## 4 Results and Analyses
### Experiment 2: Normalization--The Effect of VTLN
Nevertheless, even the VTLN model trained with adults' speech only provided a significant improvement over the baseline. For the adult test sets, results for the different VTLN models are not significantly different from baseline. **German**: The estimated warping factors in Figure 2c,d show that VTLN\({}_{\text{CV}}\) led to relatively constant warping factors (indicated by the very short whiskers), whereas VTLN\({}_{\text{JGRSTALC}}\) exhibited more varied warping factors for children's speech (much longer whisker), similar to what we observed for Dutch for VTLN\({}_{\text{Jas-DCDTNnT}}\): a VTLN training set with more diverse children's speech led to more diverse warping factors. The results in Table 7 (row d, e) show that the performance of the VTLN-based models significantly outperform the baseline model (Table 7, row a) for children's speech; however, they do not outperform the augmentation-based models. Similar to the Dutch findings, the use of the VTLN model trained on adults' speech only (VTLN\({}_{\text{CV}}\)) resulted in a modest though significant improvement (1.5% absolute WER), while VTLN\({}_{\text{KidSTALC}}\) showed a larger improvement of 5.6% absolute WER as compared to baseline. Possibly, the varying warping factors estimated for the kidsTALC test set resulted in better normalization and performance improvement. Applying VTLN to adults' speech did not lead to improvements compared to the baseline when the VTLN model was trained on adults' speech, and in fact it significantly degraded performance when the VLTN model trained on children's speech was used. **Mandarin**: Unlike what was observed for Dutch and German, the estimated warping factors in Figure 2e,f show that both VTLN\({}_{\text{SetA}}\), trained on adults' speech, and VTLN\({}_{\text{SetC1C2}}\), trained on children's speech, exhibit varying factors. The warping factors estimated for Set C2 (spontaneous speech) are in a lower range than Set C1 (read speech). This difference can likely be attributed to Set C2 containing speech of children younger, with shorter vocal tract length, than those in Set C1. The shorter vocal tract lengths result in higher frequencies in the speech spectrum, which require more frequency scaling (normalization) and thus lower warping factors. Table 8 (rows d, e) shows the results of the model trained only with VTLN. Both VTLN models significantly outperform the baseline model (row a) on children's speech. The VTLN models trained on adults' speech (VTLN\({}_{\text{SetA}}\)) and children's speech (VTLN\({}_{\text{SetC1C2}}\)) perform similarly (row d, e), giving around 4% absolute improvement on Set C2 and approximately 0.5% absolute CER improvement on Set C1 compared to the baseline. We hypothesize that the larger improvement on Set C2 is due to the presence of speech from younger children, where we expect the normalization to be more effective based on the size of the warping factors rather than due to the conversational nature of the speech. Applying the VTLN models to the adults' speech of Set A decreased performance significantly when the VTLN model trained on children's speech was applied, though a small but significant improvement was found when the VTLN model trained on adults' speech was used.