a computational method for fuzzy time series ...

2nd Reading November 23, 2012 14:44 WSPC/262-IJMSSC/S1793-9623

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International Journal of Modeling, Simulation, and Scientific Computing Vol. 4, No. 1 (2013) 1250023 (12 pages) c World Scientific Publishing Company DOI: 10.1142/S1793962312500237

Int. J. Model. Simul. Sci. Comput. Downloaded from www.worldscientific.com by Dr. Sanjay Kumar on 11/29/12. For personal use only.

A COMPUTATIONAL METHOD FOR FUZZY TIME SERIES FORECASTING BASED ON DIFFERENCE PARAMETERS

BHAGAWATI P. JOSHI∗ and SANJAY KUMAR† Department of Mathematics, Statistics & Computer Science G. B. Pant University of Agriculture & Technology Pantnagar 263 145, Uttarakhand, India ∗[email protected] †[email protected] Received 8 February 2012 Accepted 21 August 2012 Published 27 November 2012 Present study proposes a method for fuzzy time series forecasting based on difference parameters. The developed method has been presented in a form of simple computational algorithm. It utilizes various difference parameters being implemented on current state for forecasting the next state values to accommodate the possible vagueness in the data in an efficient way. The developed model has been simulated on the historical student enrollments data of University of Alabama and the obtained forecasted values have been compared with the existing methods to show its superiority. Further, the developed model has also been implemented in forecasting the movement of market prices of share of State Bank of India (SBI) at Bombay Stock Exchange (BSE), India. Keywords: Fuzzy time series; time variant; fuzzy membership grade; computational algorithm.

1. Introduction Fuzzy time series forecasting emerged as a noble approach for predicting the future values in a situation when the information is imprecise and vague. The concept of fuzzy logic having linguistic variables presented by Zadeh1,2 was successfully employed by Song and Chissom3–5 to approximate reasoning to develop the foundation of fuzzy time series forecasting. Song and Chissom4,5 implemented their developed time invariant and time variant models on the historical time series data of student enrollments of University of Alabama. Chen6 presented a simplified time invariant method for time series forecasting by using the arithmetic operations in place of max–min composition operations. A higher order fuzzy time series model for forecasting the enrollments was also proposed by Chen.7 One of the major problems lying with fuzzy time series forecasting models is the accuracy in the forecasted values. 1250023-1


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B. P. Joshi & S. Kumar

Own and Yu8 presented a heuristic higher order model by introducing a heuristic function to incorporate the heuristic knowledge to improve Taiwan Future Exchange (TAIFEX) forecast. Tsaur et al.9 made an analysis of fuzzy relations in fuzzy time series on the basis of entropy of the system, used it to determine the minimum value of invariant time index to minimize errors in the forecasted values of enrollments. Yu10 presented a refined fuzzy time series model and a weighted fuzzy time series model for TAIFEX forecasting. A ratio-based length of intervals in place of equal length intervals have been presented by Huarng and Yu11 to improve the fuzzy time series forecasting. Cheng et al.12 used the two approaches; one by using minimize entropy principle approach to partition the universe of discourse and other by using trapezoid fuzzification approach to improve the accuracy in fuzzy time series forecasting. Abd Elaal et al.13 presented a fuzzy time series model using fuzzy clustering. Kai et al.14 presented a novel concept of k-mean clustering for fuzzy time series forecasting. Singh15,16 presented difference parameters-based computational algorithm for forecasting with fuzzy time series and implemented it to forecast the enrollments of University of Albama to show its superiority over the existing methods proposed by various researchers.4,9,17,18 In this paper, we present an improved and enhanced computational algorithm based on difference parameters for fuzzy time series forecasting. In proposed computational algorithm, unique set of difference parameters is calculated for each forecast. The proposed algorithm is of linear order. It minimizes the time of generating relational equations by using complex min–max composition operations and the time consumed by the various defuzzification process. It also overcomes the difficulty of searching a suitable defuzzification procedure providing crisp output of better accuracy. The proposed algorithm has been implemented for forecasting the enrollments of University of Alabama and the results have been compared with the existing methods to show its superiority. Further, it has also been implemented on the historical time series data of the movement of market price of share of State Bank of India (SBI) at Bombay Stock Exchange (BSE), India. Rest of the paper is organized as follows. In Sec. 2, we briefly review the definition of fuzzy time series. In Sec. 3, we present an improved computational algorithm for fuzzy time series forecasting. In Sec. 4, the proposed method is implemented to forecast the enrollments at University of Albama. We also make a comparison to compare the proposed method with existing methods. In Sec. 5, we make a comparison of Singh’s15 method and proposed method by implementing it to forecast market prices of SBI share at BSE. The conclusions are discussed in Sec. 6. 2. Brief Idea about Fuzzy Time Series The various definitions and properties of fuzzy time series forecasting found in literature mentioned in Sec. 1 are summarized and are presented as: 1250023-2


