# An Analysis of the Short Utterance Problem for Speaker Characterization +
## 5 Experiments & Results
### Experimental Setup  
Our experimental work is constructed around the SRE10 \"coreext-coreext det5 female\" experiment. This experiment, part of the Speaker Recognition Evaluation 2010 [15] proposed by the National Institute of Standards and Technology (NIST), requires the scoring of more than two hundred fifty thousand trials, recorded from telephone channel in the United States. Each trial consists of two utterances, enrollment and test, with about three hundred seconds of audio per role. Each utterance is known to contain a single speaker. Evaluation rules impose no restriction about the treatment of each trial, but it is obligatory to treat trials independently, not transferring any knowledge among them.  
In this work we restrict our efforts to i-vectors. This choice was taken for illustrative purposes. Therefore, 20 MFCC feature vectors, with first and second order derivatives and Short Time Gaussianization [32] are applied. Utterances are then represented by a gender dependent 2048-Gaussian UBM trained with excerpts from SRE 04, 05, 06 and 08. Based on this UBM a gender dependent 400-dimension T matrix is trained, also using excerpts from SRE 04, 05, 06 and 08. The obtained embeddings, in this case i-vectors, are centered, whitened and length-normalized [33]. The back-end consists of a 400 dimension Simplified PLDA. No score calibration is applied, so results are measured in terms of Equal Error Rate (EER) and minDCF. This latter measure is based on the Detection Cost Function (DCF)  
\[DCF=C_{\text{Miss}}P\left(\text{Miss}\right)P\left(\text{Target}\right)+C_{ \text{F.A.}}P\left(\text{F.A.}\right)\left(1-P\left(\text{Target}\right)\right) \tag{8}\]  
This function weights the probability of missing a target (\(P\left(\text{Miss}\right)\)) and the probability of producing a false alarm (\(P\left(\text{F.A.}\right)\)) to set them in the operating point, fixed by the cost of each kind of error (\(C_{\text{Miss}}\) and \(C_{\text{F.A.}}\) respectively) and the prior probability of a target trial \(P\left(\text{Target}\right)\). For the evaluation, the defined operating point forces the values for the three parameters to be 10, 1 and 0.01 respectively.  
In this work we need to measure the relative differences in the phoneme distribution between enrollment and test. Hence we must define a metric to measure how close these two utterances are from each other. In this work we have opted for the KL2 distance [34]. This metric, based on the Kullback Leibler (KL) divergence, determines how distribution \(q\) differs from distribution \(p\). The KL divergence for discrete random variables is formulated as follows:  
\[D_{\text{KL}}(p,q)=\sum_{i}p(i)\ln\frac{p(i)}{q(i)} \tag{9}\]  
The KL divergence is not symmetric, thus we apply its symmetric version, the KL2 distance.  
\[D_{\text{KL2}}(p,q)=D_{\text{KL}}(p,q)+D_{\text{KL}}(q,p)=D_{\text{KL2}}(q,p) \tag{10}\]  
In i-vectors the phoneme distribution \(\mathbf{\alpha}\) matches the responsibility distribution. Consequently, our KL2 metric is evaluated between the responsibility distribution of enrollment and test. This distribution can be obtained from the zeroth order Baum-Welch statistic.