# Short-Term Forecasting of Ozone Concentration in Metropolitan Lima Using Hybrid Combinations of Time Series Models

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## Abstract

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## 1. Introduction

- We improve the efficiency and accuracy of one-hour-ahead ozone concentration forecasting using a proposed hybrid combination of time series models based on the seasonal trend decomposition technique and various standard time series models.
- We apply the seasonal trend decomposition method of the ozone concentration database in four districts—ATE, Campo de Marte (CDM), San Borja (SB), and Santa Anita (STA)—with severe episodes of ozone contamination between 2017 and 2019.
- We evaluate the performance of the proposed hybrid combination of time series models, by determining five different accuracy mean errors: two relative mean errors, two absolute mean errors, and one correlation measure, such as root mean square error, root mean square percentage error, mean absolute error, and mean absolute percentage error; a statistical test, the Diebold–Mariano test; and a visual evaluation.
- In this study, the results of the final best combination model are compared with the best model proposed in the literature as well as the considered baseline models and the comparative results are recorded. Based on these results, the proposed final best combination model from this work is highly accurate and efficient compared to the best models reported in the literature.
- We present a methodological proposal applicable to the environmental management system in order to mitigate ozone pollution aimed at the stakeholders of the national air quality program.
- Finally, the current work uses only the four district datasets in Lima, Peru. This can be extended to other districts of Lima, other regions of Peru, and even the world level to evaluate the performance of the proposed hybrid time series modeling and forecasting technique.

## 2. The Proposed Hybrid Time Series Forecasting Methodology

#### 2.1. Seasonal Trend Decomposition Method

#### 2.2. Modeling the Decomposed Sub-Series

#### 2.2.1. Autoregressive Model

#### 2.2.2. Nonlinear Autoregressive Model

#### 2.2.3. Autoregressive Moving Average Model

#### 2.3. Accuracy Measures

## 3. Case Study Results

#### Metropolitan Lima Stations

**${}^{\mathbf{a}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$**hybrid combination model produces the best forecasts compared to all other possible hybrid combinations of time series models. The

**${}^{\mathbf{a}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$**is the best forecasting model, which produced 4.611, 4.464, 1.711, 14.862, and 0.949 for RMSE, RMSPE, MAE, MAPE, and CC, respectively. However, the

**${}^{\mathbf{c}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$**(4.636, 4.480, 1.704, 14.985, 0.948),

**${}^{\mathbf{c}}$STLD${}_{\mathbf{b}}^{\mathbf{b}}$**(5.601, 4.817, 1.882, 16.179, 0.924), and

**${}^{\mathbf{a}}$STLD${}_{\mathbf{b}}^{\mathbf{b}}$**(5.622, 4.906, 1.871, 16.081, 0.923) models produced the second, third, and fourth best results. Similarly, in the second attempt, the case study results of the CDM station and the results of the performance accuracy measures show that the

**${}^{\mathbf{b}}$STLD${}_{\mathbf{c}}^{\mathbf{c}}$**model yields better forecasts compared to all other possible hybrid combination models. The best forecasting model,

**${}^{\mathbf{c}}$STLD${}_{\mathbf{c}}^{\mathbf{c}}$**, produced 3.637, 11.846, 2.356, 20.441, and 0.978 for RMSE, RMSPE, MAE, MAPE, and CC, respectively. However, the

**${}^{\mathbf{c}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$**(3.762, 11.689, 2.464, 20.847, 0.976),

**${}^{\mathbf{a}}$STLD${}_{\mathbf{c}}^{\mathbf{c}}$**(3.746, 11.68, 2.458, 20.882, 0.976), and

**${}^{\mathbf{c}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$**(3.794, 11.906, 2.514, 21.323) models produced the second, third, and fourth best results. In the same way, in the third attempt, the case study results of the SB station and the results of the performance accuracy measures show that the

**${}^{\mathbf{b}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$**model yields better forecasts compared to all other possible combination models. The best forecasting model is

**${}^{\mathbf{b}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$**, which gives outcomes of 1.495, 1.864, 1.078, 7.668, and 0.989 for RMSE, RMSPE, MAE, MAPE, and CC, respectively. However, the

**${}^{\mathbf{c}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$**(1.559, 1.568, 1.136, 7.897, and 0.987),

**${}^{\mathbf{a}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$**(1.535, 1.644, 1.118, 7.793, and 0.987), and

**${}^{\mathbf{b}}$STLD${}_{\mathbf{c}}^{\mathbf{c}}$**(1.721, 2.021, 1.301, 9.293, and 0.985) models produced the second, third, and fourth best results. Finally, in the fourth attempt, the case study results of the SB station and the results of the performance accuracy measures show that the

**${}^{\mathbf{c}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$**model yields better forecasts compared to all other possible combination models. The best forecasting model is

**${}^{\mathbf{c}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$**, which gives outputs of 1.969, 15.924, 1.462, 76.261, and 0.989 for RMSE, RMSPE, MAE, MAPE, and CC, respectively. However, the

**${}^{\mathbf{c}}$STLD${}_{\mathbf{c}}^{\mathbf{c}}$**(2.141, 19.925, 1.605, 88.958, and 0.988),

**${}^{\mathbf{c}}$STLD${}_{\mathbf{c}}^{\mathbf{a}}$**(2.143, 19.669, 1.603, 89.367, and 0.988), and

**${}^{\mathbf{c}}$STLD${}_{\mathbf{b}}^{\mathbf{b}}$**(3.190, 21.490, 2.298, 95.063, and 0.972) models produced the second, third, and fourth best results.

