Least-squares fitting procedure

Constants A, and c are estimated by the least squares fitting procedure. 

FIGURE 11.10 Removal of outliers points to achieve curve fits, (a) The least squares fitting procedure cannot fit a sigmoidal curve to the data points due to the ordinate value at 20j.iM. Removal of this point allows an estimate of the curve, (b) The outlier point at 2jiM causes a capricious and obviously errant fit to the complete data set. Removal of this point indicates a clearer view of the relationship between concentration and response. 

The Monod kinetic parametos were evaluated by least squares fitting procedures, for tiie single and multiple substrate systems with/without mutual inhibition, and were indicated in Table 1 [6]. The value of indicates the linear decomposition rate. It is dear that the decomposition rate for prc iionic acid is significantly lower than those for acetic add and butyric acid. 

This expression was numerically integrated and regressed to the experimental data using a nonlinear least-squares fitting procedure. The resulting integrated equation is... 

Another noteworthy example is x-ray absorption fine structure (EXAFS). EXAFS data contain information on such parameters as coordination number, bond distances, and mean-square displacements for atoms that comprise the first few coordination spheres surrounding an absorbing element of interest. This information is extracted from the EXAFS oscillations, previously isolated from the background and atomic portion of the absorption, using nonlinear least-square fit procedures. It is important in such analyses to compare metrical parameters obtained from experiments on model or reference compounds to those for samples of unknown structure, in order to avoid ambiguity in the interpretation of results and to establish error limits. 

The equation of the straight line determined from a least squares fit procedure for an experiment in which the instrument readout, R, was plotted on the y-axis and the concentration, C, in parts per million was plotted on the x-axis is... 

In this chapter, we have advocated the use of a computer for plotting the standard curve and performing the least squares fit procedure. Indeed, computers play a central role in the analytical laboratory for acquiring and manipulating data generated by instruments and for information management. 

At 25°C the partial molar volume of urea [CO(NH2)2] solution in water is found by a nonlinear least-squares fitting procedure to be the following function of m2 up to 17-molal concentration (with V in cm mol ), with experimental data from Gucker et al. [10] and the form of the equation from Stokes ... 

Figure 22. Transient TPI spectra of different vibrational bands of the Na3 C state compared with the fit function f(t) = Noe /T. The listed lifetimes t of the different vibrational bands were obtained by a least-squares fit procedure [22].

Thus, the least squares fitting procedure can be successfully applied to a large variety of ionized atoms, including fairly highly ionized ones. 

V.2 LEAST-SQUARES FITTING PROCEDURE FOR THE ANTOINE EQUATION... 

When precise vapor pressure data are available, it is desirable to employ a least-squares fitting procedure to the Antoine equation. The formulation of this problem is available in detail in the literature and will be summarized here.6... 

If the analysis of a dynamic NMR spectrum is carried out by an iterative least-squares fitting method, the results are accompanied by estimates of the errors. These are proportional to the square root of the sum of the squares of the deviations of the theoretical spectrum from the experimental one, as well as to the sensitivity of the sum to changes in the value of the parameter considered within the region where the sum attains a minimum. These estimates constitute a measure of the effects, on the resulting parameter values, of random errors. They do not include any effects due to systematic errors such as those involved in the assumed values of certain parameters. Moreover, because of the nonlinearity of the least-squares fitting procedure employed, estimates of the errors have only an approximate statistical significance (Section IV.B.2 and reference 67). 

Most recent data sets are accompanied by either estimates of errors of the measurements or observed sample-to-sample fluctuations which were entered into the library, as modern CMB programs (e.g., 13) use uncertainties of the source compositions and in ambient samples in weighted least-squares fitting procedures. Many older data sets do not include estimates of analytical errors or fluctuations of particle composition, so we attached estimates based on the... 

The parameters A, B, D and TA are determined using a least squares fitting procedure. While A, B, and D are phenomenological parameters, TA equals the so-called time to arrival which is the time between contrast agent injection and appearance of the first contrast agent molecules within the ROI. 

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