Process, continuous deterministic discrete stochastic

A computational method was developed by Gillespie in the 1970s [381, 388] from premises that take explicit account of the fact that the time evolution of a spatially homogeneous process is a discrete, stochastic process instead of a continuous, deterministic process. This computational method, which is referred to as the stochastic simulation algorithm, offers an alternative to the Kolmogorov differential equations that is free of the difficulties mentioned above. The simulation algorithm is based on the reaction probability density function defined below. 

Aside from the continuity assumption and the discrete reality discussed above, deterministic models have been used to describe only those processes whose operation is fully understood. This implies a perfect understanding of all direct variables in the process and also, since every process is part of a larger universe, a complete comprehension of how all the other variables of the universe interact with the operation of the particular subprocess under study. Even if one were to find a real-world deterministic process, the number of interrelated variables and the number of unknown parameters are likely to be so large that the complete mathematical analysis would probably be so intractable that one might prefer to use a simpler stochastic representation. A small, simple stochastic model can often be substituted for a large, complex deterministic model since the need for the detailed causal mechanism of the latter is supplanted by the probabilistic variation of the former. In other words, one may deliberately introduce simplifications or errors in the equations to yield an analytically tractable stochastic model from which valid statistical inferences can be made, in principle, on the operation of the complex deterministic process. 

As discussed in Section 1.5, the characterization of observables as random variables is ubiquitous in descriptions of physical phenomena. This is not immediately obvious in view of the fact that the physical equations of motion are deterministic and this issue was discussed in Section 1.5.1. Random functions, ordered sequences of random variable, were discussed in Section 1.5.3. The focus of this chapter is a particular class of random functions, stochastic processes, for which the ordering parameter is time. Time is a continuous ordering parameter, however in many practical situations observations of the random function z(Z) are made at discrete time 0 < Zi < t2, , < tn < T. In this case the sequence z(iz) is a discrete sample of the stochastic process z(i). 

The biomedical signal sources can be broadly classified into continuous processes and discrete-time or point processes. Each of these types of signals could be deterministic (or predictable), stochastic (or random), fractal, or chaotic. The continuous processes are typically encountered in one of the following situations. 

A great amount of stochastic physics investigates the approximation of jump processes by diffusion processes, i.e. of the master equation by a Fokker-Planck equation, since the latter is easier to solve. The rationale behind this procedure is the fact that the usual deterministic (CCD) and stochastic (CDS) models differ from each other in two aspects. The CDS model offers a stochastic description with a discrete state space. In most applications, where the number of particles is large and may approach Avogadro s number, the discreteness should be of minor importance. Since the CCD model adopts a continuous state-space, it is quite natural to adopt CCS model as an approximation for fluctuations. 

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