Startseite Mathematik Higher-order circular intuitionistic fuzzy time series forecasting methodology: Application of stock change index
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Higher-order circular intuitionistic fuzzy time series forecasting methodology: Application of stock change index

  • Shahzaib Ashraf , Muhammad Sohail , Muhammad Shakir Chohan , Siriluk Paokanta EMAIL logo und Choonkil Park EMAIL logo
Veröffentlicht/Copyright: 10. Januar 2024
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Demonstratio Mathematica
Aus der Zeitschrift Demonstratio Mathematica Band 57 Heft 1

Abstract

This article presents a higher-order circular intuitionistic fuzzy time series forecasting method for predicting the stock change index, which is shown to be an improvement over traditional time series forecasting methods. The method is based on the principles of circular intuitionistic fuzzy set theory. It uses both positive and negative membership values and a circular radius to handle uncertainty and imprecision in the data. The circularity of the time series is also taken into consideration, leading to more accurate and robust forecasts. The higher-order forecasting capability of this method provides more comprehensive predictions compared to previous methods. One of the key challenges we face when using the amount featured as a case study in our article to project the future value of ratings is the influence of the stock market index. Through rigorous experiments and comparison with traditional time series forecasting methods, the results of the study demonstrate that the proposed higher-order circular intuitionistic fuzzy time series forecasting method is a superior approach for predicting the stock change index.

Keywords: fuzzy set; circular intuitionistic fuzzy sets; score function; higher-order time series forecasting
MSC 2010: 03B52; 90B50

1 Introduction

Decision-making can be defined as the process of selecting the best option from a collection of options based on a range of factors to achieve organizational goals [1]. The main research area in the complex decision-making problem of today, with several goals or circumstances, is multi-criteria decision making (MCDM). A number of MCDM techniques, such as printer selection [2], solar power plant [3], and others, have been developed to address decisions involving a range of conflicting criteria in ambiguous situations. These classic MCDM algorithms cannot handle ambiguous or inaccurate language judgments since precise numerical values are required. To better capture this ambiguity, intuitionistic fuzzy sets, neutrosophic sets, and spherical fuzzy sets (SFSs) [4] have been added to standard fuzzy sets in MCDM techniques. In this article, a proposed method is used to forecast the data, which will be helpful for MCDM in the future. Fuzzy set theory is one of the most impressive tools for addressing the problems of handling accurate and sufficient data for real-world decision-making due to the confusion of statistics. Zadeh created a fuzzy set to support her in decision-making when faced with challenges in daily life [5]. There is only one element with a membership degree that varies from 0 to 1 for each domain set component x in a fuzzy set. Fuzzy sets are subject to a number of limitations, such as the inability to display nonmembership. The intuitionistic fuzzy set (IFS), however, was created by Atanassov [6] to comprehend general information regarding membership degree. He defined the IFS using two lists, membership degree V ( p ) and nonmembership degree M ( p ) , with the requirement that 0 ≤ V ( p ) + M ( p ) ≤ 1 . The IFS has been studied by several researchers and effectively used in a variety of real-world situations [7]. An interval-valued intuitionistic fuzzy set was analyzed by Nayagam et al. [8] following IFS. They are both a fuzzy set and an IFS tweak. They are commonly employed in a variety of decision-making circumstances [9,10].

Because they deal with situations that comprise particular attributes in which the sum of their membership and nonmembership degrees is greater than one, according to several real-life choice concepts. IFS would not be able to produce an appropriate outcome in this situation. The Pythagorean fuzzy set (PyFS) was recommended by Yager [11] as a solution to these problems. In a PyFS, the degree of membership of an element is determined by the Pythagorean theorem, which relates the length of the sides of a right triangle to the length of its hypotenuse. This allows for a more nuanced representation of uncertainty than traditional fuzzy sets, which only allow for binary membership values of either 0 or 1. We now want to determine the neutral membership independently. Cuong and Kreinovich [12] expressed the concept of a picture fuzzy set (PFS). He used three items with the limitation of PFS membership degree V ( p ) , neutral membership degree K ( p ) , and nonmembership degree M ( p ) , and 0 ≤ V ( p ) + K ( p ) + M ( p ) ≤ 1 . Weighted averaging operations for picture fuzzy sets were invented by Garg [13]. In the past, some of the researchers have looked into and used the PFS in numerous beneficial decision-making fields [14]. In practice, we encounter a variety of problems that PFS cannot solve, such as when V ( p ) + K ( p ) + M ( p ) ≥ 1 . Under these circumstances, PFS is unable to provide a good outcome. Ashraf et al. [15] suggest the SFSs, a variant of the PFS, based on these factors. The use of SFSs can lead to improved results in decision-making problems, as they allow for a more precise and accurate representation of uncertainty. Some authors [16] used SFS on t-conorms and t-norms. With the healthcare diagnostics [17], COVID-19 [18], and many more, SFS has become one of the most valuable fuzzy sets for decision-making. Ullah et al. [19] used this methodology in T-spherical fuzzy sets (T-SFSs) to solve a number of decision-making problems. A T-SFSs is a type of fuzzy set that extends the concept of SFS to handle uncertainty in higher-dimensional spaces. The implications of circular intuitionistic fuzzy sets (C-IFSs) in this article are the reason for the presence of a circular radius, which is not present in IFS.

Making predictions about upcoming events or patterns based on historical data and knowledge is the process of predicting. It is used in a variety of fields, including finance, economics, marketing, and weather prediction, among others. To solve problems that changed over time, the time series data technique was used. Song and Chissom’s definition of time series in fuzzy time series [20] outlines the notion of fuzzy time series data or its techniques. Using fuzzy sets, Song and Chissom [21,22] projected the data. Following this, Kumar and Gangwar modify the Song and Chissom’s method [23]. To estimate data in fuzzy theory, many scholars employ diverse strategies. The majority of them made use of IFSs. Kumar and Gangwar [24] and Joshi and Kumar [25] used IFSs to include the amount of uncertainty in fuzzy logical links and develop only a few temporal forecasting models [26]. Jiang et al. [27] derived the formula used to predict wind speed. Prior to this, the majority of the researchers used his proposed approach using data from the University of Alabama [28,29]. They compared the mistakes of each result with one another after determining the conclusion to determine the best planned technique.

There are several different definitions of fuzzy forecasting that have been proposed by researchers. Some of these include:

  1. Linguistic Forecasting: This definition views fuzzy forecasting as a method that uses linguistic variables to express uncertainty in the forecast. Linguistic variables are variables that can take on values from a predefined set of linguistic terms, such as “high,” “medium,” and “low” [30].

  2. Probabilistic Forecasting: This definition views fuzzy forecasting as a method that uses fuzzy sets to express the probability of events. Fuzzy sets are sets that allow for partial membership, meaning that an element can belong to a set to a certain degree [31].

  3. Interval Forecasting: This definition views fuzzy forecasting as a method that uses fuzzy intervals to express the uncertainty in the forecast. Fuzzy intervals are intervals that allow for partial membership, meaning that an element can belong to an interval to a certain degree [32].

  4. Hybrid Forecasting: This definition views fuzzy forecasting as a method that combines different forecasting techniques, such as regression analysis and time series analysis, with fuzzy set theory [33].

Cakir et al. [34] expanded on this notion in the C-IFS by including circles with centers ( j A ( x ) , ℓ A ( x ) ) rather than points. A circle with a radius of 0 ≤ r ≤ 1 and a center coordinate of ( j A ( x ) , ℓ A ( x ) ) is used to represent each C-IFS component. In this C-IFS circle, the total number of membership values is limited to one. In other words, this theory explains the constituents of X inside the intuitionistic fuzzy using circles instead of dots and offers richer models for confused and contradictory information than that of the IFS notion. At r = 0 [35], the C-IFS is converted to an IFS. The C-IFS is distinguishable from the normal IFS when r > 0 . As a complement to the C-IFS notion, decision-makers can create grades as a form of circular membership. Later, additional C-IFS research was carried out, and it was used to address the MCDM concerns [36,37]. Figure 1 shows the geometrical representation of C-IFSs.

Figure 1 Graphical representation of C-IFSs.
Figure 1

Graphical representation of C-IFSs.

Scientists used to make choices by rating and evaluating data. However, they are insufficient to estimate future values. Chen [38] began by using time series analysis to forecast enrollments. Kumar and Gangwar [39] proposed attempting to cope with forecasting by employing induced IFSs. Abhishekh et al. [40] employed this strategy on higher-order IFSs. Furthermore, if we want to check the radius of a circle in IFS, we are unable to find it. As a result, authors are thrilled to be able to meet this need. As a result, we must employ C-IFS to deal with this sort of issue. This is a transition to all predicting algorithms capable of handling any form of membership, and nonmembership includes circular radius. They are useful when we need to calculate the radius of IFS. The question arises: Why do we calculate the radius of any set? The answer is that after finding the radius, we know to check where the values of oversetting lie in this radius, which is helpful in observing our results. Hence, we implicate this technique in this article to solve any type of data for future prediction. IFS cannot handle this type of thing. We worked on filling this with circular intuitionistic fuzzy time series (C-IFTSs), which are useful for time series forecasting, particularly when the total of membership and nonmembership is less than or equal to one with a circular radius. We offer a forecasting strategy for higher-order C-IFTSs that has lower error rates than earlier studies in this work. As part of my proposed strategy, we used the example of the stock change index. The motivation for researching stock change indices is likely to be to better understand the dynamics of financial markets and to provide investors with useful information for making informed investment decisions. Stock change indices track the performance of a group of stocks and can serve as a barometer for the overall health of a given market or economy. By analyzing changes in these indices over time, researchers can gain insights into market trends, investor sentiment, and other factors that drive stock prices. In addition, the results of such research can inform investment strategies and help investors make more informed decisions about buying, holding, or selling stocks. This idea may be applied to a range of potential future predictions.

