# A Multifactor Fuzzy Time-Series Fitting Model for Forecasting the Stock Index

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## Abstract

**:**

## 1. Introduction

## 2. Related Work

#### 2.1. Fuzzy Time Series

#### 2.2. The Causality between Price and Volume

## 3. Proposed Model

#### 3.1. Proposed Computational Step

_{1}, B

_{2}, … B

_{k}in the following range:

_{ij}is the grade of u

_{j}in fuzzy set B

_{i}, 1 ≤ i ≤ k, 1 ≤ j ≤ m, and b

_{ij}∈ [0,1].

_{i}(i = 1… m). If the maximal grade is located in B

_{k}, then we mark the fuzzified stock index as B

_{k}. The following seven linguistic values were used in this study: B

_{1}(very low), B

_{2}(low), B

_{3}(slightly low), B

_{4}(normal), B

_{5}(slightly high), B

_{6}(high), and B

_{7}(very high).

_{i}(t − 1) and B

_{j}(t), we can use B

_{i}→B

_{j}to represent fuzzy logical relationships. i.e., “If part (rule condition)” is the value of the t trading day, and “Then part (rule conclusion)” is the value of the t + 1 trading day. The five TAIEX stock trading days from 2000/03/17 to 2000/3/23 are used as an example in the FLR in Table 1. For example, in the second row of Table 1, the trading day 8763.27 (t = 2000/03/17) falls into B

_{6}and the next trading day 8536.05 (t + 1 = 2000/03/20) falls into B

_{5}(please refer seven linguistic intervals in step II (1)); then, the FLR can be represented as B

_{6}(t)→B

_{5}(t + 1).

_{i}→B

_{j}, B

_{i}→B

_{k}, B

_{i}→B

_{m}can be grouped as B

_{i}→B

_{j}, B

_{k}, B

_{m}. A fluctuation-type stock index has three trends: upward trend, no change, and downward trend; these three trends are used to express the FLRs group. The range of the stock index is partitioned into seven linguistic terms in this paper. As mentioned, price fluctuation is used to group the FLRs. For example, B

_{1}→B

_{2}is grouped as an “upward” trend, B

_{1}→B

_{1}is “no change”, and B

_{2}→B

_{1}is a “downward” trend.

_{n}(t), as presented in Equation (2):

_{df}and the defuzzified matrix L

_{df}(t), which are defined as Equations (3) and (4), respectively, are applied in this step. The defuzzified value then denotes the initial stock index forecast.

_{i}is the intermediate point of each linguistic interval, L

_{i}.

_{i}based on stock market properties.

_{1}, L

_{2}, … L

_{k}in the range by using Equation (1). Then fuzzify all data into five linguistic values, defined as follows: L

_{1}(oversold), L

_{2}(sold), L

_{3}(stable), L

_{4}(bought), and L

_{5}(overbought).

_{VR}(t)) × P(t) + β × [(F(t) − P(t))] + γ × (|P(t) − N(t)|)

_{VR}(t) represents a linguistic term of the VR(t) indicator, VR(t) denotes a technical indicator of volume, M(L

_{VR}(t)) represents a signal transfer function, and α, β, and γ are the coefficient of three factors for the proposed fitting forecast model. The meanings of parameters α, β, and γ are explained as follows:

- (1)
- α represents the degree of influence of the F(t + 1) forecast from the market signals of trading volume and the actual stock index. Taiwan stock has is a volatility limitation of ±7%, whereas HSI has no such restriction; thus, to obtain accurate factors and better train the forecasting equation, we extend the range of α to between −0.15 and 0.15.
- (2)
- β represents the degree of influence of the F(t + 1) forecast based on the difference between the forecast stock index and the actual stock index. Moreover, given the volatility limitation of TAIEX (±7%) and the lack of a limit for the HSI stock, we plot the daily fluctuation of HSI as shown in Figure 2. From Figure 2, we see that the daily fluctuation is no greater than ±15%. Then, we can set the range of β from −0.15 to 0.15 to search for the optimal β.
- (3)
- γ represents the degree of influence of the F(t + 1) forecast from the daily difference of two stock indexes; the range of γ is [−1, 1], where −1 is an entire negative correlation, and 1 represents completely positive correlation.

_{VR}(t)) has been transferred into the corresponding market signals by Equation (6). Therefore, the three factors have been crisp values, and the M(L

_{VR}(t)) is an indicator signal; then M(L

_{VR}(t)) × P(t) is a linear factor. Hence, the proposed model in Equation (7) is a linear multifactor forecasting model. Next, from Step 1 to Step 4, we employed the collected stock datasets to fit the Equation (7) based on the minimal RMSE to obtain the best parameters for α, β, and γ, and each iteration step is 0.001 for α, β, and γ, the detailed computation could be referred to Algorithm 1.

