where \(\delta\) is smoothing parameter for cyclical indices. 

Transfer Function Models  

An alternative methodology based on transfer function (TF)  models was also implemented to compare two methodologies: seasonal exponential smoothing that uses the temporal dependence structure of the time series itself to reproduce the time series behavior in the future, versus the linear transfer function approach where input time series potentially impacting the drought behavior at large spatial scales are used as explanatory time serie variables in a lagged regression model. The methodology called transfer fuction models, was introduced by  \citet*{Box1976} and re-visited by  \citealp{Shumway2017} to model the impact of El Niño on fish recruitment and for \citet{Guenni2016} to forecast monthly rainfall in the coast of Ecuador based on El Niño indexes. 
In a transfer function model the output series \(Y\left(t\right)\) (in this case PDSI) can be represented as: