## 2 The Proposed Hybrid Time Series Forecasting Methodology
### Accuracy Measures  
In order to check the performance of the forecasting models in previous studies, many researchers used various performance measures and statistical tests [32; 33; 34; 35]. Hence, in this study, for model evaluation, first, we used five accuracy mean errors: two relative mean errors, two absolute mean errors, and one correlation measure for observed versus forecasted values, such as root mean square error (RMSE), root mean square percentage error (RMSPE), mean absolute error (MAE), and mean absolute percentage error (MAPE). The mathematical formula for accuracy means errors are expressed as  
\[\text{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^{n}(\mathfrak{C}_{i}-\hat{\mathfrak{ C}}_{i})^{2}}, \tag{6}\]  
\[\text{RMSPE}=\sqrt{\frac{1}{n}\sum_{i=1}^{n}\biggl{(}\frac{(\mathfrak{C}_{i}- \hat{\mathfrak{C}}_{i})^{2}}{\mathfrak{C}_{i}}\biggr{)}\times 100}, \tag{7}\]  
\[\text{MAE}=\frac{1}{n}\sum_{i=1}^{n}\bigl{(}|\mathfrak{C}_{i}-\hat{\mathfrak{ C}}_{i}|\bigr{)}, \tag{8}\]  
\[\text{MAPE}=\frac{1}{n}\sum_{i=1}^{n}\biggl{(}\frac{|\mathfrak{C}_{i}-\hat{ \mathfrak{C}}_{i}|}{|\mathfrak{C}_{i}|}\biggr{)}\times 100, \tag{9}\]  
\[\text{CC}=\text{correlation}\big{(}\mathfrak{C}_{i},\hat{\mathfrak{C}}_{i} \big{)}. \tag{10}\]  
Here, the observed value is \(\mathfrak{C}_{i}\) of the time series, and \(\hat{\mathfrak{C}}_{i}\) represent the forecasted ozone concentration value of the \(i\)th observation (\(i=1,2,\ldots,n\)), with the size of \(n\) in the testing set. Second, the Diebold and Mariano (DM) test [36] was conducted to test the significance of the differences among the performance of the forecasting models. The DM test is a broadly used statistical test for the comparison of forecasts extracted from various models [37; 38; 39]. To understand it, consider two forecasts, \(\hat{\mathfrak{C}}_{1n}\) and \(\hat{\mathfrak{C}}_{2n}\), that are available for the time series \(\mathfrak{C}_{n}\) for \(n=1,\ldots,N\). The associated forecast errors are \(\mathfrak{e}_{1n}=\mathfrak{C}_{n}-\hat{\mathfrak{C}}_{1n}\) and \(\mathfrak{e}_{2n}=\mathfrak{C}_{n}-\hat{\mathfrak{C}}_{2n}\). Let the loss associated with forecast error \(\{\mathfrak{e}_{in}\}_{i=1}^{2}\) be \(L(\mathfrak{e}_{in})\). For example, the absolute loss in time \(n\) would be \(L(\mathfrak{e}_{in})=|\mathfrak{e}_{in}|\), and the differential loss between forecast 1 and forecast 2 for time \(t\) is then \(\mathfrak{w}_{n}=L(\mathfrak{e}_{1n})-L(\mathfrak{e}_{2n})\). The null hypothesis of equal forecast accuracy for two forecasts is \(E[\mathrm{w}_{n}]=0\). The DM test needs the differential loss to be covariance stationary, i.e.,  
\[E[\mathrm{w}_{n}] = \mu,\quad\forall\;n \tag{11}\] \[\mathrm{cov}(\mathrm{w}_{n}-\mathrm{w}_{n-\mathrm{r}}) = \gamma(\tau),\quad\forall\;n\] (12) \[\mathrm{var}(\mathrm{w}_{n}) = \sigma_{\mathrm{w}},\quad 0<\sigma_{\mathrm{w}}<\infty \tag{13}\]  
Under these assumptions, the DM test of equal forecast accuracy is  
\[\mathrm{DM}=\frac{\bar{\mathrm{w}}}{\bar{\sigma}_{\mathrm{w}}}\xrightarrow[]{ d}N(0,1)\]  
where \(\bar{\mathrm{w}}=\frac{1}{N}\sum_{n=1}^{N}\mathrm{w}_{n}\) is the differential loss of the sample mean, and \(\bar{\sigma}_{\mathrm{w}}\) is a consistent estimate of standard error \(\mathrm{w}_{n}\). Finally, we verify the superiority of the proposed hybrid combination of time series forecasting models using various figures, such as the box plot, line plot, bar plot, and dot plot in this work. To conclude this section, the design of the proposed hybrid combination of time series modeling and forecasting technique is presented in Figure 2.  
Figure 2: A flowchart of the proposed forecasting methodology.