**${}^{\mathbf{a}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$**give the least values (RMSE = 4.611, RMSPE = 4.464, MAE = 1.711, MAPE = 14.862, and CC = 0.949). Therefore, it is concluded that the

**${}^{\mathbf{a}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$**is the best model among the best models as well as all twenty-seven models. In the same way, in the case of the CDM station, from Table 4, it is confirmed that the

**${}^{\mathbf{b}}$STLD${}_{\mathbf{c}}^{\mathbf{c}}$**produced the smallest values (RMSE = 3.637, RMSPE = 11.846, MAE = 2.356, MAPE = 20.441, and CC = 0.978). Hence, it is concluded that the

**${}^{\mathbf{b}}$STLD${}_{\mathbf{c}}^{\mathbf{c}}$**is the best model among the best models as well as all twenty-seven models. However, in the case of the SB station results, it is evident that the

**${}^{\mathbf{c}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$**produced the smallest values (RMSE=1.969, RMSPE = 15.924, MAE = 1.462, MAPE = 76.261, and CC = 0.989) within the final best hybrid combination models. Thus, it is concluded that the

**${}^{\mathbf{c}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$**is the best hybrid combination model among the best models as well as all twenty-seven models. Likewise, within the best hybrid combination model outcomes from the STA stations, the

**${}^{\mathbf{b}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$**produced the smallest values (RMSE = 1.495, RMSPE = 1.864, MAE = 1.078, MAPE = 7.668, and CC = 0.989). Based on these results, it is concluded that the

**${}^{\mathbf{b}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$**is the best model among the best models as well as all twenty-seven models.

**${}^{\mathbf{a}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$**) model within all four best models is statistically superior to the other best combination models at the 5% level of significance. However, in the CDM, the SB, and the STA stations, the final best combination models, the (

**${}^{\mathbf{b}}$STLD${}_{\mathbf{c}}^{\mathbf{c}}$**), the (

**${}^{\mathbf{b}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$**), and (

**${}^{\mathbf{c}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$**), are statistically superior to the other best combination models at the 5% level of significance.

**${}^{\mathbf{c}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$**model in the ATE station, the

**${}^{\mathbf{c}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$**model in the CDM station, the

**${}^{\mathbf{c}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$**model in the SB station, the

**${}^{\mathbf{c}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$**model in the STA station produce the highest accuracy measures (RMSE, RMSPE, MAE, and MAPE) in comparison to the rest of all combination models. On the other hand, from all twenty-seven models in each monitoring station, the best four hybrid combination models are selected for comparison and compared with other models in each station. The results of all these best hybrid combination models are plotted in Figure 5. For example, see the ATE station in Figure 5a, the CDM station in Figure 5b, the SB station in Figure 5c, and the STA station in Figure 5d It can be observed from these plots that the

**${}^{\mathbf{a}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$**,

**${}^{\mathbf{a}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$**, and

**${}^{\mathbf{a}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$**show the least mean errors, respectively. In addition to the above, we plot the scatter diagrams for each station using their respective best model, which were obtained previously. For instance, Figure 6 displays the scatter plots for all considered monitoring stations. This figure showed that the best model produces greater correlation coefficient values, and it indicates that the correlation between forecast and actual ozone concentration values is highly significant. In the same way, the forecasted and observed values for the supermodel in each monitoring station are plotted in Figure 7. In Figure 7, forecasts of the best models follow the observed concentration of ozone very closely; from this, we can conclude that the supermodel in each considered station has accurate and efficient forecasts. Thus, from the descriptive statistical analysis, tests, and graphical results, we can conclude that the proposed hybrid combination of time series models is highly efficient and accurate in forecasting hourly ozone concentration.

## 4. Discussion

**${}^{\mathbf{a}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$**, the

**${}^{\mathbf{b}}$STLD${}_{\mathbf{c}}^{\mathbf{c}}$**, the

**${}^{\mathbf{b}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$**, and the

**${}^{\mathbf{c}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$**for the ATE, the CDM, the SB, and the STA, respectively. However, to verify the superiority of these final best models, we compare them with some standard baseline time series models, including parametric autoregressive (PAR), nonparametric autoregressive (NPAR), and autoregressive integrated moving averages (ARIMA) models. For example, the comparative results are presented in Table 6 for all four monitoring stations. The results show that the considered baseline time series models are significantly outperformed by the best-proposed model in each station. In addition, to confirm the dominance of the best-proposed models given in Table 6 for each station, we performed a statistical DM test on each pair of models. The results (p-values) of the DM test are listed in Table 7, indicating that the baseline time series (PAR, NPAR, and ARIMA) models performed poorly in comparison to our best-proposed models in the considered stations at the 5% level of significance. To conclude, based on overall results, the performance measures of accuracy for the proposed methods of forecasting are comparatively better and more efficient than all other benchmark models in the competition.

