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).