Hedge Algebra Approach for Fuzzy Time series To Improve Result Of Time Series Forecasting . EAI Endorsed Transactions on Context-aware Systems and Applications

During the recent years, many different methods of using fuzzy time series for forecasting have been published. However, computation in the linguistic environment one term has two parallel semantics, one represented by fuzzy sets (computation-semantics) it human-imposed and the rest (context-semantic) is due to the context of the problem. If the latter semantics is not paid attention, despite the computation accomplished high level of exactly but it has been distorted about semantics. That means the result does not suitable the context of the problem. After all, the results are not accurate A new approach is proposed through a semantic-based algorithm, focus on two key steps: partitioning the universe of discourse of time series into a collection of intervals and mining fuzzy relationships from fuzzy time series, that outperforms accuracy and friendliness in computing. The experimental results, forecasting enrollments at the University of Alabama and forecasting TAIEX Index, demonstrate that the proposed method significantly outperforms the published ones about accurate level, the ease and friendliness on computing.

Thus, recent researches focus on the second method.There have been many methods of partitioning the.[12] presented a method of partitioning the universe of universe of Information granules were applied in [9,10,11] to get good intervals on the universe of discourse.By the hedge algebras approach Ho et al discourse.According to this approach, fuzziness intervals are used to quantify the values of fuzzy time series that are linguistic terms.These fuzziness intervals are employed as intervals on the universe of discourse.The rest of this paper is organized as follows: Section 2 briefly introduces some basis concepts of HA; Section 3 presents the proposed method; Section 4 presents empirical results on forecasting enrollments at University of Alabama, Forecasting AITEX Index and comment; Section 5 concludes the paper.

2.Preliminaries
In this section, we briefly recall some concepts associated with fuzzy time series and hedge algebras.

Fuzzy time series
Fuzzy time series was first introduced by Song and Chissom [1], it is considered as the set of linguistic values that is observed by the time.Linguistic values are also called linguistic terms.It can be seen that conventional time series are quantitative view about a random variable because they are the collection of real numbers.In contrast to this, as the collection of linguistic terms, fuzzy time series are qualitative view about a random variable.There are two types of fuzzy time series, timeinvariant and time-variant fuzzy time series.Because of practicality, the former is the main subject which many of researchers focus on.In most of literature, the linguistic terms are quantified by fuzzy sets.Formally, fuzzy time series are defined as following definition Definition 1.Let Y (t) (t = ...,0,1,2,...), a subset of R 1 , be the universe of discourse on which f i (t) (i = 1,2,...) are defined and F(t) is the collection of f i (t) (i = 1,2,...).Then F(t) is called fuzzy time series on Y (t) (t = ...,0,1,2,...).
Song and Chissom employed fuzzy relational equations as model of fuzzy time series.Specifically, we have following definition: Definition 2. If for any f j (t) ∈ F(t), there exists an f i (t-1) ∈ F(t-1) such that there exists a fuzzy relation R ij (t,t-1) and f j (t) = f i (t-1)∘R ij (t,t-1) where "∘" is the max-min composition, then F(t) is said to be caused by F(t-1) only.Denote this as f i (t-1) → f j (t) or equivalently F(t-1) → F(t).In [2,3], Song and Chissom proposed the method which use fuzzy time series to forecast time series.Based upon their works, there are many studies focus on this field.

