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So we talked about polynomial regression,
where you have different powers of your input.
And we also talked about seasonality where you have these sine and cosine bases.
But we can really think about any function of our single input.
So let's write down our
model a little bit more generically in terms of some set of features.
And I'm gonna denote each one of my features with this function H.
So H0 is gonna be my first feature,
H1 my second feature, H capital D, my last feature.
So we can more compactly represent this model
using the sigma notation that we introduced previously.
Where we put an index I equals one to capital D.
Saying we're summing over each of these capital D different features.
And just to be very clear h sub j of x is our jth feature,
and wj is the regression coefficient or weight associated with that feature.
So just to give some examples that we've gone through, this first feature might
just be one, this constant feature that we've used in all of the past examples.
Or when we think about our second feature, h1, maybe that's just our linear term, x.
Our third feature might be x squared or maybe it's our sine basis.
Or we could think of lots of other feature examples and
when we get to our capital Dth feature, maybe it's just our
input raised to the pth power when we're thinking about polynomial regression.
So, going back to our regression flow chart or
block diagram here, we kinda swept something under the rug before.
We never really highlighted this blue feature extraction box,
and we just said the output of it was x.
Really, now that we've learned a little bit more about regression and this
notion of features, really the output of this feature extraction is not x but h(x).
It's our features of our input x.
So x is really the input to our feature extractor, and
the output is some set of functions of x.
So for the remainder of this course,
we're gonna assume that the output of this feature extraction box is h of x.
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