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So let's go through some of the regression fundamentals, what's our data,
what's the model that we are going to use, and what's our task event interest.
Okay, so the first thing is we're going to take all of our data, so
all these houses that we looked at that sold recently, and
for each one of them we're going to record some information.
And in this case, in the case of simple regression where we're assuming that
there's just one variable that we're using to predict our house price,
specifically square feet, for every house we're going to record how many square feet
that the house had and what the price was that that house sold for.
And so we record this for each of our houses that have sold in the past.
And this variable x represents the input to our model, okay?
This is what we're going to use for our prediction.
And what's the output, what are we trying to predict?
Well, we're trying to predict the price of the house.
So that y variable is going to be the output.
So what's the difference between the input and the output?
Well the output y is our quantity of interest, this is our goal,
we're trying to predict the value of our house so we can list it for sale.
And we're going to assume that we can predict y based on x,
so we can predict the value of the house based on the square footage of the house.
Okay. So I'm going to take my data, and
I'm going to plot it x versus y,
where x is the square footage of each house, and y is the sales price.
So that's what this cloud of points represents.
And we made exactly this plot in the first course of this specialization.
So, just to be clear, this circle here represents,
let's see the ith house in my data set.
And that house had some number of
square of feet xi, and some sales price yi.
Okay, so this is my data, and what my model represents
is the expected relationship between x and y, remember that's what we're trying to
figure out, because if we have that relationship we can use it for
predicting the value of my house that I'd like to list for sale.
Okay, so
we're going to assume some relationship which is some functional relationship.
So I'm going to call this some function F of X.
And like said, what that function represents is
the expected relationship
between x and y.
So let's walk through this in a little bit more detail.
So, let's look at this house just for
to make this a little bit cleaner, lets look at another house here.
Now I've reused the letter that I like to use.
So I'm going to call this house.
Sorry, I'm going to re-annotate this.
I'm going to call this house j.
This is xj and yj.
And the reason I'm doing this is because i is a special notation for
the house that I'm interested in.
And so now, this house here that I'm looking at I'm going to say that
it sold for some value, yi.
And based on my model, what my model is saying is that I'm assuming
that yi is approximately equal to F(xi).
This functional relationship between the square footage of house i and
its sales price, yi.
But I'm assuming that my model's not 100% accurate.
You can easily imagine that there are errors in this model,
because you can have two houses that have exactly the same number of square feet,
but sell for very different prices.
They could have sold at different times.
They could have had different numbers of bedrooms, or bathrooms, or
size of the yard, or specific location, neighborhoods, school districts.
Lots of things that we might not have taken into account in our model.
So our model is just what we're using as our belief about the relationship for
prediction, but it's not 100% accurate, there's some error.
So the error, so just to be clear, this point here,
this x is exactly f(xi),
it's the function evaluated at some xi value.
And we're saying that our observations, which don't fall
exactly on this curve, defined by F, there's some error.
So we'll call this error specific to ISI, we're going to call it epsilon I.
So what our regression model's saying,
is that we're assuming that our
observation YI is equal to f(xi),
our expected relationship between x and
y, plus some error.
And, in particular, we're treating this error as a random quantity and
we're going to assume that the expected value of this error,
so this notation is the expected value.
So we're assuming the expected value of this air is equal to zero.
And what is expected value?
Well, it's just a weighted average over all possible values that air can take,
weighted by how likely the air is to take each of those values.
But what this is saying, saying that our expected error is going to be zero,
means that it's equally likely,
that we're going to have positive or negative error for any given house sale.
So, it's equally likely that our error is positive or negative.
And what does this imply?
This implies that it's equally likely that our observation,
the specific observation that we get, is above or
below the functional relationship defined by F.
So, Y-i is equally
likely to be above or
below, F of xi.
Okay.
So, I want to be clear that this is the model that we're using.
This is how we're assuming the world works.
And there's this very famous quote by George Box that says,
"Essentially, all models are wrong, but some are useful".
So what this means is no models going to be exactly how the world works.
It's not going to exactly predict how houses sell, just based on square feet,
or even if you incorporated other things as well.
There's always different idiosyncracies in how the world works.
But models are going to represent some useful extraction
of the relationship between, for example square foot and
price, that is useful for a task such as prediction.
Okay, so I want to make it very clear that everything I wrote on the last line
is just our belief about how the world is going to work.
Or maybe it's not even our belief, maybe it's just something we're going to use
because it's useful, as George Bach said, or can be useful.
And we're going to talk a lot about how we assess how useful things are in this
course, but we're going to hold off on that conversation for now.
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