4 - 1 - Multiple Features (8 min).srt

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in this video we will start
在这段视频中 我们将开始
(字幕整理：中国海洋大学 黄海广,haiguang2000@qq.com )

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to talk about a new version
介绍一种新的

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of linear regression that's more powerful.
更为有效的线性回归形式

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One that works with multiple variables
这种形式适用于多个变量

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or with multiple features.
或者多特征量的情况

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Here's what I mean.
比如说：

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In the original version of
在之前我们学习过的

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linear regression that we developed,
线性回归中

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we have a single feature x,
我们只有一个单一特征量

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the size of the house, and
房屋面积 x

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we wanted to use that to
我们希望用这个特征量

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predict why the price of
来预测

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the house and this was
房子的价格

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our form of our hypothesis.
这就是我们的假设

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But now imagine, what if
但是想象一下

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we had not only the size
如果我们不仅有房屋面积

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of the house as a feature
作为预测房屋

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or as a variable of which
价格的特征量

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to try to predict the price,
或者变量

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but that we also knew the
我们还知道

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number of bedrooms, the number
卧室的数量

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of house and the age of the home and years.
楼层的数量以及房子的使用年限

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It seems like this would give
这样就给了我们

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us a lot more information with which to predict the price.
更多可以用来

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To introduce a little bit
预测房屋价格的信息

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of notation, we sort of
先简单介绍一下记法

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started to talk about this earlier,
我们开始的时候就提到过

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I'm going to use the variables
我要用

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X subscript 1 X subscript
x 下标1

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2 and so on to
x 下标2  等等

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denote my, in this
来表示

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case, four features and I'm
这种情况下的四个特征量

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going to continue to use
然后仍然用

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Y to denote the variable,
Y来表示我们

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the output variable price that we're trying to predict.
所想要预测的输出变量

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Let's introduce a little bit more notation.
让我们来看看更多的表示方式

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Now that we have four features
现在我们有四个特征量

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I'm going to use lowercase "n"
我要用小写n

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to denote the number of features.
来表示特征量的数目

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So in this example we have
因此在这个例子中 我们的n等于4

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n4 because we have, you
因为你们看 我们有

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know, one, two, three, four features.
1 2 3 4 共4个特征量

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And "n" is different from
这里的n和我们之前

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our earlier notation where we
使用的n不同

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were using "n" to denote the number of examples.
之前我们是用的“m”来表示样本的数量

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So if you have
所以如果你有47行

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47 rows  "M" is the
那么m就是这个表格里面的行数

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number of rows on this table or the number of training examples.
或者说是训练样本数

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So I'm also
然后我要用x 上标 (i)

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going to use X superscript
来表示第i个

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"I" to denote the
训练样本的

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input features of the "I" training example.
输入特征值

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X2 is going to
x上标 (2)

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be a vector of
就是表示第二个

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the features for my second training example.
训练样本的特征向量

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And so X2 here is
因此这里

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going to be a vector 1416,
x(2)就是向量 [1416, 3, 2, 40]

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3, 2, 40 since those
因为这四个数字对应了

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are my four
我用来预测房屋价格的

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features that I have
第二个房子的

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to try to predict the price of the second house.
四个特征量

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So, in this notation, the
因此在这种记法中

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superscript 2 here.
这个上标2

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That's an index into my training set.
就是训练集的一个索引

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This is not X to the power of 2.
而不是x的2次方

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Instead, this is, you know,
这个2就对应着

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an index that says look at the second row of this table.
你所看到的表格中的第二行

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This refers to my second training example.
即我的第二个训练样本

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With this notation X2 is
x上标(2) 这样表示

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a four dimensional vector.
就是一个四维向量

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In fact, more generally, this is
事实上更普遍地来说

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an in-dimensional feature back there.
这是n维的向量

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subscript J to denote
 下标j

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the value of the J,
来表示

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of feature number J
第i个训练样本的

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and the training example.
第j个特征量

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So concretely X2 subscript 3,
因此具体的来说

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will refer to feature
x上标(2)下标3代表着

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number three in the
第2个训练样本里的第3个特征量

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x factor which is equal to 2,right?
对吧？

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That was a 3 over there, just fix my handwriting.
这个是3 我写的不太好看

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So x2 subscript 3 is going to be equal to 2.
所以说x上标(2)下标3就等于2

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Now that we have multiple features,
既然我们有了多个特征量

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let's talk about what the
让我们继续讨论一下

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form of our hypothesis should be.
我们的假设形式应该是怎样的

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Previously this was the
这是我们之前使用的假设形式

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form of our hypothesis, where x
x就是我们唯一的特征量

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was our single feature, but
但现在我们有了多个特征量

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now that we have multiple features,
我们就不能再

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we aren't going to use the simple representation any more.
使用这种简单的表示方式了

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Instead, a form
取而代之的

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of the hypothesis in linear regression
我们将把线性回归的假设

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is going to be this, can be
改成这样

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theta 0 plus theta
θ0加上

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1 x1 plus theta 2
θ1 乘以 x1 加上

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x2 plus theta 3 x3
θ2乘以x2 加上 θ3 乘以x3

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plus theta 4 X4.
加上θ4乘以x4

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And if we have N features then
然后如果我们有n个特征量

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rather than summing up over
那么我们要将所有的n个特征量相加

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our four features, we would have
而不是四个特征量

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a sum over our N features.
我们需要对n个特征量进行相加

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Concretely for a particular
举个具体的例子