Algorithm 1: Multifactor FTS model |

Input: double array ${\mathit{P}}_{\mathit{i}}$, ${\mathit{V}}_{\mathit{i}}$, ${\mathit{I}}_{\mathit{i}}$, ${\mathit{p}}_{\mathit{i}}$, ${\mathit{V}}_{\mathit{i}\mathbf{+}\mathit{h}}^{\mathit{t}}$, ${\mathit{I}}_{\mathit{i}\mathbf{+}\mathit{h}}^{\mathit{t}}$, and ${\mathit{p}}_{\mathit{i}\mathbf{+}\mathit{h}}^{\mathit{t}}$begin sum = 0 min RMSE = 999999999 // refer Equation (8) for i←−150 to 150 do for j←−150 to 150 do for k←−1000 to 1000 do for x←0 to length of factor1 do forecast train $=\frac{i}{1000}\times {\mathit{P}}_{\mathit{i}}\left[x\right]+\frac{j}{1000}\times \mathit{f}\mathit{a}\mathit{c}\mathit{t}\mathit{o}\mathit{r}\mathbf{2}\left[x\right]+\frac{k}{1000}\times \mathit{f}\mathit{a}\mathit{c}\mathit{t}\mathit{o}\mathit{r}\mathbf{3}\left[x\right]+\mathit{D}\mathit{a}\mathit{t}\mathit{a}\mathbf{1}\left[x\right]$ // refer Equation (7), and set α = $\frac{i}{1000}$, β = $\frac{j}{1000}$, and γ = $\frac{k}{1000}$ square error $={\left(\mathit{D}\mathit{a}\mathit{t}\mathit{a}\mathbf{1}\left[x+1\right]-forecast\_train\right)}^{2}$ sum = sum + square error end if (min RMSE > RMSE) best(i) = i best(j) = j best(k) = k min RMSE = RMSE endRMSE = 0 square error = 0 end end end Output: best i, best j, best k |

#### 3.2. The Pseudocode of the Proposed Model

${\mathit{P}}_{\mathit{i}}$: | stock index for training in the i-th year; |

${\mathit{V}}_{\mathit{i}}$: | trading volume for training in the i-th year; |

${\mathit{I}}_{\mathit{i}}$: | interaction between two stock markets for training in the i-th year; |

${\mathit{p}}_{\mathit{i}}$: | closing price for training in the i-th year; |

${\mathit{P}}_{\mathit{i}\mathbf{+}\mathit{h}}^{\mathit{t}}$: | next half-year stock index for testing in the i-th year; |

${\mathit{V}}_{\mathit{i}\mathbf{+}\mathit{h}}^{\mathit{t}}$: | next half-year trading volume for testing in the i-th year; |

${\mathit{I}}_{\mathit{i}\mathbf{+}\mathit{h}}^{\mathit{t}}$: | next half-year interaction between two stock markets for testing in the i-th year; |

${\mathit{p}}_{\mathit{i}\mathbf{+}\mathit{h}}^{\mathit{t}}$: | next half-year closing price for testing in the i-th year. |

## 4. Verification and Comparison

## 5. Findings and Discussion

- (1)
- From the literature review, the selected attributes (trading volume, stock index, and interaction between two stock markets) have been proved to have an impact on the forecast of the stock market, and the results have a minimal RMSE, which will lead to a higher profits for investors.
- (2)
- Table 4 and Table 5 indicate that the TAIEX is less volatile than the HSI. This is because Taiwan limits the volatility of shares to ±7%, whereas Hong Kong has no limit. From Figure 2, the daily fluctuation of HSI can help us to set the search range for quickly obtaining the optimal parameters for α and β.
- (3)
- (4)
- From Table 2 and Table 3, the maximal parameter is β = 0.057 for TAIEX (1999/07~2000/06) and β = 0.065 for HSI (2000/01~2000/12). During this period (1999/07~2000/12), we searched “2000 crisis”. The results pertaining to the Dot-com bubble (2000–2002) and the year 2000 issues carry tremendous risks of disruption in the operations of financial institutions and in financial markets. Hence, we think that the “2000 crisis” has influenced the fluctuations of the stock market.
- (5)
- The comparative results (Table 4 and Table 5) and statistical test (Table 6) show that the proposed model outperforms other models in forecast accuracy (less RMSE) because the proposed linear multifactor forecasting equation with three optimized parameters (α, β, and γ) produces an optimal prediction to match past stock index patterns and generates a more accurate forecast.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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TAIEX’s Rule for Consecutive Trading Day | FLR |
---|---|

