Forecasting model based on heuristic learning of high-order fuzzy-trend and jump rules

Abstract

We propose a heuristic learning method forecasting future performance of stock market indices based on high-order fuzzy-trend jump rules generated from historical training data. Firstly, the training time series (TSs) are fuzzified by equal intervals referencing to the whole mean differences of historical training data. Then, it generates the groups of nth-order fuzzy logical relationships (FLRs). With the knowledge of the generated relationship groups, it summarizes the probability of the jumps of the nth-order “down”, “equal” and “up” trend rules, respectively. Finally, it performs the forecasting based on the nth-order FLRs and the probabilities of their corresponding jump rules. To evaluate the outcome of the presented model with the performances of the others, we use the presented model to predict the Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX) dataset. The outcomes show that the presented model outperforms the other models using single factor and point-wise one-step ahead forecasts. Moreover, it is easily to realize by software computing without artificial participation and can be extended to deal with multiple years of dataset. We use this model to predict Shanghai Stock Exchange Composite Index (SHSECI) as well to analyze its effectiveness and universality.

Keywords

Fuzzy time series fuzzy forecasting fuzzy logical relationship probabilities of jumps fuzzy-trend logical relationship groups

1 Introduction

Since Song and Chissom [17] originally presented the fuzzy modeling for time series (TSs) [17 –19], many fuzzy forecasting models, approaches and procedures have been presented based on fuzzy time series (FTSs) [28 –30]. These models are introduced to forecast the enrollment at Alabama University [25, 26], project cost [4], electricity load demand [20] and stock market behavior [8 , 24]. Because of the dynamic and complex environment [3, 33], stock market prediction has been regarded as the most challenging application of time series prediction [16]. Authors developed models to approximately forecast the uncertainty of the future trend by analyzing the change of corresponding stock indicates over a time-window [15]. In order to provide accurate forecasts for stock market indices, some authors combine fuzzy time series with heuristic optimization methods [12, 35], and other authors combine neural networks and heuristic procedures to forecast stock market time series [21].

Concerning the early stages of FTSs methods, the most significant points are (1) the determination of the universe of discourses and suitable intervals to fuzzify TSs into FTSs, (2) the construction of fuzzy logical relationships (FLRs), (3) the defuzzification to get crisp number of the future. According to the first step, researchers usually observe the min-max values of the TSs and determine the universe of discourse and the interval-lengths. Existing fuzzy forecasting models can be divided into two categories-fixed interval-lengths [11 , 34] and variable interval-lengths [5, 14]. All of these methods require each data be in the scope of universe of discourse. Based on the second step of existing models, they can be divided into first-order fuzzy logical relationships (FLRs) [6 , 32], high-order FLRs [14], and two-factors high-order FLRs [24], etc. According to the third step, most of researchers employ the interval-lengths to forecast the crisp number [23, 24]. Although existing forecasting models can perform the future stock market indicates, however, the forecasting error rates of current approaches are pretty high. Therefore, to develop the accurate forecasts, Rubio et al. (2017) proposed a mixed method to determine the defuzzification number [8]. Other researchers propose hybrid approaches for fuzzy time series [7, 24]. These new methods introduce covariates in the FTS analysis and enhance the stability of the time series. However, the determination of covariate requires more extensive research.

This paper introduces a new heuristic learning model based on nth-order FLRs and jump rules for FTSs. Firstly, real numbers of training TSs are fuzzified into FTSs and high-order FLRs are summarized. Then, it analyzes the FLRs to obtain their fuzzy-trend and jump rules. It summarizes the probabilities of jump rules by a point-wise heuristic learning procedure. Finally, it performs the forecasting based on the current high-order fuzzy-trend and its corresponding probabilities of jump rules. To test the utility of this method, we apply it to forecast the TAIEX and compare the forecasting accuracy with existing methods. We also apply it to SHSECI to illustrate its universality.

The rest of this paper is described as the followings: Section 2 introduces basic ideas of fuzzy sets (FSs) [10], fuzzy time series (FTSs) [17 –19], fuzzy logical relationships (FLRs), fuzzy-trend and jump rules. Section 3 describes a new method of forecasting with high-order fuzzy-trend jump rules. In Section 4, the presented method is used to forecast the stock market using TAIEX and SHSECI datasets, respectively. Conclusions are assembled in Section 5.

3 Preliminaries

Song and Chissom [17 –19] originally proposed the theories of FTSs. This part we will briefly look for some related concepts of FTSs.

Definition 1. Let U be an universe of discourse, then a fuzzy set H in U can be defined by its membership function μ_H : U → [0, 1], where μ_H (u_i) represents the grade of membership of u_i, U = {u₁, u₂, … u_i, …, u_m}.

Definition 2. Let X (t) (t = 1, 2, …, T) be a TS of real numbers, where T is the number of the time series. Each element of X (t) (t = 1, 2, …, T) can be represented by a fuzzy set H (t) (t = 1, 2, …, T), respectively. H (t) (t = 1, 2, …, T) is called a FTS.

Definition 3. Let H (t) (t = 1, 2, …, T) be a FTS. Assume H (t) is determined by H (t - 1), H (t - 2), dots, and H (t - n), then the FLR is represented by $H (t - n), \dots, H (t - 2), H (t - 1) \to H (t)$ (1) where H (t - n), …, H (t - 2), H (t - 1) is named its left-hand side(LHS) of the FLR and H (t) is the right-hand side(RHS).

Definition 4. Let H (t - n) = D_ln, H (t - 2) = D_l2, H (t - 1) = D_l1, H (t) = D_l0, then the high-order FLR group could be denoted by (D_ln, … D_l2, D_l1) → D_l0, where the jump for this relationship group is k = l0-l1. This high-order FLR group can be represented by a “n-tuples” fuzzy-trend group, where each element represents the up, down or equal trend of adjacent fuzzy sets in the nth-order fuzzy sets. For example, let n = 4, the FLR groups can be shown by 27 groups shown in Table 1.

Table 1

Groups of fuzzy-trend logical relationships for n = 4

Group	Relationships
Number	Between	Between	Between
	l4 and l3	l3 and l2	l2 and l1
1	down	Down	down
2	down	Down	equal
3	down	Down	up
4	down	Equal	down
5	down	Equal	equal
6	down	Equal	up
7	down	Up	down
8	down	up	equal
9	down	up	up
10	equal	down	down
11	equal	down	equal
12	equal	down	up
13	equal	equal	down
14	equal	equal	equal
15	equal	equal	up
16	equal	up	down
17	equal	up	equal
18	equal	up	up
19	up	down	down
20	up	down	equal
21	up	down	up
22	up	equal	down
23	up	equal	equal
24	up	equal	up
25	up	up	down
26	up	up	equal
27	up	up	up

4 A novel High-order fuzzy-trend jump rules method

This paper introduces a new model based on FTSs and high-order fuzzy-trend jump rules. To compare with other researchers’ work [1 , 32], this section uses authentic Taiwan Stock Exchange(TAIEX 1999) to represent the forecasting procedure. All data for the first ten months of year 1999 are treated as training dataset and the rest data for the last two months as testing dataset. The initial steps are as follows.

Step 1. Determine the length of intervals and fuzzify original datasets

In our proposal, we define U = [0, X_max + U₀], X_max is the maximum value of original datasets and U₀ is a positive real number such that enables the universe cover the possible value of the historical data. The lengths of each adjacent intervals len are equal and determined by the whole mean of the training data.

Suppose x is a real number in the training dataset and x ∈ u_i, where $i = {\begin{matrix} trunc (\frac{x}{len}) if x can be devided \\ by len with no reminder \\ trunc (\frac{x}{len}) + 1 else \end{matrix}$ (2)

trunc(y) means the integral part of y and u_i = [len × i, len × (i + 1)), the real number x is fuzzified to D_i. For example, let len = 80 and x = 6052.45, according to Equation (2), i = 77, then x is fuzzified to D₇₇. By this way, the historical TSs can be fuzzified to FTSs.

