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Let us consider a collection of generic fuzzy sets (linguistic terms) defined in [0, 1]. As usual, we require that Given a data set of experimental outcomes (arising, e.g., as a result of expert polling), they can be arranged in the form of (c+1)- tuples, namely where dk denotes a given element of the universe of discourse whose membership grades to some linguistic categories under discussion are equal to μk1, μk2,
, μkn, respectively. Our intent is to accommodate these data to the highest extent by adapting the context of This makes the generic membership functions adjusted to the current situation conveyed by the available data - thus the context in which the frame of cognition
6.6.1. The Optimization AlgorithmThe calibration of the universe of discourse is carried out in two main steps:
The first step concerns a specification of an element in the unit interval such that the given vector μ1, μ2,
, μn, (the first index pertaining to the data point has been suppressed) matches to the highest extent the vector of the membership values in where and while ||.|| is a certain normalized distance function computed between the corresponding membership values. The result of this processing phase is concisely summarized in the form of the pairs of the corresponding elements defined in [0, 1] and [a, b], respectively Each of these discrete associations, (xk, dk), is equipped with the resulting relevance factor (coefficient) fk determined as If fk ≈ 1 then the associated correspondence is regarded as highly essential. The second step of the optimization algorithm departs from the pairs of data summarized in the above form and constructs the nonlinear mapping To properly address the core issue of context adaptation, we impose several straightforward requirements on the above mapping such as:
In light of the above properties, Finally, once this nonlinear transformation has been constructed, we locate the original fuzzy sets of namely i = 1, 2,
c. When collected together these new fuzzy sets form the required frame of cognition 6.6.2. Neural network realization of the nonlinear mappingThe nonlinear mapping is realized through a neural network with its structure shown in Fig. 6.15.
The network is composed of n nodes situated in the hidden layer and a single node placed at the output layer. The neurons in the hidden layer implement a series of local receptive fields equipped with two parameter sigmoid nonlinearities. The connections of these elements are fixed and equal to 1. Formally speaking we obtain i = 1, 2, n where mi ∈ [0, 1] α1 > 0, are the modal values and spreads of the corresponding fields. The neuron forming the output layer is described as with Concisely, the network can be written down as a single input - single output mapping of the form The learning of the network is supervised and guided via a gradient-based optimization of a specified performance index. As the training method is standard to a high degree, the details are not discussed here. Moreover, the proposed method easily generalizes to a multidimensional case. As a numerical illustration of the neural calibration of linguistic terms, we consider a data set summarized as follows This family of data consists of the elements situated in a segment of real numbers [2, 18.1] that are assigned to five linguistic categories. The fuzzy sets of
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