scholarly journals Technical Note: Review of methods for linear least-squares fitting of data and application to atmospheric chemistry problems

2008 ◽  
Vol 8 (2) ◽  
pp. 6409-6436 ◽  
Author(s):  
C. A. Cantrell
Keyword(s):  
Least Squares ◽  
Atmospheric Chemistry ◽  
Least Squares Method ◽  
Technical Note ◽  
Routine Practice ◽  
Data Sets ◽  
Straight Line ◽  
Sample Data ◽  
Straight Lines ◽  
Best Fit

Abstract. The representation of data, whether geophysical observations, numerical model output or laboratory results, by a best fit straight line is a routine practice in the geosciences and other fields. While the literature is full of detailed analyses of procedures for fitting straight lines to values with uncertainties, a surprising number of scientists blindly use the standard least squares method, such as found on calculators and in spreadsheet programs, that assumes no uncertainties in the x values. Here, the available procedures for estimating the best fit straight line to data, including those applicable to situations for uncertainties present in both the x and y variables, are reviewed. Representative methods that are presented in the literature for bivariate weighted fits are compared using several sample data sets, and guidance is presented as to when the somewhat more involved iterative methods are required, or when the standard least-squares procedure would be expected to be satisfactory. A spreadsheet-based template is made available that employs one method for bivariate fitting.

scholarly journals Technical Note: Review of methods for linear least-squares fitting of data and application to atmospheric chemistry problems

2008 ◽  
Vol 8 (17) ◽  
pp. 5477-5487 ◽  
Author(s):  
C. A. Cantrell
Keyword(s):  
Least Squares ◽  
Atmospheric Chemistry ◽  
Least Squares Method ◽  
Technical Note ◽  
Routine Practice ◽  
Data Sets ◽  
Straight Line ◽  
Sample Data ◽  
Straight Lines ◽  
Best Fit

Abstract. The representation of data, whether geophysical observations, numerical model output or laboratory results, by a best fit straight line is a routine practice in the geosciences and other fields. While the literature is full of detailed analyses of procedures for fitting straight lines to values with uncertainties, a surprising number of scientists blindly use the standard least-squares method, such as found on calculators and in spreadsheet programs, that assumes no uncertainties in the x values. Here, the available procedures for estimating the best fit straight line to data, including those applicable to situations for uncertainties present in both the x and y variables, are reviewed. Representative methods that are presented in the literature for bivariate weighted fits are compared using several sample data sets, and guidance is presented as to when the somewhat more involved iterative methods are required, or when the standard least-squares procedure would be expected to be satisfactory. A spreadsheet-based template is made available that employs one method for bivariate fitting.

Theoretical accuracy of the last squares method in the isotope dilution analysis

1984 ◽  
Vol 49 (4) ◽  
pp. 805-820
Author(s):  
Ján Klas
Keyword(s):  
Least Squares ◽  
Isotope Dilution ◽  
Least Squares Method ◽  
Isotope Dilution Analysis ◽  
Straight Line ◽  
The One ◽  
Two Parameter ◽  
Theoretical Accuracy

The accuracy of the least squares method in the isotope dilution analysis is studied using two models, viz a model of a two-parameter straight line and a model of a one-parameter straight line.The equations for the direct and the inverse isotope dilution methods are transformed into linear coordinates, and the intercept and slope of the two-parameter straight line and the slope of the one-parameter straight line are evaluated and treated.

A modified tangent-gas approximation for two-dimensional steady flow

1958 ◽  
Vol 4 (6) ◽  
pp. 600-606 ◽  
Author(s):  
G. Power ◽  
P. Smith
Keyword(s):  
Least Squares ◽  
Steady Flow ◽  
Variational Approach ◽  
Least Squares Method ◽  
Two Dimensional ◽  
Subsonic Flows ◽  
Von Karman ◽  
Straight Line ◽  
Volume Relationship ◽  
Pressure Volume Relationship

A set of two-dimensional subsonic flows past certain cylinders is obtained using hodograph methods, in which the true pressure-volume relationship is replaced by various straight-line approximations. It is found that the approximation obtained by a least-squares method possibly gives best results. Comparison is made with values obtained by using the von Kármán-Tsien approximation and also with results obtained by the variational approach of Lush & Cherry (1956).