The rest of this article is organized as follows:

  1. C-IFS and fuzzy sets are stated as being utilized to connect the other portions of the article.

  2. Following that, we gave some definitions for implementing my proposed technique, as well as a circular intuitionistic membership, nonmembership, and radius number for determining the score value.

  3. Time-variant and time-invariant C-IFTSs concepts were provided.

  4. Following that, we showed a flowchart outlining how we would anticipate data using our suggested technique, as well as specific descriptions of the proposed method.

  5. To evaluate the outcomes, we applied this approach to the stock change index and computed the findings on tables.

  6. After that, we expanded on this work to a higher-order forecasting.

  7. Finally, we presented the article’s conclusion.

2 Preliminaries

This section provides concise definitions of fuzzy sets, circular fuzzy sets, and time series analysis. They are important for linking to the following portion of this article.

Definition 2.1

The fuzzy set notion introduced by Zadeh is stated as follows: Let ξ be a set. Fuzzy set ξ in Ψ be characterized:

ξ = { ⟨ o , μ ξ ( o ) ⟩ ∣ ∀ o ∈ Ψ } ,

where μ q ( o ) is a membership function of fuzzy set ξ , μ q ( o ) : Ψ → [ 0 , 1 ] , and μ ξ ( o ) gives the membership function degree o in ξ .

Definition 2.2

[41] Suppose a nonempty set Ψ . An intuitionistic fuzzy set ξ in Ψ is an attribute having the form ξ = { ⟨ o , μ ξ ( o ) , ν ξ ( o ) ⟩ ; o ∈ Ψ } , where the function, μ ξ ( o ) , ν ξ ( o ) : → [ 0 , 1 ] , respectively, define its membership and nonmembership degree and for every aspect o ∈ Ψ , 0 ≤ μ ξ ( o ) + ν ξ ( o ) ≤ 1 .

Definition 2.3

[42] Let Ψ be a universal set. A C-IFS ξ in Ψ is an attribute having the form:

ξ = { ⟨ o , μ ξ ( o ) , ν ξ ( o ) ; r ⟩ ∣ o ∈ Ψ } ,

where

(1) 0 ≤ μ ξ ( o ) + ν ξ ( o ) ≤ 1 ,

and r ∈ [ 0 , 1 ] is the radius of a circle between each component o ∈ Ψ , which is called C-IFS, and the functions μ ξ : Ψ → [ 0 , 1 ] and ν ξ : Ψ → [ 0 , 1 ] represent the degree of membership and nonmembership, respectively, of the element o ∈ Ψ .

The level of uncertainty is calculated as follows:

(2) π ξ ( o ) = 1 − μ ξ ( o ) − ν ξ ( o ) .

Definition 2.4

[43] The following is a definition of the operations that C-IFS entails. For every Å , E ¨ ∈ C-IFS ( Ψ ) ,

  1. Å ⊆ E ¨ iff o ∈ Ψ , ( μ Å ( o ) ≤ μ E ¨ ( o ) and ν Å ( o ) ≥ ν E ¨ ( o ) );

  2. Å = E ¨ iff Å ⊆ E ¨ and E ¨ ⊆ Å ;

  3. Å c = { ( o , ν Å ( o ) , μ Å ( o ) ) } ;

    (3) d ( Å , E ¨ ) = 1 2 k ∑ j = 1 k ( μ Å ( o j ) − μ E ¨ ( o j ) ) 2 + ( ν Å ( o j ) − ν E ¨ ( o j ) ) 2 + ( π Å ( o j ) − π E ¨ ( o j ) ) 2 ,

    (4) d ( Å , E ¨ ) = 1 2 r Å − r E ¨ 2 + 1 2 k ∑ j = 1 k ( μ Å ( o j ) − μ E ¨ ( o j ) ) 2 + ( ν Å ( o j ) − ν E ¨ ( o j ) ) 2 + ( π Å ( o j ) − π E ¨ ( o j ) ) 2 .

d ( Å , E ¨ ) is the standardized shortest distance between Å and E ¨ .

Definition 2.5

[44] Let Å = ( ( μ Å ( o ) , ν Å ( o ) ) ; r Å ) and E ¨ = ( ( μ E ¨ ( o ) , ν E ¨ ( o ) ) ; r E ¨ ) be two C-IFS numbers that loop back on themselves. Since they provide the least and greatest volatility, respectively, the operations are based on the lowest and largest radii. A smaller radius indicates less ambiguity in C-IF pairings, while a bigger radius indicates more vagueness. The way they operate is as follows:

  1. Å ∩ min E ¨ = { o , min ( μ Å ( o ) , ν E ¨ ( o ) ) , max ( μ Å ( o ) , ν E ¨ ( o ) ) ; min ( r Å , r E ¨ ) ∣ o ∈ Ψ }

  2. Å ∩ max E ¨ = { o , min ( μ Å ( o ) , ν E ¨ ( o ) ) , max ( μ Å ( o ) , ν E ¨ ( o ) ) ; max ( r Å , r E ¨ ) ∣ o ∈ Ψ }

  3. Å ∪ min E ¨ = { o , max ( μ Å ( o ) , ν E ¨ ( o ) ) , min ( μ Å ( o ) , ν E ¨ ( o ) ) ; min ( r Å , r E ¨ ) ∣ o ∈ Ψ }

  4. Å ∪ min E ¨ = { o , max ( μ Å ( o ) , ν E ¨ ( o ) ) , min ( μ Å ( o ) , ν E ¨ ( o ) ) ; max ( r Å , r E ¨ ) ∣ o ∈ Ψ }

  5. Å ⊕ min E ¨ = { o , μ Å ( o ) + μ E ¨ ( o ) − μ Å ( o ) × μ E ¨ ( o ) , ν Å ( o ) × ν E ¨ ( o ) ; min ( r Å , r E ¨ ) ∣ o ∈ Ψ }

  6. Å ⊕ max E ¨ = { a , μ Å ( o ) + μ E ¨ ( a ) − μ Å ( o ) × μ E ¨ ( o ) , ν Å ( o ) × ν E ¨ ( o ) ; max ( r Å , r E ¨ ) ∣ o ∈ Ψ } .

Theorem 2.6

For any Å r 1 , E ¨ r 2 ∈ C-IFS ( Ψ ) , where r 1 , r 2 ∈ [ 0 , 2 ] , the expressions (4) are defined metrics (distance).

Proof

We must demonstrate that the formulas mentioned in expression (4) abide by the property of metric [45]. As previously demonstrated, the terms indicated in (3) are well-defined distances in IFS ( Ψ ) .

Because it is clear from the definition of C-IFSs, Å r 1 = E ¨ r 2 in C-IFS ( Ψ ) iff Å = E ¨ in IFS ( Ψ ) and r 1 = r 2 . But Å = E ¨ in IFS ( Ψ ) iff D ( Å , E ¨ ) = 0 . The sum of two nonnegative numbers is 0, and if both numbers are equal to 0, the first axiom for a distance is proven.

Because D is symmetric, the validity of the second axiom is clear. To demonstrate the third axiom’s validity, let us take a third C-IFS I ¨ r 3 and show that the triangle property

(5) D ( Å r 1 , I ¨ r 3 ) ≤ D ( Å r 1 , E ¨ r 2 ) + D ( E ¨ r 2 , I ¨ r 3 )

holds.

We know that distance without radius.

(6) D ( Å , I ¨ ) ≤ D ( Å , E ¨ ) + D ( E ¨ , I ¨ )

holds for the IFSs Å , E ¨ and I ¨ .

From well-known inequality ∣ x ∣ + ∣ y ∣ ≥ ∣ x + y ∣ for three real numbers, it follows that

(7) ∣ r 3 − r 1 ∣ ≤ ∣ r 2 − r 1 ∣ + ∣ r 3 − r 2 ∣

for all choices of , r 1 , r 2 , r 3 ∈ [ 0 , 2 ] . As a result, by adding both sides of the final two inequality expressions (6) and (7), the validity of (5), i.e., the third distance axiom, holds true.□

Definition 2.7

If ϑ ( e ) ( e = 0 , 1 , 2 , … , ) is a subset of L and the universe of discourse on which the circular intuitionistic fuzzy sets f k ( e ) = ⟨ μ ξ ( o ) , ν ξ ( o ) ; r ⟩ ( k = 1 , 2 , … , ) are defined, then F ( e ) = f 1 ( o ) , f 2 ( o ) … is a collection of f k ( e ) in order to build the C-IFTSs on ϑ ( e ) ( e = 0 , 1 , 2 , … ) .

