A Refined Approach for Forecasting based on Neutrosophic Time Series

This research introduces a neutrosophic forecasting approach based on neutrosophic time series (NTS). Historical data can be transformed into neutrosophic time series data to determine their truthiness, indeterminacy and falsity functions. The basic for the neutrosophication process is the score and accuracy functions of historical data. In addition, neutrosophic logical relationship groups (NLRG) are determined and a deneutrosophication method for NTS is presented. The objective of this research is to suggest an idea of first and high-order NTS. By comparing our approach with other approaches, we conclude that the suggested approach of forecasting gets better results over the other existing approaches of fuzzy, intuitionistic fuzzy and neutrosophic time series.


Introduction
There are different methods in the literature on fuzzy and intuitionistic fuzzy time series methods to forecast the future values.The major difference between traditional and fuzzy time series is that, the values of traditional time series are presented in numbers, but values in fuzzy time series are fuzzy sets or linguistic values with real meanings.In intuitionistic fuzzy time series the values are intuitionistic fuzzy sets or linguistic values.The first method in literature for forecasting the future values based on fuzzy time series introduced by Song and Chissom [1].They also applied time-variant and time-invariant models for forecasting the enrollments data at the University of Alabama [1,2].The identification of fuzzy relationship and the defuzzification process in both models were the main steps for calculating forecasted values.They supposed that the autocorrelation is dependent in time variant, but independent in time-invariant fuzzy time series.
The term "fuzzy relationship" means collection of fuzzy sets which caused only by other sets.Also, the "defuzzification" process means converting the fuzzy values into crisp ones.
Further, a straight forward approach for time series forecasting was presented by Chen [3] by using uncomplicated arithmetic computations.For enhancing the accuracy of forecasted outputs, some papers suggested various methods on fuzzy time series (FTS) forecasting [4][5][6][7].A high-order FTS method was also presented by Chen [8] and Singh [9], and a method of bivariate fuzzy time series analysis for the forecasting of a stock index was introduced by Hsu et al. [10].Furthermore, a framework developed for the evaluation and forecasting based on the fuzzy NEAT F-PROMETHEE method presented by Ziemba and Becker [11] for taking into account the uncertainty of input data, which is particularly burdened with the forecast values of the ICT development indicators.
The concept of fuzzy set was introduced by Zadeh [12], and it was generalized by Atanassov [13] to intuitionistic fuzzy set (IFS) for making it more suitable to handle ambiguity.The IFS considers both the membership (truthiness) and non-membership (falsity) degrees.However, the fuzzy set considers only the membership degree.Recently, the IFS was used for handling the fuzzy time series forecasting by Gangwar and Kumar [14] and Wang et al. [15].In addition, the notion of intuitionistic fuzzy time series (IFTS)was employed in forecasting, as in [16][17][18].Several researchers [19,20] proposed forecasting models using genetic algorithm, or suggested a method of forecasting based on aggregated FTS and particle swarm optimization [21].A novel method of forecasting based on hesitant fuzzy set was proposed by Bisht and Kumar [22], and fuzzy descriptor models for earthquake was introduced by Bahrami and Shafiee [23].A heuristic adaptive-order IFTS forecasting model was presented by Wang et al. [24].Further on, Abhishekh et al. [25,26] presented a weighted type 2 FTS and score function-based IFTS forecasting approach.Moreover, Abhishekh and Kumar [27] suggested an approach for forecasting rice production in the area of FTS.Since the accuracy rates of forecasting in the previous approaches are not good enough in the field of fuzzy and intuitionistic fuzzy time series, we introduce the notion of first and high-order neutrosophic time series data for this research.Additionally, with the growing need to represent vague and random information, neutrosophic sets (NSs) theory [28] is an effective extension of fuzzy and intuitionistic fuzzy set theories.Smarandache [29] suggested NSs, which consist of truth membership function, indeterminacy membership function, and falsity membership function, as a better representation of reality.
Neutrosophic sets received wide attention, as well as benefitting from various practical applications in diverse fields [30][31][32][33][34][35][36][37][38][39].However, there are only two recent research papers published in forecasting field, e.g. for stock market analysis.Guan et al. [40] proposed a new forecasting model based on multi-valued neutrosophic sets and two-factor third-order fuzzy logical relationships to forecast the stock market.Further on, Guan et al. [41] proposed a new forecasting method based on high order fluctuation trends and information entropy.
The aim of this research is to enhance accuracy rates of forecasting in the area of fuzzy, intuitionistic fuzzy, and neutrosophic time series (NTS).In this research, we present the notion of forecasting based on first and high-order NTS data by determining the suitable length of neutrosophic numbers that influence on expected values.We also suggest a neutrosophication of the historical time series data, based on the biggest score function (i.e. the maximum value of score function), and define neutrosophic logical relationship groups for obtaining forecasted outputs.The suggested approach of neutrosophic time series forecasting has been validated and compared with different existing models for showing its superiority.
The remaining parts of this research are organized as follows: The essential concepts of neutrosophic set and neutrosophic time series are briefly presented in Section 2. Section 3 presents the proposed neutrosophic time series method for forecasting process.Section 4 validates the proposed method by applying it on two numerical examples for showing its effectiveness; a comparison with other existing methods is presented.Finally, Section 5 concludes the research and determines future trends.
The stepwise method of the suggested algorithm of neutrosophic time series forecasting is depended on historical time series data.

