4 - 5 - Features and Polynomial Regression (8 min).srt

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You now know about linear regression with multiple variables.
你现在了解了多变量的线性回归
(字幕整理：中国海洋大学 黄海广,haiguang2000@qq.com )

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In this video, I wanna tell
在本段视频中 我想告诉你

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you a bit about the choice
一些用来

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of features that you have and
选择特征的方法以及

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how you can get different learning
如何得到不同的学习算法

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algorithm, sometimes very powerful
当选择了合适的特征后

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ones by choosing appropriate features.
这些算法往往是非常有效的

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And in particular I also want
另外 我也想

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to tell you about polynomial regression allows
给你们讲一讲多项式回归

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you to use the machinery of
它使得你们能够使用

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linear regression to fit very
线性回归的方法来拟合

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complicated, even very non-linear functions.
非常复杂的函数 甚至是非线性函数

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Let's take the example of predicting the price of the house.
以预测房价为例

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Suppose you have two features,
假设你有两个特征

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the frontage of house and the depth of the house.
分别是房子临街的宽度和垂直宽度

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So, here's the picture of the house we're trying to sell.
这就是我们想要卖出的房子的图片

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So, the frontage is
临街宽度

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defined as this distance
被定义为这个距离

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is basically the width
其实就是它的宽度

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or the length of
或者说是

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how wide your lot
你拥有的土地的宽度

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is if this that you
如果这块地都是你的的话

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own, and the depth
而这所房子的

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of the house is how
纵向深度就是

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deep your property is, so
你的房子的深度

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there's a frontage, there's a depth.
这是正面的宽度 这是深度

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called frontage and depth.
我们称之为临街宽度和纵深

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You might build a linear regression
你可能会 像这样 建立一个

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model like this where frontage
线性回归模型 其中临街宽度

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is your first feature x1 and
是你的第一个特征x1

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and depth is your second
纵深是你的第二个

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feature x2, but when you're
特征x2 但当我们在

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applying linear regression, you don't
运用线性回归时

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necessarily have to use
你不一定非要直接用

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just the features x1 and x2 that you're given.
给出的 x1 和 x2 作为特征

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What you can do is actually create new features by yourself.
其实你可以自己创造新的特征

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So, if I want to predict
因此 如果我要预测

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the price of a house, what I
房子的价格

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might do instead is decide
我真正要需做的 也许是

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that what really determines
确定真正能够决定

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the size of the house is
我房子大小 或者说我土地大小

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the area or the land area that I own.
的因素是什么

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So, I might create a new feature.
因此 我可能会创造一个新的特征

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I'm just gonna call this feature
我称之为

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x which is frontage, times depth.
x 它是临街宽度与纵深的乘积

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This is a multiplication symbol.
这是一个乘法符号

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It's a frontage x depth because
它是临街宽度与纵深的乘积

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this is the land area
这得到的就是我拥有的土地的面积

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that I own and I might
然后 我可以把

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then select my hypothesis
假设选择为

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as that using just
使其只使用

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one feature which is my
一个特征 也就是我的

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land area, right?
土地的面积 对吧？

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Because the area of a
由于矩形面积的

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rectangle is you know,
计算方法是

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the product of the length
矩形长和宽相乘

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of the size So, depending
因此 这取决于

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on what insight you might have
你从什么样的角度

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into a particular problem, rather than
去审视一个特定的问题 而不是

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just taking the features [xx]
只是直接去使用临街宽度和纵深

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that we happen to have started
这两个我们只是碰巧在开始时

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off with, sometimes by defining
使用的特征 有时 通过定义

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new features you might actually get a better model.
新的特征 你确实会得到一个更好的模型

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Closely related to the
与选择特征的想法

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idea of choosing your features
密切相关的一个概念

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is this idea called polynomial regression.
被称为多项式回归(polynomial regression)

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Let's say you have a housing price data set that looks like this.
比方说 你有这样一个住房价格的数据集

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Then there are a few different models you might fit to this.
为了拟合它 可能会有多个不同的模型供选择

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One thing you could do is fit a quadratic model like this.
其中一个你可以选择的是像这样的二次模型

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It doesn't look like a straight line fits this data very well.
因为直线似乎并不能很好地拟合这些数据

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So maybe you want to fit
因此 也许你会想到

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a quadratic model like this
用这样的二次模型去拟合数据

