**Fuzzy Time Series Cheng Model to Forecast the Price of Crude Oil in Malaysia**
Received Date: \*date
Accepted Date: \*date
Published Date: \*date
**HIGHLIGHTS**
- Fuzzy time series Cheng Model is suitable for forecasting the price of crude oil price in Malaysia due to the inconsistency pattern in historical price data and the method produces lower MAPE and RMSE.
- The numbers intervals used in the model which represents the linguistic variables for price of Crude oil, such higher price until lowest price, have an influence on the output forecast.
- The study finds that the use of eight intervals in the model produces more accurate forecast of price of crude oil in Malaysia.
- The finding of the study may assist the related sectors such as governments, investors and others to do better management planning for improving the economy.
**ABSTRACT**
*Crude oil is one of the important commodities to Malaysia. As a producer and exporter of oil and gas, Malaysia has gained high Gross Revenue from this sector. Crude oil is the global commodity and highly demanded. Therefore, major price changes on the commodity have a significant influence on world economy. Market sentiment, demand, and supply are some elements directly influencing the oil prices. Since crude oil is the backbone of businesses and is extremely important to the economy, it is essential to study the price of crude oil for future planning purposes. For that reason, this study proposes the use of the Fuzzy Time Series Cheng to predict crude oil price in Malaysia. In this study, Root Mean Square Error (RMSE) and Mean Absolute Percentage Error (MAPE) are used to evaluate the forecast performance. The result shows that Fuzzy Time Series Cheng is able to produce a good result in forecasting since the analyses shows that the low value of RMSE and MAPE (less than 10 percent). Although this is the fundamental study but the finding may assist many sectors in Malaysia, such as governments, enterprises, investors, and businesses to produce a better economic planning in the future especially after the pandemic covid-19 phase.*
***Keywords:** Crude oil price, Fuzzy Time Series, Fuzzy Time Series Cheng,* *RMSE, MAPE*
# INTRODUCTION
# Malaysia was the second largest oil and natural gas producer in Southeast Asia, and the world's fifth largest exporter of oil and liquefied natural gas in 2019 (U.S. Energy Information Administration, 2021). It produced about 1.7 million barrels of oil per day on average in 2018 and 596,000 barrels of oil per day in 2020. Oil and gas industries have contributed higher percentage in Malaysia Gross Revenue (Zakaria & Shamsuddin, 2017). As an oil and gas exporter, Malaysia gain more gross revenues from higher world oil price and vice versa (Jalil et al., 2009).
# The term "Oil Price" refers to the current spot price for a barrel of benchmark crude oil. The price of oil is determined by the grade, locality, and sulphur content. Besides, the price of oil may be affected by the balance between its demand and supply. Oil storage trade is a technique in which major oil corporations acquire oil at low prices for immediate storage and delivery. These huge oil firms then hold the oil in store till the price of oil rises. Because oil is a global commodity in high demand, major price changes have the potential to have a significant influence on the global economy. Market sentiment, demand, and supply are the three primary elements directly influencing oil prices. When supply falls, demand rises, and the price of oil rises, and vice versa. The supply of oil is determined by taxes, the legal framework, geological discoveries, the political status of oil-producing firms, and the cost of extracting the oil. Global macroeconomic circumstances influence oil demand. High oil prices contribute to greater inflation, which harms the economy of countries that import oil. Conversely, low oil prices may cause economic collapse and political instability in oil-producing countries by disrupting economic growth.
Another factor affecting the fluctuation of crude oil price is a natural disaster, such as COVID\_19. Since 2020, the disease has spread globally, resulting in a pandemic. This disease has a significant influence on the economy of the entire world, including Malaysia. Demand shocks in the oil market have been caused by the Movement Control Act (MCO) and international travel control. As compared to 2020, the global oil demand fell to 19.9 million barrels per day in April 2020 (IEA, 2020). However, global oil consumption is gradually rebound beginning in the second quarter of 2020. As the majority of the economy's operations had paused during the pandemic, demand for oil decreased while supply surged. As a result, BRENT oil prices have fallen from $50 USD to $20 USD since mid-March, while WTI oil prices have fallen to negative $37.63 USD for the first time in history on 20 April 2020. This makes the oil investors tremble and loss of confidence (Mensi et al., 2020). Besides, Malaysiaβs economic growth decreased severely by 3.1% in 2020 due to the impact of the policy responses implemented to prevent the spread of COVID-19 (World Bank, 2021).
