# Analyzing Parking Demand Characteristics Using a Bayesian Model Averaging
## 4 Experiments and Results
### BMA Results
These models illustrate that there is model uncertainty in describing parking duration distribution among different user groups. In summary, it is difficult to find a certain distribution form to describe the characteristics of parking duration for both temporary and long-term users. Due to the uncertainty in parking duration, it exhibits various distributional characteristics, making it challenging to characterize its properties with a specific distribution. It is important to highlight that the dissimilarity between log-normal distribution and GEV distribution for temporary users (LognMM and SIGMM for long-term users) is minimal, and one must exercise caution when considering a distribution as the best model, as a distinct goodness-of-fit function (e.g., WAIC, LOO) may yield different best models in depicting parking duration characteristics. However, the BMA approach always integrates the strengths of acceptable candidate models based on posterior model probability and provides universally reliable and robust modeling of parking duration distribution with high accuracy. According to the posterior model probability, the derived probability density function and cumulative density function of the BMA approach can be determined using Equation (1). The fitting performance of the parking duration of temporary and long-term users is illustrated in Figures 3 and 4. The histogram and dotted line (black) are the probability density function and cumulative density function of the observed data, and the dotted lines (green) and the curves (red) represent those of the best model and derived BMA approach. It can be observed that the parking duration distribution of temporary users is skewed and unimodal; the distribution of long-term users is long-tailed and exhibits a bimodal skewness. The log-normal distribution provides a precise fitting performance to parking duration data for temporary users compared with other candidate models. In contrast, SIGMM exhibits superior fitting performance compared with other models for long-term users. The derived BMA approach seamlessly integrates acceptable candidate models and provides accurate depictions of parking duration data for both temporary and long-term users. These findings illustrate the universal reliability and robustness of the BMA approach in modeling parking duration distribution. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**User Type** & **Distribution** & \begin{tabular}{c} **Log-** \\ **Likelihood** \\ \end{tabular} & **AIC 1** & **BIC 1** & **PMP 2** \\ \hline \multirow{8}{*}{\
\begin{tabular}{c} Temporary \\ users \\ \end{tabular} } & Normal & \(-3710.85\) & 7425.69 & 7436.20 & 0.000 \\  & Log-normal & \(-2884.97\) & 5773.94 & 5784.45 & 0.910 \\  & Gamma & \(-2988.53\) & 5981.06 & 5991.57 & 0.000 \\  & Weibull & \(-2992.44\) & 5988.88 & 5999.39 & 0.000 \\  & Log-logistic & \(-2941.78\) & 5887.56 & 5898.07 & 0.000 \\  & Burr & \(-2933.35\) & 5872.69 & 5881.20 & 0.000 \\  & GEV & \(-2887.29\) & 5780.58 & 5789.09 & 0.090 \\ \hline \multirow{8}{*}{
\begin{tabular}{c} Long-term \\ users \\ \end{tabular} } & GauMM & \(-4761.90\) & 9533.81 & 9560.62 & 0.000 \\  & LognMM & \(-4447.30\) & 8904.60 & 8931.41 & 0.103 \\ \cline{1-1}  & GamMM & \(-4497.93\) & 9005.86 & 9032.68 & 0.000 \\ \cline{1-1}  & WeiMM & \(-4494.62\) & 8999.23 & 9026.05 & 0.000 \\ \cline{1-1}  & LogIMM & \(-4472.90\) & 8955.80 & 8982.62 & 0.000 \\ \cline{1-1}  & SIGMM & \(-4445.13\) & 8900.27 & 8927.08 & 0.897 \\ \hline \hline \end{tabular}  
* \(AIC=2m-2 ln(L)\), \(BIC=mln(n)-2 ln(L)\), where \(m\) is the number of parameters, \(n\) denotes the number of data samples, and \(L\) represents the likelihood. 2 PMP is posterior model probability.  
\end{table} 
Table 5: Summary of BMA results for temporary users and long-term users.