**Figure 1.**
Classification of time series models. TSM, time series model; AR, autoregressive; MA, moving average; ARMA, autoregressive moving average; ARMAX, autoregressive moving average with exogenous variables; ARIMAX, autoregressive integrated moving average with exogenous variables; VARX, vector autoregressive with exogenous variables; VARMA, vector autoregressive moving average; NARX, nonlinear autoregressive with exogenous variables; NARMAX, nonlinear autoregressive moving average with exogenous variables; SETAR, self-exciting threshold autoregressive; NNNAR, neural network nonlinear autoregressive.

**Figure 1.**
Classification of time series models. TSM, time series model; AR, autoregressive; MA, moving average; ARMA, autoregressive moving average; ARMAX, autoregressive moving average with exogenous variables; ARIMAX, autoregressive integrated moving average with exogenous variables; VARX, vector autoregressive with exogenous variables; VARMA, vector autoregressive moving average; NARX, nonlinear autoregressive with exogenous variables; NARMAX, nonlinear autoregressive moving average with exogenous variables; SETAR, self-exciting threshold autoregressive; NNNAR, neural network nonlinear autoregressive.

**Figure 2.**
(**a**) The locations of air quality monitoring stations (AQMSs) in Sheffield; (**b**) annual mean NO_{2} levels (µg/m^{3}) measured by low-cost sensors (LCSs) and Automatic Urban and Rural Network (AURN) and Sheffield City Council (SCC) AQMSs in Sheffield from August 2019 to September 2020.

**Figure 2.**
(**a**) The locations of air quality monitoring stations (AQMSs) in Sheffield; (**b**) annual mean NO_{2} levels (µg/m^{3}) measured by low-cost sensors (LCSs) and Automatic Urban and Rural Network (AURN) and Sheffield City Council (SCC) AQMSs in Sheffield from August 2019 to September 2020.

**Figure 3.**
Monthly average concentrations (µg/m^{3}) of NO, NO_{2}, NOx and O_{3} in different months of the year from the Devonshire Green AQMS in Sheffield, 2014–2019.

**Figure 3.**
Monthly average concentrations (µg/m^{3}) of NO, NO_{2}, NOx and O_{3} in different months of the year from the Devonshire Green AQMS in Sheffield, 2014–2019.

**Figure 4.**
Three additive components obtained from seasonal-trend decomposition based on loess (STL), decomposition of NO_{2} concentration (µg/m^{3}) collected at the Devonshire Green AQMS, Sheffield.

**Figure 4.**
Three additive components obtained from seasonal-trend decomposition based on loess (STL), decomposition of NO_{2} concentration (µg/m^{3}) collected at the Devonshire Green AQMS, Sheffield.

**Figure 5.**
Auto-correlation function (ACF) plot of the NO_{2} (µg/m^{3}) time series (not differenced) (**a**), differenced once (**b**) and differenced twice (**c**); partial ACF plot (**d**) and once-differenced NO_{2} plot of oscillating pattern around zero with no visible strong trend (**e**) showing that the series is stationary. NO_{2} concentrations used here were collected at the Devonshire Green AQMS, Sheffield.

**Figure 5.**
Auto-correlation function (ACF) plot of the NO_{2} (µg/m^{3}) time series (not differenced) (**a**), differenced once (**b**) and differenced twice (**c**); partial ACF plot (**d**) and once-differenced NO_{2} plot of oscillating pattern around zero with no visible strong trend (**e**) showing that the series is stationary. NO_{2} concentrations used here were collected at the Devonshire Green AQMS, Sheffield.

**Figure 6.**
Different density plots of hourly NO_{2} concentrations (µg/m^{3}) measured by Envirowatch E-MOTEs, August 2019–September 2020 in Sheffield.

**Figure 6.**
Different density plots of hourly NO_{2} concentrations (µg/m^{3}) measured by Envirowatch E-MOTEs, August 2019–September 2020 in Sheffield.

**Figure 7.**
Density plots of hourly NO_{2} concentrations (µg/m^{3}) measured by AQMesh pods from August 2019 to September 2020 in Sheffield.

**Figure 7.**
Density plots of hourly NO_{2} concentrations (µg/m^{3}) measured by AQMesh pods from August 2019 to September 2020 in Sheffield.

**Figure 8.**
Density plots of NO_{2} concentrations (µg/m^{3}) measured by: (**a**) AURN sites-Barnsley road (brn), Tinsley (tin) and Devonshire Green (dg); and (**b**) SCC sites-Firvale (fv), King Ecgbert (ke), Lowfield (lf), Tinsley (tins) and Wicker (wic), from August 2019–September 2020.