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A Computational Method for Fuzzy Time Series Forecasting Based on Difference Parameters

Definition 1. A fuzzy set is a class of objects with a continuum of grade of membership. Let U be the Universe of discourse with U = {u1 , u2 , u3 , . . . , un }, where ui are possible linguistic values of U , then a fuzzy set of linguistic variables Ai of U is defined by Ai = µAi (u1 )/u1 + µAi (u2 )/u2 + µAi (u3 )/u3 + · · · + µAi (un )/un ,


where µAi is the membership function of the fuzzy set Ai , such that µAi : U → [0, 1]. Definition 2. Let Y (t)(t = . . . , 0, 1, 2, 3, . . .), is a subset of R, be the Universe of discourse on which fuzzy sets fi (t), (i = 1, 2, 3, . . .) are defined and F (t) is the collection of fi , then F (t) is defined as fuzzy time series on Y (t). As at different times, the values of F (t) can be different, F (t) is a function of time and in a similar way, the universe of discourse Y (t) may be different at different times. Definition 3. Suppose F (t) is caused only by F (t−1) and is denoted by F (t−1) → F (t); then there is a fuzzy relationship between F (t) and F (t − 1) and can be expressed as the fuzzy relational equation: F (t) = F (t − 1)oR(t, t − 1) here “o” is Max–Min composition operator. Definition 4. Suppose F (t), a fuzzy time series and R(t, t − 1) be a first-order model of F (t) such that R(t, t − 1) = R(t − 1, t − 2) for any time t, the F (t) is called time-invariant fuzzy time series. But if R(t, t − 1) is time dependent i.e., R(t, t − 1) may be different from R(t − 1, t − 2) for any time t then F (t) is called time-variant fuzzy time series. Definition 5. If F (t) is caused by more fuzzy sets, F (t−n), F (t−n+1), . . . , F (t−1), the fuzzy relationship is represented by Ai1 , Ai2 , Ai3 , . . . , Ain → Aj here F (t − n) = Ai1 , F (t − n + 1) = Ai2 , . . . , F (t − 1) = Ain . This relationship is called nth -order fuzzy time series model. In the next section, we are proposing a computational algorithm for computation of these relations and its implementation in forecasting. In the present study, the model developed uses the differences in enrollments of past three years and have been considered a fuzzy parameter in framing the fuzzy rules to impose on current year fuzzified enrollment to get forecast of next year enrollment. The computational algorithm is of order three based on different sets of difference parameters. 3. Methodology of Proposed Method for Fuzzy Time Series Forecasting In this section, we present the stepwise procedure of the proposed method of forecasting by fuzzy time series based on historical time series data. 1250023-3


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(1) Define the Universe of discourse, U based on the range of available time series data, by rule U = [Dmin − D1 , Dmax − D2 ] where D1 and D2 are two proper positive numbers. (2) Partition the Universe of discourse into equal length of intervals: u1 , u2 , . . . , um . The number of intervals will be in accordance with the number of fuzzy sets A1 , A2 , . . . , Am . (3) Construct the fuzzy sets Ai in accordance with the intervals in Step 2 and apply the triangular membership rule to each interval in each triangular fuzzy set so constructed. Further fuzzify the data with the help of maximum membership value. (4) Rules for forecasting Some notations used are defined as follows: [∗Aj ] is corresponding interval uj for which membership in Aj is supremum (i.e., 1) L[∗Aj ] is the lower bound of interval uj U [∗Aj ] is the upper bound of interval uj l[∗Aj ] is the length of the interval uj whose membership in Aj is supremum (i.e., 1) M [∗Aj ] is the mid value of the interval uj having supremum value in Aj For a fuzzy logical relation Ai → Aj : Ai is the fuzzified enrollments of year n Aj is the fuzzified enrollments of year n + 1 Ei is the actual enrollments of year n Ei−1 is the actual enrollments of year n − 1 Ei−2 is the actual enrollments of year n − 2 Fj is the crisp forecasted enrollments of the year n + 1 The proposed model utilizes the historical data of years n − 2, n − 1, n for framing rules to implement on fuzzy logical relation, Ai → Aj , where Ai , the current state, is the fuzzified enrollments of year n and Aj , the next state, is fuzzified enrollments of year n + 1. In the proposed computational algorithm, new set of difference parameters is computed for each forecast rather than using the same set of difference parameters. The proposed method for forecasting is mentioned as computational algorithms for generating the relations between the time series data of years n − 2, n − 1, n for forecasting the enrollment of year n + 1. Computational algorithm: Forecasting enrollments for year n + 1 (i.e., 1974) and onwards. F or k = 3 to K (end of time series data) Obtained fuzzy logical Relation for year k to k + 1 Ai → Aj 1250023-4