The proposed method of forecasting based on first order NTS data
Step 1: By depending on range of existing data set, determine the universe of discourse U as follows: -Select the largest   and the smallest   from all available data  , then where  1 and  2 are two proper positive numbers assigned by experts in the problem domain.So, we can define D 1 , D 2 as the values by which the range of the universe of discourse be less than the specified value of D s for the first (i.e. D 1 ) or greater than the specified value of   for the later (i.e.D 2 ) .
Step 2: Create a partition of the universe of discourse, to triangular neutrosophic numbers as it follows: -Decide the suitable length () of available time series data: o Among the value  −1 ,  , calculate all absolute differences and take average of these differences.
o Consider half the average as the initial length.
o According to obtained result, use base mapping table [42] to determine the base for length of intervals.
o Round the result to determine the appropriate length of neutrosophic numbers.
o For example: if we have these time series data 30,50,80,120,100,70 , then the absolute differences will be 20,30,40,20,30 , and the average of these values = 28.Then the half of average will be 14 and this is the initial value of length.By using base mapping table [42], the base for length = 10 because ten the result will equal 10.Here, the appropriate length of neutrosophic numbers equals 10.
-Compute the number of triangular neutrosophic numbers () as follows: Step 3:According to the numbers of triangular neutrosophic numbers on the universe of discourse and determined length (  ), begin to construct the triangular neutrosophic numbers.The triangular neutrosophic numbers are  ̃1,  ̃2, … ,  ̃.
As we illustrated in Definition 2, each triangular neutrosophic number consists of two parts which are the value of triangular neutrosophic number (lower, median, upper ) and the degree of confirmation ( truth/membership degree , indeterminacy degree  , falsity/ nonmembership degree ).The initial value of , ,  must be determined by experts according to existing problem.
Step 4:Make a neutrosophication process of the existing data: For ,  = 1,2, … ,  (the end of data): Rule 1: Use this equation to calculate score degree, and if the score degree of two neutrosophic numbers is not equal for any data, then choose the maximum value of score degree: Then, select   ̃ =  (   ̃,   ̃, … ,   ̃ ) for   ,  = 1,2 … ., , 1≤  ≤ , and assign the neutrosophic number ̃ to   .
Step 5: Construct the neutrosophic logical relationships (NLR) as follows: If  ̃, ̃ are the neutrosophication values of year and year  + 1 respectively, then the NLR is symbolized as ̃ →  ̃.
Step 6: Based on NLR, begin to establish the neutrosophic logical relationship groups (NLRG).
Step -If NLRG of  ̃ is one-to-one i.e. ̃ →  ̃, then the forecasted value is the middle value of  ̃.

