# Interpretation for Variational Autoencoder Used to Generate Financial Synthetic Tabular Data
## 3 Materials and Methods
### Sensitivity Analysis
\begin{table}
\begin{tabular}{c l} \hline \hline
**Symbol** & **Meaning** \\ \hline N & total number of data points in a dataset \\ \hline M & total number of input features \\ \hline H & total number of means or standard deviations in the latent space of \\  & a VAE where 2H will be the layer size \\ \hline \(x_{n}\) & a random data sample \\ \hline \(x_{m}\) & \(m\)th input feature \\ \hline \(x_{o}\) & Other input features except \(m\)th feature \\ \hline \(s_{m}(x_{n})\) & sensitivity of the latent space with respect to \(m\)th input feature for the \\  & data sample \(x_{n}\) \\ \hline \(s_{m,\mu}(x_{n})\) & sensitivity of means in the latent space of a VAE with respect to \(m\)th input feature for the data sample \(x_{n}\) \\ \hline \(s_{m,\sigma}(x_{n})\) & sensitivity of standard deviations in the latent space of a VAE with \\  & respect to \(m\)th input feature for the data sample \(x_{n}\) \\ \hline \(s_{hm,\mu}(x_{n})\) & sensitivity of \(h\)th mean in the latent space of a VAE (\(\mu_{h}\)) with respect to \(m\)th input feature for the data sample \(x_{n}\) \\ \hline \(s_{hm,\sigma}^{sq}(x_{n})\) & sensitivity of \(h\)th standard deviation in the latent space of a VAE (\(\sigma_{h}\)) with respect to \(m\)th input feature for the data sample \(x_{n}\) \\ \hline \(S_{m,\sigma}^{sq}\) & mean squared sensitivity of means in the latent space with respect to \(m\)th input feature for the entire dataset \\ \hline \(S_{hm,\mu}^{sq}\) & mean squared sensitivity of standard deviations in the latent space with respect to \(m\)th input feature for the entire dataset \\ \hline \(S_{hm,\mu}^{sq}\) & mean squared sensitivity of \(h\)th mean in the latent space of a VAE (\(\mu_{h}\)) with respect to \(m\)th input feature for the entire dataset \\ \hline \(S_{hm,\,\sigma}^{sq}\) & mean squared sensitivity of \(h\)th standard deviation in the latent space of a VAE (\(\sigma_{h}\)) with respect to \(m\)th input feature for the entire dataset \\ \hline \(S_{m}^{sq}\) & mean squared sensitivity of the latent space with respect to \(m\)th input feature for the entire dataset \\ \hline \(F_{m}\) & relative feature importance \\ \hline \(c\) & normalization factor \\ \hline \(S_{m}\) & \(s_{m}(x_{n})\) at a local level or \(S_{m}^{sq}\) at a global level \\ \hline \(i_{m}(x_{n})\) & interactions between the feature \(x_{m}\) and other features \(x_{o}\) for the data \\  & sample, \(x_{n}\), in a tabular dataset \\ \hline \(I_{m}\) & global interactions between \(m\)th feature and other features \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notations used in Section 3.3.  
where \(N\) is the total number of data points in the dataset. Equations (5) and (6) are designed to assess the sensitivity of \(h\)th mean or \(h\)th standard deviation in the latent space of a VAE with respect to \(m\)th input feature across the entire dataset. Then, \(S_{hm,\mu}^{sq}\) for different mean neurons, and \(S_{hm,\sigma}^{sq}\) for different standard deviation neurons in the latent space are designed to be summed up, respectively:  
\[S_{m,\mu}^{sq}=\sum_{h=1}^{H}S_{hm,\mu}^{sq}, \tag{7}\]  
\[S_{m,\sigma}^{sq}=\sum_{h=1}^{H}S_{hm,\sigma}^{sq}. \tag{8}\]  
This counts the overall effects of the feature \(x_{m}\) on means or standard deviations in the latent space of a VAE. Here, \(S_{m}^{sq}\) is defined as representing the sensitivity of the latent space with respect to a specific input feature in this paper and can be obtained by adding the sensitivity of means in the latent space with respect to that specific feature and the sensitivity of standard deviations in the latent space with respect to that specific feature together:  
\[S_{m}^{sq}=S_{m,\mu}^{sq}+S_{m,\sigma}^{sq}. \tag{9}\]  
If the sensitivity \(S_{m}^{sq}\) is close to 0, it indicates that changes in the input feature \(x_{m}\) have a negligible impact on the outputs, meaning this input feature may be irrelevant.