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f or q = k − 2 to k + 1 Compute Dq = ||(Eq − Eq−1 )| − |(Eq−1 − Eq−2 )|| Rq = Eq + Dq /2 RRq = Eq − Dq /2 Sq = Eq + Dq SSq = Eq − Dq Next q F or i = k − 2 If Ri ≥ L[∗Aj ] and Ri ≤ U [∗Aj ] T hen P1 = Ri ; m = 1 Else P1 = 0; m = 0 End if F or i = k − 1 If RRi ≥ L[∗Aj ] and RRi ≤ U [∗Aj ] T hen P2 = RRi ; n = 1 Else P2 = 0; n = 0 End if F or i = k If Si ≥ L[∗Aj ] and Si ≤ U [∗Aj ] T hen P3 = Si ; o = 1 Else P3 = 0; o = 0 End if F or i = k + 1 If SSi ≥ L[∗Aj ] and SSi ≤ U [∗Aj ] T hen P4 = SSi ; p = 1 Else P4 = 0; p = 0 End if B = P1 + P2 + P3 + P4 If B = 0 T hen Fj = M (∗Aj ) Else Fj = (B + M (∗Aj ))/(m + n + o + k + 1) End if N ext k Mean Square Error (MSE) and Average Forecasting Error are common tools to measure the accuracy in fuzzy time series forecasting. Lower the MSE or average error, better the forecasting method. The MSE and forecasting error are defined as n Mean Square Error =

i=1

(actuali − forecastedi)2 , n

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(1)


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Forecasting Error =

|forecasted − actual| × 100, actual

Average forecasting Error (in %) =

sum of forecasting error . number of errors

(2)

(3)


4. Computation of Enrollments Forecast In this section, we implement algorithm of the proposed method on the time series data of student enrollments at University of Alabama and the stepwise results obtained are: Step 1: Universe of discourse U = [13, 000, 20, 000]. Step 2: The Universe of discourse is partitioned into seven intervals: u1 = [13, 000, 14, 000], u2 = [14, 000, 15, 000], u3 = [15, 000, 16, 000], u4 = [16, 000, 17, 000], u5 = [17, 000, 18, 000], u6 = [18, 000, 19, 000], u7 = [19, 000, 20, 000]. Step 3: Seven fuzzy sets A1 , A2 , . . . , A7 on the universe of discourse U and the membership grades to these fuzzy sets are defined as: A1 = 1/u1 + 0.5/u2 + 0/u3 + 0/u4 + 0/u5 + 0/u6 + 0/u7 , A2 = 0.5/u1 + 1/u2 + 0.5/u3 + 0/u4 + 0/u5 + 0/u6 + 0/u7, A3 = 0/u1 + 0.5/u2 + 1/u3 + 0.5/u4 + 0/u5 + 0/u + 0/u7 , A4 = 0/u1 + 0/u2 + 0.5/u3 + 1/u4 + 0.5/u5 + 0/u6 + 0/u7, A5 = 0/u1 + 0/u2 + 0/u3 + 0.5/u4 + 1/u5 + 0.5/u6 + 0/u7, A6 = 0/u1 + 0/u2 + 0/u3 + 0/u4 + 0.5/u5 + 1/u6 + 0.5/u7, A7 = 0/u1 + 0/u2 + 0/u3 + 0/u4 + 0/u5 + 0.5/u6 + 1/u7 . The fuzzified historical time series data of enrollments are obtained and fuzzy logical relations are established (Table 1). Step 4: Using the proposed method, (Rule for forecasting) in Sec. 3, the computations have been carried out for the proposed model and the results obtained are placed in Table 2 along with forecasted values of other methods. MSE and average error of forecast have been computed and are placed in Table 3 to compare the accuracy in forecasted values of proposed model with other models. 1250023-6


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Table 1. Actual and fuzzified enrollments of University of Alabama.

