案例学习：预测房价

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From the course by University of Washington

机器学习：回归

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案例学习：预测房价

From the lesson

Nearest Neighbors & Kernel Regression

Up to this point, we have focused on methods that fit parametric functions---like polynomials and hyperplanes---to the entire dataset. In this module, we instead turn our attention to a class of "nonparametric" methods. These methods allow the complexity of the model to increase as more data are observed, and result in fits that adapt locally to the observations. <p> We start by considering the simple and intuitive example of nonparametric methods, nearest neighbor regression: The prediction for a query point is based on the outputs of the most related observations in the training set. This approach is extremely simple, but can provide excellent predictions, especially for large datasets. You will deploy algorithms to search for the nearest neighbors and form predictions based on the discovered neighbors. Building on this idea, we turn to kernel regression. Instead of forming predictions based on a small set of neighboring observations, kernel regression uses all observations in the dataset, but the impact of these observations on the predicted value is weighted by their similarity to the query point. You will analyze the theoretical performance of these methods in the limit of infinite training data, and explore the scenarios in which these methods work well versus struggle. You will also implement these techniques and observe their practical behavior.

- Emily FoxAmazon Professor of Machine Learning

Statistics - Carlos GuestrinAmazon Professor of Machine Learning

Computer Science and Engineering

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And this simple approach is called nearest neighbor regression.

So the idea of nearest neighbor regression is you have some set of data points,

which are shown as these blue circles.

And then, when you go to predict the value at any point in your input space.

All you do is you look for the closest observation that you have and

what its outputted value is, and

you just predict that your value is exactly equivalent to that value.

Okay, so this is really the simplest model you can think of.

So here we're showing what the resulting fit would look like.

So the green line is the fit and

we have these little arrows showing which are the closest observations.

So just to be clear, if I'm somewhere in this input space,

I have some number of square feet for my house.

I'm gonna look for the closest observation and I'm gonna choose my predicted

value for my house to be exactly the same as this other observation.

So we can see that this leads to this idea of having local

fits where these local fits are defined around each one of our observations.

And how local they are, how far they stretch is based

on where the placement of other observations are.

And this is called one nearest neighbor regression.

And this is really what people do naturally.

So if you think of some real estate agent out there and you're selling your house,

and the real estate agent wants to assess the value of your house.

What are they gonna do?

Well that agent is gonna look at all the other houses that were sold and is gonna

say, well what's the most similar house to your house and how much did it sell for?

So implicitly what this real estate agent is doing is looking for your nearest

neighbor, which house is most similar to yours, and then is gonna assess the value

of your house as being exactly equal to the sales price of this other house.

That's the best guess that your real estate agent has for

the value of your house and how much it's likely to sell for on the market.

So let's formalize this nearest neighbor method a bit.

We're gonna have some data set in our housing obligation.

It's gonna be pairs of house attributes and

values associated with each house, so the sales price of the house.

And so we're gonna denote this as (x,y) for

some set of observations, 1 to capital N.

So this is what we assume our data set is.

And then we assume that we have some query house which is not

in our training data set.

So this is some point, xk,

this is our some house that we're interested in the value.

And the first step of the nearest neighbor method is to find the closest

other house in our dataset.

So specifically let's call our x nearest neighbor

to be the house that minimizes over all of our observations,

i, the distance between

xi and our query house, xk.

So to be clear, in the picture we had on the previous slide,

this x nearest neighbors would just be that big pink house.

Then what we're gonna do is we're simply gonna predict

the value of our query house which is our big lime green house.

To be exactly the value or the sales price of this big pink house.

So sales price of big pink house.

Okay, so really, really, really straightforward algorithm.

And so just to be clear, let's say this was the square feet of our green house.

We go, and what we're gonna do is we're gonna look for our nearest neighbor.

We search along square feet and say,

which house has the most similar square feet to mine?

Happens to be this house, this is our pink house.

Which will be, if there's space I'll just write it here, x nearest neighbor.

Actually, I like to write it under, where it actually should be,

so x, nearest neighbor.

And then we say, what is the value of that house,

which is how much it sold for y, nearest neighbor.

And that's exactly what we're gonna predict for our query house.

So the key thing in our nearest neighbor method is this distance

metric which measures how similar this query house is to any other house.

And this defines our notion of in quotes closest house in the data set.

And it's a really, really key thing to how the algorithm's gonna perform.

For example, what house it's gonna say is most similar to my house.

So we're gonna talk about distance metrics in a little bit.

But first, let's talk about what one nearest neighbor looks like in

higher dimensions because so

far, we've assumed that we have just one input like square feet and in that case,

what we had to do was define these transition points.

Where we go from one nearest neighbor to the next nearest neighbor.

And thus changing our predicted values across our input space square feet.

But how do we think about doing this in higher dimensions?

Well, what we can do is look at something that's called a Voronoi diagram or

a Voronoi tessellation.

And here, we're showing a picture of such a Voronoi tessellation, but

just in two dimensions, though this idea generalizes to higher dimensions as well.

And what we do is we're just gonna divide our input space into regions.

So in the case of two dimensions,

they're just regions in this two dimensional space.

So I'm just gonna highlight one of these regions.

And each one of these regions is defined by one observation.

So here is an observation.

And what defines the region is the fact that any other point in this region,

let's call it x.

Let's call this xi, and this other point some other x.

Well, this point x is closer to xi, let me write this.

So x closer to xi than

any other xj, for

j not equal to i.

Meaning any other observation.

So in pictures, what we're saying is that the distance from x

to xi is less than the distance to any of these other observations.

In our dataset.

So what that means is, let's say that x is our query point.

So let's now put a sub q.

If x is our query point, then when we go to predict the value associated with xq.

We're just gonna look at what the value was associated with xi,

the observation that's contained within this region.

So this Voronoi diagram might look really complicated, but

we're not actually going to explicitly form all these regions.

All we have to do is be able to compute the distance and

define between any two points in our input space and define which is the closest.

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