Step 2: Build FLRs and generate jump rules for high-order fuzzy-trends

For each fuzzy set in the fuzzy time series of training dataset H (t), suppose its fuzzy-trend belongs to group p_t and the jump is s_t according to Def.3 and Def. 4. The jump rules for the high-order fuzzy-trend groups can be computed as follows: $r_{i} = \frac{\sum_{t, p_{t} = i} s_{t}}{\sum_{t, p_{t} = i} 1}$ (3)

The pseudo-code of the rule generation algorithm is shown in Fig. 1(taking n = 4 for example).

Fig.1

Pseudo-code of rule generation algorithm.

Step 3: Calculate the forecast values

Locate the left-hand sides (LHS) of the last obtained data point H(t-1) to obtain its nth-order fuzzy-trend group number. Then use the corresponding jump rule to perform the forecast as follows: $H^{'} (t) = H (t - 1) + r (g) \times len$ (4) where H’(t) is the forecast data, called the future value, H(t-1) is the observed data, called the current value, g is the n-order fuzzy-trend group number of H(t-1), r(g) is the jump rule for group g.

5 Empirical analysis

Many researches use TAIEX1999 as an example to evaluate their proposed forecasting methods [1 , 32]. We will use TAIEX1999 as an example as well. Then we collect the stock market time series datasets of SHSECI (Shanghai Stock Exchange Composite Index) to analyze its effectiveness and universality.

5.1 Forecasting TAIEX

Let the order number n = 4, interval len = 80 (approximately equal to the whole mean of the training dataset). Based on the model and forecast steps described in Section III, we use TAIEX1999 as an example to illustrate the forecasting performance.

[Step 1] According to Equation (2), the actual time series are fuzzified into fuzzy time series shown in Table 2.

Table 2
Actual dataset and the Fuzzified results of TAIEX1999

Date TAIEX Fuzzified Date TAIEX Fuzzified Date TAIEX Fuzzified Date TAIEX Fuzzified

