# Time Series Forecasting of Air Quality: A Case Study of Sofia City

^{1}

^{2}

^{*}

## Abstract

**:**

_{2}), ozone (O

_{3}) and fine particles (PM2.5). In addition, the method allows us to find out whether the pollutants’ levels exceed the limits prescribed by the World Health Organization (WHO), as well as to investigate the correlation between levels of a given pollutant measured in different air quality stations.

## 1. Introduction

_{3}), nitrogen oxide (NO), nitrogen dioxide (NO

_{2}) and carbon dioxide (CO) [18]. Hidden periodicities of the fine particle (PM10) time series were identified and used to increase the performance of the time series models [19]. The duration of cycle is calculated for observed PM10 levels in London as 365 days corresponding to a year, 7 days corresponding to a weak, 456 days corresponding to 15 months, and 183 days corresponding to 6 months and 25 days. In East Central Florida, nonlinear regression and ARIMA models were used for precipitation chemistry from 1978 to 1997 [20]. The trends in PM2.5 concentrations of Fuzhou, China between August 2014 and July 2016 were studied using the ARIMA [21]. Seasonal fluctuations of two years were identified as a result, where lower concentrations appear in warm days, while higher concentrations appear in cold days. Periodogram-based time series methodology was utilized to find the hidden periodic structure of monthly PM2.5 data, available for Paris between January 2000 and December 2019 [22].

_{2}, O

_{3}and fine particles (PM2.5). The ARIMA method is employed for predictive analytics at five local air quality monitoring stations with the aim of improving forecast accuracy at roadside, rural and urban places.

- Geographic—The city is located in a valley surrounded by mountains, which keep the pollutants over the city for prolonged periods of time, especially during the winter, when fog and thermal inversions appear.
- Domestic heating—Part of the city’s population is still using solid fuel for heating, which is mostly low-quality coal producing a lot of smoke and ash [30].

## 2. Materials and Methods

_{2}, ozone O

_{3}, PM2.5.

_{2.5}are available only for the neighborhood of Hipodruma.

#### 2.1. Data Preparation

#### 2.1.1. Quality of Raw Data

_{2}at “Orlov most” (85.3% of missing values) and “Mladost” (24.1% of missing values) air quality stations and the data related to CO at “Orlov most” (85.0% of missing values) and “Mladost” (18.0% of missing values) air quality stations are discarded. For the rest of the data, the negative values are fixed to 0.0, supposing that there were some bias or failure in the sensors measuring the level of the polluter in a small portion of the time frame.

#### 2.1.2. Imputation of Missing Values

_{2}at “Kopitoto” air quality station there are missing values for the period from 4 p.m. 20 August 2015 until 6 p.m. 14 September 2015. Thus, all the values from the same time frame for the years 2016, 2017, 2018 and 2019 are taken to substitute the missing values. Since the data for these years in this time frame may again have missing values, only the present ones are used, and the missing values are filled by averaging of the values for the corresponding slot in the selected time frames.

#### 2.1.3. Data Granularity

#### 2.2. Analytical Methods

#### 2.2.1. Autocorrelation and Partial Autocorrelation

#### 2.2.2. Stationarity

#### 2.2.3. ARIMA Method

- Number of previous observations (p);
- Degree of differencing (d);
- Size of the moving average (q).

#### 2.2.4. Evaluation Metrics

#### 2.2.5. Software Libraries

## 3. Experimental Results

#### 3.1. Analytical Methods

_{2}data aggregated by 1 h, 3 h and 12 h. The plot of Figure 5 shows a seasonality with lag 24, corresponding to the pick in the autocorrelation lower part, and that is expected because of the repeated daily activity by hour within the city. The second significant pick is observed 24 h after the first one, and namely, near the 48th tick. A similar trend is observed in Figure 6 and Figure 7 and the most significant picks in the autocorrelation part of the plots are at 24/3 = 8 and 24/12 = 2, respectively. As expected, the second significant picks shown in for Figure 6 and Figure 7 are observed at the 16th and 4th ticks, which corresponds to the 2-day lags after the first tick.

_{2}are shown only but one observes similar behavior for the rest of the pollutants.