Table 2.

Year

Actual enrollments

Fuzzified enrollments

1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992

13,055 13,563 13,867 14,696 15,460 15,311 15,603 15,861 16,807 16,919 16,388 15,433 15,497 15,145 15,163 15,984 16,859 18,150 18,970 19,328 19,337 18,876

A1 A1 A1 A2 A3 A3 A3 A3 A4 A4 A4 A3 A3 A3 A3 A3 A4 A6 A6 A7 A7 A6

Forecasted enrollments by various methods and proposed method.

Year

Actual

Proposed model

Kai et al.14

Cheng et al.19

Singh15 model

Chen6 model

Huarng17 Heuristic

Lee and Chou18

Tsaur et al.9

1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992

14,696 15,460 15,311 15,603 15,861 16,807 16,919 16,388 15,433 15,497 15,145 15,163 15,984 16,859 18,150 18,970 19,328 19,337 18,876

14,544 15,504 15,456 15,599 15,723 16,482 16,603 16,340 15,356 15,408 15,425 15,395 15,471 16,573 18,683 18,646 19,373 — —

13,997 15,461.2 15,461.2 15,461.2 15,461.2 16,861.7 17,394 17,394 15,461 15,461.2 15,461.2 15,461.5 15,461.5 16,861.7 17,394 18,932.2 18,932.2 18,932.2 18,932.2

14,242 15,474.3 15,474.3 15,474.3 15,474.3 16,146.5 16,988.3 16,988.3 16,146.5 15,474.3 15,474.3 15,474.3 15,474.3 16,146.5 16,988.3 19,144 19,144 19,144 19,144

14,286 15,361 15,468 15,512 15,582 16,500 16,361 16,362 15,744 15,560 15,498 15,302 15,442 16,558 17,187 18,475 19,382 19,487 18,744

14,000 15,500 16,000 16,000 16,000 16,000 16,833 16,833 16,833 16,000 16,000 16,000 16,000 16,000 16,833 19,000 19,000 19,000 19,000

14,000 15,500 15,500 16,000 16,000 16,000 17,500 16,000 16,000 16,000 15,500 16,000 16,000 16,000 17,500 19,000 19,000 19,500 19,000

14,568 15,654 15,654 15,654 15,654 16,197 17,283 17,283 16,197 15,654 15,654 15,654 15,654 16,197 17,283 18,369 19,454 19,454 —

14,000 15,500 15,500 16,000 16,000 16,500 16,500 15,500 15,500 15,500 15,500 15,500 16,500 18,500 19,000 19,000 19,000 19,000 —

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B. P. Joshi & S. Kumar Table 3. Comparison of MSE and average forecasting error of proposed method with other methods. Model

Proposed model

Kai et al.14

Cheng et al.19

Singh15 model

Chen6 model

Huarng17 Heuristic

Lee and Chou18

Tsaur et al.9


MSE 67,943.47 183,855.8 24,0975.5 147,015.1 483,532.4 265,190.2 256,662.2 340,660.8 Average 1.264196 1.918372 2.364449 1.830663 3.454714 2.686384 2.611228 2.581875 Forecasting Error

5. Forecasting Market Prices of SBI Share at BSE In this section, the proposed method is implemented for forecasting the market prices of SBI share at BSE. Share market system is very dynamic in nature which involves hidden uncertainties and uncontrolled parameters. The actual market prices of share of SBI from April 2008 to March 2010 are given in Table 4. Step 1: Annual report of SBI19,20 for the year 2008–2010 is used to define Dmin and Dmax . Then for this we define universe of discourse U = [1000, 2600].

Table 4.

Actual and fuzzified market prices of SBI share.

Month April 2008 May 2008 June 2008 July 2008 August 2008 September 2008 October 2008 November 2008 December 2008 January 2009 February 2009 March 2009 April 2009 May 2009 June 2009 July 2009 August 2009 September 2009 October 2009 November 2009 December 2009 January 2010 February 2010 March 2010

Actual prices at BSE (in Rs.)

Fuzzified prices

1819.95 1840.00 1496.70 1567.50 1638.90 1618.00 1569.90 1375.00 1325.00 1376.40 1205.90 1132.25 1355.00 1891.00 1935.00 1840.00 1886.90 2235.00 2500.00 2394.00 2374.75 2315.25 2059.95 2120.05

A5 A5 A3 A3 A4 A4 A3 A2 A2 A2 A2 A1 A2 A5 A5 A5 A5 A7 A8 A7 A7 A7 A6 A6

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Table 5.

