# Markov processes

I am trying to read a paper and I found it difficult to understand. This is a quote:

Standard Mel frequency cepstrum coefficient (MFCC) computation technique utilizes discrete cosine transform (DCT) for decorrelating log energies of filter bank output. The use of DCT is reasonable here as the covariance matrix of Mel filter bank log energy (MFLE) can be compared with that of highly correlated Markov-I process.

I understand how MFCCs are computed. I know DCT coefficients are computed from the filtered energy banks. I am struggling to understand how exactly is covariance matrix calculated. Another problem is that I don't fully understood what Markov-I property is. I have not found much about this specific property in internet appart from Markov property which is AFAIK a process that does not rely on preceding states. They also say, the coefficients after DCT are not really decoupled. I would be grateful for some help in explanation of what markov-I process is and how DCT is related to a covariance matrix, and what covariance matrix is. Thanks.

• Are you asking on how to compute the covariance matrix? I.e. You don't know how or understand how to implement such? – Phorce Mar 2 '15 at 15:21
• Hi. Sorry. English is not my best side. I'll come back to this post but now I think I need to read a little more. I am not even clear how to ask the question precisely. The paper simply mentions DCT and covariance matrix. They say The use of DCT is reasonable here as the covariance matrix of Mel filter bank log energy (MFLE) can be compared with that of highly correlated Markov-I process. So the Question is how covariance matrix is computed in such context. – Celdor Mar 3 '15 at 15:25
• Can you please post a link to the paper so I can check? – Phorce Mar 3 '15 at 15:37
• – Celdor Mar 4 '15 at 10:07
• I got: First, the covariance matrix of the log energies does not exactly follow Markov-I property. Meaning, if you have a matrix of the log energies, and you take the covariance matrix of such then it will not follow the Marvoc-I property. I still can't get what you're asking, which is probably why you have little replies. Can you edit the question and say what you're trying to do, and, what you're confused about? – Phorce Mar 4 '15 at 10:16

## 1 Answer

From your question, I got:

I am struggling to understand how exactly is computed the covariance matrix.

The Covariance matrix is written as:

$$COV = \frac{\sum_{i=1}^n (X_{i} - \overline{x}) (Y_{i} - \overline{y})}{n - 1}$$

You can then compute this (In English) as:

1. Calculate the mean of the matrix
2. Calculate the product of the matrix (Multiply by the mean)
3. Subtract the product matrix (Calculated in step 2) by the mean


This is therefore your Covariance matrix.

what covariance matrix is

The Covariance matrix is very similar to Correlation and unlike Correlation, the Covariance matrix is not constrained to being between -1 and 1. In simple terms, the Covariance is a measurement of how changes in one variable are linked with changes in a second variable, showing how two variables are associated with each other. (More Here

I can't really explain HOW the DCT is related to the covariance matrix, other than the example given above. The covariance matrix can be used to show if the values are dependant on each other. It could also be used to calculate the Eigen values/vectors of a particular matrix. Maybe if you provided a more specific example to the problem domain you would use this, it might be more helpful.

Hope this helps anyway.