# An Alternative Formulation of the Harrison Model
## 2 A Brief Introduction to the Harrison Model
### The Effect from Treating the \"a\" Parameter as a Random Variable  
By analogy to the Preisach formalism in which different \"hysterons\" are assigned different coercive field strengths, and thus it is natural to work with their distributions, we assume that it is possible to treat the \"_a\"_ parameter in (6) as a random variable and to monitor the effect of the varying number of hysterons and their variance on the shape of the outcome hysteresis loops. In this context, our analysis is like the one presented in [31].  
There is a direct correspondence between the values of parameter \"_a\"_ and coercive field strength \(h_{\text{crit}}\) for hysterons whose \(h=h(m)\) dependence is given with (6), namely,  
\[3\sqrt{6}\;h_{\text{crit}}=4a \tag{8}\]  
Therefore, producing a distribution of the \"_a\"_ parameter with the mean value \(a_{\text{mean}}=-0.75\sqrt{6}\) is equivalent to the introduction of hysteron distribution with coercive fields around unity.  
Figure 4 depicts the effect of increasing the number of hysterons. The mean value of coercive field strength is set to unity, the variance is set to 0.05. The seed of pseudo-random is in each case set to the same initial state (the appropriate MATLAB command is rng('default')); thus, the sequences of pseudo-random numbers drawn from the normal distribution have the same initial components. We consider the cases when there are ten hysterons; next, we increase their number to one hundred. It can be noticed that an increase in number of hysterons makes their distribution look more like the Gaussian distribution, which can be appreciated from a visual comparison of obtained histograms. On the other hand, the computations of the outcome characteristic become more cumbersome as the number of hysterons is excessively high. It is beyond the scope of the present paper to present an identification method which will enable one to resolve how many individual hysterons should be taken into account and how to assign weights to their outputs; however, we think that artificial neural networks (ANNs) might be useful for the purpose since the very principle of superposition of stop operators resembles to a large extent the architecture of classical feed-forward ANNs.  
Figure 3: The component loops (**a**) and corresponding outcome loops (**b**).  
An additional refinement to the presented approach might be based on the use of the Gaussian Mixture model for several hysteron distributions differing in mean values and variances or, e.g., some other type of kernel (Epanechnikov, triangular). In the first case mentioned, we think that the MATLAB toolbox Cupid [32] might be useful. Figure 5 depicts the probability density function (PDF) and the cumulative density function (CDF) for a mixture of two normal distributions with different mean values, variances and weights. The plot was obtained using the commands:  
myDist = Mixture(0.65, Normal(0.95, 0.05), 0.35, Normal(1.1, 0.1)) myDist.PlotDens  
Figure 4: The histograms obtained (**a**) for ten and (**b**) for one hundred hysterons with mean value of coercive field set to unity and variance set to 0.05.  
Figure 5: The PDF and the CDF for a mixture of two distributions.  
The basic statistics on the mixed distribution are obtained by typing: [myDist.Mean, myDist.SD], which, in our example, yields ans = 1.0025 0.1012.