An Improved Fuzzy Time Series Forecasting Model Ha Che-Ngoc1 , Tai Vo-Van2 , Quoc-Chanh Huynh-Le1 , Vu Ho3 , Thao Nguyen-Trang1,4(B) , and Minh-Tuyet Chu-Thi1,2,3,4 1
Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
[email protected] 2 Department of Mathematics, Can Tho University, Can Tho, Vietnam 3 Faculty of Mathematical Economics, Banking University of Ho Chi Minh City, Ho Chi Minh City, Vietnam 4 Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam
Abstract. This model is developed from the model of Abbasov and Mamedova (2003) in which the parameters are investigated by methods and algorithm to obtain the most suitable values for each data set. The experiments on Azerbaijan’s population, Vietnam’s population and Vietnam’s rice production demonstrate the feasibility and applicability of the proposed methods.
Keywords: Fuzzy time series GDP · Vietnam
1
· Abbasov-Mamedova · Population
Introduction
Forecasting is the process of making prediction for the future based on summing experiences, assembling knowledge and analyzing related problems. It is considered as the basis process, the first step for organizations as well as governments to build their policies and objectives. Because of its important role in many fields, forecasting has received much attention from scientists. Despite several discussions in the literature, the problems of forecasting have not yet been completely solved. Based on the historical data, looking for principles and rules to establish a suitable forecasting model is the major method of statistics. Time series and regression models have important roles in forecasting using statistical methods, but they have many disadvantages in practice. A regression model (Galton (1888); Pearson (1896)) requires a number of assumptions that are unsatisfactory, whereas a time series model, like ARIMA (Box and Jenkins (1976)), performs poorly when there are abnormal changes or the time series is nonstationary. To overcome the distadvantages of these two models, various models have recommended by many researches, such as (Zecchin et al. (2011); Wang and Fu (2006); Wang et al. (2001); Ren et al. (2016); Gupta and Wang (2010); Zhu and Wang (2010); Park (2010); c Springer International Publishing AG 2018 L. H. Anh et al. (eds.), Econometrics for Financial Applications, Studies in Computational Intelligence 760, https://doi.org/10.1007/978-3-319-73150-6_38
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Teo et al. (2001); Ghazali et al. (2009)). These proposals are the important contributions for forecasting problem because they have given good results in considered data sets. However, we could not obtain optimum for all cases. Some other models like Artificial Neuron Network, Supported Vector Regression (Cortes and Vapnik (1995)), Multivariable Adaptive Regression Spline (Friedman (1991)), Adaptive Spline Threshold Autoregressive (Lewis and Stevens (1991)), Autoregressive conditional heteroscedasticity (Engle (1982)) or hybrid models (Zhang (2003)); (de Oliveira and Ludermir (2014)) were also proposed; however, most of them still have many disadvantages in real forecasting applications. Based on the fuzzy theory of Zadeh (Zadeh (1965)), fuzzy time series (FTS) introduced by Song (Song and Chissom (1993)) can solve the gap mentioned above. FTS has been then interested to research and have been shown to be more efficient than traditional statistical techniques (Song and Chissom (1993)); (Tseng and Tzeng (2002)). Among them, Abbasov and Manedova (AM) proposed the model where the variations of data are represented by language level to forecast the population of Azerbaijan (Abbasov and Mamedova (2003)). Because of its better performance for some kinds of forecasting problems, AM model has been applied in many applications; for instance, Sasu utilized the AM model to forecast the population of Romanian (Sasu (2010)). Some other important studies of FTS can be listed as the models in (Chen (2004); Huarng (2001); Singh (2008)). Nonetheless, all of the above methods use only historical fuzziness data without forecasting. Moreover, the parameters in the models are not properly investigated to find the optimal values for each data set. One model is only rated as better than the others in some specific cases. As a result, there is no model that is considered optimal in all situations. To overcome the gap mentioned above, this article proposes the methods to identify the suitable parameters in the AM model. Specifically, w, the number of elements in the data set used as prior information to forecast the data is chosen through the value of partial autocorrelation function (PACF), the number of fuzzy sets n is selected through an index that can evaluating the compactness of the divided intervals. After determining suitable w and n, the optimal choice of C is searched via an efficient algorithm so that the forecasting error is the smallest. The numerical examples illustrate the proposed theories in detail and prove that this method can improve the performance in term of forecasting accuracy. The remainder of this paper is organized as follows. Section 2 reviews the AM model and some the related definitions. Section 3 proposes the modifications for AM model, in which the suitable parameters are determined by new methods and algorithm. The numerical examples are presented in Sects. 4 and 5 is the conclusion.
