次へ: Detection of s and
上へ: Multipitch Detection Algorithm
戻る: Criterion of Model Selection
Detection of the number of speakers
図 1:
An example of convergence to the true values

図 2:
Input spectrum for Fig 1

It is generally known that ML estimates obtained by EM algorithm firmly
depend on initial values
and may often converge to undesirable values.
To avoid this, we first prepare extra amount of harmonicGMMs in the model
in order to raise possibility of obtaining the true values.
Then, obviously, the model may overfit the given observed specrum.
If one Gaussian is enough for approximating the shape of one
partial^{1},
the same number of underlying harmonic structures must be enough with
the harmonicGMMs.
And this number can be detected by reducing harmonicGMM one after another
until they become the proper number on the basis of AIC.
The specific operation is as follows:
 Set initial values of
in the limited frequency
range.
 Estimate the ML model parameters by EM algorithm. However,
is constrained here as
This represents the degree of predominance of th
harmonicGMM.
In Maximizationstep, model parameters and should be
updated to
where is an integral of with respect to .
 Calculate AIC with equation (10). Since there are two free
parameters for each harmonicGMM, the model has free
parameters altogether. If the AIC increases, the number of
harmonicGMMs just before they are reduced in step4 will be the estimate of
the number of harmonic structures.
 Remove the harmonicGMM(s) which conforms either of the two conditions
as below and repeat from step 2.
 The one whose is the minimum among all. Since the
contribution to the maximum loglik
elihood must be the least.
 The one whose is smaller if the two adjacent
representative means become closer than a certain distance
(threshold). Since the two representative means are presumed to
converge to the same optimal solution.
An example of how this process actually works is shown in Fig.1 where the
observed spectrum used is depicted in Fig.2.
The broken line represents the point where the model parameters were
judged to be converged and the line graph indicates the value of AIC
at each point. Since AIC takes minimum when harmonicGMMs remain, the
detected number here is .
次へ: Detection of s and
上へ: Multipitch Detection Algorithm
戻る: Criterion of Model Selection
平成16年3月25日