, can achieve temperature field measurement in harsh environments. In order to achieve the measurement of the temperature distribution in the furnace and improve the accuracy of AT reconstruction, a temperature field reconstruction algorithm based on the radial basis function (RBF) interpolation method optimized by the evaluation function (EF-RBFI for short) is proposed. Based on a small amount of temperature data obtained by the least square method (LSM), the RBF is used for interpolation. And, the functional relationship between the parameter of RBF and the root-mean-square (RMS) error of the reconstruction results is established in this paper, which serves as the objective function for the effect evaluation, so as to determine the optimal parameter of RBF. The detailed temperature description of the entire measured temperature field is finally established. Through the reconstruction of three different types of temperature fields provided by Dongfang Boiler Works, the results and error analysis show that the EF-RBFI algorithm can describe the temperature distribution information of the measured combustion area globally and is able to reconstruct the temperature field with high precision.

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.

Research on Diagnosis Method of GIS Insulation Defects Based on Polynomial Fitting

2013 ◽  
Vol 805-806 ◽  
pp. 888-891 ◽  
Author(s):  
Yu Sheng Quan ◽  
Zong Cheng Zhang ◽  
Guang Chen ◽  
Dai Juan Wang
Keyword(s):  
Partial Discharge ◽  
Least Square Method ◽  
Development Stage ◽  
Linear Mapping ◽  
Least Square ◽  
Reference Database ◽  
Initial Stage ◽  
Diagnosis Method ◽  
Three Stages ◽  
Fitting Parameters

GIS is currently widely used in the power system and is the Developing Trends of Electrical appliances manufacturing. At present, a lot of research on GIS PD, but the study of the recognition of the PD severity is less. By analyzing a large number of partial discharge sample data, a new GIS PD severity diagnostic method is proposed in this paper. Partial discharge is divided into three stages (initial stage, development stage and dangerous stage) in the method according to the severity of the partial discharge. These three stages are fitted with non-linear mapping and get the fitting parameters. GIS partial discharge fault reference database can be dynamically generated with these fitting parameters. The similarity between the detected data and the reference data is discriminated with the least square method and Supplier division method, and then the GIS PD severity can be diagnosed. A large number of simulation results initially confirmed the effectiveness and feasibility of the proposed method.

Current Control Method in Doubly Fed Induction Generator Under Low Switching Frequency

2018 ◽  
Vol 22 (7) ◽  
pp. 1109-1113
Author(s):  
Peng Xie ◽  
Keyword(s):  
Control Method ◽  
Least Square Method ◽  
Current Control ◽  
Least Square ◽  
Induction Generator ◽  
Switching Frequency ◽  
Doubly Fed Induction Generator ◽  
On Line ◽  
Doubly Fed ◽  
Parameter Values

The present current control method in doubly fed induction generator cannot realize the segmented grid-connected current control, it’s hard to effectively control the current in doubly fed induction generator. Therefore, a current control method in doubly fed induction generator under low switching frequency is proposed in this paper. Which means to build a mathematical model of the doubly fed induction generator under low switching frequency to analyze the parameters of doubly fed induction generator filter. Then the parameter values of the filter can be obtained. The current in generator can be predicted by adopting the double-sampling predict method. And the current control in generator can be improved according to dead beat control. Then the on-line identification of current parameters by least square method is needed to finish the current control method in doubly fed induction generator under low switching frequency. The experimental results show that the proposed method realized the segmented grid-connected current control in doubly fed induction generator.

Quasi-Optimal Parameter Design of Five-Axis Tool Grinders Based on Taguchi Method

2011 ◽  
Vol 697-698 ◽  
pp. 521-524
Author(s):  
Jenn Yih Chen ◽  
Bean Yin Lee ◽  
Z.H. Huang
Keyword(s):  
Optimal Parameter ◽  
Parameter Design ◽  
Tool Geometry ◽  
Control Factors ◽  
End Mills ◽  
Machine Interface ◽  
Optimal Values ◽  
Inspection Machine ◽  
Parameter Values ◽  
Five Axis

In order to improve precision of grinding as well as accuracy of ball nose end mills, the Taguchi approach was adopted to figure out quasi-optimal parameter values of the XYZ axes drives and the controller for a five-axis tool grinder. Firstly, the backlash and pitch errors of the transmission system and rotational axes were measured via a laser interferometer, and these errors were compensated by setting compensation values on a human machine interface of the controller. Four control factors with three levels and an L9 orthogonal array were used in the experiments, and each experiment was repeated three times. Next, this parameter design was applied to obtain quasi-optimal values of the drives and the controller, and further a tool grinder was employed to grind five ball nose end mills to confirm the practicability. Finally, a tool measuring and inspection machine was utilized to measure the tool geometry of each end mill for the initial and optimal designs. Experimental results were shown to indicate the considerable improvement of the accuracy of the end mills and demonstrate the effectiveness of our proposed scheme.