## 5. Conclusions

**${}^{\mathbf{a}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$**), CDM (

**${}^{\mathbf{b}}$STLD${}_{\mathbf{c}}^{\mathbf{c}}$**), SB (

**${}^{\mathbf{b}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$**), and Santa Anita (

**${}^{\mathbf{c}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$**) stations were presented because they showed the best forecast reflected in the measurement of RMSE, RMSPE, MAE, MAPE, and CC, which were very small compared to the other models. The confirmation of the best models was statistically significant (p < 0.05), being superior to the other models. The graphical representation of the mean errors (RMSE, RMSPE, MAE, MAPE, and CC) for the twenty-seven models at each sampling station presented a better precision for the supermodels compared to the rest of all the models combined. These statistical tests and graphical results show that the proposed forecast methodology is highly accurate and efficient in predicting hourly ozone concentration, which meant that the independent AR, NPAR, and ARIMA models were outperformed by our best models (p < 0.05).

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Ozone concentration in the metropolitan area of Lima (µ$\mathrm{g}/{\mathrm{m}}^{3}$): the hourly ozone concentration of the decomposed time series by the STLD method; ATE (

**a**), Campo de Marte (

**b**), San Borja (

**c**), and Santa Anita (

**d**), in each sub-figure, the top panel shows the long-run trend (${\mathfrak{l}}_{\mathfrak{n}}$), the middle shows the seasonal (${\mathfrak{h}}_{\mathfrak{n}}$) component, and the bottom shows the residual (${\mathfrak{r}}_{\mathfrak{n}}$) component over a year.

**Figure 3.**Ozone concentration in the metropolitan area of Lima (µ$\mathrm{g}/{\mathrm{m}}^{3}$): the hourly ozone concentration time series for ATE (1st panel), Campo de Marte (2nd panel), San Borja (3rd panel), and Santa Anita (4th panel).

**Figure 4.**Ozone concentration (µ$\mathrm{g}/{\mathrm{m}}^{3}$) in four Metropolitan Lima stations: (

**a**) ATE, (

**b**) Campo de Marte, (

**c**) San Borja, and (

**d**) Santa Anita; the RMSPE (1st panel), MAPE (2nd panel), MAE (3rd panel), and RMSE (4th panel) for all twenty-seven combination models using the proposed forecasting methodology.

**Figure 5.**Ozone concentration in four Metropolitan Lima stations (µ$\mathrm{g}/{\mathrm{m}}^{3}$): (

**a**) Ate, (

**b**) Campo de Marte, (

**c**) San Borja, and (

**d**) Santa Anita evaluation measures; the barplot for the best four models among all twenty-seven models.

**Figure 6.**Correlation plots for the ozone concentration (µ$\mathrm{g}/{\mathrm{m}}^{3}$) in all four Metropolitan Lima stations using their respective best hybrid models, including (

**1st**) ATE (

**${}^{\mathbf{a}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$**), (

**2nd**) Campo de Marte (

**${}^{\mathbf{b}}$STLD${}_{\mathbf{c}}^{\mathbf{c}}$**), (

**3rd**), San Borja (

**${}^{\mathbf{b}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$**), and (

**4th**) Santa Anita (

**${}^{\mathbf{c}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$**).

**Figure 7.**Ozone concentration in four Metropolitan Lima stations (µ$\mathrm{g}/{\mathrm{m}}^{3}$): (

**a**) Ate, (

**b**) Campo de Marte, (

**c**) San Borja, and (

**d**) Santa Anita: actual and forecasted ozone concentration values for four of the best models over three weeks.

**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.

Station | ATE | CDM | SB | STA |
---|---|---|---|---|

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% |

**Note:**Campo de Marte (CDM), San Borja (SB), and Santa Anita (STA).

**Table 2.**This table contains descriptive statistics for the time series of ozone concentration and the logarithmic time series of the ozone concentration for all considered monitoring stations.

Measure | Min | Q1 | Median | Mean | Mode | Var | S.D | Skewness | Kurtosis | Q3 | Max | ADF (Statistic) |
---|---|---|---|---|---|---|---|---|---|---|---|---|

ATE | 0.80 | 5.50 | 8.50 | 28.36 | 5.20 | 1606.08 | 40.08 | 1.89 | 2.30 | 29.30 | 165.80 | −8.61 |

log(ATE) | −0.22 | 1.70 | 2.14 | 2.55 | 1.65 | 1.50 | 1.23 | 0.49 | −0.54 | 3.38 | 5.11 | −8.70 |

CDM | 0.80 | 8.98 | 24.50 | 28.13 | 1.00 | 454.02 | 21.31 | 0.53 | −0.67 | 44.03 | 117.10 | −6.03 |

log(CDM) | −0.22 | 2.19 | 3.20 | 2.86 | 0.00 | 1.37 | 1.17 | −0.89 | −0.19 | 3.78 | 4.76 | −6.53 |