**Figure 8.**
Density plots of NO_{2} concentrations (µg/m^{3}) measured by: (**a**) AURN sites-Barnsley road (brn), Tinsley (tin) and Devonshire Green (dg); and (**b**) SCC sites-Firvale (fv), King Ecgbert (ke), Lowfield (lf), Tinsley (tins) and Wicker (wic), from August 2019–September 2020.

**Figure 9.**
Time variation plots of NO_{2} concentrations (µg/m^{3}) in Sheffield at the three AURN AQMS: Barnsley, Tinsley and Devonshire Green.

**Figure 9.**
Time variation plots of NO_{2} concentrations (µg/m^{3}) in Sheffield at the three AURN AQMS: Barnsley, Tinsley and Devonshire Green.

**Figure 10.**
Time variation plots of NO_{2} concentrations (µg/m^{3}) measured at SCC sites: Firvale, King Ecgbert, Lowfield, Tinsley and Wicker, from August 2019–September 2020.

**Figure 10.**
Time variation plots of NO_{2} concentrations (µg/m^{3}) measured at SCC sites: Firvale, King Ecgbert, Lowfield, Tinsley and Wicker, from August 2019–September 2020.

**Figure 11.**
Time variation plots of NO_{2} concentrations (µg/m^{3}) measured by Envirowatch E-MOTEs.

**Figure 11.**
Time variation plots of NO_{2} concentrations (µg/m^{3}) measured by Envirowatch E-MOTEs.

**Figure 12.**
Time variation plots of NO_{2} concentrations (µg/m^{3}) measured by AQMesh pods.

**Figure 12.**
Time variation plots of NO_{2} concentrations (µg/m^{3}) measured by AQMesh pods.

**Figure 13.**
Long-term temporal trend for NO_{2} concentrations (µg/m^{3}) (2000–2019) at the Sheffield Tinsley site, one of the AURN sites. *** shows that the trend is highly significant.

**Figure 13.**
Long-term temporal trend for NO_{2} concentrations (µg/m^{3}) (2000–2019) at the Sheffield Tinsley site, one of the AURN sites. *** shows that the trend is highly significant.

**Figure 14.**
Long-term temporal trend for NO_{2} concentrations (µg/m^{3}) (2014–2019) at the Sheffield Devonshire Green site. *** shows that the trend is highly significant.

**Figure 14.**
Long-term temporal trend for NO_{2} concentrations (µg/m^{3}) (2014–2019) at the Sheffield Devonshire Green site. *** shows that the trend is highly significant.

**Figure 15.**
Predicted vs. observed daily average NO_{2} concentrations (µg/m^{3}) using the ARIMA (1,1,1) model. Model performance was assessed against held-out testing data, which was not used for model fitting. No exogenous variable was used in the model. The solid line represents the 1:1 relationship, whereas the dashed lines represent the 1:0.5 and 1:2 relationships, between observed and predicted concentrations. The dashed lines show the points that are within a factor of two (FAC2).

**Figure 15.**
Predicted vs. observed daily average NO_{2} concentrations (µg/m^{3}) using the ARIMA (1,1,1) model. Model performance was assessed against held-out testing data, which was not used for model fitting. No exogenous variable was used in the model. The solid line represents the 1:1 relationship, whereas the dashed lines represent the 1:0.5 and 1:2 relationships, between observed and predicted concentrations. The dashed lines show the points that are within a factor of two (FAC2).

**Figure 16.**
Scatter plot showing the association of NO_{2} with NO, NOx and O_{3} concentrations (µg/m^{3}) at the Devonshire Green AQMS in Sheffield.

**Figure 16.**
Scatter plot showing the association of NO_{2} with NO, NOx and O_{3} concentrations (µg/m^{3}) at the Devonshire Green AQMS in Sheffield.

**Table 1.**
Names, IDs and annual mean NO_{2} concentrations (µg/m^{3}) measured by low-cost sensors (AQMesh and Envirowatch E-MOTEs) and AURN and Sheffield City Council sites from August 2019 to September 2020.

**Table 1.**
Names, IDs and annual mean NO_{2} concentrations (µg/m^{3}) measured by low-cost sensors (AQMesh and Envirowatch E-MOTEs) and AURN and Sheffield City Council sites from August 2019 to September 2020.