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setting of our parameters we
在我们的设置的参数中

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may have H of
我们可能有h(x)等于

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X 80 + 0.1 X1 +  0.01x2 + 3x3 - 2x4.
80 + 0.1 x1 + 0.01x2 + 3x3  -  2x4

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This would be one
这就是一个

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example of a hypothesis
假设的范例

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and you remember a
别忘了

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hypothesis is trying to predict
假设是为了预测

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the price of the house in
大约以千刀为单位的房屋价格

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thousands of dollars, just saying
就是说

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that, you know, the base
一个房子的价格

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price of a house
可以是

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is maybe 80,000 plus another open
80 k加上

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1, so that's an extra,
0.1乘以x1

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what, hundred dollars per square feet,
也就是说 每平方尺100美元

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yeah, plus the price goes up
然后价格

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a little bit for each
会随着楼层数的增加

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up further for each additional
随着卧室数的增加

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bedroom the house has, because
因为x3是

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X three was the number
卧室的数量

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of bedrooms, and the price
但是呢

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goes down a little bit
房子的价格会

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with each additional age of the house.
随着使用年数的增加

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With each additional year of the age of the house.
而贬值

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Here's the form of a hypothesis rewritten on the slide.
这是重新改写过的假设的形式

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And what I'm gonna do is
接下来

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introduce a little bit of
我要来介绍一点

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notation to simplify this equation.
简化这个等式的表示方式

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For convenience of notation, let
为了表示方便

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me define x subscript 0 to be equals one.
我要将x下标0的值设为1

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Concretely, this means that for
具体而言 这意味着

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every example i I
对于第i个样本

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have a feature vector X superscript
都有一个向量x上标(i)

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I and X superscript
并且x上标(i)

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I subscript 0 is going to be equal to 1.
下标0等于1

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You can think of this as defining
你可以认为我们

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an additional zero feature.
定义了一个额外的第0个特征量

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So whereas previously I had
因此 我过去有n个特征量

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n features because x1, x2
因为我们有x1 x2

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through xn, I'm now defining
直到xn 由于我另外定义了

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an additional sort of zero
额外的第0个特征向量

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feature vector that always takes
并且它的取值

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on the value of one.
总是1

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So now my feature vector
所以我现在的特征向量x

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X becomes this N+1 dimensional
是一个从0开始标记的

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vector that is zero index.
n+1维的向量

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So this is now a n+1
所以现在就是一个

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dimensional feature vector, but
n+1维的特征量向量

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I'm gonna index it from
但我要从0开始标记

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0 and I'm also going
同时

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to think of my
我也想把我的参数

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parameters as a vector.
都看做一个向量

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So, our parameters here, right
所以我们的参数就是

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that would be our theta zero,
我们的θ0

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theta one, theta two, and so
θ1 θ2 等等

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on all the way up to theta n,
直到θn

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we're going to gather
我们要把

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them up into a parameter
所有的参数都写成一个向量

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vector written theta 0, theta
θ0

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1, theta 2, and so
θ1 θ2

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on, down to theta n.
直到θn

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This is another zero index vector.
这里也有一个从0开始标记的矢量

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It's of index signed from zero.
下标从0开始

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That is another n plus 1 dimensional vector.
这是另外一个

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So, my hypothesis cannot be
所以我的假设

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written theta 0x0 plus
现在可以写成θ0乘以x0

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theta 1x1+ up to
加上θ1乘以x1直到

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theta n Xn.
θn 乘以xn

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And this equation is
这个等式

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the same as this on
和上面的等式是一样的

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top because, you know,
因为你看

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eight zero is equal to one.
x0等于1

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Underneath and I now
下面 我要

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take this form of the
把这种形式假设等式

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hypothesis and write this
写成

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as either transpose x,
θ转置乘以X

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depending on how familiar
取决于你对

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you are with inner products of
向量内积有多熟悉

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vectors if you
如果你展开

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write what theta transfers x
θ转置乘以X

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is what theta transfer and
那么就得到

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this is theta zero,
θ0

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theta one, up to theta
θ1直到θn

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N. So this
这个就是θ转置

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thing here is theta transpose
实际上

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and this is actually a N
这就是一个

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plus one by one matrix.
n+1乘以1维的矩阵

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It's also called a row vector
也被称为行向量

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and you take that and
用行向量

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multiply it with the
与X向量相乘

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vector X which is X
X向量是

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zero, X one, and so
x0 x1等等

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on, down to X n.
直到xn

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And so, the inner product
因此内积就是

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that is theta transpose X
θ转置乘以X

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is just equal to this.
就等于这个等式

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This gives us a convenient way
这就为我们提供了一个

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to write the form of the
表示假设的

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hypothesis as just the inner
更加便利的形式

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product between our parameter
即用参数向量θ以及

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vector theta and our theta
特征向量X的内积

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vector X. And it
这就是改写以后的

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is this little bit of notation,
表示方法

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this little excerpt of the
这样的表示习惯

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notation convention that let
就让我们

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us write this in this compact form.
可以以这种紧凑的形式写出假设

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So that's the form of a hypthesis when we have multiple features.
这就是多特征量情况下的假设形式

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And, just to give this another
起另一个名字

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name, this is also
就是

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called multivariate linear regression.
所谓的多元线性回归

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And the term multivariable that's just
多元一词

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maybe a fancy term for saying
也就是用来预测的多个特征量

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we have multiple features, or
或者变量

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multivariables with which to try to predict the value Y.
就是一种更加好听的说法罢了