Stochastic Nets

In the previous section we discussed how a Hopfield net can sometimes converge to a local minimum that docs not correspond to any of the desired stored patterns. The problem is that while the dynamics embodied by equation 10.7 steadily decreases the net s energy (equation 10.9), because of the general bumpiness of the energy landscape (see figure 10.5), whether or not such a steady decrease eventually lands the system at one of the desired minima depends entirely on where the system begins its descent, or on its initial state. There is certainly no general assurance that the system will evolve towards the desired minimum. 

T for the Hopfield net and replacing the deterministic threshold dynamics with a stochastic rule [hinton83]  

We should stress that the temperature T has nothing to do with the real temperature of either a brain or neural circuit. Its sole purpose is to act as a control parameter regulating the amount of noise in the stochastic system. 

The form of the stochastic transfer function p x) is shown in figure 10.7. Notice that the steepness of the function near a - 0 depends entirely on T. Notice also that this form approaches that of a simple threshold function as T — 0, so that the deterministic Hopfield net may be recovered by taking the zero temperature limit of the stochastic system. While there are a variety of different forms for p x) satisfying this desired limiting property, any of which could also have been chosen, this sigmoid function is convenient because it allows us to analyze the system with tools borrowed from statistical mechanics. 

The main diflierence between looking at stochastic nets rather than their deterministic counterparts is stochastic nets force us to shift our focus of attention since under a stochastic rule the same initial states generally evolve into diflierent final states, we are in the stochastic case not so much interested in the final state of the 

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Chapter 10 covers another important field with a great overlap with CA neural networks. Beginning with a short historical survey of what is really an independent field, chapter 10 discusses the Hopfield model, stochastic nets, Boltzman machines, and multi-layered perceptrons. 

Applying exactly the same reasoning to our stochastic net, but using equation 10.9 for the Hopfield energy in place of the Ising Hamiltonian, we obtain the analogous expression... 

Figure 10.11 shows a smooth sigmoidal threshold function that is often used in practice. It has the same form as the transition probability function used for stochastic nets ... 

Wlren the door is open, the optimal net flux into the store isgiven by equation (C2.14.7). It may be that the stochastically gated diffusion treated by Szabo et aJ [47], see also [48] is a good representation of typical biological storage reactions (C2.14.8). 

The Boltzman Machine generalizes the Hopfield model in two ways (1) like the simple stochastic variant discussed above, it t(>o substitutes a stochastic update rule for Hopfield s deterministic dynamics, and (2) it separates the neurons in the net into sets of visible and hidden units. Figure 10.8 shows a Boltzman Machine in which the visible neurons have been further subdividetl into input and output sets. 

Ecell (http //ecell.sourceforge.net/) offers stochastic and deterministic time course simulation (including stiff solvers). It only runs on Windows and offers a user-friendly GUI. 

It is not difficult to observe that, using this simple stochastic model of liquid flow inside the porosity, we obtain that the parameters of the model, such as the net flow velocity (w) and the dispersion coefficient (D), are determined by the porous structure. This last parameter is considered here through the value of Ax (length of one pore which is not in contact with nearby pores). 

Licht, T., Dohmen, L., Schmitz, P., Schmidt, L., Luczak, H. Person-centered simulation of product development processes using timed stochastic coloured petri nets. In Proceedings of the European Simulation and Modelling Conference, EUROSIS-ETI, Ghent, Belgium, pp. 188-195 (2004)... 

The correlation function, <-P2[am(0) ( )]>. provides a measure of the internal motions of particular residues in the protein.324 333 Figure 46 shows the results obtained for Trp-62 and Trp-63 from the stochastic boundary molecular dynamics simulations of lysozyme used to analyze the displacement and velocity autocorrelation functions. The net influence of the solvent for both Trp-62 and Trp-63 is to cause a slower decay in the anisotropy than occurs in vacuum. In vacuum, the anisotropy decays to a plateau value of 0.36 to 0.37 (relative to the initial value of 0.4) for both residues within a picosecond. In solution there is an initial rapid decay, corresponding to that found in vacuum, followed by a slower decay (without reaching a plateau value) that continues beyond the period (10 ps) over which the correlation function is ex-... 

As diseussed above, expected utility can be reconciled with the two-stage stochastic framework. For example, if one uses the nonlinear coordinate transformation of real value into utility value given by the utility function (Figure 12.3), one can modify the view of the risk curve, as shown in Figure 12.12. If such utility function can be constructed based more on quantitative relations to shareholder value, then one does not need to perform any risk management at all. One could speculate that it suffices to maximize utility value, but only if one has identified the ultimate objective function associated with the company s optimum financial path. It is worth noting that anything less, like the net present value, which can be considered a utility function too, will require the analysis of different curves before a final choice is made. 

University, Germany, in collaboration with SAP AG. It is the key component of SAP R/3 s modeling concepts for business engineering and customizing. It is based on the concepts of stochastic networks and Petri nets. Simple versions exclude conditions and messages and include only E(vent)/A(ction) illustrations. 

Kelling, C., Henz, J., and Hommel, G. (1995), Design of a Communication Scheme for a Distributed Controller Architecture Using Stochastic Petri Nets, in Proceedings of the 3rd Workship on Parallel and Distributed Real-Time Systems (Santa Barbara, CA, April 25), pp. 147-154. 

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