Linear-regression parameters, comparison

Table III. Comparison of Linear-Regression Parameters for the Equation n = au + b from Various Sea-Salt Aerosol Studies...

For comparison purposes, regression parameters were computed for the model defined by Equations 6, 7, 8, and 10 and the model obtained by replacing In (1/R) in those equations by R. The dependent variable (y) is particulate concentration because it is desired to predict particulate content from reflectance values. Data from Tables I and II were also fitted to exponential and power functions where the independent variable (x) was reflectance but the fits were found to be inferior to that of the linear relationship. 

First, the simplest model which includes the minimum number of compartments and model parameters must be defined. For this model, the parameters are estimated from a set of measurements obtained by non-linear regression or curve-fitting (Section 13.2.8.3).The purpose of this process is to find a set of model parameters which best fits the measurements (Section 13.2.8.2). If the goodness-of-fit is acceptable (Section 13.2.8.5), the model can be evaluated by comparison with other models (Section 13.2.8.6). 

Comparisons made below refer to kinetic data obtained for processes proceeding under similar conditions. All available values of (log A, E) within each group of related reactions were included in the linear regression analysis (Appendix II) and the compensation line was calculated using these formulas. Unless otherwise stated, the units of A are always molecules m-2 sec-1 at 1 Torr pressure of reactants and those of E are kJ mole-1. The compilation of Arrhenius parameters referred to identical reaction conditions is not always easy (or, indeed, possible in some instances) and it may be necessary to recalculate data from literature sources using an extrapolation. Not all details of the necessary corrections are recorded below, but such estimations were always minimized to preserve the objectivity of the conclusions reached. 

For evaluation of the PLS model and for comparison with multiple linear regression the independent parameters were varied in the calibration range and predictions were made. Tab. 8-15 illustrates the comparison of the predicted and the measured values. 

Figure 9. Comparison between the traditional BET linear regression of the first ten points when assuming k = 1 ( ) and the BET linear regression assuming k as an additional best-fit parameter (O). The values of the parameters are collected in Table 1 in the first and second rows for the homogeneous surface model.

Figure 12.8. Comparison of the experimental values [16] of the steric hindrance parameters c of 53 polymers with the calculated values obtained by a four-parameter linear regression.

The quantities Jj(t) are integrals over pure measured signals. The Zj are the parameters to be determined. Taking the eq. (5.23), a linear regression will yield Zj- This procedure has the following advantage in comparison to the evaluation of difference equations ... 

Problems, however, arise if the intervals between the knots are not narrow enough and the spline begins to oscillate (cf. Figure 3.13). Also, in comparison to polynomial filters, many more coefficients are to be estimated and stored, since in each interval, different coefficients apply. An additional disadvantage is valid for smoothing splines, where the parameter estimates are not expectation-true. The statistical properties of spline functions are, therefore, more difficult to describe than in the case of linear regression (cf. Section 6.1). ... 

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