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All right.

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Let's do this.

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Logistic regression intuition.

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And you can probably already
tell by my voice that I'm pretty excited.

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There's
some very interesting slides coming up.

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And this is quite an important topic.

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But at the same time
it is quite challenging.

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So a quick heads up,
there will be some math.

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And I've done a few run
through of this presentation already.

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And I really I will try my best to convey
everything in the simplest way possible.

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So let's get into it.

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We already know about the linear
regression.

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We know that there is a simple linear
regression,

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and it has a very short formula
with one independent variable.

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And we also have looked
into the multiple linear regression

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which has many independent variables.

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So we already know how to deal
with this type of challenge.

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So when we have a scatterplot like that
where on the horizontal axis

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we've got the independent variable

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on the vertical axis
we've got the dependent variable.

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And this is an example
we looked at salary versus experience.

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How do we create a model here.

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So we use a simple linear regression.

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It puts a line through our data.

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And that line
is modeling our observations.

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So we can basically forecast things and

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compare our actual observations
to our model and so on.

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But so we know how to deal with challenges
like that or problems like that.

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But your company had hired you
as a data scientist.

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What they do
is they send out email offers to customers

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with like a proposal
to buy certain products.

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it might be a clothing store, it might be
grocery store or something like that.

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So, what they do
is basically they send out

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a, offer in, in email,

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to a lot of customers
to purchase certain products.

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And here you've got a sample of those
customers that they contacted recently.

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You've got their age.

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And also you have a variable
whether or not they took action.

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So did the person take up an action,
perform an action.

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Did they take up an offer?
Did they buy a product?

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Did they open up an email, respond
to our email and so on.

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So was the action taken or not?

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And the very black and white
very different.

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But at the same time,
like even though we don't know what to do,

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we don't know what's going on here. It's
not what we're expecting.

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But at the same time,

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intuitively, we can see that
there is some sort of correlation.

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We can see that
the observations on the bottom

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there are a bit more to the left.

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Observations on the top are a bit more
right, implying kind of that probably

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older
people are more likely to take action,

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based on this offer, and the younger
people are more likely to ignore it.

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So can we somehow model this?

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How about we try our, the existing method

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in our toolkit,
which is the linear regression?

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Let's run a linear regression.

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And that's what it looks like.

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As you can tell,

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it doesn't look like the best approach
doesn't look like the best method.

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to solve this problem.

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So let's look into
this in a bit more detail.

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We're going to draw

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the other horizontal line over here.

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Instead of trying to predict
exactly what's going to happen for

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for any given person, let's imagine
a person and let's say we want to predict

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for that person knowing their age,
we want to predict whether they will

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take up the offer or not.

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But instead of predicting exactly whether
they're going to take it up or not,

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how about instead we will,

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predict the probability we will
we will state a probability

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or a likelihood of that person
taking up that offer.

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And if you think of it in that way
right away, things start becoming clearer.

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Right away.

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You can see that, okay,
so this chart is actually from 0 to 1.

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And I also know
that probabilities are from 0 to 1.

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Oh that's interesting.

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So basically I could fit in probabilities
between 0 and 1.

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The fact that the red dots

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the red observations are already either
0 or 1 and nowhere in between.

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Well, that's simply
because we already know the results.

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We already know that
they're either there or there.

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But for something that we're predicting,
it kind of makes sense to say, well,

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I don't know for sure.

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I don't know
100%. He'll take it up or not.

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But I know maybe, maybe with an 80% chance
he'll take it up or not.

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Right.

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And when you think of it that way,
the linear regression line,

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or at least that part that's in the middle
between 0 and 1, it makes sense, right?

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Well, it makes some sense
because that is basically

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it's telling you
that anybody between those ages of,

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for instance, where it's crossing
the horizontal line for the first time,

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it might be where it's crossing
the horizontal axis, it might be like 25,

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or let's say 35,
and we're crossing the vertical,

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the horizontal axis for one,
it might be, let's say 55.

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So those people between 35 and 55,

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they,

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anything in between,
any person that falls in between that age,

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there's a probability of them
taking up this offer.

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And that probability is increasing
as we move to the right,

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as we take more and more older people,
that probability is increasing.

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So the part of the linear regression
in the middle kind of makes sense.

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And we can do something with it.

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But the parts that don't make sense at all
are the ones at the top

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and at the bottom.

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Because a probability can never be
less than zero, I can never be above one.

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So what is the linear regression?

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Trying to give us a hint about here?

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Well, what it's probably saying
what we could interpret as is that

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people above that age that

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nominal age,
we said 55 above that age, they

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they are very, very likely to take off
or actually more than, more than 100%.

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So basically,
they're definitely taking it up.

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Anybody below
35 on the other side, on the left,

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they're definitely not taking it.

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So essentially what we're saying is

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if we ever take that approach,
then we would have to replace this linear

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regression line with a line
that looks like that.

