On: “THE USE OF LINEAR PROGRAMMING TO FILTER DIGITIZED MAP DATA,” BY E. L. DOUGHERTY AND S. T. SMITH (GEOPHYSICS, FEBRUARY 1966, PP. 253–259)

Geophysics ◽  
1966 ◽  
Vol 31 (4) ◽  
pp. 828-829
Author(s):  
Norman S. Neidell
Keyword(s):  
Linear Programming ◽  
Least Squares ◽  
Least Squares Method ◽  
Short Note ◽  
Least Square ◽  
Local Polynomial ◽  
Polynomial Fit ◽  
Map Data

I wish to comment on the short note by E. L. Dougherty and S. T. Smith titled “The Use of Linear Programming to Filter Digitized Map Data” which appeared in the February, 1966, issue of Geophysics. First, I would like to congratulate the authors for showing explicitly how Taylor’s theory can be used to justify a local polynomial fit. I think, however, that they should have made clear that this same justification is valid for the least squares method. In essence these authors have chosen to make an approximation in the [Formula: see text] Norm instead of the [Formula: see text] or Least Square Norm (see Rice, 1964).

Research on the Wire Drawing Die Pass Parameter Calculation Based on the Least Square Algorithm

2012 ◽  
Vol 591-593 ◽  
pp. 850-853
Author(s):  
Huai Xing Wen ◽  
Yong Tao Yang
Keyword(s):  
Pore Structure ◽  
Least Squares ◽  
Least Squares Method ◽  
Piecewise Linear ◽  
Wire Drawing ◽  
Least Square ◽  
Structure Characteristics ◽  
Drawing Die ◽  
The Wire ◽  
Drawing Dies

Drawing Dies meter A / D acquisition module will be collected from the mold hole contour data to draw a curve in Matlab. According to the mold pore structure characteristics of the curve, the initial cut-off point of each part of contour is determined and iteratived optimization to find the best cut-off point, use the least squares method for fitting piecewise linear and fitting optimization to find the function of the various parts of the curve function, finally calculate the pass parameters of drawing mode. Parameters obtained compare with the standard mold, both of errors are relatively small that prove the correctness of the algorithm. Also a complete algorithm flow of pass parameters is designed, it can fast and accurately measure the wire drawing die hole parameters.

Fuzzy C-Regression Models

2013 ◽  
Vol 278-280 ◽  
pp. 1323-1326
Author(s):  
Yan Hua Yu ◽  
Li Xia Song ◽  
Kun Lun Zhang
Keyword(s):  
Linear Regression ◽  
Least Squares ◽  
Linear Models ◽  
Least Squares Method ◽  
Least Square Method ◽  
Least Square ◽  
Estimation Methods ◽  
Statistical Regression ◽  
Fuzzy Linear Regression ◽  
Fuzzy Degree

Fuzzy linear regression has been extensively studied since its inception symbolized by the work of Tanaka et al. in 1982. As one of the main estimation methods, fuzzy least squares approach is appealing because it corresponds, to some extent, to the well known statistical regression analysis. In this article, a restricted least squares method is proposed to fit fuzzy linear models with crisp inputs and symmetric fuzzy output. The paper puts forward a kind of fuzzy linear regression model based on structured element, This model has precise input data and fuzzy output data, Gives the regression coefficient and the fuzzy degree function determination method by using the least square method, studies the imitation degree question between the observed value and the forecast value.

A least squares method for earthquake mechanism determination using S-wave data

1964 ◽  
Vol 54 (6A) ◽  
pp. 2037-2047
Author(s):  
Agustin Udias
Keyword(s):  
Least Squares ◽  
Statistical Evaluation ◽  
Least Squares Method ◽  
Numerical Approach ◽  
Focal Mechanisms ◽  
Least Square ◽  
S Wave ◽  
Double Couple ◽  
Earthquake Mechanism

abstract In this paper a numerical approach to the determination of focal mechanisms based on the observation of the polarization of the S wave at N stations is presented. Least-square methods are developed for the determination of the orientation of the single and double couple sources. The methods allow a statistical evaluation of the data and of the accuracy of the solutions.

THE USE OF LINEAR PROGRAMMING TO FILTER DIGITIZED MAP DATA

Geophysics ◽  
1966 ◽  
Vol 31 (1) ◽  
pp. 253-259 ◽  
Author(s):  
E. L. Dougherty ◽  
S. T. Smith
Keyword(s):  
Linear Programming ◽  
Least Squares ◽  
Error Detection ◽  
Linear Programming Problem ◽  
Mineral Resources ◽  
Least Squares Approximation ◽  
Data Errors ◽  
Order Polynomial ◽  
Data Points ◽  
Map Data

The procedure used to discover subsurface formations where mineral resources may exist normally requires the accumulation and processing of large amounts of data concerning the earth’s fields. Data errors may strongly affect the conclusions drawn from the analysis. Thus, a method of checking for errors is essential. Since the field should be relatively smooth locally, a typical approach is to fit the data to a surface described by a low‐order polynomial. Deviations of data points from this surface can then be used to detect errors. Frequently a least‐squares approximation is used to determine the surface, but results could be misleading. Linear programming can be applied to give more satisfactory results. In this approach, the sum of the absolute values of the deviations is minimized rather than the squares of the deviations as in least squares. This paper describes in detail the formulation of the linear programming problem and cites an example of its application to error detection. Through this formulation, once errors are removed, the results are meaningful physically and, hence, can be used for detecting subsurface phenomena directly.

