# An Analysis of the Short Utterance Problem for Speaker Characterization +
## 4 Effects of the Short Utterances in I-Vectors  
The phonetic distribution in an utterance has important implications during the embedding extraction. Embedding shifts due to incorrect contributions \(G_{i}(\mathbf{A})\) are complementary to those created by the mismatch in the phonetic distribution. In this section we illustrate their impact with i-vectors. This choice of well-known embeddings makes the study of both problems more illustrative in a simple way. For this purpose we propose a small dimension i-vector experiment to test the effects of short utterances in some artificial controlled data. Given an evaluation UBM i-vector pipeline, we compare the i-vectors obtained from an original utterance and those obtained from the same utterance after undergoing controlled short-utterance modifications. These modifications affect both the acoustic unit distribution \(\boldsymbol{\alpha}\) and their contributions \(G_{i}(\mathbf{A})\). We make use of the following experimental setup: We first sample a large artificial data pool from a UBM i-vector pipeline. This data pool consists of more than ten thousand independent utterances, with one hundred two-dimension samples each. The UBM is a 4-Gaussian GMM whose components are located in \((0,0)\), \((0,10)\), \((10,0)\) and \((10,10)\), all of them with the identity matrix as covariance. The generative i-vector extractor has a 3-dimension hidden variable subspace. With then thousand of these utterances we train our evaluation pipeline, an alternative UBM i-vector system. For simplicity we share the generative UBM. Regarding the i-vector extractor, we train a model with only a two-dimension latent subspace. This dimension reduction between generation and evaluation has been considered to imitate real life, where the generation of data is a too complex process that we only can approximate. From the remaining data pool we choose two extra utterances, unseen during the model training, for evaluation purposes. Because these two utterances are independent, we assume them to represent two different speakers. In Figure 1 we represent them, red and blue respectively. The representation includes three parts: In the first part we show the original feature domain, that is, the utterance set of feature vectors. Each ellipse in the figure represents the distribution of each Gaussian in their GMMs. The image also includes in green the representation of the UBM model. The second part in Figure 1 represents the same red and blue utterances in the latent space by means of the posterior distribution of the latent variable \(\mathbf{w}\). The third part of Figure 1 illustrates the location of the particular estimations per component \(G_{i}(\mathbf{A})\) for the two utterances in the latent space. Reddish estimations correspond to the red speaker while bluish ellipses represent the phonemes for the blue speaker. Following the described setup we can carry out an analysis of degradation in short utterances. First we illustrate the phoneme dependent estimation error due to limited data. For this reason we estimate the posterior distribution of the embeddings for multiple utterances only differing the number of samples. The distribution of phonemes \(\boldsymbol{\alpha}\) remains unaltered. Theoretically, the embeddings should not suffer any bias but its uncertainty should get larger as long as the utterances contain less data. In Figure 2 we compare the original utterances to those obtained with one fifth of the data and one tenth of the data. Figure 2 illustrates the posterior distribution of the latent variable for the short utterances (dashed-line red and blue ellipses) as well as the original utterances (red and blue ellipses with continuous line respectively). The location of the ellipse represents the mean of the posterior distribution while its contour the uncertainty. As expected, the original reference utterance and their shorter versions present very reduced shifts among themselves, with almost concentric ellipses.