# COVID-19: A Comparison of Time Series Methods to Forecast Percentage of Active Cases per Population
## 3 Description of Time Series Models
### ARIMA: Auto-Regressive Integrated Moving Average  
One of the most well-known and widely used families of time-series models includes the auto-regressive integrated moving average (ARIMA) models--originally developed for economics applications [14; 41]. Their statistical properties, the implementation of the well-known Box-Jenkins methodology during model training process [42] and their ability to implement various exponential smoothing models have all contributed to their popularity and widespread adoption [41; 43].  
ARIMA models assume a linear correlation between the time-series values and attempt to exploit these linear dependencies in observations, in order to extract local patterns, while removing high-frequency noise from the data [24].  
Such an approach comes with two clear benefits. Firstly, it offers a high level of interpretability, as, based on the assumptions of the model, the relationship between the independent variables and the dependent variables are well-understood and therefore easily explained. This enables researchers to gain a deep understanding not only of the relationship between the current state as a function of the past states (endogenous variables), but also of any influence inputs outside the state of the series might have (exogenous variables). The second benefit concerns model selection, which for ARIMA models can be performed in an automated way to maximize prediction accuracy [44].  
Another benefit of ARIMA models is their ability to accommodate systems governed by dynamics that change over time by updating the model based on recent events to predict the future state of the system [44].  
However, since the ARIMA models cannot deal with nonlinear patterns or relationships, their approximation of complex real-world problems and dynamics is not always satisfactory [45].  
Holt-Winters Additive Model (HWAAS): Exponential Smoothing with Additive Trend and Additive Seasonality  
The Holt-Winters additive model is an extension of [PERSON]'s exponential smoothing, a time series forecasting method for univariate data. It is extended so that it adds the seasonality factor to the trended forecast, being to model data with a systematic trend or seasonal component. It is a simple yet powerful and widely used forecasting method which, as mentioned, can cope with trend and seasonal variation and may be used as an alternative to the popular Box-Jenkins ARIMA family of methods. However, empirical studies have tended to show that the method is not as accurate on average as the more complicated Box-Jenkins procedure [15].  
Exponential smoothing is the procedure of continuously revising a prediction after taking into account the more recent observations. In practice, this is achieved by exponentially diminishing the older observations' importance for forecasting by decreasing their weights. In other words, when it comes to predicting a new value, more recent observations matter more than older ones [46].  
The Holt-Winters additive model is best for data with trend and seasonality that do not increase over time and results in a curved forecast that shows the seasonal changes in the data. Practical issues in implementing the method include the choice of initial values, their sensitivity to unusual events or outliers, the choice of smoothing parameters and the normalization of seasonal indices [47; 48].