0:07

explicit to real value outcomes and real value predictors,

Â so take as a possible example that y looks something like this.

Â y looks like a function over let's say time from time zero to one.

Â And x also looks like a function, over time from say zero to one.

Â So let's consider the space L^2[0,1] which is this space of square

Â rootable functions on zero one where the interproduct of f and

Â g on the space Is integral from zero to one.

Â Half of t, g f, t, dt.

Â So, what we want to do, consider regression to the origin.

Â We wanna explain, why with the scale of multiple times this function x.

Â 1:15

So we can just do the same thing we did before.

Â Y minus beta hat x plus beta hat x

Â Minus beta x quantity squared,

Â and we can expand that out just using the standard products of squares and

Â get y minus beta hat x squared plus

Â inner product of y minus beta hat x.

Â X, and beta hat x minus beta of x plus twice that,

Â and then plus the inner product beta hat

Â x minus beta of x quantity squared.

Â Now, this quantity Is positive, so if we get rid of it, we only get larger.

Â 2:07

And I'd like for you to go through the notes or do it as homework,

Â to show that that quantity is 0,

Â so that we get that, or if we plug in an arbitrary value of beta,

Â we're always gonna be bigger than if we plug in the specific value of beta hat.

Â So what we see is that the least squares equation works, in this case,

Â in this generalized space.

Â 2:28

So that if we want to explain an observed outcome y, we wanna explain

Â it as a function of the other function x, as a scale or multiple of x.

Â Then, we get the same answer.

Â 2:52

Which of course has a finite interval on that range.

Â Then the beta hat.

Â Works out to be in a product of y and x over the interproduct of x.

Â The self denominator is just one, the numerator is integral y of t,

Â times the one.

Â Which I'm gonna omit dt over zero one.

Â So we could just call that function y bar the average of

Â the function over the range domain 0 to 1.

Â So we could define say y tilde as the centered version of y,

Â y minus y bar times the function.

Â Let me call J the function, let constant one, between zero and one.

Â Now let's consider linear regression and let's x tilde

Â the x minus x bar time J, the standard versions.

Â 4:01

Let me define the covariance between two functions y and

Â x as the integral 0 to 1 of

Â (y- y bar times j) (x- x bar times j).

Â And I'm omitting the the dt.

Â I'm omitting the fact that this is J of t, this is y of t.

Â Okay I guess I'm not omitting it cuz I'm writing it right now, Okay.

Â so I'm gonna define that as the covariance and

Â then define the variance of say y as the covariance of y by itself.

Â So now we've just extended things like the covariance and

Â the variance, too and of course we can then also extend the correlation and

Â the standard deviation to, now, functions on the square integral space.

Â Now let's suppose we wanted to minimize y,

Â our function y minus beta not times j minus beta 1 times x.

Â We can do our same old trick if we hold Beta 1 fixed, right?

Â And think of Y minus beta 1 x as a single function,

Â then we're gonna get at the solution beta not, as it depends on beta 1,

Â has to be equal to y bar minus beta 1 x bar.

Â Plugging that back in.

Â We're going to then just get a regression to the origin case again.

Â And we're gonna get the beta 1 is equal to beta 1 hat is equal to the correlation.

Â The correlation between the function y and the function x times the functional

Â standard deviation of y divided by the functional standard deviation of x.

Â 5:43

In addition, then we're gonna get to beta not hat,

Â plug it in back into it is y bar minus beta one hat, x bar.

Â So more than anything what I wanted to show is that we can extend

Â our equations that we've written out for linear regression and

Â the same thing is true to multivariable aggression.

Â We can extend them to these more complex spaces.

Â This is an example of a so called Hilbert space and you can just have linear

Â regression and multi variable regression for a general Hilbert space you basically

Â just need the inner product and to define concepts like the correlation and

Â the covariance like this but all of the results basically turn out to be the same

Â