Least-Squares Fit of a Straight Line

The variance of Oj and 02 is obtained by using the principle of propagation of error presented in Chap. 2. 

In many cases, the standard deviation of y, defines the weighting functions, and specifically, analysts use 

---

Williamson JH (1968) Least-squares fitting of a straight line. Can J Phys 46 1845-1847 York D (1966) Least-squares fitting of a straight line. Can J Phys 44 1079-1086... 

York, D. (1969). Least squares fitting of a straight line with correlated errors. Earth Planet. Sci. Letters, 5, 320-24. 

Figure 9. The ln[l - (A A )] versus time for a reduced AgPFSA film exposed to air after 19 hours of reduction in H<>. Dots represent points calculated from experimental data. The solid line shows the least squares fitting to a straight line on the long time scale. Reproduced with permission from Ref. 20, Copyright 1985, Academic Press, Inc.

In order to separate the intermolecular and intramolecular contributions to cross-relaxation, values of the cross-relaxation rate were plotted as a function of mole fraction of EHB in EHB-daa- experimental points were least-squares fit to a straight line and the y-intercept was taken to be the intramolecular contribution to the total cross-relaxation rate. This analysis neglects any cross-relaxation due to deuterium. 

When the computation is done with finite precision, using normal equations is not always recommended because for some matrices A, higher precision of computation is required in order to solve the system of normal equations correctly. For example, let us consider the problem of least-squares fitting by a straight line, Cq - - cix, to the set of data, shown in the accompanying tabulation. 

This equation can be solved by a least-squares fit to a straight line, in which case the group contributions G,> become the regression coefficient for the parameters A FORTRAN program to run Free-Wilson anaijrsis has been provided by Purcell et al. However, any least-squares analysis program can be used for the analysis provided sufficient observations are available. The minimum number of observations required to solve this equation is given by... 

It is a straightforward matter to obtain expressions for the slope and y-intercept of a weighted least-squares fit to a straight line by solving the partial differential of the value. The resulting expression for the slope (m) is... 

A number of statistical techniques exist for fitting a function to a set of scattered data. The application of the most common of these techniques—linear regression or the method of least squares—to the fitting of a straight line to a series of y versus jc data points is outlined and illustrated in Appendix A.l, and the use of this technique is required for the solution of Problems 2.39 through 2.42 at the end of this chapter. 

Linear Regression and the Method of Least Squares. We seek the best fit of a straight line. 

Expand the three detemiinants D, Dt, and for the least squares fit to a linear function not passing through the origin so as to obtain explicit algebraic expressions for b and m, the y-intercept and the slope of the best straight line representing the experimental data. 

In Fig. 2.2, the critical deposition temperature of NbCl5 as a function of its initial pressure, is shown from experimental data from Blocher and the author. There are two temperature-pressure regions, which are separated by a straight line. The metal is deposited only in the region below the line. Above, there is no deposition. The line is a least-square fit of the data. Its position was confirmed using the SOLGASMIX computer program. 

Fig. 34. Arrhenius plot of In k versus 1/T for PSS-doped [Fe(6-Mepy)2(py)tren](CI04)2 for the temperature range 50 to 300 K. Here k is the relaxation rate constant, the straight line representing a least squares fit of the 150-300 K data producing AE = 823 cm". The insert shows k versus T between 4.2 and 50 K. According to Ref. [138]...

PB28-PU1 at a temperature of — 20°C. This value was determined to be 0.72 by calculating the limiting slope of the data near the origin. The dashed, straight line represents the least-squares fit of the data in the linear region near the origin. 

A calibration curve is a model used to predict the value of an independent variable, the analyte concentration, when only the dependent variable, the analytical response, is known. The normal procedure used to establish a calibration curve is based on a linear least-squares fit of the best straight line for a linear regression, as indicated in... 

---