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So let's just cut those bits off
and replace them with horizontal parts.

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And that would

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be a very

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basic, but it still would be an attempt
at creating a model for this situation.

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So we would still be able to use this
to make some sort of predictions

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and assumptions.

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that's a lot that's, talk about

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the correlation between the action
and the age of a person.

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So that's a very basic, understanding.

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And that's kind of the start of our,

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understanding of intuition
behind logistic regression.

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So let's see what
the actual scientific approaches.

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So here
we've got the line that we looked at

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and it is described by this equation.

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Now this part is going to be
this is the most fun part.

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So bear with me

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if you apply to this equation
a sigmoid function which looks like that.

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So you put the y into the

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sigmoid function in purple,
and then you solve

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for y from the purple box
and you put y back into the blue box.

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Then you will get the green box.

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So basically your linear regression
will start to look like this.

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And this is the formula
for logistic regression.

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And what that will do to a chart
which is most importantly

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this visual part.

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It will convert it from

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the chart
that we see at the top to this new chart,

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which is actually the logistic regression
function.

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So if at this stage you're asking yourself

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what just happened, then you're not alone.

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The first time I saw this or I learned
this, this was the expression on my face,

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if you if you told me if you told
you come through with all of that,

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that's super great.

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That means
you'll fly through this section.

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But if you're confused right
now, not a problem.

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I was the same when I was in your shoes.

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So let's, take this step by step.

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Let's look at it step by step.

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Exactly
what happened. So there's our graph.

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There's our independent variable.

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There's our outcome.

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Yes or no.

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So that's the y the dependent variable.

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And there are our observations in our data
set based on these observations.

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And plus using this formula

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which we're going to take as given
this is the logistic regression formula.

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Using this formula.

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And these observations
we come up with this line.

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And what is important to understand here.

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It's not a magical line.

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This line for the logistic regression
is the same

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as a slope
or a trend line for a linear regression.

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So basically what this line is doing is

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it is using
the formula is following the formula.

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And it's the best fitting line
that can fit these data sets.

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So basically we're doing exactly
the same thing as a for linear regression.

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But it just looks different that's all.

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So there there are heaps of these lines
that can you can draw that look like that.

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But only one of them is the best
fitting line.

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So the point of the logistic regression
is to find that best fitting line.

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And this is it.

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So we found the best fitting line
that follows that equation. And

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it fits these

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variables that we or these observations
that we had in our data set.

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After that
we can forget about the equation.

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We forget about the variables.
We've got our line.

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So this is our logistic regression
function

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we found in same thing
as with the linear regression.

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We've created the model.
We've built the model. You can see it.

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This is the model in front of you
right there.

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Now what can we do with this
logistic regression.

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Well we can use it
to predict probabilities.

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And we've already touched on probabilities
that the lie between 0 and 1.

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And that instead of predicting for sure
that something will or will not happen,

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how will we predict probability?

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So let's look at Oh by the way,

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probability here is called p hat.

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So that that's a little sign about the p.

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It gives it the name p hat.

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And anything you see in the info
hat in this section

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just basically means that it's,

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something we're predicting.

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So, and that's, that's the way to remember
it, that that picture P hat.

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So we're predicting this probability

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okay.

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So let's take four random values
for the independent variable for x.

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We're going to say 2030 4050.

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Let's see what happens with the variables.

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So let's put them on the X line.

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Those are the dots.

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And I specifically put dots
not x's or crosses

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because it doesn't mean
that they're on the horizontal.

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The bottom line doesn't mean
that the probability is zero or that

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their dependent variable is zero.

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No, they're just there
because they're on the x axis.

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We just plotted them.

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There has nothing to do
with the vertical axis.

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Now let's what you need to do to find
the probabilities is

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you need to project these values
onto your curve.

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Once you project them you get these blue

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light blue blue dots or blue

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observations which plotted basically.

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So these are the fitted values.

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As you remember in Gretel you have in red.

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You have the actual
and in blue you have the fitted values.

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So these are your fitted values.

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And now if you project them,

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if you want the probabilities you need
to project them to the left like that.

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And let's have a look
at these probabilities.

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So the person who's 20 years old
the probability of taking up

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this offer is very low, perhaps 0.7%.

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So less than one percentage.

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To take up this offer, the person
who's 30 years old, the percent.

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The probability is higher.

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It's about 23%. To take up this offer.

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The person who's 40 years
old, their probability take up.

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This offers 85% according to this model,

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and the person who's 50 years old,
their probability is 99.4%.

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So that's the first thing that you can
get out of a logistic regression.

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That's what we're going to be using.

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Very we're going to be using it
very actively when we're talking

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about building geo
demographic segmentations, because you use

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this probability as a score.

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And we'll talk about this more.

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So you can actually rank people who is the
most likely to take up your friend.

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Who's the least likely
to take your offer up.