2 2.1
Related Definitions and Abbasov-Mamadova Model Related Definitions
Definition 1. Let U be a universe (domain), with a generic element of U denoted by u. A fuzzy set A on universe U is a set defined by the membership function μA (u) which is a mapping from the universe U into the unit interval: μA (u) : U → [0, 1]
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The value of the membership function of a specific u, μA (u) is called as the membership degree or grade of membership. If the grade of membership equals one, u belongs completely to the fuzzy set. If the grade of membership equals zero, u does not belong to the set. If the grade of membership is between 0 and 1, u is a partial member of the fuzzy set. ⎧ ⎪ u is a full member of A ⎨= 1 μA (u) ∈ (0, 1) u is a partial member of A ⎪ ⎩ =0 u is not member of A In above definition, a fuzzy set A in U is characterized by a membership function μA (u). There are several ways to define the membership function μA (u). Some of forms of membership functions which are often used such as trapezoidal membership function, triangular membership function, Gaussian membership function, etc. An example of some membership functions with different shapes is presented in Fig. 1. We next examine a special case of Definition 1 where the universe is a time series and introduce the definition proposed by (Song and Chissom (1993)) as follows.
Fig. 1. Algorithm to determine C.
Definition 2. Let Y (t) ∈ R, t = 0, 1, 2, . . . be a time series, with a generic element denoted by yt . If μA (yt ) is the membership function which is a mapping from the universe Y (t) into [0, 1] and F (t) = {μA (y0 ), μA (y1 ), μA (y2 ), . . .} is a collection of μA (yt ) then F (t) is called a fuzzy time series.
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ˆi , i = Definition 3. Given the actual data {Xi } and predictive value X 1, 2, . . . , m, respectively, we have the popular indexes to evaluate established model as follows: Mean absolute error:
m 1 ˆ X i − X i . m i=1
(1)
2 1 ˆ Xi − Xi . m i=1
(2)
M AE = Mean squared error:
m
M SE = Mean absolute percentage error:
⎛ ⎞ ˆ i − Xi m X 1 ⎝ M AP E = .100⎠ . m i=1 Xi 2.2
(3)
Abbasov-Mamedova Model
Given the historical data Xt corresponded to year t = 1, 2, ..., m. The AM model consists of the following six steps. – Step 1: Compute the variation Vt between every next and previous year by Formula (4). Then define the universal set U by Formula (5). Vt = Xt − Xt−1
(4)
U = [Vmin − D1 , Vmax + D2 ]
(5)
where Vmin is the smallest variation, Vmax is the greatest variation, D1 and D2 are positive numbers. – Step 2: Divide the universal set U into n equal-length intervals ui , i = 1, 2, . . . , n, such that each interval ui contains at least one variation value. Then find the middle points uim of each interval. – Step 3: Define the fuzzy set Ai , i = 1, 2, . . . , n, on the universal set U by the following formula: 1 μAi (u) = (6) 2, 1 + [C × (u − uim )] where u is a generic element of universal set U , uim is the middle point of the corresponding interval ui , (i = 1, 2, . . . , n) and C is a constant. – Step 4: Convert the input data, time-point variations, into fuzzy values by Formula 6. – Step 5: Select an integer w, 1 < w < l, where l is the number of years, prior to the current year included in experimental evaluation. Based on the chosen w and Mamdani inference system, we establish an operation matrix Ow (t) of size i × j (here i is the number of rows, which conforms to the sequence of
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years t − 2, t − 3, . . . , t − w, j is the number of columns, which conforms to the number of variation intervals) and a criteria matrix K(t) of size 1 × j (a row matrix corresponding to fuzzy variation in total population for the year t − 1). After that, the relationship matrix R(t) is calculated as follows. R(t) [i, j] = Ow [i, j] ∩ K(t) [j] , ⎡
or
R11 ⎢ R 21 R(t) = Ow (t) ⊗ K(t) = ⎢ ⎣ ... Ri1
R12 R22 ... Ri2
⎤ ... R1j ... R2j ⎥ ⎥, ... ... ⎦ ... Rij
where Ow (t) is the operation matrix, K(t) is the criteria matrix, ⊗ is the min operator (∩). Define F (t), the fuzzy forecasting of variations for the year t, in a fuzzy form as follows. F (t) = [max(R11 , . . . , Ri1 ), . . . , max(R1j , . . . , Rij )] = [μA1 (Vt ), μA2 (Vt ), . . . , μAm (Vt )]. – Step 6: Defuzzify the obtained results of the 5-th step according to the Formula 7. m μAi (Vt ) × uim i=1 V (t) = , (7) m μAi (Vt ) i=1