SB | 0.20 | 8.30 | 15.10 | 17.09 | 6.50 | 122.05 | 11.05 | 0.83 | 0.51 | 24.00 | 83.90 | −13.02 |

log(SB) | −1.61 | 2.12 | 2.71 | 2.56 | 1.87 | 0.78 | 0.88 | −1.46 | 3.38 | 3.18 | 4.43 | −10.35 |

STA | 0.10 | 1.80 | 6.20 | 10.56 | 0.40 | 149.09 | 12.21 | 1.94 | 5.54 | 14.80 | 152.60 | −16.17 |

log(STA) | −2.30 | 0.59 | 1.82 | 1.59 | −0.92 | 2.02 | 1.42 | −0.44 | −0.61 | 2.69 | 5.03 | −14.38 |

**Table 3.**Ozone concentration in four Metropolitan Lima stations (µ$\mathrm{g}/{\mathrm{m}}^{3}$): out-of-sample one-hour ahead mean forecast error for all models combined with the STL decomposition method.

Station | ATE | Campo de Marte | San Borja | Santa Anita | |||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

S.No | Models | RMSE | RMSPE | MAE | MAPE | CC | RMSE | RMSPE | MAE | MAPE | CC | RMSE | RMSPE | MAE | MAPE | CC | RMSE | RMSPE | MAE | MAPE | CC |

1 | ${}^{\mathbf{a}}$STLD${}_{\mathbf{a}}^{\mathbf{a}}$ | 5.529 | 5.414 | 2.209 | 20.827 | 0.932 | 5.073 | 16.53 | 3.329 | 25.735 | 0.957 | 2.115 | 2.600 | 1.587 | 11.217 | 0.975 | 5.279 | 40.464 | 3.958 | 196.406 | 0.916 |

2 | ${}^{\mathbf{a}}$STLD${}_{\mathbf{b}}^{\mathbf{a}}$ | 5.699 | 4.828 | 2.076 | 18.005 | 0.921 | 5.145 | 16.406 | 3.354 | 24.908 | 0.955 | 2.081 | 2.417 | 1.547 | 10.817 | 0.976 | 5.338 | 40.070 | 3.959 | 188.568 | 0.913 |

3 | ${}^{\mathbf{a}}$STLD${}_{\mathbf{c}}^{\mathbf{a}}$ | 4.675 | 4.628 | 1.913 | 18.257 | 0.947 | 3.993 | 12.117 | 2.719 | 22.96 | 0.973 | 1.818 | 1.854 | 1.376 | 9.768 | 0.982 | 3.958 | 33.264 | 2.965 | 158.543 | 0.954 |

4 | ${}^{\mathbf{a}}$STLD${}_{\mathbf{a}}^{\mathbf{b}}$ | 5.410 | 4.976 | 1.950 | 17.562 | 0.937 | 4.889 | 16.474 | 3.136 | 25.196 | 0.96 | 1.974 | 2.497 | 1.448 | 10.279 | 0.979 | 5.224 | 36.786 | 3.878 | 179.823 | 0.917 |

5 | ${}^{\mathbf{a}}$STLD${}_{\mathbf{b}}^{\mathbf{b}}$ | 5.622 | 4.906 | 1.871 | 16.081 | 0.923 | 4.921 | 16.336 | 3.108 | 24.118 | 0.959 | 1.973 | 2.333 | 1.439 | 10.174 | 0.979 | 5.310 | 36.972 | 3.907 | 172.380 | 0.914 |

6 | ${}^{\mathbf{a}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$ | 4.611 | 4.464 | 1.711 | 14.862 | 0.949 | 3.774 | 11.89 | 2.504 | 21.293 | 0.976 | 1.535 | 1.644 | 1.118 | 7.793 | 0.987 | 3.909 | 30.357 | 2.937 | 148.290 | 0.955 |

7 | ${}^{\mathbf{a}}$STLD${}_{\mathbf{a}}^{\mathbf{c}}$ | 5.529 | 5.414 | 2.209 | 20.827 | 0.932 | 4.717 | 16.474 | 2.848 | 26.253 | 0.963 | 2.115 | 2.600 | 1.587 | 11.217 | 0.975 | 5.277 | 40.431 | 3.959 | 197.407 | 0.916 |

8 | ${}^{\mathbf{a}}$STLD${}_{\mathbf{b}}^{\mathbf{c}}$ | 5.699 | 4.828 | 2.076 | 18.005 | 0.921 | 4.685 | 16.271 | 2.764 | 24.623 | 0.963 | 2.081 | 2.417 | 1.547 | 10.817 | 0.976 | 5.337 | 40.071 | 3.963 | 190.219 | 0.913 |

9 | ${}^{\mathbf{a}}$STLD${}_{\mathbf{c}}^{\mathbf{c}}$ | 4.675 | 4.628 | 1.913 | 18.258 | 0.947 | 3.746 | 11.68 | 2.458 | 20.882 | 0.976 | 1.818 | 1.854 | 1.376 | 9.767 | 0.982 | 3.958 | 33.365 | 2.970 | 159.531 | 0.954 |