_{2}measurements in Drujba air quality station. It can be seen that for one 1 h aggregated data there is a direct relationship of the members of the series at and immediately before the 24th tick, which is expected since the granularity of the data by hour and the daily activities withing the city. Similar behavior can be observed in the plots corresponding to the 3 h and 12 h granularity of the data. There are direct relationships of the first lag and the lags on and immediately before the 24/3 = 8th lag for 3 h granularity and the first lag and the lags on and immediately before the 24/12 = 2nd lag for the 12 h granularity. Similarly, as in the autocorrelation part, further picks are observed in the partial autocorrelation plots, which are located immediately before the 48th, 6th and 4th lags.

- If p-value > 0.05 (test value > 5% test critical value), one fails to reject the null hypothesis (H0) and the data is considered as non-stationary;
- If p-value ≤ 0.05 (test value ≤ 5% test critical value), one rejects the null hypothesis (H0) and the data is considered as stationary.

- Value of the test statistic
- The p-value
- Number of lags considered for the test
- The critical value cut-offs

_{2}, for all levels of granularity except Kopitoto for KPSS, all other air quality stations show significant level of signal stationarity. Additionally, that significant stationarity is confirmed by both tests. That is, for all considered air quality stations, the ADF p-value < 0.05, and again for all air quality stations except Kopitoto, the KPSS p-value > 0.05. Furthermore, the last consideration is valid not only for the 5% critical values of the corresponding tests, but also for the further 10% critical values. However, for the station obtained from Kopitoto and the KPSS test, we observe that for all levels of granularity, the test statistic > critical% value and p-value < 0.05, which rejects the null hypothesis, and we conclude that the signal is rather non-stationary.

_{2}dataset for granularities: 1 h with lag difference 24, 3 h with lag difference 8 and 12 h with lag difference 2. In Table 6, only the corresponding test values are shown, where the improvement consists of the reduction of the values of ADF and KPSS tests for the corresponding data granularities and lag differences. For Kopitoto air quality station, the KPSS test, having shown non-stationarity with critical p-value 0.05 before, after the lag difference, has a test value significantly lower than the critical test value. Therefore, the lag difference turned the signals corresponding to Kopitoto from non-stationary to stationary for all granularity levels.

#### 3.2. ARIMA Models

_{2}for optimally chosen parameters p, d and q for granularity 1 h and 3 h, respectively. Table 10 shows the corresponding numerical values.

## 4. Discussion

_{2}measurements for all air quality stations except “Kopitoto”, where ADF shows stationarity and KPSS shows non-stationarity. The stationarity levels are improved after applying the corresponding optimal lag differences for all granularity levels, obtained from the Python’s library pmdarima and optimizing the ADF statistics. The inter- and intra-annual fluctuations in data could be further explored by the least-squares wavelet analysis (LSWA), which avoids interpolation and/or gap filling and decomposes the time series into the time-frequency domain [32]. The SHP representing both inter- and intra-annual fluctuations can be successfully determined in the time series [31]

## 5. Conclusions

_{2}, O

_{3}and PM2.5 at five local air quality monitoring stations. The proposed approach aims to improve forecast accuracy at roadside, rural and urban places. ADF and KPSS statistical tests are successfully applied to improve the stationarity levels of the data.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Premature deaths attributed to PM 2.5 for European countries in 2019, normalized by population (source: European Environmental Agency).

**Figure 3.**Interaction relationships—scatterplot (lower left triangular area) and Pearson correlation values (upper right triangular area) and histogram (on the diagonal) of the O

_{3}at each air quality station measured per hour.

**Figure 4.**Interaction relationships—scatterplot (lower left triangular area) and Pearson correlation values (upper right triangular area) and histogram (on the diagonal) of the NO

_{2}at each air quality station measured per hour.

**Figure 5.**The signal for the different stations corresponding to NO

_{2}measured by 1 h (

**a**) and the corresponding autocorrelation (

**b**).

**Figure 6.**The signal for the different stations corresponding to NO

_{2}measured by 3 h (

**a**) and the corresponding autocorrelation (

**b**).

**Figure 7.**The signal for the different stations corresponding to NO

_{2}measured by 12 h (

**a**) and the corresponding autocorrelation (

**b**).

**Figure 8.**Partial autocorrelation of NO

_{2}in “Drujba” air quality station measured at 1 h granularity (

**a**), 3 h granularity (

**b**) and 12 h granularity (

**c**).

**Figure 9.**The signal for the different stations corresponding to NO

_{2}for granularity 1 h with lag difference 1 (

**a**) and its autocorrelation (

**b**).

**Figure 10.**The signal for the different stations corresponding to NO

_{2}for granularity 3 h with lag difference 1 (

**a**) and its autocorrelation (

**b**).

**Figure 11.**The signal for the different stations corresponding to NO

_{2}for granularity 12 h with lag difference 1 (

**a**) and its autocorrelation (

**b**).

**Figure 12.**Partial autocorrelation of NO

_{2}in “Drujba” air quality station measured at 1 h granularity (

**a**), 3 h granularity (

**b**) and 12 h granularity (

**c**) with lag difference 1.

**Figure 13.**ARIMA model with predictions for NO

_{2}, 1 h granularity, based on grid search and ACF, PACF lags consideration.