Some basis concepts of Hedge Algebras
" In the HAapproach, it seems to be essential that the fuzziness measure of words of a variable, which is a quantitative characteristic expressing an essential and key semantic aspect of the fuzzy linguistic information, does play a centric role in the determination of other quantitative characteristics of words, such as the fuzziness intervals of words, the similarity intervals and the semantically quantifying mappings (or the numeric semantics of) words, when providing the values the fuzziness parameters of the variable.In summary, this approach is developed based on a convincing logical and mathematical foundation, as the inherent word semantics and its fuzziness are defined and formalized in an axiomatization manner" [14] In this section, we briefly introduce some basis concepts in HA, these concepts are employed as basis to build our proposed method.HA are created by Ho Cat Nguyen et al. in 1990.This theory is a new approach to quantify the linguistic terms differing from the fuzzy set approach.The HA denoted by AX = (X,G,C,H,≤ ), where, G = {c + ,c − } is the set of primary generators, in which c + and c − are, respectively, the negative primary term and the positive one of a linguistic variable X, C = {0,1,w} a set of constants, which are distinguished with elements in X,H is the set of hedges, "≤" is a semantically ordering relation on X.For each x ∈ X in HA, H(x) is the set of hedge u ∈ X that generated from x by applying the hedges of H and denoted u = h n ,...,h 1 x, with h n ,...,h 1 ∈ H. H = H + ∪H − , in which H − is the set of all negative hedges and H + is the set of all positive ones of X.The positive hedges increase semantic tendency and vise versa with negative hedges.Without loss of generality, it can be assumed that If X and H are linearly ordered sets, then AX = (X,G,C,H,≤ ) is called linear hedge algebra, furthermore, if AX is equipped with additional operations Σ and Φ that are, respectively, the infimum and supremum of H(x), then it is called complete linear hedge algebra (ClinHA) and denoted AX = (X,G,C,H,Σ,Φ,≤).Complete linear hedge algebra(ClinHA) There are also following important concepts and properties are present in [12]: -Fuzziness interval of terms in X and its properties.

It is user's Linguistic
 is a set of similar fuzzy space of all grades from and two similar fuzzy space close to g(x) as follows: Where y,z are two grades defining two similar fuzzy space neighbors left and right of g(x).(See the Figure 2.)

Calculations on the language value apply to the forecast
In Figure 2   are member functions representing the following similar fuzzy intervals, where in the order U 1 corresponds to A 1 and ...... U 7 corresponds to A 7 Here the set of 3 linguistic in order from left to right only the left end, the semantic core, and the right end of a similarity Interval for semantic.According to Definition 1. [12], Definition 2. [12] and on Figure 2 Show: 0 U , , U 1 ,…….U 7 , 1 U are fuzzy triangular sets created Similar fuzzy space of elements {Very Low(V.Low), Low(Lw), LitleLow(L.Lw), W, LitleHigh(L.Hi), High(Hi), VeryHigh(V.Hi) }, It is also the membership function in the order of similar fuzziness interval A 1 ,……A 7 In Where y,z are two grades defining two similar fuzzy space neighbors left and right of x .this is membership funtion of similar fuzzy interval A i (i=1,….7)Easy to deduce that: if () In [15] we have clearly stated how to build a HA that matches the context of the problem.In this section we introduce additional expressions for calculating linguistic values according to two HA parameters.This works for problem solving.More importantly, it solves the problem by using the neural network method or the Ge 3.1.2.Specifying some expressions.Similarly, for cases we have the numerical result in Figure 2.
"There is an induced about the trend change in the forecasting in the discourse space into the space of [0,1] where there is a trend change to the quantitative semantics value due to the impact on the terms of the hedges.That is the basic to we construct the mathematics model for forecasting time series by (HA) approach" [15].We want to say about µ(h), the single operator impact on the operand (language  -Change of each value "Continue to increase or decrease" or "change in the opposite direction".as a bridge between the two semantics: "inherent" and "represented by fuzzy set imposed by the user" of each word On that basis, we propose the following new time series forecasting algorithm.The algorithm emphasizes semantics generated by the problematic context and focuses on two steps: -Adjust spacing partitioning the universe of discourse -Set up a logical relationship group

Forecasting Algorithm (Algorithm Based on semantics)
-For convenience to present proposed method, we name the linguistic values of fuzzy time series as the variables A i with i ∈ N. Revυ(x) and Rev f m (x) are reversed mapping of υ(x) and f m (x), respectively, from [0,1] to the universe of discourse of fuzzy time series U. Denote I k , on U, is the interval corresponding to A k .
-7 basic language values: Very Low (V.Low), Low, L.Low, W, Little High(L.Hi), Hi, V.Hi.-Adjust the position of the historical values near the boundaries of the divisors intervals to reach the optimal divise method.
-The logical relation group is established as follows for mining fuzzy relationships from fuzzy time series In A j →A i Who: -if j> i then the group is forecasting down, -if j = i then the group forecasting is equal, -if j < i then the group is forecasting increase.Hedge Algebra Approach for Fuzzy Time series To Improve Result Of Time Series Forecasting.

