# Bayesian Proxy Modelling for Estimating Black Carbon Concentrations using White-Box and Black-Box Models

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## Abstract

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## 1. Introduction

#### 1.1. Motivation

#### 1.2. Data-Driven Air Pollution Models

## 2. Case Study: Jordan Air Pollution Measurement Campaign

## 3. Methods: Bayesian Modelling

#### 3.1. Features Analysis

#### 3.2. Bayesian Model: White Box

#### 3.2.1. Prior Distribution

#### 3.2.2. Likelihood Function

#### 3.2.3. Posterior and Predictive Distributions

#### 3.3. Bayesian Model: Black Box

## 4. Results

#### 4.1. Modelling Process

#### 4.2. Performance Analysis

#### 4.3. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ADVI | Automatic Differentiation to Variational Inference |

AR | Auto Regressive |

ARX | Auto Regressive eXogenous |

BB | Black Box |

BC | Black Carbon |

BNN | Bayesian Neural Network |

CO | Carbon Monoxide |

MAE | Mean Absolute Error |

MCMC | Markov Chain Monte Carlo |

MENA | Middle East and North Africa |

NO | Nitrogen Oxides |

NUTS | No-U-Turn Sampler |

O${}_{3}$ | Ozone |

PM | Particulate Matter |

PN | Particle Number |

ReLU | Rectified Linear Unit |

RMSE | Root Mean Squared Error |

SO${}_{2}$ | Sulfur Dioxide |

tanh | hyperbolic tangent function |

UFP | Ultra-Fine Particle |

WB | White Box |

WHO | World Health Organization |

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**Figure 1.**Scatter plots between black carbon (BC) mass concentrations and size-fractionated particulate matter (PM) mass/number concentrations measured throughout Jordan. Pearson (r${}_{p}$) and Spearman (r${}_{s}$) correlation coefficients and mutual information (MI) values are shown on the top of each subplot. The red data points represent the used features for the BC proxies’ inputs whereas the blue data points indicate the remaining unused features.

**Figure 2.**The estimated posterior distributions. The left-hand side is the marginal posterior distribution and the right-hand side is the sampling chain of each model parameter. The multi-process sampling is done in parallel and both demonstrate similar results.

**Figure 3.**The procedure of the modelling processes for the white-box (WB) and black-box (BB) proxies.

**Figure 4.**Time-series data of BC mass concentrations in the Amman city center. The blue dot is the measured BC, whereas the red dashed line and the light green region are the measured BC and its $2\sigma $ uncertainty.Two numerical solutions

**Figure 5.**Regression plots between measured and estimated BC mass concentrations and error histogram between measured and estimated BC mass concentrations. (

**a**) Regression plot.(

**b**) Error histogram.

**Figure 6.**Bar chart of the percentage of measured BC data points within three levels of the confidence interval ($\sigma $) of the estimated BC.

**Table 1.**List of aerosol instrumentation and measured variables involved in the mobile air pollution measurement campaign in Jordan.

Measured Variable | Instrument | Measurement Range | Maximum Concentration |
---|---|---|---|

Submicron particle number concentration (cm${}^{-3}$) | CPC 3007-2 (TSI Inc.) P-Trak 8525 (TSI Inc.) | 0.01 $\mathsf{\mu}$m–1 $\mathsf{\mu}$m 0.02 $\mathsf{\mu}$m–1 $\mathsf{\mu}$m | 4 × ${10}^{5}$ cm${}^{-3}$ |

Particle number size distribution (cm${}^{-3}$) | AeroTrak 9306-V2 (TSI Inc.) | 0.3 $\mathsf{\mu}$m–25 $\mathsf{\mu}$m (6 channels) | 210 cm${}^{-3}$ |

PM${}_{\mathrm{x}}$ ($\mathsf{\mu}$g/m${}^{3}$) | DustTrak DRX 8533 (TSI Inc.) | PM${}_{1}$, PM${}_{2.5}$, PM${}_{10}$ | 150 mg/m${}^{3}$ |

Black carbon, BC ($\mathsf{\mu}$g/m${}^{3}$) | microAeth AE51 aethalometer (AethLabs) | Fine fraction | 1 mg/m${}^{3}$ |

**Table 2.**The performance metrics used for evaluation of the developed BC proxies. The real measurement value, the mean of the measurement data points, and the predicted proxy value are symbolized by y, $\overline{y}$, and $\widehat{y}$, respectively. The notations of i and n are the point number and the total predicted values from the proxies, respectively.

Performance Metrics | Formulation |
---|---|

Mean Absolute Error | $\mathrm{MAE}=\frac{{\sum}_{i=1}^{n}|{\widehat{y}}_{i}-{y}_{i}|}{n}$ |

Root Mean Squared Error | $\mathrm{RMSE}=\sqrt{\frac{{\sum}_{i=1}^{n}{({\widehat{y}}_{i}-{y}_{i})}^{2}}{n}}$ |

Coefficient of Determination | ${\mathrm{R}}^{2}=\frac{{\sum}_{i=1}^{n}{({\widehat{y}}_{i}-{y}_{i})}^{2}}{{\sum}_{i=1}^{n}{({y}_{i}-\overline{y})}^{2}}$ |

**Table 3.**Mean absolute error (MAE), root mean squared error (RMSE), and coefficient of determination (R${}^{2}$) of the proposed BC proxies evaluated on BC measured at urban areas across Jordan.

Measurement Locations | MAE ($\mathsf{\mu}$g/m${}^{3}$) | RMSE ($\mathsf{\mu}$g/m${}^{3}$) | R${}^{2}$ | |||
---|---|---|---|---|---|---|

WB | BB | WB | BB | WB | BB | |

Urban (Amman and Zarqa) | 1.834 | 1.777 | 2.111 | 2.061 | 0.76 | 0.77 |

Jordan (including urban) | 1.945 | 1.893 | 2.414 | 2.358 | 0.77 | 0.78 |

Proxy Usage Type | MAE ($\mathsf{\mu}$g/m${}^{3}$) | RMSE ($\mathsf{\mu}$g/m${}^{3}$) | R${}^{2}$ |
---|---|---|---|

Low-cost sensor use (one input) | 2.328 | 2.950 | 0.54 |

“Real" instrument use (two inputs) | 1.945 | 2.414 | 0.77 |

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**MDPI and ACS Style**

Zaidan, M.A.; Wraith, D.; Boor, B.E.; Hussein, T. Bayesian Proxy Modelling for Estimating Black Carbon Concentrations using White-Box and Black-Box Models. *Appl. Sci.* **2019**, *9*, 4976.
https://doi.org/10.3390/app9224976

**AMA Style**

Zaidan MA, Wraith D, Boor BE, Hussein T. Bayesian Proxy Modelling for Estimating Black Carbon Concentrations using White-Box and Black-Box Models. *Applied Sciences*. 2019; 9(22):4976.
https://doi.org/10.3390/app9224976

**Chicago/Turabian Style**

Zaidan, Martha A., Darren Wraith, Brandon E. Boor, and Tareq Hussein. 2019. "Bayesian Proxy Modelling for Estimating Black Carbon Concentrations using White-Box and Black-Box Models" *Applied Sciences* 9, no. 22: 4976.
https://doi.org/10.3390/app9224976