# Few-Shot Wideband Tympanometry Classification in
## 2 Material and Method  
### Background  
**Classification of WBT in otosclerosis**. The WBT analysis for diagnosis of otosclerosis can be regarded as a binary image classification scenario, and the solution generally consists of two steps: feature extraction and SoftMax classification. Feature extraction is the key to the pipeline, and therefore, the existing works regularly focus on it. The common extracted features include interpretable univariate feature [2], interpretable multivariate features [4], abstract multivariate features [20], and abstract CNN-based features [5], where the first two are manually extracted features, the third is obtained by classical machine learning methods such as principal components analysis, and the last is usually based on deep-learning approaches.  
**Domain adaption**. The data and its corresponding labels from a dataset could be formalized as a sampling of a specific joint distribution. To achieve an effective transfer from the pre-trained task to the concerned task, a domain adaption method should accurately measure the difference between these two corresponding joint distributions and guide the adjustment of the network parameters on the target dataset with this knowledge. The existing methods usually employ the maximum mean discrepancy (MMD) distance [21; 22; 23] or Wasserstein distance [24; 25; 26] to measure the difference between two distributions. Generally, the MMD distance only considers the first-order moment, while the Wasserstein distance is relatively difficult to calculate numerically.  
**GP framework**. When given the mean function \(m(\mathbf{x})\) and the covariance (or kernel) function \(k(\mathbf{x},\mathbf{x}^{\prime})\) of \(f(\mathbf{x})\) as  
\[\begin{split} m(\mathbf{x})&=\mathbb{E}[f(\mathbf{x})],\\ k(\mathbf{x},\mathbf{x}^{\prime})&=\mathbb{E}[(f(\mathbf{x})- m(\mathbf{x}))(f(\mathbf{x}^{\prime})-m(\mathbf{x}^{\prime}))],\end{split} \tag{1}\]  
the GP can be written as,  
\[f(\mathbf{x})\sim\mathcal{GP}(m(\mathbf{x}),k(\mathbf{x},\mathbf{x}^{\prime})). \tag{2}\]The sampling value of GP is a Gaussian random variable, and their collection of those obeys a joint Gaussian distribution. The realistic observation value \(y\) can be modeled as  
\[y=f(\mathbf{x})+\epsilon,\quad\epsilon\sim\mathcal{N}(0,\sigma_{\epsilon}^{2}), \tag{3}\]  
where \(\epsilon\) is the potential error and \(\sigma_{\epsilon}^{2}\) is its variance.  
If given a set of observed data \(\{(\mathbf{x},y)\}\), the conditional probability \(p(y^{\prime}|\{\mathbf{x}\},\{y\},\mathbf{x}^{\prime})\) of the sampling value \(y^{\prime}\) of the new observation point \(\mathbf{x}^{\prime}\) obeys Gaussian distribution \(\mathcal{N}(\hat{\mu},\hat{\Sigma})\)[27] with  
\[\hat{\mu}= K(\mathbf{x}^{\prime},\{\mathbf{x}\})[K(\{\mathbf{x}\},\{\mathbf{x}\})+\sigma_{ \epsilon}^{2}I]^{-1}(\{y\}-\{m(\mathbf{x})\})+m(\mathbf{x}^{\prime}), \tag{4}\] \[\hat{\Sigma}= K(\mathbf{x}^{\prime},\mathbf{x}^{\prime})-K(\mathbf{x}^{\prime},\{\mathbf{x} \})[K(\{\mathbf{x}\},\{\mathbf{x}\})+\sigma_{\epsilon}^{2}I]^{-1}\cdot K(\{\mathbf{x}\}, \mathbf{x}^{\prime}),\]  
where \(K(\cdot)\) is a matrix in which each element consists of the corresponding kernel \(k(\cdot)\).  
For our work, \(m(\mathbf{x})\) is 0 (regular setting) and \(k(\mathbf{x},\mathbf{x}^{\prime})\) is the radial basis function (RBF) kernel:  
\[k(\mathbf{x},\mathbf{x}^{\prime})=\exp(-\frac{1}{2l^{2}}\|\mathbf{x}-\mathbf{x}^{\prime}\|^{ 2}). \tag{5}\]