Site Name | Sensor Type | Sensor ID | NO_{2} (µg/m^{3}) |
---|

Brightside Lane | AQMesh | 2003150 | 41.3 |

Saville Street | AQMesh | 2005150 | 46.9 |

Cundy Street | AQMesh | 2007150 | 19.8 |

off Endcliffe Crescent | AQMesh | 2008150 | 12.7 |

Sharrow Vale Rd | AQMesh | 2009150 | 25.1 |

Abbeydale Rd | AQMesh | 2450206 | 43.7 |

London Rd | AQMesh | 2001150 | 14.6 |

Prince of Wales Rd | AQMesh | 2006150 | 28.2 |

Maltravers Rd | AQMesh | 2004150 | 20.8 |

Hunter’s Bar School | AQMesh | 2450204 | 29.0 |

Malin Bridge PS | AQMesh | 1999150 | 22.6 |

Broad Lane | AQMesh | 1998150 | 19.2 |

Carter Knowle Bridge | AQMesh | 2450205 | 37.6 |

Tinsley | AURN | SHE | 22.9 |

Devonshire Green | AURN | SHDG | 19.1 |

Barnsley Road | AURN | SHBR | 32.1 |

Regent Court, E-camp | E_MOTE | 711 | 41.7 |

Leavygreave Road, E-camp | E_MOTE | 712 | 34.2 |

Gell Street, E-camp | E_MOTE | 701 | 21.1 |

Upper Hanover/Henderson’s building | E_MOTE | 702 | 28.7 |

Behind Jessop West | E_MOTE | 703 | 23.7 |

Diamond/Bio-incubator | E_MOTE | 704 | 34.2 |

Broad Lane/St George’s Terrace | E_MOTE | 713 | 44.9 |

Portobello Street, Mappin Street | E_MOTE | 714 | 37.6 |

28 Portobello Street, EC | E_MOTE | 705 | 20.8 |

Howard Street, CC | E_MOTE | 731 | 41.2 |

Arundel Gate/Genting Club | E_MOTE | 732 | 115.6 |

Arundel Gate/Surrey Street | E_MOTE | 733 | 43.5 |

Harmer Lane/Pond Street | E_MOTE | 901 | 44.2 |

Harmer Lane/Sheaf Street | E_MOTE | 902 | 49.9 |

Pond Street/Sheaf Building | E_MOTE | 736 | 38.3 |

Howard Street/Science Park | E_MOTE | 734 | 39.0 |

Paternoster Rows | E_MOTE | 735 | 42.4 |

Sheaf Street/Sheaf Square | E_MOTE | 903 | 107.2 |

Railway Station Taxi rank | E_MOTE | 904 | 136.8 |

Upper Hanover St/Info. Commons | E_MOTE | 707 | 26.6 |

Leavygreave Road/Favell Road | E_MOTE | 708 | 25.7 |

Hounsfield Rd/Hicks Building | E_MOTE | 709 | 24.9 |

Sheffield Children’s Hospital | E_MOTE | 710 | 28.2 |

Robert Hadfield Building | E_MOTE | 706 | 25.2 |

Brook Hill/Firth Court1 | E_MOTE | 737 | 39.3 |

Brook Hill/Firth Court2 | E_MOTE | 738 | 42.9 |

Arts Tower Concourse | E_MOTE | 739 | 34.3 |

Arts Tower Concourse/Library | E_MOTE | 740 | 33.2 |

Firvale | SSC | GH1 | 25.0 |

Tinsley | SSC | GH2 | 24.1 |

Lowfield | SSC | GH3 | 24.6 |

Wicker | SSC | GH4 | 25.9 |

King Ecgbert | SSC | GH5 | 8.1 |

**Table 2.**
Autoregressive integrated moving average (ARIMA) model specification and corresponding Akaike’s information criterion (AIC) values of NO_{2} time series (µg/m^{3}), where p represents the order of the autoregressive, d represents the difference and q represents the MA.

**Table 2.**
Autoregressive integrated moving average (ARIMA) model specification and corresponding Akaike’s information criterion (AIC) values of NO_{2} time series (µg/m^{3}), where p represents the order of the autoregressive, d represents the difference and q represents the MA.

AIC | p | d | q |
---|

8910.88 | 1 | 1 | 1 |

9104.28 | 1 | 2 | 1 |

8910.53 | 2 | 1 | 1 |

8912.42 | 2 | 1 | 2 |

8944.8 | 1 | 0 | 0 |

8941.53 | 1 | 0 | 1 |

8904.29 | 3 | 1 | 1 |

8905.83 | 4 | 1 | 1 |

**Table 3.**
Comparing the performances of different models, including both linear and nonlinear models, using the testing dataset (cross validation). MBE, MAE and RMSE are expressed in µg/m^{3}. r is the value of correlation coefficient.

