So, this might be a

Other things you might try are to well maybe try a smaller set of features.

But taking, you know, this max minus min would be fine.

So that was for when we had only one feature.

So that sigma 1 and

You can think of this as defining an additional zero feature.

lot of features, maybe a hundred different features of different houses.

X subscript 1 X subscript 2 and so on to denote my, in this case, four features and I'm going to continue to use

I'm going to use lowercase n to denote the number of features.

M aircraft engines being manufactured.

like it's maybe a good idea, because that is a

There's one little difference which is that when we previously had only one feature, we would call that feature x(i) but now in our new notation we would of course call this x(i)_1 to denote our one feature.

line of the ellipses like these, but you certainly cannot fit a more complex data set like that shown here.

Or maybe you need to get additional features.

And obviously we want to apply this to the future x zero, because the future x zero is always equal to one, so it cannot have an average value of zero.

But if you implement the algorithm written up here then you have a working implementation of linear regression with multiple features.

So, that's already in that range, but you may end up dividing other features by different numbers to get them to this range.

load on this machine divided by the amount of network traffic on this machine?

Here you have only 100 such quadratic features, but this is not enough features and certainly won't let you fit the data set like that on the upper left.

But for many interesting machine learning problems would have a

I to denote the input features of the I training example.

linear regression that we developed, we have a single feature x, the size of the house, and we wanted to use that to predict why the price of the house and this was our form of our hypothesis.

So including all the quadratic features doesn't seem

These comics are now going to the New York festival.

Concretely if you instead define the feature X one to be the size of the house divided by two thousand, and define X two to be maybe the number of bedrooms divided by five, then the count well as of the cost function J can become much more, much less skewed so the contours may look more like circles.

Let's say we have this data set plotted on the upper left of this slide.

In that case maybe we'll call these features x1, x2, x3, and x4.

In these settings, a useful thing to do is to scale the features.

Previously this was the form of our hypothesis, where x was our single feature, but now that we have multiple features, we aren't going to use the simple representation any more.

So x2 subscript 3 is going to be equal to 2. Now that we have multiple features,

In this video, let's talk about how to fit the parameters of that hypothesis.

If you want to apply logistic regression to this problem, one thing you could do is apply

The numbers 1 and 1 aren't too important.

So, if we were to try to learn a nonlinear hypothesis by including all the quadratic features, that is all the terms of the form, you know,

And as we saw we can come up with quite a

lot of features and you might up overfitting the training set, and it can also be computationally expensive, you know, to be working with that many features.

We can also try adding polynomial features things like x2 square x2 square and product features x1, x2.

In fact, they are going to be order and cube such features and if any is 100 you can compute that, you end up with on the order of about 170,000 such cubic features and so including these higher auto polynomial features when your original feature set end is large this really dramatically blows up your feature space and this doesn't seem like a good way to come up with additional features with which to build none many classifiers when n is large.

And so a matrix is a block of numbers that lets me take all of my data, all of my x's.

More generally, when we're performing feature scaling, what we often want to do is get every feature into approximately a 1 to 1 range and concretely, your feature x0 is always equal to 1.

In this simple example we could plot the hypothesis h of x and just see what was going on.

Let's introduce a little bit more notation.

Let's say that we have an unlabeled training set of M examples, and each of these examples is going to be a feature in Rn so your training set could be, feature vectors from the last

But in general for problems with more features than just one feature, for problems with a large number of features like these it becomes hard or may be impossible to plot what the hypothesis looks like and so we need some other way to evaluate our hypothesis.

But now imagine, what if we had not only the size of the house as a feature or as a variable of which to try to predict the price, but that we also knew the number of bedrooms, the number of house and the age of the home and years.