Definition 2.8

If circular intuitionistic logical relationship exist L ( e − 1 , e ) , then V ( e ) = V ( e − 1 ) × L ( e − 1 , e ) and V ( e ) is caused by V ( e − 1 ) a connection between V ( e ) and V ( e − 1 ) is indicated by V ( e − 1 ) → V ( e ) .

Definition 2.9

Suppose V ( e ) is caused V ( e − 1 ) and symbolize by V ( e − 1 ) → V ( e ) sequentially, a circular intuitionistic relationship exists, which is between V ( e ) and V ( e − 1 ) , which is represented as V ( e ) = V ( e − 1 ) × L ( e − 1 , e ) , since L is a first-order model of V ( e ) . Then V ( e ) is a time-invariant circular intuitionistic time series if L ( e − 1 , e ) is independent of time e , L ( e , e − 1 ) = L ( e − 1 , e − 2 ) for all e . Likewise, V ( e ) is called a time-variant circular intuitionistic time series.

Definition 2.10

Suppose V ( e − 1 ) = G a and ( e ) = G b , a circular intuitionistic logical relationship is described as G a → G b , where G a , G b are the current and coming state of circular intuitionistic logical relation (C-ILRs). Since V ( e ) is happened by more than one circular intuitionistic fuzzy set V ( e − n ) , V ( e − n + 1 ) , … V ( e − 1 ) , then the circular intuitionistic fuzzy set is represented by G a 1 , G a 2 , … , G a n → G b , where V ( e − n ) = G a 1 , V ( e − n + 1 ) = G a 2 . This kind of relationship is called a higher-order circular intuitionistic time series.

3 An algorithm of handling circular intuitionistic time series forecasting

There are three parts to our algorithm (A, B, and C). To cope with situations of this nature in C-IFTSs, we created a strategy. The first portion establishes the C-ILR and its groups; the second section employs the circular intuitionistic forecasting technique to determine the forecasted value of our issue; and the last section informs us of the approach’s flaws. Figure 2 shows the flow chart of the proposed technique.

Figure 2 A flowchart that shows how the proposed method works.
Figure 2

A flowchart that shows how the proposed method works.

A. First-order C-IFTSs proposed method of forecasting The following will guide you through all the procedures necessary to establish a C-ILR and its groups using the score formula.

Step I: The universe of discourse is determined by time series data to the stated range Ψ . Ψ = [ A min − A 1 , A max − A 2 ] , where A min and A max are the smallest and largest data points in the time series data, respectively, and A 1 and A 2 are any reasonable positive values that can manage the data time series in its whole.

Step II: A subset of Ψ divides the discourse universe into intervals of similar length.

Step III: Compute the n th circular intuitionistic fuzzy membership and nonmembership value ρ v according to the intervals that were erected in Step II.

(8) μ ( ϱ ; [ ξ , o , θ ] ) = ( ϱ − ξ ) ( o − ξ ) − ε , if ξ < ϱ ≤ o ( θ − ϱ ) ( θ − o ) − ε , if o < ϱ ≤ θ 0 , otherwise

(9) ν ( ϱ ; [ ξ , o , θ ] ) = 1 − μ ( ϱ ; [ ξ , o , θ ] ) .

Step IV: Calculate the radius of a circular intuitionistic fuzzy set by using equations (10) and (11).

In an IFS N i , let intuitionistic fuzzy pairs have the form { ⟨ c i , 1 , d i , 1 ⟩ ⟨ c i , 2 , d i , 2 ⟩ , … } , where i is the number of IFS N i each including λ i . First, the arithmetic average of the intuitionistic fuzzy pairs is calculated as follows:

(10) ⟨ μ ( N i ) , ν ( N i ) ⟩ = Σ j = 1 λ i c i , j λ i , Σ j = 1 λ i d i , j λ i ,

where λ i is the number of intuitionistic fuzzy pairs N i .

The radius of the ⟨ μ ( N i ) , ν ( N i ) ⟩ is the highest of the Euclidian distances.

(11) r i = max 1 ≤ j ≤ λ i ( μ ( N i ) − c i , j ) 2 + ( ν ( N i ) − d i , j ) 2 .

Step V: To calculate the score degree, utilize the following equation, and then choose the score degree that has the highest possible value [46]:

(12) ζ ( s ) = 1 3 ( μ ( s ) − ν ( s ) + 2 r ( 2 p − 1 ) ) where ζ ( s ) ∈ [ − 1 , 1 ] ,

p is any value between 0 and 1.

Step VI: Construct the circular intuitionistic fuzzy logical relationships (C-IFLRs): If ρ a is the C-IFS given of year y and ρ b is the C-IFS coming year of y + 1 , then the C-IFLRs are represented by ρ a → ρ b . Furthermore, ρ a is called the current state, and ρ b is the coming next state.

Step VII: Depending on the C-IFLRs, create circular intuitionistic fuzzy logical relationship groups (C-IFLRGs).

B. Measure the forecasted value of circular intuitionistic fuzzy sets

The following is the technique that is used to determine the predicted values:

If circular intuitionistic value of data ℘ a is not formed by any other circular intuitionistic values, then look forward to the C-IFLRG of the same value. If you are unable to determine the value that relies upon ℘ a , then the circular intuitionistic value will stay the same (null). If the circular intuitionistic value of data ℘ a is generated by ℘ b ( ℘ b → ℘ a ) , then see C-IFLRG of ℘ b .

  1. If C-IFLRGs of ℘ b is vacant ( ℘ b → ℘ b ), then the value of forecasted is the center one of ℘ b .

  2. If C-IFLRGs of ℘ b is one-to-one ( ℘ b → ℘ a ), then the value of forecasted is the center one value of ℘ a .

  3. If C-IFLRGs of ℘ b is not one-to-one ( ℘ b → ℘ a 1 , ℘ a 2 , … , ℘ a n ), then value of forecasted is the average of center one value of ℘ a 1 , ℘ a 2 , … , ℘ a n .

C. Measurement of error by using root mean square error (RMSE) and average forecasting error (AFE)

Techniques such as RMSE and AFE are utilized rather frequently to evaluate the precision of time series forecasting. The following definitions are applicable to all types of error measures.

RMSE = ∑ i = 1 n ( O i − F i ) 2 τ ,

Forecasting percentage error (fpe) = ∣ F i − O i ∣ O i × 100 ,

AFE = ∑ ( fpe ) τ .

In this case, the predicted and observed data of the time series are denoted by F i and O i , respectively, and τ is the number of observations contained in the time series. When the RMSE or AFE is lower, it indicates that the forecasting method is becoming more accurate.

4 Case study

The values of specific shares are used to derive a stock market index, which is a measurement of the worth of a portion of the stock market. This value is computed based on the index. It is a technique that investors employ to provide a description of the market and to compare the returns generated by various investments. An index provides a fast assessment of a market’s condition. Active funds are a reduced method of investing that offers greater returns than most fund managers and aids investors in more consistently achieving their objectives. In this article, we picked the data from the stock change index (Taiwan).

Forecasting the change in the Taiwan stock exchange index can bring several benefits for investors and traders. Some of these benefits include:

  1. Improved decision-making: By having an accurate forecast, investors and traders can make informed decisions on when to buy or sell their investments, which can lead to better returns.

  2. Risk management: Accurate forecasting can help investors and traders manage their risk by allowing them to make informed decisions on when to enter or exit the market.

  3. Increased confidence: With a better understanding of the future progress of the stock change index, investors and traders can have increased confidence in their investment decisions, which can help them make more informed and profitable trades.

  4. Improved portfolio diversification: By forecasting the stock change index, investors and traders can better understand the potential impact of their investments on their overall portfolio, which can help them diversify their investments for better risk management.

4.1 Implementation of the proposed method of stock change index

It is important to note that stock market forecasts are not always accurate and that past performance does not guarantee future results. Therefore, investors and traders should always use a combination of forecasting tools and fundamental analysis when making investment decisions.

We apply the method using data from the stock index in 2001 and explain the findings step-by-step to make the process of understanding and verifying the suggested model as simple as possible. The following is a list of the steps:

Step I : The universe of discourse Ψ for the stock indexes for the year 2001 is defined as follows:

Universe of discourse Ψ = [ 3,920 , 5,600 ] . By taking two proper positive numbers, A 1 = 9.69 and A 2 = 48.76 . A min and A max are taken from the Table 1.

