[MUSIC]
Okay, so up to this point we just considered fitting the data with a line
and the question is, what does a good choice?
So I'm actually feeling pretty good about my analysis to be pretty truthful here.
I fit this line, I minimize the residual sum of squares, I made a prediction for
my house value.
In doing so I leveraged all these observations that I went through and
recorded of all the recent house sales.
And I go and I go to Carlos and I say, hey, look at my analysis.
This is my estimate of the value of our house.
And he goes, well, I'm not so sure.
Because really, to me, I'm not sure that this is a linear trend.
He actually says >> It's not linear.
>> He says it's not linear.
Or according to my cartoon, he says, dude, that's not a linear relationship.
Dude.
No, I guess Carlos would not say dude.
But, anyway. >> He says bro.
Okay.
Bro.
He always refers to me as bro, of course.
Okay.
But anyway, the point is Carlos doesn't think that it's a linear relationship.
He thinks, you know maybe it's quadratic.
He said did you try a quadratic fit?
Well.
Now I look at the plot that he just put up here and
I say actually that looks pretty good.
And what do I have to do?
I have to figure out which is the best quadratic fit to this data.
And how am I gonna do that?
I'm gonna go again and I'm gonna minimize my residual sum of squares.
So I'm just about to go minimize my residual sum of squares.
So let's talk about what that would involve because when I'm
looking at a quadratic function I now have three parameters here.
I have my still my intercept, which is just where is this curve?
Up and down on the y axis.
And then I have this linear term of x, and
then I also have this extra term here, which is now the square of x.
That's where I get that quadratic component.
But I wanna make one quick comment here, so a little aside,
that this is actually still called linear regression.
And the reason is because we think of x squared just as another feature.
And what we see is that the w's always appear just as w's,
not w squared or other functions of w.
And we're going to discuss this in more detail in the regression course.
But remember, even though we're talking about a quadratic function fit to
the data, this is still called linear regression.
Okay but the point I want to make here is we have three parameters
when I'm going to minimize my residual sum of squares,
I'm going to have to search over the space of three different things now.
I have to minimize over the combination of best w zero, w one and
w two and finding the quadratic fit that minimizes my residual sum of squares.
Okay, so I'm just about to go and
do this computation, which actually turns out to also be efficient and again we
are going to discuss the generality of that in the regression course.
But then Carlos has a brilliant idea.
He says wait, wait wait!
I told you about that quadratic, but did you try a 13th order polynomial?
And I go, no I didn't.
>> It makes sense.
>> It does, it makes a lot of sense.
Look at this, this is pretty good.
This is the fit that Carlos gets with his 13th order polynomial.
He says, I just minimized your residual sum of squares.
Pretty good, right?
My residual sum of squares here are basically zero.
But I'm personally not feeling so great about this.
Cuz I'm looking and I'm saying my house isn't worth so little.
I know that, I do.
Yes, we talked about residual sum of squares as being this cost of the fit.
And yes, Carlos seems to have really really really
minimized my residual sum of squares, but something's not sitting right with me.
This function just looks crazy.
[MUSIC]