S referred to as the function A : U x [0, 1] and defined as A ( x ) = sup Jx , x U x . GS-626510 Purity Type-2 fuzzy set A will be interval if A ( x, u) = 1 x U x , u Jx . Time series modeling demands to define interval fuzzy sets and their shape. Figure 1 shows the look in the sets.Figure 1. The shape of your upper and lower 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 is the maximum worth from the membership function of of type-2 interval fuzzy set A the Nimbolide Purity & Documentation element ai (for the upper and reduce membership functions, respectively), implies that ( A)i depends of height of triangle.Mathematics 2021, 9,5 ofAn operation of combining fuzzy sets of variety 2 is needed when operating having a rule base depending on 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 ))); 2 two 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 dilemma domain, will probably be described by two sets of type-2 fuzzy labels: ts = ( A, AC ),(6)exactly where A–a set of type-2 fuzzy sets describing the tendencies of your time series obtained in the analysis from the points from the time series, | A| = l – 1; AC –a set of type-2 fuzzy sets describing the trends of the time series obtained from the context of your problem domain of the time series, | AC | l – 1. The component A of model (six) is extracted from time series values by fuzzifying all numerical representations of the time series tendencies. By the representation of information granules within the form of fuzzy tendencies from the time series (1), the numerical values of the tendencies are fuzzified: At = Tendt ) = tst – tst-1 ), t 0. C of model (6) by professional or analytical methods is formed along with the element A describes probably the most general behavior in the time series. This component is necessary for solving difficulties: Justification on the option from the boundaries of the type-2 fuzzy set intervals when modeling a time series. Analysis and forecasting of a time series having a lack of information or after they are noisy. Therefore, the time series context, represented by the element AC of model (6), is determined by the following parameters: C Price of tendency alter At . Number of tendency modifications | AC |.4. Modeling Algorithm The modeling procedure consists of the following actions: 1. two. three. Verify the constraints of your time series: discreteness; length getting extra than two values. Calculate the tendencies Tendt from the time series by (3) at each and every moment t 0. Ascertain the universe for the fuzzy values of your time series tendencies: U = Ai , i are offered by N is the quantity of fuzzy sets inside the universe. Type-2 fuzzy sets A membership functions of a triangular type, and in the second level, they are intervals; see Figure 1. By an expert or analytical process, get the guidelines 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 really a pair ( Ai , AC ), Ai is k C is the consequent from the guidelines and i, k would be the indices the antecedent of th.