# A Tri-Model Prediction Approach for COVID-19 ICU Bed Occupancy: A Case Study
## 3 Research Design
### Models & Algorithms
#### 3.2.3 Multivariate Regression  
This model takes the average of two predictive scenarios. The first is pessimistic and the second optimistic, following a Multivariate Regression (MR- or multiple LR) model. The MR model is an enhancement of the simple LR equation by increasing the complexity and adding more independent variables in the model. In addition, it considers \(R_{0}\) and lockdown variables. The mathematical formula for this model is presented by Equation (4):  
\[ICUlbeds=b_{0}+b_{1}\times PositiveCases+b_{2}\times R_{0}-b_{3}\times lockdown+e \tag{4}\]  
where, \(b_{x}\): vector/scalar, \(PositiveCases\): an independent variable, \(R_{0}\) and \(lockdown\): independent variables and \(e\): the statistical error of the equation.  
According to Equation (4), the ICU Beds give a forecast based on PositiveCases (dependent variable), \(R_{0}\) and lockdown (which are the independent parameters of the equation) considering the statistical error \(e\) and the vector \(b_{x}\) that follows the trend of the phenomenon. Concerning this research study, the output is two datasets of prediction, the pessimistic and the optimistic. Their average is the result of the Multivariate Regression forecasting. Regression analysis is a method which tries to identify and quantify the relationships between multiple variables. The outcome can be adjusted according to the impact of other factors. The advantages of regression are the ease of variable control and isolation by keeping them constant in case of need [52; 53; 54].  
Figure 4: ARTXP and ARIMA process.  
Moreover, the maximum likelihood estimate was utilised for this model in order to maximise the likelihood of the model's accuracy based on the observed data. Consequently, the optimal values were attained, and this estimate was chosen on the basis of the independent and identical data distribution [55; 56].  
This method attempts to identify the best fit in their linear multivariate relationship with all variables of the model. Regression is used by quantifying relationships in case of lockdown. When the country was under strict lockdown the independent binary variable _lockdown_ was activated in the equation as a decreasing coefficient. Regression analysis has the capacity to quantify relationships for ICU bed prediction in the short and mid-term. Based on regression analysis by the Multivariate Regression model, there is a relationship between ICU beds (the predicted value), positive COVID-19 cases and the metrics of \(R_{0}\) and lockdown. Experimental trials influenced feature selection with the purpose to eliminate noise of other independent variables that do not affect ICU Beds and finalizing independent parameters (\(R_{0}\) and _lockdown_) of the multivariate model. By selecting different features and checking the correlation of dependent and independent variables separately as for hospitalized, recovered, deaths, \(R_{0}\), means of transport mobility, this study concluded to using positive COVID-19 cases, \(R_{0}\) and the existence of lockdown as independent variables for the regression model.