Original subtitles

In order to implement linear regression

the first key step is first to

define something called a cost function.

This is something we'll build in this video,

and the cost function will tell us how well

the model is doing so that

we can try to get it to do better.

Let's look at what this means.

Recall that you have a training set that contains

input features x and output targets y.

The model you're going to use to fit this training set

is this linear function f_w,

b of x equals to w times x plus b.

To introduce a little bit more terminology the w

and b are called the parameters of the model.

In machine learning parameters of the model are

the variables you can adjust during

training in order to improve the model.

Sometimes you also hear the parameters w and b

referred to as coefficients or as weights.

Now let's take a look at what

these parameters w and b do.

Depending on the values you've chosen for w and

b you get a different function f of x,

which generates a different line on the graph.

Remember that we can write f of x as

a shorthand for f_w, b of x.

We're going to take a look at some plots

of f of x on a chart.

Maybe you're already familiar

with drawing lines on charts,

but even if this is a review for you,

I hope this will help you build intuition

on how w and b the parameters

determine f. When w is equal to 0 and b is equal to 1.5,

then f looks like this horizontal line.

In this case, the function f of x is 0 times x

plus 1.5 so f is always a constant value.

It always predicts 1.5 for

the estimated value of y. Y hat is always equal to b

and here b is also called

the y intercept because that's where it

crosses the vertical axis or the y axis on this graph.

As a second example,

if w is 0.5 and b is equal 0,

then f of x is 0.5 times x.

When x is 0,

the prediction is also 0,

and when x is 2,

then the prediction is 0.5 times 2, which is 1.

You get a line that looks like this and notice that

the slope is 0.5 divided by 1.

The value of w gives you the slope

of the line, which is 0.5.

Finally, if w equals 0.5 and b equals 1,

then f of x is 0.5 times x plus 1 and when x is 0,

then f of x equals b,

which is 1 so the line intersects

the vertical axis at b, the y intercept.

Also when x is 2,

then f of x is 2,

so the line looks like this.

Again, this slope is 0.5 divided by

1 so the value of w gives you the slope which is 0.5.

Recall that you have

a training set like the one shown here.

With linear regression,

what you want to do is to choose

values for the parameters w and

b so that the straight line you get from

the function f somehow fits the data well.

Like maybe this line shown here.

When I see that the line fits the data visually,

you can think of this to mean that the line

defined by f is roughly passing

through or somewhere close to the training examples

as compared to other possible lines

that are not as close to these points.

Just to remind you of some notation,

a training example like this point

here is defined by x superscript i,

y superscript i where y is the target.

For a given input x^i,

the function f also makes a predictive value for

y and a value that it predicts to

y is y hat i shown here.

For our choice of a model f of x^i is w times x^i plus b.

Stated differently, the prediction y hat i

is f of wb of x^i where

for the model we're using f

of x^i is equal to wx^i plus b.

Now the question is how do you find values for

w and b so that the prediction y hat i is

close to the true target y^i for

many or maybe all training examples x^i, y^i.

To answer that question,

let's first take a look at how to

measure how well a line fits the training data.

To do that, we're going to construct a cost function.

The cost function takes the prediction y hat and compares

it to the target y by taking y hat minus y.

This difference is called the error,

we're measuring how far off to

prediction is from the target.

Next, let's computes the square of this error.

Also, we're going to want to compute this term for

different training examples i in the training set.

When measuring the error,

for example i,

we'll compute this squared error term.

Finally, we want to measure

the error across the entire training set.

In particular, let's sum up the squared errors like this.

We'll sum from i equals 1,2,

3 all the way up to

m and remember that m is the number of training examples,

which is 47 for this dataset.

Notice that if we have more training examples m is

larger and your cost function

will calculate a bigger number.

This is summing over more examples.

To build a cost function that

doesn't automatically get bigger

as the training set size gets larger by convention,

we will compute the average squared error instead of

the total squared error and we do

that by dividing by m like this.

We're nearly there. Just one last thing.

By convention,

the cost function that machine learning people use

actually divides by 2 times m. The extra division

by 2 is just meant to make some of

our later calculations look neater,

but the cost function still works whether you

include this division by 2 or not.

This expression right here is

the cost function and we're going to write

J of wb to refer to the cost function.

This is also called the squared error cost function,

and it's called this because you're taking

the square of these error terms.

In machine learning different people

will use different cost functions

for different applications,

but the squared error cost function is by far the most

commonly used one for

linear regression and for that matter,

for all regression problems where it

seems to give good results for many applications.

Just as a reminder, the prediction y hat

is equal to the outputs of the model f at x.

We can rewrite the cost function J of

wb as 1 over 2m times

the sum from i equals 1 to m of f

of x^i minus y^i the quantity squared.

Eventually we're going to want to find values of

w and b that make the cost function small.

But before going there,

let's first gain more intuition about what

J of wb is really computing.

At this point you might be thinking we've done

a whole lot of math to define the cost function.

But what exactly is it doing?

Let's go on to the next video where we'll step

through one example of what the cost function

is really computing that I hope will

help you build intuition about what it

means if J of wb is large versus if the cost j is small.

Let's go on to the next video.