# Sleep Apnea Classification Using the Mean Euler-Poincare Characteristic and AI Techniques
## 2 Mathematical Framework
### Euler-Poincare Characteristic  
Let \(\phi(A_{\lambda}(\theta,\mathcal{D}))\) represent the EPC of the set \(A_{\lambda}(\theta,\mathcal{D})\). For multiple trials, if the expected value \(\mathcal{E}\{\phi(A_{\lambda}(\theta,\mathcal{D}))\}\) is computable, the random field theory imposes specific regularity conditions on \(\theta(X,\xi)\). These conditions ensure that the realizations of \(\theta\) are smooth and that the boundary \(\partial\mathcal{D}\) is smooth as well. Thus, \(\mathcal{D}\) is regarded as a regular \(C^{2}\) domain within a compact subset of \(\mathcal{R}^{2}\), bounded by a smooth, one-dimensional \(C^{2}\) manifold \(\partial\mathcal{D}\). Consider \(\theta(\mathcal{X},\xi)\), where \(\mathcal{X}=[\chi_{1},\chi_{2}]\subset\mathbb{R}^{2}\), as a stationary, non-isotropic random field. Define \(\dot{\theta}_{i}(\mathcal{X},\xi)=\frac{\partial\theta(\mathcal{X},\xi)}{ \partial\dot{\chi}_{i}}\) and \(\ddot{\theta}_{ij}(\mathcal{X},\xi)=\frac{\partial^{2}\theta(\mathcal{X}, \xi)}{\partial(\mathcal{X},\partial\dot{\chi}_{i})}\), where \(i,j=1,2\). The moduli of continuity of \(\dot{\theta}_{j}(\mathcal{X},\xi)\), and \(\ddot{\theta}_{jk}(\mathcal{X},\xi)\) inside \(\mathcal{D}\), are given by the following:  
\[\xi_{j}(h)=\sup_{\|\mathcal{X}-\mathcal{Y}\|<h,\xi}\big{|}\dot{\theta}_{j}( \mathcal{X},\xi)-\dot{\theta}_{j}(\mathcal{Y},\xi)\big{|},\]  
and  
\[\xi_{jk}(h)=\sup_{\|\mathcal{X}-\mathcal{Y}\|<h,\xi}\big{|}\dot{\theta}_{jk}( \mathcal{X},\xi)-\ddot{\theta}_{jk}(\mathcal{Y},\xi)\big{|},\]  
respectively. To ensure that the realizations of \(\theta(\mathcal{X},\xi)\) are sufficiently smooth, the following conditions must be met:  
* **C1** : The maximum of \(\xi_{j}(h)\) and \(\xi_{jk}(h)\) should be asymptotically negligible, specifically \(o(h^{N})\), as \(h\to 0\). * **C2:** The Hessian \(\bar{\theta}\) of \(\dot{\theta}_{jk}\) must have a finite variance conditional on \(\theta\), where \(\dot{\theta}\) is the gradient of \(\dot{\theta}_{j}\). * **C3:** The densities of \(\theta\) and \(\dot{\theta}\) must be uniformly bounded for all \(\mathcal{X}\in\mathcal{D}\). At a boundary point \(\mathcal{X}\) on \(\partial\mathcal{D}\):
* \(\dot{\theta}_{\perp}\) is the gradient of \(\theta\) in the direction perpendicular to \(\partial\mathcal{D}\). * \(\dot{\theta}_{\mathcal{D}}\) is the gradient in the tangent direction within the plane of \(\partial\mathcal{D}\). * \(\ddot{\theta}_{\mathcal{D}}\) denotes the \(1\times 1\)-Hessian matrix in the tangent plane. * \(r\) is the \(1\times 1\) matrix representing the internal curvature of \(\partial\mathcal{D}\). Using the sign function, denoted as \(sign\), and following [PERSON]'s notation [44], which assigns a value of \(1\) to a true expression in parentheses, and \(0\) if false, the EPC is then given by the following: \[ \begin{split}\Psi(\mathcal{A}_{\lambda}(\theta,\mathcal{D}))& =\Sigma_{\mathcal{X}\in\mathcal{D}}(\theta\geq\lambda)(\dot{\theta}=0) sign[\det(-\bar{\theta})]\\ &+\Sigma_{\mathcal{X}\in\mathcal{D}}(\theta\geq\lambda)(\dot{ \theta}_{\mathcal{D}}=0)(\dot{\theta}_{\perp}\leq 0)\times\text{sign}[\det(-\bar{ \theta}_{\mathcal{D}})-\dot{\theta}\perp r)],\end{split} \] (7) with a probability of one. In \(\mathcal{R}^{2}\), integral geometry characterizes \(\varphi(\mathcal{A}_{\lambda}(\theta,\mathcal{D}))\) as the difference between the count of connected components and the number of voids in \(\mathcal{A}_{\lambda}(\theta,\mathcal{D})\). Moreover, the expected value of \(\varphi(\mathcal{A}_{\lambda}(\theta,\mathcal{D}))\) over multiple realizations is as follows: \[ \begin{split}\mathbb{E}\{\Psi(\mathcal{A}_{\lambda}(\theta, \mathcal{D}))\}&=\int_{\mathcal{D}}\mathbb{E}\{(\theta\geq \lambda)\det(-\bar{\theta})|\dot{\theta}=0\}\beta(0)dX\\ &\qquad+\int_{\partial\mathcal{D}}\mathbb{E}\{(\theta\geq \lambda)(\dot{\theta}_{\perp}<0)\det(-\bar{\theta}_{\mathcal{D}}-\dot{\theta} \perp r)|\dot{\theta}_{\mathcal{D}}=0\}\times\beta_{\mathcal{D}}(0)d\mathcal{ X},\end{split} \] (8) where \(\beta(.