Straight line fit

The least-square straight line fit through points at — 1, 0, 1 is... 

Cununing GL, Rollett JS, Rosotti FJC, Whewell RJ (1972) Statistical methods for the computation of stability constants, I. Straight-line fitting of points with correlated errors. J Chem Soc Dalton Trans 23 2652-2658... 

Fig. 4.10. (A) Schematic of percentage weights of glycerol in composite solvents corresponding to array of fluorescein solutions of varying viscosity (B) fluorescence lifetime (C) rotational correlation time images of this array and (D) plot of the rotational correlation time as a function of viscosity for this sample array exited at 470 nm the straight line fit yields a fluorophore radius of 0.54 nm for fluorescein. Adapted from Fig. 2 of Ref. [64].

Performing the calculations on function fitted to the raw data has another ramification the differences, and therefore the sums of squares, will depend on the units that the K-values are expressed in. It is preferable that functions with similar appearances give the same computed value of nonlinearity regardless of the scale. Therefore the sum-of-squares of the differences between the linear and the quadratic functions fitted to the data is divided by the sum-of-squares of the T-values that fall on the straight line fitted to the data. This cancels the units, and therefore the dependency of the calculation on the scale. 

X-axis. It presents the coefficients of the linear models (straight lines) fitted to the several curves of Figure 67-1, the coefficients of the quadratic model, the sum-of-squares of the differences between the fitted points from the two models, and the ratio of the sum-of-squares of the differences to the sum-of-squares of the X-data itself, which, as we said above, is the measure of nonlinearity. Table 67-1 also shows the value of the correlation coefficient between the linear fit and the quadratic fit to the data, and the square of the correlation coefficient. 

The best straight line fit is obtained when the sum of the squares of the individual y-axis value deviations (deviations between the plotted y values and the values on the proposed line) is at a minimum. 

And now we repeat what we have done earlier with the straight line fit, i.e. calculating and plotting the sum of squares, ssq, as a function of a range of parameters. 

Comparing Figure 4-5 with the corresponding plot from the straight line fit in Figure 4-3, an important difference is that the landscape is no longer parabolic. There is a flat region and a very steep increase at the back comer. Nevertheless, the contour lines clearly indicate that there is a minimum near the correct position. 

Note that the term Linear Regression is somewhat misleading. It is not restricted to the task of just fitting a straight line to some data. While this task is an example of linear regression, the expression covers much more. However, to start with, we return to the task of the straight line fit. 

Rather than writing a short program in Matlab for this result, we demonstrate how to perform the task of a straight line fit in Excel. Excel actually provides several ways of performing the job of fitting the best line through a set of data pairs. The most convenient is probably the Add Trendline. .. tool which delivers the result in a few clicks. 

There are several ways to derive the equations for the computation of the optimal vector a. One option would be to generalise the procedure we used for the straight line fit in equations (4.5) - (4.13), which would be rather cumbersome. In the following we use a different approach. 

As indicated in Chapter 4.2.1, Straight Line Fit, the Excel function LINEST is a more general function than the rRENDLINE. In addition to allowing the... 

From a straight-line fit of 13C signal heights vs. contact times of 1.5-... 

Alternatively, the sum can be computed from the Birge-Sponer graph by measuring the area under the straight-line fit to the graph of AEVj or vj from vj = 0 to vj = Vj inax ... 

A second important application of CMD has been to study the dynamics of the hydrated proton. This study involved extensive CMD simulations to determine the proton transport rate in on our Multi-State Empirical Valence Bond (MS-EVB) model for the hydrated proton. = Shown in Fig. 4 are results for the population correlation function, (n(t)n(O)), for the Eigen cation, HsO, in liquid water. Also shown is the correlation function for D3O+ in heavy water. It should be noted that the population correlation function is expected to decay exponentially at long times, the rate of which reflects the excess proton transport rate. The straight line fits (dotted lines) to the semi-log plots of the correlation functions give this rate. For the normal water case, the CMD simulation using the MS-EVB model yields excellent agreement with the experimental proton hopping... 

Figure 1.16 Relative complex viscosity ( > / f)ol) versus calculated conversion for polymerization at 120 and 150°C. Linear, straight-line fit to the phenomenological equation >/ / r 0 = exp(19.6 X) is also shown...

Figure 12.1 shows how well the predicted results agree with the experimental values listed in Table 12.2. The predictions consistently overestimate experimental dielectric constant values for this series of samples. The difference varies from 12 to 27% of the experimental values. Qualitatively, predictions follow the same trend as experiment. The calculated slope for the best straight-line fit through the data points is equal to 0.822 with a standard deviation of 0.088. The correlation coefficient is equal to 0.881. For several of the polyimides tested here the method... 

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