# A Gaussian Process Method with Uncertainty Quantification for Air Quality Monitoring
## 1 Introduction  
It is generally believed that urban areas provide better opportunities in terms of economic, political, and social facilities compared to rural areas. As a result, more and more people are migrating to urban areas. At present, more than fifty percent of people worldwide live in urban areas, and this percentage is increasing with time. This has led to several environmental issues in large cities, such as air pollution [1]. Landrigan reported that air pollution caused 6.4 million deaths worldwide in 2015 [2]. According to World Health Organization (WHO) statistical data, three million premature deaths were caused by air pollution worldwide in 2012 [3]. Air pollution has a strong link with dementia, causing 850,000 people to suffer from dementia in the UK [4]. Children growing up in residential houses near busy roads and junctions have a much higher risk of developing various respiratory diseases, including asthma, due to high levels ofair pollution [5]. Polluted air, especially air with high levels of NO, NO\({}_{2}\), and SO\({}_{2}\) and particulate matter (PM\({}_{2.5}\)), is considered the most serious environmental risk to public health in urban areas [6]. Therefore, many national and international organisations are actively working on understanding the behaviour of various air pollutants [7]. This eventually leads to the development of air quality forecasting models so that people can be alerted in time [8]. Essentially, being like a time series, air quality data can be easily processed by models that are capable of time series data processing. For instance, [PERSON] applies an autoregressive moving average (ARMA) model in PM\({}_{2.5}\) concentration prediction in a few Chinese cities [9]. Filtering techniques like Kalman filter are also applied to adjust data biases to improve air quality prediction accuracy [10]. These methods, though with good results reported, are limited by the requirement of a prior model before data processing. Machine learning methods, on the other hand, can learn a model from the data directly. This has enabled them to attract wide attention in recent decades in the field of air quality forecasting. For instance, [PERSON] et al. propose the support vector regression with logarithm preprocessing procedure and immune algorithms (SVRLIA) method, which outperforms general regression neural networks (GRNN) [11] and BackPropagation neural networks (BPNN) [12] in Taiwan air quality forecasting [13]. Recently, inspired by the fact that large scale data are accumulated, deep learning models have been applied in air quality prediction [14]. Some work has added these deep learning models with the ability to quantify uncertainties introduced by inputs. For instance, [PERSON] et al. endow a deep convolutional network with uncertainty quantification, by taking it as an equivalent of a Gaussian processes (GPs) model [15]. This is because GPs predictions are accompanied by confidence intervals, which are usually taken as a metric to measure prediction uncertainties. Applications of GPs in air quality forecasting can be found in [16; 17]. However, the involvement of matrix inversion in GPs limits their application in large-scale datasets [18]. This has inspired research on improving the efficiency of GP models, and a series of efficient GP models have been published [19]. We also proposed an efficient GP model with application in air quality forecasting [17]. Despite the rich number of GP models published, there lacks work that investigates how noise level, hyperparameters, etc. affect the performance of GP models. It is necessary because air quality data vary due to seasonal variations and sensor degradations. A well-trained GP model may not work when fed with new data, simply due to measurement noise level change. By knowing how the variation of GPs performance can be attributed to noise level and hyperparameters, etc., we will still be able perform analysis when noise level or hyperparameters vary. Aiming at this, a general solution is proposed in this paper.