# An Analysis of the Short Utterance Problem for Speaker Characterization +
## 4 Effects of the Short Utterances in I-Vectors
While the blue speaker suffers almost no degradation, the red speaker biases are more noticeable. Besides, the illustration shows that the less data in the utterance, the bigger the uncertainty of the estimation. Now we study the impact of the distribution of acoustic units \(\boldsymbol{\alpha}\) on the embedding. In the reference utterances this distribution was uniform, this is, 25% of the samples came from each component. We now modify this distribution for both utterances, red and blue. In Figure 3 we show the posterior distributions of the original utterances (red and blue ellipses with continuous line) as well as the altered short utterances (dashed-line red and blue ellipses). In the illustrated example half of the feature  
Figure 1: Scenario of interest. (**a**) Utterances red and blue in the feature domain, with the Universal Background Model (UBM) components in green; (**b**) Utterances red and blue in the i-vector domain; (**c**) Projections of the Gaussian Mixture Model (GMM) components in the i-vector domain for utterances red (reddish ellipses) and blue (bluish ellipses). vectors are sampled from a single component of the GMM while the remaining data is evenly sampled along the other components. We have studied the effect with the four components in the GMM. Illustrated results in Figure 3 reveal the relevance of the distribution of phonemes \(\mathbf{\alpha}\) for its proper modelling. The modification of the distribution of weights makes the red speaker to offer four different representations of the same embedding. Besides, these representations are not overlapped among  
Figure 3: Comparison of posterior distribution of i-vectors with modifications in the phoneme distribution \(\mathbf{\alpha}\). Figure 2: Comparison of posterior distribution of the i-vectors with reference phoneme distribution. Continuous line ellipse represents the original utterance while dashed-lined ellipses illustrate utterances with the limited data. themselves, beyond the uncertainty region from the original utterance. Therefore, these alternative embeddings are likely to fail. Nevertheless, not all speakers behave equally. Whilst red speaker is degraded, our blue speaker has suffered the same alterations without any visible shift on his/her embeddings. The scenario with a distorted phoneme distribution can be taken to the limit. In this situation some components do not contribute to the final embedding. This scenario is the most adverse, significantly modifying the distribution of patterns \(\mathbf{\alpha}\) and some estimations per phoneme \(G_{i}(\mathbf{A})\) being set to zero. In this experiment we have disturbed the distribution of acoustic units \(\mathbf{\alpha}\) forcing two of the components to zero. In Figure 4 we illustrate the six possible scenarios in terms of the non-contributing components. The results are shown for the two test speakers red and blue, with continuous line ellipses for the reference utterances and dashed-line ellipses for their altered versions. According to the representations shown in Figure 4, embeddings from utterances with missing components experiment large biases with respect to the reference embeddings. These shifts are more significant than those previously seen with less extreme distortions in the phoneme distribution \(\mathbf{\alpha}\). Some of the hypothesized embeddings are far beyond the uncertainty from the original utterance. The biases suffered by the utterances are not the same for both speakers. Again the blue speaker suffers no relevant degradation. This behaviour fits in our hypothesis because the missing components scenario is the limit case of phoneme distribution degradation. In all our experiments the red speaker has suffered from strong degradations while the blue speaker has remained almost unaltered. This different behaviour is a consequence of the locations of the phonetic estimations \(G_{i}(\mathbf{A})\) for each speaker.