## 4 Autocorrelation Function and Stochastic Memory in Malaria Empirical Series  
The autocorrelation function (\(\rho_{k}\)) behavior has been used with reasonable success in different research fields, from finance to hydrology, and climate data time series. It measures the linear relationship between two sequential values of a time series with a specific time lag \(k\). The autocorrelation function (\(\rho_{k}\)) expresses the magnitude of that correlation between \(k\) lagged values:  
\[\rho_{k}=\frac{\sum_{t=1}^{N-k}(x_{t}-\overline{x})(x_{t+k}-\overline{x})}{ \sum_{t=1}^{N}(x_{t}-\overline{x})^{2}} \tag{5}\]  
Figure 4 shows the autocorrelation function for each empirical case. The algebraic decay of the empirical autocorrelation function \(\rho_{k}\) is strongly connected to the memory of stochastic processes, such as long memory in the form of _Long-Range Dependence_ (LRD). The existence of LRD assumes the presence of stationarity in the time series. A memory parameter \(d\) is defined in relation to the slope of the autocorrelation function (\(\rho\)) decay. When \(d>0\) the term _persistent_ defines the time series, with progressively larger values in time. In the opposite case, we have \(d<0\) with the presence of _anti-persistence_, where positive values will tend to alternate with negative values and vice versa. Here, we borrow the concept as defined in Ref. [34]: _For \(d<0\) we have anti-persistence; i.e., positive values tend to be followed by negative values and vice versa._ In the special case of \(d=0\), the process will correspond to the presence of white noise, without evidence of autocorrelation and corresponding to a pure Markovian process [34]. \begin{table}
\begin{tabular}{l c c c c c c} \hline
**Average** & \multicolumn{2}{c}{**24 Months**} & \multicolumn{2}{c}{**36 Months**} & \multicolumn{2}{c}{**48 Months**} \\ \hline
**Pearson coeff.** & \(\mathbf{r^{2}(I,H)}\) & \(\mathbf{r^{2}(I,S)}\) & \(\mathbf{r^{2}(I,H)}\) & \(\mathbf{r^{2}(I,S)}\) & \(\mathbf{r^{2}(I,H)}\) & \(\mathbf{r^{2}(I,S)}\) \\ \hline Aregawi, 2014 [22] & 0.048 & 0.216 & 0.323 & 0.223 & **0.727** & 0.012 \\ \hline Alhassan, 2017 [21] & 0.528 & **0.657** & **0.638** & **0.879** & 0.530 & **0.746** \\ \hline Bedane, 2016 [23] & 0.006 & 0.107 & 0.094 & 0.027 & 0.000 & 0.181 \\ \hline Appiah, 2015 [16] & 0.336 & **0.620** & 0.222 & 0.353 & 0.299 & 0.534 \\ \hline Landoh, 2012 [25] & 0.310 & 0.339 & 0.436 & **0.633** & **0.722** & **0.743** \\ \hline Muwanika, 2017 [26] & 0.043 & 0.547 & 0.175 & **0.748** & 0.449 & **0.903** \\ \hline Elipe, 2007 [24] & 0.074 & **0.886** & 0.012 & **0.655** & 0.590 & **0.683** \\ \hline [PERSON], 2008 [27] & 0.068 & **0.603** & 0.414 & 0.069 & 0.013 & **0.949** \\ \hline \end{tabular}
\end{table} 
Table 3: Pearson correlation coefficient \(r^{2}\), between pairs among the three metrics malaria incidence \(I\), the Hurst exponent \(H\) and Shannon entropy \(S\). The three metrics were computed for three different time-windows of 24, 36 and 48 months. See also Appendix B. Results with \(r^{2}>0.600\) are highlighted in bold. The decay in time of the autocorrelation function \(\rho_{k}\) in a stochastic process correlates with the presence of memory persistence of past events in the present state of the system. In the case of fast \(\rho_{k}\) the exponential decay the system memory will be short. With slower \(\rho_{k}\) decays (corresponding to power law processes) memory will be longer in relation to the presence of LRD. For the present eight empirical time series the autocorrelation function \(\rho_{k}\) revealed similar decay patterns. The \(\rho_{k}\) decay seems to have deviated from exponential decay in most of the cases, which would have been expected in the case of a pure-white-noise Markovian process, thus suggesting the presence of memory persistence in all the empirical series shown. The slow \(\rho_{k}\) decay is usually related to the presence of time series' non-stationarity.