So now let's go on with our lecture and to show you the next ideas I want to repeat in

a short way the story with the dynamical systems of money for it.

So the vision of this was to say a dynamical system might live in a very high dimensional

space but the dynamics itself is only on a surface or on a subset of this high dimensional

space.

Examples of this are examples like this geometric picture on the left side or simply the idea

of the side conditions and on the right side here where you have equations as side conditions,

energy conservation, impulse conservation and such things.

And so the point here is to solve such a thing we have to find data driven analysis how we

can cut down a high dimensional description down to something which is low dimensional

and the framework for this obviously is something like an autoencoder.

Then the question is how can I combine the autoencoder story with the dynamical system

which means the description over time.

And the answer to this in the years 2004 around was to say let's take the autoencoder and

then let's take a narrow correctional network and whenever a new thing here is observed,

a new high dimensional description is observed.

First of all the compression part of the autoencoder is used to make it to a smaller description

and then the dynamical analysis here is only acting on the smaller description and then

later when we do a forecast we have only a forecast for the low dimensional description

then we apply the second part of the autoencoder to pump it up to the full description of what

we want to do there.

So this is a combination of two points here the subnetworks for the compression and then

the dynamical part here.

This looks like two different models but it's not.

Please have in mind that the matrices E and E slash here they are also part here which

means from the viewpoint of shared weights this is simply an extended shared weight story

where the matrices which you have here are shared weights again used here in this additional

part here.

Now what is the trick with the additional part here, the trick is that you say here

I have a simple problem because this is a problem without any time in it.

It's always the same time point and even put in the output side.

Why the complicated stuff is the dynamical system here.

This is something which is acting then in the lower dimensional description and the

one example I have explained yesterday was the one with the project on the German Railway

energy forecast and the other example I did not explain yesterday.

I told you that I will explain it today.

It's another example which from the mathematical strategy here is absolutely similar to what

we have seen before but from the application view you would say it's a completely different

world.

So what is the idea?

The idea is to say we want to do interest rate forecasts and there is not only one interest

rate in economics you have different interest rates.

You have an interest rate for one year, you have an interest rate for two years, three,

four, five, six, seven, eight, nine, ten.

You have even more but let's focus on the interest rates from one year up to ten years.

So again the subject of interest rates they are all together you call it you call this

a yield curve which means the interaction of all these different interest rates is a

yield curve and then you can say let's forecast every individual interest rate.

You can do this in interest rate forecast.

Maybe you have not daily data but you have things like weekly data and then if you do