Sugeno Intuitionistic Fuzzy Generator Based Computational Technique for Crude Oil Price Forecasting

Crude oil being a significant source of energy, change of crude oil price can affect the global economy. In this paper, a new approach based on the intuitionistic fuzzy set theory has been implemented to predict the crude oil price. This paper presents the intuitionistic fuzzy time series forecasting algorithm to enhance the efficacy of time series forecasting which includes fuzzy c-means clustering to obtain the optimal cluster centers. Further, a computational technique is proposed for the construction of triangular fuzzy sets and these fuzzy sets are converted to intuitionistic fuzzy sets with the help of Sugeno type intuitionistic fuzzy generator. The popular benchmark dataset of West Texas Intermediate crude oil spot price is used for the validation process. The numerical results when compared with existing methods notify that the proposed method enhances the accuracy of the crude oil price forecasts.


Introduction
Crude oil price is one of the major factors affecting the world's economy. Forecasting of crude oil prices has been gaining a lot of attention as these forecasts are very useful to industries, investors, government, household expenses and so on. Several prominent researches have been made in this field due to the explosive nature of the oil price. It depends on many factors which makes the price varying from time to time in an unpredictable manner. Crude oil is one of the main input and most traded commodity in the world. The uncertainty in crude oil price affects different sectors in different ways. Crude oil is one of the inputs that cost the firm and variation in its price creates difficulty for the decision maker to take important investment decisions (Henrique and Sadorsky, 2011). Also, it affects the consumers in various situations for instance, when petroleum and gas prices vary or when there is variation in the cost of products whose production requires crude oil. When the price of oil increases, it is in favor of oil producers but against the oil consumers and vice-a-versa (Phan et al., 2015). Thus, the prediction of crude oil prices carries huge importance in the market as well as corporate sectors.
Significant researches are made on the qualitative and quantitative bases in the literature. Xie et al. (2006) proposed a method based on support vector machine (SVM), including pre-processing, training and learning. An adaptive neuro-fuzzy inference system (ANFIS) was also employed for the forecasting process (Rubinstein et al., 2015). A method integrating ensemble empirical mode decomposition (EEMD), adaptive particle swarm optimization (APSO), and relevance vector machine (RVM) was proposed to predict the price based on decomposition and ensemble framework (Li et al., 2016). Also, using different machine learning techniques a hybrid model was proposed using multi-layer perception, support vector regression (SVR) and Holt Winter exponential smoothing techniques (Kanchymalay et al., 2017) in which it was observed that SVR performs better than other two techniques. A new reconstruction rule based on fuzzy clustering and dynamic time wrapping (DTW) was proposed by Chai et al. (2019) using complete integration empirical mode decomposition algorithm and autoregressive integrated moving average (ARIMA). These traditional techniques lack when there is uncertainty in the historical data due to which Song and Chissom (1993) proposed the concept of fuzzy time series (FTS). After which FTS has been used in various applications (Bisht et al., 2013;Jain and Bisht, 2015;Jain et al., 2017;Jain et al., 2018;Goyal and Bisht, 2019). Zhang et al. (2010) proposed a model to predict the short-term crude oil price using fuzzy time series.
The theory of intuitionistic fuzzy set (IFS) was proposed by Atanassov (1986) as an extension of the fuzzy set (FS) theory to deal with uncertainty and vagueness. In IFS, the non belongingness of an element is also taken in consideration with the belongingness of an element to a set. Kumar and Gangwar (2015) proposed the concept of intuitionistic fuzzy time series combining IFS and FTS. In this paper, an improved method of fuzzy time series forecasting based on IFS theory is proposed for the crude oil price prediction. This method is applied for crude oil price prediction. The methodology starts with the pre-processing of data in which outliers are identified and are replaced by linear interpolation technique. In the process, fuzzy c-means (FCM) clustering is used to obtain the optimal cluster centers which are then used in the process of interval construction with a new computational approach and Sugeno type intuitionistic fuzzy generator (STIFG) is used for the evaluation of non-membership degree. Then hesitancy degree is used for the process of fuzzification and for different order of fuzzy logical relationships (FLRs), the process is evaluated. Also, new rules are formed for the defuzzification process. The model is applied to the benchmark dataset of West Texas Intermediate (WTI) crude oil price and with the help of rootmean squared error (RMSE) and mean average percent error (MAPE), the accuracy of the model is determined.

Preliminaries 2.1 Fuzzy c-means Clustering
FCM is a clustering method which allows each data point to belong to more than one cluster with different membership degree which is based on minimization of the objective function Jm , where D is the total number of data points (yi's), Cj is the center of j th cluster and N is the number of clusters and m is the exponent of fuzzy partition matrix.
Steps 2 to 4 are repeated until a termination criterion is satisfied.

Intuitionistic Fuzzy Set and Intuitionistic Fuzzy Time Series
An intuitionistic fuzzy set (IFS) I on universe of discourse  is defined as eq. (4) where is known as the degree of non determinacy (hesitancy factor) of y to the IFS I. Let IFS Is(t) be defined on (t) (t=..., 0, 1, 2, ...), then collection of such IFSs is known as Intuitionistic fuzzy time series (IFTS) (Kumar and Gangwar, 2015).