The proposed method of forecasting based on high order NTS data
We can also apply the proposed method of forecasting based on high order NTS data: -All steps from 1 to 4 are the same as previously, but in step 5 we begin to construct the neutrosophic logical relationships (NLR) of  th order NTS where  ≥ 2.
-Based on NLR of  th order, NTS begin to establish the neutrosophic logical relationship groups (NLRG).
-Calculate the forecasted values as follows: o

Numerical examples
We solved in this section two numerical examples and compared outputs with other existing methods for verifying the applicability and superiority of the suggested method.

Numerical example 1
In this example, the suggested approach is implemented on the bench marking time series data of student enrollments at the University of Alabama from year 1971-1992 adopted from [26].The steps are as follows: Step 1: Let the two proper positive numbers  1 and  2 be 5 and 13, determined by the expert.By selecting the largest and the smallest observation from all available data which presented in Table 1, then   = 19337 ,   = 13055 respectively.Consequently, the universe of discourse U = [13055 − 5, 19337 + 13] = [13050,19350].
Step 2: Create a partition of the universe of discourse, to  triangular neutrosophic numbers, as it follows: -Determine the suitable length () of available time series data: o From Table 1, the average of absolute differences = 510.o By using base mapping table [42], the base for length of intervals = 100, since Step 4: Make a neutrosophication of the available time series data: The first value of actual enrollments is 13055 which is located only in the range of triangular neutrosophic number  ̃1, then the neutrosophication value of 13055 is  ̃1 as appears in Table 1.
In this case, we must calculate the score degree of 13563 in both N ̃1 and N ̃2 and select the maximum value.Since the score degree of 13563 in N ̃2 is greater than N ̃1, then the neutrosophication value of 13563 is N ̃2, as in Table 1.
We will apply the previous steps on the remaining data as follows: The value 13867 locates in the range of  ̃2 = 〈13350,13650,13950; 0.80,0.Step 5: Construct the neutrosophic logical relationships (NLR) as in Table 2: Step 6: Based on NLR, begin to establish the neutrosophic logical relationship groups (NLRG) as in Table 3.
Step 7: Calculate the forecasted values as in Table 4: For calculating forecasted value of 13055 in year 1971, do the following: -Look at the neutrosophication value of 13055 in year 1971 which is  ̃1 as it appears in Table 1.
-Go to NLRG which is presented in The other forecasted values are calculated in the same manners.The forecasted enrollment data obtained with the suggested method, along with the forecasted data obtained with the models in [43], [44], [45], [46], [14] and [17] are presented in Table 5.If we plan to find the second order neutrosophic logical relationships of previous example by applying the proposed method of forecasting based on the second order NTS, it will be as in Table 7.The second order neutrosophic logical relationships groups of previous example will be as in Table 8.We compared forecasted values of enrollments based on second order of neutrosophic logical relationship groups of the proposed method with the method of second order of Gautam and Singh [47].The results are presented in Table 9.The MSE and AFE of the two methods are presented in Table 10.10, it appears that our proposed method of second order is also better than the proposed method of second order of Gautam and Singh [47].
In addition, the third order neutrosophic logical relationship groups of previous example is constructed and presented in Table 11.We also compared the forecasted values of enrollments based on third order of neutrosophic logical relationship groups of proposed method with proposed methods of third order of [47], [8],and [9],and presented the results in Table 12.

Numerical example 2
We verified the proposed method by solving the TAIEX2004 example [40], and by putting  1 ,  2 equal 56, and 61 respectively, then U = [5600.17,6200.69].TAIEX2004 is used as a baseline to compare our method with other competitive methods.The objective is to compare and identify how all the methods can manage error reduction, in which RMSE is a common approach used in financial analysis.For example, Chang [55] has developed a pioneering business intelligence approach in financial stock analysis and used RMSE for error reduction and measurement.To suit our approach, we have devised it, with the aim to calculate the suitable length as illustrated previously and found that it is equal to 40.
Therefore, the number of triangular neutrosophic numbers is equal to 12.For these neutrosophic numbers, the decision makers determined the truth, indeterminacy and falsity degrees equal 0.9,0.1,0.1 respectively.The actual and forecasted values of TAIEX2004 example are presented in Table 14 and Fig. 3.The RMSE and AFE of the proposed method presented in Table 15.[48] 73.57 Chen and Kao's method [49] 58.17 Cheng et al's method [50] 54.24 Chen et al's method [51] 56.16 Chen and Chang's method [52] 60.48 Chen and Chen's method [53] 61.94 Yu and Huarng's method [54] 55.91 Proposed method 42.05By comparing the proposed method with other existing methods as appears in Fig. 4 the RMSE tools confirms that our proposed method is better than others.

