Box and Jenkins [23] first developed the Autoregressive Integrated Moving Average (ARIMA) model. As the name implies, Autoregressive (AR) and the Moving Average (MA) models were combined on stationary data. The order of the ARIMA model is usually denoted as ARIMA (p, d, q) where “d” represents the frequency of differencing done to make the data stationary, “p” is the number of spikes that crosses the significant line of the Partial Autocorrelation Function (PACF) plot and “q” is the number of spikes that crosses the significant line of the Autocorrelation Function (ACF) plot. The process of identifying the model order was explained in [24]. The Augmented Dickey Fuller test with a trend and an intercept is used to test for stationarity.

Given a time series data x t , the ARIMA (p, d, q) model is given as:

ϕ ( B ) ( 1 − B ) d X t = θ ( B ) Z t (1)

where

ϕ ( B ) is the characteristic polynomial of order “p” for the autoregressive component of the model;

θ ( B ) is the characteristic polynomial of order “q” for the moving average component of the model;

( 1 − B ) d is the differencing of order “d” of the data;

X t is the observed value at time t;

Z t is the random error associated with observation at timet.

The model obtained by inspecting the ACF and PACF plots is compared to models with parameters close to it in order to identify a better model using the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC). The model identified to have the least values of both AIC and BIC is suggested and subjected to residual checks using the ACF plot, time series plot and normality histogram plot of the residuals to test if the residuals are white noise. White noise is very crucial in time series forecasting and if the residuals are not white noise, then the model should be improved before it can be used for prediction purposes. The concept of identifying a white noise residual is explained in section 3.1 of this study.

The ANN model as invented by Frank Rosenblatt in [25] is a machine learning method used in modelling complex nonlinear relationships between the response and explanatory variables. ANN has been used and explained by [26].

The mathematical representation of the ANN model is given as:

y ⌢ ( x i , w ) = Φ 0 ( α + ∑ h = 1 H w h Φ h ( α h + ∑ j = 1 J w j h x i j ) ) (2)

where

y ⌢ ( x i , w ) is the estimated response variable;

w j h is the weight from the input to hidden nodes;

w h is the weight from the hidden to output node;

x i j is the input node;

α and α h are bias that can be interpreted as the intercept in a linear regression;

Φ 0 and Φ h are activation functions.

The transmission from input layer to the hidden layer is done using the logistic activation function while the linear activation function is used for the transmission from the hidden layer to the output layer. The connection between the nodes is assigned weights. The quadratic error function given in Equation (3) is used in this study to determine the weights.

E Q = ∑ k = 1 K ∑ i = 1 n ( y ^ ( x i , w ) − y i k ) 2 (3)

where

y ⌢ k ( x i , w ) is the estimated response variable;

y i k is the response variable.

The importance of each input node is estimated using the Olden method [27].

The explanatory variables usually constitute the input node and were normalized using the min-max normalization method to help the neural network to converge quickly [24] [28].

Random forest is an easy-to-use machine learning method. It is an ensemble of decision trees for regression and classification. The output is the mode of the classes (for classification) or mean prediction (for regression) of the individual trees [29].

Random forest searches for the best feature among a random subset of features when splitting a node. This results in obtaining a better model. When splitting a node, only a random subset of the features is taken into consideration by the algorithm. The steps used to grow each tree are given in [30]. If two trees are correlated, it will increase the forest error rate. Figure 1 is a plot of a random forest.

Usually, Time series and regression models are used in forecasting and prediction using statistical methods but these models have many drawbacks in practice. The drawbacks arise because of some required number of assumptions that are unsatisfactory in regression models and poor performance when there are abnormal changes or series is non-stationary when using time series models. In order to overcome these drawbacks, various models have been recommended such as the random forest [30], artificial neural network [31], support vector regression [32], multivariate adaptive regression spline [33], adaptive spline threshold autoregressive model [34], etc.