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-Assume Set the group of fuzzy logical relationships is established in the Step 2 having the same left side : -Suppose that the value of the time series at (t-1 )have known according to above logical relationship groups if f(t) belong to Revfm(At), then The forecasting value at t is .Experiment Result and Comment.

Enrollment Forecasting.
The proposed approach is applied to forecast the enrollments at the University of Alabama from year 1971 to 1992 (n = 22).The result will then be compared with different published methods.To measure the accuracy of the forecasting methods, the following metrics are used for comparison that Defined in [15] RMSE: The Root Mean Square Error; NE(%):The Numerical Error (NE) percentage

NEE(%):The Normalized Numerical Error (NNE) percentage
According to [15] we have (W) =  =0.4563 and µ(V) ==0.4563 are parameter values use to constructing 7 similar fuzzy intervals for partitioning the universe of discourse are illustrated in Figure 2 .
However, in this division at A4 there are two levels of value between the "low" and the "high" as the "elements of meaning" so that the semantic difference in this range is greater than the difference in numeric value.Therefore, these values must be adjusted accordingly.After adjusting the reasonable divisions we have (VL.Lw replace L.Low) Table 2. Shown left, right and Rve. of A i The following is the result of our statistics together with the results of other authors for omparison.

Table 3. shown Metrics of results of the methods
The details are shown in Table 4 below

Forecasting AITEX Index
In this section, our proposed method is compared to [13].Chen et.al.[13] suggested a method consisting of 6 steps to calculate the forecast TAIEX Index, these steps are listed below: Propose a method to fuzzify the historical training data of TAIEX into fuzzy sets to from fuzzy logical relationships.
• Grouped the logiccal relationships into fuzzy logical relationship groups (FLRGs) based on the fuzzy variations of secondary factor.
• Evaluate the leverage of the fuzzy variations between the main factor and the secondary factor to construct fuzzy variation groups.

A i Left and right of A i Rve(A i )
A • Get the statistics of the fuzzy variations appearing in each fuzzy variation group.
• Calculate the weight of the statistics of the fuzzy varitions appearing in each fuzzy variation group, respectively.
• Use the weights of the statistics of the fuzzy variations appearing in the fuzzy variation groups and the FLRGs to perform the forecasting the daily TAIEX.
Chen et.al.[13] have applied their proposed method on the experimental data sets TAIEX Index of November and December 2004.The data set consists of 44 items.
In the first step, the historical training data of TAIEX is fuzzified into 9 fuzzy sets (h = 9 form A1 to A9), the accuracy metrics of the result are Hedge Algebra Approach for Fuzzy Time series To Improve Result Of Time Series Forecasting.
Our proposed method is applied to the same TAIEX datasets.The process is as follows According to Algorithm Based on semantics.
According to [15] we have (W) =  =0.52 and µ(V) ==0.29 are parameter values use to constructing 7 similar fuzzy intervals for partitioning the universe of discourse.Then perform the remaining steps of the algorithm.Forecast results according to the metrics as shown below with previous results for Compare.

Comment
The First of all, this proposed method is an improvement of the method already in [15], so it has advantages over the methods of the previously published authors, we briefly recall : We compare our approach with the method Wei Lu et.al.published in [11] to illustrate our superior efficiency.