Table 1

Fuzzy stock change index

Date True value C-Intuitionistic value Date True value C-Intuitionistic value
01-11-2001 3929.69 ℘ 1 03-12-2001 4646.61 ℘ 6
02-11-2001 3998.48 ℘ 1 04-12-2001 4766.43 ℘ 7
05-11-2001 4080.51 ℘ 1 05-12-2001 4924.56 ℘ 8
06-11-2001 4082.92 ℘ 1 06-12-2001 5208.86 ℘ 11
07-11-2001 4158.15 ℘ 2 07-12-2001 5333.93 ℘ 12
08-11-2001 4135.03 ℘ 2 10-12-2001 5321.28 ℘ 12
09-11-2001 4123.78 ℘ 2 11-12-2001 5273.97 ℘ 11
12-11-2001 4172.63 ℘ 2 12-12-2001 5539.31 ℘ 13
13-11-2001 4136.54 ℘ 2 13-12-2001 5407.54 ℘ 12
14-11-2001 4277.70 ℘ 3 14-12-2001 5486.73 ℘ 13
15-11-2001 4403.59 ℘ 4 17-12-2001 5456.15 ℘ 13
16-11-2001 4446.62 ℘ 4 18-12-2001 5329.19 ℘ 12
19-11-2001 4548.63 ℘ 5 19-12-2001 5221.96 ℘ 11
20-11-2001 4455.80 ℘ 4 20-12-2001 5309.10 ℘ 12
21-11-2001 4533.37 ℘ 5 21-12-2001 5109.24 ℘ 10
22-11-2001 4450.02 ℘ 4 24-12-2001 5164.73 ℘ 10
23-11-2001 4519.08 ℘ 5 25-12-2001 5372.81 ℘ 12
26-11-2001 4608.32 ℘ 6 26-12-2001 5392.43 ℘ 12
27-11-2001 4580.33 ℘ 6 27-12-2001 5332.98 ℘ 12
28-11-2001 4447.58 ℘ 4 28-12-2001 5398.28 ℘ 12
29-11-2001 4465.83 ℘ 5 31-12-2001 5551.24 ℘ 14
30-11-2001 4441.12 ℘ 4

Step II: The universe of discourse is divided into 14 intervals by Ψ . ℏ v = [ 3,920 + ( v − 1 ) p , 3,920 + v p ] , where v = 1 , 2 , 3 , … , 14 and p = 120 .

Step III: Following 14 C-IFTS, ℘ v ( v = 1 , 2 , 3 , … , 12 ) according to the interval ℏ v , which are described in the universe of the discourse Ξ :

℘ v = [ 3,920 + ( v − 1 ) p , 3,920 + v p , 3,920 + ( i + 1 ) p ] for v = 1 , 2 , 3 , … , 13 , where p = 120 .

℘ v = [ 3,920 + ( v − 1 ) p , 3,920 + v p , 3,920 + i p ] for v = 14 where, p = 120 .

By using equations (8) and (9), the grades of membership and nonmembership in circular intuitionistic fuzzy sets are calculated as follows: Suppose ε = 0.001 .

℘ 1 = { ( 3929.69 , 0.08 , 0.92 ) , ( 3998.48 , 0.65 , 0.35 ) , ( 4080.51 , 0.66 , 0.34 ) , ( 4082.92 , 0.64 , 0.36 ) , ( 4158.15 , 0.01 , 0.99 ) , ( 4135.03 , 0.21 , 0.79 ) , ( 4123.78 , 0.30 , 0.70 ) , ( 4136.54 , 0.19 , 0.81 ) }

℘ 2 = { ( 4080.51 , 0.34 , 0.66 ) , ( 4082.92 , 0.36 , 0.64 ) , ( 4158.15 , 0.97 , 0.03 ) , ( 4135.03 , 0.79 , 0.21 ) , ( 4123.78 , 0.70 , 0.30 ) , ( 4172.63 , 0.89 , 0.11 ) , ( 4136.54 , 0.80 , 0.20 ) , ( 4277.70 , 0.02 , 0.98 ) }

℘ 3 = { ( 4172.63 , 0.10 , 0.90 ) , ( 4277.70 , 0.98 , 0.02 ) }

℘ 4 = { ( 4403.59 , 0.97 , 0.03 ) , ( 4446.62 , 0.61 , 0.39 ) , ( 4455.80 , 0.53 , 0.47 ) , ( 4450.02 , 0.58 , 0.42 ) , ( 4159.08 , 0.01 , 0.99 ) , ( 4447.58 , 0.60 , 0.40 ) , ( 4465.83 , 0.45 , 0.55 ) , ( 4441.12 , 0.66 , 0.34 ) }

℘ 5 = { ( 4403.59 , 0.03 , 0.97 ) , ( 4446.62 , 0.39 , 0.61 ) , ( 4548.63 , 0.76 , 0.53 ) , ( 4455.80 , 0.46 , 0.54 ) , ( 4533.37 , 0.89 , 0.11 ) , ( 4450.02 , 0.42 , 0.58 ) , ( 4519.08 , 0.99 , 0.01 ) , ( 4608.32 , 0.26 , 0.74 ) , ( 4580.33 , 0.50 , 0.50 ) , ( 4447.58 , 0.40 , 0.60 ) , ( 4465.83 , 0.55 , 0.45 ) , ( 4441.61 , 0.35 , 0.65 ) }

℘ 6 = { ( 4548.63 , 0.24 , 0.76 ) , ( 4533.37 , 0.11 , 0.89 ) , ( 4608.32 , 0.73 , 0.27 ) , ( 4580.33 , 0.50 , 0.50 ) , ( 4646.61 , 0.94 , 0.06 ) }

℘ 7 = { ( 4646.61 , 0.05 , 0.95 ) , ( 4766.43 , 0.95 , 0.05 ) }

℘ 8 = { ( 4766.43 , 0.05 , 0.95 ) , ( 4924.56 , 0.63 , 0.37 ) }

℘ 9 = { ( 4924.56 , 0.37 , 0.63 ) , ( 5109.24 , 0.09 , 0.91 ) }

℘ 10 = { ( 5208.86 , 0.26 , 0.74 ) , ( 5221.96 , 0.15 , 0.85 ) , ( 5109.24 , 0.91 , 0.09 ) , ( 5164.73 , 0.63 , 0.37 ) }

℘ 11 = { ( 5208.86 , 0.74 , 0.26 ) , ( 5333.93 , 0.22 , 0.78 ) , ( 5329.19 , 0.26 , 0.74 ) , ( 5221.96 , 0.85 , 0.15 ) , ( 5309.10 , 0.42 , 0.58 ) , ( 5164.73 , 0.37 , 0.63 ) , ( 5332.98 , 0.22 , 0.78 ) , ( 5273.97 , 0.72 , 0.28 ) , ( 5321.28 , 0.32 , 0.68 ) }

℘ 12 = { ( 5333.93 , 0.78 , 0.22 ) , ( 5407.54 , 0.60 , 0.40 ) , ( 5456.15 , 0.20 , 0.80 ) , ( 5329.19 , 0.74 , 0.26 ) , ( 5309.10 , 0.57 , 0.43 ) , ( 5372.81 , 0.89 , 0.11 ) , ( 5392.43 , 0.73 , 0.27 ) , ( 5332.98 , 0.77 , 0.23 ) , ( 5398.28 , 0.68 , 0.32 ) , ( 5321.28 , 0.68 , 0.32 ) , ( 5273.97 , 0.28 , 0.72 ) }

℘ 13 = { ( 5407.54 , 0.40 , 0.60 ) , ( 5486.73 , 0.94 , 0.06 ) , ( 5456.15 , 0.80 , 0.20 ) , ( 5372.81 , 0.11 , 0.89 ) , ( 5392.43 , 0.27 , 0.73 ) , ( 5398.28 , 0.32 , 0.68 ) , ( 5551.24 , 0.41 , 0.59 ) , ( 5539.31 , 0.50 , 0.50 ) }

℘ 14 = { ( 5486.73 , 0.06 , 0.94 ) , ( 5551.24 , 0.59 , 0.41 ) , ( 5539.31 , 0.49 , 0.51 ) }

Step IV: Calculate the radius of circular intuitionistic fuzzy set by using equations (10) and (11).

℘ 1 = { ( 3929.69 ( 0.08 , 0.92 ; 0.37 ) ) , ( 3998.48 ( 0.65 , 0.35 ; 0.44 ) ) , ( 4080.51 ( 0.66 , 0.34 ; 0.45 ) ) , ( 4082.92 ( 0.64 , 0.36 ; 0.42 ) ) , ( 4158.15 ( 0.01 , 0.99 ; 0.47 ) ) , ( 4135.03 ( 0.21 , 0.79 ; 0.19 ) ) , ( 4123.78 ( 0.30 , 0.70 ; 0.06 ) ) , ( 4136.54 ( 0.19 , 0.81 ; 0.21 ) ) }

℘ 2 = { ( 4080.51 ( 0.34 , 0.66 ; 0.39 ) ) , ( 4082.92 ( 0.36 , 0.64 ; 0.36 ) ) , ( 4158.15 ( 0.97 , 0.03 ; 0.52 ) ) , ( 4135.03 ( 0.79 , 0.21 ; 0.26 ) ) , ( 4123.78 ( 0.70 , 0.30 ; 0.12 ) ) , ( 4172.63 ( 0.89 , 0.11 ; 0.40 ) ) , ( 4136.54 ( 0.80 , 0.20 ; 0.28 ) ) , ( 4277.70 ( 0.02 , 0.98 ; 0.84 ) ) }

℘ 3 = { ( 4172.63 ( 0.10 , 0.90 ; 0.62 ) ) , ( 4277.70 ( 0.98 , 0.02 , 0.62 ) ) }

℘ 4 = { ( 4403.59 ( 0.97 , 0.03 ; 0.59 ) ) , ( 4446.62 ( 0.61 , 0.39 ; 0.08 ) ) , ( 4455.80 ( 0.53 , 0.47 ; 0.02 ) ) , ( 4450.02 ( 0.58 , 0.42 ; 0.04 ) ) , ( 4159.08 ( 0.01 , 0.99 ; 0.77 ) ) , ( 4447.58 ( 0.60 , 0.40 ; 0.07 ) ) , ( 4465.83 ( 0.45 , 0.55 ; 0.14 ) ) , ( 4441.12 ( 0.66 , 0.34 : 0.15 ) ) }

