Constant error structure

Figure 3.7. Mean residual analysis for the experimental data set. The pattens obtained suggest a homogeneous, or constant, error structure in the data.

Analysis of the rotational fine structure of IR bands yields the moments of inertia 7°, 7°, and 7 . From these, the molecular structure can be fitted. (It may be necessary to assign spectra of isotopically substituted species in order to have sufficient data for a structural determination.) Such structures are subject to the usual errors due to zero-point vibrations. Values of moments of inertia determined from IR work are less accurate than those obtained from microwave work. However, the pure-rotation spectra of many polyatomic molecules cannot be observed because the molecules have no permanent electric dipole moment in contrast, all polyatomic molecules have IR-active vibration-rotation bands, from which the rotational constants and structure can be determined. For example, the structure of the nonpolar molecule ethylene, CH2=CH2, was determined from IR study of the normal species and of CD2=CD2 to be8... 

A quarter of a century ago Behnken [224] as well as Tidwell and Mortimer [225] pointed out that the linearization transforms the error structure in the observed copolymer composition with the result that such errors after transformation have no longer zero mean and constant variances. It means that such transformed variables do not meet the requirements for the least-squares procedure. The only statistically accurate means of estimation of the reactivity ratios from the experimental data is based on the non-linear least-squares procedure. An effective computing program for this purpose has been published by Tidwell and Mortimer (TM) [225]. Their method is considered to be such a modification of the curve-fitting procedure where the sum of the squares of the difference between the observed and computed polymer compositions is minimized. 

If 0 = 0, sf is not dependent on the magnitude of the y values, and w = K for all data points. This is the case for an error that is constant throughout the data (homogeneous or constant error). Thus, if the error structure is homogeneous, weighting of the data is not required. A value... 

These residuals will be referred to as mean residuals. It is important to realize that the criterion used to judge whether a weighted regression analysis should be carried out is the error structure of the experimental data, not the error structure of the fit of the model to the data. The mean-residuals plot depicted in Fig. 3.7 suggests that the error stmcture of the data is homogeneous, or constant. This being the case, weighting is not necessary. A more quantitative analysis of the error structure of... 

A correct knowledge of the error structure is needed in order to have a correct summary of the statistical properties of the estimates. This is a difficult task. Measurement errors are usually independent, and often a known distribution, for example, Gaussian, is assumed. Many properties of least squares hold approximately for a wide class of distributions if weights are chosen optimally, that is, equal to the inverse of the variances of the measurement errors, or at least inversely proportional to them if variances are known up to a proportionality constant, that is, is equal or proportional to Zy, the N x N covariance matrix of the measurement error v. Under these circumstances, an asymptotically correct approximation of the covariance matrix of the estimation error 0 = 0 — 0 can be used to evaluate the precision of parameter estimates ... 

The theoretical equation of state for an ideal rubber in tension, Eq. (44) or (45), equates the tension r to the product of three factors RT, a structure factor (or re/Eo, the volume of the rubber being assumed constant), and a deformation factor a—l/a ) analogous to the bulk compression factor Eo/E for the gas. The equation of state for an ideal gas, which for the purpose of emphasizing the analogy may be written P = RT v/Vq) Vq/V), consists of three corresponding factors. Proportionality between r and T follows necessarily from the condition dE/dL)Ty=0 for an ideal rubber. Results already cited for real rubbers indicate this condition usually is fulfilled almost within experimental error. Hence the propriety of the temperature factor... 

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