## 2 Data and Method
### The Gaussian Mixture Model  
Gaussian mixture model (GMM) is a probabilistic function that tends to group multiple Gaussian distributions into a single distribution and can be defined as:  
\[p(x_{N}\vee\alpha_{M},\mu_{M},\sigma_{M})=\sum_{m=1}^{M}\alpha_{m}p \tag{3}\]  
where \(x_{N}\) is the data sample, \(\alpha_{M}\) represents the weight of each component (mixture) \(m=\dots\) 1, \(\dots\), \(M\). The components, also referred to as the modes, are represented as a normal distribution with parameters' mean \(\mu_{M}\) and standard deviation \(\sigma_{M}\) values. GMM are widely applied for fitting as well as to identify subgroups that compose a given distribution. In meteorology, GMM has been considered, for instance, for forecast verification [45], polarimetric radar-based rainfall-rate estimation [46], predicting precipitation events [47], density estimation and the feature identification of Atmospheric Lagrangian Particle Dispersion Models [48]. Several studies have been conducted so far on the development and improvement of an evolutionary GMM algorithm that automatically fits the distribution by minimizing the given distribution error [49; 50; 51]. Here, the GMM method was implemented thanks to the evolutionary \"Distribution Optimization\" algorithm [51] which separates the \(M\) (modes) that composes the mixture of gaussians. The algorithm involves multiple and sequential steps (i.e., creation of GMMs' population, fitness calculation, filter, mutation, recombination and final check for GMM selection). Finally, the algorithm is complete after a fixed number of iterations by returning the GMM distribution with the highest fitness and its correspondent parameters (i.e., \(\mu\), \(\sigma\), \(\alpha\) and Bayesian decision borders) which can be independently used as input parameters. The final algorithm output includes both the individual gaussian elements (\(M\)) and the associated statistics responsible for describing the overall GMM distribution, which is derived from the sum of all individuals M. In order to find the GMM with the highest fitness, the Distribution Optimization algorithm considers, as a core, an adjustable error function \(X^{2}\) based on chi-square statistics and the probability density function (PDF). To minimize the associated distribution error, the fitness function considers multiple statistics, such as best root-mean-square (RMS), the standard deviation, the overlap error, etc. Thus, for each individual GMM element (\(M=\) 1, 2,... ), the \(\mu\), \(\sigma\), \(\alpha\), the Bayesian decision borders and other associated goodness-of-fit statistics (e.g., Similarity Error, Overlap Error, Mixed Distribution Error) can be retrieved via the Distribution Optimization algorithm. The automated genetic machine-learning GMM \"Distribution Optimization\" algorithm is available in the R library \"DistributionOptimization\" ([[https://cran.r-project.org/package=DistributionOptimization](https://cran.r-project.org/package=DistributionOptimization)]([https://cran.r-project.org/package=DistributionOptimization](https://cran.r-project.org/package=DistributionOptimization)), accessed on 1 February 2021) and a detailed description is provided by [PERSON] et al. [51]. For this study, the GMM distribution fit parameters (i.e., \(\mu\), \(\sigma\), \(\alpha\) and Bayesian decision borders) were retrieved to reconstruct and assess the uncertainties using two Gaussians (\(M=\) 2). The results were contrasted with a single gaussian model (\(M=\) 1). The number of iterations adopted was 100 (sensitivity tests were performed on the number of iterations starting from 100, with \(M=\) 1,..., 5, 10 gaussian modes, and revealed no significant gains on the goodness-of-fit results). The goodness-of-fit statistics, such as the Akaike Information Criterion (AIC, [52]), were also computed, in order to demonstrate the final GMM distribution performances across the precipitation bins.