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Ordinary least squares.

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So we have our data points.

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Now the question is 
that we're answering in this tutorial is

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how do we know
which of the sloped lines is the best one?

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Is it
this one? Is it this one? Is it this one?

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Is it this one or is it this one?

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As we can see,
there can be multiple slope lines

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that we can draw through our data points.

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And how do we know
which one is the best one.

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Which is the best linear regression.

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And in fact
how do we even define the best one.

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So in order to answer those questions,
we need to look at a method

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called the ordinary least squares method.

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And the way it works in a visual sense
is we need to take our data points

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and project them vertically onto,

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our linear regression line.

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Now we would need to do this
for every single linear regression line

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that we're considering.

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But for simplicity's sake, in

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this tutorial we're just going to do it
with this, line here in the middle.

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Now for each pair of points
we have two values y I and y I hat.

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So what are these values?

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Y is the actual, amount

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of, potatoes in our case, in our example,
potatoes yielded from the farm

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when that, specific amount of nitrogen
fertilizer was used.

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So let's say 15kg of nitrogen
fertilizer were used

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and the farm yielded two tons of potatoes.

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Y I hat, on the other hand,
is what this linear regression

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that we're considering what it predicts
the yield,

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to be or two would have been.

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So in this case, let's say I
again, 15kg of nitrogen were used.

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But the, linear regression
that we're looking at predicts that only

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one and a half tonnes of potatoes,
would have been yielded from the farm.

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As you can see,
there's a slight difference between the,

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actual value, actual yield
and the predicted.

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And that's normal.

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It's never going to be perfectly.

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This line's never going to go perfectly
through every single data point.

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That's simply impossible.

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but what we want to do
is we want to find the best line,

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and it will be related to how small

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these differences are, as we can imagine.

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So let's have a look here.

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There's our y and y hat.

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The difference
between them is called the residual.

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here's our equation.

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And the best equation is such equation
A where b or where

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such an equation where the parameters b0
and b1 are such

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that the sum
of the squares of the residuals

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is minimized, and that's why it's called
the ordinary least squares method.

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So we need to take all of these, 
residuals, these differences.

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We need to square them for every single 
data point.

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and then we need to add up the sum
and whenever we find the smallest value.

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So for whichever regression line,
this value is going to be the smallest,

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that will be the best regression line.

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And that will guarantee that the line
is going nicely through our data points.

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And it is the best line or the best linear

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regression
to use for modeling our problem.

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So there you go.

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That's in a nutshell how the ordinary
least squares method works.

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And I look forward
to seeing you next time.

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And till then enjoy machine learning.