Construction of Information Granules: From Data and Perceptions to Information Granules
We focus on the development of fuzzy sets by presenting various ways of designing fuzzy sets and determining their membership functions.
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If Hansel cannot find information from outside, say because it is clouded and he cannot see how the pebble sequence continues, he must find it inside his brain. We speak in general of experience or intuit or creativity in these cases, to mean that this information comes from a sound synthesis of previous data, eventually processed with the millennia by the human kind and stored in the genome, possibly compressed in relatively recent time in the synapses of a single mind, or what else. This means that Hansel has some rules available to make decision exactly from the few pebbles he sees in its surrounding, where generally these rules lead to a spectrum of decisions and some criterion to differentiate between them. Probability too could represent a criterion, but the peculiarity of the criteria we will discuss in this part is that they associate instances within a set to a given decision, say the appropriateness of possible slopes of the pebble ground to a given direction of the next move, rather than a single instance to different decisions, thus the preference of the next move directions in respect to a specific ground slope. The fact is that at the end of the day we need the latter criterion to make our final decision, but dispose only of the former. Hence drawing in our mind in order to asses the allowed criterion – that we will denote as the membership function of the set of instances vaguely associated with a given decision A, hence to the fuzzy set A – we must endow it with features that prove to be mostly suitable to arrive to a concrete decision-making tool. Next chapter will be devoted to the task of identifying and discussing these features and characterizing families of fuzzy sets accordingly. In Chapter 5, we will discuss basic methods of adapting some of these families to the data we have available, a task that we are used to denote as membership function estimation. The process in which we start from data and arrive at decisions will be treated in Part V.
Degrees to many elements of a single attribute
in order to assign degrees to many attributes for a single element.
4 Construction of Information Granules: From Data and Perceptions to Information Granules
We focus on the development of fuzzy sets by presenting various ways of designing fuzzy sets and determining their membership functions. From a logical perspective, moving from Definition 1.4 we have no many other conceptual tools than the following elementary relationships:
Fuzziness is a non univocal category with a slim axiomatic basis,
Definition 4.1. For the fuzzy sets A and B in the universe of discourse X , the following relationships are defined: A ⊆ B if μA (x) ≤ μB (x), for each x A = B if μA (x) = μB (x), for each x A = B if μA (x) = 1 − μB (x), for each x
(inclusion) (identity)
(4.1) (4.2)
(complementation) (4.3)
and a few obvious corollaries such as the fact that if A = B then A = B. Therefore, the subject of elicitation and interpretation of fuzzy sets is of paramount relevance from the conceptual, algorithmic, and application-driven
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