Equations least squares regression line

The least squares regression line is the line which minimizes the sum of the square or the error of the data points. It is represented by the linear equation y = ax + b. The variable x is assigned the independent variable, and variable j is assigned the dependent variable. The term b is the y-intercept or regression constant (the value of when x = 0), and the term a is the slope or regression coefficient. 

Equations (2.12) and (2.13) enable the best-fit calibration line to be drawn through the experimental x,y points. Once the slope m and the y intercept b for the least squares regression line are obtained, the calibration line can be drawn by any one of several graphical techniques. A commonly used technique is to use Excel to graphically present the calibration curve. These equations can also be incorporated into computer programs that allow rapid computation. 

The uncertainty in both the slope and intercept of the least squares regression line can be found using the following equations to calculate the standard deviation in the slope, and the standard deviation in the y intercept, sy. 

Equation (2.25) is an important relationship if one desires to plot the confidence intervals that surround the least squares regression line. The upper confidence limit for the particular case where x = 0 is obtained from Eq. (2.25) in which a 1 — a probability exists that the Normal distribution of instrument responses falls to within the mean y at x = 0. Mathematically, this instrument response y, often termed a critical response, is defined as... 

To conclude this discussion on IDLs, it is useful to compare Eqs. (2.19) and (2.29). Both equations relate Xp to a ratio of a standard deviation to the slope of the least squares regression line multiplied by a factor. In the simplified case, Eq. (2.19), the standard deviation refers to detector noise found in the baseline of a blank reference standard, whereas the standard deviation in the statistical case, Eq. (2.29), refers to the uncertainty in the least squares regression itself. This is an important conceptual difference in estimating IDLs. This difference should be understood and incorporated into all future methods. 

These equations generate the least squares regression line. Leptocurtical Distribution n A distribution that is sharper or more bunched in the middle than the normal distribution. It has a negative excess kurtosis. See Kurtosis. 

The most widely used method for fitting a straight line to integrated rate equations is by linear least-squares regression. These equations have only two variables, namely, a concentration or concentration ratio and a time, but we will develop a more general relationship for use later in the book. 

MLR is based on classical least squares regression. Since known samples of things like wheat cannot be prepared, some changes, demanded by statistics, must be made. In a Beer s law plot, common in calibration of UV and other solution-based tests, the equation for a straight line... 

Figure 1. Rate of Cs uptake from batch solutions at 25°C. Straight lines are least square regressions to Equation 1.

This equation of a straight line takes into account a slight difference of the base lines by the constant a. By applying least-squares regression the constants a and b are determined. The variance of A(R) indicates the degree of conformity between the two spectra. If sample and reference spectrum differ only slightly, the spectra should be compared blockwise, and the standard deviations of the different blocks should be used. This method is useful for spectra which show a sufficient number of sharp bands. It may fail if there are broad bands, in which case it is necessary to compare the second derivatives of the spectra. The algorithm is shown in Fig. 5.1-15. 

When a linear relationship is observed between two variables, the correlation is quantified by a method such as linear least-squares regression. This method determines the equation for the best straight line that fits the experimental data. 

The principle of the method will first be illustrated using the data that were shown in figure 4.1, with the fitted line for the model coefficients (equation 4.1) calculated by least squares regression ... 

In Figure 2.3 a comparison of regression lines is shown. Using the variables Fc203 and CaO from Table 2.2 the ordinary least squares (regressing both x ony andy on x), and the reduced major axis methods are used to fit straight lines to the data. The equation for each line is given. 

Three dUTerent regression lines drawn for the same data with their regression equations (data taken from Table 2.2). The regression lines are ordinary least squares regression of a- on jr (x on — slope and intercept calculated from Eqns [2,5] and [2.6] reduced major axis (RMA) — slope and intercept calculated from Eqns [2.7] and [2,5] ordinary least squares regression of y on X. (y on x) slope and intercept calculated from Eqns [2.5] and [2.6]. 

When the enhanced solvatochromic shift are next compared with solvent j(3i g values, it is seen that the relationship is linear and very nearly directly proportional (the broken line in Fig. 1 lb). The least squares regression equation for 14 data points (13 HBA solvents and one zero/zero point representing all non-HBA solvents) is... 

Again the data points coalesce to cluster around a single regression line when plotted against (tt -F d5), d = —0.174 (Fig. 23fe). The all data least-squares regression equation becomes... 

The intercept Ayis.a" pertains to the analyses of the VIS in the VIS-spiked calibration solutions and can not be determined since these solutions all contained the same quantity Qvis.a- Instead, it is assumed that this parameter is equal to Ayis determined from analysis of the series of solutions of only VIS (Equation [8.76b]). (Of course it is always preferable to ensure that intercepts of calibration curves are statistically zero, see Section 8.3.2.) The quantity [(5 . /a)/(5 vis-/vis)]/Qvis,a" is determined as the slope of the least-squares regression of the values of the left side of Equation [8.77] (i.e., the ratio of signals for analyte to VIS) against Q this regression line should have a zero intercept (to within the confidence limits determined as before) if the two intercepts were measured properly. This completes the calibration procedure. 

A technique called least squares regression is used to solve for the model equation using the peak height/area data and the known constituent quantities. This mathematical technique calculates the coefficients of a given equation such that the differences between the known spectral responses (peak areas or heights) and the predicted spectral responses are minimized. (The predicted spectral responses are calculated by reading the measurements off the calibration line at the known concentrations see Fig. 1.) As... 

We can choose different coefficients for B in Equation (5.40) provided they sum to 1. An approach which does this is known as the least squares filter. It gets its name from the least squares regression technique used to fit a line to a set of points plotted on an XY (scatter) chart. Its principle, as shown in Figure 5.19 is to fit a straight fine to the lastiVpoints. The end of this line is 7 . 

There is a reasonable agreement among these three estimates of the H diffusion coefficient in Mg in Fig. 8.25, so that it is possible to put a line through all these data, and extrapolate to 23 °C. This yields an estimate of the diffusion coefficient at ambient temperature to be 10 m /s-10 cm /s. This value of the H diffusion coefficient is sufficient to allow significant H transport ahead of a stress corrosion crack in Mg at ambient temperature. A least squares regression through the data [184—186] of Fig. 8.25 yielded the following equation for the diffusion coefficient of H in Mg,... 

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 ... 

Concentrations of terbacil and its Metabolites A, B and C are calculated from a calibration curve for each analyte run concurrently with each sample set. The equation of the line based on the peak height of the standard versus nanograms injected is generated by least-squares linear regression analysis performed using Microsoft Excel. 

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