All these models, both the traditional and the recommended models were developed primarily for solving forecasting problems. In the same approach towards solving these forecasting problems but now in which the historical data are presented as linguistic values, the fuzzy time series (FTS) model was proposed by [32] and tested with Enrollment data from the University of Alabama (EnrollmentUA). The model consists of two major processes: 1) fuzzification and 2) the establishment of fuzzy relationships and forecasting. Lengths of intervals will result in various forecasting results during fuzzification process Therefore; effective lengths of intervals should be used [35]. The forecasting results that were based on the effective lengths of intervals were found to outperform those based on arbitrary ones. So many authors have study and even extended the FTS model (Chen [36]; Huarng [37]; Huarng and Yu [38]; Singh [39]; Teoh et al. [40]; Liu et al. [41]; Yu and Huarng [42]; Khashei et al. [43]; Bas et al. [44] and Egrioglu et al. [45] ).

Chen [36] algorithm is simple, does not require complex matrix operations in the establishment step of fuzzy relationships and the algorithm produced better results with the same enrollment data [46]. This advantage encouraged the usage of Chen’s algorithm for this research. The algorithm of Chen’s method is summarized in Table 1.

The Chen’s algorithm and other models in this work were implemented using R programming software version 4.05.

A simple performance measure to include: RMSE, MAPE and NSE were calculated. The model with the least RMSE and MAPE values and the highest NSE value are chosen as the best model. The modified Diebold-Mariano test was also implemented to test the hypothesis that model 2 is a better model than model 1 at 5% level of significance.

The time series plot of the Bonny light crude oil price for the periods (Jan. 2006 to Dec. 2020) is displayed in Figure 2. As illustrated in [47], the parameters (p, d, q) of the ARIMA model were estimated using the train set (Crude oil price between Jan. 2006 and Jun. 2016). The Augmented Dickey-Fuller test of lag 4 shown in Table 2 suggests that the train set is not stationary (p-value (0.6387) > 0.05) and this is also evident in the time series plot of Figure 3. The stationary data was obtained after the first differencing as shown in the time series plot in Figure 4 and the Augmented Dickey-Fuller test of lag order 4 is now significant (p-value (0.000) < 0.05). The first order of differencing suggests a “1” value for the “d” parameter of the ARIMA model. The lags die out quickly in the ACF plot

in Figure 5 while there is a sharp cutoff after the first lag in the PACF plot in Figure 6. This suggests an order “1” and “0” for the “p” and “q” parameters respectively. To choose the optimal ARIMA model, the suggested ARIMA model of order (1, 1, 0) was compared to models with parameters similar to it, as indicated in [47], using the AIC and BIC. As shown in Table 3, the ARIMA (1, 1, 0) model was chosen as the best order of ARIMA model since it had the lowest AIC and BIC values. The parameter of the model is therefore displayed in Table 4. Because there was no trend, no spikes cut through the significant line of the ACF plot, and the residuals are normally distributed, the ARIMA (1, 1, 0) errors are white noise, as seen in the time series plot of the residual in Figure 7. The residuals are then white noise, and the model can now be used for prediction.

Here, the best model is selected by trying different node sizes in the hidden layer of the ANN and the resulting model with the best trade off values for RMSE, MAPE and NSE is selected. Therefore, different node sizes ranging from 1 to 8 were used in the hidden layer of the ANN model. Then the model with node size of one in the hidden layer is chosen for the ANN model considering the comparison values in Table 5. The ANN plot with one hidden node, four input nodes and one output node is therefore shown in Figure 8. Thus, the ANN is of order ANN (4, 1, 1). The weights of the neural network were estimated by minimizing the quadratic loss function. Figure 9 shows that Lag 1 values of the crude oil price has more importance in estimating the current crude oil price in Nigeria, followed by quantity produced, time and quantity exported.

Here, the best model is selected by trying different number of explanatory

variables and as in ANN, the model with the best trade-off in the RMSE, MAPE and NSE is selected. The performance measures of the different choices of the explanatory variables selected are given in Table 6. It is evident that using the four explanatory variables (lag 1, production, time and export) at each node for splitting the node had the best tradeoff in the RMSE, MAPE and NSE values in both the training and test sets. Again, the RF model shows that the first lag variable of the crude oil price had more importance in estimating the Nigerian Bonny Light crude oil price. The mean square error (%IncMSE) and the node impurity (%IncNodePurity) increases the most when the “Lag1” variable is randomly permuted, followed by the “Time” variable as shown in Figure 10.

