The line of best fit

2019 ◽  
pp. 26-37
Author(s):  
M. D. Edge
Keyword(s):  
Least Squares ◽  
Statistical Estimation ◽  
Best Fit

One way to visualize a set of data on two variables is to plot them on a pair of axes. A line that “best fits” the data can then be drawn as a summary. This chapter considers how to define a line of “best” fit—there is no sole best choice. The most commonly chosen line to summarize the data is the “least-squares” line—the line that minimizes the sum of the squared vertical distances between the points and the line. One reason for the least-squares line’s popularity is convenience, but, as will be seen later, it is also related to some key ideas in statistical estimation. The derivations of expressions for the intercept and slope of the least-squares line are discussed.

Spreadsheet Method for Evaluation of Biochemical Reaction Rate Coefficients and Their Uncertainties by Weighted Nonlinear Least-Squares Analysis of the Integrated Monod Equation

1998 ◽  
Vol 64 (6) ◽  
pp. 2044-2050 ◽  
Author(s):  
Laurence H. Smith ◽  
Perry L. McCarty ◽  
Peter K. Kitanidis
Keyword(s):  
Least Squares ◽  
Mixed Culture ◽  
Substrate Concentration ◽  
Reaction Rate ◽  
Initial Rate ◽  
Weighted Least Squares ◽  
Nonlinear Least Squares ◽  
Rate Coefficients ◽  
Monod Equation ◽  
Best Fit

ABSTRACT A convenient method for evaluation of biochemical reaction rate coefficients and their uncertainties is described. The motivation for developing this method was the complexity of existing statistical methods for analysis of biochemical rate equations, as well as the shortcomings of linear approaches, such as Lineweaver-Burk plots. The nonlinear least-squares method provides accurate estimates of the rate coefficients and their uncertainties from experimental data. Linearized methods that involve inversion of data are unreliable since several important assumptions of linear regression are violated. Furthermore, when linearized methods are used, there is no basis for calculation of the uncertainties in the rate coefficients. Uncertainty estimates are crucial to studies involving comparisons of rates for different organisms or environmental conditions. The spreadsheet method uses weighted least-squares analysis to determine the best-fit values of the rate coefficients for the integrated Monod equation. Although the integrated Monod equation is an implicit expression of substrate concentration, weighted least-squares analysis can be employed to calculate approximate differences in substrate concentration between model predictions and data. An iterative search routine in a spreadsheet program is utilized to search for the best-fit values of the coefficients by minimizing the sum of squared weighted errors. The uncertainties in the best-fit values of the rate coefficients are calculated by an approximate method that can also be implemented in a spreadsheet. The uncertainty method can be used to calculate single-parameter (coefficient) confidence intervals, degrees of correlation between parameters, and joint confidence regions for two or more parameters. Example sets of calculations are presented for acetate utilization by a methanogenic mixed culture and trichloroethylene cometabolism by a methane-oxidizing mixed culture. An additional advantage of application of this method to the integrated Monod equation compared with application of linearized methods is the economy of obtaining rate coefficients from a single batch experiment or a few batch experiments rather than having to obtain large numbers of initial rate measurements. However, when initial rate measurements are used, this method can still be used with greater reliability than linearized approaches.

AN AUTOMATIC LEAST‐SQUARES MULTIMODEL METHOD FOR MAGNETIC INTERPRETATION

Geophysics ◽  
1973 ◽  
Vol 38 (2) ◽  
pp. 349-358 ◽  
Author(s):  
Peter H. McGrath ◽  
Peter J. Hood
Keyword(s):  
Least Squares ◽  
Magnetic Anomalies ◽  
Magnetic Data ◽  
Model Parameters ◽  
Hudson Bay ◽  
Northern Ontario ◽  
Magnetic Interpretation ◽  
Hudson Bay Lowlands ◽  
Sverdrup Basin ◽  
Best Fit

The magnetic anomalies caused by such diverse model shapes as the finite strike length thick dike, the vertical prism, thes loping step, the parallelepiped body, etc., may be obtained through an appropriate numerical integration of the expression for the magnetic effect produced by a finite thin plate. Using models generated in this manner, an automatic computer method has been developed at the Geological Survey of Canada for the interpretation of magnetic data. Because the magnetic anomalies produced by the various model shapes are nonlinear in parameters of shape and position, it is necessary to use an iterative procedure to obtain the values for the various model parameters which yield a least‐squares best‐fit anomaly curve to a set of discrete observed data. The interpretation method described in this paper uses the Powell algorithm for this purpose. The procedure can sometimes be made more efficient using a Marquardt modification to the Powell algorithm. Examples of the use of the method are presented for an elongated anomaly in the Moose River basin of the Hudson Bay lowlands in northern Ontario, and for an areally large elliptical anomaly in the Sverdrup basin of the Canadian Arctic Islands.

A Task Driven Approach to Unified Synthesis of Planar Four-Bar Linkages Using Algebraic Fitting of a Pencil of G-Manifolds

2013 ◽  
Author(s):  
Q. J. Ge ◽  
Ping Zhao ◽  
Anurag Purwar
Keyword(s):  
Least Squares ◽  
Singular Values ◽  
Geometric Constraints ◽  
Motion Generation ◽  
Motion Approximation ◽  
Value Decomposition ◽  
Four Bar Linkage ◽  
The Given ◽  
Best Fit ◽  
Additional Constraints

This paper studies the problem of planar four-bar motion approximation from the viewpoint of extraction of geometric constraints from a given set of planar displacements. Using the Image Space of planar displacements, we obtain a class of quadrics, called Generalized- or G-manifolds, with eight linear and homogeneous coefficients as a unified representation for constraint manifolds of all four types of planar dyads, RR, PR, and PR, and PP. Given a set of image points that represent planar displacements, the problem of synthesizing a planar four-bar linkage is reduced to finding a pencil of G-manifolds that best fit the image points in the least squares sense. This least squares problem is solved using Singular Value Decomposition. The linear coefficients associated with the smallest singular values are used to define a pencil of quadrics. Additional constraints on the linear coefficients are then imposed to obtain a planar four-bar linkage that best guides the coupler through the given displacements. The result is an efficient and linear algorithm that naturally extracts the geometric constraints of a motion and leads directly to the type and dimensions of a mechanism for motion generation.

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.

Analysis of Chesapeake Bay Racing Results 1972 and 1973 Seasons

1974 ◽  
Author(s):  
Robert W. Peach
Keyword(s):  
Least Squares ◽  
Chesapeake Bay ◽  
Power Curve ◽  
Average Performance ◽  
Best Fit

The analysis of racing on the Chesapeake Bay was performed covering the three handicapping rules used for the 1972 and 1973 seasons. The actual average speeds of each yacht were determined and the normalized to obtain season average performances. This average performance with the yacht’s rating was used to determine the least squares best fit for a power curve to show how each rule actually worked when corrected for the time allowance used by each class.

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