Sugeno Type Intuitionistic Fuzzy Generator
To construct the IFS from the FS, intuitionistic fuzzy generators are used. Fuzzy generator φ is a type of fuzzy complement satisfying the property given by eq. (5). Sugeno type fuzzy complement is defined as N((y)).

Algorithm of Proposed Method for Forecasting Using IFTS
This section describes the proposed methodology in two phases comprising of pre-processing, fuzzification, construction of FLRs and defuzzification.

Phase I: Pre-processing and Fuzzification
Step 1. Using the generalized extreme studentized deviate (ESD) test (Grubbs, 1950), the data set is processed to find out the outliers in the dataset and is replaced by linear interpolation technique. Step 2. Now using fuzzy c-means clustering, the dataset is divided into the desired number of clusters after which obtained cluster centers are then used for the construction of intervals. Length of these cluster centers are calculated and further partitioning is done using the following procedure: (i) Test for outliers is done for a series of the length of cluster centers. (ii) To remove the outlier length obtained, the average length of the remaining series is calculated. (iii) Then the length as outlier is divided by average length to obtain q partitioning points and outlier length is equally divided into q parts.
length Average length Outlier q  (iv) Also, the universal set is defined as  = [L, U] where L = lowest value of cluster centeraverage length and U = highest value of cluster center + average length. (v) Now from the obtained final series of partitioning points pk (k=1, 2, ..., n), the intervals vl (l=1, 2, ..., n-1) are constructed.
Step 4. Construct IFS Is corresponding to TFS Ts in which the membership degree is the same as of TFS and non-membership degree is evaluated using Sugeno type intuitionistic fuzzy complement as eq. (11). Therefore, IFS I is defined as Step 5. Evaluate the hesitancy factor  of each yi corresponding to each Is as eq. (12

Phase II: Construction of Intuitionistic Fuzzy Logical Relationships and Defuzzification
Step 7. For the construction of intuitionistic fuzzy logical relationships (IFLR), the IFS of time t-2, t-1, t and t+1 are considered i.e., It-2, It-1, It and It+1. Hence, the IFLR formed is It-2, It-1, It  It+1. Then on the basis of same antecedent, these IFLR are grouped and rule base is constructed.
Step 8. To evaluate the forecasted output at time t, rules given below are considered assuming intuitionistic fuzzified value at time t-3, t-2, t-1 as I, Iβ, Iγ:

Numerical Illustration
The proposed method is applied to the popular benchmark dataset for international oil prices, i.e., West Texas Intermediate (WTI). The crude oil monthly price from January 1986 to August 2019 is collected from the site https://www.eia.gov/. The data set is divided into two sets: 70% for the training set (Jan 1986 to Jul 2009) and 30% for the testing test (Aug 2009 to Aug 2019). The dataset is processed under ESD test and outliers found at 1% level of significance are replaced by linear interpolation technique. Next using fuzzy c-means clustering, the 14 cluster centers are obtained as Table 1.

Evaluation Parameters
To measure the accuracy of the results of proposed intuitionistic time series forecasting model, RMSE and MAPE are used defined by eq. (14) and (15)  where (t) is the actual price of crude oil at time t and (t) is the predicted price at time t and D is the number of data points. Smaller the RMSE and MAPE, higher is the accuracy rate.

Experimental Results
The proposed method is modeled to predict the price of the crude oil and the monthly dataset of WTI is used with a total of 404 data points. The proposed model is evaluated on the dataset for different orders of FLRs and different values of . In the experiment, it is observed that considering different values of  does not affect the forecasting process. However, the order of FLRs affects the accuracy of the model. In order to show the superiority of the proposed model, results obtained from the fuzzy time series Chen's model (Chen, 1996) are compared with the results of the intuitionistic fuzzy time series model (proposed model). Table 2 and Table 3 show the comparative results of training and testing dataset and graphical representation is shown in Figure 1.

Conclusion
In this paper, a new approach based on intuitionistic fuzzy time series has been proposed to predict the price of crude oil. The proposed model is explained in two phases, firstly preprocessing and fuzzification in which FCM clustering is used with a new computational approach to construct the TFS and STIFG is used for the evaluation of non-membership degree. Then hesitancy degree is used for the process of fuzzification. Second is the construction of rule base and defuzzification in which the proposed model is carried out for first order FLR, second order FLR and third order FLR separately and new rule are formed for the defuzzification process. The model is applied on the benchmark dataset of WTI crude oil price and with the help of evaluation parameters RMSE and MAPE; it is observed that proposed intuitionistic fuzzy set theory based model forecasted better results in comparison to the model based on the fuzzy set theory and moving average method. Further, this work can be extended using the different intuitionistic fuzzy generator for the conversion of fuzzy set to intuitionistic fuzzy set and nature-inspired optimization can be employed to enhance the process.