# Markov Chain Monte Carlo Based Energy Use Behaviors Prediction of Office Occupants
## 4 Experimental Evaluation
### Comparison between MCMC Estimation and MLE  
The Maximum Likelihood Estimate (MLE) is a statistical method based on the principle of maximum likelihood, which is the application of probability theory in statistics. It provides a way to estimate a model for a given observation. The steps of solving the maximum likelihood estimation value mainly include: writing out the likelihood function \(\ell\left(\theta\right)\) at first, taking the logarithm of the likelihood function, and finally, solving the maximum likelihood function estimation value \(\widehat{\theta}\). \(\ell\left(\theta\right)\) and \(\widehat{\theta}\) can be described as:  
\[\ell(\theta)=p(x_{1},x_{2},...,x_{N}\left|\theta\right.)=\prod_{i=1}^{N}p(x_{ i}\left|\theta\right.) \tag{11}\]  
\[\widehat{\theta}=argmax_{\theta}\prod_{i=1}^{N}p(x_{i}\left|\theta\right.) \tag{12}\]  
where \(x_{1},x_{2},...,x_{N}\) are samples; \(\ell\) is the parameter vector. We compared the MCMC and MLE algorithm for the analysis of the energy use behavior models corresponding to the optimal parameters \(\beta\) and \(\alpha\). For the convenience of the statistics, the time periods of [6:00-9:00] and [14:00-17:00] were mapped respectively to the numerical interval of [\(-\)85,95] and [\(-\)60,120]. The comparative analysis results are shown in Table 2, for the desktop computer power-on model; the model interval difference (\(d\)) fitted by the MLE algorithm and the MCMC algorithm was 1.6074 and 0.9780, respectively. The smaller the d was, the more the data in the confidence interval, and the better the fitting result. Therefore, the MCMC algorithm was better than the MLE algorithm. By comparing the mean (\(v\)), the 90% credible interval (\(L\)), and the difference of the interval (\(d\)) for the model, it could be obtained that while the mean value performed similarly in estimating the energy use model, the proposed method outperformed  
\begin{table}
\begin{tabular}{c r r r r} \hline \hline \multirow{2}{*}{**Equipment Action Model**} & \multicolumn{2}{c}{**Optimal Parameter**} & \multicolumn{2}{c}{**Evaluation Index**} \\ \cline{2-5}  & \(\alpha\) & \(\beta\) & **RMSE** & **MAE** \\ \hline Desktop computer Power-on & 3.2269 & \(-\)0.0578 & 0.0823 & 0.0517 \\ Desktop computer Power-off & \(-\)0.8263 & 0.0331 & 0.0732 & 0.0415 \\ Light Power-on & 3.3025 & \(-\)0.0693 & 0.0536 & 0.0346 \\ Light Power-off & \(-\)0.4510 & 0.0345 & 0.0634 & 0.0238 \\ Water dispenser Power-on & 4.3892 & \(-\)0.0705 & 0.0815 & 0.0591 \\ Water dispenser Power-off & \(-\)1.0008 & 0.0362 & 0.0924 & 0.0612 \\ Air-conditioner heating-on & 2.8954 & \(-\)0.0873 & 0.0687 & 0.0491 \\ Air-conditioner heating-off & 3.7358 & \(-\)0.0492 & 0.0824 & 0.0574 \\ Air-conditioner cooling-on & 5.1029 & \(-\)0.0617 & 0.0635 & 0.0329 \\ Air-conditioner cooling-off & 4.8796 & \(-\)0.0584 & 0.0744 & 0.0482 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Comparisons of three prediction models in the first experiment.  
the Maximum Likelihood Estimation (MLE) algorithm on uncertainty quantification with relatively narrower confidence intervals.