## 2 Materials and Methods
### Processing and Selection of Prosodic Features  
The acoustic features from each recording in the corpus were extracted using the openSmile toolkit [29]. For feature extraction, the GeMAPS feature set [30] was chosen due to its comprehensive collection of acoustic and prosodic features. This feature set was used in automatic emotion detection [31] and in our previous works [9]. GeMAPS encompasses frequency-related features, energy-related features, and temporal features. To capture the variation along the utterance, the arithmetic mean and coefficient of variation were calculated for each feature. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Speaker** & **Gender** & **CA** & **VA** & **STVM** & **NVCL** & **MPercT** & **MProdT** & **OLRight** & **\#Audios** \\ \hline
022 & f & 195 & 84 & 94 & 17 & 69.7\% & 48.30\% & 72\% & 120 \\
023 & m & 204 & 99 & 134 & 18 & 76.04\% & 72.10\% & 76\% & 106 \\
024 & f & 178 & 96 & 78 & 20 & 73.96\% & 74.65\% & 80\% & 97 \\
025 & m & 190 & 60 & below 74 & 10 & 60.42\% & 49.76\% & 57\% & 131 \\
026 & m & 223 & 69 & below 74 & 13 & 56.25\% & 45.70\% & 51\% & 151 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Description of the informants analyzed in this study. The table displays the following information for each speaker: chronological age (CA), verbal mental age (VA), short-term verbal memory (STVM), and non-verbal cognitive level (NVCL). The ages are denoted in months. Furthermore, it includes the mean percentage of success in perception (MPercT) and production (MProdT) for PEPS-C tasks. The table is completed with the percentage of sentences classified as right in the offline evaluation (OLRight) and the total number of audio recordings of each speaker (\#Audios). Additionally, four extra temporal features were included: silence percentage, sounding percentage, silences per second, and the mean length of silences. These additional features were derived using the silences and sounding intervals identified by the Praat software (2006) [32], which utilizes an intensity threshold along with minimum silent and sounding interval durations to detect these intervals (default values were used). In total, a set of 92 features was employed, including 10 from the frequency domain, 10 from the energy domain, 11 from the temporal domain, and 61 from the spectral domain. A detailed description of these features can be found in [9]. \(U\) is the set of utterances in the corpus, with \(U=\theta_{TD}\cup\theta_{R}\cup\theta_{W}\), where \(\theta_{TD}\) are the utterances of typical speakers and \(\theta_{R}\) or \(\theta_{W}\) the utterances of speakers with Down syndrome, judged as right or wrong by the evaluator. Every \(u\in U\) is characterized as \(u=(f_{1},f_{2},f_{3},\ldots f_{N})\), where \(f_{i}\) is a feature computed in the previous stage. The distribution of values of the feature \(f_{i}\) in a subset \(s\) of \(U\) is referred to as \(pdf(s,f_{i})\), and its corresponding mean as \(\mu(s,f_{i})\), where \(s\) could be the samples of a single speaker or the ones of a subset \(\theta_{TD}\), \(\theta_{W}\), or \(\theta_{R}\). Algorithm 1 describes the process of selecting the most appropriate features for visualizing the particular problems of the speakers. For a feature \(f\) to be included in the set of selected features, two conditions should be fulfilled:  
1. Separation: Statistical analysis using the Mann-Whitney test with a p-value threshold of less than 0.01 is applied to determine if there are significant differences between the values of \(f\) in the groups \(\theta_{R}\) (right utterances) and \(\theta_{W}\) (wrong utterances). This criterion ensures that clear distinctions between right and wrong utterances are observed. 2. Consistency: Let \(\mu(\theta_{TD},f)\) represent the mean value of feature \(f\) in the group \(\theta_{TD}\) (typical speakers), \(\mu(\theta_{R},f)\) represent the mean value in the group \(\theta_{R}\), and \(\mu(\theta_{W},f)\) represent the mean value in the group \(\theta_{W}\).