**Predicting the price of crude oil is essential in providing information for policy makers to the government and investors. Many researchers have put some effort into developing various mathematical forecasting models to predict oil price with minimal error such as Variational Mode Decomposition (VMD) (Huang and Deng, 2021; Li,2019) and New Text-Based and Big Driven Models (Wu, 2021), Autoregressive Model Moving Average (ARIMA) (Jain and Gupta, 2018) and many more.**
**Although there are many forecasting models, Fuzzy Time Series (FTS) is one of the suitable methods for forecasting. This method is capable of dealing with fluctuation data, imprecise environment, uncertainty and subjectivity in the data, as compared to the classical statistics (Song and Chissom, 1993). FTS has been implemented in many studies such as in students enrolment prediction (Song and Chissom,1993), oil price prediction (Zhang et al., 2010), stock market prediction (Cai, et al.,2013) and many more. FTS model has been extended for more accurate prediction and some of the models are Chen Method, Cheng Method, Marcov Chain Method and others. In this study, FTS Cheng Method will be implemented to predict Malaysian crude oil price. The model performance will be tested using Root Mean Square Error (RMSE) and Mean Average Percentage Error (MAPE).**
#
# METHODOLOGY
**Data Collection Method**
The monthly data of crude oil price in Malaysia from September 2011 to September 2021 as stated in Table 1 were collected from the IndexMundi Website Page. The maximum price was RM359.59 per barrel on February 2012 and the minimum price is RM91.64 per barrel on April 2020.
**Table 1**: Price of Crude Oil in Malaysia
| Month | Price of crude oil per barel (RM) |
| ------ | --------------------------------- |
| Sep-11 | 311.68 |
| Oct-11 | 313.55 |
| Nov-11 | 331.98 |
| Dec-11 | 329.66 |
| Jan-12 | 333.18 |
| Feb-12 | 340.88 |
| β¦ | β¦ |
| Apr-21 | 259.61 |
| May-21 | 273.99 |
| Jun-21 | 296.88 |
| Jul-21 | 307.68 |
| Aug-21 | 290.7 |
| Sep-21 | 303.58 |
| Aug-21 | 290.7 |
| Sep-21 | 303.58 |
**Fuzzy Time Series Cheng**
Fuzzy time series (FTS) was first introduced by Song and Chissom by using fuzzy relations equations and approximate reasoning to predict the number of student enrolments (1993). Then the models was further developed by Chen by utilizing arithmetic operations to solve the same problems. Cheng's algorithm is a form of improvement from the lack of model Chen. The Chen model does not emphasize repetition and does not have a smaller weight on more extended observations. One of Chen's model improvements was the Cheng model (2008).
**The definition of FTS of Song and Chissom (1993) are still relevant and they are as the following:**
**Definition 1: Let with , a subset of be the universe of discourse in which fuzzy set are defined. If consists of with then is called a Fuzzy time series defined on .**
**Definition 2: If and , the relationship between 2 consecutive observation, and written as is called Fuzzy Logical Relationship (FLR) and is the known as Left Hand Side (LHS) and is the Right Hand Side (RHS).**
**Definition 3: All FLRs of the same LHS are grouped together into a Fuzzy Logical Relationship Group (FLRG). For example, the two FLS and are grouped as .**
***Implementation Fuzzy Time Series Cheng to Forecasting Crude Oil Price in Malaysia***
In this study, Fuzzy time series Cheng method is used to predict the price of crude oil prices in Malaysia, and the detail algorithms are shown below:
**Step 1**: Define the universe of discourse and establish fuzzy intervals.
The universe is defined as
(1)
where and represent the minimum and maximum actual oil prices. Both *D1* and *D2* are positive numbers. *D1 *and *D2* are any positive numbers chosen to divide the interval evenly.
The number of linguistic intervals can be calculated by using this Sturges formula:
(2)
where is the number of interval and is the number of historical data of crude oil price.
The calculated number of the intervals is 8 and the intervals are denoted as π΄1, π΄2, π΄3, π΄4, π΄5, π΄6, π΄7 and π΄8 as in Table 2.
**Table 2**: Linguistic Description
| Fuzzy Value | Linguistic Variable |
| ----------- | ------------------- |
| π΄1 | Lowest price |
| π΄2 | Very low price |
| π΄3 | Little low price |
| π΄4 | Regular price |
| π΄5 | Little high price |
| π΄6 | Moderate high price |
| π΄7 | Very high price |
| π΄8 | Highest price |
The length of linguistic intervals, *l* can be calculated using the formula
(3)
> **Step 2**: Fuzzify the crude oil prices into linguistic values and define the fuzzy set based on the universe of discourse and intervals.
