Spreadsheet least squares

By constructing a plot of S(t,) versus Xvdt, we can visually identify distinct time periods during the culture where the specific uptake rate (qs) is "constant" and estimates of qs are to be determined. Thus, by using the linear least squares estimation capabilities of any spreadsheet calculation program, we can readily estimate the specific uptake rate over any user-specified time period. The estimated... 

Note in Table 5.10 that many of the integrals are common to different kinetic models. This is specific to this reaction where all the stoichiometric coefficients are unity and the initial reaction mixture was equimolar. In other words, the change in the number of moles is the same for all components. Rather than determine the integrals analytically, they could have been determined numerically. Analytical integrals are simply more convenient if they can be obtained, especially if the model is to be fitted in a spreadsheet, rather than purpose-written software. The least squares fit varies the reaction rate constants to minimize the objective function ... 

The viscosity is the shear stress at the bob, as given by Eq. (3-10), divided by the shear rate at the bob, as given by Eq. (3-12). The value of n in Eq. (3-12) is determined from the point slope of the log T versus log rpm plot at each data point. Such a plot is shown Fig. 3-3a. The line through the data is the best fit of all data points by linear least squares (this is easily found by using a spreadsheet) and has a slope of 0.77 (with r2 = 0.999). In general, if the... 

Enter data into a spreadsheet and obtain a graph of absorbance versus concentration of the standards. Obtain the least-squares line and its equation (Fig. 5.6). 

The response of many instruments is linear as a function of the measured variable, if variations due to experimental conditions or the instrument are taken into account. The objective is to determine the parameters of the linear equation that best represents the observations. The primary hypothesis in using the method of least squares is that one of the two variables should be without error while the second one is subject to random errors. This is the most frequently applied method. The coefficients a and b of the linear equation y = ax + b, as well as the standard deviation on a and on the estimation of y have been obtained in the past using a variety of similar equations. The choice of which formula to use depended on whether calculations were carried out manually, with calculator or using a spreadsheet. However, appropriate computer software is now widely used. 

Figure 4-13 Spreadsheet for linear least-squares analysis.

The method of least squares is used to determine the equation of the best straight line through experimental data points. Equations 4-16 to 4-18 and 4-20 to 4-22 provide the least-squares slope and intercept and their standard deviations. Equation 4-27 estimates the uncertainty in x from a measured value of y with a calibration curve. A spreadsheet greatly simplifies least-squares calculations. 

Figure 7-12 Spreadsheet for efficient experimental design uses Excel UNEST routine to fit the function y = mAxA + nv% + /Tfcxfc to experimental data by a least-squares procedure.

B. HI The spreadsheet lists molar absorptivities of three dyes and the absorbance of a mixture of the dyes at visible wavelengths. Use the least-squares procedure in Figure 19-3 to find the concentration of each dye in the mixture. 

An Excel spreadsheet illustrating the use of the Solver tool for nonlinear least-squares analysis of a fluorescent decay curve of a ruby crystal. The sum of the squares of residuals is calculated in cell C14 and is minimized in Solver by iterative variation of the parameters in cells CIO, Cll, and C12. 

Most of the exercises in Chapter II can be done using a spreadsheet program, and you are encouraged to do so. It would also be worthwhile to duplicate the examples in Chapter II, and in Chapter XXI on least-squares procedures, using spreadsheet functions. 

Use the data shown in Fig. 2 and Eqs. (4) to (11) to confirm the least-squares results from the spreadsheet linear regression of Fig. 2. 

The normal-equations algorithm described here is generally the fitting method of choice when (1) the errors in the observations conform to a normal distribution (see discussion of this distribution in Chapter II), and (2) the observational equations are linear in the adjustable parameters. As a matter of convenience, this algorithm is often used (especially in spreadsheet and other least-squares computer programs) when one or both of these conditions is not fulfilled. This is not always bad practice, but one should be aware of the hazards discussed below. 

Carry out the least-squares minimization of the quantity in Eq. (7) according to an appropriate algorithm (presumably normal equations if the observational equations are linear in the parameters to be determined otherwise some other such as Marquardf s ). The linear regression and Solver operations in spreadsheets are especially useful (see Chapter HI). Convergence should not be assumed in the nonlinear case until successive cycles produce no significant change in any of the parameters. 

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