10 | ${}^{\mathbf{b}}$STLD${}_{\mathbf{a}}^{\mathbf{a}}$ | 5.607 | 5.313 | 2.277 | 21.015 | 0.933 | 5.485 | 16.957 | 3.697 | 26.817 | 0.949 | 2.213 | 2.872 | 1.664 | 11.793 | 0.974 | 5.319 | 41.263 | 3.977 | 199.487 | 0.915 |

11 | ${}^{\mathbf{b}}$STLD${}_{\mathbf{b}}^{\mathbf{a}}$ | 5.730 | 4.845 | 2.067 | 17.830 | 0.922 | 5.579 | 16.845 | 3.776 | 26.481 | 0.947 | 2.231 | 2.711 | 1.680 | 11.819 | 0.973 | 5.375 | 40.769 | 3.979 | 191.434 | 0.912 |

12 | ${}^{\mathbf{b}}$STLD${}_{\mathbf{c}}^{\mathbf{a}}$ | 4.709 | 4.683 | 2.033 | 19.601 | 0.947 | 4.187 | 12.464 | 2.984 | 24.15 | 0.971 | 1.721 | 2.021 | 1.301 | 9.293 | 0.985 | 3.991 | 33.742 | 2.979 | 160.368 | 0.953 |

13 | ${}^{\mathbf{b}}$STLD${}_{\mathbf{a}}^{\mathbf{b}}$ | 5.509 | 4.895 | 2.047 | 17.770 | 0.937 | 5.247 | 16.859 | 3.458 | 26.089 | 0.954 | 2.132 | 2.803 | 1.597 | 11.384 | 0.976 | 5.257 | 37.440 | 3.894 | 182.865 | 0.916 |

14 | ${}^{\mathbf{b}}$STLD${}_{\mathbf{b}}^{\mathbf{b}}$ | 5.672 | 4.951 | 1.909 | 16.143 | 0.924 | 5.305 | 16.733 | 3.507 | 25.446 | 0.952 | 2.182 | 2.661 | 1.643 | 11.671 | 0.975 | 5.339 | 37.506 | 3.927 | 175.424 | 0.913 |

15 | ${}^{\mathbf{b}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$ | 4.669 | 4.552 | 1.893 | 16.799 | 0.949 | 3.886 | 12.183 | 2.687 | 22.163 | 0.975 | 1.495 | 1.864 | 1.078 | 7.668 | 0.989 | 3.933 | 30.607 | 2.941 | 148.480 | 0.954 |

16 | ${}^{\mathbf{b}}$STLD${}_{\mathbf{a}}^{\mathbf{c}}$ | 5.607 | 5.313 | 2.277 | 21.015 | 0.933 | 4.921 | 16.764 | 2.979 | 26.353 | 0.959 | 2.213 | 2.872 | 1.664 | 11.793 | 0.974 | 5.317 | 41.231 | 3.978 | 200.433 | 0.915 |

17 | ${}^{\mathbf{b}}$STLD${}_{\mathbf{b}}^{\mathbf{c}}$ | 5.730 | 4.845 | 2.067 | 17.830 | 0.922 | 4.921 | 16.575 | 2.995 | 25.119 | 0.959 | 2.231 | 2.711 | 1.680 | 11.820 | 0.973 | 5.374 | 40.771 | 3.984 | 193.141 | 0.912 |

18 | ${}^{\mathbf{b}}$STLD${}_{\mathbf{c}}^{\mathbf{c}}$ | 4.709 | 4.683 | 2.033 | 19.601 | 0.947 | 3.637 | 11.846 | 2.356 | 20.441 | 0.978 | 1.721 | 2.021 | 1.301 | 9.293 | 0.985 | 3.991 | 33.843 | 2.985 | 161.576 | 0.953 |

19 | ${}^{\mathbf{c}}$STLD${}_{\mathbf{a}}^{\mathbf{a}}$ | 5.545 | 5.581 | 2.197 | 20.797 | 0.932 | 5.092 | 16.544 | 3.34 | 25.732 | 0.956 | 2.124 | 2.506 | 1.606 | 11.322 | 0.975 | 3.267 | 25.624 | 2.460 | 125.732 | 0.973 |

20 | ${}^{\mathbf{c}}$STLD${}_{\mathbf{b}}^{\mathbf{a}}$ | 5.678 | 4.753 | 2.081 | 17.964 | 0.922 | 5.166 | 16.42 | 3.363 | 24.898 | 0.955 | 2.075 | 2.316 | 1.554 | 10.821 | 0.976 | 3.289 | 25.472 | 2.435 | 117.338 | 0.971 |

21 | ${}^{\mathbf{c}}$STLD${}_{\mathbf{c}}^{\mathbf{a}}$ | 4.700 | 4.659 | 1.900 | 18.266 | 0.946 | 4.013 | 12.134 | 2.73 | 22.965 | 0.973 | 1.858 | 1.807 | 1.421 | 10.163 | 0.981 | 2.143 | 19.669 | 1.603 | 89.367 | 0.988 |

