Algorithms for descent along nodal straight lines in the problem of estimating regression equations using the least absolute deviations method

2021 ◽  
Vol 87 (5) ◽  
pp. 68-75
Author(s):  
A. N. Tyrsin
Keyword(s):  
Objective Function ◽  
Gradient Descent ◽  
Least Square Method ◽  
Least Square ◽  
Regression Equations ◽  
Exact Methods ◽  
Computational Costs ◽  
Least Absolute Deviations ◽  
Performance Reduction ◽  
Straight Lines

A problem of estimating linear regression equations by the least absolute deviations method is considered. The exact methods of implementation of the method are significantly inferior in performance to the least square method. The fastest algorithm based on coordinate descent along nodal straight lines has a computational complexity proportional to the square of the number of observations, which limits the practical application of the method to monitoring and diagnostic tasks. The goal of the study is to describe a faster version of the descent along the nodal straight lines, as well as to evaluate the performance. Reduction of the computational costs is achieved due to the fact that instead of calculating the values of the objective function at nodal points, we find the derivative of the objective function in the vicinity of these points along the nodal line. The computational efficiency of gradient descent along nodal straight lines is estimated. For a typical computer, a comparative analysis of the average calculation time for various algorithms of descent along nodal straight lines is performed. A simple example is given to illustrate the implementation of a gradient descent procedure. Along with reduction of the computational costs, we also eliminated the possibility of accumulating computational errors when determining the values of the objective function for large samples. Moreover, gradient descent is quite simple for implementation. This makes it possible to use the method of least absolute deviations as an alternative to the least square method in various practical applications.

Electrochemical Properties of GaCl3 in Ionic Liquid [C2mim][GaCl4]

2013 ◽  
Vol 634-638 ◽  
pp. 2596-2599
Author(s):  
Yi Jin
Keyword(s):  
Ionic Liquid ◽  
Reference Electrode ◽  
Reduction Process ◽  
Least Square Method ◽  
Least Square ◽  
Diffusion Activation Energy ◽  
Cyclic Voltammogram ◽  
Diffusion Controlled ◽  
Straight Lines ◽  
Diffusion Activation

Under dry argon atmosphere, the colourless and transparent ionic liquid [C2mim][GaCl4] was synthesized. Using [C2mim][GaCl4] as solvent, a solution of GaCl3with molarity 5.227×10-3 mol/cm3 was prepared at 303.15 K and there is a equilibrum between the solute and the solvent: GaCl3+ [C2mim][GaCl4] → [C2mim][Ga2Cl7]. The cyclic voltammogram of the solution had been carried out relative to Al/Al3+reference electrode, with GC working electrode and Ga counter electrode, with various scan rates from 30 to 700mV/s, in a temperature range of 303.15 to 343.15 K, respectively. The results indicated that (1) there were two processes: Ga(Ⅲ)→Ga(I)→Ga(0), which corresponded to the double reduction peaks in the cyclic voltammogram; (2) the first reduction process was a diffusion controlled and quasi-reversible process. Plotting current density of the peak vs the square root of scan rate, a series of straight lines were obtained, then the diffusion coefficients were obtained according to Randles-Sevcik equation in various temperature. The values of diffusion coefficient were fitted to Arrhenius equation with least-square method and value of diffusion activation energy, ED = 41.5 kJ/mol, was obtained from Arrhenius slope.

Optimization of Vehicle Handing and Stability Based on RSM

2012 ◽  
Vol 591-593 ◽  
pp. 733-736
Author(s):  
Dong Shan Sun ◽  
Fang Wang ◽  
Yan Pin He
Keyword(s):  
Objective Function ◽  
Simulation Analysis ◽  
Target Function ◽  
Least Square Method ◽  
Linear Interpolation ◽  
Simulation Software ◽  
Least Square ◽  
Lateral Acceleration ◽  
Slip Angle ◽  
Vehicle Model

Based on the theory of multi-body system dynamics and simulation software ADAMS/Car, the whole vehicle model was established. Afterwards by means of pylon course slalom, the accuracy of modeling was tested. The analysis and evaluation were later made to draw a conclusion that the handling stability of model needed improved. So as to improve the handling stability, the index lateral acceleration, yaw rate, side slip angle were taken as target function while suspension stiffness parameters were designed variables. Optimization of vehicle handing stability was practiced by applying second-order Response Surface Methodology (RSM) model. The relationship could be obtained by least square method obviously. Moreover applying with linear interpolation the final objective function was decided. The minimum of final objective function was the optimal result. Simulation analysis was performed again for the whole vehicle model by parameters modification. Subsequently, the results showed that this method greatly improved handling stability.

scholarly journals Time Scale in Least Square Method

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Özgür Yeniay ◽  
Öznur İşçi ◽  
Atilla Göktaş ◽  
M. Niyazi Çankaya
Keyword(s):  
Least Squares ◽  
Time Scale ◽  
Least Squares Method ◽  
Least Square Method ◽  
Forward Direction ◽  
Ordinary Least Squares ◽  
Least Square ◽  
Regression Equations ◽  
Nabla Derivative ◽  
Derivative Concept

