In the previous video, we gave the mathematical definition of the cost
function. In this video, let's look at some examples, to get back to intuition
about what the cost function is doing, and why we want to use it. To recap, here's
what we had last time. We want to fit a straight line to our data, so we had this
formed as a hypothesis with these parameters theta zero and theta one, and
with different choices of the parameters we end up with different straight line
fits. So the data which are fit like so, and there's a cost function, and
that was our optimization objective. [sound] So this video, in order to better
visualize the cost function J, I'm going to work with a simplified hypothesis
function, like that shown on the right. So I'm gonna use my simplified hypothesis,
which is just theta one times X. We can, if you want, think of this as setting the
parameter theta zero equal to 0. So I have only one parameter theta one and
my cost function is similar to before except that now H of X that is now equal
to just theta one times X. And I have only one parameter theta one and so my
optimization objective is to minimize j of theta one. In pictures what this means is
that if theta zero equals zero that corresponds to choosing only hypothesis
functions that pass through the origin, that pass through the point (0, 0). Using
this simplified definition of a hypothesizing cost function let's try to understand the cost
function concept better. It turns out that two key functions we want to understand.
The first is the hypothesis function, and the second is a cost function. So, notice
that the hypothesis, right, H of X. For a face value of theta one, this is a
function of X. So the hypothesis is a function of, what is the size of the house
X. In contrast, the cost function, J, that's a function of the parameter, theta
one, which controls the slope of the straight line. Let's plot these functions
and try to understand them both better. Let's start with the hypothesis. On the left,
let's say here's my training set with three points at (1, 1), (2, 2), and (3, 3). Let's
pick a value theta one, so when theta one equals one, and if that's my choice for
theta one, then my hypothesis is going to look like this straight line over here.
And I'm gonna point out, when I'm plotting my hypothesis function. X-axis, my
horizontal axis is labeled X, is labeled you know, size of the house over here.
Now, of temporary, set theta one equals one, what I want to do is
figure out what is j of theta one, when theta one equals one. So let's go ahead
and compute what the cost function has for. You'll devalue one. Well, as usual,
my cost function is defined as follows, right? Some from, some of 'em are training
sets of this usual squared error term. And, this is therefore equal to. And this.
Of theta one x I minus y I and if you simplify this turns out to be. That. Zero
Squared to zero squared to zero squared which is of course, just equal to zero. Now,
inside the cost function. It turns out each of these terms here is equal to zero. Because
for the specific training set I have or my 3 training examples are (1, 1), (2, 2), (3,3). If theta
one is equal to one. Then h of x. H of x i. Is equal to y I exactly, let me write
this better. Right? And so, h of x minus y, each of these terms is equal to zero,
which is why I find that j of one is equal to zero. So, we now know that j of one Is
equal to zero. Let's plot that. What I'm gonna do on the right is plot my cost
function j. And notice, because my cost function is a function of my parameter
theta one, when I plot my cost function, the horizontal axis is now labeled with
theta one. So I have j of one zero zero so let's go ahead and plot that. End
up with. An X over there. Now lets look at some other examples. Theta-1 can take on a
range of different values. Right? So theta-1 can take on the negative values,
zero, positive values. So what if theta-1 is equal to 0.5. What happens then? Let's
go ahead and plot that. I'm now going to set theta-1 equals 0.5, and in that case my
hypothesis now looks like this. As a line with slope equals to 0.5, and, lets
compute J, of 0.5. So that is going to be one over 2M of, my usual cost function.
It turns out that the cost function is going to be the sum of square values of
the height of this line. Plus the sum of square of the height of that line, plus
the sum of square of the height of that line, right? ?Cause just this vertical
distance, that's the difference between, you know, Y. I. and the predicted value, H
of XI, right? So the first example is going to be 0.5 minus one squared.
Because my hypothesis predicted 0.5. Whereas, the actual value was one. For my
second example, I get, one minus two squared, because my hypothesis predicted
one, but the actual housing price was two. And then finally, plus. 1.5 minus three
squared. And so that's equal to one over two times three. Because, M when trading
set size, right, have three training examples. In that, that's times
simplifying for the parentheses it's 3.5. So that's 3.5 over six which is about
0.68. So now we know that j of 0.5 is about 0.68. Lets go and plot that. Oh
excuse me, math error, it's actually 0.58. So we plot that which is maybe about over
there. Okay? Now, let's do one more. How about if theta one is equal to zero, what
is J of zero equal to? It turns out that if theta one is equal to zero, then H of X
is just equal to, you know, this flat line, right, that just goes horizontally
like this. And so, measuring the errors. We have that J of zero is equal to one
over two M, times one squared plus two squared plus three squared, which is, One
six times fourteen which is about 2.3. So let's go ahead and plot as well. So it
ends up with a value around 2.3 and of course we can keep on doing this
for other values of theta one. It turns out that you can have you know negative
values of theta one as well so if theta one is negative then h of x would be equal
to say minus 0.5 times x then theta one is minus 0.5 and so that corresponds
to a hypothesis with a slope of negative 0.5. And you can
actually keep on computing these errors. This turns out to be, you know, for 0.5,
it turns out to have really high error. It works out to be something, like, 5.25. And
so on, and the different values of theta one, you can compute these things, right?
And it turns out that you, your computed range of values, you get something like
that. And by computing the range of values, you can actually slowly create
out. What does function J of Theta say and that's what J of Theta is. To recap, for
each value of theta one, right? Each value of theta one corresponds to a different
hypothesis, or to a different straight line fit on the left. And for each value
of theta one, we could then derive a different value of j of theta one. And for
example, you know, theta one=1, corresponded to this straight line
straight through the data. Whereas theta one=0.5. And this point shown in magenta
corresponded to maybe that line, and theta one=zero which is shown in blue that corresponds to
this horizontal line. Right, so for each value of theta one we wound up with a
different value of J of theta one and we could then use this to trace out this plot
on the right. Now you remember, the optimization objective for our learning
algorithm is we want to choose the value of theta one. That minimizes J of theta one.
Right? This was our objective function for the linear regression. Well, looking at
this curve, the value that minimizes j of theta one is, you know, theta one equals
to one. And low and behold, that is indeed the best possible straight line fit
through our data, by setting theta one equals one. And just, for this particular
training set, we actually end up fitting it perfectly. And that's why minimizing j
of theta one corresponds to finding a straight line that fits the data well. So,
to wrap up. In this video, we looked up some plots. To understand the cost
function. To do so, we simplify the algorithm. So that it only had one
parameter theta one. And we set the parameter theta zero to be only zero. In the next video.
We'll go back to the original problem formulation and look at some
visualizations involving both theta zero and theta one. That is without setting theta
zero to zero. And hopefully that will give you, an even better sense of what the cost
function j is doing in the original linear regression formulation.