Conclusion and future directions
The objective of this research was to enhance the accuracy rates of forecasting, since the accuracy rates of forecasting in the existing approaches of fuzzy and intuitionistic fuzzy time series were not good enough.Thus, in this research we introduced the notion of first and high-order neutrosophic time series data via defining the fitting length of intervals and proposing a novel method for calculating forecasted values that affect actually in the obtained results.For obtaining truth, indeterminacy and falsity membership degrees of each historical data, we defined triangular neutrosophic numbers.The neutrosophication process of historical time series data depends on the biggest score function of the triangular neutrosophic numbers.For the deneutrosophication process of first and high-order NTS, we used simple arithmetic computations.The suggested approach of first and high-order neutrosophic time series proved its superiority against other existing methods in the field of fuzzy, intuitionistic fuzzy and neutrosophic time series.In the future, we plan to apply meta-heuristic optimization techniques for improving accuracy of the suggested method.
We will apply this model for predicting other time series, such as demand forecasting, electricity consumption, etc.Furthermore, we may consider using other approaches for comparing similarities of historical data, like information entropy.

Fig. 4 .
Fig.4.The RMSE of different methods that solved the TAIEX2004 example 2 ,  3 ∈  ,and being the lower, median, and upper values of the triangular neutrosophic number.

7 :
Calculate the forecasted values as follows:Rule 1: If the neutrosophication value of   is  ̃ and it is not caused by any other neutrosophication values, and by looking at the NLRG of this value, you cannot find the value which it depends on ( i.e. ≠→  ̃ ), then the forecasted value in this case will equal − (i.e.leave it empty).The ≠ symbol means no value.Rule 2: If the neutrosophication value of   is  ̃ and it is caused by ̃( ̃ →  ̃), then look at NLRG of  ̃ , and -If NLRG of  ̃ is empty i.e. ̃ → ∅, or  ̃ →  ̃ , then the forecasted value is the middle value of  ̃.
cannot find the values which it depends on ( i.e. ≠→  ̃ ) then the forecasted value in this case will equal − (i.e.leave it empty).The ≠ symbol means no value.o

Table 3 .
The neutrosophic logical relationship groups of enrollments

Table 3
, and because  ̃1 is the first neutrosophication value of data, then it is not caused by any other value (i.e.≠→  ̃1) as in Table3.Therefore, the forecasted value of 13055 is  ̃2 as it appears in Table1, and because  ̃2 is caused by  ̃1 (i.e.Go to Table3, and look at the NLRG which starts with  ̃1, and we noted that it is

Table 4 .
Actual and forecasted values of enrollments

Table 5 .
Forecasted values by suggested method and other methods

Table 9 .
Actual and forecasted values of enrollments based on order 2 of proposed method vs.

Table 10 .
[47]r measures of proposed method and Gautam and Singh method[47]

Table 12 .
Actual and forecasted values of enrollments based on order 3 of proposed method vs.

Table 14 .
The actual and forecasted values of TAIEX2004

Table 15 .
Error measures of proposed methodFor confirming the performance of the suggested method, we compared it with other existing methods and presented the results in Table16.Compared with the existing methods, our proposed method can offer the least presence of errors since it has the most minimized RMSE.In other words, our method appears to be performing the best in reducing errors and ensuring all our analyses are accurate with insights.This may provide a new insight for business intelligence with artificial intelligence, cloud computing and neutrosophic research.

Table 16 .
Error measures of proposed method and other existing methods which solved