22 | ${}^{\mathbf{c}}$STLD${}_{\mathbf{a}}^{\mathbf{b}}$ | 5.427 | 5.143 | 1.940 | 17.535 | 0.936 | 4.909 | 16.487 | 3.146 | 25.199 | 0.96 | 1.965 | 2.384 | 1.441 | 10.090 | 0.979 | 3.125 | 20.593 | 2.277 | 99.420 | 0.975 |

23 | ${}^{\mathbf{c}}$STLD${}_{\mathbf{b}}^{\mathbf{b}}$ | 5.601 | 4.817 | 1.882 | 16.179 | 0.924 | 4.942 | 16.35 | 3.118 | 24.128 | 0.959 | 1.947 | 2.212 | 1.415 | 9.869 | 0.979 | 3.190 | 21.490 | 2.298 | 95.063 | 0.972 |

24 | ${}^{\mathbf{c}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$ | 4.636 | 4.480 | 1.704 | 14.985 | 0.948 | 3.794 | 11.906 | 2.514 | 21.323 | 0.976 | 1.559 | 1.568 | 1.136 | 7.897 | 0.987 | 1.969 | 15.924 | 1.462 | 76.261 | 0.989 |

25 | ${}^{\mathbf{c}}$STLD${}_{\mathbf{a}}^{\mathbf{c}}$ | 5.545 | 5.581 | 2.197 | 20.797 | 0.932 | 4.734 | 16.481 | 2.855 | 26.204 | 0.962 | 2.124 | 2.506 | 1.606 | 11.322 | 0.975 | 3.262 | 25.637 | 2.460 | 126.306 | 0.973 |

26 | ${}^{\mathbf{c}}$STLD${}_{\mathbf{b}}^{\mathbf{c}}$ | 5.678 | 4.753 | 2.081 | 17.963 | 0.922 | 4.704 | 16.28 | 2.771 | 24.576 | 0.963 | 2.075 | 2.316 | 1.554 | 10.821 | 0.976 | 3.286 | 25.540 | 2.432 | 116.842 | 0.971 |

27 | ${}^{\mathbf{c}}$STLD${}_{\mathbf{c}}^{\mathbf{c}}$ | 4.700 | 4.659 | 1.900 | 18.267 | 0.946 | 3.762 | 11.689 | 2.464 | 20.847 | 0.976 | 1.858 | 1.807 | 1.421 | 10.163 | 0.981 | 2.141 | 19.925 | 1.605 | 88.958 | 0.988 |

**Table 4.**Ozone concentration in four Metropolitan Lima stations (µ$\mathrm{g}/{\mathrm{m}}^{3}$): mean forecast error of one-hour-ahead post-sample for the best four models among all twenty-seven models.

ATE Station | |||||
---|---|---|---|---|---|

Models | RMSE | RMSPE | MAE | MAPE | CC |

${}^{\mathbf{a}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$ | 4.611 | 4.464 | 1.711 | 14.862 | 0.949 |

${}^{\mathbf{c}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$ | 4.636 | 4.480 | 1.704 | 14.985 | 0.948 |

${}^{\mathbf{c}}$STLD${}_{\mathbf{b}}^{\mathbf{b}}$ | 5.601 | 4.817 | 1.882 | 16.179 | 0.924 |

${}^{\mathbf{a}}$STLD${}_{\mathbf{b}}^{\mathbf{b}}$ | 5.622 | 4.906 | 1.871 | 16.081 | 0.923 |

Campo de Marte Station | |||||

Models | RMSE | RMSPE | MAE | MAPE | CC |

${}^{\mathbf{b}}$STLD${}_{\mathbf{c}}^{\mathbf{c}}$ | 3.637 | 11.846 | 2.356 | 20.441 | 0.978 |

${}^{\mathbf{c}}$STLD${}_{\mathbf{c}}^{\mathbf{c}}$ | 3.762 | 11.689 | 2.464 | 20.847 | 0.976 |

${}^{\mathbf{a}}$STLD${}_{\mathbf{c}}^{\mathbf{c}}$ | 3.746 | 11.68 | 2.458 | 20.882 | 0.976 |

${}^{\mathbf{c}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$ | 3.794 | 11.906 | 2.514 | 21.323 | 0.976 |

San Borja Station | |||||

Models | RMSE | RMSPE | MAE | MAPE | CC |

${}^{\mathbf{b}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$ | 1.495 | 1.864 | 1.078 | 7.668 | 0.989 |

${}^{\mathbf{c}}$STLD${}_{\mathbf{c}}^{\mathbf{c}}$ | 1.559 | 1.568 | 1.136 | 7.897 | 0.987 |

${}^{\mathbf{a}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$ | 1.535 | 1.644 | 1.118 | 7.793 | 0.987 |

${}^{\mathbf{b}}$STLD${}_{\mathbf{c}}^{\mathbf{c}}$ | 1.721 | 2.021 | 1.301 | 9.293 | 0.985 |

