## 4 Results and Analyses
### Experiment 2: Normalization--The Effect of VTLN  
To study the impact of VTLN, we first calculated and analyzed the warping factors for the children's speech test sets in each language. These factors were estimated using the VTLN models in Table 5. The distribution of the warping factors for each test set is visually represented in the box plots in Figure 2. In each plot, the lower and upper ends of the whiskers correspond to the minimum and maximum values of the warping factors. The straight solid orange horizontal line inside the box represents the median, while the green triangular marker indicates the mean of the warping factors. The dotted red line in each plot is the average value of the warping factors for adults' speech in that language. **Dutch:** The warping factor plots are shown for the VTLN\({}_{\text{CGN}}\) and the VTLN\({}_{\text{Jas-DCDTNnT}}\) models (note that the VTLN\({}_{\text{Jas-DCDTNnT}}\) model gave almost the same warping factors as the VTLN\({}_{\text{Jas-DCDTNnT}}\) model). In both Dutch plots, the left three warping factors in each plot correspond to read speech (non-shaded boxes), while the right three warping factors correspond to HMI speech (shaded boxes). With the VTLN\({}_{\text{CGN}}\) model, i.e., VTLN trained on only CGN adults' speech, all speaker groups have almost the same warping factors, \(\alpha<0.9\) (Figure 2a), which indicates that the warping factors have not been estimated well. This may be due to the fact that the model is trained with only adults' speech from CGN rather than also with children's speech whose warping factors we aimed to estimate. However, when the VTLN model is trained with children's speech, VTLN\({}_{\text{Jas-DCDTNnT}}\), the warping factors are estimated well (children's speech \(\alpha<1\) and adults' speech \(\alpha\approx 1\)) (Figure 2b). These warping factors are similar to those reported for other studies in English [73]. The warping factors for read and HMI speech from both Dutch VTLN models are highly similar, only slightly higher for HMI speech than for read speech. The effect of VTLN on ASR system performance without any augmentations is shown in Table 6 (rows d-f). With VTLN\({}_{\text{CGN}}\), despite the not-so-great estimation of the warping factors, the WER is lower than the baseline (row a) for almost all speaker groups but only significantly so for the native child speaker and teenager speaker groups. This is probably due to the warping factors having \(\alpha<0.9\), which means that the children's speech frequencies are lowered, making them more similar to adults' speech, hence leading to a smaller mismatch with the adult training speech and improved recognition performance. The better  
Figure 2: Box plots of the warping factors estimated across different child speaker groups using different VTLN models. Dutch: (**a**) VTLN\({}_{\text{CGN}}\), (**b**) VTLN\({}_{\text{Jas-DCDTNnT}}\); German: (**c**) VTLN\({}_{\text{CGN}}\), (**d**) VTLN\({}_{\text{JidsTALC}}\); and Mandarin: (**e**) VTLN\({}_{\text{SetA}}\), (**f**) VTLN\({}_{\text{SetC1C2}}\). For each children’s speech test set, the orange horizontal line within the box is the median. The triangular marker indicates the mean of the warping factors (also indicated in text at the top of the Figure). The top and bottom range represent the minimum and maximum values of the warping factors. Red dotted line: warping factor for adult speakers. estimated warping factors of VTLN\({}_{\text{Jas-DCDT}}\) (not shown in Figure 2) and VTLN\({}_{\text{Jas-DCDTNnT}}\) (see Figure 2b) led to a further improvement for children's speech (rows e, f), with the best results for the VTLN model trained on speech data that also included the non-native children's speech, yielding a significant improvement over baseline for all speaker groups. For DC-Read, the performance is even better than when SP and SpecAug are applied (compare Table 6 rows (c) and (f)). On average, the performance with VTLN\({}_{\text{Jas-DCDTNnT}}\) is better than VTLN\({}_{\text{CGN}}\).