# Precipitation Products' Inter-Comparison over East and Southern Africa 1983-2017
## 2 Materials and Methods
### Cluster Analysis for Identification of Precipitation Seasonality Zones  
The marked spatial variability of precipitation seasonality in EA and SA suggested a partition of these regions into sub-areas characterized by a predominant precipitation annual cycle evolution. The adopted approach takes advantage of the procedure already used by [PERSON] et al. (2017) for the EA region. According to that approach, the non-hierarchical k-means clustering algorithm of [PERSON] (2018) was applied to the monthly normals of the Global Precipitation Climatology Center data set (GPCC_Clim version 2018) at a 0.25\({}^{\circ}\) spatial resolution ([PERSON] et al., 2018), transforming monthly data into empirical cumulative distribution functions at each grid cell ([PERSON] et al., 2018). The use of rain gauge-based climatology to accomplish this task is apparently in contrast with the decision to investigate the intercomparisons among the satellite precipitation products without the direct involvement of rain gauge data as a reference, considering the motivations discussed at the end of the Introduction. However, the availability of a trusted approach already applied to EA and the necessity to have a subdivision of the two regions independent of a specific precipitation product in order to simplify the satellite product intercomparison suggested using the GPCC_Clim data set for the initial partitioning phase. In this way, a single subdivision for each region was determined through the GPCC_Clim data; however, the quantitative evaluation of the monthly climatology for each sub-area was carried out by mutually contrasting only the satellite products and excluding the GPCC data. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Name** & **Definition** & **Perfect Score** \\ \hline Probability of Detection & \(POD=\frac{H}{H+M}\) & 1 \\ False Alarm Ration & \(FAR=\frac{H+FA}{H+M}\) & 0 \\ BIAS Score & \(BIAS=\frac{H+FA}{H+M}\) & 1 \\ Hanssen and Kuipers discriminant & \(HK=\frac{H}{H+M}-\frac{FA}{A++N}\) & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters for the dichotomous statistics. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Name** & **Definition** & **Perfect Score** \\ \hline Correlation Coeff. & \(\frac{\mathbb{CC}}{\Sigma\left(\mathit{SPE_{1}}-\mathit{SPE_{1}}\right)\left( \mathit{SPE_{2}}-\mathit{\overline{SPE_{2}}}\right)}\) & 1 \\  & \(\sqrt{\mathbb{C}\left(\mathit{SPE_{1}}-\mathit{\overline{SPE_{1}^{*}}}\right) }\sqrt{\mathbb{C}\left(\mathit{SPE_{2}}-\mathit{\overline{SPE_{2}^{*}}} \right)}\) & \\ Mean Error & \(ME=\frac{1}{N}\sum\limits\left(\mathit{SPE_{1}}-\mathit{SPE_{2}}\right)\) & 0 \\ Mean Absolute Error & \(\frac{1}{N}\sum\limits\left|\left(\mathit{SPE_{1}}-\mathit{SPE_{2}}\right)\right|\) & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Statistical parameters used for the quantitative intercomparisons. The non-hierarchical k-means clustering algorithm allows subdividing of a set of objects, in this case the annual precipitation cycle of each 0.25\({}^{\circ}\) grid cell, into k groups based on their characteristics, in this case precipitation seasonality. The goal is to minimize the total intra-cluster variance. The result of the algorithm depends on the number of clusters initially chosen; for this reason, the algorithm is normally executed several times by varying the k value. The analysis of the sum of the squared distances between each member of a cluster and the cluster centroid is the common method to select the appropriate value for k, ending the iterative procedure when increasing k values does not induce any significant decrease in the sum of the squared distances. Thus, a certain degree of subjectivity is involved in the final decision of the cluster number [41], and it is necessary to find a balance between obtaining good homogeneity between the elements belonging to the same cluster without an excessive partition of the regions and consequent reduction in the number of elements in each cluster.