## 3 Results
### Experiments with Dimensionality and Sample Size  
Initial trials with latent space found \(\mathbb{R}^{50}\) to produce the best results on average, especially in the case of DS2 (results of which have already been reported in Table 2). Nevertheless, interesting patterns emerged when experimenting with larger and smaller latent vector inputs. Starting with the case of the smallest target domain (DS1, Table 4), a significant negative correlation was detected when observing rTOST \(p\) values compared with the size of the latent vector over numerous runs ([PERSON]'s \(\tau=-0.44\), \(p=0.001\)). This is also true when considering rTOST absolute difference values (\(\tau=-0.41\), \(p=0.003\)). This correlation highlights larger \(\mathbb{R}^{n}\)s to work best when working with smaller target domains. When training on larger feature spaces (e.g., DS3, Table 5), correlations proved insignificant for both rTOST \(p\) values ([PERSON]'s \(\tau=-0.21\), \(p=0.13\)) and absolute difference calculations (\(\tau=-0.29\), \(p=0.15\)). Nevertheless, while correlations remain insignificant, smaller latent vectors were seen to create more predictable and stable data (Figure 5). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**LSGAN**} & \multicolumn{2}{c}{**WGAN**} & \multicolumn{2}{c}{**WGAN-GP**} \\ \cline{2-7}  & \(|\boldsymbol{|d|}\) & \(\boldsymbol{p}\) & \(|\boldsymbol{|d|}\) & \(\boldsymbol{p}\) & \(|\boldsymbol{|d|}\) & \(\boldsymbol{p}\) \\ \hline \(\mathbb{R}^{25}\) & 0.337 & 0.726 & 0.179 & 0.014 & 0.199 & 0.017 \\ \(\mathbb{R}^{50}\) & 0.116 & 2.0\(\times 10^{-4}\) & 0.157 & 0.004 & 0.203 & 0.026 \\ \(\mathbb{R}^{75}\) & 0.187 & 0.0105 & 0.075 & \(2.9\times 10^{-6}\) & 0.169 & 0.0013 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Best obtained absolute difference and \(p\) value calculations for robust equivalency testing. Values calculated comparing the original target distribution (DS1) with synthetic data generated by different GANs with different sized latent vectors as generator input. Figure 4: Descriptive figure presenting the optimal GAN architecture for geometric morphometric data augmentation. Input (\(I\)) and output (\(O\)) neurons are represented in black with bias (\(b\)) in green. The output of the generator and the input of the discriminator is represented by the n number of dimensions in the \(\mathbb{R}^{n}\) dimensional target feature space. The latent vector (\(L\)) input for the generator must be adjusted according to the dimensionality of the target feature space. Hidden neurons (\(h\)) in layers \(h^{n}\) have a density (\(d\)) that is also conditioned by the shape of the target distribution. Finally, the discriminator has an additional dropout layer (red) with a threshold of \(p=0.4\). In most experiments, 400 epochs were considered enough for GANs to produce realistic data. Moreover, the best results of each GAN began appearing after approximately 100-130 epochs. Performance significantly decreased, however, when trained on the same number of epochs using less data. To test this, subsets of each dataset were taken for experimentation (e.g., 30 out of 60 samples from DS2, Table 6). On all accounts, significant correlations were detected, finding smaller datasets to need more training time in order to obtain optimal results ([PERSON]'s \(r=0.65\), \(p=0.0005\)). Likewise, _LSGAN_ appeared to be the model least affected by dataset size, producing the most realistic distributions in each of the cases (Table 6). While training GANs using 400 epochs is still able to produce realistic data on small datasets, when considering the optimal number of epochs, increasing this number to 1000 produces a significant improvement in results.