Best matching unit

Once a dimensionality for the map and the type of local measure to be used have been chosen, training can start. A sample pattern is drawn at random from the database and the sample pattern and the weights vector at each unit are compared. As in a conventional SOM, the winning node or BMU is the unit whose weights vector is most similar to the sample pattern, as measured by the squared Euclidean distance between the two. 

After the winning unit has been identified, say uh, the local measure at that unit is updated. If the local measure is a signal counter, the signal counter at the BMU is incremented by 1  

If the local measure is the error, this is increased by adding to it the Euclidean distance between the sample pattern and the weights at the unit. 

The next step is to update the network weights. The weights at the winning unit are updated by an amount Awja, as in a standard SOM  

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. 

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A model of some multidimensional observations, possibly a vector consisting of features (variables), is associated with each unit. The map attempts to represent all available observations with optimal accuracy using a restricted set of models. At the same time the models become ordered on the grid so that similar models are close to each other and dissimilar models far from each other. Fitting of the model vectors is usually carried out by a sequential regression process, where t = 1, 2,... is the step index. For each sample x(t), the winner index c (best matching unit—BMU) is first identified by the condition... 

Step 1. Selecting the winner prototype. Given an input data vector, the competitive units compete each other to select the winner neuron comparing their prototypes with the input. This winner unit, also called Best Matching Unit (BMU) is selected in MSCL as the one that minimizes the product of a user-defined Magnitude Function and the distance of the unit prototypes to the input data vector. This differs from other usual competitive algorithms where BMU is determined only by distance. MSCL is implemented by a two-step competition global and local, as it is explained in next section. 

M unit weights are initialized with data inputs randomly selected from the dataset, and their initial value of its magnitude is equal to the magnitude function at these samples. We also initialize to ones the value of a counter rii of the number of times that each unit has been the best matching unit. 

A function calculated depending on the unit weights and/or the values of data samples belonging to its Voronoi region. An example of this function is the used for Homogeneous Color Quantization. In that case mf t) is the value of the mean quantization error of the best matching unit of x(t). 

In the on-line application, the feed data is 8-hour mean values achieved from the history database at the plant. The interface shows changes in bed stability by a five-days path (5x3 observations) of the best-matching unit (BMU) on the map. The BMU is the unit representing the prototype vector with shortest distance to the input vector. Also, the application outputs a plot of the quantization error (Euclidean distance between BMU and input vector) for the same period, which can be used as an indicator of model reliability. 

Competition in which a competition between all the I neurons for each input vector x is done, with the winning neuron being called the Best Matching Unit (BMU) for which the Euclidean distance x - Wy is minimized. The BMU index for the corresponding input vector X is denoted by i(x). 

An analysis of geometrical matching between bulk RujSis and nanocrystalline RuSi2 has been performed for three types of its lattice, namely for a-, (3-, and y-phase. The best matching has been found for a-RuSii. This is characterized by the common unit cell area of 1.86 nm and... 

A low temperature of approach for the network reduces utihties but raises heat-transfer area requirements. Research has shown that for most of the pubhshed problems, utility costs are normally more important than annualized capital costs. For this reason, AI is chosen eady in the network design as part of the first tier of the solution. The temperature of approach, AI, for the network is not necessarily the same as the minimum temperature of approach, AT that should be used for individual exchangers. This difference is significant for industrial problems in which multiple shells may be necessary to exchange the heat requited for a given match (5). The economic choice for AT depends on whether the process environment is heater- or refrigeration-dependent and on the shape of the composite curves, ie, whether approximately parallel or severely pinched. In cmde-oil units, the range of AI is usually 10—20°C. By definition, AT A AT. The best relative value of these temperature differences depends on the particular problem under study. 

The unit wins the competition to be the BMU only rarely and, when it does win, the match between its weights and the sample pattern is poor. Although the unit rarely is the BMU, the poor match when it is chosen means that its local error grows to a large value. If the match at this unit is the best for a particular pattern, even though the match is poor, it is evident that no unit in the network is capable of representing the pattern adequately. 

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