Conclusion of his methodological
advantages of semantic assurance due to the context of Wei Lu stated "Interval information granules are always run through the whole process of finding optimal intervals, which make the partitioned intervals carry apparent semantics" [11].As so , in the method of Wei Lu also pay attention to the balance between accuracy and semantics which is suitable with context in the calculation that is why we compare it with our approach.Our comparison focuses on two aspects: the calculation convenience and the forecasting accuracy.
• First, the convenience in calculations: only with the simple calculations using Method for partitioning the universe of discourse and Algorithm for forecasting, we have obtained results about group of logical relationship like the results from Wei Lu [11].However, our calculation is • Second, the forecasting accuracy: Table 3.
shows our proposed method is about 10% better in term of accuracy (all metrics) compared to Wei Lu et.al.approach [11].[15] The Second, As its name implies, semantic-based algorithms for performing two important steps are for partitioning the universe of discourse and data mining through logical relational grouping.Thus, a more convenient and efficient method of adjusting the divide interval and predicting more accurately than similar functional methods show in [15].In terms of accuracy, the greater the number of divisions, the higher the accuracy.In both empirical problems forecasting enrollments at the University of Alabama forecating (forecasting enrollments) and forecasting TAIEX Index (forecasting TAIEX) we used the divisor of 7 and compared the results of other methods with the number of divisions equal (7 for forecasting enrollments) and larger (9 for forecasting TAIEX).The accuracy of the method proposed (for metrics RMSE: The Root Mean Square Error) -for the forecasting enrollments problem was 20.23% higher than that of Lu and 15.25% higher than our results at [15] -for the forecasting TAIEX problem The accuracy of the proposed approach to the problem was 52.72% higher than that of Chen and 36.12%higher than our results at [15].The numbers are very impressive and very convincing demonstrates the superiority of the accuracy of the proposed method!1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 Chart Title  6.

5.Conclusion
This paper presents a novel technique based on the hedge algebras (HA) approach.It is highlights in proposed method:  Analyze the data of forecasting problem, special for historical values and their relationship to determine the fuzzy parameter set of hedge algebras. Thereby the context-semantic of terms has preserved in the calculation.which concentrated on key steps: partitioning the universe of discourse of time series into a collection of intervals, mining fuzzy relationships from fuzzy time series,  Forecasting outputsit do not have to choose fuzzy sets for linguistics terms and defuzzifying output, which are the required steps in method based on fuzzy set theory.This is subjective imposition and so it is the reason for separating the two types of semantics mentioned above.
Frame of Cognitive and is basis of calculations on the word makes the calculation method are simple and accurate Here we just add two properties related to (Clin.HA) which have two factors (one negative and one positive the similar fuzzy space g(x) we have triangular fuzzy sets: Here the set of 3 linguistic values for example the very little low-denominated (VL.Lw) are the vertices of the triangle 0 U :=(0, 0, ( )); 1 U := ( ( ),1,1)

Figure 1 .Figure 2 .
Figure 1.Shown Partition [0,1] by the similar fuzziness iterval sets of the Hedge algebras) For Similar fuzzy space and similar fuzziness interval.

5 value)
generate new semantics for it -to create an upward or downward direction of of the operand, corresponding to the change of time series at a timeis an important factor for the time series forecasting."According to the context, semantics of ̅( ) denotes number of the enrollment students at the medium level and W is the normalization value of ̅( ), they are calculated according formulas" :[16] Determining the two parameters of the HA through the analysis of the relationship of historical values in the properties: -The average value -the boundary between the main semantic value: what is "high" and what is "low".

Step 1 :Step 2
Constructing the Hedge algebra (HA) Constructing the Hedge algebra (HA) is consistent with the context of the forecasting problem by determining the fuzzy parameters set of HA based on the historical values relationship analysis of the time series .Specifically -Determine the U, the universe of discourse of fuzzy time series F(t).U = [min(F(t))−D 1 ,max(F(t))+D 2 ], where D 1 and D 2 are proper positive numbers.with x i (1≤ i ≤ n) are historical values If S + ≥ S − then h := h +1 else h := h −1 Method for partitioning the universe of discourse , fuzzifying historical data of time series and mining fuzzy relationships from fuzzy time series -Based on the explanation in Figure 2, constructing 7 similar fuzzy intervals correspond to 7 basic language values to partitioning the universe of discourse -Based upon the distribution of historical values, put them into the corresponding linguistic term fuzziness interval for fuzzifying historical data of time series.

Table 1 .
The value of the fuzzy spaces for the calculation EAI Endorsed Transactions on Context-aware Systems and Applications 03 2018 -06 2018 | Volume 4 | Issue 14 | e2

Table 4 .
Shown the detailed results of the proposed method and the preceding results.

Table 5 .
shown Metrics of results of the methods .The details are shown in Table6 below

Table 6 .
Shown the detailed results of the proposed method and the preceding results.