## 2 The Proposed Hybrid Time Series Forecasting Methodology  
Before starting the modeling, it often makes sense to prepare the data. The goal of preprocessing is usually to simplify the modeling of the data. To do this, the database is sorted, classified, and analyzed for each monitoring station, taking into account the winter period of the city, which runs from 21 June to 22 September, for ozone. From 2017 to 2019, four monitoring stations located at strategic points in the capital of Lima were considered. It should be noted that the number of monitoring stations in the capital of Lima is ten; however, four were selected due to a lack of data in the registry. The hourly ozone concentrations were measured with a Teledyne analyzer (an instrument with about 15 sensing technologies used in the monitoring and manufacturing of gas, liquid analysis, and medical fields). Analyzer operations include zero and span testing, calibration, and leak detection. Data are transmitted by telemetry to SENAMHI (National Meteorology and Hydrology Service of Peru) for validation after correcting zeros, duplicates, and/or anomalies. Similarly, SENAMHI has a systematic network of stations that normally and automatically monitor and report the variables studied to a processing center. These stations use high-quality instruments and sensors to measure temperature, relative humidity, wind speed, and direction on an hourly scale. In addition, an inductive algorithm called Multiple Imputation by Chained Equations was applied. This algorithm is based on a fully conditional specification, where each incomplete variable is specified by a separate model [25]. This performs multiple assignments to replace missing values in a dataset, in this case, for hourly rate records details (see Table 1).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Station** & **ATE** & **CDM** & **SB** & **STA** \\ \hline Total hours & 6768 & 6768 & 6768 & 6768 \\ Available hours & 6654 & 6634 & 6614 & 6613 \\ Imputed hours & 114 & 134 & 154 & 155 \\ Imputed\% & 1.68\% & 1.98\% & 2.27\% & 2.29\% \\ \hline \hline \end{tabular}  
* **Note:** Campo de Marte (CDM), San Borja (SB), and Santa Anita (STA).  
\end{table}
Table 1: This table is based on 6768 observations taken throughout the winter season encompassing three years (2017, 2018, and 2019). It includes the percentage of imputation for each monitoring site.  
After obtaining the imputed ozone time series (free from missing values), we then proceed with the imputed ozone series and achieve a one-hour-ahead ozone concentration using the proposed hybrid combination of time series models. As explained previously, the hourly time series of ozone contains specific properties, such as a nonlinear long-run trend, an hourly cycle, and a different mean and variance. Considering these particular features in the model improves forecast accuracy significantly. To get these results, the ozone concentration in time series (\(\mathfrak{C}_{n}\)) is divided into three new sub-series: the first is a long-run trend (\(\mathfrak{t}_{n}\)), the second is a seasonal series (\(\mathfrak{b}_{n}\)), and the third is a residual (\(\mathfrak{r}_{n}\)) series. The mathematical description of the decomposed subsequence is given by  
\[\mathfrak{C}_{n}=\mathfrak{t}_{n}+\mathfrak{b}_{n}+\mathfrak{r}_{n} \tag{1}\]  
however, these sub-series are obtained using the seasonal trend decomposition method described in the following subsection.