TAIEX TAIEX TAIEX TAIEX

1/5/1999 6152.43 D₇₇ 4/9/1999 7265.7 D₉₁ 7/8/1999 8592.43 D₁₀₈ 10/6/1999 7501.63 D₉₄

1/6/1999 6199.91 D₇₈ 4/12/1999 7242.4 D₉₁ 7/9/1999 8550.27 D₁₀₇ 10/7/1999 7612 D₉₆

1/7/1999 6404.31 D₈₁ 4/13/1999 7337.85 D₉₂ 7/12/1999 8463.9 D₁₀₆ 10/8/1999 7552.98 D₉₅

1/8/1999 6421.75 D₈₁ 4/14/1999 7398.65 D₉₃ 7/13/1999 8204.5 D₁₀₃ 10/11/1999 7607.11 D₉₆

1/11/1999 6406.99 D₈₁ 4/15/1999 7498.17 D₉₄ 7/14/1999 7888.66 D₉₉ 10/12/1999 7835.37 D₉₈

1/12/1999 6363.89 D₈₀ 4/16/1999 7466.82 D₉₄ 7/15/1999 7918.04 D₉₉ 10/13/1999 7836.94 D₉₈

1/13/1999 6319.34 D₇₉ 4/17/1999 7581.5 D₉₅ 7/16/1999 7411.58 D₉₃ 10/14/1999 7879.91 D₉₉

1/14/1999 6241.32 D₇₉ 4/19/1999 7623.18 D₉₆ 7/17/1999 7366.23 D₉₃ 10/15/1999 7819.09 D₉₈

1/15/1999 6454.6 D₈₁ 4/20/1999 7627.74 D₉₆ 7/19/1999 7386.89 D₉₃ 10/16/1999 7829.39 D₉₈

1/16/1999 6483.3 D₈₂ 4/21/1999 7474.16 D₉₄ 7/20/1999 7806.85 D₉₈ 10/18/1999 7745.26 D₉₇

1/18/1999 6377.25 D₈₀ 4/22/1999 7494.6 D₉₄ 7/21/1999 7786.65 D₉₈ 10/19/1999 7692.96 D₉₇

1/19/1999 6343.36 D₈₀ 4/23/1999 7612.8 D₉₆ 7/22/1999 7678.67 D₉₆ 10/20/1999 7666.64 D₉₆

1/20/1999 6310.71 D₇₉ 4/26/1999 7629.09 D₉₆ 7/23/1999 7724.52 D₉₇ 10/21/1999 7654.9 D₉₆

1/21/1999 6332.2 D₈₀ 4/27/1999 7550.13 D₉₅ 7/26/1999 7595.71 D₉₅ 10/22/1999 7559.63 D₉₅

1/22/1999 6228.95 D₇₈ 4/28/1999 7496.61 D₉₄ 7/27/1999 7367.97 D₉₃ 10/25/1999 7680.87 D₉₇

1/25/1999 6033.21 D₇₆ 4/29/1999 7289.62 D₉₂ 7/28/1999 7484.5 D₉₄ 10/26/1999 7700.29 D₉₇

1/26/1999 6115.64 D₇₇ 4/30/1999 7371.17 D₉₃ 7/29/1999 7359.37 D₉₂ 10/27/1999 7701.22 D₉₇

1/27/1999 6138.87 D₇₇ 5/3/1999 7383.26 D₉₃ 7/30/1999 7413.11 D₉₃ 10/28/1999 7681.85 D₉₇

1/28/1999 6063.41 D₇₆ 5/4/1999 7588.04 D₉₅ 7/31/1999 7326.75 D₉₂ 10/29/1999 7706.67 D₉₇

1/29/1999 5984 D₇₅ 5/5/1999 7572.16 D₉₅ 8/2/1999 7195.94 D₉₀ 10/30/1999 7854.85 D₉₉

1/30/1999 5998.32 D₇₅ 5/6/1999 7560.05 D₉₅ 8/3/1999 7175.19 D₉₀ 11/1/1999 7814.89 D₉₈

2/1/1999 5862.79 D₇₄ 5/7/1999 7469.33 D₉₄ 8/4/1999 7110.8 D₈₉ 11/2/1999 7721.59 D₉₇

2/2/1999 5749.64 D₇₂ 5/10/1999 7484.37 D₉₄ 8/5/1999 6959.73 D₈₇ 11/3/1999 7580.09 D₉₅

2/3/1999 5743.86 D₇₂ 5/11/1999 7474.45 D₉₄ 8/6/1999 6823.52 D₈₆ 11/4/1999 7469.23 D₉₄

2/4/1999 5514.89 D₆₉ 5/12/1999 7448.41 D₉₄ 8/7/1999 7049.74 D₈₉ 11/5/1999 7488.26 D₉₄

2/5/1999 5474.79 D₆₉ 5/13/1999 7416.2 D₉₃ 8/9/1999 7028.01 D₈₈ 11/6/1999 7376.56 D₉₃

2/6/1999 5710.18 D₇₂ 5/14/1999 7592.53 D₉₅ 8/10/1999 7269.6 D₉₁ 11/8/1999 7401.49 D₉₃

2/8/1999 5822.98 D₇₃ 5/15/1999 7576.64 D₉₅ 8/11/1999 7228.68 D₉₁ 11/9/1999 7362.69 D₉₃

2/9/1999 5723.73 D₇₂ 5/17/1999 7599.76 D₉₅ 8/12/1999 7330.24 D₉₂ 11/10/1999 7401.81 D₉₃

2/10/1999 5798 D₇₃ 5/18/1999 7585.51 D₉₅ 8/13/1999 7626.05 D₉₆ 11/11/1999 7532.22 D₉₅

2/20/1999 6072.33 D₇₆ 5/19/1999 7614.6 D₉₆ 8/16/1999 8018.47 D₁₀₁ 11/15/1999 7545.03 D₉₅

2/22/1999 6313.63 D₇₉ 5/20/1999 7608.88 D₉₆ 8/17/1999 8083.43 D₁₀₂ 11/16/1999 7606.2 D₉₆

2/23/1999 6180.94 D₇₈ 5/21/1999 7606.69 D₉₆ 8/18/1999 7993.71 D₁₀₀ 11/17/1999 7645.78 D₉₆

2/24/1999 6238.87 D₇₈ 5/24/1999 7588.23 D₉₅ 8/19/1999 7964.67 D₁₀₀ 11/18/1999 7718.06 D₉₇

2/25/1999 6275.53 D₇₉ 5/25/1999 7417.03 D₉₃ 8/20/1999 8117.42 D₁₀₂ 11/19/1999 7770.81 D₉₈

2/26/1999 6318.52 D₇₉ 5/26/1999 7426.63 D₉₃ 8/21/1999 8153.57 D₁₀₂ 11/20/1999 7900.34 D₉₉

3/1/1999 6312.25 D₇₉ 5/27/1999 7469.01 D₉₄ 8/23/1999 8119.98 D₁₀₂ 11/22/1999 8052.31 D₁₀₁

3/2/1999 6263.54 D₇₉ 5/28/1999 7387.37 D₉₃ 8/24/1999 7984.39 D₁₀₀ 11/23/1999 8046.19 D₁₀₁

3/3/1999 6403.14 D₈₁ 5/29/1999 7419.7 D₉₃ 8/25/1999 8127.09 D₁₀₂ 11/24/1999 7921.85 D₁₀₀

3/4/1999 6393.74 D₈₀ 5/31/1999 7316.57 D₉₂ 8/26/1999 8097.57 D₁₀₂ 11/25/1999 7904.53 D₉₉

3/5/1999 6383.09 D₈₀ 6/1/1999 7397.62 D₉₃ 8/27/1999 8053.97 D₁₀₁ 11/26/1999 7595.44 D₉₅

3/6/1999 6421.73 D₈₁ 6/2/1999 7488.03 D₉₄ 8/30/1999 8071.36 D₁₀₁ 11/29/1999 7823.9 D₉₈

3/8/1999 6431.96 D₈₁ 6/3/1999 7572.91 D₉₅ 8/31/1999 8157.73 D₁₀₂ 11/30/1999 7720.87 D ₉₇

3/9/1999 6493.43 D₈₂ 6/4/1999 7590.44 D₉₅ 9/1/1999 8273.33 D₁₀₄ 12/1/1999 7766.2 D₉₈