where μAi (Vt ) is the value of membership function of the forecast variation in interval i, V (t) is the defuzzified forecast variation. In orders to estimate the forecast value X(t) for year t, the following formula is utilized: X(t) = X(t − 1) + V (t),
(8)
where X(t − 1) is the forecast value for year t − 1, V (t) is the variation for year t.
3
The Proposed Method
In AM model, there are three parameters including the number of equal-length intervals n, the positive integer w and the constant C have effects on the forecasting result. However, in the studies of (Abbasov and Mamedova (2003); Sasu (2010)), these parameters were only identified according to the experiences. Hence, this method is not suitable when dealing with various types of time series. For w, Song and Chissom conducted a survey on the specific data and pointed out that w = 2 is the best. They also concluded that the forecasting
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result is better if we utilize a less complex model, with smaller w (Song and Chissom (1993)). However, this conclusion is only drawn from a few specific surveys, so lose generality. In fact, w is number of previous times that have a strong influence on current value of time series (it is similar to the partial autocorrelation p in autoregressive integrated moving average, ARIMA). Therefore, it is not reasonable if we utilize a model, with a fixed value of w, for all type of time series. For instance, when dealing with a monthly or quarterly data, w = 7 is consider as an unreasonable parameter. According to above remarks, it is certain that the forecasting performance of the AM model can be significantly improved if its parameters are determined in reasonable ways. To overcome the limitations mentioned above, this section proposes a method called MAM (Modified Abbasov-Mamedova model), which can identify the parameters n, w and C in a reasonable way. Details of the proposed method are presented as follows. 3.1
Determine the Number of Interval n
In the fuzzification step, the middle point ui0 is used as the representative element of ith interval. Therefore, if the data in each interval are well-represented by ui0 , the forecasting performance can be improved. In general, we can evaluate whether data are well-represented by ui0 or not according to the compact measure between this middle point and elements in the interval i. Figure 2 illustrates a few cases of representative elements. It can be seen that the universal set U is defined as the interval (0, 3) and divided into three equal-length intervals. The distance between middle points (red points) and elements belonging to the intervals (0, 1) and (1, 2) are really large; therefore, using u10 and u20 as the representative elements can lead to a low forecasting performance. Conversely, u30 is close to the elements in 3rd interval, and it can lead to a good measure of compactness as well as a high forecasting performance. Therefore, it is important to point out the number of intervals n so that the measure of compactness is optimized. According to mentioned idea, this paper proposes a measure denoted as MMSE to evaluate the compactness of algorithm. MMSE is computed by (9): 2 n 1 vt − ui0 , (9) M M SE = n i=1 v ∈u ni t
i
where n is the number of equal-intervals, vt is the variation, ui0 and ni is the middle point and the number of elements belonging to the interval i, respectively. Clearly, the smaller distances between middle point ui0 and variations vt are, the smaller of numerator as well as the MMSE criterion. Therefore, MMSE can be used to evaluate the compactness of time series model with different number of intervals n. In addition, when the number of intervals n is extended up to a specific number, the empty intervals containing no variation values are created. Meanwhile, the denominator in (9) is equal to 0, and MMSE converges to infinity. Hence, the choice of n in this case is not suitable. This is also entirely consistent with the constraint mentioned in (Abbasov and Mamedova (2003)).
