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So, we saw in the previous
video that the house prices

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were distributed over the
area in an interesting way,

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certainly not the
kind of linear way.

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And we wouldn't necessarily
expect linear regression

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to do very well at
predicting house price,

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just given latitude
and longitude.

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We can kind of develop
an intuition more

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by plotting the relationship
between latitude and house

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prices-- which doesn't look
very linear-- or the longitude

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and the house prices, which
also looks pretty nonlinear.

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So, we'll try fitting it in
a linear regression anyway.

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So, let's call it latlonlm.

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And we'll use the LM
command, linear model,

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to predict house prices based
on latitude and longitude using

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the Boston data set.

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If we take a look at
our linear regression,

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we see that r squared is
around 0.1, which is not great.

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The latitude is not
significant, which

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means the north-south
differences aren't

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going to be really used at all.

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Longitude is significant,
and it's negative.

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Which we can interpret as,
as we go towards the oceans--

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we go towards the east-- house
prices decrease linearly.

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So this all seems
kind of unlikely,

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but let's work with it.

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So let's see how this
linear regression

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model looks on a plot.

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So let's plot the
census tracts again.

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OK.

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Now, remember before, we had--
from the previous video--

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we plotted the
above-median house prices.

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So we're going to do
that one more time.

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Median was 21.2.

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We had-- the color was red.

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And we used solid dots.

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Ha.

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Oops.

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See what I did there?

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I used the plot command,
instead of the points command,

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and it plotted just
the new points.

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I meant to plot
the original points

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and use the points
command to plot it

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on top of the existing plot.

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OK.

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So that's more like it.

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So now we have the median values
with the above median value

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census tracts.

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So, OK, we want to
see, now, the question

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we're going to
ask, and then plot,

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is what does a linear regression
model think is above median.

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So we could just do
this pretty easily.

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We have latlonlm$fitted.values
and this is what the linear

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regression model predicts for
each of the 506 census tracts.

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So we'll plot these on top.

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Boston$LON-- take all
the census tracts,

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such that the latlonlm's fitted
values are above the median.

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Take the latitudes, too.

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And I'm going to make them blue,
but let's pause for a moment

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and think.

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If we use the dots again,
we'll cover up the red dots

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and cover up some
of the black dots.

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What we won't be
able to see is where

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the red dots and the
blue dots match up.

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You know, we're
interested in seeing

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how the linear regression
matches up with the truth.

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So it'd be ideal
if we could plot

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the linear regression blue
dots on top of the red dots,

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in some way that we can
still see the red dots.

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It turns out that
you can actually

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pass in characters
to this PCH option.

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So since we're
talking about money,

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let's plot dollar signs
instead of points.

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And there you have it.

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So, the linear regression
model has plotted a dollar sign

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for every time it
thinks the census

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tract is above median value.

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And you can see
that, indeed, it's

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almost as-- you can
see the sharp line

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that the linear
regression defines.

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And how it's pretty
much vertical,

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because remember before,
the latitude variable

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was not very significant
in the regression.

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So that's interesting
and pretty wrong.

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One thing that
really stands out is

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how it says Boston is
mostly above median.

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Even knowing-- we saw it
right from the start--

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there's a big
non-red spot, right

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in the middle of
Boston, where the house

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prices were below the median.

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So the linear regression model
isn't really doing a good job.

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And it's completely
ignored everything

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to the right side
of the picture.