# Mixture Density Conditional Generative Adversarial Network Models (MD-CGAN)
## 5 Conclusions  
In this paper, we present the MD-CGAN model, which offers extensions to the CGAN [2] methodology, particularly to allow for GAN inference of a (multi-modal) posterior distribution over forecast values. Our approach uses a CGAN-like approach to infer the hyperparameters of a Gaussian Mixture model posterior, and we find that the MD-CGAN approach outperforms other methods on all datasets in which noise is prevalent, including all the financial time series investigated, over long-term forecast horizons. Although the method performs well across all tested datasets, in cases in which strong correlations exist in the time series and little noise is expected at test time, alternative forecasting methods (including neural networks and a range of parametric models) may perform equally well. As a GAN model, our approach retains \"adversarial\" robustness to data perturbations when forecasting. We find this is most notable when noise is extensively present in data. Our method is, thus, particularly well suited to dealing with financial data. Furthermore, MD-CGAN can effectively estimate a flexible posterior distribution, in contrast to standard GAN models which (almost without exception) produce point value outputs. Exploiting this rich, multi-model, posterior distribution is not reported in detail here but will be featured in follow-up work. In the experiments reported here, \"adversarial\" perturbations of the observed test data are due to stochastics in the data, rather than carefully planned malicious attacks. Assessing the performance of the model in data with optimized adversarial attacks is an interesting avenue of future research that might be of value in finance, where \"spoofing\" is often prevalent. In this paper, we do not optimize the lookback window or the structure of the MD-CGAN model for each dataset. Optimization of lookback, and performing neural architecture search, is an interesting avenue for future research, likely to improve performance on a dataset by dataset basis. In summary, the MD-CGAN model combines the advantageous features of both flexible probabilistic forecasting and GAN methods. We see this as a particularly useful approach for dealing with time series in \begin{table}
\begin{tabular}{l c c c c c c c c c c} \hline  & **USJC** & **EURUSD FX Rate** & **WTI** & **Nat Gas** & **VIX Index** & **Heating Oil** & **USD Index** & **EM ETF** & **DBQ** & **MDT** \\ \hline \(m=1\) & \(-1.01\) & \(-\)**1.79** & \(-\)**1.75** & \(-1.28\) & \(-\)**1.50** & \(-\)**1.63** & \(-\)0.65 & \(-\)**1.26** & \(-\)0.62 & \(-\)0.82 \\  & (0.02) & (0.08) & (0.09) & (0.04) & (0.05) & (0.06) & (0.05) & (0.05) & (0.01) & (0.02) \\ \(m=2\) & \(-1.05\) & \(-0.98\) & \(-\)1.24 & \(-\)**1.33** & \(-\)1.39 & \(-\)1.37 & \(-\)0.83 & \(-\)1.10 & \(-\)**0.69** & \(-\)0.85 \\  & (0.04) & (0.05) & (0.02) & (0.02) & (0.04) & (0.03) & (0.05) & (0.03) & (0.02) & (0.03) \\ \(m=3\) & \(-\)**1.09** & \(-\)0.67 & \(-\)1.33 & \(-\)1.25 & \(-\)1.48 & \(-\)1.35 & \(-\)**0.86** & \(-\)1.09 & \(-\)0.67 & \(-\)**0.91** \\  & (0.02) & (0.04) & (0.06) & (0.02) & (0.03) & (0.02) & (0.07) & (0.05) & (0.02) & (0.03) \\ \hline \end{tabular}
\end{table} 
Table 4: Negative log-likelihood variation with \(m\), with standard deviations in brackets. Figure 5: Estimated distributions for (**left**) Nat Gas with \(m=2\) and (**right**) VIX index with \(m=1\) over out-of-sample test datasets. True samples are shown as white dots. Red indicates high data likelihood, and blue data low likelihood, under the MD-CGAN model. which noise is significant and for providing robust, long-term forecasts beyond simple point estimates. Conceptualization, J.Z., S.R. ; methodology, J.Z., S.R. ; software, J.Z. ; validation, J.Z., S.R. ; formal analysis, J.Z, S.R. ; Investigation, J.Z., S.R. ; data curation J.Z. ; writing--original draft preparation, J.Z. ; writing--review and editing, S.R. ; visualization, J.Z., S.R. ; supervision, S.R. ; project administration, S.R.. All authors have read and agreed to the published version of the manuscript.