3/10/1999 6486.61 D₈₂ 6/5/1999 7639.3 D₉₆ 9/2/1999 8226.15 D₁₀₃ 12/2/1999 7806.26 D₉₈

3/11/1999 6436.8 D₈₁ 6/7/1999 7802.69 D₉₈ 9/3/1999 8073.97 D₁₀₁ 12/3/1999 7933.17 D₁₀₀

3/12/1999 6462.73 D₈₁ 6/8/1999 7892.13 D₉₉ 9/4/1999 8065.11 D₁₀₁ 12/4/1999 7964.49 D₁₀₀

3/15/1999 6598.32 D₈₃ 6/9/1999 7957.71 D₁₀₀ 9/6/1999 8130.28 D₁₀₂ 12/6/1999 7894.46 D₉₉

3/16/1999 6672.23 D₈₄ 6/10/1999 7996.76 D₁₀₀ 9/7/1999 7945.76 D₁₀₀ 12/7/1999 7827.05 D₉₈

3/17/1999 6757.07 D₈₅ 6/11/1999 7979.4 D₁₀₀ 9/8/1999 7973.3 D₁₀₀ 12/8/1999 7811.02 D₉₈

3/18/1999 6895.01 D₈₇ 6/14/1999 7973.58 D₁₀₀ 9/9/1999 8025.02 D₁₀₁ 12/9/1999 7738.84 D₉₇

3/19/1999 6997.29 D₈₈ 6/15/1999 7960 D₁₀₀ 9/10/1999 8161.46 D₁₀₃ 12/10/1999 7733.77 D₉₇

3/20/1999 6993.38 D₈₈ 6/16/1999 8059.02 D₁₀₁ 9/13/1999 8178.69 D₁₀₃ 12/13/1999 7883.61 D₉₉

3/22/1999 7043.23 D₈₉ 6/17/1999 8274.36 D₁₀₄ 9/14/1999 8092.02 D₁₀₂ 12/14/1999 7850.14 D₉₉

3/23/1999 6945.48 D₈₇ 6/21/1999 8413.48 D₁₀₆ 9/15/1999 7971.04 D₁₀₀ 12/15/1999 7859.89 D₉₉

3/24/1999 6889.42 D₈₇ 6/22/1999 8608.91 D₁₀₈ 9/16/1999 7968.9 D₁₀₀ 12/16/1999 7739.76 D₉₇

3/25/1999 6941.38 D₈₇ 6/23/1999 8492.32 D₁₀₇ 9/17/1999 7916.92 D₉₉ 12/17/1999 7723.22 D₉₇

3/26/1999 7033.25 D₈₈ 6/24/1999 8589.31 D₁₀₈ 9/18/1999 8016.93 D₁₀₁ 12/18/1999 7797.87 D₉₈

3/29/1999 6901.68 D₈₇ 6/25/1999 8265.96 D₁₀₄ 9/20/1999 7972.14 D₁₀₀ 12/20/1999 7782.94 D₉₈

3/30/1999 6898.66 D₈₇ 6/28/1999 8281.45 D₁₀₄ 9/27/1999 7759.93 D₉₇ 12/21/1999 7934.26 D₁₀₀

3/31/1999 6881.72 D₈₇ 6/29/1999 8514.27 D₁₀₇ 9/28/1999 7577.85 D₉₅ 12/22/1999 8002.76 D₁₀₁

4/1/1999 7018.68 D₈₈ 6/30/1999 8467.37 D₁₀₆ 9/29/1999 7615.45 D₉₆ 12/23/1999 8083.49 D₁₀₂

4/2/1999 7232.51 D₉₁ 7/2/1999 8572.09 D₁₀₈ 9/30/1999 7598.79 D₉₅ 12/24/1999 8219.45 D₁₀₃

4/3/1999 7182.2 D₉₀ 7/3/1999 8563.55 D₁₀₈ 10/1/1999 7694.99 D₉₇ 12/27/1999 8415.07 D₁₀₆

4/6/1999 7163.99 D₉₀ 7/5/1999 8593.35 D₁₀₈ 10/2/1999 7659.55 D₉₆ 12/28/1999 8448.84 D₁₀₆

4/7/1999 7135.89 D₉₀ 7/6/1999 8454.49 D₁₀₆ 10/4/1999 7685.48 D₉₇

4/8/1999 7273.41 D₉₁ 7/7/1999 8470.07 D ₁₀₆ 10/5/1999 7557.01 D₉₅

Date	TAIEX	Fuzzified	Date	TAIEX	Fuzzified	Date	TAIEX	Fuzzified	Date	TAIEX	Fuzzified
1/5/1999	6152.43	D₇₇	4/9/1999	7265.7	D₉₁	7/8/1999	8592.43	D₁₀₈	10/6/1999	7501.63	D₉₄
1/6/1999	6199.91	D₇₈	4/12/1999	7242.4	D₉₁	7/9/1999	8550.27	D₁₀₇	10/7/1999	7612	D₉₆
1/7/1999	6404.31	D₈₁	4/13/1999	7337.85	D₉₂	7/12/1999	8463.9	D₁₀₆	10/8/1999	7552.98	D₉₅
1/8/1999	6421.75	D₈₁	4/14/1999	7398.65	D₉₃	7/13/1999	8204.5	D₁₀₃	10/11/1999	7607.11	D₉₆
1/11/1999	6406.99	D₈₁	4/15/1999	7498.17	D₉₄	7/14/1999	7888.66	D₉₉	10/12/1999	7835.37	D₉₈
1/12/1999	6363.89	D₈₀	4/16/1999	7466.82	D₉₄	7/15/1999	7918.04	D₉₉	10/13/1999	7836.94	D₉₈
1/13/1999	6319.34	D₇₉	4/17/1999	7581.5	D₉₅	7/16/1999	7411.58	D₉₃	10/14/1999	7879.91	D₉₉
1/14/1999	6241.32	D₇₉	4/19/1999	7623.18	D₉₆	7/17/1999	7366.23	D₉₃	10/15/1999	7819.09	D₉₈
1/15/1999	6454.6	D₈₁	4/20/1999	7627.74	D₉₆	7/19/1999	7386.89	D₉₃	10/16/1999	7829.39	D₉₈
1/16/1999	6483.3	D₈₂	4/21/1999	7474.16	D₉₄	7/20/1999	7806.85	D₉₈	10/18/1999	7745.26	D₉₇
1/18/1999	6377.25	D₈₀	4/22/1999	7494.6	D₉₄	7/21/1999	7786.65	D₉₈	10/19/1999	7692.96	D₉₇
1/19/1999	6343.36	D₈₀	4/23/1999	7612.8	D₉₆	7/22/1999	7678.67	D₉₆	10/20/1999	7666.64	D₉₆
1/20/1999	6310.71	D₇₉	4/26/1999	7629.09	D₉₆	7/23/1999	7724.52	D₉₇	10/21/1999	7654.9	D₉₆
1/21/1999	6332.2	D₈₀	4/27/1999	7550.13	D₉₅	7/26/1999	7595.71	D₉₅	10/22/1999	7559.63	D₉₅
1/22/1999	6228.95	D₇₈	4/28/1999	7496.61	D₉₄	7/27/1999	7367.97	D₉₃	10/25/1999	7680.87	D₉₇
1/25/1999	6033.21	D₇₆	4/29/1999	7289.62	D₉₂	7/28/1999	7484.5	D₉₄	10/26/1999	7700.29	D₉₇
1/26/1999	6115.64	D₇₇	4/30/1999	7371.17	D₉₃	7/29/1999	7359.37	D₉₂	10/27/1999	7701.22	D₉₇
1/27/1999	6138.87	D₇₇	5/3/1999	7383.26	D₉₃	7/30/1999	7413.11	D₉₃	10/28/1999	7681.85	D₉₇
1/28/1999	6063.41	D₇₆	5/4/1999	7588.04	D₉₅	7/31/1999	7326.75	D₉₂	10/29/1999	7706.67	D₉₇
1/29/1999	5984	D₇₅	5/5/1999	7572.16	D₉₅	8/2/1999	7195.94	D₉₀	10/30/1999	7854.85	D₉₉
1/30/1999	5998.32	D₇₅	5/6/1999	7560.05	D₉₅	8/3/1999	7175.19	D₉₀	11/1/1999	7814.89	D₉₈
2/1/1999	5862.79	D₇₄	5/7/1999	7469.33	D₉₄	8/4/1999	7110.8	D₈₉	11/2/1999	7721.59	D₉₇