℘ 5 = { ( 4403.59 ( 0.03 , 0.97 ; 0.65 ) ) , ( 4446.62 ( 0.39 , 0.61 ; 0.14 ) ) , ( 4548.63 ( 0.76 , 0.53 ; 0.26 ) ) , ( 4455.80 ( 0.46 , 0.54 ; 0.04 ) ) , ( 4533.37 ( 0.89 , 0.11 ; 0.57 ) ) , ( 4450.02 ( 0.42 , 0.58 ; 0.10 ) ) , ( 4519.08 ( 0.99 , 0.01 ; 0.71 ) ) , ( 4608.32 ( 0.26 , 0.74 ; 0.32 ) ) , ( 4580.33 ( 0.50 , 0.50 ; 0.02 ) ) , ( 4447.58 ( 0.40 , 0.60 : 0.13 ) ) , ( 4465.83 ( 0.55 , 0.45 ; 0.09 ) ) , ( 4441.61 ( 0.35 , 0.65 ; 0.20 ) ) }

℘ 6 = { ( 4548.63 ( 0.24 , 0.76 ; 0.38 ) ) , ( 4533.37 ( 0.11 , 0.89 ; 0.56 ) ) , ( 4608.32 ( 0.73 , 0.27 ; 0.32 ) ) , ( 4580.33 ( 0.50 , 0.50 ; 0.01 ) ) , ( 4646.61 ( 0.94 , 0.06 ; 0.62 ) ) }

℘ 7 = { ( 4646.61 ( 0.05 , 0.95 ; 0.63 ) ) , ( 4766.43 ( 0.95 , 0.05 ; 0.63 ) ) }

℘ 8 = { ( 4766.43 ( 0.05 , 0.95 ; 0.41 ) ) , ( 4924.56 ( 0.63 , 0.37 ; 0.41 ) ) }

℘ 9 = { ( 4924.56 ( 0.37 , 0.63 ; 0.20 ) ) , ( 5109.24 ( 0.09 , 0.91 ; 0.20 ) ) }

℘ 10 = { ( 5208.86 ( 0.26 , 0.74 ; 0.32 ) ) , ( 5221.96 ( 0.15 , 0.85 ; 0.48 ) ) , ( 5109.24 ( 0.91 , 0.09 ; 0.60 ) ) , ( 5164.73 ( 0.63 , 0.37 ; 0.20 ) ) }

℘ 11 = { ( 5208.86 ( 0.74 , 0.26 ; 0.40 ) ) , ( 5333.93 ( 0.22 , 0.78 ; 0.34 ) ) , ( 5329.19 ( 0.26 , 0.74 ; 0.29 ) ) , ( 5221.96 ( 0.85 , 0.15 ; 0.55 ) ) , ( 5309.10 ( 0.42 , 0.58 ; 0.05 ) ) , ( 5164.73 ( 0.37 , 0.63 ; 0.12 ) ) , ( 5332.98 ( 0.22 , 0.78 ; 0.33 ) ) , ( 5273.97 ( 0.72 , 0.28 ; 0.37 ) ) , ( 5321.28 ( 0.32 , 0.68 ; 0.19 ) ) }

℘ 12 = { ( 5333.93 ( 0.78 , 0.22 ; 0.13 ) ) , ( 5407.54 ( 0.60 , 0.40 ; 0.13 ) ) , ( 5456.15 ( 0.20 , 0.80 ; 0.70 ) ) , ( 5329.19 ( 0.74 , 0.26 ; 0.07 ) ) , ( 5309.10 ( 0.57 , 0.43 ; 0.17 ) ) , ( 5372.81 ( 0.89 , 0.11 ; 0.28 ) ) , ( 5392.43 ( 0.73 , 0.27 ; 0.05 ) ) , ( 5332.98 ( 0.77 , 0.23 ; 0.12 ) ) , ( 5398.28 ( 0.68 , 0.32 ; 0.02 ) ) , ( 5321.28 ( 0.68 , 0.32 ; 0.02 ) ) , ( 5273.97 ( 0.28 , 0.72 ; 0.58 ) ) }

℘ 13 = { ( 5407.54 ( 0.40 , 0.60 ; 0.10 ) ) , ( 5486.73 ( 0.94 , 0.06 ; 0.67 ) ) , ( 5456.15 ( 0.80 , 0.20 ; 0.47 ) ) , ( 5372.81 ( 0.11 , 0.89 ; 0.51 ) ) , ( 5392.43 ( 0.27 , 0.73 ; 0.28 ) ) , ( 5398.28 ( 0.32 , 0.68 ; 0.21 ) ) , ( 5551.24 ( 0.41 , 0.59 ; 0.09 ) ) , ( 5539.31 ( 0.50 , 0.50 ; 0.05 ) ) }

℘ 14 = { ( 5486.73 ( 0.06 , 0.94 ; 0.46 ) ) , ( 5551.24 ( 0.59 , 0.41 ; 0.30 ) ) , ( 5539.31 ( 0.49 , 0.51 ; 0.16 ) ) }

Step V: Use (12) in your computation to obtain the C-IFS score value.

The actual value of 4080.51 is between the two values of the circular intuitionistic membership and nonmembership of ℘ 1 and ℘ 2 . Then, we determined the highest score degree of 4080.51 as follows: The degrees of membership, nonmembership, and the radii of these values are given in step IV. Suppose the value of p is 0.5.

μ ℘ 1 ( 4080.51 ) = 0.66 , ν ℘ 1 ( 4080.51 ) = 0.34 , r ℘ 1 ( 4080.51 ) = 0.45 .

Similarly,

μ ℘ 2 ( 4080.51 ) = 0.34 , ν ℘ 2 ( 4080.51 ) = 0.66 , r ℘ 2 ( 4080.51 ) = 0.39 .

We must calculate the degree of score 4080.51 ℘ 1 and ℘ 2 and then select the larger value.

ζ ℘ 1 ( 4080.51 ) = 1 3 ( 0.66 − 0.34 + 2 × 0.45 ( 2 × 0.5 − 1 ) ) = 0.11 ,

ζ ℘ 1 ( 4080.51 ) = 1 3 ( 0.34 − 0.66 + 2 × 0.39 ( 2 × 0.5 − 1 ) ) = − 0.11 .

So the degree of score 4080.51 in ℘ 1 is greater than ℘ 2 , so the circular intuitionistic value of 4080.51 is ℘ 1 . The remaining computed data are as follows:

Step VI: Circular intuitionistic logical relationships are in presented Table 2.

Table 2

Circular intuitionistic logical relationships of order I

℘ 1 → ℘ 1 ℘ 1 → ℘ 1 ℘ 1 → ℘ 1 ℘ 1 → ℘ 2 ℘ 2 → ℘ 2 ℘ 2 → ℘ 2 ℘ 2 → ℘ 2
℘ 2 → ℘ 2 ℘ 2 → ℘ 3 ℘ 3 → ℘ 4 ℘ 4 → ℘ 4 ℘ 4 → ℘ 5 ℘ 5 → ℘ 4 ℘ 4 → ℘ 5
℘ 5 → ℘ 4 ℘ 4 → ℘ 5 ℘ 5 → ℘ 6 ℘ 6 → ℘ 6 ℘ 6 → ℘ 4 ℘ 4 → ℘ 5 ℘ 5 → ℘ 4
℘ 4 → ℘ 6 ℘ 6 → ℘ 7 ℘ 7 → ℘ 8 ℘ 8 → ℘ 11 ℘ 11 → ℘ 12 ℘ 12 → ℘ 12 ℘ 12 → ℘ 11
℘ 11 → ℘ 13 ℘ 13 → ℘ 12 ℘ 12 → ℘ 13 ℘ 13 → ℘ 13 ℘ 13 → ℘ 12 ℘ 12 → ℘ 11 ℘ 11 → ℘ 12
℘ 12 → ℘ 10 ℘ 10 → ℘ 10 ℘ 10 → ℘ 12 ℘ 12 → ℘ 12 ℘ 12 → ℘ 12 ℘ 12 → ℘ 12 ℘ 12 → ℘ 4

Step VII: We developed a collection of C-IFLRGs based on C-IFLRs, which are presented in Table 3.

Table 3

Circular intuitionistic logical relationship groups of order I

℘ 1 → ℘ 1 ℘ 1 → ℘ 1 ℘ 1 → ℘ 1 ℘ 1 → ℘ 2
℘ 2 → ℘ 2 ℘ 2 → ℘ 2 ℘ 2 → ℘ 2 ℘ 2 → ℘ 2 ℘ 2 → ℘ 3
℘ 3 → ℘ 4
℘ 4 → ℘ 4 ℘ 4 → ℘ 5 ℘ 4 → ℘ 5 ℘ 4 → ℘ 5 ℘ 4 → ℘ 5 ℘ 4 → ℘ 6
℘ 5 → ℘ 4 ℘ 5 → ℘ 4 ℘ 5 → ℘ 6 ℘ 5 → ℘ 4
℘ 6 → ℘ 6 ℘ 6 → ℘ 4 ℘ 6 → ℘ 7
℘ 7 → ℘ 8
℘ 8 → ℘ 11
℘ 10 → ℘ 10 ℘ 10 → ℘ 12
℘ 11 → ℘ 12 ℘ 11 → ℘ 13 ℘ 11 → ℘ 12
℘ 12 → ℘ 12 ℘ 12 → ℘ 11 ℘ 12 → ℘ 13 ℘ 12 → ℘ 11 ℘ 12 → ℘ 12 ℘ 12 → ℘ 12 ℘ 12 → ℘ 12 ℘ 12 → ℘ 12 ℘ 12 → ℘ 14
℘ 13 → ℘ 12 ℘ 13 → ℘ 13 ℘ 13 → ℘ 12

B. Measure the forecasted value of circular intuitionistic fuzzy sets

The computed forecasted values are shown in Table 4. We are unable to predict what the anticipated value for November 1, 2001, will be since we do not have access to the beginning value for that day. The estimated value for the next day was computed similarly.

