Experiments and discussion
Methods
Computational models of music styles
Assuming that composer A and composer B each has a unique style that
can be recognized in a certain repertoire,
one can select a feature that looks interesting, define the state space,
and build up the Markov chain models
,
, for composer A and B, respectively.
Based on these two Markov models, for each given unknown work U,
two-way composer identification can be achieved by the following
hypothesis test,
where
is the transition matrix of the repertoire that
contains only the unknown work U,
is
the composer that is more likely than the other one to have written
the work, and is a distance metric on the
space of
probability matrice. In this
research, we use the Kullback-Leibler distance metric,
Although the Kullback-Leibler distance metric is not symmetric and
does not obey the trangular
inequality, it is useful to interpret of the metric as the distance between
distributions [#!Cover!#]. Accepting this interpretation, the above
hypothesis test reads ``if the transition matrix of the unknown work is
closer to that of composer A's repertoire, then it is more likely to have
been written by A.'' This hand-waving statement can be justified
by the following theorem.
THEOREM (likelihood ratio interpretation)
For the two-way composer identification test, if the two marginal
distributions are identical,
then, the difference of the two distances has a
likelihood ratio interpretation,
where
UA is the a priori probability that
U
is generated
according to
, and is the length of U.
Proof:
where denotes how many times the transition between
and
states occurs in U. Therefore,
|
|
(1) |
|
|
(identical marginals) |
(2) |
|
|
END |
(3) |
The above theorem states that, for the purposes of composer identification,
if the marginals of are identical and hence don't help, then
- it is necessary to apply Markov modeling;
- the above mentioned hypothesis test is a maximum likelihood test.
For example, if two composers use as frequently each of
the notes as one another but concatenate them differently, then
Kullback-Leibler distance is the right metric to choose for the purposes
of composer identification.
Experiments and discussion
Methods
Computational models of music styles
Copyright © 2002-06-11
Center for Computer Research in Music and Acoustics,
Stanford University