Numerical methods least squares curve fitting

The data in Table 8.4 [4] represent the vapor pressure of mercury as a function of temperature. Plot In P as a function of 1/T to a scale consistent with the precision of the data. If the resultant plot is linear, calculate AH Iz from the slope obtained by a least-squares fit to the line. If the plot is curved, use a numerical differentiation procedure to obtain the value of AHmjZ as a function of T, and calculate ACpm- See Appendix A for methods. 

Figure 2. Gel permeation data for polypeptide linear random coils plotted according to the method of Porath (8) M0,555 is plotted vs. Kd1/3. Lines drawn through the data from each column are lines of best fit determined by linear least-squares analysis. Numerical designation for each curve represents the agarose...

Figure 3. Gel permeation data for linear randomly coiled polypeptides on various agarose resins, plotted according to the method of Ackers (9). M0 555 is plotted vs. the inverse error function complement of Kd (erfc 1 Kd). Lines drawn through the data points represent best fits obtained from linear least-squares analysis of the data. Numerical designation of each curve represents the percent agarose composition for the resin used. Filled triangles on the curve for the 6% resin, and the filled squares on the curve for the 10% resin are points determined using fluorescent proteins. Data for the labeled polypeptides were not included in the least-squares analysis.

Both equations are useful to obtain well-defined D values in each experiment based on a fitting method. Although we understand that the form in Eq. (33.9) is more general, the numerical data from FCS measurement is not sufficient to obtain the full lineshape of D(t) in Eq. (33.9). Seki et al. obtained an analytical solution of autocorrelation curves for D(L) in a step function [39]. They proved that the solution lineshape is different from that of normal diffusion with a non-linear least square algorithm if the deviation from Eq. (33.17) is too small. Even in this case of moderate anomalous diffusion, the observed value of D changes sensitively,depending on f or I. 

Section 11.5 Numerical Curve Fitting The Method of Least Squares (Regression)... 

A straightforward extension of the three-point technique is to utilize a larger number of measured AE-i data pairs, and to analyze the data by using some kind of numerical curve-fitting procedure, usually a nonlinear least-squares method. This increases... 

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