2/2/1999	5749.64	D₇₂	5/10/1999	7484.37	D₉₄	8/5/1999	6959.73	D₈₇	11/3/1999	7580.09	D₉₅
2/3/1999	5743.86	D₇₂	5/11/1999	7474.45	D₉₄	8/6/1999	6823.52	D₈₆	11/4/1999	7469.23	D₉₄
2/4/1999	5514.89	D₆₉	5/12/1999	7448.41	D₉₄	8/7/1999	7049.74	D₈₉	11/5/1999	7488.26	D₉₄
2/5/1999	5474.79	D₆₉	5/13/1999	7416.2	D₉₃	8/9/1999	7028.01	D₈₈	11/6/1999	7376.56	D₉₃
2/6/1999	5710.18	D₇₂	5/14/1999	7592.53	D₉₅	8/10/1999	7269.6	D₉₁	11/8/1999	7401.49	D₉₃
2/8/1999	5822.98	D₇₃	5/15/1999	7576.64	D₉₅	8/11/1999	7228.68	D₉₁	11/9/1999	7362.69	D₉₃
2/9/1999	5723.73	D₇₂	5/17/1999	7599.76	D₉₅	8/12/1999	7330.24	D₉₂	11/10/1999	7401.81	D₉₃
2/10/1999	5798	D₇₃	5/18/1999	7585.51	D₉₅	8/13/1999	7626.05	D₉₆	11/11/1999	7532.22	D₉₅
2/20/1999	6072.33	D₇₆	5/19/1999	7614.6	D₉₆	8/16/1999	8018.47	D₁₀₁	11/15/1999	7545.03	D₉₅
2/22/1999	6313.63	D₇₉	5/20/1999	7608.88	D₉₆	8/17/1999	8083.43	D₁₀₂	11/16/1999	7606.2	D₉₆
2/23/1999	6180.94	D₇₈	5/21/1999	7606.69	D₉₆	8/18/1999	7993.71	D₁₀₀	11/17/1999	7645.78	D₉₆
2/24/1999	6238.87	D₇₈	5/24/1999	7588.23	D₉₅	8/19/1999	7964.67	D₁₀₀	11/18/1999	7718.06	D₉₇
2/25/1999	6275.53	D₇₉	5/25/1999	7417.03	D₉₃	8/20/1999	8117.42	D₁₀₂	11/19/1999	7770.81	D₉₈
2/26/1999	6318.52	D₇₉	5/26/1999	7426.63	D₉₃	8/21/1999	8153.57	D₁₀₂	11/20/1999	7900.34	D₉₉
3/1/1999	6312.25	D₇₉	5/27/1999	7469.01	D₉₄	8/23/1999	8119.98	D₁₀₂	11/22/1999	8052.31	D₁₀₁
3/2/1999	6263.54	D₇₉	5/28/1999	7387.37	D₉₃	8/24/1999	7984.39	D₁₀₀	11/23/1999	8046.19	D₁₀₁
3/3/1999	6403.14	D₈₁	5/29/1999	7419.7	D₉₃	8/25/1999	8127.09	D₁₀₂	11/24/1999	7921.85	D₁₀₀
3/4/1999	6393.74	D₈₀	5/31/1999	7316.57	D₉₂	8/26/1999	8097.57	D₁₀₂	11/25/1999	7904.53	D₉₉
3/5/1999	6383.09	D₈₀	6/1/1999	7397.62	D₉₃	8/27/1999	8053.97	D₁₀₁	11/26/1999	7595.44	D₉₅
3/6/1999	6421.73	D₈₁	6/2/1999	7488.03	D₉₄	8/30/1999	8071.36	D₁₀₁	11/29/1999	7823.9	D₉₈
3/8/1999	6431.96	D₈₁	6/3/1999	7572.91	D₉₅	8/31/1999	8157.73	D₁₀₂	11/30/1999	7720.87	D ₉₇
3/9/1999	6493.43	D₈₂	6/4/1999	7590.44	D₉₅	9/1/1999	8273.33	D₁₀₄	12/1/1999	7766.2	D₉₈
3/10/1999	6486.61	D₈₂	6/5/1999	7639.3	D₉₆	9/2/1999	8226.15	D₁₀₃	12/2/1999	7806.26	D₉₈
3/11/1999	6436.8	D₈₁	6/7/1999	7802.69	D₉₈	9/3/1999	8073.97	D₁₀₁	12/3/1999	7933.17	D₁₀₀
3/12/1999	6462.73	D₈₁	6/8/1999	7892.13	D₉₉	9/4/1999	8065.11	D₁₀₁	12/4/1999	7964.49	D₁₀₀
3/15/1999	6598.32	D₈₃	6/9/1999	7957.71	D₁₀₀	9/6/1999	8130.28	D₁₀₂	12/6/1999	7894.46	D₉₉
3/16/1999	6672.23	D₈₄	6/10/1999	7996.76	D₁₀₀	9/7/1999	7945.76	D₁₀₀	12/7/1999	7827.05	D₉₈
3/17/1999	6757.07	D₈₅	6/11/1999	7979.4	D₁₀₀	9/8/1999	7973.3	D₁₀₀	12/8/1999	7811.02	D₉₈
3/18/1999	6895.01	D₈₇	6/14/1999	7973.58	D₁₀₀	9/9/1999	8025.02	D₁₀₁	12/9/1999	7738.84	D₉₇
3/19/1999	6997.29	D₈₈	6/15/1999	7960	D₁₀₀	9/10/1999	8161.46	D₁₀₃	12/10/1999	7733.77	D₉₇
3/20/1999	6993.38	D₈₈	6/16/1999	8059.02	D₁₀₁	9/13/1999	8178.69	D₁₀₃	12/13/1999	7883.61	D₉₉
3/22/1999	7043.23	D₈₉	6/17/1999	8274.36	D₁₀₄	9/14/1999	8092.02	D₁₀₂	12/14/1999	7850.14	D₉₉
3/23/1999	6945.48	D₈₇	6/21/1999	8413.48	D₁₀₆	9/15/1999	7971.04	D₁₀₀	12/15/1999	7859.89	D₉₉
3/24/1999	6889.42	D₈₇	6/22/1999	8608.91	D₁₀₈	9/16/1999	7968.9	D₁₀₀	12/16/1999	7739.76	D₉₇
3/25/1999	6941.38	D₈₇	6/23/1999	8492.32	D₁₀₇	9/17/1999	7916.92	D₉₉	12/17/1999	7723.22	D₉₇
3/26/1999	7033.25	D₈₈	6/24/1999	8589.31	D₁₀₈	9/18/1999	8016.93	D₁₀₁	12/18/1999	7797.87	D₉₈
3/29/1999	6901.68	D₈₇	6/25/1999	8265.96	D₁₀₄	9/20/1999	7972.14	D₁₀₀	12/20/1999	7782.94	D₉₈
3/30/1999	6898.66	D₈₇	6/28/1999	8281.45	D₁₀₄	9/27/1999	7759.93	D₉₇	12/21/1999	7934.26	D₁₀₀
3/31/1999	6881.72	D₈₇	6/29/1999	8514.27	D₁₀₇	9/28/1999	7577.85	D₉₅	12/22/1999	8002.76	D₁₀₁
4/1/1999	7018.68	D₈₈	6/30/1999	8467.37	D₁₀₆	9/29/1999	7615.45	D₉₆	12/23/1999	8083.49	D₁₀₂
4/2/1999	7232.51	D₉₁	7/2/1999	8572.09	D₁₀₈	9/30/1999	7598.79	D₉₅	12/24/1999	8219.45	D₁₀₃
4/3/1999	7182.2	D₉₀	7/3/1999	8563.55	D₁₀₈	10/1/1999	7694.99	D₉₇	12/27/1999	8415.07	D₁₀₆
4/6/1999	7163.99	D₉₀	7/5/1999	8593.35	D₁₀₈	10/2/1999	7659.55	D₉₆	12/28/1999	8448.84	D₁₀₆
4/7/1999	7135.89	D₉₀	7/6/1999	8454.49	D₁₀₆	10/4/1999	7685.48	D₉₇
4/8/1999	7273.41	D₉₁	7/7/1999	8470.07	D ₁₀₆	10/5/1999	7557.01	D₉₅

