That's what everyone uses.

It's I think, you can prove,

it's about as fast as you can take the Fourier transform.

It's much faster than doing the linear model, and that's why, if you were a sound

engineer, you wouldn't think of it as a linear model problem, you would think of

this as a Fourier transform problem, and you would take fast Fourier transforms.

So, that's in the end of the class, but I hope in this lecture,

what you've seen is, one, a case where we can fit pretty complicated functions

with a couple of lines of code of a couple of lines of our code very easily.

Secondly, an instance where we can do something that maybe at first blush,

we would've thought maybe that's not really in the scope of what linear models

can do in diagnosing what are the various notes of a chord.

That's something we just wouldn't naturally think a linear model could do.

We showed that not only are both of those things possible with linear models,

they're actually kind of easy with linear models, it's not

reams of code we had to do, there was, you know six or seven lines at the most for

each of these examples which includes generating the data.

So that just goes to show how powerful these techniques are and so

I hope you've enjoyed the class and

I especially hope that you take the knowledge from this class and build on it.

If you were to build on it my suggestion would be to

learn a little bit more about generalizing your models.

because we only touched on them in this class.

And the second thing would be to approach correlated data and longitudinal data.

Another divergent aspect of linear models that's quite important.

Everything we've done is independent errors and

data that were exchangeable at some level.

A very important subset is when that doesn't happen.

If we measure, take measurements,

on a large collection of siblings, the measurements between the siblings

are going to be closer together than a different pair of siblings.

So handling that correlation is a big part of generalizing

linear models to a broad collection of important topics.

So if I were to suggest two ways to go further with this stuff, that would be it,

generalized linear models, and longitudinal multi-level data.