scholarly journals DESIGN SPACE EXPLORATION AND OPTIMIZATION OF SUPER SCALAR PROCESSOR

2013 ◽  
pp. 29-32
Author(s):  
K. NAVATHA ◽  
PROF.G. KRISHNA MURTHY
Keyword(s):  
Architectural Design ◽  
Design Space Exploration ◽  
Design Space ◽  
Optimal Parameter ◽  
Space Exploration ◽  
Instruction Level Parallelism ◽  
Optimal Values ◽  
Target Environment ◽  
Parameter Values ◽  
Level Parallelism

Designing a microprocessor involves determining the optimal microarchitecture for a given objective function and a given set of constraints. Superscalar processing is the latest in along series of innovations aimed at producing ever-faster microprocessors. By exploiting instruction-level parallelism, superscalar processors[1] are capable of executing more than one instruction in a clock cycle.The architectural design of super scalar processor involves a lot of trade off issues when selecting parameter values for instruction level parallelism.The use of critical quantitative analysis based upon the Simple Scalar simulations is necessary to select optimal parameter values for the processor aimed at specific target environment. This paper aims at finding optimal values for the super scalar processor and determines which processor parameters have the greatest impact on the simulated execution time.

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.

Data-driven rational feedforward tuning: With application to an ultraprecision wafer stage

2019 ◽  
Vol 234 (6) ◽  
pp. 748-758 ◽  
Author(s):  
Weicai Huang ◽  
Kaiming Yang ◽  
Yu Zhu ◽  
Xin Li ◽  
Haihua Mu ◽  
...  
Keyword(s):  
High Performance ◽  
Optimal Parameter ◽  
Least Square Method ◽  
Parameter Tuning ◽  
Basis Functions ◽  
Least Square ◽  
Data Driven ◽  
Estimation Accuracy ◽  
Rational Basis ◽  
Noisy Condition

Rational basis functions are introduced into iterative learning control to enhance the flexibility towards nonrepeating tasks. At present, the application of rational basis functions either suffers from nonconvex optimization problem or requires the predefinition of poles, which restricts the achievable performance. In this article, a new data-driven rational feedforward tuning approach is developed, in which convex optimization is realized without predefining the poles. Specifically, the optimal parameter which eliminates the reference-induced error is directly solved using the least square method. No parametric model is involved in the parameter tuning process and the optimal parameter is estimated using the measured data. In the noisy condition, it is proved that the estimated optimal parameter is unbiased and the estimation accuracy in terms of variance is analysed. The performance of the proposed approach is tested on an ultraprecision wafer stage. The experimental results confirm that high performance is achieved using the proposed approach.

Direct Assessment of Effective Renal Plasma Flow from Renoscintigraphy with a Gamma Camera and an On-Line Computer

1981 ◽  
Vol 20 (06) ◽  
pp. 274-278
Author(s):  
J. Liniecki ◽  
J. Bialobrzeski ◽  
Ewa Mlodkowska ◽  
M. J. Surma
Keyword(s):  
Count Rate ◽  
Renal Plasma Flow ◽  
Early Period ◽  
Least Square Method ◽  
Least Square ◽  
Uptake Coefficient ◽  
Positive Linear Correlation ◽  
On Line ◽  
Best Fit ◽  
Plasma Activity

A concept of a kidney uptake coefficient (UC) of 131I-o-hippurate was developed by analogy from the corresponding kidney clearance of blood plasma in the early period after injection of the hippurate. The UC for each kidney was defined as the count-rate over its ROI at a time shorter than the peak in the renoscintigraphic curve divided by the integral of the count-rate curve over the "blood"-ROI. A procedure for normalization of both curves against each other was also developed. The total kidney clearance of the hippurate was determined from the function of plasma activity concentration vs. time after a single injection; the determinations were made at 5, 10, 15, 20, 30, 45, 60, 75 and 90 min after intravenous administration of 131I-o-hippurate and the best-fit curve was obtained by means of the least-square method. When the UC was related to the absolute value of the clearance a positive linear correlation was found (r = 0.922, ρ > 0.99). Using this regression equation the clearance could be estimated in reverse from the uptake coefficient calculated solely on the basis of the renoscintigraphic curves without blood sampling. The errors of the estimate are compatible with the requirement of a fast appraisal of renal function for purposes of clinical diagknosis.

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