## 4 Results
### HMM Analysis Results  
In this section, the results of the HMM analysis for Region 3 are presented. The HMM analyses on the total sum ozone time-series in each region are performed for two different cases _with_ aerosol as a covariate in the HMM. Outputs of AIC and BIC in testing for 2, 3, 4 and five state HMM model candidates are given, with the selection of the optimal model based on the minimum AIC and BIC achieved. Also presented is the transmission matrix of the transition probabilities, and the mean and standard deviation estimates for each state means for both ozone and aerosol where an adjustment is made in the HMM for aerosol as a covariate. This section reports the results of the intervention analysis for Region 3 (NSW). The corresponding results for the optimal HMM models for Region 1 (Pacific Ocean) are given in Appendix B.1 and for Region 2 (Vic) shown in Appendix B.2. HMM Analysis of Ozone for Region 3 with Adjustment by Aerosol as a Covariate  
Tables 2-4, and Figure 20 report the HMM models when aerosol levels are added as a covariate (the so-called bivariate model) to better explain the ozone changes observed in Region 3. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Model Name** & **AIC** & **BIC** \\ \hline HMM (with Aerosol)—with five states & 3957.754 & 4098.487 \\ \hline HMM (with Aerosol)—with four States & 4013.133 & 4112.286 \\ \hline HMM (with Aerosol)—with three States & 4020.225 & 4084.195 \\ \hline HMM (with Aerosol)—with two States & 4088.763 & 4123.947 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: AIC and BIC criteria for HMM model selection with differing number of states for total sum ozone with total sum aerosol as a covariate in Region 3. Figure 19: CPs for TSA and TSO for Region 3 (NSW) as shown by vertical grey dashed lines. According to the intercept of the parameters of the HMM model, the intercept value of ozone of State 3 is the largest with an average mean ozone level of 7738.53 DU and lowest aerosol = 6.11, and State 1 has the smallest average state-specific mean ozone level of 6563.19 DU aligned with an increased aerosol of 15.29 (Table 4). From the beginning of the series to day 66 (5 November 2019) the stochastic process is identified as being in State 3, with the highest mean ozone intercept parameter =7738.53 DU, and lowest mean aerosol = 6.11. Then, after day 66 (5 November 2019) to day 82 (21 November 2019) ozone transitions between States 4 and 5, with State 5 (mean ozone intercept of 7327.05 DU, aerosol level = 32.36). Between day 82 (21 November 2019) and day 91 (30 November 2019), ozone levels change between States 1 and 2. After day 91 (30 November 2019) to day 97 (6 December 2019), \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{3}{c}{**Ozone (DU)**} & \multicolumn{3}{c}{**Aerosol**} \\ \cline{2-5}  & **Mean** & **Std. Deviation** & **Mean** & **Std. Deviation** \\ \hline State 1 & 6563.193 & 197.722 & 15.293 & 7.648 \\ \hline State 2 & 6802.999 & 181.552 & 36.726 & 18.652 \\ \hline State 3 & 7738.528 & 386.539 & 6.113 & 6.335 \\ \hline State 4 & 7624.158 & 407.595 & 14.421 & 2.57 \\ \hline State 5 & 7327.054 & 295.041 & 32.355 & 6.97 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameter estimates of all five states from HMM for total sum ozone and total sum aerosol Region 3. Figure 20: HMM bivariate state changes in total sum ozone time-series with respect to total sum aerosol time-series as a covariate for Region 3. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Transmission Matrix** & **State 1** & **State 2** & **State 3** & **State 4** & **State 5** \\ \hline State 1 & 0.979 & 0 & 0 & 0.021 & 0 \\ \hline State 2 & 0.043 & 0.957 & 0 & 0 & 0 \\ \hline State 3 & 0 & 0 & 0.985 & 0.015 & 0 \\ \hline State 4 & 0 & 0 & 0 & 0.364 & 0.636 \\ \hline State 5 & 0 & 0.133 & 0 & 0.068 & 0.8 \\ \hline \hline \end{tabular}
\end{table}