[Step 2] According to the fuzzified training data from January 5, 1999 to October 30, 1999 in Table 2, the fuzzy logical relationships, fuzzy-trend groups and the corresponding jumps can be generated, as shown in Table 3.

Table 3

Logical relationships, fuzzy-trend groups and jumps of the training fuzzy datasets

FLRs	Fuzzy-	Jump	FLRs	Fuzzy-	Jump	FLRs	Fuzzy-	Jump
	trend			trend			trend
	Group			Group			Group
(D₇₇,D₇₈,D₈₁,D₈₁)⟶D₈₁	26	0	(D₉₅,D₉₆,D₉₆,D₉₄)⟶D₉₄	22	0	(D₉₇,D₉₅,D₉₃,D₉₄)⟶D₉₂	3	–2
(D₇₈,D₈₁,D₈₁,D₈₁)⟶D₈₀	23	–1	(D₉₆,D₉₆,D₉₄,D₉₄)⟶D₉₆	11	2	(D₉₅,D₉₃,D₉₄,D₉₂)⟶D₉₃	7	1
(D₈₁,D₈₁,D₈₁,D₈₀)⟶D₇₉	13	–1	(D₉₆,D₉₄,D₉₄,D₉₆)⟶D₉₆	6	0	(D₉₃,D₉₄,D₉₂,D₉₃)⟶D₉₂	21	–1
(D₈₁,D₈₁,D₈₀,D₇₉)⟶D₇₉	10	0	(D₉₄,D₉₄,D₉₆,D₉₆)⟶D₉₅	17	–1	(D₉₄,D₉₂,D₉₃,D₉₂)⟶D₉₀	7	–2
(D₈₁,D₈₀,D₇₉,D₇₉)⟶D₈₁	2	2	(D₉₄,D₉₆,D₉₆,D₉₅)⟶D₉₄	22	–1	(D₉₂,D₉₃,D₉₂,D₉₀)⟶D₉₀	19	0
(D₈₀,D₇₉,D₇₉,D₈₁)⟶D₈₂	6	1	(D₉₆,D₉₆,D₉₅,D₉₄)⟶D₉₂	10	–2	(D₉₃,D₉₂,D₉₀,D₉₀)⟶D₈₉	2	–1
(D₇₉,D₇₉,D₈₁,D₈₂)⟶D₈₀	18	–2	(D₉₆,D₉₅,D₉₄,D₉₂)⟶D₉₃	1	1	(D₉₂,D₉₀,D₉₀,D₈₉)⟶D₈₇	4	–2
(D₇₉,D₈₁,D₈₂,D₈₀)⟶D₈₀	25	0	(D₉₅,D₉₄,D₉₂,D₉₃)⟶D₉₃	3	0	(D₉₀,D₉₀,D_89,D₈₇)⟶D₈₆	10	–1
(D₈₁,D₈₂,D₈₀,D₈₀)⟶D₇₉	20	–1	(D₉₄,D₉₂,D₉₃,D₉₃)⟶D₉₅	8	2	(D₉₀,D₈₉,D₈₇,D₈₆)⟶D₈₉	1	3
(D₈₂,D₈₀,D₈₀,D₇₉)⟶D₈₀	4	1	(D₉₂,D₉₃,D₉₃,D₉₅)⟶D₉₅	24	0	(D₈₉,D₈₇,D₈₆,D₈₉)⟶D₈₈	3	–1
(D₈₀,D₈₀,D₇₉,D₈₀)⟶D₇₈	12	–2	(D₉₃,D₉₃,D₉₅,D₉₅)⟶D₉₅	17	0	(D₈₇,D₈₆,D₈₉,D₈₈)⟶D₉₁	7	3
(D₈₀,D₇₉,D₈₀,D₇₈)⟶D₇₆	7	–2	(D₉₃,D₉₅,D₉₅,D₉₅)⟶D₉₄	23	–1	(D₈₆,D₈₉,D₈₈,D₉₁)⟶D₉₁	21	0
(D₇₉,D₈₀,D_78,D₇₆)⟶D₇₇	19	1	(D₉₅,D₉₅,D₉₅,D₉₄)⟶D₉₄	13	0	(D₈₉,D₈₈,D₉₁,D₉₁)⟶D₉₂	8	1
(D₈₀,D₇₈,D₇₆,D₇₇)⟶D₇₇	3	0	(D₉₅,D₉₅,D₉₄,D₉₄)⟶D₉₄	11	0	(D₈₈,D₉₁,D₉₁,D₉₂)⟶D₉₆	24	4
(D₇₈,D₇₆,D₇₇,D₇₇)⟶D₇₆	8	–1	(D₉₅,D₉₄,D₉₄,D₉₄)⟶D₉₄	5	0	(D₉₁,D₉₁,D₉₂,D₉₆)⟶D₁₀₁	18	5
(D₇₆,D₇₇,D₇₇,D₇₆)⟶D₇₅	22	–1	(D₉₄,D₉₄,D₉₄,D₉₄)⟶D₉₃	14	–1	(D₉₁,D₉₂,D₉₆,D₁₀₁)⟶D₁₀₂	27	1
(D₇₇,D₇₇,D₇₆,D₇₅)⟶D₇₅	10	0	(D₉₄,D₉₄,D₉₄,D₉₃)⟶D₉₅	13	2	(D₉₂,D₉₆,D₁₀₁,D₁₀₂)⟶D₁₀₀	27	–2
(D₇₇,D₇₆,D₇₅,D₇₅)⟶D₇₄	2	–1	(D₉₄,D₉₄,D₉₃,D₉₅)⟶D₉₅	12	0	(D₉₆,D₁₀₁,D₁₀₂,D₁₀₀)⟶D₁₀₀	25	0
(D₇₆,D₇₅,D₇₅,D₇₄)⟶D₇₂	4	–2	(D₉₄,D₉₃,D₉₅,D₉₅)⟶D₉₅	8	0	(D₁₀₁,D₁₀₂,D₁₀₀,D₁₀₀)⟶D₁₀₂	20	2
(D₇₅,D₇₅,D₇₄,D₇₂)⟶D₇₂	10	0	(D₉₃,D₉₅,D₉₅,D₉₅)⟶D₉₅	23	0	(D₁₀₂,D₁₀₀,D₁₀₀,D₁₀₂)⟶D₁₀₂	6	0
(D₇₅,D₇₄,D₇₂,D₇₂)⟶D₆₉	2	–3	(D₉₅,D₉₅,D₉₅,D₉₅)⟶D₉₆	14	1	(D₁₀₀,D₁₀₀,D₁₀₂,D₁₀₂)⟶D₁₀₂	17	0
(D₇₄,D₇₂,D₇₂,D₆₉)⟶D₆₉	4	0	(D₉₅,D₉₅,D₉₅,D₉₆)⟶D₉₆	15	0	(D₁₀₀,D₁₀₂,D₁₀₂,D₁₀₂)⟶D₁₀₀	23	–2
(D₇₂,D₇₂,D₆₉,D₆₉)⟶D₇₂	11	3	(D₉₅,D₉₅,D₉₆,D₉₆)⟶D₉₆	17	0	(D₁₀₂,D₁₀₂,D₁₀₂,D₁₀₀)⟶D₁₀₂	13	2
(D₇₂,D₆₉,D₆₉,D₇₂)⟶D₇₃	6	1	(D₉₅,D₉₆,D₉₆,D₉₆)⟶D₉₅	23	–1	(D₁₀₂,D₁₀₂,D₁₀₀,D₁₀₂)⟶D₁₀₂	12	0
(D₆₉,D₆₉,D₇₂,D₇₃)⟶D₇₂	18	–1	(D₉₆,D₉₆,D₉₆,D₉₅)⟶D₉₃	13	–2	(D₁₀₂,D₁₀₀,D₁₀₂,D₁₀₂)⟶D₁₀₁	8	–1
(D₆₉,D₇₂,D₇₃,D₇₂)⟶D₇₃	25	1	(D₉₆,D₉₆,D₉₅,D₉₃)⟶D₉₃	10	0	(D₁₀₀,D₁₀₂,D₁₀₂,D₁₀₁)⟶D₁₀₁	22	0
(D₇₂,D₇₃,D₇₂,D₇₃)⟶D₇₆	21	3	(D₉₆,D₉₅,D₉₃,D₉₃)⟶D₉₄	2	1	(D₁₀₂,D₁₀₂,D₁₀₁,D₁₀₁)⟶D₁₀₂	11	1
(D₇₃,D₇₂,D₇₃,D₇₆)⟶D₇₉	9	3	(D₉₅,D₉₃,D₉₃,D₉₄)⟶D₉₃	6	–1	(D₁₀₂,D₁₀₁,D₁₀₁,D₁₀₂)⟶D₁₀₄	6	2
(D₇₂,D₇₃,D₇₆,D₇₉)⟶D₇₈	27	–1	(D₉₃,D₉₃,D₉₄,D₉₃)⟶D₉₃	16	0	(D₁₀₁,D₁₀₁,D₁₀₂,D₁₀₄)⟶D₁₀₃	18	–1
(D₇₃,D₇₆,D₇₉,D₇₈)⟶D₇₈	25	0	(D₉₃,D₉₄,D₉₃,D₉₃)⟶D₉₂	20	–1	(D₁₀₁,D₁₀₂,D₁₀₄,D₁₀₃)⟶D₁₀₁	25	–2
(D₇₆,D₇₉,D₇₈,D₇₈)⟶D₇₉	20	1	(D₉₄,D₉₃,D₉₃,D₉₂)⟶D₉₃	4	1	(D₁₀₂,D₁₀₄,D₁₀₃,D₁₀₁)⟶D₁₀₁	19	0
(D₇₉,D₇₈,D₇₈,D₇₉)⟶D₇₉	6	0	(D₉₃,D₉₃,D₉₂,D₉₃)⟶D₉₄	12	1	(D₁₀₄,D₁₀₃,D₁₀₁,D₁₀₁)⟶D₁₀₂	2	1
(D₇₈,D₇₈,D₇₉,D₇₉)⟶D₇₉	17	0	(D₉₃,D₉₂,D₉₃,D₉₄)⟶D₉₅	9	1	(D₁₀₃,D₁₀₁,D₁₀₁,D₁₀₂)⟶D₁₀₀	6	–2
(D₇₈,D₇₉,D₇₉,D₇₉)⟶D₇₉	23	0	(D₉₂,D₉₃,D₉₄,D₉₅)⟶D₉₅	27	0	(D₁₀₁,D₁₀₁,D₁₀₂,D₁₀₀)⟶D₁₀₀	16	0