A Direct Least Squares Method for Camera Pose Estimation Based on Straight Line Segment Correspondences

2015 ◽  
Vol 35 (6) ◽  
pp. 0615003
Author(s):  
李鑫 Li Xin ◽  
张跃强 Zhang Yueqiang ◽  
刘进博 Liu Jinbo ◽  
张小虎 Zhang Xiaohu ◽  
于起峰 Yu Qifeng
Keyword(s):  
Least Squares ◽  
Pose Estimation ◽  
Line Segment ◽  
Least Squares Method ◽  
Straight Line Segment ◽  
Camera Pose Estimation ◽  
Straight Line ◽  
Camera Pose

The least-squares intersection of a family of lines and its application to phase equilibria

1982 ◽  
Vol 60 (15) ◽  
pp. 1978-1981 ◽  
Author(s):  
John W. Lorimer
Keyword(s):  
Phase Equilibria ◽  
Least Squares ◽  
Least Squares Method ◽  
Solid Phase ◽  
Generalized Least Squares ◽  
First Approximation ◽  
Generalized Least Squares Method ◽  
Point Of Intersection ◽  
Straight Lines ◽  
General Method

A generalized least-squares method is described for finding the point of intersection of a family of straight lines, each of which is defined by two experimental points. It is shown that the method of the least-squares triangle (Can. J. Chem. 59, 3076 (1981)) is a good first approximation to the general method. An example demonstrates the method of iteration of both parameters and observations for a problem involving evaluation of solid phase compositions from solubility measurements.

scholarly journals RANSAC FOR OUTLIER DETECTION

2012 ◽  
Vol 31 (3) ◽  
pp. 83-87 ◽  
Author(s):  
Birutė Ruzgienė ◽  
Wolfgang Förstner
Keyword(s):  
Image Processing ◽  
Image Analysis ◽  
Image Matching ◽  
Data Sets ◽  
Digital Photogrammetry ◽  
Huge Data ◽  
Straight Line ◽  
Sample Data ◽  
Random Sample Consensus ◽  
Classical Image

Up-to-date digital photogrammetry involves operations on huge data sets, and with classical image processing procedures it might be time consuming to find out the best solution. One of the key tasks is to detect outliers in given data, eg for curve fitting or image matching. The problem is hard as the number of outliers is usually large, possibly larger than 50%, thus powerful estimation techniques are needed. We demonstrate one of these techniques, namely Random Sample Consensus (RANSAC), for fitting a model to sample data, especially for fitting a straight line through a set of given points. Experiments with up to 80% outliers prove the efficiency of RANSAC. The results are representative for image analysis in digital photogrammetry

scholarly journals Fitting limit lines (envelope curves) to spreads of geoenvironmental data

2021 ◽  
pp. 030913332110599
Author(s):  
Paul A Carling ◽  
Philip Jonathan ◽  
Teng Su
Keyword(s):  
Least Squares ◽  
Lower Limit ◽  
Statistical Models ◽  
Marked Reduction ◽  
Ad Hoc ◽  
Data Sets ◽  
Least Squares Regression ◽  
Sample Data ◽  
Considerable Spread

