# Analyzing Parking Demand Characteristics Using a Bayesian Model Averaging
## 3 Methodology
### Model Space Determination  
The initial phase in executing the BMA approach involves the meticulous selection of a set of candidate models as the model space. According to the empirical histograms of the observed data, the parking duration of temporary users shows a unimodal trend, while that of long-term users displays a bimodal trend. Therefore, several commonly used single distributions were chosen as the candidate models to analyze the characteristics of the parking duration of temporary users, including normal, log-normal, gamma, Weibull, log-logistic, Burr, and generalized extreme value (GEV) distributions, as shown in Table 2.  
In addition, several mixture models were chosen as candidate models to depict the parking duration characteristics of long-term users. The general formulation of the mixture model encompassing two components is formulated as  
\[f(x|\theta)=\sum_{i=1}^{2}\omega_{1}\cdot f_{1}(x|\theta_{1})+\omega_{2}\cdot f _{2}(x|\theta_{2}) \tag{6}\]  
where \(f(x|\theta)\), \(f_{1}(x|\theta_{1})\), and \(f_{2}(x|\theta_{2})\) denote the probability density function of the mixture model and the two components under the corresponding parameter vectors (i.e., \(\theta\), \(\theta_{1}\), and \(\theta_{2}\)), respectively. \(\omega_{1}\) and \(\omega_{2}\) are the weights for the two components, and \(\omega_{1}+\omega_{2}=1\). Please note the parameter vector \(\theta\), which is represented as \(\theta=[\omega_{1},\omega_{2},\theta_{1},\theta_{2}]\). In this study, Gaussian (normal), log-normal, gamma, Weibull, log-logistic, and generalized inverse Gaussian (GIG) distributions were chosen as components (the probability density functions are provided in Table 1), and the corresponding mixture models are denoted as the Gaussian  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Distribution** & \multicolumn{1}{c}{**Probability Density**} & **Parameter** \\  & \multicolumn{1}{c}{**Function**} & \\ \hline Normal & \(f(x|\mu,\sigma)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(1-x)^{2}}{2\sigma^{2}}}\) & \(\mu\in(-\infty,\infty)\), \(\sigma>0\) \\ Log-normal & \(f(x|\mu,\sigma)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(1-x)^{2}}{2\sigma^{2}}}\) & \(\mu\in(-\infty,\infty)\), \(\sigma>0\) \\ Gamma 1 & \(f(x|\alpha,\beta)=\frac{1}{\beta\cdot\Gamma(x)}x^{-1}e^{-\frac{1}{2}}\) & \(\alpha>0\), \(\beta>0\) \\ Weibull & \(f(x|\alpha,\beta)=\frac{\theta}{\beta}\left(\frac{x}{\beta}\right)^{-\frac{1}{ 2}}e^{-(\frac{1}{2})^{\alpha}}\) & \(\alpha>0\), \(\beta>0\) \\ Log-logistic & \(f(x|\mu,\sigma)=\frac{1}{\sigma x}\cdot\frac{e^{\frac{1}{2}\sigma-2}}{\left(1- e^{\frac{1}{2}\sigma-2}\right)^{2}}\) & \(\mu>0\), \(\sigma>0\) \\ Burr & \(f(x|\alpha,\beta,\gamma)=\frac{\beta\gamma\left(\frac{x}{\beta}\right)^{\delta} }{x\left[1+\left(\frac{1}{2}\right)^{\delta}\right]^{\gamma+1}}\) & \(\alpha>0\), \(\beta>0\), \(\gamma>0\) \\  & \(f(x|k,\mu,\sigma)=\) & \\ GEV & \(\frac{1}{\sigma}\left(1+k^{\frac{x-x}{\sigma}}\right)^{-1-\frac{1}{2}}\) & \(k\
eq 0,1+k^{\frac{x-x}{\sigma}}>0\) \\ GIG 2 & \(f(x|\mu,\sigma,\
u)=\) & \(\mu>0\),\(\sigma>0\), \\  & \(\left(\frac{\varepsilon}{\mu}\right)^{\
u}\cdot\left[\frac{x^{\
u-1}}{2\sigma x \left(\frac{1}{2}\right)}\right]e^{-\frac{1}{2\sigma}\left(\frac{1}{\sigma}+ \frac{x}{\sigma}\right)}\) & \(\
u\in(-\infty,\infty)\) \\ \hline \hline \end{tabular} \({}^{\dagger}\Gamma(\kappa)\) represents the gamma function. 2\(\kappa_{p}\) is the modified Bessel function of the third kind with order \(\
u\), and \(\epsilon=\left[\kappa_{p+1}\left(\frac{1}{\sigma^{2}}\right)\right]\left[\kappa _{p}\left(\frac{1}{\sigma^{2}}\right)\right]^{-1}\).  
\end{table}
Table 2: Summary of candidate models.  
mixture model (GauMM), log-normal mixture model (LognMM), gamma mixture model (GamMM), Weibull mixture model (WeiMM), log-logistic mixture model (LogIMM), and GIG mixture model (GIGMM).