Table 4

Forecasted value of stock change index of order I

Date True value Forecasted value Years True value Forecasted value
01-11-2001 3929.69 — 03-12-2001 4646.61 4,520
02-11-2001 3998.48 4,100 04-12-2001 4766.43 4,600
05-11-2001 4080.51 4,100 05-12-2001 4924.56 4,880
06-11-2001 4082.92 4,100 06-12-2001 5208.86 5,240
07-11-2001 4158.15 4,100 07-12-2001 5333.93 5,360
08-11-2001 4135.03 4,220 10-12-2001 5321.28 5,420
09-11-2001 4123.78 4,220 11-12-2001 5273.97 5,420
12-11-2001 4172.63 4,220 12-12-2001 5539.31 5,420
13-11-2001 4136.54 4,220 13-12-2001 5407.54 5,420
14-11-2001 4277.70 4,220 14-12-2001 5486.73 5,420
15-11-2001 4403.59 4,400 17-12-2001 5456.15 5,420
16-11-2001 4446.62 4,520 18-12-2001 5329.19 5,420
19-11-2001 4548.63 4,520 19-12-2001 5221.96 5,420
20-11-2001 4455.80 4,520 20-12-2001 5309.10 5,420
21-11-2001 4533.37 4,520 21-12-2001 5109.24 5,420
22-11-2001 4450.02 4,520 24-12-2001 5164.73 5,240
23-11-2001 4519.08 4,520 25-12-2001 5372.81 5,240
26-11-2001 4608.32 4,520 26-12-2001 5392.43 5,420
27-11-2001 4580.33 4,600 27-12-2001 5332.98 5,420
28-11-2001 4447.58 4,600 28-12-2001 5398.28 5,420
29-11-2001 4465.83 4,520 31-12-2001 5551.24 5,420
30-11-2001 4441.12 4,520

Figure 3 presents a graph that illustrates the true value as well as the observed value that make up the stock change index.

Figure 3 True value and observed value of order I.
Figure 3

True value and observed value of order I.

4.2 Circular intuitionistic logical relationship of order II

Now we will compute the circular intuitionistic logical relationship and its groups according to the provided approach to order II stock change index.

The circular intuitionistic logic relations of order II are in Table 5. Table 6 displays the circular intuitionistic logic relation groups that we created based on C-ILRs. These C-ILRs are of order II.

Table 5

Circular intuitionistic logical relationship of order II

℘ 1 , ℘ 1 → ℘ 1 ℘ 1 , ℘ 1 → ℘ 1 ℘ 1 , ℘ 1 → ℘ 2 ℘ 1 , ℘ 2 → ℘ 2 ℘ 2 , ℘ 2 → ℘ 2 ℘ 2 , ℘ 2 → ℘ 2
℘ 2 , ℘ 2 → ℘ 2 ℘ 2 , ℘ 2 → ℘ 3 ℘ 2 , ℘ 3 → ℘ 4 ℘ 3 , ℘ 4 → ℘ 4 ℘ 4 , ℘ 4 → ℘ 5 ℘ 4 , ℘ 5 → ℘ 4
℘ 5 , ℘ 4 → ℘ 5 ℘ 4 , ℘ 5 → ℘ 4 ℘ 5 , ℘ 4 → ℘ 5 ℘ 4 , ℘ 5 → ℘ 6 ℘ 5 , ℘ 6 → ℘ 6 ℘ 6 , ℘ 6 → ℘ 4
℘ 6 , ℘ 4 → ℘ 5 ℘ 4 , ℘ 5 → ℘ 4 ℘ 5 , ℘ 4 → ℘ 6 ℘ 4 , ℘ 6 → ℘ 7 ℘ 6 , ℘ 7 → ℘ 8 ℘ 7 , ℘ 8 → ℘ 11
℘ 8 , ℘ 11 → ℘ 12 ℘ 11 , ℘ 12 → ℘ 12 ℘ 12 , ℘ 12 → ℘ 11 ℘ 12 , ℘ 11 → ℘ 13 ℘ 11 , ℘ 13 → ℘ 12 ℘ 13 , ℘ 12 → ℘ 13
℘ 12 , ℘ 13 → ℘ 13 ℘ 13 , ℘ 13 → ℘ 12 ℘ 13 , ℘ 12 → ℘ 11 ℘ 12 , ℘ 11 → ℘ 12 ℘ 11 , ℘ 12 → ℘ 10 ℘ 12 , ℘ 10 → ℘ 10
℘ 10 , ℘ 10 → ℘ 12 ℘ 10 , ℘ 12 → ℘ 12 ℘ 12 , ℘ 12 → ℘ 12 ℘ 12 , ℘ 12 → ℘ 12 ℘ 12 , ℘ 12 → ℘ 14
Table 6

Circular intuitionistic logical relationship groups of order II

℘ 1 , ℘ 1 → ℘ 1 ℘ 1 , ℘ 1 → ℘ 1 ℘ 1 , ℘ 1 → ℘ 2 ℘ 1 , ℘ 2 → ℘ 2
℘ 2 , ℘ 2 → ℘ 2 ℘ 2 , ℘ 2 → ℘ 2 ℘ 2 , ℘ 2 → ℘ 2 ℘ 2 , ℘ 2 → ℘ 3 ℘ 2 , ℘ 3 → ℘ 4
℘ 3 , ℘ 4 → ℘ 4 ℘ 4 , ℘ 4 → ℘ 5
℘ 4 , ℘ 5 → ℘ 4 ℘ 4 , ℘ 5 → ℘ 4 ℘ 4 , ℘ 5 → ℘ 6 ℘ 4 , ℘ 5 → ℘ 4 ℘ 5 , ℘ 4 → ℘ 5 ℘ 5 , ℘ 4 → ℘ 5 ℘ 5 , ℘ 4 → ℘ 6
℘ 5 , ℘ 6 → ℘ 6 ℘ 6 , ℘ 6 → ℘ 4
℘ 6 , ℘ 4 → ℘ 5 ℘ 4 , ℘ 6 → ℘ 7
℘ 6 , ℘ 7 → ℘ 8 ℘ 7 , ℘ 8 → ℘ 11
℘ 8 , ℘ 11 → ℘ 12 ℘ 11 , ℘ 12 → ℘ 12 ℘ 11 , ℘ 12 → ℘ 10
℘ 12 , ℘ 12 → ℘ 11 ℘ 12 , ℘ 12 → ℘ 12 ℘ 12 , ℘ 12 → ℘ 12 ℘ 12 , ℘ 12 → ℘ 14 ℘ 12 , ℘ 11 → ℘ 13 ℘ 12 , ℘ 11 → ℘ 12
℘ 11 , ℘ 13 → ℘ 12 ℘ 13 , ℘ 12 → ℘ 13 ℘ 13 , ℘ 12 → ℘ 11
℘ 12 , ℘ 13 → ℘ 13 ℘ 13 , ℘ 13 → ℘ 12
℘ 12 , ℘ 10 → ℘ 10 ℘ 10 , ℘ 10 → ℘ 12
℘ 10 , ℘ 12 → ℘ 12

B. Measure the forecasted value of circular intuitionistic fuzzy sets.

Table 7 presents the results of the forecasting calculations that were performed with the help of the defuzzified criteria described in Section B. Figure 4 is a graph that depicts the true value, as well as the observed value, of the stock change index for order II.

Table 7

Forecasted value of stock change index of order II

Date True value Forecasted value Years True value Forecasted value
01-11-2001 3929.69 — 03-12-2001 4646.61 4,580
02-11-2001 3998.48 — 04-12-2001 4766.43 4,760
05-11-2001 4080.51 4,100 05-12-2001 4924.56 4,880
06-11-2001 4082.92 4,100 06-12-2001 5208.86 5,240
07-11-2001 4158.15 4,100 07-12-2001 5333.93 5,360
08-11-2001 4135.03 4,160 10-12-2001 5321.2 5,240
09-11-2001 4123.78 4,220 11-12-2001 5273.97 5,400
12-11-2001 4172.63 4,220 12-12-2001 5539.31 5,420
13-11-2001 4136.54 4,220 13-12-2001 5407.54 5,360
14-11-2001 4277.70 4,220 14-12-2001 5486.73 5,360
15-11-2001 4403.59 4,400 17-12-2001 5456.15 5,480
16-11-2001 4446.62 4,400 18-12-2001 5329.19 5,360
19-11-2001 4548.63 4,520 19-12-2001 5221.96 5,360
20-11-2001 4455.80 4,520 20-12-2001 5309.10 5,420
21-11-2001 4533.37 4,580 21-12-2001 5109.24 5,420
22-11-2001 4450.02 4,520 24-12-2001 5164.73 5,120
23-11-2001 4519.08 4,580 25-12-2001 5372.81 5,360
26-11-2001 4608.32 4,520 26-12-2001 5392.43 5,360
27-11-2001 4580.33 4,640 27-12-2001 5332.98 5,400
28-11-2001 4447.58 4,400 28-12-2001 5398.28 5,400
29-11-2001 4465.83 4,520 31-12-2001 5551.24 5,400
30-11-2001 4441.12 4,520
Figure 4 Graph of the actual and proposed value of order II.
Figure 4

Graph of the actual and proposed value of order II.