Geoscientists frequently are interested in defining the overall trend in x- y data clouds using techniques such as least-squares regression. Yet often the sample data exhibits considerable spread of y-values for given x-values, which is itself of interest. In some cases, the data may exhibit a distinct visual upper (or lower) ‘limit’ to a broad spread of y-values for a given x-value, defined by a marked reduction in concentration of y-values. As a function of x-value, the locus of this ‘limit’ defines a ‘limit line’, with no (or few) points lying above (or below) it. Despite numerous examples of such situations in geoscience, there has been little consideration within the general geoenvironmental literature of methods used to define limit lines (sometimes termed ‘envelope curves’ when they enclose all data of interest). In this work, methods to fit limit lines are reviewed. Many commonly applied methods are ad-hoc and statistically not well founded, often because the data sample available is small and noisy. Other methods are considered which correspond to specific statistical models offering more objective and reproducible estimation. The strengths and weaknesses of methods are considered by application to real geoscience data sets. Wider adoption of statistical models would enhance confidence in the utility of fitted limits and promote statistical developments in limit fitting methodologies which are likely to be transformative in the interpretation of limits. Supplements, a spreadsheet and references to software are provided for ready application by geoscientists.

scholarly journals A Least Squares Estimation Method for the Linear Learning Model

1978 ◽  
Vol 15 (1) ◽  
pp. 145-153
Author(s):  
Berend Wierenga
Keyword(s):  
Least Squares ◽  
Simulation Study ◽  
Goodness Of Fit ◽  
Least Squares Method ◽  
Estimation Method ◽  
Learning Model ◽  
New Method ◽  
Data Sets ◽  
Goodness Of Fit Test ◽  
Chi Square

The author presents a new method for estimating the parameters of the linear learning model. The procedure, essentially a least squares method, is easy to carry out and avoids certain difficulties of earlier estimation procedures. Applications to three different data sets are reported, as well as results from a goodness-of-fit test. A simulation study was carried out to validate the method. The outcomes are compared with those obtained from the minimum chi square estimation method. The results of the new method appear to be satisfactory.

Calibration of stormwater quality regression models: a random process?

2010 ◽  
Vol 62 (4) ◽  
pp. 875-882 ◽  
Author(s):  
A. Dembélé ◽  
J.-L. Bertrand-Krajewski ◽  
B. Barillon
Keyword(s):  
Experimental Data ◽  
Least Squares ◽  
Regression Models ◽  
Linear Models ◽  
Least Squares Method ◽  
Weighted Least Squares ◽  
Ordinary Least Squares ◽  
Data Sets ◽  
Data Set ◽  
Urban Catchments

Regression models are among the most frequently used models to estimate pollutants event mean concentrations (EMC) in wet weather discharges in urban catchments. Two main questions dealing with the calibration of EMC regression models are investigated: i) the sensitivity of models to the size and the content of data sets used for their calibration, ii) the change of modelling results when models are re-calibrated when data sets grow and change with time when new experimental data are collected. Based on an experimental data set of 64 rain events monitored in a densely urbanised catchment, four TSS EMC regression models (two log-linear and two linear models) with two or three explanatory variables have been derived and analysed. Model calibration with the iterative re-weighted least squares method is less sensitive and leads to more robust results than the ordinary least squares method. Three calibration options have been investigated: two options accounting for the chronological order of the observations, one option using random samples of events from the whole available data set. Results obtained with the best performing non linear model clearly indicate that the model is highly sensitive to the size and the content of the data set used for its calibration.

A Discrete Data Fitting Models Fusing Genetic Algorithm

2011 ◽  
Vol 267 ◽  
pp. 427-432
Author(s):  
Tong Rang Fan ◽  
Yong Bin Zhao ◽  
Lan Wang
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Discrete Distribution ◽  
Data Fitting ◽  
Small Sample ◽  
Load Testing ◽  
Accurate Data ◽  
Sample Data ◽  
Testing Data ◽  
Dynamic Measuring

To address problems of Least squares method (LSM) fitting curves in application domains, the essay attempts to build a new model by using LMS (Least Median Squares) to analyze the error points, and pretreating the dynamic measuring errors and then getting the fitting curves of testing data. This model is used for electromotor parameters testing which includes load testing and unload testing. Experiments show that the model can erase the influence of outline points, while improving the effects of data curve fitting and reflecting the characteristic of the motor, provide more accurate data fitting curve with small sample data that is in discrete distribution compared with Least squares method.

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