(D₇₉,D₇₉,D₇₉,D₇₉)⟶D₈₁	14	2	(D₉₃,D₉₄,D₉₅,D₉₅)⟶D₉₆	26	1	(D₁₀₁,D₁₀₂,D₁₀₀,D₁₀₀)⟶D₁₀₁	20	1
(D₇₉,D₇₉,D₇₉,D₈₁)⟶D₈₀	15	–1	(D₉₄,D₉₅,D₉₅,D₉₆)⟶D₉₈	24	2	(D₁₀₂,D₁₀₀,D₁₀₀,D₁₀₁)⟶D₁₀₃	6	2
(D₇₉,D₇₉,D₈₁,D₈₀)⟶D₈₀	16	0	(D₉₅,D₉₅,D₉₆,D₉₈)⟶D₉₉	18	1	(D₁₀₀,D₁₀₀,D₁₀₁,D₁₀₃)⟶D₁₀₃	18	0
(D₇₉,D₈₁,D₈₀,D₈₀)⟶D₈₁	20	1	(D₉₅,D₉₆,D₉₈,D₉₉)⟶D₁₀₀	27	1	(D₁₀₀,D₁₀₁,D₁₀₃,D₁₀₃)⟶D₁₀₂	26	–1
(D₈₁,D₈₀,D₈₀,D₈₁)⟶D₈₁	6	0	(D₉₆,D₉₈,D₉₉,D₁₀₀)⟶D₁₀₀	27	0	(D₁₀₁,D₁₀₃,D₁₀₃,D₁₀₂)⟶D₁₀₀	22	–2
(D₈₀,D₈₀,D₈₁,D₈₁)⟶D₈₂	17	1	(D₉₈,D₉₉,D₁₀₀,D₁₀₀)⟶D₁₀₀	26	0	(D₁₀₃,D₁₀₃,D₁₀₂,D₁₀₀)⟶D₁₀₀	10	0
(D₈₀,D₈₁,D₈₁,D₈₂)⟶D₈₂	24	0	(D₉₉,D₁₀₀,D₁₀₀,D₁₀₀)⟶D₁₀₀	23	0	(D₁₀₃,D₁₀₂,D₁₀₀,D₁₀₀)⟶D₉₉	2	–1
(D₈₁,D₈₁,D₈₂,D₈₂)⟶D₈₁	17	–1	(D₁₀₀,D1₀₀,D₁₀₀,D₁₀₀)⟶D₁₀₀	14	0	(D₁₀₂,D₁₀₀,D₁₀₀,D₉₉)⟶D₁₀₁	4	2
(D₈₁,D₈₂,D₈₂,D₈₁)⟶D₈₁	22	0	(D₁₀₀,D₁₀₀,D₁₀₀,D₁₀₀)⟶D₁₀₁	14	1	(D₁₀₀,D₁₀₀,D₉₉,D₁₀₁)⟶D₁₀₀	12	–1
(D₈₂,D₈₂,D₈₁,D₈₁)⟶D₈₃	11	2	(D₁₀₀,D₁₀₀,D₁₀₀,D₁₀₁)⟶D₁₀₄	15	3	(D₁₀₀,D₉₉,D₁₀₁,D₁₀₀)⟶D₉₇	7	–3
(D₈₂,D₈₁,D₈₁,D₈₃)⟶D₈₄	6	1	(D₁₀₀,D₁₀₀,D₁₀₁,D₁₀₄)⟶D₁₀₆	18	2	(D₉₉,D₁₀₁,D₁₀₀,D₉₇)⟶D₉₅	19	–2
(D₈₁,D₈₁,D₈₃,D₈₄)⟶D₈₅	18	1	(D₁₀₀,D₁₀₁,D₁₀₄,D₁₀₆)⟶D₁₀₈	27	2	(D₁₀₁,D₁₀₀,D₉₇,D₉₅)⟶D₉₆	1	1
(D₈₁,D₈₃,D₈₄,D₈₅)⟶D₈₇	27	2	(D₁₀₁,D₁₀₄,D₁₀₆,D₁₀₈)⟶D₁₀₇	27	–1	(D₁₀₀,D₉₇,D₉₅,D₉₆)⟶D₉₅	3	–1
(D₈₃,D₈₄,D₈₅,D₈₇)⟶D₈₈	27	1	(D₁₀₄,D₁₀₆,D₁₀₈,D₁₀₇)⟶D₁₀₈	25	1	(D₉₇,D₉₅,D₉₆,D₉₅)⟶D₉₇	7	2
(D₈₄,D₈₅,D₈₇,D₈₈)⟶D₈₈	27	0	(D₁₀₆,D₁₀₈,D₁₀₇,D₁₀₈)⟶D₁₀₄	21	–4	(D₉₅,D₉₆,D₉₅,D₉₇)⟶D₉₆	21	–1
(D₈₅,D₈₇,D₈₈,D₈₈)⟶D₈₉	26	1	(D₁₀₈,D₁₀₇,D₁₀₈,D₁₀₄)⟶D₁₀₄	7	0	(D₉₆,D₉₅,D₉₇,D₉₆)⟶D₉₇	7	1
(D₈₇,D₈₈,D₈₈,D₈₉)⟶D₈₇	24	–2	(D₁₀₇,D₁₀₈,D₁₀₄,D₁₀₄)⟶D₁₀₇	20	3	(D₉₅,D₉₇,D₉₆,D₉₇)⟶D₉₅	21	–2
(D₈₈,D₈₈,D₈₉,D₈₇)⟶D₈₇	16	0	(D₁₀8,D₁₀₄,D₁₀₄,D₁₀₇)⟶D₁₀₆	6	–1	(D₉₇,D₉₆,D₉₇,D₉₅)⟶D₉₄	7	–1
(D₈₈,D₈₉,D₈₇,D₈₇)⟶D₈₇	20	0	(D₁₀₄,D₁₀₄,D₁₀₇,D₁₀₆)⟶D₁₀₈	16	2	(D₉₆,D₉₇,D₉₅,D₉₄)⟶D₉₆	19	2
(D₈₉,D₈₇,D₈₇,D₈₇)⟶D₈₈	5	1	(D₁₀₄,D₁₀₇,D₁₀₆,D₁₀₈)⟶D₁₀₈	21	0	(D₉₇,D₉₅,D₉₄,D₉₆)⟶D₉₅	3	–1
(D₈₇,D₈₇,D₈₇,D₈₈)⟶D₈₇	15	–1	(D₁₀₇,D₁₀₆,D₁₀₈,D₁₀₈)⟶D₁₀₈	8	0	(D₉₅,D₉₄,D₉₆,D₉₅)⟶D₉₆	7	1
(D₈₇,D₈₇,D₈₈,D₈₇)⟶D₈₇	16	0	(D₁₀₆,D₁₀₈,D₁₀₈,D₁₀₈)⟶D₁₀₆	23	–2	(D₉₄,D₉₆,D₉₅,D₉₆)⟶D₉₈	21	2
(D₈₇,D₈₈,D₈₇,D₈₇)⟶D₈₇	20	0	(D₁₀₈,D₁₀₈,D₁₀₈,D₁₀₆)⟶D₁₀₆	13	0	(D₉₆,D₉₅,D₉₆,D₉₈)⟶D₉₈	9	0
(D₈₈,D₈₇,D₈₇,D₈₇)⟶D₈₈	5	1	(D₁₀₈,D₁₀₈,D₁₀₆,D₁₀₆)⟶D₁₀₈	11	2	(D₉₅,D₉₆,D₉₈,D₉₈)⟶D₉₉	26	1
(D₈₇,D₈₇,D₈₇,D₈₈)⟶D₉₁	15	3	(D₁₀₈,D₁₀₆,D₁₀₆,D₁₀₈)⟶D₁₀₇	6	–1	(D₉₆,D₉₈,D₉₈,D₉₉)⟶D₉₈	24	–1
(D₈₇,D₈₇,D₈₈,D₉₁)⟶D₉₀	18	–1	(D₁₀₆,D₁₀₆,D₁₀₈,D₁₀₇)⟶D₁₀₆	16	–1	(D₉₈,D₉₈,D₉₉,D₉₈)⟶D₉₈	16	0
(D₈₇,D₈₈,D₉₁,D₉₀)⟶D₉₀	25	0	(D₁₀₆,D₁₀₈,D₁₀₇,D₁₀₆)⟶D₁₀₃	19	–3	(D₉₈,D₉₉,D₉₈,D₉₈)⟶D₉₇	20	–1
(D₈₈,D₉₁,D₉₀,D₉₀)⟶D₉₀	20	0	(D₁₀₈,D₁₀₇,D₁₀₆,D₁₀₃)⟶D₉₉	1	–4	(D_99,D₉₈,D₉₈,D₉₇)⟶D₉₇	4	0
(D₉₁,D₉₀,D₉₀,D₉₀)⟶D₉₁	5	1	(D₁₀₇,D₁₀₆,D₁₀₃,D₉₉)⟶D₉₉	1	0	(D₉₈,D₉₈,D₉₇,D₉₇)⟶D₉₆	11	–1
(D₉₀,D₉₀,D₉₀,D₉₁)⟶D₉₁	15	0	(D₁₀₆,D₁₀₃,D₉₉,D₉₉)⟶D₉₃	2	–6	(D₉₈,D₉₇,D₉₇,D₉₆)⟶D₉₆	4	0
(D₉₀,D₉₀,D₉₁,D₉₁)⟶D₉₁	17	0	(D₁₀₃,D₉₉,D₉₉,D₉₃)⟶D₉₃	4	0	(D₉₇,D₉₇,D₉₆,D₉₆)⟶D₉₅	11	–1
(D₉₀,D₉₁,D₉₁,D₉₁)⟶D₉₂	23	1	(D₉₉,D₉₉,D₉₃,D₉₃)⟶D₉₃	11	0	(D₉₇,D₉₆,D₉₆,D₉₅)⟶D₉₇	4	2
(D₉₁,D₉₁,D₉₁,D₉₂)⟶D₉₃	15	1	(D₉₉,D₉₃,D₉₃,D₉₃)⟶D₉₈	5	5	(D₉₆,D₉₆,D₉₅,D₉₇)⟶D₉₇	12	0
(D₉₁,D₉₁,D₉₂,D₉₃)⟶D₉₄	18	1	(D₉₃,D₉₃,D₉₃,D₉₈)⟶D₉₈	15	0	(D₉₆,D₉₅,D₉₇,D₉₇)⟶D₉₇	8	0
(D₉₁,D₉₂,D₉₃,D₉₄)⟶D₉₄	27	0	(D₉₃,D₉₃,D₉₈,D₉₈)⟶D₉₆	17	–2	(D₉₅,D₉₇,D₉₇,D₉₇)⟶D₉₇	23	0
(D₉₂,D₉₃,D₉₄,D₉₄)⟶D₉₅	26	1	(D₉₃,D₉₈,D₉₈,D₉₆)⟶D₉₇	22	1	(D₉₇,D₉₇,D₉₇,D₉₇)⟶D₉₇	14	0
(D₉₃,D₉₄,D₉₄,D₉₅)⟶D₉₆	24	1	(D₉₈,D₉₈,D₉₆,D₉₇)⟶D₉₅	12	–2	(D₉₇,D₉₇,D₉₇,D₉₇)⟶D₉₉	14	2
(D₉₄,D₉₄,D₉₅,D₉₆)⟶D₉₆	18	0	(D₉₈,D₉₆,D₉₇,D₉₅)⟶D₉₃	7	–2
(D₉₄,D₉₅,D₉₆,D₉₆)⟶D₉₄	26	–2	(D₉₆,D₉₇,D₉₅,D₉₃)⟶D₉₄	19	1