Santa Anita Station | |||||

Models | RMSE | RMSPE | MAE | MAPE | CC |

${}^{\mathbf{c}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$ | 1.969 | 15.924 | 1.462 | 76.261 | 0.989 |

${}^{\mathbf{c}}$STLD${}_{\mathbf{c}}^{\mathbf{c}}$ | 2.141 | 19.925 | 1.605 | 88.958 | 0.988 |

${}^{\mathbf{c}}$STLD${}_{\mathbf{c}}^{\mathbf{a}}$ | 2.143 | 19.669 | 1.603 | 89.367 | 0.988 |

${}^{\mathbf{c}}$STLD${}_{\mathbf{b}}^{\mathbf{b}}$ | 3.190 | 21.490 | 2.298 | 95.063 | 0.972 |

**Table 5.**Ozone concentration in four Metropolitan Lima stations (µ$\mathrm{g}/{\mathrm{m}}^{3}$): results (p-value) of the DM test for the best four models given in Table 4.

ATE Station | ||||
---|---|---|---|---|

Models | ${}^{\mathbf{a}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$ | ${}^{\mathbf{c}}$STLD${}_{\mathbf{c}}^{\mathbf{c}}$ | ${}^{\mathbf{c}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$ | ${}^{\mathbf{a}}$STLD${}_{\mathbf{b}}^{\mathbf{b}}$ |

${}^{\mathrm{a}}$STLD${}_{\mathrm{c}}^{\mathrm{b}}$ | - | 0.229 | 0.988 | 0.992 |

${}^{\mathrm{c}}$STLD${}_{\mathrm{c}}^{\mathrm{c}}$ | 0.771 | - | 0.991 | 0.993 |

${}^{\mathrm{c}}$STLD${}_{\mathrm{b}}^{\mathrm{b}}$ | 0.012 | 0.009 | - | 0.332 |

${}^{\mathrm{a}}$STLD${}_{\mathrm{b}}^{\mathrm{b}}$ | 0.008 | 0.007 | 0.668 | - |

Campo de Marte Station | ||||

Models | ${}^{\mathbf{b}}$STLD${}_{\mathbf{c}}^{\mathbf{c}}$ | ${}^{\mathbf{c}}$STLD${}_{\mathbf{c}}^{\mathbf{c}}$ | ${}^{\mathbf{a}}$STLD${}_{\mathbf{c}}^{\mathbf{c}}$ | ${}^{\mathbf{c}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$ |

${}^{\mathbf{b}}$STLD${}_{\mathbf{c}}^{\mathbf{c}}$ | - | 0.965 | 0.944 | 0.963 |

${}^{\mathbf{c}}$STLD${}_{\mathbf{c}}^{\mathbf{c}}$ | 0.036 | - | 0.000 | 0.716 |

${}^{\mathbf{a}}$STLD${}_{\mathbf{c}}^{\mathbf{c}}$ | 0.056 | 1.000 | - | 0.806 |

${}^{\mathbf{c}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$ | 0.037 | 0.284 | 0.194 | - |

San Borja Station | ||||

Models | ${}^{\mathbf{b}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$ | ${}^{\mathbf{c}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$ | ${}^{\mathbf{a}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$ | ${}^{\mathbf{b}}$STLD${}_{\mathbf{c}}^{\mathbf{c}}$ |

${}^{\mathbf{b}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$ | - | 0.989 | 0.945 | 1.000 |

${}^{\mathbf{c}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$ | 0.011 | - | 0.005 | 1.000 |

${}^{\mathbf{a}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$ | 0.055 | 0.996 | - | 1.000 |

${}^{\mathbf{b}}$STLD${}_{\mathbf{c}}^{\mathbf{c}}$ | 0.000 | 0.000 | 0.000 | - |

Santa Anita Station | ||||

Models | ${}^{\mathbf{b}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$ | ${}^{\mathbf{c}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$ | ${}^{\mathbf{a}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$ | ${}^{\mathbf{b}}$STLD${}_{\mathbf{c}}^{\mathbf{c}}$ |

${}^{\mathbf{b}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$ | - | 1.000 | 1.000 | 1.000 |

${}^{\mathbf{c}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$ | 0.000 | - | 0.704 | 1.000 |

${}^{\mathbf{a}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$ | 0.000 | 0.296 | - | 1.000 |

${}^{\mathbf{b}}$STLD${}_{\mathbf{c}}^{\mathbf{c}}$ | 0.000 | 0.000 | 0.000 | - |

**Table 6.**Ozone concentration in four Metropolitan Lima stations (µ$\mathrm{g}/{\mathrm{m}}^{3}$): mean accuracy measures of the proposed versus the baseline models.

