Neighborhood function

The size of the adjustment made to the node weights is determined by a neighborhood function. Using the simplest plausible function, the amount of adjustment could be chosen to fall off linearly with distance from the node (Figure 3.17). Beyond some cut-off distance from the winning node, no changes are made to the weights if this function is used. 

A linear neighborhood function x denotes the number of nodes to the right or left of the winning node. 

Back-to-back exponentials used as a neighborhood function. 

This simpler definition of a neighborhood means that there is no need to define a neighborhood function of the sort used in the SOM because all of the small number of neighbors in a GCS are treated identically. Because the neighborhood is of only limited size, updating of weights in that neighborhood is rapid. 

Fewer parameters are needed to define evolution of the map In particular, decay schedule parameters are not needed because the size and the shape of the neighborhood vary as the algorithm runs. Nor is it necessary to decide what form of neighborhood function to use, as all neighbors of the BMU are in an identical position. 

Fig. 10.7 Principle of self-organizing map (Kohonen map) with two typical neighborhood functions (Hierlemann et al., 1996). The information content increases from the bottom up during the self-assembly...

Here, hc x) i is the neighborhood function, a decreasing function of the distance between the /th and cth nodes on the map grid. This regression is usually reiterated over the available objects. 

The preservation of topology results from the introduction of neighborhood relationships of neurons in the learning algorithm. These relationships are described by a neighborhood function with the distance measure d as the independent variable. This function characterizes the distance to the winning neuron ... 

Figure 8.14 Neighborhood function in the form of a triangle (a) and a Mexican hat (b).

Rigorous mathematical proof of the SOM algorithm is very difficult in general. In the case of a discrete data set and a fixed neighborhood function, the error function of the system may be defined as... 

For turbulent flow of a fluid past a solid, it has long been known that, in the immediate neighborhood of the surface, there exists a relatively quiet zone of fluid, commonly called the Him. As one approaches the wall from the body of the flowing fluid, the flow tends to become less turbulent and develops into laminar flow immediately adjacent to the wall. The film consists of that portion of the flow which is essentially in laminar motion (the laminar sublayer) and through which heat is transferred by molecular conduction. The resistance of the laminar layer to heat flow will vaiy according to its thickness and can range from 95 percent of the total resistance for some fluids to about I percent for other fluids (liquid metals). The turbulent core and the buffer layer between the laminar sublayer and turbulent core each offer a resistance to beat transfer which is a function of the turbulence and the thermal properties of the flowing fluid. The relative temperature difference across each of the layers is dependent upon their resistance to heat flow. 

The values in Figures 2-11 and 2-12 are not entirely typical of all composite materials. For example, follow the hints in Exercise 2.6.7 to demonstrate that E can actually exceed both E., and E2 for some orthotropic laminae. Similarly, E, can be shown to be smaller than both E. and E2 (note that for boron-epoxy in Figure 2-12 E, is slightly smaller than E2 in the neighborhood of 6 = 60°). These results were summarized by Jones [2-6] as a simple theorem the extremum (largest and smallest) material properties do not necessarily occur in principal material coordinates. The moduli Gxy xy xyx exhibit similar peculiarities within the scope of Equation (2.97). Nothing should, therefore, be taken for granted with a new composite material its moduli as a function of 6 must be examined to truly understand its character. 

Product composition can be controlled to a considerable extent by the molar ratio of reactants alkylation tends to become more extensive as the molar ratio of carbonyl to amine increases. Product distribution is influenced also by the catalyst and by steric hindrance with the amount of higher alkylate formed being inversely proportional to the steric hindrance in the neighborhood of the function (60 2). Cyclic ketones tend to alkylate ammonia or amines to a further extent than do linear ketones of comparable carbon number 36). 

Totalistic Rules Totalistic (T) rules, 0tot, are functions of the sums of values of all sites in a particular neighborhood 0tot = 0tot(Zlj=, . Here we define the... 

Outer-Totalistic Rules Outer-totalistic (OT) rules, out-tot, are functions of both the value of a given site and the sum of values of all remaining sites in the neighborhood of that site. Thus, 0oui.-tot = 0out-tot(<7, - ai), with... 

Additive Rules Rules belonging to the special class of additive rules, add , are linear functions of neighborhood sites ... 

Defining a distance function d i,j) = miiipat/is of links (i,j path) and a d -size neighborhood of f as Afd i) = /c such that d i, k) < d, we wish to study the discrete time-evolution of S(f) in which the site variables undergo transitions of the general form... 

A trivial reversible CA consists of a collection of completely isolated systems each site contains only itself in its neighborhood. Since a7 = 1, the rule table and site value sets have the same cardinality. In particular, if the function 21,—> Z is invertible (i.e. if 4> serves merely to permute the elements of the set Zk) then the global CA system is itself reversible. More formally, writing < = tt H/ci where... 

Margolus [marg84] points out that second-order reversible CA may also be constructed by using operations other than the subtraction modulok we used in our example. The actual operation could in fact be a function of the neighbor s values at time f . In the most general case, the neighborhood at time C can be used to choose a permutation on the set of allowed site values. The permutation is then applied to the site value at time t-V to obtain its next state. 

CML / ([chate89a], [chate89b]. When / is replaced by another discrete function /, taking on one of a finite site of distinct values, the CML defined by / and equation 8.44 effectively becomes a fc-state CA defined on the same lattice and local neighborhood structure. Of course, we are entirely free to choose any / that we desire, provided that it preserves the critical dynamical features of the original function in particular, / must preserve the absorbing character of the laminar state. 

---