Forecasted prices by proposed model and Singh model.

Year

Actual

Proposed model

Proposed by Singh15

April 2008 May 2008 June 2008 July 2008 August 2008 September 2008 October 2008 November 2008 December 2008 January 2009 February 2009 March 2009 April 2009 May 2009 June 2009 July 2009 August 2009 September 2009 October 2009 November 2009 December 2009 January 2010 February 2010 March 2010

1819.95 1840.00 1496.70 1567.50 1638.90 1618.00 1569.90 1375.00 1325.00 1376.40 1205.90 1132.25 1355.00 1891.00 1935.00 1840.00 1886.90 2235.00 2500.00 2394.00 2374.75 2315.25 2059.95 2120.05

— — — 1499.58 1682.71 1669.60 1526.03 1325.53 1310.11 1349.53 1278.025 1091.91 1290.225 1900.00 1876.60 1895.825 1876.11 2317.50 2496.20 2313.16 2328.95 2283.625 — —

— — — 1499.58 1665.94 1665.94 1499.58 1300.00 1300.00 1300.00 1300.00 1100.00 1300.00 1900.00 1900.00 1900.00 1900.00 2300.00 2400.00 2300.00 2300.00 2300.00 2100.00 2100.00

Step 2: The Universe of discourse is partitioned into eight intervals of linguistic values: u1 = [1000, 1200], u2 = [1200, 1400], u3 = [1400, 1600], u4 = [1600, 1800], u5 = [1800, 2000], u6 = [2000, 2200], u7 = [2200, 2400], u8 = [2400, 2600]. Step 3: Eight fuzzy sets A1 , A2 , A3 , . . . , A8 , on the universe of discourse U and membership grades to these fuzzy sets are defined as: A1 = 1/u1 + 0.5/u2 + 0/u3 + 0/u4 + 0/u5 + 0/u6 + 0/u7 + 0/u8 , A2 = 0.5/u1 + 1/u2 + 0.5/u3 + 0/u4 + 0/u5 + 0/u6 + 0/u7 + 0/u8 , A3 = 0/u1 + 0.5/u2 + 1/u3 + 0.5/u4 + 0/u5 + 0/u6 + 0/u7 + 0/u8 , A4 = 0/u1 + 0/u2 + 0.5/u3 + 1/u4 + 0.5/u5 + 0/u6 + 0/u7 + 0/u8 , A5 = 0/u1 + 0/u2 + 0/u3 + 0.5/u4 + 1/u5 + 0.5/u6 + 0/u7 + 0/u8 , A6 = 0/u1 + 0/u2 + 0/u3 + 0/u4 + 0.5/u5 + 1/u6 + 0.5/u7 + 0/u8 , 1250023-9


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B. P. Joshi & S. Kumar Table 6.

MSE and average forecasting error of market prices forecast.

Model

Proposed model

MSE Average forecasting error

Proposed by Singh15

2221.773 2.561074

3608.053 3.18418

20000

Forecasted enrollment

18000

Forecasted

Actual Proposed Kai.... Cheng... Singh Chen Huarng Lee... Tsaur...

17000

16000

15000

1992

1991

1990

1988

1989

1987

1985

1986

1984

1983

1981

1982

1980

1979

1978

1976

1977

1974

1975

1973

14000

1972


19000

Years Fig. 1.

Actual enrollments versus forecasted enrollments.

A7 = 0/u1 + 0/u2 + 0/u3 + 0/u4 + 0/u5 + 0.5/u6 + 1/u7 + 0.5/u8, A8 = 0/u1 + 0/u2 + 0/u3 + 0/u4 + 0/u5 + 0/u6 + 0.5/u7 + 1/u8. Now the market prices of SBI share at BSE are fuzzified with triangular membership function and are placed in Table 4. Step 4: Using the proposed algorithm, (Rule for forecasting) given in Sec. 3, the computations have been carried out for the market prices of share of SBI at BSE. The forecasted prices have been obtained by using the proposed algorithm in Sec. 3. The forecasted prices have also been obtained for the market prices by algorithm given by Singh.15 The forecasted market prices of SBI share at BSE obtained by these two methods are placed in Table 5. The MSE and average error of forecast have been computed and are placed in Table 6 for comparison of accuracy in forecasted values of our proposed model with the method given by Singh.15 1250023-10


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Forecasted enrolment of SBI share prices

2600 2400

2000 1800

Actual Proposed Singh

1600 1400 1200

Jan. 10

Dec. 09

Oct. 09

Nov. 09

Sept. 09

July 09

Aug. 09

May 09

June 09

Mar. 09

April 09

Jan. 09

Feb. 09

Dec. 08

Oct. 08

Nov. 08

Sept. 08

July 08

1000 Aug. 08


Forecasted

2200

Months Fig. 2.

