# Mixture Density Conditional Generative Adversarial Network Models (MD-CGAN)
## 4 Experiments
### Multi-Modal Posterior Predictions  
Finally, we compare the performance of MD-CGAN over varying numbers of mixture components. In all the previous experiments, we set \(m=1\) (hence, the model produced a single predictive Gaussian posterior). Here, we briefly present the results for the finance datasets with \(m\in\{1,2,3\}\). We report (negative) log-likelihood measures (as we do not compare against point-value models in this section) and consider one-step forecasts on all the datasets. Table 4 presents the performance across datasets for varying numbers of mixture components in the posterior prediction. Figure 5 shows the predicted distribution for the test datasets for Nat Gas and Vix Index.  
We note performance improvement for some datasets for \(m>1\). We do not attempt to infer \(m\), though choosing its value based on performance on a set of cross-validation data would be an option, as would enforcing regularization over the mixture model  
\begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline  & **0\% Noise** & **5\% Noise** & **10\% Noise** & **15\% Noise** & **20\% Noise** & **25\% Noise** & **30\% Noise** \\ \hline AR(0) & 0.0080 & 0.0109 & 0.0200 & 0.0262 & 0.0494 & 0.0795 & 0.0974 \\  & (0.0000) & (0.0000) & (0.0000) & (0.0000) & (0.0000) & (0.0000) & (0.0000) \\ AR(5) & **0.0068** & **0.0081** & 0.0137 & 0.0138 & 0.0226 & 0.0314 & 0.0409 \\  & (0.0000) & (0.0000) & (0.0000) & (0.0000) & (0.0000) & (0.0000) & (0.0000) \\ SNN & 0.0114 & 0.0132 & 0.0154 & 0.0201 & 0.0335 & 0.0401 & 0.0536 \\  & (0.0004) & (0.0006) & (0.0008) & (0.0009) & (0.0014) & (0.0022) & (0.0025) \\ CGAN & 0.0137 & 0.0143 & 0.0149 & 0.0161 & 0.0228 & 0.0269 & 0.0266 \\  & (0.0002) & (0.0008) & (0.0012) & (0.0010) & (0.0022) & (0.0021) & (0.0023) \\ MDN & 0.0105 & 0.0140 & 0.0161 & 0.0263 & 0.0384 & 0.0592 & 0.0758 \\  & (0.0000) & (0.0008) & (0.0013) & (0.0013) & (0.0018) & (0.0023) & (0.0017) \\ MD-CGAN & 0.0093 & 0.0096 & **0.0113** & **0.0126** & **0.0159** & **0.0194** & **0.0203** \\  & (0.0001) & (0.0002) & (0.0002) & (0.0002) & (0.0003) & (0.0006) & (0.0007) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Sunspot data: MSE variation with added noise level, with standard deviations in parentheses.  
\begin{table}
\begin{tabular}{l c c c c c c c c c} \hline \hline  & **USIJC** & **EURUSD FX Rate** & **WTI** & **Nat Gas** & **VIX Index** & **Heating Oil** & **USD Index** & **EM ETF** & **DBQ** & **MDT** \\ \hline AR(5) & 0.78 & 1.91 & 0.85 & 1.01 & 0.71 & 0.82 & 1.24 & 0.89 & 0.56 & 0.65 \\ SNN & 0.79 & 1.25 & 0.89 & 0.94 & 0.71 & 0.93 & 1.34 & 0.82 & 0.41 & 0.64 \\ CGAN & 0.77 & 0.85 & 1.53 & 1.07 & 0.91 & **0.54** & 1.37 & 0.69 & 0.66 & 0.69 \\ MDN & 0.84 & 3.48 & 1.48 & 1.13 & 0.77 & 0.89 & 0.68 & 0.81 & 0.45 & 0.62 \\ MD-CGAN & **0.73** & **0.76** & **0.80** & **0.82** & **0.66** & 0.59 & **0.54** & **0.65** & **0.38** & **0.61** \\ \hline \hline \end{tabular}
\end{table}
Table 3: Ratio of model MSE to martingale, AR(0), baseline model over 10-week forecast horizon.  
posterior, through more extensive use of Bayesian inference. We leave these extensions for future research.