案例学习：预测房价

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来自 University of Washington 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

Simple Linear Regression

Our course starts from the most basic regression model: Just fitting a line to data. This simple model for forming predictions from a single, univariate feature of the data is appropriately called "simple linear regression".<p> In this module, we describe the high-level regression task and then specialize these concepts to the simple linear regression case. You will learn how to formulate a simple regression model and fit the model to data using both a closed-form solution as well as an iterative optimization algorithm called gradient descent. Based on this fitted function, you will interpret the estimated model parameters and form predictions. You will also analyze the sensitivity of your fit to outlying observations.<p> You will examine all of these concepts in the context of a case study of predicting house prices from the square feet of the house.

- Emily FoxAmazon Professor of Machine Learning

Statistics - Carlos GuestrinAmazon Professor of Machine Learning

Computer Science and Engineering

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So let's see what happens if we remove this observation.

And this observation here, this is the observation for Center City,

that downtown region.

So, not surprisingly, that's where a lot of crimes happen, but

it's also where there's a mixture of low value and very high value homes.

So on average, the value is higher than one might expect for

the amount of crime that occurs in that region.

Okay, so what we're gonna do now,

is just get down to this line here,

we're gonna simply remove Center City from our data sites.

Okay, and I know that Center City is the town that is zero miles.

If we go back to this column that we discussed before,

it's zero miles to Center City, because it is Center City.

So we're just removing that row of our data and

then we're gonna redo our scatter plot.

If we scroll down we see that what we have is our cloud of points, but

for a much smaller range of crime now that outlying center city has been removed.

Okay, so now what we're gonna do is we're gonna go and

refit our simple regression model.

But on this data set where Center City has been removed.

So I'm calling this Crime Model_NoCC meaning for no Center City observation.

Now let's look at the fit associated with this new model.

Well actually it's the same model, but just on a revised dataset.

And what we see again is this downward trend

with house value with increasing crime rate.

But we see a much better fit to the observations that are remaining

in our dataset.

But to make this a little more explicit, let's actually compare the coefficients,

between the fit that we had when Center City was in our dataset, and

the fit that we got when we removed Center City.

Okay, so here are the coefficients, our intercept, and

our slope when we had Center City.

And here are the coefficients when we remove Center city.

So let's talk about this slope term.

When Center city was in our dataset, we said that the average house value

decreased by an amount of $576 per unit increase in crime rate.

Remember we know how to interpret these coefficients and

that's what I'm doing right now.

In contrast, when I remove Center City,

what is the predicted decrease in crime rate I'm get, I mean sorry, predicted

decrease in house of value that I'm getting per unit increase in crime rate?

Now, just removing one observation,

my predicted decrease is $2,287.

That's significantly different.

So when I'm going and I'm making an interpretation about

how much crime rate affects drops in house value.

I have significantly different interpretations when I include Center City

in the dataset versus removing it.

So now let's just discuss a little bit about why this is, and

this brings us to two points.

I've put a little paragraph of text.

We're gonna share these notebooks with you.

You can go through rerun everything we're doing, do different analysis, and

also read the comments that I've put here.

But let's discuss this idea of what are called High Leverage Points and

Influential Observations.

So, a high leverage point is a point that along the x-axis,

along our input axis is an outlier.

So, it's very extreme value, either extremely large or

extremely small, relative to where we have other observations.

So, if we go back up to the plot that has Center City in it,

we see clearly that the crime rate associated with Center City is very,

very different than the crime rates we see for other towns.

So what that means is that point is a high leverage point, because, if we go back and

think about our closed form solution for

our simple regression model, for estimating the coefficients of this model.

Well if you go and look at those equations you'll see that there's a term that

relates to the center of mass of our X values, so the average X value.

And so including a point that's very far out is gonna

strongly influence where the center of mass of this line is.

So that's gonna dramatically change the fit as well as

another term that depends on the value of this observation,

which is gonna have this line trying to get close to this observation.

Remember, we're trying to minimize residuals on the squares.

So if it ignored it and it just draw a line very steeply going down.

We'd have a very massive residual sum of squares for this point here.

So it's gonna try and hit this point.

And thus, the influence of that point can be very large.

Okay, so this gets us to a point of influential observations.

Now, let's just return to this little text I have here,

where just because an observation is a high leverage point,

meaning that it's outlined, either very small or very large X.

It doesn't mean that it's going to strongly influence the fit,

because if that observation follows the trend of the other data.

Then it might not influence things very much at all.

Removing that observation you might get a very similar fit, had Center City

had a similar kind of trend to what we saw for the other observations.

However, it has the potential to strongly influence the fit

as we've seen in this demo.

So an influential observation is an observation where if you

remove it from the dataset you get a very different fit.

But I also wanna emphasize that points that are not high leverage points.

So points that are actually within our typical X range

can be influential observations.

So in particular you can think of an observation that's very outlined in the Y

direction in our response.

So for example, a town that has an extremely high value relative to what you

might see from other observations, well that can also strongly influence the fit.

But the potential for doing so

is much less when it's in the typical X range if you have dense observations,

because the fit will be controlled by all these other observations.

Whereas if it's an outlying point,

you can just think of the control it has as being much, much greater.

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