# Mutual Information Input Selector and Probabilistic Machine Learning Utilisation for Air Pollution Proxies

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## Abstract

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## 1. Introduction

_{x}), sulphur dioxide (${\mathrm{SO}}_{2}$), ozone (${\mathrm{O}}_{3}$) and particulate matter (PM) [7]. In practice, a comprehensive measurement covering a large area may not always be possible as air pollutant analysers are generally complicated, bulky and labour-intensive [7,8]. Furthermore, instrument failure, fault in data acquisition or data corruption often result in missing data during research campaigns or continuous measurements [9,10,11]. If the data gap is relatively large, interpolation methods become ineffective. As a result, these problems pose a significant obstacle for comprehensive air pollution data analysis and time series prediction scheme.

## 2. Methods

#### 2.1. Input Selector: Mutual Information

#### 2.2. Probabilistic Machine Learning: Bayesian Neural Networks

## 3. Case Study

## 4. Results and Discussion

#### 4.1. Data Analysis

#### 4.2. Performance Analysis

#### 4.3. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

BNN | Bayesian Neural Network |

MAE | Mean Absolute Error |

ML | Machine Learning |

MI | Mutual Information |

NN | Neural Network |

$\mathrm{NO}$ | Nitric oxide |

${\mathrm{NO}}_{2}$ | Nitrogen dioxide |

${\mathrm{NO}}_{\mathrm{x}}$ | Nitrogen oxides |

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

PCC | Pearson Correlation Coefficient |

ppb | part per billion |

RMSE | Root Mean Squared Error |

SVM | Support Vector Machine |

TNX | $m,p-$Xylene |

WS | Wind Speed |

WHO | World Health Organization |

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**Figure 2.**Venn diagram of entropy properties. $I(X,Y)$ is the mutual information between entropy X and Y, symbolised by $H(X)$ and $H(Y)$, respectively.

**Figure 3.**The difference between standard NN and Dropout NN [53]. Dropout method randomly drops units (along with their connections) from the NN during training, which prevents units from learning too much and thus potentially avoid overfitting. The dropout NN can also be seen as BNN.

**Figure 4.**Map of Jeddah with the sampling site marked with a star [54]. Map data ©Google, 2013 Terra Metrics.

**Figure 5.**Matrix plot correlation analysis between ozone (${\mathrm{O}}_{3}$) and other measured variables. It can be seen that linear correlation (a) does not capture well the relationship between ${\mathrm{O}}_{3}$ and other measured variables, such as m,p-xylene (TNX), $\mathrm{NO}$ and ${\mathrm{NO}}_{\mathrm{x}}$.

**Figure 6.**The scatter plot between ${\mathrm{O}}_{3}$ concentration and several selected variables, including ${\mathrm{NO}}_{\mathrm{x}}$, wind speed, m,p-Xylene (TNX) and ${\mathrm{NO}}_{2}$. R${}^{2}$ is Pearson correlation coefficient (PCC), whereas MI is mutual information score. The concentration unit of trace gases and TNX is in part per billion (ppb) whereas the unit of wind speed is in m/s.

**Figure 7.**Mutual information level between ozone concentration and other measured variables on full day (

**a**), day only (

**b**) and night only (

**c**).

**Figure 8.**Proxy performance metrics and the percentage of data loss evaluated on testing data. In the subplot (

**a**), the green, blue and red constitute the metrics of Mean Absolute Error (MAE), Root-Mean-Squared Error (RMSE) and coefficient of determination (R${}^{2}$), respectively. The subplot (

**b**) presents the percentage of data loss due to the involvement of more input numbers. For both subplots as shown the legend in subplot (

**b**), the symbols ◯, × and ⋆ denotes the performances of single, day and night models, respectively.

**Figure 9.**The time series data of real ${\mathrm{O}}_{3}$ measurement (red), the ${\mathrm{O}}_{3}$ estimation (blue) produced by the BNN and its confidence interval (light green).

**Figure 10.**Regression plot (

**a**) and the prediction bias (

**b**) between the test data of ${\mathrm{O}}_{3}$ concentration measurement and the estimated ${\mathrm{O}}_{3}$ concentration generated by BNN.

**Figure 11.**The average of the measurement and estimation of Ozone concentration from two different models on weekday and weekend.

**Table 1.**The above table shows that the combination of the best four variables to understand the impact of proxy performance by reducing the required instruments for computing ${\mathrm{O}}_{3}$ proxy.

Proxy Inputs | MAE | RMSE | R^{2} |
---|---|---|---|

${\mathrm{NO}}_{2}$–${\mathrm{NO}}_{\mathrm{x}}$ | 7.154 | 86.621 | 0.703 |

${\mathrm{NO}}_{2}$–${\mathrm{NO}}_{\mathrm{x}}$–WS | 5.594 | 50.409 | 0.83 |

${\mathrm{NO}}_{2}$–${\mathrm{NO}}_{\mathrm{x}}$–WS–TNX | 5.192 | 48.564 | 0.84 |

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## Share and Cite

**MDPI and ACS Style**

Zaidan, M.A.; Dada, L.; Alghamdi, M.A.; Al-Jeelani, H.; Lihavainen, H.; Hyvärinen, A.; Hussein, T.
Mutual Information Input Selector and Probabilistic Machine Learning Utilisation for Air Pollution Proxies. *Appl. Sci.* **2019**, *9*, 4475.
https://doi.org/10.3390/app9204475

**AMA Style**

Zaidan MA, Dada L, Alghamdi MA, Al-Jeelani H, Lihavainen H, Hyvärinen A, Hussein T.
Mutual Information Input Selector and Probabilistic Machine Learning Utilisation for Air Pollution Proxies. *Applied Sciences*. 2019; 9(20):4475.
https://doi.org/10.3390/app9204475

**Chicago/Turabian Style**

Zaidan, Martha A., Lubna Dada, Mansour A. Alghamdi, Hisham Al-Jeelani, Heikki Lihavainen, Antti Hyvärinen, and Tareq Hussein.
2019. "Mutual Information Input Selector and Probabilistic Machine Learning Utilisation for Air Pollution Proxies" *Applied Sciences* 9, no. 20: 4475.
https://doi.org/10.3390/app9204475