Virtually, forecasting plays everywhere a major role in human life, especially in making future decisions such as weather forecasting, university enrollment, production, sales and finance, etc. Based on these forecasting results, we can prevent damages to occur or get benefits from the forecasting activities.
Up to now, many qualitative and quantitative forecasting models were proposed. However, these models are unable to deal with problems in which historical data take form of linguistic constructs instead of numerical values. In recent years, many methods have been proposed to deal with forecasting problems using fuzzy time series. In this paper, we present a new method to predict the calendar day for average Arabian Gulf Oil Company using fuzzy time series approach based on average lengths of intervals. A visual-based programming is used in the implementation of the proposed model.
Results obtained demonstrate that the proposed forecasting model can forecast the data effectively and efficiently Keywords: Fuzzy time series, Forecasting, Fuzzy sets, Average-based lengthl . Forecasting the size of any phenomenon in future is important and helpful for understanding behavior of phenomenon along time. It is impossible to make future plans to face the phenomenon without defining its future dimensions and identifying shape and modes of complicated process, especially when it is related to future forecasting. Making decisions depends completely on accuracy of forecasting.
It is evident that forecasting plays major role in our daily life. The accurate and the most efficient forecasting may support making correct decisions to raise accuracy of our expectations up to 100%. This may be impossible, yet we try to reduce forecasting errors. In order to solve forecasting problems, many researchers proposed several methods and different models. One of these models is traditional time series analysis, uni-variant and multi-variant. However, traditional time series has wide applications, but it must satisfy proper conditions to be successful.
For example, 50 up to 100 bservations at least are needed to achieve Autoregressive and Integrated Moving Average Models (ARIMA) and average zero is needed to achieve autoregressive. Traditional time series has been applied in many fields such as pollution monitoring, blood pressure estimation, etc. This problem has been studied widely in statistics areas and neural networks. However, in practical life, there are regression models in which the uncertainty accompanied to the model is because of vagueness, not because of neither randomness nor both of them.
In these models, probability theory cannot be pplied and fuzzy sets theory is applied, where variables are foggy i. e. declarative variables are not fixed and scaling of these variables is not expressed by a point, but by an interval or linguistic variables [1, 3]. 2. FUZZY LOGIC Fuzzy logic [1 1], is a form of logic which has used in some expert systems and artificial intelligence applications. It was first proposed in 1965 by the Iranian scientist Lutfi Zadeh, at University of California, where he developed it as a better method of data processing.
However, his theory didn’t find a wide interest until 1974, where fuzzy logic was used to control a steam engine. Since then, applications of fuzzy logic kept developing until the manufacturing of fuzzy logic chip which have been used in many products such this science. There are many reasons for scientists to improve fuzzy logic. For example, development of computers and software founded the need to invent or program systems that are capable of dealing with ambiguous information to mimic human thinking.
However, this created a problem since computers can only deal with exact and accurate data. This problem caused occurrence of expert systems and artificial intelligence. Fuzzy logic is a theory for building such systems. Fuzzy set theory has many useful achievements in different fields and it aims at approximation of professional knowledge that contains vagueness in human thinking. Figure 1 illustrates the difference between traditional and fuzzy set theories. Fuzzy logic simply reflects how do people think and try to represent our feelings by words, decisions making and our common sense.
So, fuzzy logic models are being increasingly used in time series analysis, where they are important for dealing with linguistic values and other models in order to yield better forecasting results. Time Series is defined as a sequence of events easured in successive times at definite intervals. It was widely used in economic systems such as stock index and interest. Also, it was used in metrology, especially in wind speed, temperature, pressure, Figure 1: Traditional and fuzzy sets 3.
FUZZY TIME SERIES Fuzzy time series is another concept to solve forecasting problems in which the historical data are linguistic values. Fuzzy time series based on Zadeh’s works [1 1], Song and Chissom , first proposed a forecasting model called Fuzzy Time Series, which provided a theoretic framework to model a special dynamic process hose observations are linguistic values. The main difference between the traditional time series and fuzzy time series is that the observed values of the former are real numbers while the observed values of the latter are fuzzy sets or linguistic values.
In the following, some basic concepts of fuzzy time series are briefly reviewed Definition 1: Let U ,u2 un } be a universe of discourse (universal set); a fuzzy set A of U is defined fA (u ) / u fA (u ) / un ,where fA is a membership function of a given set A , fA [0,1]. Definition 2 If there exists a fuzzy relationship R(t – 1, t), such that F(t) =F(t ), where is an arithmetic operator, then F(t) is said to be caused by F(t – 1). The relationship between F(t) and – 1) can be denoted by – 1) Definition 3 Suppose F(t) is calculated by F(t – 1) only, and – 1) R(t,t-l).
