Squares Fit to a Straight Line

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.

Figures 5 and 6 show plots of the integral on the left side of Equation 5 vs. time for chars A and B, respectively. The parameter b is chosen to minimize the sum of the squares of the errors of a last-square fit to a straight line through the data points. The value of K is then evaluated from the slope of the straight line.

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. 

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... 

These data are fitted to a straight line by least squares. From the intercept,... 

Fig. 6.7 Plots of the molar conductivity against the square root of the electrolyte concentration for three strong electrolytes ( ) and one weak electrolyte ( ). The data for the strong electrolytes were fitted to a straight line at lower concentrations in order to obtain the limiting molar conductivity Aq.

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. 

Figure 4. (above, right) Arrhenius plots [In k vs. 10 /T] for P HMX decomposition at four different pressures (least-squares fits). The fits for the 3.6 GPa and 4.6 GPa data both have linear correlation coefficients of 0.996. The 5.5 GPa and 6.5 GPa data are less well-fitted to a straight line with correlation coefficients of 0.91 and 0.96, respectively. 

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. 

Using a computer programme for a fit of data to a straight line, with a least squares procedure, gives... 

The plots of the mean square displacements along the x axis versus the time interval r are also represented in Fig. 4. The experimental points are well fitted by a straight line with a slope corresponding to a diffusion coefficient Dx = 1.438 10 nr/s. A similar treatment along the y axis gives Dy = 1.412 10 trr/s. This difference gives an idea of the uncertainty, which is of order 2ck for nearly 20,000 independent positions. Note that the correlation time of the velocities, r - with t] the viscosity of the suspending fluid, is less than... 

Referring to Equations (B.1.1) and (B.1.6), if there is no correlation between x and y, then there are no trends for y to either increase or decrease with increasing x. Therefore, the least-squares fit must yield 2 = 0- Now, consider the question of whether the data correspond to a straight line of the form ... 

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. 

At the least, a chart of the data should be produced for visual inspection of the fit, as illustrated in Figure 11-2. You can also use the F SQ(known ys, known xs) worksheet function to return the square of the correlation coefficient, to provide information about the goodness of fit to the straight line. 

The most probable value of a quantity is obtained from a set of measurements by choosing the value that minimizes the sum of the squares of the deviations of all the measurements from that chosen value (see Equation 10.5). The simple example of a fitting of points to a straight line is given in Figure 10.2. 

Table 9-2 shows the values of R computed from the data, with K = 73. The most probable values of the constants are those corresponding to a least-mean-square fit of the data to a straight line of the general form... 

Least-squares fitting to a general straight line... 

If the specific form of the absorption rate equation is inserted into Eqn. (12), these two equations may be-arranged to yield a relationship between B. and B°. Now it is possible to plot B. against B and the to construct McCabe-Thiele type steps, the actual number of plates required for the operation may then be found graphically (see, for example, (65)). Juvekar and Sharma (50) proposed, however, that the plot of Bfj. versus B may be approximated to a straight line by a least square fit. Thus ... 

Corrosion scientists seldom have enough replicate data to analyze distributions using such techniques as histograms and tests of fit. Sometimes, however, enough data are available to order and plot as a function of linearized forms of the different kinds of probability distributions. Also, nonlinear least squares curve fitting may be used. The best approach is to choose the type of distribution that cleeirly produces the best approximation to a straight line. If an extreme value distribution is only slightly better than a normal distribution, a normal distribution may be assumed because the statistical techniques are easier to use and more universally understood. 

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