**Table 3.**
Comparing the performances of different models, including both linear and nonlinear models, using the testing dataset (cross validation). MBE, MAE and RMSE are expressed in µg/m^{3}. r is the value of correlation coefficient.

Model | FAC2 | MBE | MAE | RMSE | r |
---|

SETAR | 0.90 | −0.13 | 8.28 | 10.56 | 0.44 |

NNET | 0.89 | −0.29 | 8.12 | 10.45 | 0.45 |

ARIMA | 0.91 | −0.26 | 6.46 | 8.61 | 0.59 |

**Table 4.**
Estimating the parameters of the ARIMA (1,1,1) model for the NO_{2} concentrations (µg/m^{3}) training dataset.

**Table 4.**
Estimating the parameters of the ARIMA (1,1,1) model for the NO_{2} concentrations (µg/m^{3}) training dataset.

ARIMA Model Applied to Log_NO_{2} with p, d and q, Order of 1, 1, 1 |
---|

Coefficients | AR1 (ϕ) | MA1 (θ) |

| 0.5362 | −0.9511 |

S.E. | 0.0296 | 0.0117 |

Sigma square (σ^{2}) estimated as 0.148: log likelihood = −556.75, AIC = 1117.5 |

**Table 5.**
Estimating the parameters of the ARIMAX (1,1,1) model for the NO_{2} concentrations (µg/m^{3}) training dataset, with NO as exogenous variable.

**Table 5.**
Estimating the parameters of the ARIMAX (1,1,1) model for the NO_{2} concentrations (µg/m^{3}) training dataset, with NO as exogenous variable.

ARIMAX Model Applied to Log_NO_{2} with p, d and q, Order of 1, 1, 1 and Xreg as NO |
---|

Coefficients | AR1 (ϕ) | MA1(θ) | XREG |

| 0.2533 | −0.9689 | 0.4133 |

Sigma square (σ^{2}) estimated as 0.056: log likelihood = 20.82, AIC = −31.64 |

**Table 6.**
Model statistics showing the value of several metrics in assessed the model performance by comparing observed and predicted NO_{2} concentrations (µg/m^{3}) for testing and training data, using NO as exogenous variables. MBE, MAE and RMSE are expressed in µg/m^{3}.

**Table 6.**
Model statistics showing the value of several metrics in assessed the model performance by comparing observed and predicted NO_{2} concentrations (µg/m^{3}) for testing and training data, using NO as exogenous variables. MBE, MAE and RMSE are expressed in µg/m^{3}.

Statistics | FAC2 | MBE | MAE | RMSE | r |
---|

Training data | 0.96 | 0.07 | 5.53 | 7.11 | 0.85 |

Testing data | 0.88 | −6.84 | 7.81 | 10.15 | 0.70 |

**Table 7.**
Estimating the parameters of the ARIMAX (1,1,1) model for the NO_{2} concentrations (µg/m^{3}) training dataset, with NO and O_{3} as exogenous variables.

**Table 7.**
Estimating the parameters of the ARIMAX (1,1,1) model for the NO_{2} concentrations (µg/m^{3}) training dataset, with NO and O_{3} as exogenous variables.

ARIMAX MODEL Applied to log_NO_{2} with p, d and q, Order of 1, 1, 1 and Xreg as NO and O_{3} |
---|

Coefficients | AR1 (ϕ) | MA1(θ) | XREG2 (NOx) | XREG3 (O_{3}) |

| 0.21 | −0.985 | 0.839 | −0.108 |

Sigma square (σ^{2}) estimated as 0.014: log likelihood = 866.86, AIC = −1721.73 |

**Table 8.**
Statistical metrics assessing the model performance by comparing observed and predicted concentrations for both the training and testing datasets, using NO and O_{3} as exogenous variables. MBE, MAE and RMSE are expressed in µg/m^{3}.

**Table 8.**
Statistical metrics assessing the model performance by comparing observed and predicted concentrations for both the training and testing datasets, using NO and O_{3} as exogenous variables. MBE, MAE and RMSE are expressed in µg/m^{3}.

Statistics | FAC2 | MBE | MAE | RMSE | r |
---|

Training data | 0.65 | −10.25 | 10.45 | 11.75 | 0.90 |

Testing data | 0.73 | −7.94 | 9.34 | 9.90 | 0.84 |