Based on the fuzzy-trend groups and the corresponding jump rules generated from training data, we can perform the forecast. For example, the 4th-order fuzzy-trend group number of November 1, 1999 is (D97, D97, D97, D99), belongs to group 15 (see Table 1). From Table 3, the jump rule for group 15 is 0.625 (the jumps are –1, –1, 3, 0, 1, 0, 3, 0, respectively). Then according to Eq.(4), the forecasting value of the TAIEX on for November 1, 1999 is obtained as follows: $\begin{matrix} H^{'} (t) = H (t - 1) + r (g) \times len = 7854.85 \\ + 80 \times 0.625 = 7904.85 \end{matrix}$

The other outcomes are shown in Table 4.

Table 4

Predicting results from November 1,1999 to December 30, 1999

Date	Actual	Forecast	(Forecast-actual)²	Date	Actual	Forecast	(Forecast-actual)²
11/1/1999	7814.89	7904.85	8092.80	12/1/1999	7766.20	7706.33	3584.42
11/2/1999	7721.59	7824.89	10670.89	12/2/1999	7806.26	7736.20	4908.40
11/3/1999	7580.09	7710.16	16918.20	12/3/1999	7933.17	7817.69	13335.63
11/4/1999	7469.23	7596.09	16093.46	12/4/1999	7964.49	7978.88	207.07
11/5/1999	7488.26	7485.23	9.18	12/6/1999	7894.46	7937.83	1880.96
11/6/1999	7376.56	7408.26	1004.89	12/7/1999	7827.05	7860.17	1096.93
11/8/1999	7401.49	7392.56	79.74	12/8/1999	7811.02	7792.76	333.43
11/9/1999	7362.69	7472.60	12080.21	12/9/1999	7738.84	7731.02	61.15
11/10/1999	7401.81	7490.69	7899.65	12/10/1999	7733.77	7754.84	443.94
11/11/1999	7532.22	7458.95	5368.49	12/13/1999	7883.61	7804.88	6198.41
11/15/1999	7545.03	7582.22	1383.10	12/14/1999	7850.14	7895.91	2094.89
11/16/1999	7606.20	7518.37	7714.11	12/15/1999	7859.89	7823.48	1325.69
11/17/1999	7645.78	7651.91	37.58	12/16/1999	7739.76	7811.89	5202.74
11/18/1999	7718.06	7619.12	9789.12	12/17/1999	7723.22	7753.10	892.81
11/19/1999	7770.81	7763.77	49.56	12/18/1999	7797.87	7794.33	12.53
11/20/1999	7900.34	7807.17	8680.65	12/20/1999	7782.94	7810.17	741.47
11/22/1999	8052.31	7920.34	17416.08	12/21/1999	7934.26	7756.28	31676.88
11/23/1999	8046.19	8072.31	682.25	12/22/1999	8002.76	7979.97	519.38
11/24/1999	7921.85	8056.19	18047.24	12/23/1999	8083.49	8039.12	1968.70
11/25/1999	7904.53	7887.56	287.98	12/24/1999	8219.45	8103.49	13446.72
11/26/1999	7595.44	7870.24	75515.04	12/27/1999	8415.07	8239.45	30842.38
11/29/1999	7823.90	7611.44	45139.25	12/28/1999	8448.84	8435.07	189.61
11/30/1999	7720.87	7757.24	1322.78	Root Mean Square Error(RMSE)			92.53

We can evaluate the forecasting performance by some common difference comparison methods. The following indicators are usually applied in the comparisons of TSs forecasting models, including mean squared error (MSE), root of the mean squared error (RMSE), mean absolute error (MAE), mean percentage error (MPE), etc. The definitions of these indicators are detailed by Equations (5–8): $MSE = \frac{\sum_{i = 1}^{m} (f_value (i) - r_value (i))^{2}}{m}$ (5) $RMSE = \sqrt{\frac{\sum_{i = 1}^{m} (f_value (i) - r_value (i))^{2}}{m}}$ (6) $MAE = \frac{\sum_{i = 1}^{m} | (f_value (i) - r_value (i)) |}{m}$ (7) $MPE = \frac{\sum_{i = 1}^{m} | (f_value (i) - r_value (i)) | / r_{v} alue (i)}{m}$ (8) where m represents the total number of forecasting values, f_value(i) and r_value (i) represent the forecasting value and actual value at time i, respectively. As to our presented model, the indicators of MSE, RMSE, MAE, MPE are 8561.03, 92.53, 70.15, 0.01, respectively. This model is also applied to predict the TAIEX from 1997 to 2005. Corresponding results are shown in Fig. 2 and Table 5.

Fig.2

The stock market fluctuation for TAIEX test dataset (1997–2005).

Table 5

RMSEs of forecast errors for TAIEX 1997 to 2005

Year	1997	1998	1999	2000	2001	2002	2003	2004	2005
RMSE	139.62	115.54	92.53	129.67	115.21	66.63	55.12	54.76	53.34

The following Table 6 describes the RMSEs of different methods when predicting TAIEX1999. Obviously, the accuracy of presented method outperforms many existing methods. The result of Chen & Chen’s method [13] is slight better than the proposed method in this paper. However, Chen & Chen’s method employed complex discretization partitioning method and extra adjustment deviation methods. The method presented in this paper is more easily to be automatically carried out by a computer.

Table 6

A comparison of RMSEs for different methods for forecasting TAIEX1999

Methods	RMSE
Yu ‘s Method [6]	145
Hsieh et al. ‘s Method [32]	94
Chang et al. ‘s Method [9]	100
Cheng et al. ‘s Method [2]	103
Chen et al. ‘s Method [24]	102.11
Chen &Chen ‘s Method [22]	103.9
Chen &Chen ‘s Method [13]	92
Zhao et al. ‘s Method [1]	110.85
The Proposed Method	92.53

5.2 Forecasting SHSECI

In this paper we use SHSECI (Shanghai Stock Exchange Composite Index) as our second example. The following section are composed of using the presented method to forecast the SHSECI from the year of 2001 till 2015 during which in each year the actual datasets from January to October are training data and the rests of the year are the testing data. The forecasting results and the RMSEs of forecast errors are shown in Fig. 3 and Table 7, respectively.

Fig.3

The stock market fluctuation for SHSECI test dataset (2001–2015).

From Fig. 3 and Table 7, we can realize that the proposed method can successfully predict the stock market. The maximum RMSEs of forecast errors occurred when forecasting the November to December stock market in 2007 and 2008. In these two years, because of the significant changes in economic environment, the fluctuation rules of stock market from January to October are not the same as from November to December. For example, in 2007, the SHSECI kept rising from January to October and remained consolidation from November to December. For 2008, as everyone knows, the financial crisis caused the abnormal fluctuation of stock market and led to a more RMSE of forecast errors.

Table 7

RMSEs of predict errors for SHSECI from 2001 to 2015

Year	2001	2002	2003	2004	2005	2006	2007	2008	2009	2010	2011	2012	2013	2014	2015
RMSE	24.86	21.75	26.57	19.07	9.63	28.98	129.2	79.77	59.96	49.48	29.7	23.14	22.13	44.11	58.89

6 Conclusion

In this paper, we have described a novel method for the prediction of the TAIEX and SHSECI based on high-order fuzzy-trend and the corresponding jump rules. The jump rules forecasting the change of future are generated by heuristic learning from the high-order FLRs and their fuzzy-trend of the training FTSs. The experiment results show that the proposed method both outperforms the existing method and it is effective in stock market forecasting. In this paper, jump rules are generated from the stock indicate and the training datasets are always from January to October for each year. In the future, we will use all of the historical data to generate the jump rules. Also, we will consider the influences of surrounding stock markets and other factors of the stock market itself.

Footnotes

Acknowledgments

Thanks reviewers for providing so insightful comments and suggestions, which led to a better quality of this paper. This work is supported by the Fund of China Nation Tourism Administration (15TACK003), the Fund of Ministry of education of Humanities and Social Sciences (14YJAZH025), the Foundation program of Jiangsu University (16JDG005), and the Natural Science Foundation of Shandong Province (ZR2013GM003).

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