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Well, we basically, yeah, basically we do not look at the semantics, so not the words.

What we are looking at are the signals and the random measures associated with the signals that are assumed to be independent.

Right, that's right.

Well, I mean, well, statistical dependency has different levels. For instance, you can have also conditionally statistically independent measurements.

Conditionally independent means given a certain environment, given a certain context.

So if we say, okay, we have two sentences spoken within this lecture, that's a precondition.

And we can say, given the fact that we have the same topic, pattern recognition, based on that, the measurements are statistically independent.

So in terms of math, it's written like this.

We have P of X, Y given Z, and these two variables are conditionally independent if we can write it this way.

P of Y given Z.

That's something that we use, for instance, in Bayesian network modeling a lot. But you are right, there is a borderline.

Let's assume that we have a measurement A, a measurement B, and the joint density is the factorization in a way that we multiply P of A and P of B to get the joint density.

And the second thing we have considered yesterday was data normalization.

We talked about the whitening transform. What is the whitening transform actually doing?

What's your name again?

Radish.

Radish?

I'm asking this again and again and again, I know, but my brain is limited.

So Radish, what was the idea of the whitening? Where are you from?

India.

India? Mumbai? No, I have been to Mumbai, that's the only city I know.

In whitening, basically, we remove the coordination between the elements.

Okay, that's one point and the other point.

We normalize the mean in a way that we have zero mean, so the signal is just jumping around the zero, the origin of the coordinate system.

It's just a rigid transformation in a way that we translate the data, just the translation.

So you subtract the mean from all the measurements and you get a zero mean random measure.

And the second is that you enforce an identity matrix to be the covariance matrix.

And what is the trick, how do we do that?

SVD.

SVD? What do you answer if I ask you a question and you have no clue?

What is the answer?

SVD.

SVD, so he's smart.

Okay, and by using the SVD, we can split up the singular value decomposition in a way that we get a linear transform for the random measures that in between, we basically end up with an identity matrix.

So what linear mappings do to random measures that are normally distributed?

Well, the linear transform given by a matrix is just multiplied with a mean vector.

If the mean vector is zero, I multiply the matrix with a mean vector zero.

What happens?

Zero.

Zero, yeah.

And if I multiply a random variable with a matrix, what effect has this transformation on the covariance matrix?

It's matrix A times previous covariance matrix times matrix A transpose.

Did I show to you a proof why this is true?

No, I did not, because this proof is a little bit tricky.

We use generating functions to show that, actually.

So for the theoreticians, that might be interesting to know.

But for us, it's completely sufficient if you are aware of this rule.

Just accept it as a rule.

Okay, so we can do this type of normalization of the data.