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where you think the size, where
你可能会考量

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you think the price is a quadratic
是关于价格的一个二次函数

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function and maybe that'll
也许这样做

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give you, you know, a fit
会给你一个

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to the data that looks like that.
像这样的拟合结果

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But then you may decide that your
但是 然后你可能会觉得

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quadratic model doesn't make sense
二次函数的模型并不好用

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because of a quadratic function, eventually
因为 一个二次函数最终

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this function comes back down
会降回来

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and well, we don't think housing
而我们并不认为

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prices should go down when the size goes up too high.
房子的价格在高到一定程度后 会下降回来

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So then maybe we might
因此 也许我们会

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choose a different polynomial model
选择一个不同的多项式模型

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and choose to use instead a
并转而选择使用一个

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cubic function, and where
三次函数 在这里

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we have now a third-order term
现在我们有了一个三次的式子

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and we fit that, maybe
我们用它进行拟合

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we get this sort of
我们可能得到这样的模型

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model, and maybe the
也许这条绿色的线

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green line is a somewhat better fit
对这个数据集拟合得更好

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to the data cause it doesn't eventually come back down.
因为它不会在最后下降回来

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So how do we actually fit a model like this to our data?
那么 我们到底应该如何将模型与我们的数据进行拟合呢？

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Using the machinery of multivariant
使用多元

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linear regression, we can
线性回归的方法 我们可以

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do this with a pretty simple modification to our algorithm.
通过将我们的算法做一个非常简单的修改来实现它

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The form of the hypothesis we,
按照我们以前假设的形式

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we know how the fit
我们知道如何对

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looks like this, where we say
这样的模型进行拟合 其中

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H of x is theta zero
?θ(x) 等于 θ0

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plus theta one x one plus x two theta X3.
+θ1×x1 + θ2×x2 + θ3×x3

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And if we want to
那么 如果我们想

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fit this cubic model that
拟合这个三次模型

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I have boxed in green,
就是我用绿色方框框起来的这个

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what we're saying is that
现在我们讨论的是

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to predict the price of a
为了预测一栋房子的价格

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house, it's theta 0 plus theta
我们用 θ0 加 θ1

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1 times the size of the house
乘以房子的面积

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plus theta 2 times the square size of the house.
加上 θ2 乘以房子面积的平方

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So this term is equal to that term.
因此 这个式子与那个式子是相等的

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And then plus theta 3
然后再加 θ3

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times the cube of the
乘以

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size of the house raises that third term.
房子面积的立方

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In order to map these
为了将这两个定义

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two definitions to each other,
互相对应起来

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well, the natural way
为了做到这一点

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to do that is to set
我们自然想到了

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the first feature x one to
将 x1 特征设为

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be the size of the house, and
房子的面积

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set the second feature x two
将第二个特征 x2 设为

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to be the square of the size
房屋面积的平方

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of the house, and set the third feature x three to
将第三个特征 x3 设为

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be the cube of the size of the house.
房子面积的立方

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And, just by choosing my
那么 仅仅通过将

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three features this way and
这三个特征这样设置

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applying the machinery of linear
然后再应用线性回归的方法

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regression, I can fit this
我就可以拟合

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model and end up with
这个模型 并最终

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a cubic fit to my data.
将一个三次函数拟合到我的数据上

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I just want to point out one
我还想再说一件事

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more thing, which is that
那就是

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if you choose your features
如果你像这样选择特征

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like this, then feature scaling
那么特征的归一化

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becomes increasingly important.
就变得更重要了

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So if the size of the
因此 如果

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house ranges from one to
房子的大小范围在

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a thousand, so, you know,
1到1000之间 那么

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from one to a thousand square
比如说

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feet, say, then the size
从1到1000平方尺 那么

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squared of the house will
房子面积的平方

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range from one to one
的范围就是

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million, the square of
一到一百万 也就是

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a thousand, and your third
1000的平方 而你的第三个特征

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feature x cubed, excuse me
x的立方 抱歉

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you, your third feature x
你的第三个特征 x3

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three, which is the size
它是房子面积的

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cubed of the house, will range
立方 范围会扩大到

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from one two ten to
1到10的9次方

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the nine, and so these
因此

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three features take on very
这三个特征的范围

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different ranges of values, and
有很大的不同

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it's important to apply feature
因此 如果你使用梯度下降法

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scaling if you're using gradient
应用特征值的归一化是非常重要的