C. Measurement of error by using RMSE and AFE

The formulas for determining errors are provided in Table 8 and are used to calculate RMSE and AFE, respectively. Comparing our results with other forecasting methods using RMSE and AFE can give an indication of the accuracy of your forecast. If the RMSE and AFE values are lower than those of other forecasting methods, it means that the proposed method is more accurate because of the lower error.

Table 8

RMSE and AFE

Tools Chen [38] NASDAQ & Dow Jones [47] Proposed method order I Proposed method order II
RMSE 147.84 124.02 98.03 84.61
AFE Not calculated Not calculated 1.61 1.34

5 Discussion

Table 8 shows the difference in predicted performance between first- and second-order C-IFTSs. Compared to first-order C-IFTSs forecasting, second-order C-IFTSs forecasting uses error calculation formulas. When we calculate third-order C-IFTS forecasting, the error of the third order is less than the error of the second. Use the same formula to obtain the n th order. Higher-orders tend to have fewer errors. It is important to note that the accuracy of the forecasts depends not only on the choice of forecasting method but also on the quality and completeness of the data.

The value of RMSE clearly shows the validity of my proposed algorithm to deal with this kind of problem in future prediction. As we discussed earlier, errors show the accuracy of your method. Researchers use this technique to invest the amount in stock index problems. A proposed algorithm is used to handle all these types of problems in fuzzy logic, in which a radius is also given for membership and nonmembership. In the past, no one could give any algorithm to solve this type of problem where the radius was also given. This algorithm is helpful for dealing with this kind of problem.

Radius tells us where the value of membership and nonmembership lies. The radius in C-IFS is important because it can impact the overall shape and size of the circular fuzzy set, which in turn can affect its ability to represent complex and uncertain information in a flexible and intuitive way.

6 Conclusion

With the circular radius, C-IFS membership and nonmembership degree totals are all equal to one or fewer, and they are rapidly gaining favor. It is obvious that utilizing intuitionistic fuzzy sets cannot address this issue; as a result, we employ C-IFS instead of IFSs if the total of the membership value and degree of the nonmembership value are equal. The suggested C-IFS approach was less complex and more straightforward since it employed a simple scoring formula. We used our method to verify the number of stock change indexes, and now we can forecast our data using the given criteria. In addition, we applied this approach to higher-order predictions and displayed the error, which indicates that higher-order forecasts have fewer errors and are thus more helpful for calculating future values. By using the advised strategy, we came up with a prediction for the next few years. Researchers may provide further solutions to the forecasting process in the future. One potential direction for future research could be to apply C-IFTS to various time series forecasting problems and compare their performance to existing methods.

  1. Funding information: The authors received no financial support for the research of this article.

  2. Author contributions: All authors have contributed equally to this article.

  3. Conflict of interest: The authors declare that they have no conflict of interest regarding the publication of the research article.

  4. Ethical approval: This article does not contain any studies with human participants or animals performed by any of the authors.

  5. Data availability statement: The data used and analyzed during the current study available from the corresponding author on reasonable request.

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Received: 2023-03-21
Revised: 2023-08-08
Accepted: 2023-09-05
Published Online: 2024-01-10

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Artikel in diesem Heft

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  3. L-Fuzzy fixed point results in ℱ -metric spaces with applications
  4. Solutions of a coupled system of hybrid boundary value problems with Riesz-Caputo derivative
  5. Nonparametric methods of statistical inference for double-censored data with applications
  6. LADM procedure to find the analytical solutions of the nonlinear fractional dynamics of partial integro-differential equations
  7. Existence of projected solutions for quasi-variational hemivariational inequality
  8. Spectral collocation method for convection-diffusion equation
  9. New local fractional Hermite-Hadamard-type and Ostrowski-type inequalities with generalized Mittag-Leffler kernel for generalized h-preinvex functions
  10. On the asymptotics of eigenvalues for a Sturm-Liouville problem with symmetric single-well potential
  11. On exact rate of convergence of row sequences of multipoint Hermite-Padé approximants
  12. The essential norm of bounded diagonal infinite matrices acting on Banach sequence spaces
  13. Decay rate of the solutions to the Cauchy problem of the Lord Shulman thermoelastic Timoshenko model with distributed delay
  14. Enhancing the accuracy and efficiency of two uniformly convergent numerical solvers for singularly perturbed parabolic convection–diffusion–reaction problems with two small parameters
  15. An inertial shrinking projection self-adaptive algorithm for solving split variational inclusion problems and fixed point problems in Banach spaces
  16. An equation for complex fractional diffusion created by the Struve function with a T-symmetric univalent solution
  17. On the existence and Ulam-Hyers stability for implicit fractional differential equation via fractional integral-type boundary conditions
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  22. On Berezin norm and Berezin number inequalities for sum of operators
  23. Geometric invariants properties of osculating curves under conformal transformation in Euclidean space ℝ3
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  25. A novel numerical approach to solutions of fractional Bagley-Torvik equation fitted with a fractional integral boundary condition
  26. Holomorphic curves into projective spaces with some special hypersurfaces
  27. On Periodic solutions for implicit nonlinear Caputo tempered fractional differential problems
  28. Approximation of complex q-Beta-Baskakov-Szász-Stancu operators in compact disk
  29. Existence and regularity of solutions for non-autonomous integrodifferential evolution equations involving nonlocal conditions
  30. Jordan left derivations in infinite matrix rings
  31. Nonlinear nonlocal elliptic problems in ℝ3: existence results and qualitative properties
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  33. Novel results for two families of multivalued dominated mappings satisfying generalized nonlinear contractive inequalities and applications
  34. Global in time well-posedness of a three-dimensional periodic regularized Boussinesq system
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https://doi.org/10.1515/dema-2023-0115
Schlagwörter für diesen Artikel
fuzzy set; circular intuitionistic fuzzy sets; score function; higher-order time series forecasting
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Artikel in diesem Heft

