Parameters, relationships among statistical

A key factor in modeling is parameter estimation. One usually needs to fit the established model to experimental data in order to estimate the parameters of the model both for simulation and control. However, a task so common in a classical system is quite difficult in a chaotic one. The sensitivity of the system s behavior to the initial conditions and the control parameters makes it very hard to assess the parameters using tools such as least squares fitting. However, efforts have been made to deal with this problem [38]. For nonlinear data analysis, a combination of statistical and mathematical tests on the data to discern inner relationships among the data points (determinism vs. randomness), periodicity, quasiperiodicity, and chaos are used. These tests are in fact nonparametric indices. They do not reveal functional relationships, but rather directly calculate process features from time-series records. For example, the calculation of the dimensionality of a time series, which results from the phase space reconstruction procedure, as well as the Lyapunov exponent are such nonparametric indices. Some others are also commonly used ... 

Equations 27—29 with two parameters seem to show reasonable correlation. Although Equation 28 would be unacceptable because of unusually large parameter coefficients, the correlations are completely equivalent statistically. As far as compounds used for the analyses are concerned, it is impossible to choose the two significant parameters. In this case, besides a significant correlation between parameters nr and E8, there are mutual relationships among three parameters. Each parameter is expressed as a linear combination of the other two and is not separated from others. 

This section gives some of the more elementary statistical parameters that may be used to charaeterize and analyze data and discover the underlying relationships among variables that may be hidden by the overlaid variation or noise. Although everything discussed in this section is available in standard texts, a review of the more elementary statistical principles is presented to address the basic problems of measurement quality. This is given... 

However, the mentioned above single models have some shortages unavoidably. To remedy the defects of single models, hybrid models are actively researched recently. One kind of hybrid models (Qi, 1999) combines part of first principle equations with ANN, in which ANN is used to determine parameters of the first principle models. Fuzzy logic approach (Qian, 1999) is used for representing imprecision and approximation of the relationship among process variables. It is successfully incorporated into conventional process simulators. Several efforts (Baffi, 1999) have been made to combine statistical analysis with non-linear regression, which are polynomial, spline function and ANN. 

The theorem of correspondii states can therefore be applied to the mixture as well as to pure compounds, when the average molecular parameters of the mixture are known. A simple graphical discussion of the excess properties is then possible (cf. 4). One can read the values of the excess functions directly from a diagram representing the reduced properties of pure compounds as functioJ3s of reduced temperature and pressure. This is the most remarkable feature of the theory, if one keeps in mind the complex relationship among the excess functions themselves and between excess functions and intermolecular forces. In the present treatment the behaviour of the excess functions is directly related to the behaviom of pure components which may be either obtained experimentally or deduced from some statistical model. 

Among the approaches proposed so far, we recall here single-parameter models [102-111, 115, 118-120, 122, 123, 125, 126, 129], and multi-parametric correlation equations (either based on the combination of two or more existing scales or on the use of specific parameters to account for distinct types of effects) [112, 113, 116, 117, 121, 124]. Additional popular models are the Abraham s scales of solute hydrogen-bond acidity and solute hydrogen-bond basicity [127, 128], and the Catalan et al. solvatochromic scales [130,132, 133]. Methods based on quantitative stmcture-property relationships (QSPR) with solvent descriptors derived from the molecular structure [131, 134], and on principal component analysis (PCA) [135, 136] have been also proposed. An exhaustive review concerning the quantification of the solvent polarity has been recently published [138-140], including a detailed list of solvent scales, interrelations between parameters and statistical approaches. 

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