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Fig. 2. Illustrate for representing elements.
3.2
Determine w
In literature, the parameter w is chosen according to experiences. Although Song and Chissom conducted a survey on specific datasets and pointed out that w = 2 is the best, this conclusion is only drawn from a few specific surveys and does lose generality. Here, an effective method that can determine w is proposed, based on the partial autocorrelation function (PACF) of the time series. Let φkk be PACF at lag k (k = 1, 2, . . .), φkk can be obtained according to the recursive formula (Durbin (1960)) as follows: φp+1,j = φp,j − φp+1,p+1 φp,p−j+1 , rp+1 − φp+1,p+1 =
p j=1 p
1−
(10)
φp,j rp+1−j
j=1
,
(11)
φp,j rj
where rk is the autocorrelation function (ACF) at lag k, φ1,1 = r1 . The PACF at lag k considers only the direct correlation between vt and vt−k , with the linear dependences between the intermediate variables are removed. Therefore, it can accurately reflect the number of the previous years, on which the current year depends. Based on this result, we can determine the appropriate w.
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Note that, when PACF presents the largest value at lag 1, w is considered as 2 so that the AM model conditions are fitted. In practice, based on statistical programs including R, Matlab, etc., it is possible to calculate the PACF and determine the reasonable w for fuzzy time series model. Determine the Constant C
3.3
Given AM model with specific parameters w and n, the value of μAi (ui ) as well as the forecasting result is strongly affected by the constant C. However, the previous studies of (Abbasov and Mamedova (2003); Sasu (2010)) did not offer guidance on how to determine the reasonable C for each specific dataset. This subsection proposes an algorithm to determine optimum C (DOC), using the following five steps: Step 1. Initialize an integer k (k > 499), a very small positive number , where k is the number points which divided for each iteration and is the error of C. Step 2. When t = 0, assign values: a(0) = 0, b(0) = 1, ΔC (0) = 12 , n(0) = 1. Step 3. When t = i, i ≥ 1, calculate the values a(t) = a(t−1) + n(t−1) − 1 ΔC (t−1) b(t) − a(t) b(t) = a(t−1) + n(t−1) + 1 ΔC (t−1) , ΔC (t) = k If If If If
a=0 a=0 a = 0 a = 0
and and and and
b = 1, b = 1, b = 1, b = 1,
then then then then
(t)
Ci (t) Ci (t) Ci (t) Ci
= a(t) + iΔC (t) , = a(t) + iΔC (t) , = a(t) + iΔC (t) , = a(t) + iΔC (t) ,
i = 1, 2, . . . , k − 1. i = 1, 2, . . . , k. i = 1, 2, . . . , k − 1. i = 1, 2, . . . , k. (t)
Step 4. Run the Abbasov-Mamedova model with all the values Ci (t) Find Cn at which the criterion CEF is the current best.
in Step 3. (m)
Step 5. With the new n, repeat the Step 3 and Step 4 to find C = Cn a(m) + nΔC (m) until b(m) − a(m) < ε. Note that,
=
(i) In each iteration, (k + 1) values of C are considered. In the numerical examples in this paper, k = 1000 is chosen. (ii) is a very small number and is chosen arbitrarily. The smaller is, the more iterations and computer time it are required. In fact, the optimum value of C can be determined with an acceptable error depending on the value of . In numerical examples, = 10−6 is chosen. (iii) There are a many criterions considered to evaluate the forecasting model (CEF). In this article, we use MAE, MAPE and MSE presented in Subsect. 2.1 to compare established models.
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The DOC algorithm is illustrated by Fig. 3.
Fig. 3. Algorithm to determine C.
