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So, to start with, let's look at the machine learning model.
This multiple regression model.
And let's just recall where we left off in the last module, where we were talking
about simple linear regression, where our goal was just to fit a line to the data.
And we just had a single input.
And in our example, we always talked about having square feet and
trying to model the relationship between square feet of a house and
the output, which was the value of the house.
But as the name implies, this simple linear regression model is really simple.
And in a lot of cases,
we're gonna be interested in more complex functions of our input.
So one example of this is something called polynomial regression.
And we actually saw this back in the first course in the specialization.
And in that case,
what we did was we took our simple linear regression model and our fit of that.
Of course at that time, when we were in the first course of the specialization,
we didn't have all this terminology that we learned in the last module, but
now we know that this is a simple linear regression model.
And we take this fit and we show it to our friend and we say, hey, look,
this is so cool.
I have this line that I fit to my data, and
now I can predict the value of my house.
And your friend is a little bit skeptical and says, dude,
it's not a linear relationship between square feet and the value of a house.
He's looking at the data and saying he just doesn't believe it.
Instead, he thinks it's a quadratic fit.
So, what your friend is saying is that he doesn't believe the model that you use.
He doesn't believe that it's just this linear relationship, of course plus error.
He thinks that there's this quadratic function, which has the following
equation, underlying the relationship between square feet and house value.
And again,
our regression model is gonna assume that there's some noise around that.
But of course, you could consider even higher order polynomials.
For example, here, I'm showing some pth order polynomial that you might choose to
be your model of the relationship between square feet and the value of the house.
So here's our generic polynomial regression model,
where we take our observation, yi, and model it as this polynomial in terms of,
for example, square feet of our house, which is just some input x.
And then we assume that there's some error, epsilon i.
So that's the error associated with the ith observation.
And what we see is that in this model, in contrast to our simple linear regression
model, we have all these powers of x that are now appearing in the model.
And what we can do, is we can treat these different powers of x as features.
Okay, so now we're introducing this new word features.
And what features are, they're just some function of your input.
So, in this case, in particular, our features, just to be very explicit,
our first feature of the model is just the number 1,
which is called the constant feature.
Then the second feature of our model is x.
So, that's just the linear term, just like we had in simple linear regression.
And then our third feature is x squared.
And we keep going up to our p+1 feature, which is x to the power p.
And associated with each one of these features in our model is a parameter.
So we have some p+1 parameters, w0, which is just the intercept term.
All the way up to wp, the coefficient associated with pth power of our input.
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