Independent variables justification

Justification of the choice of independent variables. All reasonable parameters must be validated by an appropriate statistical procedure (e.g., by stepwise regression analysis). The best equation is normally the one with the lowest standard deviation, all terms being significant (indicated by the 95% confidence intervals or by a sequential F test). Alternatively, the equation with the highest overall F value may be selected as the best model (nowadays crossvalidation and/or F-scrambling are recommended as validation tools). 

If a random experimental error arises as the sum of many contributions, the central limit theorem of statistics gives some justification for assuming that our experimental error will be governed by the Gaussian distribution. This theorem states that if a number of random variables (independent variables) x, X2,..., x are governed by some probability distributions with finite means and finite standard deviations, then a linear combination (weighted sum) of them... 

At one time, scientists selected variables for inclusion in a regression model based on some type of strong justification for including them. That justification would have been based on some a priori idea or hypothesis that the variables chosen were the critical factors in establishing the variability in the response variable. Although this approach may still be used by scientists, more often today researchers have no prior knowledge about the relative importance of the various independent variables. Consequently, many regression... 

In the early papers [4,8], the development of the CMD method was guided in part by the effective harmonic analysis and, in part, by physical reasoning. In Paper III, however, a mathematical justification of CMD was provided. In the latter analysis, it was shown that (1) CMD always yields a mathematically well-defined approximation to the quantum Kubo-transformed position or velocity correlation function, and (2) the equilibrium path centroid variable occupies an important role in the time correlation function because of the nature of the preaveraging procedure in CMD. Critical to the analysis of CMD and its justification was the phase-space centroid density formulation of Paper III, so that the momentum could be treated as an independent dynamical variable. The relationship between the centroid correlation function and the Kubo-transformed position correlation function was found to be unique if the centroid is taken as a dynamical variable. The analysis of Paper III will now be reviewed. For notational simplicity, the equations are restricted to a two-dimensional phase space, but they can readily be generalized. 

---