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descent to get them into
这样才能将他们的

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comparable ranges of values.
值的范围变得具有可比性

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Finally, here's one last example
最后 这里是最后一个例子

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of how you really have
关于如何使你

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broad choices in the features you use.
真正选择出要使用的特征

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Earlier we talked about how a
此前我们谈到

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quadratic model like this might
一个像这样的二次模型

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not be ideal because, you know,
并不是理想的 因为 你知道

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maybe a quadratic model fits the
也许一个二次模型能很好地拟合

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data okay, but the quadratic
这个数据 但二次

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function goes back down
函数最后会下降

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and we really don't want, right,
这是我们不希望的

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housing prices that go down,
就是住房价格往下走

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to predict that, as the size of housing freezes.
像预测的那样 出现房价的下降

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But rather than going to
但是 除了转而

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a cubic model there, you
建立一个三次模型以外

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have, maybe, other choices of
你也许有其他的选择

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features and there are many possible choices.
特征的方法 这里有很多可能的选项

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But just to give you another
但是给你另外一个

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example of a reasonable
合理的选择的例子

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choice, another reasonable choice
另一种合理的选择

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might be to say that the
可能是这样的

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price of a house is theta
一套房子的价格是

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zero plus theta one times
θ0 加 θ1 乘以

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the size, and then plus theta
房子的面积 然后

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two times the square root of the size, right?
加 θ2 乘以房子面积的平方根 可以吧？

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So the square root function is
平方根函数是

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this sort of function, and maybe
这样的一种函数

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there will be some value of theta
也许θ1 θ2 θ3

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one, theta two, theta three, that
中会有一些值

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will let you take this model
会捕捉到这个模型

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and, for the curve that looks
从而使得这个曲线看起来

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like that, and, you know,
是这样的

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goes up, but sort of flattens
趋势是上升的 但慢慢变得

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out a bit and doesn't ever
平缓一些 而且永远不会

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come back down.
下降回来

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And, so, by having insight into, in
因此 通过深入地研究

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this case, the shape of a
在这里我们研究了平方根

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square root function, and, into
函数的形状 并且

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the shape of the data, by choosing
更深入地了解了选择不同特征时数据的形状

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different features, you can sometimes get better models.
有时可以得到更好的模型

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In this video, we talked about polynomial regression.
在这段视频中 我们探讨了多项式回归

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That is, how to fit a
也就是 如何将一个

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polynomial, like a quadratic function,
多项式 如一个二次函数

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or a cubic function, to your data.
或一个三次函数拟合到你的数据上

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Was also throw out this idea,
除了这个方面

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that you have a choice in what
我们还讨论了

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features to use, such as
在使用特征时的选择性

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that instead of using
例如 我们不使用

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the frontish and the depth
房屋的临街宽度和纵深

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of the house, maybe, you can
也许 你可以

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multiply them together to get
把它们乘在一起 从而得到

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a feature that captures the land area of a house.
房子的土地面积这个特征

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In case this seems a little
实际上 这似乎有点

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bit bewildering, that with all
难以抉择 这里有这么多

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these different feature choices, so how do I decide what features to use.
不同的特征选择 我该如何决定使用什么特征呢

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Later in this class, we'll talk
在之后的课程中 我们将

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about some algorithms were automatically
探讨一些算法 它们能够

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choosing what features are used,
自动选择要使用什么特征

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so you can have an
因此 你可以使用一个算法

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algorithm look at the data
观察给出的数据

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and automatically choose for you
并自动为你选择

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whether you want to fit a
到底应该选择

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quadratic function, or a cubic function, or something else.
一个二次函数 或者一个三次函数 还是别的函数

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But, until we get to
但是 在我们

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those algorithms now I just
学到那种算法之前

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want you to be aware that
现在我希望你知道

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you have a choice in
你需要选择

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what features to use, and
使用什么特征

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by designing different features
并且通过设计不同的特征

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you can fit more complex functions
你能够用更复杂的函数

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your data then just fitting a
去拟合你的数据 而不是只用

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straight line to the data and
一条直线去拟合

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in particular you can put polynomial
特别是 你也可以使用多项式

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functions as well and sometimes
函数 有时候

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by appropriate insight into the
通过采取适当的角度来观察

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feature simply get a much
特征就可以

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better model for your data.
得到一个更符合你的数据的模型