4
Numerical Examples
Section 3 proposes the methods of determining w, n and C in order to improve the forecasting performance of AM model. In Sect. 4, this paper presents two examples to illustrate and test the forecasting performance of proposed method. Specifically, Example Sect. 4.1 presents in detail the proposed method when dealing with the well-known data, Azerbaijan’s population. This example, in addition to clarifying the proposed algorithm, can test its forecasting performance. In Example Sect. 4.2, the new method is applied to forecast the GDP per capita in Vietnam. In each example, we compare the forecasting results of proposed method with those of the AM model (Abbasov and Mamedova (2003)), the Chen model (Chen (1996)) and the Huarng model (Huarng (2001)). Furthermore, to present that the proposed method is more efficient in predicting time series than the traditional statistical methods, MAM is also compared with the auto-regressive model AR(p) where p is choose based on AIC criterion.
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Example 1
This example forecasts the annual population (thousand persons) of Azerbaijan from 1980 to 2001 to clarify the proposed method and test its performance. This is a well-known dataset presented in (Abbasov and Mamedova (2003)). The detailed procedure is presented by six following steps. Step 1. Table 1 presents the annual populations over 1980–2001 and variations in all given years. Variation for the current year is the difference between the population values in current year and previous year. For example, variation for 1990 is equal to 7131900 − 7021200 = 1110700. To define the universal set U , first of all, the smallest and greatest variation values must be found over the interval [1980, 2001], later, to ensure the smoothness of boundaries of the interval, adequate non-negative numbers D1 , D2 are selected. After that, the universal set U can be defined as U :U = [Vmin − D1 , Vmax + D2 ], where Vmin = 62800 is the smallest variation, Vmax = 115900 is the greatest variation, D1 = 0, D2 = 0. Thus, the universal set U is defined as: U = [62800, 115900]. Based on the variation in Table 1, MMSE and PACF that are computed and presented in Fig. 4 are considered as the criterion to find the suitable n and w. From Fig. 4, it can be seen that MMSE is inversely correlated with the number of equal intervals n. It proves that the more intervals we have, the better representation the middle points make. However, the split of intervals must stop at a specific level. Specifically, in Fig. 4, the empty intervals, which is associated with n, are created when n is greater or equal to 10. As a result, MMSE is unspecified, and the method is considered to be unreasonable in those cases, which stands for the suitable number of equal intervals is 9. For w, according to Fig. 4, it can be observed that PACF reaches the maximum value at lag 1. As mentioned in Sect. 3, w is chosen as 2 in this case. It is also suitable with the survey of (Song and Chissom (1993)). In summary, based on MMSE and PACF, n = 9 and w = 2 are utilized in the AM model. With n = 9 and w = 2, performing the DOC algorithm, the optimum value of C is reach at 0.3197. Series variation
Partial ACF
20
15
−0.4
10
5
0
0.0 0.2 0.4 0.6
25
2 2
3
4
5
6
7
8
9
10
11
4
6
8
12
Lag
Fig. 4. MMSE according to n and the PACF.
10
12
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Step 2. The universal set U must be divided into 9 equal intervals: u1 = [62800, 68700], u2 = [68700, 74600] u3 = [74600, 80500], u4 = [80500, 86400] u5 = [86400, 92300], u6 = [92300, 98200] u7 = [98200, 104100], u8 = [104100, 110000] u9 = [110000, 115900] u2m u8m
The middle points of the intervals are determined as follows: u1m = 65750, = 71650, u3m = 77550, u4m = 83450, u5m = 89350, u6m = 95250, u7m = 101150, = 107050, u9m = 112950.