  1. Regular Articles
  2. On the p-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity
  3. L-Fuzzy fixed point results in ℱ -metric spaces with applications
  4. Solutions of a coupled system of hybrid boundary value problems with Riesz-Caputo derivative
  5. Nonparametric methods of statistical inference for double-censored data with applications
  6. LADM procedure to find the analytical solutions of the nonlinear fractional dynamics of partial integro-differential equations
  7. Existence of projected solutions for quasi-variational hemivariational inequality
  8. Spectral collocation method for convection-diffusion equation
  9. New local fractional Hermite-Hadamard-type and Ostrowski-type inequalities with generalized Mittag-Leffler kernel for generalized h-preinvex functions
  10. On the asymptotics of eigenvalues for a Sturm-Liouville problem with symmetric single-well potential
  11. On exact rate of convergence of row sequences of multipoint Hermite-Padé approximants
  12. The essential norm of bounded diagonal infinite matrices acting on Banach sequence spaces
  13. Decay rate of the solutions to the Cauchy problem of the Lord Shulman thermoelastic Timoshenko model with distributed delay
  14. Enhancing the accuracy and efficiency of two uniformly convergent numerical solvers for singularly perturbed parabolic convection–diffusion–reaction problems with two small parameters
  15. An inertial shrinking projection self-adaptive algorithm for solving split variational inclusion problems and fixed point problems in Banach spaces
  16. An equation for complex fractional diffusion created by the Struve function with a T-symmetric univalent solution
  17. On the existence and Ulam-Hyers stability for implicit fractional differential equation via fractional integral-type boundary conditions
  18. Some properties of a class of holomorphic functions associated with tangent function
  19. The existence of multiple solutions for a class of upper critical Choquard equation in a bounded domain
  20. On the continuity in q of the family of the limit q-Durrmeyer operators
  21. Results on solutions of several systems of the product type complex partial differential difference equations
  22. On Berezin norm and Berezin number inequalities for sum of operators
  23. Geometric invariants properties of osculating curves under conformal transformation in Euclidean space ℝ3
  24. On a generalization of the Opial inequality
  25. A novel numerical approach to solutions of fractional Bagley-Torvik equation fitted with a fractional integral boundary condition
  26. Holomorphic curves into projective spaces with some special hypersurfaces
  27. On Periodic solutions for implicit nonlinear Caputo tempered fractional differential problems
  28. Approximation of complex q-Beta-Baskakov-Szász-Stancu operators in compact disk
  29. Existence and regularity of solutions for non-autonomous integrodifferential evolution equations involving nonlocal conditions
  30. Jordan left derivations in infinite matrix rings
  31. Nonlinear nonlocal elliptic problems in ℝ3: existence results and qualitative properties
  32. Invariant means and lacunary sequence spaces of order (α, β)
  33. Novel results for two families of multivalued dominated mappings satisfying generalized nonlinear contractive inequalities and applications
  34. Global in time well-posedness of a three-dimensional periodic regularized Boussinesq system
  35. Existence of solutions for nonlinear problems involving mixed fractional derivatives with p(x)-Laplacian operator
  36. Some applications and maximum principles for multi-term time-space fractional parabolic Monge-Ampère equation
  37. On three-dimensional q-Riordan arrays
  38. Some aspects of normal curve on smooth surface under isometry
  39. Mittag-Leffler-Hyers-Ulam stability for a first- and second-order nonlinear differential equations using Fourier transform
  40. Topological structure of the solution sets to non-autonomous evolution inclusions driven by measures on the half-line
  41. Remark on the Daugavet property for complex Banach spaces
  42. Decreasing and complete monotonicity of functions defined by derivatives of completely monotonic function involving trigamma function
  43. Uniqueness of meromorphic functions concerning small functions and derivatives-differences
  44. Asymptotic approximations of Apostol-Frobenius-Euler polynomials of order α in terms of hyperbolic functions
  45. Hyers-Ulam stability of Davison functional equation on restricted domains
  46. Involvement of three successive fractional derivatives in a system of pantograph equations and studying the existence solution and MLU stability
  47. Composition of some positive linear integral operators
  48. On bivariate fractal interpolation for countable data and associated nonlinear fractal operator
  49. Generalized result on the global existence of positive solutions for a parabolic reaction-diffusion model with an m × m diffusion matrix
  50. Online makespan minimization for MapReduce scheduling on multiple parallel machines
  51. The sequential Henstock-Kurzweil delta integral on time scales
  52. On a discrete version of Fejér inequality for α-convex sequences without symmetry condition
  53. Existence of three solutions for two quasilinear Laplacian systems on graphs
  54. Embeddings of anisotropic Sobolev spaces into spaces of anisotropic Hölder-continuous functions
  55. Nilpotent perturbations of m-isometric and m-symmetric tensor products of commuting d-tuples of operators
  56. Characterizations of transcendental entire solutions of trinomial partial differential-difference equations in ℂ2#
  57. Fractional Sturm-Liouville operators on compact star graphs
  58. Exact controllability for nonlinear thermoviscoelastic plate problem
  59. Improved modified gradient-based iterative algorithm and its relaxed version for the complex conjugate and transpose Sylvester matrix equations
  60. Superposition operator problems of Hölder-Lipschitz spaces
  61. A note on λ-analogue of Lah numbers and λ-analogue of r-Lah numbers
  62. Ground state solutions and multiple positive solutions for nonhomogeneous Kirchhoff equation with Berestycki-Lions type conditions
  63. A note on 1-semi-greedy bases in p-Banach spaces with 0 < p ≤ 1
  64. Fixed point results for generalized convex orbital Lipschitz operators
  65. Asymptotic model for the propagation of surface waves on a rotating magnetoelastic half-space
  66. Multiplicity of k-convex solutions for a singular k-Hessian system
  67. Poisson C*-algebra derivations in Poisson C*-algebras
  68. Signal recovery and polynomiographic visualization of modified Noor iteration of operators with property (E)
  69. Approximations to precisely localized supports of solutions for non-linear parabolic p-Laplacian problems
  70. Solving nonlinear fractional differential equations by common fixed point results for a pair of (α, Θ)-type contractions in metric spaces
  71. Pseudo compact almost automorphic solutions to a family of delay differential equations
  72. Periodic measures of fractional stochastic discrete wave equations with nonlinear noise
  73. Asymptotic study of a nonlinear elliptic boundary Steklov problem on a nanostructure
  74. Cramer's rule for a class of coupled Sylvester commutative quaternion matrix equations
  75. Quantitative estimates for perturbed sampling Kantorovich operators in Orlicz spaces
  76. Review Articles
  77. Penalty method for unilateral contact problem with Coulomb's friction in time-fractional derivatives
  78. Differential sandwich theorems for p-valent analytic functions associated with a generalization of the integral operator
  79. Special Issue on Development of Fuzzy Sets and Their Extensions - Part II
  80. Higher-order circular intuitionistic fuzzy time series forecasting methodology: Application of stock change index
  81. Binary relations applied to the fuzzy substructures of quantales under rough environment
  82. Algorithm selection model based on fuzzy multi-criteria decision in big data information mining
  83. A new machine learning approach based on spatial fuzzy data correlation for recognizing sports activities
  84. Benchmarking the efficiency of distribution warehouses using a four-phase integrated PCA-DEA-improved fuzzy SWARA-CoCoSo model for sustainable distribution
  85. Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part II
  86. A study on a type of degenerate poly-Dedekind sums
  87. Efficient scheme for a category of variable-order optimal control problems based on the sixth-kind Chebyshev polynomials
  88. Special Issue on Mathematics for Artificial intelligence and Artificial intelligence for Mathematics
  89. Toward automated hail disaster weather recognition based on spatio-temporal sequence of radar images
  90. The shortest-path and bee colony optimization algorithms for traffic control at single intersection with NetworkX application
  91. Neural network quaternion-based controller for port-Hamiltonian system
  92. Matching ontologies with kernel principle component analysis and evolutionary algorithm
  93. Survey on machine vision-based intelligent water quality monitoring techniques in water treatment plant: Fish activity behavior recognition-based schemes and applications
  94. Artificial intelligence-driven tone recognition of Guzheng: A linear prediction approach
  95. Transformer learning-based neural network algorithms for identification and detection of electronic bullying in social media
  96. Squirrel search algorithm-support vector machine: Assessing civil engineering budgeting course using an SSA-optimized SVM model
  97. Special Issue on International E-Conference on Mathematical and Statistical Sciences - Part I
  98. Some fixed point results on ultrametric spaces endowed with a graph
  99. On the generalized Mellin integral operators
  100. On existence and multiplicity of solutions for a biharmonic problem with weights via Ricceri's theorem
  101. Approximation process of a positive linear operator of hypergeometric type
  102. On Kantorovich variant of Brass-Stancu operators
  103. A higher-dimensional categorical perspective on 2-crossed modules
  104. Special Issue on Some Integral Inequalities, Integral Equations, and Applications - Part I
  105. On parameterized inequalities for fractional multiplicative integrals
  106. On inverse source term for heat equation with memory term
  107. On Fejér-type inequalities for generalized trigonometrically and hyperbolic k-convex functions
  108. New extensions related to Fejér-type inequalities for GA-convex functions
  109. Derivation of Hermite-Hadamard-Jensen-Mercer conticrete inequalities for Atangana-Baleanu fractional integrals by means of majorization
  110. Some Hardy's inequalities on conformable fractional calculus
  111. The novel quadratic phase Fourier S-transform and associated uncertainty principles in the quaternion setting
  112. Special Issue on Recent Developments in Fixed-Point Theory and Applications - Part I
  113. A novel iterative process for numerical reckoning of fixed points via generalized nonlinear mappings with qualitative study
  114. Some new fixed point theorems of α-partially nonexpansive mappings
  115. Generalized Yosida inclusion problem involving multi-valued operator with XOR operation
  116. Periodic and fixed points for mappings in extended b-gauge spaces equipped with a graph
  117. Convergence of Peaceman-Rachford splitting method with Bregman distance for three-block nonconvex nonseparable optimization
  118. Topological structure of the solution sets to neutral evolution inclusions driven by measures
  119. (α, F)-Geraghty-type generalized F-contractions on non-Archimedean fuzzy metric-unlike spaces
  120. Solvability of infinite system of integral equations of Hammerstein type in three variables in tempering sequence spaces c 0 β and ℓ 1 β
  121. Special Issue on Nonlinear Evolution Equations and Their Applications - Part I
  122. Fuzzy fractional delay integro-differential equation with the generalized Atangana-Baleanu fractional derivative
  123. Klein-Gordon potential in characteristic coordinates
  124. Asymptotic analysis of Leray solution for the incompressible NSE with damping
  125. Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part I
  126. Long time decay of incompressible convective Brinkman-Forchheimer in L2(ℝ3)
  127. Numerical solution of general order Emden-Fowler-type Pantograph delay differential equations
  128. Global smooth solution to the n-dimensional liquid crystal equations with fractional dissipation
  129. Spectral properties for a system of Dirac equations with nonlinear dependence on the spectral parameter
  130. A memory-type thermoelastic laminated beam with structural damping and microtemperature effects: Well-posedness and general decay
  131. The asymptotic behavior for the Navier-Stokes-Voigt-Brinkman-Forchheimer equations with memory and Tresca friction in a thin domain
  132. Absence of global solutions to wave equations with structural damping and nonlinear memory
  133. Special Issue on Differential Equations and Numerical Analysis - Part I
  134. Vanishing viscosity limit for a one-dimensional viscous conservation law in the presence of two noninteracting shocks
  135. Limiting dynamics for stochastic complex Ginzburg-Landau systems with time-varying delays on unbounded thin domains
  136. A comparison of two nonconforming finite element methods for linear three-field poroelasticity
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Heruntergeladen am 4.2.2026 von https://www.degruyterbrill.com/document/doi/10.1515/dema-2023-0115/html
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