Study of dynamic equations in time scale is a new area in mathematics. Time scale tries to build a bridge between real numbers and integers. Two derivatives in time scale have been introduced and called as delta and nabla derivative. Delta derivative concept is defined as forward direction, and nabla derivative concept is defined as backward direction. Within the scope of this study, we consider the method of obtaining parameters of regression equation of integer values through time scale. Therefore, we implemented least squares method according to derivative definition of time scale and obtained coefficients related to the model. Here, there exist two coefficients originating from forward and backward jump operators relevant to the same model, which are different from each other. Occurrence of such a situation is equal to total number of values of vertical deviation between regression equations and observation values of forward and backward jump operators divided by two. We also estimated coefficients for the model using ordinary least squares method. As a result, we made an introduction to least squares method on time scale. We think that time scale theory would be a new vision in least square especially when assumptions of linear regression are violated.

scholarly journals A Comparison of the Accuracy of Semi-empirical PM3, PDDG and PM6 methods in Predicting Heats of Formation for Organic Compounds

2017 ◽  
Vol 58 (2) ◽  
Author(s):  
Yang-Yang Wu ◽  
Feng-Qi Zhao ◽  
Xue-Hai Ju
Keyword(s):  
Organic Compounds ◽  
Least Square Method ◽  
Least Square ◽  
Heats Of Formation ◽  
Regression Equations ◽  
Chemical Calculation ◽  
Average Deviation ◽  
Experimental Values ◽  
Isomerization Energy ◽  
Semi Empirical

<p>Gas phase heats of formation (HOF) of 18 kinds of 390 organic compounds were calculated by quantum chemical calculation using semi-empirical PM3, PDDG and PM6 methods. The calculated HOFs were compared with the experimental data to illustrate the accuracy for different kinds of organics. Furthermore, the calculated values were linearly fitted with experimental values using the least square method, and were afterward substituted into the fitted regression equations to obtain the calibrated ones. The results show that, for 10 kinds of the selected organics, PM6 is more accurate, and PDDG is more accurate for 7 kinds of organics, while PM3 is only good for amino acid. As a whole, PM6 predicts the HOFs more accurately, with its weighted total mean average deviation (WTMAD) being 0.4 kJ/mol and 2.4 kJ/mol smaller than those of PM3 and PDDG, respectively. On the other hand, our results show that PDDG is the best to differentiate the isomers, with its mean average deviation (MAD) for isomerization energy being 7.8 kJ/mol and 11.0 kJ/mol smaller than PM6 and PM3, respectively. After the calibration, the values of MADs from the PM3, PDDG and PM6 results for most organics are reduced by 0.1 to 18.2 kJ/mol, with exceptions of the PM3 for amines, PDDG for carboxylic acids, and PM6 for ethers.</p>

Study on paste pumping characteristics based on double pumps in parallel

2015 ◽  
Vol 12 (3) ◽  
pp. 255-260 ◽  
Author(s):  
Guimei Wang ◽  
Shaohua Jin ◽  
Buqin Zhang ◽  
Fugui Yang ◽  
Xiaoli Ji
Keyword(s):  
Objective Function ◽  
Optimal Solution ◽  
Least Square Method ◽  
Least Square ◽  
Pumping System ◽  
Filling Station ◽  
Unit Energy ◽  
Parallel Forms ◽  
Coal Reserves ◽  
Flow Power

The coal reserves overlain by buildings, water bodies and railways are estimated to be about 14 billion tons in China. Based on the concept of green mining, in order to save the unit energy, Double pumps in parallel forms of pumping are adopted in 0the paste filling of the pumping system. The optimal solution of the objective function could be obtained by taking the unit flow power of the pumping system as the objective function, using the quadratic polynomial by the least square method curve fitting and by the application of MATLAB. Thereby it is possible to reduce power consumption and increase the efficiency of the operating economy in the paste filling station.

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.

Method of Automobile-Sensitive Component of Safety Belt's Dimensional Inspection Based on Machine Vision

2013 ◽  
Vol 442 ◽  
pp. 397-404
Author(s):  
Bao Jun Yu ◽  
Ming Yang ◽  
Guo Qing Zhao
Keyword(s):  
Machine Vision ◽  
Edge Detection ◽  
Ccd Camera ◽  
Least Square Method ◽  
Morphological Processing ◽  
Least Square ◽  
Dimensional Inspection ◽  
Edge Points ◽  
Canny Algorithm ◽  
Straight Lines

The purpose of this paper is to measure the open size of automobile-sensitive component of safety belt by the method of machine vision. At first ,use the CCD camera to get images of the component. Then the software (NI Labview) can help to process those images easily. The needed edge points of the component in the picture can be found after image has been processed by filtering, gray-scale morphological processing, binaryzation and edge detection which has been implemented by Canny algorithm. Through those points we can get the result of the beeline equation by least square method, and finally detected the angle between the two straight lines which stand for different straight edges of the component .And the detected angle values is the open size we need .

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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