案例学习：预测房价

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來自 University of Washington 的課程

机器学习：回归

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案例学习：预测房价

從本節課中

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

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Okay, so that was rewriting the model for just a single observation.

But it's gonna be very, very helpful in our deriving these algorithms

to write the equation for all the observations stacked up together.

Okay so, we're gonna have to dig through a little bit of linear algebra

in order to have a really nice closed form solution for fitting this model.

So let's start by just stacking up all our observations,

taking all these pink squares that I showed on this previous slide here and

putting them into one big vector.

Where this vector is my first house sale, my second house sale,

my third house sale, all the way up to how many observations do I have in my dataset?

Capital N, that's my capital Nth house sale and I can write

the model for all these observations in this big matrix vector notation,

where here again I have this W vector that we talked about before.

Where I have W0, W1, W2, to W capital D.

And I know that each one of my observations has an error

associated with it.

There's the error for the first observation, error for

the second observation, error for the third observation.

All the way up to the error for this nth observation.

And I'm gonna call this vector, this capital N dimensional vector epsilon.

Okay, so now, let's describe this big green box, this matrix.

Well, how do I form my first observation?

I multiply the weights by the features of that

first observation, so the features of that

first Observation are H0 of X1, H1 of X1,

H2 of X1, all the way up to H capital D of X1.

So if I just forget the fact that it's in a big matrix, if I just look at that row,

that looks exactly like my H transpose, my little H transpose.

Okay so this row is this

little H transpose of X1.

And so just for

that row I get the same result as what I showed on the previous slide.

I multiplied W by H transpose and

that forms that functional fit for that observation.

And then I just add that noise term.

And so then I do this for each one of my observations.

The same is true for the second observation, but when I go and

plug into my features.

Of course I have to plug in the input, the x associated with that second observation.

All the way to, well let me just be very clear,

let me write one more so it's clear.

H1, X2, all the way up to H capital D of X2.

And I do this for every row of this big matrix until I get to my

nth observation and I plug in XN into my set of features,

all the way up to H capital D of XN.

Okay, I'm gonna call this really big matrix here capital H.

And what we're saying, is we're saying that we can write our entire

regression model for capital N observations as this y vector.

I'm sorry I should annotate the fact that this pink vector here I'm calling bold Y.

So this Y vector is equal

to this H matrix times this W vector.

I'm still sticking with the bolds, but I'm gonna give up very soon, okay?

Plus this epsilon vector that represents all the errors in my model.

Okay, so this is our matrix notation for

our model of N observations.

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