## 4 Results
### Modelling Process  
As discussed in Section 3.1, the PM\({}_{2.5}\) and PN\({}_{0.3-0.025}\) data sets are used as inputs to the BC proxy, whereas BC data is assigned as the proxy's output. K-fold cross-validation is used to select the training and testing data. The cross-validation is repeated many times with different randomization in each repetition, where the method is known as repeated k-fold cross-validation [67].  
For the WB model, the proposed proxy structure, shown in Equation (1), is first established. Then the model coefficients of \(\beta\) and noise variance (\(\sigma^{2}\)) can be estimated using the NUTS inference, as explained in Section 3.2. Multi-process sampling is performed on two core processors simultaneously (i.e., 2 chains in 2 jobs). Figure 2 shows an example of the posterior distribution of our parameters and the individual samples drawn. These include all model parameters that are \(\beta\) and \(\sigma\). The colors blue and orange show two different samplings performed in parallel. It can be seen that the sampling chains for the individual parameters, shown in the subplots on the right hand side, converge well and are stationary (e.g., no large drifts or other odd patterns). As mentioned previously, the predictive distributions can then be obtained by generating data from the models using the estimated posterior distributions.  
For the BB model, the data needs to first be normalized. In this way, the weight-input product is guaranteed to be small when initialising the network weights to small random values. The magnitudes of the weights also have a consistent meaning [59]. The next step is to setup the many possibilities of the BNN structures, including the different number of hidden layers, different activation functions on hidden layers, such as rectified linear unit (ReLU), sigmoid and hyperbolic tangent function (tanh) [68], and the number of neurons on each layer. We start the training from the simplest BNN structure first, then the results are validated using the testing data and the performance is recorded. The training and validation processes are performed iteratively by increasing the complexity of the BNN structure to find the best BNN structure.  
For the BB model, it is not possible to run an MCMC sampler, such as NUTS, because the sampling will become very slow as the model is scaled up to deeper architectures with more layers and/or the number of neurons increases. Instead, the ADVI variational inference algorithm is used as mentioned in Section 3.3. The ADVI is based on a mean-field approximation such that the correlations in the  
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.  
posterior are neglected, with the advantage of being computationally much faster and scaling well to higher dimensions. The \"brute-force\" method is applied to find the best BNN structure using performance metrics, which are also used for model evaluation, as described in Section 4.2. It is then found that the most optimal BNN structure is one single hidden layer network with a ReLU function with 100 neurons.  
Figure 3 shows the procedure of the WB and BB model developments. The database is established using the measurement campaign data as explained in Section 2. The data is then divided into training and testing data using repeated k-fold cross-validation. The parameter \(k\) is chosen between 2 and 10 and each process is repeated 50 times with different randomisation. The key differences between WB and BB development are in the normalisation method for the data sets applied to the BB model and the search of optimal structure for the BB model. The approaches utilised are computationally demanding, therefore these modelling process are performed in a super-cluster, provided by CSC-IT Center for Science Ltd. [69].