Detection of the number of speakers

図 1: An example of convergence to the true values
図 2: Input spectrum for Figure 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 tied-GMMs in the model in order to raise possibility of obtaining the true values. Then, obviously, the model may over-fit the given observed specrum. If one Gaussian is enough for approximating the shape of one partial, the same number of underlying harmonic structures must be enough with the tied-GMMs. And this number can be detected by reducing tied-GMM one after another until they become the proper number on the basis of AIC. The specific operation is as follows:

1. Set initial values of $\{\mu_1,\cdots,\mu_K\}$ in the limited frequency range.
2. Estimate the ML model parameters by EM algorithm. However, $w_n^k$ is constrained here as
   

   $$w_1^k=w_2^k=\cdots=w_{N_k}^k (=w^k).$$ (10)

   
   This $w^k$ represents the degree of predominance of $k$th tied-GMM. In Maximization-step, model parameters $\mu_k$ and $w^k$ should be updated to
   
   

   $$\bar{\mu}_k =\!\!\!\! \displaystyle\frac{\displaystyle\sum_{n=1}^{N_k}\!\int_{-\infty}^... ...tyle\sum_{n=1}^{N_k}\int_{-\infty}^{\infty}\!\!\!P_{\theta}(n,k\vert x)f(x)dx},$$ (11)

   $$\bar{w}^k =\!\!\!\! \displaystyle\frac{1}{FN_k}\sum_{n=1}^{N_k}\int_{-\infty}^{\infty}P_{\theta}(n,k\vert x)dx,$$ (12)

   
   
   
   where $F$ is an integral of $f(x)$ with respect to $x$.
3. Calculate AIC with equation (9). Since there are two free parameters for each tied-GMM, the model has $2\times K$ free parameters altogether. If the AIC increases, the number of tied-GMMs just before they are reduced in step4 will be the estimate of the number of harmonic structures.
4. Remove the tied-GMM(s) which conforms either of the two conditions as below and repeat from step 2.
   • The one whose $w^k$ is the minimum among all. Since the contribution to the maximum log-lik-
     elihood must be the least.
   • The one whose $w^k$ 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 circled value indicates the value of AIC at each point. Since AIC takes minimum when $3$ tied-GMMs remain, the detected number here is $3$.