# Locally Scaled and Stochastic Volatility Metropolis-Hastings Algorithms
## 4 Results and Discussion
NUTS outperforms on the USDZAR dataset due to the lower kurtosis on this dataset as seen in Table 1. NUTS appears to perform poorly on the datasets that have a high kurtosis. Figure 2 shows that the SVMH algorithm converges to the same level as NUTS on the majority of the datasets. On the other hand, the MH and LSMH algorithms converge to higher (i.e., less optimal) negative log-likelihoods. Figure 3 shows that NUTS produces the lowest autocorrelations on all the targets except on the jump-diffusion datasets. MH produces the highest autocorrelations on the majority of the targets. The autocorrelations on the SVMH algorithm are higher than NUTS and MALA but approach zero quicker than for MH across the targets considered. Note that these are auto correlations after the burn-in period. Table 3 and 4 show that, as expected, MH produces the lowest execution time with SVMH being a close second. LSMH is slower than SVMH due to the computation of the Hessian matrix at each state. The NUTS algorithm has the highest execution time on the jump diffusion datasets. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{5}{c}{**MTN dataset**} \\ \hline  & \(m\overline{\text{ESS}}\) & \(t\) (in sec) & \(m\overline{\text{ESS}/t}\) & **NLL (Train)** & **NLL (Test)** \\ \hline NUTS & **314** & 65 & 4.77 & \(-\)**2153** & \(-\)174 \\ MALA & 45 & 11 & 4.14 & \(-\)**2153** & \(-\)174 \\ MH & 0 & **5** & 0.00 & \(-\)1981 & \(-\)173 \\ LSMH & 36 & 45 & 0.82 & \(-\)2056 & \(-\)**180** \\ SVMH & 37 & **5** & 7.56 & \(-\)2144 & \(-\)174 \\ \hline  & \multicolumn{5}{c}{**S\&P 500 dataset**} \\ \hline NUTS & **326** & 209 & 1.56 & \(-\)**2942** & \(-\)278 \\ MALA & 30 & 11 & 2.64 & \(-\)2910 & \(-\)286 \\ MH & 0 & **5** & 0.73 & \(-\)2544 & \(-\)283 \\ LSMH & 37 & 51 & 0.00 & \(-\)2782 & \(-\)**300** \\ SVMH & 35 & 6 & **6.27** & \(-\)2911 & \(-\)286 \\ \hline  & \multicolumn{5}{c}{**Bitcoin dataset**} \\ \hline NUTS & **247** & 426 & 0.58 & \(-\)2387 & \(-\)286 \\ MALA & 47 & 11 & 4.11 & \(-\)2315 & \(-\)**291** \\ MH & 0 & **5** & 0.00 & \(-\)2282 & \(-\)286 \\ LSMH & 39 & 50 & 0.78 & \(-\)2286 & \(-\)289 \\ SVMH & 34 & 6 & **5.94** & \(-\)2325 & \(-\)**291** \\ \hline  & \multicolumn{5}{c}{**USDZAR dataset**} \\ \hline NUTS & **1302** & 118 & **52.76** & \(-\)4457 & \(-\)**489** \\ MALA & 54 & 11 & 4.61 & \(-\)**4272** & \(-\)475 \\ MH & 0 & **5** & 0.00 & \(-\)3 978 & \(-\)446 \\ LSMH & 37 & 52 & 0.72 & \(-\)**4272** & \(-\)475 \\ SVMH & 36 & 6 & 6.48 & \(-\)**4272** & \(-\)474 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Results averaged over thirty runs of each method for the jump diffusion process datasets. Each row corresponds to the results for a specific method for each performance metric. NLL is the negative log-likelihood. Tables 3 and 4 as well as Figures 4 and 5 show that, in terms of predictive performance using the negative log-likelihood on test data and AUC, the SVMH algorithm outperforms all the algorithms on the bulk of the real world datasets. Note that the SVMH method outperforms all the methods or has similar performance on a test negative log-likelihood basis on the logistic regression datasets. The results also show that the MH algorithm progressively becomes worse as the dimensions increase. The predictive performance of the LSMH algorithm is not stable as dimensions increase, with its performance at times being close to the MH algorithm. It is worth noting that although LSMH and SVMH outperform the MH on several metrics, the LSMH and SVMH methods still produce low effective sample sizes compared with the required number of samples. This can potentially be improved by centering the proposal distribution at the current point plus first-order gradient--which should assist in guiding the exploration (and decrease the autocorrelations in the generated samples) as in Metropolis Adjusted Langevin Algorithm [4], and Hamiltonian Monte Carlo [3; 14].