ATE Station | |||||
---|---|---|---|---|---|

Models | RMSE | RMSPE | MAE | MAPE | CC |

${}^{\mathbf{a}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$ | 4.611 | 4.464 | 1.711 | 14.862 | 0.949 |

PAR | 5.607 | 5.313 | 2.277 | 21.015 | 0.933 |

NPAR | 5.730 | 4.845 | 2.067 | 17.830 | 0.922 |

ARIMA | 4.709 | 4.683 | 2.033 | 19.601 | 0.947 |

Campo de Marte Station | |||||

Models | RMSE | RMSPE | MAE | MAPE | CC |

${}^{\mathbf{b}}$STLD${}_{\mathbf{c}}^{\mathbf{c}}$ | 3.637 | 11.846 | 2.356 | 20.441 | 0.978 |

PAR | 5.485 | 16.957 | 3.697 | 26.817 | 0.949 |

NPAR | 5.579 | 16.845 | 3.776 | 26.481 | 0.947 |

ARIMA | 4.187 | 12.464 | 2.984 | 24.150 | 0.971 |

San Borja Station | |||||

Models | RMSE | RMSPE | MAE | MAPE | CC |

${}^{\mathbf{b}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$ | 1.495 | 1.864 | 1.078 | 7.668 | 0.989 |

PAR | 2.213 | 2.872 | 1.664 | 11.793 | 0.974 |

NPAR | 2.231 | 2.711 | 1.680 | 11.819 | 0.973 |

ARIMA | 1.721 | 2.021 | 1.301 | 9.293 | 0.985 |

Santa Anita Station | |||||

Models | RMSE | RMSPE | MAE | MAPE | CC |

${}^{\mathbf{c}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$ | 1.969 | 15.924 | 1.462 | 76.261 | 0.989 |

PAR | 5.319 | 41.263 | 3.977 | 199.487 | 0.915 |

NPAR | 5.375 | 40.769 | 3.979 | 191.434 | 0.912 |

ARIMA | 3.991 | 33.742 | 2.979 | 160.368 | 0.953 |

**Table 7.**Ozone concentration in four Metropolitan Lima stations (µ$\mathrm{g}/{\mathrm{m}}^{3}$): results (p-value) of the DM test for the final best-proposed model versus the baseline models given in Table 6.

ATE Station | ||||
---|---|---|---|---|

Models | ${}^{\mathbf{a}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$ | PAR | NPAR | ARIMA |

${}^{\mathbf{a}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$ | - | 0.999 | 0.995 | 0.927 |

PAR | 0.001 | - | 0.662 | 0.001 |

NPAR | 0.005 | 0.338 | - | 0.006 |

ARIMA | 0.073 | 0.999 | 0.995 | - |

Campo de Marte Station | ||||

Models | ${}^{\mathbf{b}}$STLD${}_{\mathbf{c}}^{\mathbf{c}}$ | PAR | NPAR | ARIMA |

${}^{\mathbf{b}}$STLD${}_{\mathbf{c}}^{\mathbf{c}}$ | - | 1.000 | 1.000 | 1.000 |

PAR | 0.000 | - | 0.926 | 0.000 |

NPAR | 0.000 | 0.074 | - | 0.000 |

ARIMA | 0.000 | 1.000 | 1.000 | - |

San Borja Station | ||||

Models | ${}^{\mathbf{b}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$ | PAR | NPAR | ARIMA |

${}^{\mathbf{b}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$ | - | 1.000 | 1.000 | 1.000 |

PAR | 0.000 | - | 0.907 | 0.000 |

NPAR | 0.000 | 0.093 | - | 0.000 |

ARIMA | 0.000 | 1.000 | 1.000 | - |

Santa Anita Station | ||||

Models | ${}^{\mathbf{b}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$ | PAR | NPAR | ARIMA |

${}^{\mathbf{b}}$STLD${}_{\mathbf{c}}^{\mathbf{b}}$ | - | 1.000 | 1.000 | 1.000 |

PAR | 0.000 | - | 0.995 | 0.000 |

NPAR | 0.000 | 0.005 | - | 0.000 |

ARIMA | 0.000 | 1.000 | 1.000 | - |

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## Share and Cite

**MDPI and ACS Style**

Carbo-Bustinza, N.; Iftikhar, H.; Belmonte, M.; Cabello-Torres, R.J.; De La Cruz, A.R.H.; López-Gonzales, J.L.
Short-Term Forecasting of Ozone Concentration in Metropolitan Lima Using Hybrid Combinations of Time Series Models. *Appl. Sci.* **2023**, *13*, 10514.
https://doi.org/10.3390/app131810514

**AMA Style**

Carbo-Bustinza N, Iftikhar H, Belmonte M, Cabello-Torres RJ, De La Cruz ARH, López-Gonzales JL.
Short-Term Forecasting of Ozone Concentration in Metropolitan Lima Using Hybrid Combinations of Time Series Models. *Applied Sciences*. 2023; 13(18):10514.
https://doi.org/10.3390/app131810514

**Chicago/Turabian Style**

Carbo-Bustinza, Natalí, Hasnain Iftikhar, Marisol Belmonte, Rita Jaqueline Cabello-Torres, Alex Rubén Huamán De La Cruz, and Javier Linkolk López-Gonzales.
2023. "Short-Term Forecasting of Ozone Concentration in Metropolitan Lima Using Hybrid Combinations of Time Series Models" *Applied Sciences* 13, no. 18: 10514.
https://doi.org/10.3390/app131810514