We will denote as ``X'' a population of m registered speakers:
For scoring purposes, we will consider a set of test utterances. We will use the term genuine test utterances for those which correspond to registered speakers, and the term impostor test utterances for those which are treated as belonging to impostor speakers. Note that the same speech segment can be used as a genuine utterance and as an impostor utterance, in some test configurations. Therefore, different notations can correspond to the same speech utterance.
Each registered speaker is supposed to have produced genuine test utterances , the set of which will be denoted as ``'':
where superscript k denotes the test utterance of speaker .
In the rest of this chapter, we will denote as c the total number of
genuine test utterances , and as the proportion of utterances
belonging to speaker in the test set , that is:
With the convention:
we will denote:
Integer is the number of registered speakers for which there is at least one genuine test utterance .
Finally, we will denote as M the set of male registered speakers, as F the set of female registered speakers and as and the respective number of male and female registered speakers for which there is at least one genuine test utterance .
In the most general case, the whole set of impostor test utterances can be divided in subsets corresponding to one among n impostors using the system with a claimed identity . Hence the general notation
to denote the set of impostor test utterances produced by impostor claiming he is .
Similarly to genuine test utterances, we will denote as d the total number of impostor utterances, and as the proportion of impostor tests by impostor against registered speaker , that is:
We will also write:
When the identity of impostors is not a relevant factor (or is unknown), subscript j can be dropped, and denotes the impostor attempt against registered speaker (). Conversely, in open-set speaker identification , impostors do not claim a particular identity; they just try to be identified as one of the registered speakers , whoever this speaker may be. In this case, subscript can be dropped, and denotes the impostor attempt by impostor (). If, moreover, the impostor's identity does not matter, subscript j can also be dropped, and simply denotes the impostor attempt ().
With the conventions:
we will write:
Integer is the number of impostors for which there is at least one test utterance against registered speaker , integer represents the number of registered speakers against which there is at least one test utterance from impostor , is the number of impostors from which there is at least one impostor test utterance, is the number of registered speakers against which there is at least one impostor test utterance , and denotes the total number of couples for which there is at least one impostor test utterance (from against ).
Finally, we will denote as the set of male impostor speakers, as the set of female impostor speakers, and