\(\)The commonly used RMSE \citep{Fox1981} quantifies the differences between predicted and observed values, and thus indicates how far the forecasts are from actual data. A few major outliers in the series can skew the RMSE statistic substantially because the effect of each deviation on the RMSE is proportional to the size of the squared error. The overall, nondimensional measure of the accuracy of forecasts MASE   \citep*{Hyndman2006} is less sensitive to outliers than the RMSE. The MASE is recommended for determining comparative accuracy of forecasts \citep{Franses2016} because it examines the performance of forecasts relative to a benchmark forecast. It is calculated as the average of the absolute value of the difference between the forecast and the actual value divided by the scale determined by using a random walk model (naïve reference model on the history prior to the period of data held back for model  training). MASE<1 indicates that the forecast model is superior to a random walk.
The correlation coefficient between estimates and observations \cite{Addiscott_1987}, (-1 anti-correlation) \(-1\le R\le1\) (1 perfect correlation), assesses linear relationships, in that forecasted values may show a continuous increase or decrease as actual values increase or decrease. Its extent is not consistently related to the accuracy of the estimates. Forecasts online means were used to employ simulation model with support of Excel spreadhseet. The statistics were assessed interactively using Statistics Software STATGRAPHIC Online and WESSA R–JAVA web \citep{Wessa2012}.
In order to quantify long-range dependence and appraise the cyclical-trend patterns in the series, we estimated the Hurst \(H\) exponent (rate of chaos)(add reference) , which is related to the fractal dimension \((D=2-H)\) of the series. Long memory occurs when \(0.5<H<1.0\), that is, events that are far apart are correlated because correlations tend to decay very slowly. On the contrary, short-range dependence \(0.5<H<1.0\) is characterized by quickly decaying correlations, i.e. past trends tend to revert in the future (an up value is more likely followed by a down value). Calculating the Hurst exponent is not straightforward because it can only be estimated and several methods are available to estimate it, which often produce conflicting estimates \citep{Karagiannis2002}. Using SELFIS, we have referred to two methods, which are both credited to be good enough to estimate H \cite{2006}  the widely used  \citep{Yin2009} rescaled range analysis (R/S method) and the ratio variance of residuals method, which is known to be unbiased almost through all Hurst range \citep*{Sheng2009}. Long-memory in the occurrence of PDSI values was also analyzed to see if the memory characteristic is correlated with the length of the time series. To determine whether this characteristic changes over time, the Hurst exponent was not only estimated for the full time series (1801-2014), but also for a shorter series starting in 1901 (the most recent period, which is also the period held out of the calibration process).