Steps 3 and 4. Define the fuzzy sets A1 , A2 , . . . , A9 on the universal set U and convert the input data into fuzzy values by formula 6. An exemplary growth of the continuous membership functions of fuzzy sets Ai is shown in Fig. 5. For the sake of briefly, the results of fuzzification for all the given years with last two digits are shown in Table 1. Table 1. Population of Azerbaijan T
Nt
Vt
Fuzzy time series Ft
1980 6114.3 0 1981 6206.7 92.4
0.00 0.02 0.04 0.11 0.51 0.55 0.11 0.04 0.02
1982 6308.8 102.1 0.01 0.01 0.02 0.03 0.06 0.17 0.92 0.29 0.08 1983 6406.3 97.5
0.01 0.01 0.02 0.05 0.13 0.66 0.42 0.10 0.04
1984 6513.3 107.0 0.01 0.01 0.01 0.02 0.03 0.07 0.22 1.00 0.22 1985 6622.4 109.1 0.01 0.01 0.01 0.01 0.02 0.05 0.13 0.70 0.40 1986 6717.9 95.5
0.01 0.02 0.03 0.06 0.21 0.99 0.23 0.07 0.03
1987 6822.7 104.8 0.01 0.01 0.01 0.02 0.04 0.10 0.42 0.66 0.13 1988 6928.0 105.3 0.01 0.01 0.01 0.02 0.04 0.09 0.36 0.76 0.14 1989 7021.2 93.2
0.01 0.02 0.04 0.09 0.40 0.70 0.13 0.05 0.02
1990 7131.9 110.7 0.00 0.01 0.01 0.01 0.02 0.04 0.10 0.42 0.66 1991 7218.5 86.6
0.02 0.04 0.11 0.50 0.56 0.12 0.04 0.02 0.01
1992 7324.1 105.6 0.01 0.01 0.01 0.02 0.04 0.08 0.33 0.82 0.15 1993 7440.0 115.9 0.00 0.00 0.01 0.01 0.01 0.02 0.04 0.11 0.53 1994 7549.6 109.6 0.01 0.01 0.01 0.01 0.02 0.05 0.12 0.60 0.47 1995 7643.5 93.9
0.01 0.02 0.04 0.08 0.32 0.84 0.16 0.05 0.03
1996 7726.2 82.7
0.03 0.07 0.27 0.95 0.18 0.06 0.03 0.02 0.01
1997 7799.8 73.6
0.14 0.72 0.39 0.09 0.04 0.02 0.01 0.01 0.01
1998 7879.7 79.9
0.05 0.13 0.64 0.44 0.10 0.04 0.02 0.01 0.01
1999 7953.4 73.7
0.13 0.70 0.40 0.09 0.04 0.02 0.01 0.01 0.01
2000 8016.2 62.8
0.53 0.11 0.04 0.02 0.01 0.01 0.01 0.00 0.00
2001 8081.0 64.8
0.92 0.17 0.06 0.03 0.02 0.01 0.01 0.01 0.00
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Fig. 5. Membership functions of 9 fuzzy sets.
Step 5. Apply the min − max operator to forecast the population in 1990, we have the results: O2 (1990) = [0.01 0.01 0.01 0.02 0.04 0.09 0.36 0.76 0.14] K(1900) = [0.01 0.02 0.04 0.09 0.40 0.70 0.13 0.05 0.02] R(1990) = [0.01 0.01 0.01 0.02 0.04 0.09 0.13 0.05 0.02] Hence, the fuzzy forecasting of the variation for the year 1990, F (1990), is [0.01 0.01 0.01 0.02 0.04 0.09 0.13 0.05 0.02] Step 6. Finally, compute the variations in 1990 by the Formula 7. V (1990) =
0.01 ∗ 65750 + . . . + 0.02 ∗ 112950 = 97181 0.01 + . . . + 0.02
Hence, the forecasting population in 1990 is: X (1990) = X (1989) + V (1990) = 7021200 + 97181 = 7118381 Perform in a similar way for the remainder, the forecasting results are presented in Table 2 and Fig. 6. As shown in Table 3, in addition to the proposed method, the performance of models presented in (Abbasov and Mamedova (2003)); Chen (1996); Huarng (2001)) are examined for comparison purpose. It can be observed that MAM model outperforms others in term of accuracy for all cases of criterion. The result verifies that the proposed method is suitable at first and need to be retested in actual application as follows.
H. Che-Ngoc et al. Table 2. Actual and forecast population (thousand person) Years Actual Forecasted Total Variation Total Variation 1988
6928.0 105.3
6921.096
1989
7021.2
7031.659 103.659
1990
7131.9 110.7
7118.381
97.181
1991
7218.5
86.6
7230.382
98.482
1992
7324.1 105.6
7314.037
95.537
1993
7440.0 115.9
7418.289
94.189
1994
7549.6 109.6
7545.407 105.407
1995
7643.5
93.9
7658.468 108.868
1996
7726.2
82.7
7742.128
98.628
1997
7799.8
73.6
7814.826
88.626
1998
7879.7
79.9
7879.700
79.900
1999
7953.4
73.7
7958.389
78.689
2000
8016.2
62.8
8032.098
78.698
2001
8081.0
64.8
8089.909
73.709
93.2
98.396
8500
Actual series vs forecated series by Abbasov−Mamedova model of 9 fuzzy set with w = 2 and C = 0.319363409408 Forecasted
7500
Actual
6500
data
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1980
1985
1990
1995
2000
2005
point
Fig. 6. The forecasting result of proposed model. Table 3. The performance of comparative methods MAM MAE MAPE MSE
AM
Chen (1996) Huarng (2001)
9.989
15.007
77.756
77.756
0.136
0.197
1.099
1.099
127.751 290.459 8835.054
8835.054
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Example 2
GDP is gross domestic product converted to international dollars using purchasing power parity rates. It is one of the major measures of nation’s economic health. Therefore, GDP forecasting has an essential role for countries all over the world. In this example, the new method is applied to forecast the GDP per capita in Vietnam. In particular, the GDP per capita (USD) from 1990 to 2015 are collected (http://data.worldbank.org). Similar to Example 1, the performance of the proposed method are compared with those of (Abbasov and Mamedova (2003); Chen (1996); Huarng (2001)) (Fig. 7 and Table 4). In addition, the forecasts for the 5-year period 2016–2020 are presented (Table 5). Table 4 and Fig. 7 show that the proposed method has good forecasting results. The established model fits well almost all the actual data, with the mean of absolute error is less than 26USD% per year and the mean of absolute percentage error is less than 8% in comparison with actual data. In addition, Table 4 and Fig. 7 demonstrate Table 4. Forecasting results of comparative methods GDP per capita (USD) MAM model AM model The Chen mode The Huarng model MAE
25.669
42.942
201.552
0.744
1.224
7.86
7.86
2834.763
62332.37
62332.37
MAPE MSE
1117.06
201.552
4000 2000
USD
Actual GDP vs forecated GDP by models
Actual GDP Abbasov−Mamedova New model Chen Huarng
1990
1995
2000
2005
2010
2015
year Fig. 7. Vietnam GDP per capita forecasting results of comparative models
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H. Che-Ngoc et al. Table 5. Out-of-sample forecasting results GDP Year Forecast 2016 5908.166 2017 6148.920 2018 6372.591 2019 6588.328 2020 6796.071
the superiority of proposed method over comparative models, when it always shows the best results. For out-of-sample forecasting, it can be seen from Table 5 that Vietnam’s GDP continues to increase steadily, with average growth rates of over 200 USD each year. GDP will reach over 6500 USD per capita by 2019. 4.3
On the Comparison Between MAM and the Traditional Statistical Method
As mentioned earlier, we resolve two above experiments in which MAM is compared with the auto-regressive model AR(p), (auto-regressive order p is choose based on AIC criterion). The brief summary of results in Table 6 present that the MAM model outperforms the AR model. Based on the above examples, at first, we can be see that MAM is a good and competitive model in comparison with traditional statistical method as well as other fuzzy time series model. It is feasible and capable of practical problems, particularly of population and GDP forecasting. Table 6. The results of MAM and AR models for Example 1 and Example 2 Example 1 MAM AR MAE MSE MAPE
5
9.989
Example 2 MAM AR
93.908
25.669
127.751 8995.331 1117.06 0.136
1.329
0.744
165.612 30881.76 5.323
Conclusion
This study proposes an improved fuzzy time series forecasting model based on the methods to determine the suitable parameters for each data set in the AM model. The numerical examples prove that the proposed method is more feasible and capable of practical problems. In future, a program will be written in the R statistical software to apply the proposed model in many different practice problems.
An Improved Fuzzy Time Series Forecasting Model
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