So our next chapter here is something which is, let us say, uncorrelated or very different

to what we have done before.

And the story is to think about causal versus retrocausal and a mixture of causal and retrocausal

explanations of dynamical systems.

Now to do so, first of all, let's go back to our HCNN.

So this is a fancy thing because it includes the symmetricity between past and future.

And we do not have a lot of problems that we have with small and open dynamical systems.

Nevertheless, it's a causal system.

You explain the next time point by what was there before.

And then you do it over and over and over again.

So you have to develop it through time.

In 2010, one of our large applications was the copper price forecast, which means metal

prices.

And so therefore, this is a thing which was not only academic fun, but it was a practical

question.

How can you do such a thing?

And I told you the original one was unsolvable, but including the teacher forcing, then you

can solve it.

And an example of an outcome of this forecast at this time point here was for the EX, the

energy market in Germany.

Here you have for the 20 days ahead forecast, which means day by day from the beginning

time point to the next 20 days.

You have the black lines as real world behavior and you have the green lines for the forecast

model.

This was an HCNN.

Not additional tricks, simply HCNN with this architectural teacher forcing at all time

points.

So upside there you have an energy model, where especially here I show the energy price,

the electrical power price.

And downside there you have from the London Metal Exchange, you have a copper price model,

which shows the development of copper.

So the black lines are real world behavior and the green lines were the forecast of an

HCNN done before the generalization passed that you see here.

And if you do so, then you can see there at this time, by the way, the time intervals

upside and downside in the pictures are always the same.

And so here you have two different time intervals.

The one time interval here is the, you can read it down there.

And then the next thing is another time interval a little bit later.

And so you could say, okay, HCNN was the most modern development at this time.

And for the copper price indeed upside there, given the viewpoint of a purchase department,

this is a wonderful forecast because the green line says the minimum price you can have in

the next days is at the point where you have the minimum of the green point here.

And indeed, this is also a perfect price given the real world development that you have there.

So if there are more differences here, that's not important.

If your customer is a purchase department, they are interested in first of all, what

is the trend and second, where is the minimum of the prices so that you can take this time

point as a purchase time point.

And as I told you here, wonderful example, big success.

Same time interval here, we do not have only the energy model.

We also have the metal model here.