The SES is applied when the data pattern is nearly horizontal, and shows no particular trend or seasonal variation exists in previous data sets. For the series ϕ 1 , ϕ 2 , ⋯ , ϕ t the forecast for the preceding value ϕ t + 1 , say #Math_5#, is based on the weights 1 − α and α to the recent observation ϕ t and forecast ϕ ¯ t respectively. Where α is the smoothing constant called alpha, ϕ t is the actual value for period t, ϕ ¯ t is the forecast value for period t. The model is of the form

ϕ ¯ t + 1 = ϕ ¯ t + α ( ϕ t − ϕ ¯ t ) ,     0 < α < 1   and   t > 0. (1)

The value of α is subjectively such that a value close to zero is for smoothing out unwanted cyclical and irregular components and a value close to one is for forecasting.

This technique is used when the data exhibits a trend in its pattern. If you have a time series that can be described using an additive model with increasing or decreasing trend and no seasonality. The model is

ϕ ¯ t = α ϕ t + ( 1 − α ) ( ϕ ¯ t − 1 + β t − 1 ) ,     0 < α < 1 , (2)

β t = θ ( ϕ ¯ t − ϕ ¯ t − 1 ) + ( 1 − θ ) β t − 1 ,     0 < θ < 1 , (3)

ϕ ^ t + m = ϕ ¯ t + β t m (4)

where ϕ t is the actual value in time t, ϕ ¯ t is the level of series at time t, β t is the slope (trend) of the time series at time t. α and β ( = 0.1 , 0.2 , ⋯ , 0.9 ) are the smoothing coefficient for level and smoothing coefficient for trend respectively. The best values of α and β correspond to the minimum mean square error (MSE).

The TES model is applied when time series data exhibit seasonality. It incorporates three smoothing equations; first for the level, second for trend and third for seasonality. The Triple exponential smoothing model is:

ϕ ¯ t = α ϕ t S t − p + ( 1 − α ) ( ϕ ¯ t − 1 + β t − 1 ) ,     0 < α < 1 , (5)

β t = θ ( ϕ ¯ t − ϕ ¯ t − 1 ) + ( 1 − θ t − 1 ) β t − 1 ,     0 < θ < 1 , (6)

S t = γ ϕ t ϕ ¯ t + ( 1 − γ ) S t − p ,     0 < γ < 1 , (7)

So we have our prediction for time period T + τ :

ϕ ^ T + τ = ( ϕ ¯ T + τ θ T ) S T (8)

where: ϕ ¯ T is the smoothed estimate of the level at time T, θ T is the smoothed estimate of the change in the trend value at time T, S T is the smoothed estimate of the appropriate seasonal component at T. α, β and γ are the level, trend and seasonal smoothing parameters respectively. ϕ ¯ t is the smoothed level at time t, θ t is the change in the trend at time t, S t is the seasonal smooth at time t and p is the number of seasons per year.

The R output for the SES model was as shown in Table 2. The alpha level was

estimated at α = 0.9104 with initial state, l = 5521.3676 and AIC = 578.5190

And the fitted model for this result took the form of

ϕ ¯ t + 1 = 0.9104 ϕ t + 0.0896 ϕ ¯ t (12)

The R output for the DES model was as shown in Table 3. The level and trend components were estimated at α = 0.8006 and β = 0.0004 respectively, with initial states, l = 6033.9228 and b = − 118.9081 . AIC = 581.2877 .

The following equations constituted the fitted DES model for SHC importation using Equations ((2) and (3)).

ϕ ¯ t = 0.8006 ϕ t + 0.1994 ( ϕ ¯ t − 1 + β t − 1 ) , (13)

β t = 0.0004 ( ϕ ¯ t − ϕ ¯ t − 1 ) + 9.9996 β t − 1 ,

β t = 0.0004 ( ϕ ¯ t − ϕ ¯ t − 1 ) + 9.9996 β t − 1 , (14)

Table 4 shows the R output for the HW model. The smoothing parameters level, trend and gamma were estimated at α = 0.8006 , β = 0 and γ = 1 respectively. The coefficients are; a = 2158.93914 and.

Using Equations (5)-(7), we fitted the HW model for SHC imports as;

ϕ ¯ t = 0.7706147 ϕ t S t − p + 0.2293853 ( ϕ ¯ t − 1 + β t − 1 ) , (15)

β t = β t − 1 , (16)

S t = ϕ t ϕ ¯ t (17)

Figure 3 shows the plots for three fitted ES techniques and original data models for easy of comparisons and choosing.

Figure 3(a), which shows TES, is eliminated easily because, by graphical inspection, it does not closely mimic the time plot for observed data as good as the other two. Now the choice was to be made between Figure 3(b) and Figure 3(c) which clearly look so closely alike and both were mimicking the observed data plot quite well. Therefore, to make a good choice the AIC for both were calculated and compared with the model giving a smaller AIC being chosen.

Clearly the AICs in Table 5 show that the SES was a better fit than DES. Hence the appropriate ES technique of the three ES compared was chosen to be SES.

To model an ARIMA, a time plot is the first step. Figure 4 shows a time plot of the SHC imports for d = 0 and d = 1. Figure 4(b) is as a result of non-stationarity nature of the observed data as evidenced by Figure 4(a) and the ACF and PACF plots in Figure 5. ARIMA modelling requires that observed data be stationary and if not, it must be made stationary.

Hence Figure 4(b) and Figure 5(c) and Figure 5(d) which are as a result of first difference, that is d = 1.

Model selection requires that the ACF and PACF plots for d = 1 in Figure 4 be examined to establish the most suitable ARIMA. But the ACF and PACF plots did not give clear indication of significant spikes at any one lag. Hence, several tentative ARIMA models and their respective AICs were examined as shown in Table 6. Table 6 shows ARIMA (2, 1, 2) was chosen as the best fit of the tentative ARIMA models examined. Although the first six had smaller AICs, their parameters were found to be insignificant. ARIMA (2, 1, 2) had all its parameters estimated significant as Table 7 shows.

When estimating the parameters, R gave the following output for ARIMA (2, 1, 2) in Table 7(a). Then their significance was tested by use of p-value (see Table 7(b) for p-values of each parameter).

The parameters found significant were AR (1), AR (2), MA (1), and MA (2) at

Note: *Number of significant parameters and lesser prediction errors

Note: *implies p-value < 0.05 hence significant coefficient.

5% significance level. Hence the fitted ARIMA (2, 1, 2) using equation 11 was;

X ⌢ t = 1.0907 ε t − 1 + 0.9983 ε t − 2 − 1.0536 X t − 1 − 0.9947 X t − 2 (18)

Figure 6 is a plot of the fitted model to the observed SCH imports which shows that the model fluctuates so closely to the actual SHC imports.

The model with best fit was identified by analysis of residuals to ensure they form a white noise process. The ACF of residual, the Q-Q plot and the histogram of residuals were used to show that the residuals of the fit form a white noise process. Figure 7 below shows that the residual are white noise and all p-values of the Ljung Box test are greater than 0.05. Hence ARIMA (2, 1, 2) is indeed the best fit model.