## 3 Machine Learning Models: Results and Interpretation
The slope is slightly steeper in this plot versus the previous, with under-contribution of LEX-Out-State on the low end and over-contribution on the high end of the variable range. The third-ranked variable is Population Density, which shows a slightly positive trend in plot (Figure 3c). The higher the Population Density, the greater the number of cases. The model tracks the actual values very closely. The contribution of Population Density increases gradually and then tapers off at the high end, suggesting a ceiling effect. The fourth most important variable is LEX-Total, as seen in the partial dependence plot (Figure 3d). The plot shows that LEX-Total--the people traveling in-state or out-of-state--contributes steadily to the number of cases (log). The trend is similar to that of LEX-Out-State. LEX-Total makes an under-contribution at the low end and over-contribution at the higher end of the variable range. There is a correlational relationship between travel and cases: increased travel is strongly and negatively associated with cases. Causually, the lower the number of cases, the more travel is deemed to be safe. Conversely, the greater the number of cases, the less travel is deemed to be safe. The more one travels, the more one is exposed to possible infection. In addition, higher Population Density compounds that risk. State-by-state differences may be attributable to certain states having greater population density or a lower level of infection but greater duration of spreading or an occasional superspreader event. Residents who live close to another state's border are also at risk because people cross state borders when it is allowed and also when it is prohibited. Table 4 shows the accuracy generally held when going from training to test partition, regardless of the method used, in the Reopen Regime. The different accuracy measures are in agreement, with Gradient Boosted Machine always having the lowest error, indicated in gray, and there is very little error. The adjusted R\({}^{2}\) = 99.2% and RMSE = 0.063. We choose Gradient Boosted Machine as the best method for the Lockdown Regime and show its lift and residual plots in Figure 4.  
The residuals are very small for the entire range of COVID-19 cases (logged), as can be seen in Figure 4.  
Figure 5 shows the relative importance of the variables from most to least impact: Population Density, State, LEX-Out-State, Political Color, LEX-Total, Climate Temperature,  
Figure 3: Gradient Boosted Machine-Plots for Lockdown Regime. Legend: - Partial Dependence \(\mid\circ\) Actual Value \(\mid\) + Model Prediction. (**a**) Log(cases)–Non-Home Dwell Time. (**b**) Log(cases)–LEX-Out-State. (**c**) Log(Cases)–Log(Population Density). (**d**) Log(Cases)–LEX-Total. In the Reopen Regime, governors were hoping that the worst of the pandemic had passed. The lockdowns were approximately one month old, and the cases were beginning to plateau. Many people had died, but more had recovered. Economic distress was growing, and businesses were looking to re-open. Jobless claims had totaled by the end of April more than 1 million each week for thirteen weeks. People needed or wanted to get back to work. Governors started to lift lockdown orders and also relax social distancing/mask mandates. In Figure 5, the partial dependence plot (Figure 5a) shows that cases are closely tracked by population density. As population density increases, the exposure to infection from other people increases. The second-ranked variable is State. The partial dependence plot (Figure 5b) shows State by State variation, highlighting the 19 states with the most observations and grouping the remainder under All Other. The plot shows strong predictive accuracy, even in states with fewer cases, e.g., Hawaii and Maine. The contribution of the state variable is sometimes optimal, e.g., for California and Maryland, but can be too small, e.g., Virginia, or too large, e.g., Nebraska. The third-ranked variable is LEX-Out-State.