# Type 2 Fuzzy Inference-Based Time Series Model

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## Abstract

**:**

_{2}FTS) forecasting model. The T

_{2}FTS model was used to exploit more information in time series forecasting. The concepts of sliding window method (SWM) and fuzzy rule-based systems (FRBS) were incorporated in the utilization of T

_{2}FTS to obtain forecasting values. A sliding window method was proposed to find a proper and systematic measurement for predicting the number of class intervals. Furthermore, the weighted subsethood-based algorithm was applied in developing fuzzy IF–THEN rules, where it was later used to perform forecasting. This approach provides inferences based on how people think and make judgments. In this research, the data sets from previous studies of crude palm oil prices were used to further analyze and validate the proposed model. With suitable class intervals and fuzzy rules generated, the forecasting values obtained were more precise and closer to the actual values. The findings of this paper proved that the proposed forecasting method could be used as an alternative for improved forecasting of sustainable crude palm oil prices.

## 1. Introduction

_{2}FTS) model, was suggested to get the benefit of the related element and solve the forecasting problem indirectly.

_{2}FTS) forecasting that is systematic and flexible, together with a reasoning-based model, which is the sliding window method and weighted subsethood-based algorithm, was applied to address this uncertainty. Moreover, to improve the forecasting value, this research extended the observation using T

_{2}FTS. By utilizing extra observations in the proposed forecasting method, it was hypothesized that the forecasting results would be improved.

_{2}FTS model to forecast accurate future data values with minimum forecasting error. Specifically, this research suggests a new approach of sliding window method in determining the number of the class intervals of the universe of discourse of FTS. Secondly, this research develops a fuzzy rule-based system using weighted subsethood-based algorithm (WSBA) in FTS forecasting. Third, this paper exploited more variables of observations in forecasting using a new T

_{2}FTS model. All the three objectives were utilized to refine the optimum numbers of intervals and created fuzzy ruled based relationships. Thereby, forecasting performance could be improved. The detailed explanation is given in Section 2.

## 2. Methodology

#### 2.1. Collection and Selection of Data

#### 2.2. Proposed Forecasting Model

_{2}FTS) model that were implemented. The further illustrations are as follows.

#### 2.3. Evaluation of the Performance

_{t}is the actual price, defuzzification(t) is the defuzzified forecast and there were n forecasts as shown in Equation (7).

## 3. Empirical Analysis

## 4. Algorithm of the Proposed Method

_{2}FTS) model, as follows.

- Step 1:
- The class interval of the universe of discourse is determined by using the sliding window method.
- Step 2:
- The observations are fuzzified into corresponding fuzzy sets.
- Step 3:
- Fuzzy logical relationship groups (FLRGs) are obtained.
- Step 4:
- Out-of-sample observations are mapped to FLRGs.
- Step 5:
- Operators and fuzzy rules obtained by using weighted subsethood-based algorithm are applied to the FLRGs for all the observations and obtain forecasts.
- Step 6:
- The forecasts are defuzzified.
- Step 7:
- Forecast values are computed for all data individually.
- Step 8:
- The method is compared with the previous method.

## 5. Results and Discussion

## 6. Conclusions

_{2}FTS) models to exploit an extra observation. To increase the level of efficiency of this method, the sliding window method and the weighted subsethood-based algorithm were implemented in this model. The forecast values obtained from the use of the proposed method were then compared to Chen’s model. The forecast error was tested through the use of root mean squared error (RMSE) for both methods. The outcome of the RMSEs using the proposed method is less than that for Chen’s model. This demonstrates that the proposed method is capable of giving a superior forecast compared to Chen’s model. Hence, the employment of the proposed method will lead to the creation of an efficient approach in forecasting application which will support decisions made by alternative methods indirectly. This research proposes an extension of the current research in achieving a universal view of suitable combination of factors as well as the classification of the class interval. Thus, this method could enhance the capability of the proposed type 2 fuzzy time series (T