8763.27 (t = 2000/03/17)→8536.05 (t + 1 = 2000/03/20) | B_{6} (t)→B_{5} (t + 1) |

8536.05 (t = 2000/03/20)→9004.48 (t + 1 = 2000/03/21) | B_{5} (t)→B_{6} (t + 1) |

9004.48 (t = 2000/03/21)→9069.39 (t + 1 = 2000/03/22) | B_{6} (t)→B_{6} (t + 1) |

9069.39 (t = 2000/03/22)→9533.87 (t + 1 = 2000/03/23) | B_{6} (t)→B_{7} (t + 1) |

Training | Testing | α | β | γ | RMSE |
---|---|---|---|---|---|

1997/01~1997/12 | 1998/01~1998/06 | −0.004 | 0.002 | 0.002 | 112 |

1997/07~1998/06 | 1998/07~1998/12 | −0.006 | 0.005 | 0.002 | 104 |

1998/01~1998/12 | 1999/01~1999/06 | −0.008 | −0.001 | 0.002 | 97 |

1998/07~1999/06 | 1999/07~1999/12 | −0.008 | 0.001 | 0.003 | 122 |

1999/01~1999/12 | 2000/01~2000/06 | −0.009 | 0.049 | 0.003 | 168 |

1999/07~2000/06 | 2000/07~2000/12 | 0.0 | 0.057 | −0.001 | 147 |

2000/01~2000/12 | 2001/01~2001/06 | −0.006 | −0.003 | −0.002 | 94 |

2000/07~2001/06 | 2001/07~2001/12 | −0.01 | 0.016 | −0.003 | 91 |

2001/01~2001/12 | 2002/01~2002/06 | −0.013 | −0.003 | 0.001 | 98 |

2001/07~2002/06 | 2002/07~2002/12 | −0.009 | 0.007 | 0.001 | 82 |

2002/01~2002/12 | 2003/01~2003/06 | −0.005 | 0.024 | 0.0 | 71 |

2002/07~2003/06 | 2003/07~2003/12 | −0.008 | −0.002 | 0.002 | 61 |

2003/01~2003/12 | 2004/01~2004/06 | −0.005 | −0.01 | 0.003 | 112 |

2003/07~2004/06 | 2004/07~2004/12 | −0.006 | −0.004 | 0.002 | 59 |

Training | Testing | α | β | γ | RMSE |
---|---|---|---|---|---|

1997/01~1997/12 | 1998/01~1998/06 | −0.013 | −0.021 | −0.001 | 226 |

1997/07~1998/06 | 1998/07~1998/12 | 0.0 | −0.07 | −0.003 | 228 |

1998/01~1998/12 | 1999/01~1999/06 | −0.019 | 0.024 | −0.001 | 192 |

1998/07~1999/06 | 1999/07~1999/12 | 0.0 | −0.014 | 0.002 | 208 |

1999/01~1999/12 | 2000/01~2000/06 | −0.013 | 0.017 | 0.0 | 329 |

1999/07~2000/06 | 2000/07~2000/12 | −0.014 | 0.041 | −0.001 | 238 |

2000/01~2000/12 | 2001/01~2001/06 | −0.016 | 0.065 | −0.001 | 217 |

2000/07~2001/06 | 2001/07~2001/12 | 0.0 | −0.015 | −0.001 | 199 |

2001/01~2001/12 | 2002/01~2002/06 | −0.01 | 0.013 | 0.0 | 116 |

2001/07~2002/06 | 2002/07~2002/12 | −0.009 | 0.012 | 0.0 | 119 |

2002/01~2002/12 | 2003/01~2003/06 | −0.008 | −0.003 | 0.0 | 88 |

2002/07~2003/06 | 2003/07~2003/12 | −0.009 | 0.017 | 0.001 | 115 |

2003/01~2003/12 | 2004/01~2004/06 | 0.0 | 0.006 | 0.001 | 153 |

2003/07~2004/06 | 2004/07~2004/12 | −0.01 | −0.017 | 0.0 | 106 |

Testing | RMSE | |||||
---|---|---|---|---|---|---|

[4] | [9] | [7] | SVR | GRNN | Proposed | |

1998/01~1998/06 | 209 | 139 | 207 | 275 | 1208 | 112 ^{a} |

1998/07~1998/12 | 339 | 160 | 361 | 393 | 1964 | 104 ^{a} |

1999/01~1999/06 | 324 | 211 | 352 | 897 | 2381 | 97 ^{a} |

1999/07~1999/12 | 195 | 162 | 205 | 919 | 1624 | 122 ^{a} |

2000/01~2000/06 | 404 | 231 | 496 | 469 | 2508 | 168 ^{a} |