For any t, if R(t – 1, t) is independent of t, then F(t) is considered a timeinvariant fuzzy time series. Otherwise, F(t) is timevariant. Definition 4 Suppose – 1) and F(t)= A], a fuzzy logical relationship can be defined as Ai AJ where Ai and AJ are called the left-hand side and the right-hand side of the fuzzy logical relationship, respectively. 4. REVIEW OF RELATED WORKS Many studies have interested in fuzzy time series and have been applied in various fields including university enrollment.
Fuzzy time series had proved its efficiency in forecasting as a good new method for predicting linguistic values. Song and Chisson [9, 10] first introduced the method of fuzzy time series, humidity and rainfall. In addition, time series was used in geophysical records including indexed measurements, times of earthquake, radiological activities, industrial production, rates of idleness, etc. therefore, they are considered as founders of fuzzy time series science. Also, in 1994, they introduced a eries.
Chen  presented a new method for forecasting university enrollment using fuzzy time series historical data enrollments of the university of Alabama from 1971 to 1992, the proposed method is more efficient than the proposed method by Song and Chissom, due to the fact that the proposed method uses simplified arithmetic operation rather than the complicated MaxMin composition operation. Hwang  proposed a new method on fuzzification to revise Song and Chissom’s method. He used a different triangle fuzzification method to Fuzzify crisp values.
His method involved determining an interval of xtension from both sides of crisp value in triangle membership function to get a variant degree of membership. The result got a better average forecasting error, in addition, the influences of factors and variables in a fuzzy time series model such as definition area, number and length of intervals and the interval of extension in triangle membership function were discussed in details
Step 2: Define the universe of discourse U. Find the maximum Dmax and the minimum Dmin among all Dh. For easy partitioning of U, choose two small numbers Dl and D2 as two proper positive numbers. The purpose of Dl and D2 is to make the lower and upper bounds of U become multiple of hundreds, thousands, etc. The universe of discourse U is then defined by: U = Dt-ntn -Dl , Drnax+D2 Step 3: Determine the appropriate length of interval L. Here, the average-based length method (Huarng, 2001 b) can be applied to determine the appropriate L.
The length of interval L is computed according to the Table 1: Base mapping table Range Base 0. 1-1. 0 0. 1 1. -10 11-100 10 101-1000 a) Calculate all the absolute differences between the values Dh-l and Dh as the first differences, and then compute the average of the first differences. b) Take one-half of the average as the length. c) Find the located range of the length and determine the base from Table 1 d) According to the assigned base, round the length as the appropriate L.
Then the number of intervals m, is computed by: D max+D2-D Then U can be partitioned into equal-length intervals Assume that the m intervals are Step4: Define fuzzy sets from the universe of discourse: f(un)(3) Ai=A11+A22+….. +Ai l Then fuzzify the time series. First determine some linguistic values A1, A2, … , An. Second, defined fuzzy sets on U. The fuzzy sets Ai are expressed as follows: 10. 500 0. 510 . 50 00. 51 0. 5 Step 5: fuzzify the historical data. If the value of Dh is located in the range of ui, then it belongs to fuzzy sets A’.
All Dh must be classified into the corresponding fuzzy sets. However, fuzzify the historical data and give fuzzy set to each year’s historical data. If the historical data belongs to Ai at year t, the historical data of that year can be written by A’. But usually one historical data to ifferent A1, the need to find out maximum degree of each year’s historical data belonging to each A1. Step 6: Establish fuzzy logical relationships (FLRs) for all fuzzified data, derive the fuzzy logical relationships based on Definition (3).
The fuzzy logical relationship which have the same left-hand sides is like Ai Ak, which denotes that if the Dh-lvalue of time t-1 is AJ then that of time t is Ak Table 2: Fuzzy relationship Ak Ar A1 Am 0. 5 um -2 um -1 um Where ui n) is the element and the number below ‘/’is the membership of ui to Then follow the rules for determining the degree of the membership of the istorical data Yi belonging to interval u’. The general triangular membership function is expressed as below: Step 7: establish the fuzzy logic relationship groups (FLRG).
The derived fuzzy logical relationships can be arranged into fuzzy logical relationship groups based on the same fuzzy numbers on the left-hand sides of the fuzzy logical relationships. The fuzzy logical relationship groups are like the following: AJI Step 8: The forecasting of the historical data is based on heuristic rules proposed by chen (1996) and outlined as follows.
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