# A Risk Assessment for Ozone Regulation Based on Statistical Rollback
## 2 Statistical Rollback Approach
### Quantile Matching Approach  
Suppose that current ozone level, denoted by \(x\), is a random variable having a distribution function \(F_{\mathbf{\theta}}\). We assume that adjusted ozone level corresponding to \(x\) under a new scenario follows the same distribution with different parameters, \(F_{\mathbf{\theta}^{*}}\), and there exists a mapping \(m\) such that a random variable \(m(x)\) has the distribution function \(F_{\mathbf{\theta}^{*}}\). Thus,  
Figure 1: QQ-plots of high ozone levels in 1992 under baseline conditions against low ozone levels in 1999 under attainment conditions at six different locations.  
the adjusted ozone level can be obtained using \(m(x)\). That is, \(\theta^{*}\) is the adjusted model parameter based on the relationship between current levels and adjusted levels under the new scenario. Moreover, let \(F(x;\theta)\) and \(G(z;\gamma)\) be the ozone distribution functions based on the baseline condition and attainment condition, respectively, (e.g., \(G\prec F\)). That is, \(m(x)=F_{\theta^{*}}^{-1}(F_{\theta}(x))\) or \(m(x)=G^{-1}(F(x))\). Note that \(F\) and \(G\) can be estimated by baseline and attainment years' ozone data, respectively. Attainment year based on its AQI values or the condition \(G\prec F\) is required. \(G\) may require adjustment based on \(\rho(s)=\frac{\mathbf{x}(p)}{\mathbf{x}(s)}\), where \(\mathbf{x}(b)\) is current design values and \(\mathbf{x}(s)\) is design values under scenario \(s\).  
Let \(f_{\theta}\) be a density function of current ozone levels \(x\). Then adjusted rollback values \(z\) corresponding to \(x\) are obtained as follows:  
1. Based on current ozone levels \(\mathbf{x}=\{x_{i}\}_{i=1}^{n}\), estimate parameters \(\theta\) (i.e., \(\hat{\theta}\)).
2. Compute \(q_{i}^{x}\), the quantile of \(x_{i}\) such that \[q_{i}^{x}=\int_{-\infty}^{x_{i}}f_{\hat{\theta}}(z)dz,\] for \(i=1,\ldots,n\).
3. Estimate parameters \(\theta^{*}\) under new scenario (i.e., \(\hat{\theta}^{*}\)).
4. Determine the corresponding \(z_{i}\) satisfying \[q_{i}^{x}=\int_{-\infty}^{z_{i}}f_{\hat{\theta}^{*}}(z)dz,\] for \(i=1,\ldots,n\).
5. \(\{z_{i}\}_{i=1}^{n}\) are adjusted (rollback) values of \(\mathbf{x}\).  
We extend this approach through the Bayesian framework by putting prior distributions of parameters, \(\pi(\theta)\). By generating samples from posterior distribution, \(\pi(\theta|\mathbf{x})\propto f_{\theta}\pi(\theta)\), we can generate rollback ensembles in the same fashion.  
In general, the distribution of \(x\) can be expressed as a mixture of the form  
\[f(x)=\left\{\begin{array}{cc}(1-w)h(x|\theta_{1})&x<u\\ wg(x|\theta_{2})&x\geq u\end{array}\right.,\]  
where \(u\) is a threshold, \(h\) can be any parametric distribution such as Weibull, truncated Gamma or Gaussian distribution, and \(g\) is a Generalized Pareto distribution (GPD) with parameters \(\theta_{2}=(\sigma,\xi)\).  
The quantile matching approach is simple and straightforward but causes various problems in adjusting multiple ozone processes in large dimensions. Now we consider more feasible approaches in the next section.