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So it's actually even better
than just having a one or a zero.

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You have a probability.

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So you can, order people
by this probability.

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Anyway, the you might want to say, well,
I don't want the probability,

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I want a prediction
as because this is a regression.

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I want a prediction for the,
the y value.

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So okay. We can do that.

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Can we get let's get rid of those,
probabilities.

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Now can we get the Y the actual obviously
we can't get the actual

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because the actual is something
that we can only observe in, in our data

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set or in real life.

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We can only get a prediction
for the actual.

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So y hat, as it has
suggests, is a predicted value

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for the dependent variable.

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How did you get y hat?

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Well, the approach is very arbitrary.

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You have to select a line.

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Let's wait for that okay.

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So you have to select a line.

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In this case we're going to take 50%.

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you can select it anywhere,
but 50% is usually selected

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because it's in the middle.

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And it's therefore you have symmetry
and anything below this line.

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So anything that falls on the curve below
this line will be projected

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downwards onto the zero line,
which, which makes sense.

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So it's basically saying
if your probability,

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your predicted probability of taking up
this offer is less than 50%, let's say

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it's 40% or 20%,

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then we're just going to say that you're

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not you're
probably not going to take up this offer.

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And so that's what's happening.

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The person with 0.7%, the person
with whatever it was, 27%, 23%,

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their, predicted their probabilities
are not zero, but they're below 50.

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So you are if you're if you're

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if you do require a Y hat, so a predicted

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value, a yes or no value,
then make sense that

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if something is below
50%, you're probably going to say that

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they're not
going to take up the offer now.

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And you think about,
oh yeah, so there you go.

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Both of them. Y hat so zero.

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Now anything above the horizontal line
that we've selected, the 50% line.

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It is agreed that all of those values

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that fall onto the curve above that
line are projected upwards.

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They're projected onto the yes line,
the one line.

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So the person that's had a probability of,
85% is projected

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upwards, and the person had the
probability of 99.7% is projected upwards.

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Also makes sense. Right.

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So if, if, somebody has got a chart,
you're predicting

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that somebody is probability
of taking up an offer is 85%.

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Then if you have to say yes or no,
then you're probably going to say yes.

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You're going to say yes.

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This person will take up the offer

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if you just
if you have to choose one of the two.

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So those are our predicted y hat values.

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In this case
they're both one for those two variables.

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And those are the two things,

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you can get out of,
the logistic regression.

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So you get them,

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probabilities, which are important.

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Also, you can get the Y hat.

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So the predicted values for the dependent
variables, once again, it's important

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00:15:26,400 --> 00:15:30,600
to think of it as it's doing exactly
the same thing as a linear regression.

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It's

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It's fitting this line
even though it's not a straight line.

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And, and the values are not scattered.

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00:15:41,366 --> 00:15:46,900
Everything looks bizarre in its uniformity
or in the way it is structured.

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00:15:46,900 --> 00:15:52,200
Its structure makes it look very bizarre,
but still, it's it's a pretty much

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the same way we agreed on a line
or a formula for a curve,

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and we're trying to fit the best curve
to our data.

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Once we've done
that, we've got we've got a model,

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we've got the, coefficients,
which we'll talk about later,

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00:16:05,466 --> 00:16:08,366
and we can start drawing conclusions

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00:16:08,366 --> 00:16:11,366
or insights from, this model.

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00:16:11,400 --> 00:16:16,100
And some of the insights are we can get
a probability of somebody taking action

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00:16:16,100 --> 00:16:21,600
or on of the event occurring
and or basically of the answer being yes.

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00:16:21,600 --> 00:16:23,633
So it's not a yes no, it's a probability.

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00:16:23,633 --> 00:16:26,066
So 85% or 20% or whatever.

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00:16:26,066 --> 00:16:29,233
So that's when we projected
to the left onto the y axis.

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00:16:29,533 --> 00:16:32,933
And also we can get a predicted value
for the dependent variable

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based on where we select this arbitrary
line 50%.

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00:16:36,333 --> 00:16:38,000
You can select it anywhere you like.

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00:16:38,000 --> 00:16:40,300
You can select higher or lower.

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00:16:40,300 --> 00:16:43,866
depends on your knowledge
about the problem at hand.

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00:16:44,200 --> 00:16:47,100
And as you as you understand,
depending on where you selected,

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00:16:47,100 --> 00:16:50,100
that will significantly affect, 
your variables.

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00:16:50,333 --> 00:16:53,333
So I really hope this, explanation was,

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00:16:54,900 --> 00:16:58,166
trivial enough and yet insightful enough

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00:16:58,166 --> 00:17:02,166
for you to gain an intuitive understanding
of logistic regression.

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00:17:02,633 --> 00:17:03,900
I look forward to seeing you then.

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00:17:03,900 --> 00:17:05,933
And until next time, happy analyzing.