**Figure 14.**ARIMA model with predictions for NO

_{2}, 3 h granularity, based on grid search and ACF, PACF lags consideration.

**Table 1.**Availability of polluters regarding districts of and near Sofia City. The available measurements are marked with black circles.

Neighborhood/Polluter | NO_{2} | CO | O_{3} | PM_{2.5} |
---|---|---|---|---|

Hipodruma | ● | ● | ● | ● |

Pavlovo | ● | ● | ● | ○ |

Kopitoto | ● | ● | ● | ○ |

Mladost | ● | ● | ○ | ○ |

Drujba | ● | ○ | ● | ○ |

Nadejda | ● | ○ | ● | ○ |

Orlov most | ● | ● | ○ | ○ |

NO_{2} | Min | Mean | Max | Missing | Negatives |
---|---|---|---|---|---|

Hipodruma | −0.59 | 32.99 | 212.45 | 0.7% | 0.1% |

Pavlovo | −2.93 | 32.01 | 275.2 | 6.0% | 0.4% |

Kopitoto | −1.51 | 4.97 | 72.3 | 9.8% | 0.4% |

Mladost | 0.0 | 30.31 | 229.13 | 24.1% | 0.0% |

Drujba | 0.45 | 25.10 | 185.91 | 0.6% | 0.0% |

Nadejda | 0.0 | 24.56 | 226.55 | 5.0% | 0.0% |

Orlov most | 1.88 | 43.33 | 240.04 | 85.3% | 0.0% |

CO | Min | Mean | Max | Missing | Negatives |
---|---|---|---|---|---|

Hipodruma | 0.0 | 0.63 | 7.86 | 1.9% | 0.0% |

Pavlovo | −0.14 | 0.66 | 7.1 | 3.1% | 0.2% |

Kopitoto | −0.24 | 0.32 | 2.46 | 8.9% | 3.4% |

Mladost | 0.0 | 0.60 | 6.28 | 18.0% | 0.0% |

Drujba | - | - | - | - | - |

Nadejda | - | - | - | - | - |

Orlov most | 0.14 | 0.82 | 7.59 | 85.0% | 0.0% |

O_{3} | Min | Mean | Max | Missing | Negatives |
---|---|---|---|---|---|

Hipodruma | 0.01 | 34.84 | 152.23 | 0.5% | 0.0% |

Pavlovo | −4.43 | 45.30 | 199.3 | 2.8% | 0.5% |

Kopitoto | −7.32 | 79.24 | 195.22 | 10.8% | 0.2% |

Mladost | - | - | - | - | - |

Drujba | 0.0 | 42.07 | 254.0 | 2.5% | 0.0% |

Nadejda | 0.0 | 43.55 | 184.16 | 1.4% | 0.0% |

Orlov most | - | - | - | - | - |

PM2.5 | Min | Mean | Max | Missing | Negatives |
---|---|---|---|---|---|

Hipodruma | 0.0 | 23.95 | 580.27 | 12.0% | 0.0% |

NO_{2}/1 h | ADF | p-Value | Lags | KPSS | p-Value | Lags |
---|---|---|---|---|---|---|

Hipodruma | −15.455 | 0.0 | 52 | 0.165 | 0.1 | 109 |

Pavlovo | −17.265 | 0.0 | 55 | 0.174 | 0.1 | 110 |

Kopitoto | −19.228 | 0.0 | 50 | 5.618 | 0.01 | 82 |

Drujba | −10.814 | 0.0 | 40 | 0.30 | 0.1 | 109 |

Nadejda | −15.580 | 0.0 | 52 | 0.337 | 0.1 | 111 |

NO_{2}/3 h | ADF | p-Value | Lags | KPSS | p-Value | Lags |
---|---|---|---|---|---|---|

