## 3 System Modeling
#### Training the Model  
Let the pair \(\left(x_{input},y_{output}\right)\) represent the values of units in the input and output layers, respectively. Let \(w_{ij}\) represent the weight of the link between units \(i\) and \(j\), and \(b_{i}\) represent the bias of unit \(i\). Let \(w\) be the matrix of the weights of all links and \(b\) be the matrix of all the bias values; then the training of the deep belief architecture (DBA) comprises two steps, namely forward propagation and back propagation processes. Here, forward propagation is used for constructing the structure and activating the output, whilst back propagation is used for adapting the structure and fine-tuning the values of the weight and bias matrices. The forward propagation process can be modeled as a log-likelihood function and is given as:  
\[l\left(w,b,x_{input},y_{output}\right)=\sum_{t=1}^{m}\log\ p\left(v^{\left(t \right)}\right), \tag{2}\]  
where \(v^{\left(t\right)}\) denotes the \(t\)th training data. Here, the DBA training can be seen as a log-linear Markov random field (MRF). As such, \(p\left(v^{\left(t\right)}\right)\) represents the probability of \(\left(v^{\left(t\right)}\right)\). \(m\) is the total amount of training data.  
The purpose of the training process is to minimize \(l\left(w,b,x_{input},y_{output}\right)\), in the back propagation process. Back propagation is an efficient way to compute the partial derivatives of the gradient. It is a computation derived from the chain rule of calculus, and it operates by passing values backwards through the network to compute how the loss is affected by each weight. The link weight \(w\) and the bias \(b\) are adjusted using the gradient descent method. We represent \(w\) and \(b\) as:  
\[w=w+\xi\frac{\partial\left(w,b,x_{input},y_{output}\right)}{\partial w}, \tag{3}\]  
\[b=b+\xi\frac{\partial\left(w,b,x_{input},y_{output}\right)}{\partial b}, \tag{4}\]  
where \(\xi\) is the learning rate of the training process. The Gradient descent is a first-order optimization method, this means it uses only information of the first derivative of the error; hence, it can be used in combination with error back propagation. The challenge with this method is the difficulty of choosing the learning rate so as to get fast learning but at the same time avoid oscillation.  
As the input layer increases (i.e., gets larger) and becomes less connected in high dimensions, the DBA structure becomes inefficient as it fails to capture the spatial features efficiently. The CNNs tend to be powerful in this regard. The convolution operation extracts features of the input, and the  
Figure 2: A centralized system architecture—the dotted circle defines a network of hospitals, whilst the solid circle defines a catchment area of a particular hospital.  
parameters of the convolution operation comprise a set of learnable filters. Let \(W_{l_{1}}\) denote the filters and the \(k\)th filter be represented by \(W_{k}^{l_{i}}\), then the convolution operation outputs a feature map given as:  
\[\begin{split} u_{i,j,k}^{(l_{1})}&=\left(U^{l_{1}-1} \ast W_{k}^{l_{1}}\right)(i,j)+w_{kk}^{l_{1}}\\ &=\sum_{p=1}^{P}\sum_{m=1}^{M^{\prime}}\sum_{n=1}^{N^{\prime}}~{}w _{m,n,p}a_{i+m,j+n,p}^{(l-1)}+w_{kk}^{(l1)}\\ a_{i,j,k}^{(l_{1})}&=f\left(u_{i,j,k}^{l_{1}}\right),\end{split} \tag{5}\]  
where \(f(\cdot)\) is the activation function and \(a_{i,j,k}^{(l_{1})}\) is the activated value of the unit in the \(i\)th row and \(j\)th column of the feature map. Therefore, \(u_{i,j,k}^{(l_{1})}\) is the value before activation. \(w_{kk}^{(l_{1})}\) denotes the bias of the \(k\)th filter and is usually a single numeric value. \(a_{i+m,j+n,p}^{(l-1)}\) is the activated value of the unit in the \((i+m)\)th row and \((j+n)\)th column. The rectified linear units (ReLU) has gained popularity among activation functions. Introduced by [76], ReLU works by thresholding values at 0, i.e., \(f(x)=max(0,x)\)[77].