Best fit line

The regression equation is the best-fit line through the data that minimises the sum of the deviations. 

The Least Squares or Best-fit Line. The simplest type of approximating curve is a straight line, the equation of which can be written as in form number 1 above. It is customary to employ the above definition when X is the independent variable and Y is the dependent variable. 

The scatter of the points around the calibration line or random errors are of importance since the best-fit line will be used to estimate the concentration of test samples by interpolation. The method used to calculate the random errors in the values for the slope and intercept is now considered. We must first calculate the standard deviation Sy/x, which is given by ... 

Equation 19 utilizes the Y-residuals, 1) — Y, where 1) are the points on the calculated best-fit line or the fitted 1) values. The appropriate number of degrees of freedom is A — 2 the minus 2 arises from the fact that linear calibration lines are derived from both a slope and an intercept which leads to a loss of two degrees of freedom. 

Example 14 For the best-fit line found in Example 13, express the result in terms of confidence intervals for the slope and intercept. We will choose 95% for the confidence interval. 

To conclude the discussion about the best-fit line, the following relationship can be shown to exist among Y, Y, and Y ... 

FIGURE 6.6 Schilcl regression for pirenzepine antagonism of rat tracheal responses to carbachol. (a) Dose-response curves to carbachol in the absence (open circles, n = 20) and presence of pirenzepine 300 nM (filled squares, n = 4), 1 jjM (open diamonds, n=4), 3j.lM (filled inverted triangles, n = 6), and 10j.iM (open triangles, n = 6). Data fit to functions of constant maximum and slope, (b) Schild plot for antagonism shown in panel A. Ordinates Log (DR-1) values. Abscissae logarithms of molar concentrations of pirenzepine. Dotted line shows best line linear plot. Slope = 1.1 + 0.2 95% confidence limits = 0.9 to 1.15. Solid line is the best fit line with linear slope. pKB = 6.92. Redrawn from [5],... 

Overdetermination of the system of equations is at the heart of regression analysis, that is one determines more than the absolute minimum of two coordinate pairs (xj/yi) and xzjyz) necessary to calculate a and b by classical algebra. The unknown coefficients are then estimated by invoking a further model. Just as with the univariate data treated in Chapter 1, the least-squares model is chosen, which yields an unbiased best-fit line subject to the restriction ... 

In order to extract the contributions and dynamics of the ketyl radical and fluoranil anion from the TR spectra obtained with the 416 nm probe wavelength, a deconvolution of the Raman bands were done using a fitting procedure employing a Lorentzian lineshape for the Raman bands of the two intermediates. Figure 3.20 shows a comparison of the best-fit (lines) to the experimental TR spectra (dots) in the left-side spectra and the deconvolution extracted from this best fit for the ketyl radical spectra... 

Figure 6. A portion of the concordia curve for initial =150 and four data points (Edwards et al. 1997) for coral samples from the last interglacial terrace in Barbados. The points lie along a hne, and illustrate the possibility of using best-fit lines through discordant points to extrapolate to concordia and possibly the tme age of a set of samples (after Cheng et al. 1998).

Find the slope and y-intercept of the best-fit line on the frequency versus energy graph. To calculate the slope, choose two points (iq. j ) and (v2, E2) that are well separated on the... 

The value of the intercept is the value of the best-fit line where it crosses the vertical... 

By fitting a straight line ( best-fit line) through the data points in Fig. 3 (this can be done by eye using a transparent straight edge) and extrapolating to t = 0,... 

Here m is the slope value and [ii]app is the apparent total enzyme concentration, typically estimated from protein assays and other methods (Copeland, 1994). Note from Equation (7.13) that when our estimate of enzyme concentration is incorrect, the slope of the best fit line of IC50 as a function of [E] will not be 1/2, as theoretically expected. Nevertheless, the v-intercept estimate of K pp is unaffected by inaccuracies in [ ]. In fact we can combine Equations (7.12) and (7.13) to provide an accurate determination of [ /]T from the slope of plots such as those shown in Figure 7.2. The true value of [ii]T is related to the apparent value [ TPP as... 

Fig. 25 Plot of log I and log V against log Mn for ordered and disordered phases where FCCs and ECCs are formed, respectively, from which a common power law of I and V for PE, J, V oc D(Mn) oc M H, is proposed. Solid and broken lines are the best-fit lines of the experimental power laws. It should be noted that H of the ordered phase is larger than that of the disordered phase...

It is important to recognize that all of the terrestrial isotopic data lie along the best-fit line (nearly always within 2o) independent of sample mineralogy and Mg content. We observed no systematic deviations from this line due to molecular... 

We see m[>L. is the reciprocal slope of the best-fit line in a plot of Se04 / >L Se07 versus flse04, and K is the ratio of the line s intercept to its slope. The sorption capacity, in mol g 1 soil, then, is given nw ntT LJns. 

Plot the reciprocal of the A ion concentration on the ordinate against the H30+ ion concentration on the abscissa. Draw a straight best-fit line (see Figure 1). 

Figure 5. Percentage illite layers versus layer charge for K-smectites subjected to 100 WD cycles in water at 60°C and 1 Sr-exchange. Numbers in parentheses refer to percentage of octahedral charge. Best fit line is for montmorillonites having 69% or more octahedral charge. Data from Tables III and IV.

Fig. 12.4 The results of the determination of the rotational diffusion tensor of the /7ARK PH domain A fit of the orientational dependence of the experimental values ofp for the /iARK PH domain and B a ribbon representation of the 3D structure of the protein, with the orientation of the diffusion axis indicated by a rod. Shown in A are the experimental (symbols) and the best fit (line) values ofp (Eq.

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