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BACKGROUND

MFCCs are coefficients which represent the most important parts of speech, and about 12 of them are used to model a one 512 points long frame (of speech). Along with them you would use delta coeffients, which track the change of MFCCs over time (adjecent frames).

I want to use these feature vectors (dimensions 12+12=24) as data for a speech recognizer, which is based on a Hidden Markov Model, whose states are mixtures of Gaussians.I will use training data to re-estimate parameters of a HMM, such as:

transition matrix $A$,

starting state probabilities $\pi$,

weights for each of the mixtures inside a state $c_{jk}$ (jth state, kth mixture),

means of the mixtures $\mu_{jk}$

covariances of the mixtures $\Sigma_{jk}$

I'm using Baum–Welch algorithm to do this.

PROBLEM

I have found two suitable libraries online for the re-estimation, but when I feed them my data, one sequence (audio file of a particular keyword), they complete very quickly and when I try to get log-probabilities they return positive values. But, one algorithm has informed me immediately that during an iteration, when it's time to update a particular covariance matrix, on the nth line I have a badly scaled or a singular matrix. When I look at that line I see that it's doing an inverse of a $\Sigma_{jk}$, which is defined as $\Sigma_{jk} = \frac{\sum\limits_{t=1}^T\gamma_t(j,k) (o_t - \mu_j)(o_t - \mu_j)'}{\sum\limits_{t=1}^T \gamma_t(j,k)}$

After seome tinkering with the MFCCs and deltas, this other implementation of BM algorithm tells me that I am dividing by 0 when assigning new covariances (step above).

Gammas are obtained from:

$\gamma_t(j,k) = \frac{\alpha_t(j)\beta_t(j)}{\sum\limits_{j=1}^N\alpha_t(j)\beta_t(j)} \frac{c_{jk}\mathcal{N}(o_t, \mu_{jk}, \Sigma_{j,k})}{\sum\limits_{k=1}^M c_{jk}\mathcal{N}(o_t, \mu_{jk}, \Sigma_{j,k})}$

Can someone more experienced in data science see what is wrong with gamma? I have tried normalazing feature vectors, but it did not help. Below I will list how I have calculated MFCCs, in case it is of help.

MFCC CALCULATION

No noise cancelling (audio is quite clear), no windows functions, frames overlap by 50%.

I find the FFT of a 512 long sample. Then I find its absolute value.

There are 26 filters in the filterbank, I dot product every one of them with the previously found absolute value, and I get 26 mel cepstrum coefficients.

Then I do the DCT2 on these coefficients and I keep 12 of them, 2-13

Delta coefficients are calculated: $d_t=\frac{\sum\limits_{n=1}^N n(c_{t+n} - c_{t-n})}{2\sum\limits_{n=1}^Nn^2}$

Help is appreciated. Here is the work I have used as a guide.

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The very same work I cited offers instruction on how to deal with this! Section 4.6.3. When you have high dimensional inputs, 24 in my case, exponents in normal distributions can easily become large negative numbers. Further explanation in said work.

$-\frac{1}{2}(o_t - \mu_{jk})^T\Sigma_{jk}(o_t-\mu_{jk})$

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