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[MUSIC]

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So let's go through some of the regression
fundamentals, what's our data,

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what's the model that we are going to use,
and what's our task event interest.

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Okay, so the first thing is we're
going to take all of our data, so

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all these houses that we looked
at that sold recently, and

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for each one of them we're going
to record some information.

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And in this case, in the case of simple
regression where we're assuming that

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there's just one variable that we're
using to predict our house price,

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specifically square feet, for every house
we're going to record how many square feet

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that the house had and what the price
was that that house sold for.

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And so we record this for each of our
houses that have sold in the past.

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And this variable x represents
the input to our model, okay?

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This is what we're going to use for
our prediction.

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And what's the output,
what are we trying to predict?

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Well, we're trying to predict
the price of the house.

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So that y variable is
going to be the output.

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So what's the difference between
the input and the output?

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Well the output y is our quantity
of interest, this is our goal,

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we're trying to predict the value of
our house so we can list it for sale.

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And we're going to assume that
we can predict y based on x,

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so we can predict the value of the house
based on the square footage of the house.

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Okay.
So I'm going to take my data, and

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I'm going to plot it x versus y,

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where x is the square footage of each
house, and y is the sales price.

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So that's what this cloud
of points represents.

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And we made exactly this plot in
the first course of this specialization.

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So, just to be clear,
this circle here represents,

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let's see the ith house in my data set.

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And that house had some number of

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square of feet xi, and
some sales price yi.

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Okay, so this is my data,
and what my model represents

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is the expected relationship between x and
y, remember that's what we're trying to

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figure out, because if we have that
relationship we can use it for

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predicting the value of my house
that I'd like to list for sale.

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Okay, so

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we're going to assume some relationship
which is some functional relationship.

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So I'm going to call this
some function F of X.

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And like said,
what that function represents is

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the expected relationship

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between x and y.

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So let's walk through this
in a little bit more detail.

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So, let's look at this house just for

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to make this a little bit cleaner,
lets look at another house here.

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Now I've reused the letter
that I like to use.

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So I'm going to call this house.

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Sorry, I'm going to re-annotate this.

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I'm going to call this house j.

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This is xj and yj.

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And the reason I'm doing this is
because i is a special notation for

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the house that I'm interested in.

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And so now, this house here that I'm
looking at I'm going to say that

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it sold for some value, yi.

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And based on my model, what my model
is saying is that I'm assuming

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that yi is approximately equal to F(xi).

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This functional relationship between
the square footage of house i and

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its sales price, yi.

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But I'm assuming that my
model's not 100% accurate.

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You can easily imagine that
there are errors in this model,

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because you can have two houses that have
exactly the same number of square feet,

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but sell for very different prices.

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They could have sold at different times.

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They could have had different numbers
of bedrooms, or bathrooms, or

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size of the yard, or specific location,
neighborhoods, school districts.

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Lots of things that we might not have
taken into account in our model.

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So our model is just what we're using as
our belief about the relationship for

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prediction, but it's not 100% accurate,
there's some error.

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So the error, so just to be clear,
this point here,

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this x is exactly f(xi),

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it's the function
evaluated at some xi value.

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And we're saying that our observations,
which don't fall

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exactly on this curve,
defined by F, there's some error.

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So we'll call this error specific to ISI,
we're going to call it epsilon I.

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So what our regression model's saying,

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is that we're assuming that our

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observation YI is equal to f(xi),

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our expected relationship between x and

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y, plus some error.

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And, in particular, we're treating
this error as a random quantity and

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we're going to assume that
the expected value of this error,

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so this notation is the expected value.

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So we're assuming the expected
value of this air is equal to zero.

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And what is expected value?

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Well, it's just a weighted average over
all possible values that air can take,

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weighted by how likely the air
is to take each of those values.

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But what this is saying, saying that
our expected error is going to be zero,

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means that it's equally likely,

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that we're going to have positive or
negative error for any given house sale.

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So, it's equally likely that our
error is positive or negative.

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And what does this imply?

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This implies that it's equally
likely that our observation,

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the specific observation that we get,
is above or

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below the functional
relationship defined by F.

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So, Y-i is equally

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likely to be above or

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below, F of xi.

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Okay.

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So, I want to be clear that this
is the model that we're using.

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This is how we're
assuming the world works.

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And there's this very famous
quote by George Box that says,

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"Essentially, all models are wrong,
but some are useful".

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So what this means is no models going
to be exactly how the world works.

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It's not going to exactly predict how
houses sell, just based on square feet,

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or even if you incorporated
other things as well.

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There's always different
idiosyncracies in how the world works.

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But models are going to
represent some useful extraction

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of the relationship between,
for example square foot and

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price, that is useful for
a task such as prediction.

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Okay, so I want to make it very clear
that everything I wrote on the last line

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is just our belief about how
the world is going to work.

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Or maybe it's not even our belief, maybe
it's just something we're going to use

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because it's useful,
as George Bach said, or can be useful.

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And we're going to talk a lot about how
we assess how useful things are in this

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course, but we're going to hold
off on that conversation for now.

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[MUSIC]