_{2}FTS) models. The use of crude palm oil is dependent upon its price. Therefore, the price of crude palm oil determines its usage for plantation activities. For instance, the price of crude palm oil influences its use in mills as well as feedstock for biodiesel. The price of crude palm oil also determines other plantation activities including the preparation of plantation land. Failure to forecast crude palm oil prices may cause plantations to use fire as a low-cost solution. The resulting environmental impacts include deforestation, biodiversity loss, water and air pollution, such as haze, and emission of greenhouse gases. The price of crude palm oil has social impacts such as land use rights; smallholders including livelihoods, income, and wellbeing; forced and child labor, and terms and conditions of labor including wages and health and safety. Thus, this research offers a sustainable palm oil price forecasting model which helps the government and palm oil industries in making business decisions and to understand strategies of major players in the industry.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Performance comparison: (

**a**) Year 2012; (

**b**) Year 2013; (

**c**) Year 2014; (

**d**) Year 2015; (

**e**) Year 2016.

Date (yyyy/mm/dd) | CPO Price | Fuzzy Sets |
---|---|---|

2012/10/1 | 424.20 | A_{7} |

2012/10/2 | 424.20 | A_{7} |

2012/10/3 | 407.40 | A_{3} |

2012/10/4 | 395.20 | A_{1} |

2012/10/5 | 411.00 | A_{4} |

2012/10/6 | 412.50 | A_{4} |

2012/10/7 | 412.50 | A_{4} |

2012/10/8 | 406.80 | A_{3} |

2012/10/9 | 418.50 | A_{5} |

2012/10/10 | 418.00 | A_{5} |

2012/10/11 | 430.00 | A_{8} |

2012/10/12 | 423.00 | A_{6} |

2012/10/13 | 415.50 | A_{5} |

2012/10/14 | 415.50 | A_{5} |

2012/10/15 | 417.40 | A_{5} |

2012/10/16 | 417.00 | A_{5} |

2012/10/17 | 414.90 | A_{5} |

2012/10/18 | 419.60 | A_{6} |

2012/10/19 | 425.80 | A_{7} |

2012/10/20 | 424.00 | A_{6} |

2012/10/21 | 424.00 | A_{6} |

2012/10/22 | 435.10 | A_{9} |

2012/10/23 | 424.70 | A_{7} |

2012/10/24 | 424.70 | A_{7} |

2012/10/25 | 435.20 | A_{9} |

2012/10/26 | 435.20 | A_{9} |

2012/10/27 | 434.30 | A_{9} |

2012/10/28 | 434.30 | A_{9} |

2012/10/29 | 428.60 | A_{7} |

2012/10/30 | 426.00 | A_{7} |

2012/10/31 | 424.70 | A_{7} |

${A}_{7}\to {A}_{7},{A}_{7}\to {A}_{3},{A}_{3}\to {A}_{1},{A}_{1}\to {A}_{4}$ |

${A}_{4}\to {A}_{4},{A}_{4}\to {A}_{3},{A}_{3}\to {A}_{5},{A}_{5}\to {A}_{5}$ |

${A}_{5}\to {A}_{8},{A}_{8}\to {A}_{6},{A}_{6}\to {A}_{5},{A}_{5}\to {A}_{6}$ |

${A}_{6}\to {A}_{7},{A}_{7}\to {A}_{6},{A}_{6}\to {A}_{6},{A}_{6}\to {A}_{9}$ |

${A}_{9}\to {A}_{7},{A}_{7}\to {A}_{9},{A}_{9}\to {A}_{9}$ |

${A}_{1}\to {A}_{4}$ |

${A}_{3}\to {A}_{1},{A}_{5}$ |

${A}_{4}\to {A}_{4},{A}_{3}$ |

${A}_{5}\to {A}_{5},{A}_{8},{A}_{6}$ |

${A}_{6}\to {A}_{5},{A}_{7},{A}_{6},{A}_{9}$ |

${A}_{7}\to {A}_{7},{A}_{3},{A}_{6},{A}_{9}$ |

${A}_{8}\to {A}_{6}$ |

${A}_{9}\to {A}_{7},{A}_{9}$ |

Date (mm/dd) | Closing | High | Low |
---|---|---|---|

… | … | … | … |

10/11 | 430.00 $\left({A}_{8}\right)$ | 431.10 $\left({A}_{8}\right)$ | 419.20 $\left({A}_{6}\right)$ |

10/12 | 423.00 $\left({A}_{6}\right)$ | 428.80 $\left({A}_{7}\right)$ | 412.80 $\left({A}_{4}\right)$ |

