A closed-set identification system can be viewed as a function which assigns, to any test utterance z, an estimated speaker index , corresponding to the identified speaker in the set of registered speakers .
In closed-set identification, all test utterances belong to one of the registered speakers . Therefore, a misclassification error occurs for test utterance number k produced by speaker when:
where denotes the Kronecker function, which is 1 if the two arguments are the same and 0 otherwise.
The most natural figure that indicates the performance of a speaker identification system is the relative number of times the system fails in correctly identifying an applicant speaker ; in other words, how often a test utterance will be assigned an erroneous identity. Whereas it is straightforward to calculate a performance figure on a speaker-by-speaker basis, care should be taken when deriving a global score.
With our notation, and assuming that , we define the misclassification rate for speaker as:
If we denote as the probability that the system under test identifies another speaker (with index j) than the actual speaker , the quantity provides an estimate of this probability, whereas provides an estimate of . However, it is preferable to report error scores rather than success scores, and performance improvements should be measured as relative error rate reduction. If , is undefined but .
We suggest using the term dependable speaker to qualify a registered speaker with a low misclassification rate , and the term unreliable speaker for a speaker with a high misclassification rate .
From speaker-by-speaker figures, the average misclassification rate can be derived as:
and by computing separately:
the gender-balanced misclassification rate can be obtained as:
The previous scores are different from the test set misclassification rate , calculated as:
Scores and are formally identical if and only if does not depend on i, i.e. when the test set contains an identical number test utterances for each speaker. As it is usually observed that speaker recognition performances may vary with the speaker's gender, the comparison of and can show significant differences, if the registered population is not gender-balanced . Therefore, we believe that an accurate description of the identification performance requires the three numbers , , and to be provided.
Taking another point of view, performance scores can be designed to measure how reliable the decision of the system is when it has assigned a given identity; in other words, to provide an estimate of the probability , i.e. the probability that the speaker is not really when the system under test has output as the most likely identity.
To define the mistrust rate, we have to introduce the following notation:
By definition, and are respectively the number and proportion of test utterances identified as over the whole test set , while is the number of registered speakers whose identity was assigned at least once to a test utterance.
The mistrust rate for speaker can then be computed (for ) as:
Here again, if , is undefined, but .
We suggest that the term resistant speaker could be used to qualify a registered speaker with a low mistrust rate, and the term vulnerable speaker for a speaker with a high mistrust rate.
From this speaker-by-speaker score, the average mistrust rate can be derived as:
and by computing separately:
the gender-balanced mistrust rate is defined as:
By noticing now that:
there appears to be no need to define a test set mistrust rate . In other words: the test set mistrust rate is equal to the test set misclassification rate.
From a practical point of view, misclassification rates and mistrust rates can be obtained by the exact same scoring programs, operating successively on the confusion matrix and on its transpose.
Most speaker identification systems use a similarity measure between a test utterance and all training patterns to decide, by a nearest neighbour decision rule, which is the identity of the test speaker. In this case, for a test utterance x, an ordered list of registered speakers can be produced:
where, for all index j, is judged closer to the test utterance than is.
The identification rank of the genuine speaker of utterance can then be expressed as:
In other words, is the position at which the correct speaker appears in the ordered list of neighbours of the test utterance . Note that a correct identification of corresponds to .
Under the assumption that , let us now denote:
The confidence rank for speaker , which we
will denote here as can then be defined
as the smallest integer number for which of the test
utterances belonging to speaker are part of the
nearest neighbours in the ordered list of candidates. Hence the
Then, the average confidence rank can be computed as the average of over all registered speakers (for which ):
Though a gender-balanced confidence rank could be defined analogously to gender-balanced misclassification and mistrust rates, the relevance of this figure is not clear.
If finally we denote:
the test set confidence rank is defined as:
Average scores, gender-balanced scores and
test set scores all fall under the
same formalism. If we denote as a certain quantity which we will
call the relative representativity of speaker and which
satisfies , and if we now consider the linear combination:
It is clear that:
Therefore , and correspond to different estimates of a global score, under various assumptions on the relative representativity of each speaker. For average scores, it is assumed that each speaker is equally representative, irrespectively of its sex group, but if the test population is strongly unbalanced this hypothesis may not be relevant (unless there is a reason for it). For gender-balanced scores , each test speaker is supposed to be representative of its sex group, and each sex group is supposed to be equiprobable. Test set scores make the assumption that each test speaker has a representativity which is proportional to its number of test utterances , which is certainly not always a meaningful hypothesis.
Test set scores can therefore be used as an overall performance measure if the test data represent accurately the profile and behaviour of the user population, both in terms of population composition and individual frequency of use. If only the composition of the test set population is representative of the general user population, average scores allow neutralisation of the possible discrepancies in number of utterances per speaker. If finally the composition of the test set speaker population is not representative, gender-balanced scores provide a general purpose estimate.
If there is a way to estimate separately the relative representativity for each test registered speaker , a representative mi sclassification rate can be computed as in equation (11.30). Conversely, some techniques such as those used in public opinion polls can be resorted to in order to select a representative test population when setting up an evaluation experiment.