>
> By referring to the value of *l,* the result for grouping up the intervals is shown in Table 3.
**Table 3**: Fuzzy Sets Intervals and Midpoints
| Fuzzification | Interval | Midpoint, | | |
| ------------- | -------- | --------- | ------ | ------- |
| π΄1 | 90 | \- | 123.75 | 106.875 |
| π΄2 | 123.75 | \- | 157.50 | 140.625 |
| π΄3 | 157.50 | \- | 191.25 | 174.375 |
| π΄4 | 191.25 | \- | 225.00 | 208.125 |
| π΄5 | 225.00 | \- | 258.75 | 241.875 |
| π΄6 | 258.75 | \- | 292.50 | 275.625 |
| π΄7 | 292.50 | \- | 326.25 | 309.375 |
| π΄8 | 326.25 | \- | 360.00 | 343.125 |
Fuzzy sets, on the universe of discourse is defined according to (4), where represents the membership of in the fuzzy set . In fuzzy sets theory also has . In this study, the fuzzy sets \(A_{1}\text{\ until\ }A_{n}\) are defined as represented in Equation (4).
(4)
Then find each crude oil price's membership belongs to . If the maximum membership of one crude oil price is in , then we fuzzy this price as .
**Step 3**: Create fuzzy logical relationship (FLR). If there are two consecutives and , then they are written as fuzzy logic relationship, .
**Table** 4: Fuzzification and Fuzzy Logical Relationship (FLR)
| Month/year | Price | Fuzzification | | FLR | |
| ---------- | ------ | ------------- | --- | --- | -- |
| Sep-11 | 311.68 | A7 | N/A | | A7 |
| Oct-11 | 313.55 | A7 | A7 | | A7 |
| Nov-11 | 331.98 | A8 | A7 | | A8 |
| Dec-11 | 329.66 | A8 | A8 | | A8 |
| Jan-12 | 333.18 | A8 | A8 | | A8 |
| Feb-12 | 340.88 | A8 | A8 | | A8 |
| β¦ | β¦ | β¦ | β¦ | β¦ | β¦ |
| Apr-21 | 259.61 | A6 | A6 | | A6 |
| May-21 | 273.99 | A6 | A6 | | A6 |
| Jun-21 | 296.88 | A7 | A6 | | A7 |
| Jul-21 | 307.68 | A7 | A7 | | A7 |
| Aug-21 | 290.7 | A6 | A7 | | A6 |
| Sep-21 | 303.58 | A7 | A6 | | A7 |
| Aug-21 | 290.7 | A6 | A7 | | A6 |
| Sep-21 | 303.58 | A7 | A6 | | A7 |
**Step 4**: Create Fuzzy Logical Relationship Group (FLRG) based on the FLRβs Left-Hand Side(LHS).
For example, if there are 4 FLRs with the same LHS, let say
we can group them as FLRG .
To easily view the relationship of each LHS to RHS, the frequency of FLGS in Matrix form is created as in in Table 5.
> **Table 5**: Fuzzy Logical Relationship (FLR) Frequency Matrix
| | A1 | A2 | A3 | A4 | A5 | A6 | A7 | A8 | Total |
| -- | -- | -- | -- | -- | -- | -- | -- | -- | ----- |
| A1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| A2 | 1 | 3 | 2 | 0 | 0 | 0 | 0 | 0 | 6 |
| A3 | 0 | 1 | 8 | 5 | 0 | 0 | 0 | 0 | 14 |
| | | | | | | | | | |
| A4 | 0 | 1 | 4 | 9 | 4 | 0 | 0 | 0 | 18 |
| A5 | 0 | 0 | 0 | 4 | 15 | 5 | 0 | 0 | 24 |
| A6 | 0 | 0 | 0 | 0 | 5 | 6 | 5 | 0 | 16 |
| A7 | 0 | 0 | 0 | 0 | 0 | 5 | 10 | 4 | 19 |
| A8 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 18 | 22 |
**Step 5**: Find the normalized weighting matrix by
(5)
where are the fuzzy relationshipβs occurrences.