Actual versus forecasted market price of SBI share at BSE.

Table 6 shows the superiority of the proposed models over the Singh’s15 model as it provides forecast of higher accuracy. The forecasted values obtained by the proposed method are significantly in closed accordance to the actual prices (Fig. 2).

6. Conclusion In this paper we have proposed an improved and versatile method for fuzzy time series forecasting. The algorithm of the proposed method is simple as it minimizes the complicated computations of fuzzy relational matrices and search for a suitable defuzzification process and provides the forecasted values of better accuracy. The method has been implemented on the historical time series data of enrollments of University of Alabama to have a comparative study with existence methods. Further the method has also been implemented to forecast the market prices of SBI share at BSE, India. In both applications, the proposed method is found to be superior in terms of MSE, average forecasting error (Tables 3 and 6) and in terms of close trend of time series data with actual values (Figs. 1 and 2.). The forecasted prices obtained by the proposed method show its suitability in fuzzy time series forecasting without any price governing parameters. 1250023-11


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References 1. Zadeh L. A., The concept of a linguistic variable and its application to approximate reasoning. Part I, Inform. Sci. 8:199–249, 1975. 2. Zadeh L. A., Fuzzy set, Inform. Control 8:338–353, 1965. 3. Song Q., Chissom B., Fuzzy time series and its models, Fuzzy Set. Syst. 54:269–277, 1993. 4. Song Q., Chissom B., Forecasting enrollments with fuzzy time series — Part I, Fuzzy Set. Syst. 54:1–9, 1993. 5. Song Q., Chissom B., Forecasting enrollments with fuzzy time series — Part II, Fuzzy Set. Syst. 64:1–8, 1994. 6. Chen S. M., Forecasting enrollments based on fuzzy time series, Fuzzy Set. Syst. 81:311–319, 1996. 7. Chen S. M., Forecasting enrollments based on high-order fuzzy time series, Cybernet. Syst. 33:1–16, 2002. 8. Own C. M., Yu P. T., Forecasting fuzzy time series on a heuristic high-order model, Cybernet. Syst. 36:705–717, 2005. 9. Tsaur R. C., Yang J. C. O., Wang H. F., Fuzzy relation analysis in fuzzy time series model, Comput. Math. Appl. 49:539–548, 2005. 10. Yu H.-K., Weighted fuzzy time series model for TAIEX forecasting, Physica A 349:609–624, 2005. 11. Huarng K., Yu T. H. K., Ratio-based lengths of intervals to improve fuzzy time series forecasting, IEEE Trans. Syst., Man, Cybern. B, Cybern. 36(2):328–340, 2006. 12. Cheng C. H., Chang R. J., Yeh C. A., Entropy-based and trapezoid fuzzificationbased fuzzy time series approach for forecasting IT project cost, Technol. Forecast. Soc. Change 73:524–542, 2006. 13. Abd Elaal A. K., Hefny H. A., Abd-Elwahab A. H., Constructing fuzzy time series model based on fuzzy clustering for a forecasting, J. Comput. Sci. 7:735–739, 2010. 14. Kai C., Fang-Ping F., Wen-Gang C., A novel forecasting model of fuzzy time series based on K-means clustering, IWETCS, IEEE, pp. 223–225, 2010. 15. Singh S. R., A simple method of forecasting based on fuzzy time series, Appl. Math. Comput. 186:330–339, 2007. 16. Singh S. R., A robust method of forecasting based on fuzzy time series, Appl. Math. Comput. 188:472–484, 2007. 17. Hurang K., Heuristic models of fuzzy time series for forecasting, Fuzzy Set. Syst. 123:369–386, 2001. 18. Lee H. S., Chou M. T., Fuzzy forecasting based on fuzzy time series, Int. J. Comput. Math. 81(7):781–789, 2004. 19. Cheng C. H., Cheng G. W., Wang J. W., Multi attribute fuzzy time series method based on fuzzy clustering, Expert Systems with Applications, 34:1235–1242, 2008. 20. Annual Report of State Bank of India, 2010, http://www.statebankofindia.com.

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a computational method for fuzzy time series ...

a computational method for fuzzy time series ...

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