2000/07~2000/12 | 319 | 293 | 563 | 742 | 4159 | 147 ^{a} |

2001/01~2001/06 | 245 | 418 | 368 | 468 | 2813 | 94 ^{a} |

2001/07~2001/12 | 368 | 823 | 536 | 441 | 2032 | 91 ^{a} |

2002/01~2002/06 | 215 | 264 | 186 | 657 | 1797 | 98 ^{a} |

2002/07~2002/12 | 155 | 237 | 157 | 672 | 2077 | 82 ^{a} |

2003/01~2003/06 | 160 | 150 | 157 | 159 | 1582 | 71 ^{a} |

2003/07~2003/12 | 150 | 459 | 246 | 733 | 1473 | 61 ^{a} |

2004/01~2004/06 | 188 | 534 | 314 | 360 | 1924 | 112 ^{a} |

2004/07~2004/12 | 106 | 166 | 96 | 392 | 1357 | 59 ^{a} |

Average | 241 | 303 | 303 | 541 | 2064 | 101^{a} |

^{a}The best performance among 6 models, SVR: support vector regression, and GRNN: general regression neural network.

Testing | RMSE | |||||
---|---|---|---|---|---|---|

[4] | [9] | [7] | SVR | GRNN | Proposed | |

1998/01~1998/06 | 620 | 491 | 1506 | 591 | 10192 | 226 ^{a} |

1998/07~1998/12 | 434 | 294 | 776 | 251 | 3844 | 228 ^{a} |

1999/01~1999/06 | 415 | 327 | 1197 | 758 | 6546 | 192 ^{a} |

1999/07~1999/12 | 728 | 357 | 856 | 729 | 8309 | 208 ^{a} |

2000/01~2000/06 | 678 | 460 | 924 | 353 | 5975 | 329 ^{a} |

2000/07~2000/12 | 372 | 304 | 503 | 726 | 3703 | 238 ^{a} |

2001/01~2001/06 | 589 | 466 | 662 | 504 | 4422 | 217 ^{a} |

2001/07~2001/12 | 626 | 415 | 1099 | 994 | 9973 | 199 ^{a} |

2002/01~2002/06 | 318 | 172 | 402 | 230 | 2761 | 116 ^{a} |

2002/07~2002/12 | 232 | 192 | 225 | 496 | 3380 | 119 ^{a} |

2003/01~2003/06 | 261 | 157 | 284 | 433 | 2091 | 88 ^{a} |

2003/07~2003/12 | 325 | 256 | 329 | 509 | 7329 | 115 ^{a} |

2004/01~2004/06 | 376 | 212 | 596 | 393 | 7913 | 153 ^{a} |

2004/07~2004/12 | 313 | 159 | 262 | 119 | 2572 | 106 ^{a} |

Average | 449 | 304 | 687 | 506 | 5644 | 181^{a} |

TAIEX | [4] | [9] | [7] | SVR | GRNN |

Proposed | + * | + * | + * | + * | + * |

GRNN | − * | − * | − * | − * | |

SVR | − * | − * | − * | ||

[7] | − * | + | |||

[9] | − * | ||||

HSI | [4] | [9] | [7] | SVR | GRNN |

Proposed | + * | + * | + * | + * | + * |

GRNN | − * | − * | − * | − * | |

SVR | − | − * | + | ||

[7] | − * | − * | |||

[9] | + |

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## Share and Cite

**MDPI and ACS Style**

Tsai, M.-C.; Cheng, C.-H.; Tsai, M.-I.
A Multifactor Fuzzy Time-Series Fitting Model for Forecasting the Stock Index. *Symmetry* **2019**, *11*, 1474.
https://doi.org/10.3390/sym11121474

**AMA Style**

Tsai M-C, Cheng C-H, Tsai M-I.
A Multifactor Fuzzy Time-Series Fitting Model for Forecasting the Stock Index. *Symmetry*. 2019; 11(12):1474.
https://doi.org/10.3390/sym11121474

**Chicago/Turabian Style**

Tsai, Ming-Chi, Ching-Hsue Cheng, and Meei-Ing Tsai.
2019. "A Multifactor Fuzzy Time-Series Fitting Model for Forecasting the Stock Index" *Symmetry* 11, no. 12: 1474.
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