Using the Equation (5), the normalized weighting matrix in Table 6 is obtained. **able 6**: Weighting Matrix
| | A1 | A2 | A3 | A4 | A5 | A6 | A7 | A8 | Total |
| -- | --- | ---- | ---- | ---- | ----- | ---- | ----- | ----- | ----- |
| A1 | 0 | 1/1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| A2 | 1/6 | 3/6 | 2/6 | 0 | 0 | 0 | 0 | 0 | 6 |
| A3 | 0 | 1/14 | 8/14 | 5/14 | 0 | 0 | 0 | 0 | 14 |
| A4 | 0 | 1/18 | 4/18 | 9/18 | 4/18 | 0 | 0 | 0 | 18 |
| A5 | 0 | 0 | 0 | 4/24 | 15/24 | 5/24 | 0 | 0 | 24 |
| A6 | 0 | 0 | 0 | 0 | 5/16 | 6/16 | 5/16 | 0 | 16 |
| A7 | 0 | 0 | 0 | 0 | 0 | 5/19 | 10/19 | 4/19 | 19 |
| A8 | 0 | 0 | 0 | 0 | 0 | 0 | 4/22 | 18/22 | 22 |
**Step 6**: Calculate the forecast price of crude oil by using the formula
> (6)
where
is the midpoint of each interval (Refer to Table 1) and it is called as the defuzzified matrix .
For example
The calculated forecasted values of the crude oil price are shown in Table 7.
> **Table 7**: The Defuzzified Matrix for Each Interval
| Variables | Bdf(t) | | WT(t) | |
| --------- | ----------------- | | ---------------- | ------------ |
| A1 | 0 | | 0 | 0 |
| | 106.875 | | 0 | 0 |
| | 140.625 | | 1 | 140.625 |
| | | | | **140.625** |
| A2 | 106.875 | | 0.166667 | 17.8125 |
| | 140.625 | | 0.5 | 70.3125 |
| | 174.375 | | 0.333333 | 58.125 |
| | | | | **146.25** |
| **β¦** | β¦ | | β¦ | β¦ |
| A7 | 275.625 | | 0.263158 | 72.53289 |
| | 309.375 | | 0.526316 | 162.8289 |
| | 343.125 | | 0.210526 | 72.23684 |
| | | | | **307.5987** |
| A8 | 309.375 | | 0.181818 | 56.25 |
| | 343.125 | | 0.818182 | 280.7386 |
| | 0 | | 0 | 0 |
| | | | | **336.9886** |
**Step 7**: Use an adaptive forecasting model to control our forecast outcome because of the crude oil market's clustering phenomenon. One period error is used to update the forecast. Otherwise, a cumulative variance will produce a poor forecast result. If the current fuzzy price is *P(t),* the following period's price is calculated as
> (8)
where denotes forecasting errors of period .
The defuzzification of forecast price of crude oil are listed in Table 8.
**Table 8: The Forecast Price of Crude Oil and Forecast Error**
| Sep-11 |
311.68 |
0 |
0 |
| Oct-11 |
313.55 |
307.5986842 |
-5.951315789 |
| Nov-11 |
331.98 |
336.9886364 |
5.008636364 |
| Dec-11 |
329.66 |
336.9886364 |
7.328636364 |
| Jan-12 |
333.18 |
336.9886364 |
3.808636364 |
| Feb-12 |
340.88 |
336.9886364 |
-3.891363636 |
| β¦ |
β¦ |
β¦ |
β¦ |
| Apr-21 |
259.61 |
275.625 |
16.015 |
| May-21 |
273.99 |
275.625 |
1.635 |
| Jun-21 |
296.88 |
307.5986842 |
10.71868421 |
| Jul-21 |
307.68 |
307.5986842 |
-0.081315789 |
| Aug-21 |
290.7 |
275.625 |
-15.075 |
| Sep-21 |
303.58 |
307.5986842 |
4.018684211 |
**This study also uses several different number of intervals with different class widths since the midpoint gives the influence on the defuzzified forecast result.**
**Model Evaluation Method**
To determine the accuracy of the crude oil price prediction, the evaluation of the result obtained from the FTS Cheng method is analysed by using Root Mean Square Error (RMSE) and Mean Absolute Percentage Error (MAPE).
(9)
(10)
where
and are the actual and forecasting price of crude oil, respectively and
is the number of observation.
RMSE is a good measure of how accurately the model predicts the response, and it is the most important criterion to fit the original data for prediction purposes. Lower values of RMSE indicates a better fit. For MAPE, the assessment results are indicated as excellent significance if the result is less than 10 percent, good significance if the result is 10 percent to 20 percent, moderate significance if the result is 20 percent to 50 percent and low significance if the result is greater than 50 percent. In conclusion, the smaller values in MAPE indicates the more accurate the prediction is.