Hipodruma | −12.864 | 0.0 | 40 | 0.119 | 0.1 | 68 |

Pavlovo | −12.160 | 0.0 | 40 | 0.123 | 0.1 | 67 |

Kopitoto | −12.449 | 0.0 | 41 | 3.586 | 0.01 | 64 |

Drujba | −10.814 | 0.0 | 40 | 0.191 | 0.1 | 69 |

Nadejda | −10.479 | 0.0 | 40 | 0.220 | 0.1 | 69 |

NO_{2}/12 h | ADF | p-Value | Lags | KPSS | p-Value | Lags |
---|---|---|---|---|---|---|

Hipodruma | −7.306 | 0.0 | 30 | 0.087 | 0.1 | 31 |

Pavlovo | −6.745 | 0.0 | 27 | 0.084 | 0.1 | 32 |

Kopitoto | −6.511 | 0.0 | 25 | 2.361 | 0.01 | 30 |

Drujba | −5.763 | 0.0001 | 28 | 0.116 | 0.1 | 34 |

Nadejda | −6.334 | 0.0 | 28 | 0.141 | 0.1 | 33 |

**Table 9.**Values of the ADF and KPSS tests applied on the NO

_{2}dataset for granularities: 1 h with lag difference 1, 3 h with lag difference 1 and 12 h with lag difference 1.

NO_{2} | ADF 1 h/1 | ADF 3 h/1 | ADF 12 h/1 | KPSS 1 h/1 | KPSS 3 h/1 | KPSS 12 h/1 |
---|---|---|---|---|---|---|

Hipodruma | −33.392 | −25.186 | −17.470 | 0.002 | 0.002 | 0.031 |

Pavlovo | −33.471 | −24.916 | −16.753 | 0.001 | 0.001 | 0.025 |

Kopitoto | −38.502 | −27.869 | −18.243 | 0.003 | 0.008 | 0.055 |

Drujba | −32.502 | −25.718 | −16.040 | 0.002 | 0.001 | 0.021 |

Nadejda | −32.888 | −25.421 | −16.512 | 0.003 | 0.002 | 0.026 |

**Table 10.**Mean absolute error of NO

_{2}for the chosen parameters p, d, q, for 1 h and 3 h granularity.

NO_{2} | MAE 1 h | RMSE 1h | p | d | q | MAE 3 h | RMSE 1h | p | d | q |
---|---|---|---|---|---|---|---|---|---|---|

Hipodruma | 5.12 | 8.17 | 2 | 0 | 2 | 12.2 | 16.23 | 5 | 0 | 1 |

Pavlovo | 5.77 | 8.39 | 3 | 1 | 1 | 14.12 | 19.18 | 3 | 1 | 2 |

Kopitoto | 1.52 | 2.57 | 5 | 1 | 3 | 2.53 | 4.29 | 2 | 1 | 1 |

Drujba | 4.69 | 6.84 | 5 | 1 | 1 | 9.80 | 7.56 | 5 | 0 | 5 |

Nadejda | 4.42 | 7.11 | 2 | 1 | 1 | 9.45 | 7.11 | 4 | 1 | 3 |

NO_{2} | MAE 1 h | RMSE 1h | p | d | q | MAE 3 h | RMSE 1h | p | d | q |
---|---|---|---|---|---|---|---|---|---|---|

Hipodruma | 0.046 | 0.074 | 3 | 0 | 2 | 0.23 | 0.37 | 3 | 0 | 1 |

Pavlovo | 0.044 | 0.064 | 3 | 0 | 1 | 0.25 | 0.40 | 4 | 0 | 1 |

Kopitoto | 0.018 | 0.029 | 4 | 1 | 3 | 0.029 | 0.045 | 3 | 1 | 1 |

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**MDPI and ACS Style**

Marinov, E.; Petrova-Antonova, D.; Malinov, S.
Time Series Forecasting of Air Quality: A Case Study of Sofia City. *Atmosphere* **2022**, *13*, 788.
https://doi.org/10.3390/atmos13050788

**AMA Style**

Marinov E, Petrova-Antonova D, Malinov S.
Time Series Forecasting of Air Quality: A Case Study of Sofia City. *Atmosphere*. 2022; 13(5):788.
https://doi.org/10.3390/atmos13050788

**Chicago/Turabian Style**

Marinov, Evgeniy, Dessislava Petrova-Antonova, and Simeon Malinov.
2022. "Time Series Forecasting of Air Quality: A Case Study of Sofia City" *Atmosphere* 13, no. 5: 788.
https://doi.org/10.3390/atmos13050788