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[MUSIC]

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Okay.

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Well instead of the analysis
that we just did,

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we can instead think about
modeling the relationship

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between the square footage of
the house and the house sales price.

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And to do this, we're gonna use something
that's called linear regression.

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Okay, so to leverage all the observations
that we've collected, what we wanna do is

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be able to understand the relationship
between the square foot of the house and

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the house sales price.

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And so
the simplest model we can use for this,

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is just fitting a line through the data.

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So here's an example of
a line fit to this data.

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And this line is defined by
an intercept W0 and a slope W1.

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And so often we'll talk about W1
being the weight on the feature X or

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its called the regression coefficient.

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And what this weight has
an interpretation of,

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is as we vary X,
the square footage of the house,

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how much of an impact does that have on
changes in the observed house sales price?

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Okay, so these two things,
our intercept and

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our slope are the parameters of our model.

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And so, to be very explicit here,
we're gonna write this function,

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this linear function here,
with the subscript W,

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that indicates that this function
is specified by parameters.

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W being the set of W0 and W1.

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Okay, so this is the line
that we fit through the data.

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But a question is, which line is
the right line or a good line to use for

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a given data set.

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So maybe we could draw this line or
this one instead.

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And each one of those is given by
a different set of parameters w.

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And a question is which w do we
want to choose for our model?

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Okay, well, to think about this,
let's talk about defining a cost for

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a given line.

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And one very common cost
associated with a specific fit to

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the data is something called
the residual sum of squares.

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So, the residual sum of squares,
we take our fitted line and

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we look at each of our observations.

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And we look at how far is that observation

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from what the model would predict.

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Which is just the point on the line.

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So we look at all of these distances here,
and

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we actually look at the square
of these distances.

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That's why it's called the residual.

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The residual is the difference of your
prediction and your actual observation.

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We're gonna look at the square and
we're going to sum over them.

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Okay, so this is the equation, here more
explicitly, where we have the price,

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this is our observed house sales price for
the first house.

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And what is this term here?

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So, if this is what I'm
calling dollar house one,

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then what this point here is,

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is if this is square footage of house one.

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Then this x that I've drawn here

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represents exactly this term here.

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This is that value, the point on the line.

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So this dollar sign
minus this term here is

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the difference between our observed
house price versus what our model,

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just this line, predicts for
this given house square foot.

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And we're squaring that and

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we're summing over all the different
houses in our data set.

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Okay, so we think about trying to find the
best line according to this metric that

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I've defined here,
this residual sum of squares,

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what we do is we research over
all possible W 0 and W 1.

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So all possible lines,

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and we choose the one that minimizes
the residual sum of squares.

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So we're gonna denote
the resulting W as W hat.

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So remember that's gonna be the set
of w hat zero, and w hat one.

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Our intercept and our slope.

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Okay.

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So there exists really nice and
efficient algorithms for

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computing this w hat,
this search over all possible ws.

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We could look at parameters of this model.

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But we're gonna discuss these algorithms
more in the regression course.

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Ok, so now let's talk about how to take
our estimated model parameters and

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use them to predict the value of my house.

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So I've gone through, sorry,
this star shouldn't be there.

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This should be a hat, just to be clear.

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And I've gone through and plotted the line

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associated with our estimated w0 hat and
w1 hat.

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And now, here's my house.

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This is its square footage.

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And the best guess of my house price,
is simply what the line predicts.

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So I go, and I compute what is the value
for this square footage of my house.

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Which is W0 hat plus W1 hat
times the square footage

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of my house okay very,
very straightforward.

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