案例学习：预测房价

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来自 华盛顿大学 的课程

机器学习：回归

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案例学习：预测房价

从本节课中

Multiple Regression

The next step in moving beyond simple linear regression is to consider "multiple regression" where multiple features of the data are used to form predictions. <p> More specifically, in this module, you will learn how to build models of more complex relationship between a single variable (e.g., 'square feet') and the observed response (like 'house sales price'). This includes things like fitting a polynomial to your data, or capturing seasonal changes in the response value. You will also learn how to incorporate multiple input variables (e.g., 'square feet', '# bedrooms', '# bathrooms'). You will then be able to describe how all of these models can still be cast within the linear regression framework, but now using multiple "features". Within this multiple regression framework, you will fit models to data, interpret estimated coefficients, and form predictions. <p>Here, you will also implement a gradient descent algorithm for fitting a multiple regression model.

- Emily FoxAmazon Professor of Machine Learning

Statistics - Carlos GuestrinAmazon Professor of Machine Learning

Computer Science and Engineering

[MUSIC]

Okay, so now we're onto the final important step of the derivation,

which is taking the gradient.

Because as we saw in the simple regression case, the gradient was important both for

our closed form solution as well as, of course, for

the gradient descent algorithm.

So what's the gradient of our residual sum of squares in this multiple

regression case?

Well, it's the gradient of this matrix notation that we use for

representing the residual sum of squares.

And if you know gradients of vectors and matrices,

which we're not assuming you do, so please don't think that you need to know this,

but the result is -2H transpose, so

taking that big grain matrix and turning it on its side, times y-Hw,

which again is that vector, of residuals.

And why is this the result?

Well, I'm not gonna give a complete proof of this.

I'm just gonna give some motivation.

I'm going to walk through an analogy to 1D case, and we'll see some patterns, and

maybe you'll believe that that's the result of the matrix case.

So, in particular, if we think about taking the derivative with respect to w

of a function that is y-hw times

y-hw where these things are all scalars.

So this is the 1D analog to this equation here,

where the gradient is just this derivative of this one parameter w.

That arrow is not quite pointing to w.

Well what's the derivative of this?

It's equivalent to the derivative with respect to w of Y minus hw squared.

And, like we've done multiple times in this course now,

when I take the derivative with respect to w of some function raised to the power,

by the chain rule, I bring that power down.

Then I'm gonna multiply by the function Hw raised to the power minus 1.

And then I'm gonna take the derivative of the inside.

And what's the derivative of this function with respect to w?

It's minus h.

And so the result here is -2h(y-Hw).

So we have the -2 in both cases,

this little scalar H is this big matrix in our case,

and y- Hw in the scalar case, this big vector matrix notation here.

Okay, so just believe that this is the gradient.

We didn't wanna bog you down in too much linear algebra, or

too much in terms of derivatives.

But if we have this notation, then we can derive everything we need to for

our two different solutions to fitting this model.

[MUSIC]