10/13 | 415.50 $\left({A}_{5}\right)$ | 420.30 $\left({A}_{6}\right)$ | 413.80 $\left({A}_{4}\right)$ |

… | … | … | … |

Date (mm/dd) | Forecasts | $\mathbf{Forecasts}\text{}\mathbf{After}\text{}{\wedge}_{\mathit{m}}$ | |
---|---|---|---|

Closing | ${A}_{8}\to {A}_{6}$ | ||

10/12 | High | ${A}_{8}\to {A}_{6}$ | ${A}_{6}$ |

Low | ${A}_{6}\to {A}_{5},{A}_{6},{A}_{7},{A}_{9}$ | ||

Closing | ${A}_{6}\to {A}_{5},{A}_{6},{A}_{7},{A}_{9}$ | ||

10/13 | High | ${A}_{7}\to {A}_{3},{A}_{6},{A}_{7},{A}_{9}$ | ${A}_{6}$ |

Low | ${A}_{4}\to {A}_{3},{A}_{4}$ | ||

Closing | ${A}_{5}\to {A}_{5},{A}_{6},{A}_{8}$ | ||

10/14 | High | ${A}_{6}\to {A}_{5},{A}_{6},{A}_{7},{A}_{9}$ | ${A}_{5}$ |

Low | ${A}_{4}\to {A}_{3},{A}_{4}$ |

Date (mm/dd) | Forecasts | $\mathbf{Forecasts}\text{}\mathbf{After}\text{}{\vee}_{\mathit{m}}$ | |
---|---|---|---|

Closing | ${A}_{8}\to {A}_{6}$ | ||

10/12 | High | ${A}_{8}\to {A}_{6}$ | ${A}_{5},{A}_{6},{A}_{7},{A}_{9}$ |

Low | ${A}_{6}\to {A}_{5},{A}_{6},{A}_{7},{A}_{9}$ | ||

Closing | ${A}_{6}\to {A}_{5},{A}_{6},{A}_{7},{A}_{9}$ | ||

10/13 | High | ${A}_{7}\to {A}_{3},{A}_{6},{A}_{7},{A}_{9}$ | ${A}_{3},{A}_{4},{A}_{5},{A}_{6},{A}_{7},{A}_{9}$ |

Low | ${A}_{4}\to {A}_{3},{A}_{4}$ | ||

Closing | ${A}_{5}\to {A}_{5},{A}_{6},{A}_{8}$ | ||

10/14 | High | ${A}_{6}\to {A}_{5},{A}_{6},{A}_{7},{A}_{9}$ | ${A}_{3},{A}_{4},{A}_{5},{A}_{6},{A}_{7},{A}_{8},{A}_{9}$ |

Low | ${A}_{4}\to {A}_{3},{A}_{4}$ |

Year | Mean Square Error (MSE) | Root Mean Square Error (RMSE) | ||
---|---|---|---|---|

Proposed Method | Chen’s model | Proposed Method | Chen’s model | |

2012 | 0.0010 | 0.0010 | 0.518 | 0.520 |

2013 | 0.00067 | 0.0011 | 0.457 | 0.635 |

2014 | 0.0019 | 0.0027 | 0.637 | 0.855 |

2015 | 0.00069 | 0.00079 | 0.414 | 0.462 |

2016 | 0.0009 | 0.0015 | 0.567 | 0.754 |

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## Share and Cite

**MDPI and ACS Style**

Rahim, N.F.; Othman, M.; Sokkalingam, R.; Abdul Kadir, E.
Type 2 Fuzzy Inference-Based Time Series Model. *Symmetry* **2019**, *11*, 1340.
https://doi.org/10.3390/sym11111340

**AMA Style**

Rahim NF, Othman M, Sokkalingam R, Abdul Kadir E.
Type 2 Fuzzy Inference-Based Time Series Model. *Symmetry*. 2019; 11(11):1340.
https://doi.org/10.3390/sym11111340

**Chicago/Turabian Style**

Rahim, Nur Fazliana, Mahmod Othman, Rajalingam Sokkalingam, and Evizal Abdul Kadir.
2019. "Type 2 Fuzzy Inference-Based Time Series Model" *Symmetry* 11, no. 11: 1340.
https://doi.org/10.3390/sym11111340