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

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.

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

So that sigma 1 and

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

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

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

M aircraft engines being manufactured.

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

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.

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, this might be 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.

And as we saw we can come up with quite a

Or maybe you need to get additional features.

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

So including all the quadratic features doesn't seem

So now we have, you know, four features. We want to use these four features to predict why the price of the house.

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

Given my patient has a particular tumor size represented by my features X, and this probability is parametrized by theta.

These comics are now going to the New York festival.

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

And, asymptotically, the number of quadratic features grows roughly as order n squared, where n is the number of the original features,

But for many interesting machine learning problems would have a

We'll talk about its advantages and disadvantages later. Second, we'll also talk about algorithms for learning with a larger number of features.

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

So rather than just having x our features we now have x1 through x4.

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,

The first step is to choose features, or come up with features xi that we think might be indicative of anomalous examples.

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

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

The numbers 1 and 1 aren't too important.

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

In fact, if you include only these quadratic features together with the original x1, and so on, up to x100 features, then you can actually fit very interesting hypotheses.

And you want to use that to predict the price of the houses.

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.

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.

less well skilled feature and similarly, if your features take on a very, very small range of values so if X 4 takes on values between minus 0.0001 and positive 0.0001, then again this takes on a much smaller range of values than the minus one to plus one range.

that's just not a good way to learn complex nonlinear hypotheses when n is large because you just end up with too many features.

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,

Let's introduce a little bit more notation.

Choose features X I, that you think might take on unusually

And that's just too large to be reasonable the computation would be very expensive to find and to represent all of these three million features per training example.

It's only if you have a different feature, say X 3 that is between, that ranges from 100 tp 100 , then, this is a very different values than minus 1 and plus 1.