**FINDINGS AND DISCUSSIONS**
Figure 1 shows the actual historical price of crude oil in Malaysia. There is no such regular fluctuation pattern. For the period of 10 years, the highest price of crude oil per barrel was RM358.59 in March 2012 while the lowest price was RM91.94 in April 2020 due to Covid-19 pandemic. Luckily, the price starts to increase beginning May 2020 onwards though the small fluctuation is still there.
**Figure 1**: Price of Crude Oil in Malaysia
Several intervals are being tested in the model but majority show a very high MAPE and RMSE. The ability to predict are categorized as very poor. Only the FTS Cheng model with 7, 8 and 10 intervals of crude oil price shows more accurate output as shown in Table 9 and Figure 2.
**Table 9**: Forecast Result for 7, 8 and 10 Intervals
| Month/year | Price | Forecast value 7 intervals | Forecast value 8 intervals | Forecast value 10 intervals |
| ---------- | ------ | -------------------------- | -------------------------- | --------------------------- |
| Sep-11 | 311.68 | 0 | 0 | 0 |
| Oct-11 | 313.55 | 301.673956 | 305.7286842 | 301.655 |
| Nov-11 | 331.98 | 285.113956 | 318.5586364 | 285.095 |
| Dec-11 | 329.66 | 305.863956 | 339.3086364 | 305.845 |
| Jan-12 | 333.18 | 300.023956 | 333.4686364 | 336.6270588 |
| Feb-12 | 340.88 | 295.843956 | 329.2886364 | 332.4470588 |
| β¦ | β¦ | β¦ | β¦ | β¦ |
| Apr-21 | 259.61 | 244.1002747 | 278.325 | 266.7789474 |
| May-21 | 273.99 | 286.0714286 | 261.245 | 249.6989474 |
| Jun-21 | 296.88 | 286.0714286 | 284.7086842 | 274.5190909 |
| Jul-21 | 307.68 | 303.543956 | 296.7986842 | 292.725 |
| Aug-21 | 290.7 | 286.0714286 | 292.605 | 314.3890909 |
| Sep-21 | 303.58 | 303.543956 | 294.7186842 | 284.5290909 |
Figure 2 shows the comparison of monthly forecast results crude oil price in Malaysia using 7, 8 and 10 intervals. There is slightly small difference between the actual price and forecast results using 8 intervals compared to forecast results using 7 and 10 intervals. Moreover, the forecast result using 8 intervals had accurate line with actual crude oil price intervals compare to forecast result using 7 and 10 intervals.
> **Figure 2**: Forecast Results of Crude Oil Price in Malaysia Using 7,8 And 10 Intervals
Table 10 shows the value of RMSE and MAPE for 7, 8 and 10 intervals. The forecast 8 intervals have the smallest value of RMSE and MAPE with 23.44669 and 8.09% respectively. However, the forecast 7 intervals have the highest value of RMSE and MAPE with 29.1195617 and 9.90% respectively. To conclude, the forecast using 8 intervals in FTS Cheng is the best fit model because its RMSE is the smallest and its percentage error is also the smallest.
**Table 10**: RMSE and MAPE for 7, 8 and 10 intervals
| Fuzzy Time Series | RMSE | MAPE |
| ----------------- | ---------- | ----- |
| 7 intervals | 29.1195617 | 9.90% |
| 8 intervals | 23.44669 | 8.09% |
| 10 intervals | 26.41379 | 9.16% |
**CONCLUSION AND RECOMMENDATIONS**
FTS series Cheng model has been proposed and implemented to predict the price of crude oil in Malaysia. The actual data are partitioned into several number of intervals and class widths are being tested in the same model. The finding shows that only 7, 8 and 10 intervals are fit the original data and the model shows good performance since their MAPE \< 10%. Among these three, FTS Cheng with 8 intervals is the best performance for crude oil price prediction.
The finding of this study shows that the number of intervals used would give great influence on the prediction and this may cause higher percentage error in forecasting. More variations in linguistic values affects the forecast output and need to further studied. It is advisable to do the right partitioning of the universe in order to improve the forecasting result. On some cases, better accuracy can be achieved with a shorter interval length (Huarng, 2001).
This preliminary study is only focusing on FTS Cheng methods where there is no comparison made with other forecasting models. To find a better prediction model, there is a need to use and compare with other models of forecasting techniques not only other fuzzy time series models but also to use econometric models, artificial intelligence models and statistical models as well.
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