Solitary waves of the RLW equation via least squares method

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Ozlem Ersoy Hepson ◽  
Idris Dag ◽  
Bülent Saka ◽  
Buket Ay
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Least Square Method ◽  
Least Square ◽  
Spline Functions ◽  
Algebraic Equations ◽  
B Spline ◽  
B Splines ◽  
Rlw Equation ◽  
Set Up

Abstract Integration using least squares method in space and Crank–Nicolson approach in time is managed to set up an algorithm to solve the RLW equation numerically. Trial functions in the least square method consist of a combination of the quartic B-spline functions. Integration of the RLW equation gives a system of algebraic equations. The solutions consisting of a combination of the quartic B-splines are given for some initial and boundary value problems of RLW equation.

A New Method to Identify the Preisach Distribution Function of Hysteresis

2005 ◽  
Vol 475-479 ◽  
pp. 2107-2110 ◽  
Author(s):  
Fan Li ◽  
Jian Qin Mao ◽  
Hai Shan Ding ◽  
Wen Bo Zhang ◽  
Hui Bin Xu ◽  
...  
Keyword(s):  
Distribution Function ◽  
Least Squares ◽  
Fuzzy Inference System ◽  
Fuzzy Inference ◽  
Least Squares Method ◽  
Input Sequence ◽  
Least Square Method ◽  
Least Square ◽  
New Method ◽  
Inference System

In this paper, a new method which combines the least square method with Tree-Structured fuzzy inference system is presented to approximate the Preisach distribution function. Firstly, by devising the input sequence and measure the output, discrete Preisach measure can be identified by the use of the least squares method. Then, the Preisach function can be obtained with Tree-Structured fuzzy inference system without any special smoothing means. So, this new method is not sensitive to noise, and is a universal approximator of the Preisach function. It collect the merit and overcome the deficiency of the existing methods.

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 Comparative Analysis on Some Estimators of Parameters of Linear Regression Models in Presence of Multicollinearity

2018 ◽  
pp. 1-8
Author(s):  
Warha, Abdulhamid Audu ◽  
Yusuf Abbakar Muhammad ◽  
Akeyede, Imam
Keyword(s):  
Linear Regression ◽  
Least Squares ◽  
Regression Models ◽  
Mean Squared Error ◽  
Least Squares Method ◽  
Ordinary Least Squares ◽  
Least Square ◽  
Linear Regression Models ◽  
Independent Variables ◽  
Different Levels

Linear regression is the measure of relationship between two or more variables known as dependent and independent variables. Classical least squares method for estimating regression models consist of minimising the sum of the squared residuals. Among the assumptions of Ordinary least squares method (OLS) is that there is no correlations (multicollinearity) between the independent variables. Violation of this assumptions arises most often in regression analysis and can lead to inefficiency of the least square method. This study, therefore, determined the efficient estimator between Least Absolute Deviation (LAD) and Weighted Least Square (WLS) in multiple linear regression models at different levels of multicollinearity in the explanatory variables. Simulation techniques were conducted using R Statistical software, to investigate the performance of the two estimators under violation of assumptions of lack of multicollinearity. Their performances were compared at different sample sizes. Finite properties of estimators’ criteria namely, mean absolute error, absolute bias and mean squared error were used for comparing the methods. The best estimator was selected based on minimum value of these criteria at a specified level of multicollinearity and sample size. The results showed that, LAD was the best at different levels of multicollinearity and was recommended as alternative to OLS under this condition. The performances of the two estimators decreased when the levels of multicollinearity was increased.

Estimation of blood flow distribution in skeletal muscle from inert gas washout

1989 ◽  
Vol 66 (4) ◽  
pp. 1942-1955 ◽  
Author(s):  
J. Gronlund ◽  
G. M. Malvin ◽  
M. P. Hlastala
Keyword(s):  
Skeletal Muscle ◽  
Blood Flow ◽  
Linear Programming ◽  
Least Squares ◽  
Least Squares Method ◽  
Flow Distribution ◽  
Volume Ratio ◽  
Gracilis Muscle ◽  
Inert Gas ◽  
Washout Curves

A new method is evaluated for the estimation of blood flow-to-volume distribution in skeletal muscle from inert gas washout kinetics. Acetylene washout from the isolated, blood-perfused canine gracilis muscle was measured continuously with a blood gas catheter in combination with a mass spectrometer. The washout curves were transformed to flow-to-volume ratio distributions by means of a 50-compartment model. The algorithm fits the expression for the washout curve derived from the model by a least-squares method with enforced smoothing. The algorithm was evaluated using computer simulations in which artificial washout curves were generated by a multicompartment model with a known flow distribution. A wide range of given flow distributions could be recovered from the simulated data. The data were also analyzed using a linear programming technique. Analysis of the experimental data with the least-squares method showed that there is considerable heterogeneity in the distribution of perfusion in resting gracilis muscle. The distribution is characterized by at least two modes and a single compartment with a very low perfusion-to-volume ratio. Experimental noise made it impossible to obtain feasible flow distributions by means of linear programming.

An Application of Least Squares Method to the Solution of the Inverse Problem of Heat Conduction

1963 ◽  
Vol 85 (4) ◽  
pp. 378-379 ◽  
Author(s):  
Irving Frank
Keyword(s):  
Experimental Data ◽  
Inverse Problem ◽  
Heat Conduction ◽  
Least Squares ◽  
Heat Input ◽  
Least Squares Method ◽  
Least Square Method ◽  
The Body ◽  
Least Square

When the temperature of a body at some point is known, it is generally possible to determine the rate of heat input to the surface of the body. However, when the temperatures are determined experimentally, it will be found that there is some uncertainty in the solution for the rate of heat input. It is suggested that a least square method be used to determine the rate of heat input which best fits the experimental data.

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