With the steps in Table 1, the following results for the test set are obtained:

1) The universal set U = [ D ′ − D 1 , D ″ + D 2 ] where D ′ and D ″ are minimum and maximum value of time series data with D 1 and D 2 are fixed numbers with positive value. For this paper, we have chosen D 1 = 8 and D 2 = 10 for the modeling of the test set. Given that the minimum and maximum values of the test set are 14.28 and 79.59 respectively, then

U = [ 6.28 , 89.59 ]

2) The method [48] was used to get the number of subintervals for the linguistic interval and the size of the subinterval was determined by dividing the range by the number of subintervals.

3) The subintervals are hereby defined as:

u 1 = [ 6.28 , 18.18 ] ; u 2 = [ 18.18 , 30.08 ] ; u 3 = [ 30.08 , 41.98 ] ; u 4 = [ 41.98 , 53.88 ] ; u 5 = [ 53.88 , 65.79 ] ; u 6 = [ 65.79 , 77.68 ] ; u 7 = [ 77.68 , 89.59 ] .

4) The fuzzification vector is given thus:

fuzzify = ( 4       4       4       4       4       5       4       4         4       4       4       4       4       5       5       5         5     6       6     6       6         6       6         6         6       7     7     6     5       5       5       6         6       6       6       6         5       5         5         5       6       6       5       3         1         2       3       3       4         4       3         3       4       4 )

where the n^th value of the fuzzification vector represents the n^th fuzzified subinterval (linguistic values).

5) The fuzzy relationship created based on the fuzzification vector is given in Table 7.

6) The transition matrix obtained from the fuzzy relationship table in Table 7 is given by:

Transition   Matrix = ( 0 .00 1 .00 0 .00 0 .00 0 .00 0 .00 0 .00 0 .00 0 .00 1 .00 0 .00 0 .00 0 .00 0 .00 0 .20 0 .00 0 .40 0 .40 0 .00 0 .00 0 .00 0 .00 0 .00 0 .07 0 .80 0 .13 0 .00 0 .00 0 .00 0 .00 0 .08 0 .08 0 .62 0 .23 0 .00 0 .00 0 .00 0 .00 0 .00 0 .19 0 .75 0 .06 0 .00 0 .00 0 .00 0 .00 0 .00 0 .50 0 .50 )

It is important to note that the transition matrix is calculated as State matrix divided by sum of row values. The state matrix is given by:

Statematrix = ( 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 2 2 0 0 0 0 0 1 12 2 0 0 0 0 1 1 8 3 0 0 0 0 0 3 12 1 0 0 0 0 0 .00 1 1 )

For instance, the value at row 4 column 4 of the transition matrix is obtained as: 12 0 + 0 + 1 + 12 + 2 + 0 + 0 = 12 15 = 0.80 , the same method is applied to get the remaining values of the transition matrix.

6) The forecast fuzzy output is the same as the defuzzified crisp values which is the fuzzy predicted values of the Bonny Light Crude oil price.

The ARIMA (1, 1, 0), ANN (4, 1, 1), RF and FTS models were compared in both the training and the test sets using the RMSE, MAPE and NSE performance measures to get a high-performance model. As an additional comparison measure, the modified Diebold-Mariano test was also used to test the hypothesis that model 2 is the same as model 1. When comparing (ARIMA-ANN), the ARIMA is model 1, while the ANN is model 2. The alternative hypothesis is that the ANN is a better model than the ARIMA model. This implies that a significant p-value shows that model 2 is better than model 1. This test revealed that ANN, RF and TFS are better than ARIMA in the test sets, FTS is better than RF and ANN while ANN and RF performed likely in the test set. This is also verified by examining the RMSE, MAPE and NSE values for these models. This result is in agreement with the works of [49] in China and [50] using New York Stock exchange data.