Stochastic analysis correlation function

This statement is not self-evident and needs some comments. A role of concentration degrees of freedom in terms of the formally-kinetic description was discussed in Section 2.1.1. Stochastic approach adds here a set of equations for the correlation dynamics where the correlation functions are field-type values. Due to very complicated form of the complete set of these equations, the analytical analysis of the stationary point stability is hardly possible. In its turn, a numerical study of stability was carried out independently for the correlation dynamics with the fixed particle concentrations. 

Stochastic identification techniques, in principle, provide a more reliable method of determining the process transfer function. Most workers have used the Box and Jenkins [59] time-series analysis techniques to develop dynamic models. An introduction to these methods is given by Davies [60]. In stochastic identification, a low amplitude sequence (usually a pseudorandom binary sequence, PRBS) is used to perturb the setting of the manipulated variable. The sequence generally has an implementation period smaller than the process response time. By evaiuating the auto- and cross-correlations of the input series and the corresponding output data, a quantitative model can be constructed. The parameters of the model can be determined by using a least squares analysis on the input and output sequences. Because this identification technique can handle many more parameters than simple first-order plus dead-time models, the process and its related noise can be modeled more accurately. 

There are many different ways to treat mathematically uncertainly, but the most common approach used is the probability analysis. It consists in assuming that each uncertain parameter is treated as a random variable characterised by standard probability distribution. This means that structural problems must be solved by knowing the multi-dimensional Joint Probability Density Function of all involved parameters. Nevertheless, this approach may offer serious analytical and numerical difficulties. It must also be noticed that it presents some conceptual limitations the complete uncertainty parameters stochastic characterization presents a fundamental limitation related to the difficulty/impossibility of a complete statistical analysis. The approach cannot be considered economical or practical in many real situations, characterized by the absence of sufficient statistical data. In such cases, a commonly used simplification is assuming that all variables have independent normal or lognormal probability distributions, as an application of the limit central theorem which anyway does not overcome the previous problem. On the other hand the approach is quite usual in real situations where it is only possible to estimate the mean and variance of each uncertainty parameter it being not possible to have more information about their real probabilistic distribution. The case is treated assuming that all uncertainty parameters, collected in the vector d, are characterised by a nominal mean value iJ-dj and a correlation =. In this specific... 

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