S known as the function A : U x [0, 1] and defined as A ( x ) = sup Jx , x U x . Type-2 fuzzy set A might be interval if A ( x, u) = 1 x U x , u Jx . Time series Nimbolide Technical Information modeling demands to define interval fuzzy sets and their shape. Figure 1 shows the look with the sets.Figure 1. The shape with the upper and decrease membership functions.Triangular fuzzy sets are defined as follows:u u u l l l l Ai = ( AU , AiL ) = (( ai1 , ai2 , ai3 , h( AU )), ( ai1 , ai2 , ai3 , h( Ai ))). i i(5)u u u l l l where AU and AiL are triangular type-1 fuzzy sets, ai1 , ai2 , ai3 , ai1 , ai2 , ai3 are reference points i i , and h could be the maximum worth of your membership function of of type-2 interval fuzzy set A the element ai (for the upper and reduce membership functions, respectively), means that ( A)i depends of height of triangle.Mathematics 2021, 9,five ofAn operation of combining fuzzy sets of form two is essential when working with a rule base determined by the values of a time series. The combined operation is defined as follows: L L A1 A2 = ( AU , A1 ) ( AU , A2 ) 2u u u u u u = (( a11 a21 , a12 a22 , a13 a23 ; min(h1 ( AU ), h1 ( AU ))), min(h2 ( AU ), h2 ( AU ))); two 2 1 1 l l l l l l ( a11 a21 , a12 a22 , a13 a23 ; L L L L min(h1 ( A1 ), h1 ( A2 )), min(h2 ( A1 ), h2 ( A2 )));Proposition 1. A fuzzy time series model, reflecting the context on the difficulty domain, is going to be described by two sets of type-2 fuzzy labels: ts = ( A, AC ),(6)where A–a set of type-2 fuzzy sets describing the tendencies from the time series obtained in the analysis of the points in the time series, | A| = l – 1; AC –a set of type-2 fuzzy sets describing the trends in the time series obtained in the context in the trouble domain of the time series, | AC | l – 1. The GS-626510 Autophagy component A of model (6) is extracted from time series values by fuzzifying all numerical representations with the time series tendencies. By the representation of data granules within the kind of fuzzy tendencies on the time series (1), the numerical values of the tendencies are fuzzified: At = Tendt ) = tst – tst-1 ), t 0. C of model (6) by specialist or analytical solutions is formed as well as the element A describes probably the most general behavior on the time series. This element is essential for solving troubles: Justification of the option from the boundaries from the type-2 fuzzy set intervals when modeling a time series. Evaluation and forecasting of a time series having a lack of information or once they are noisy. As a result, the time series context, represented by the element AC of model (6), is determined by the following parameters: C Price of tendency change At . Number of tendency changes | AC |.4. Modeling Algorithm The modeling procedure consists of the following steps: 1. 2. 3. Check the constraints in the time series: discreteness; length getting much more than two values. Calculate the tendencies Tendt on the time series by (three) at every moment t 0. Figure out the universe for the fuzzy values of the time series tendencies: U = Ai , i are offered by N could be the quantity of fuzzy sets inside the universe. Type-2 fuzzy sets A membership functions of a triangular kind, and in the second level, they’re intervals; see Figure 1. By an professional or analytical technique, acquire the rules in the time series as a set of C C C C pairs of type-2 fuzzy sets: RulesC = Rr , r N, exactly where Rr is often a pair ( Ai , AC ), Ai is k C will be the consequent in the rules and i, k are the indices the antecedent of th.
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