14 - 3 - Principal Component Analysis Problem Formulation (9 min).srt

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For the problem of dimensionality
对于降维问题来说
(字幕整理：中国海洋大学 黄海广,haiguang2000@qq.com )

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reduction, by far the
目前

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most popular, by far the
最流行

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most commonly used algorithm is
最常用的算法是

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something called principal components analysis or PCA.
主成分分析方法(Principal Componet Analysis, PCA）

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It In this video, I would
在这段视频中

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like to start talking about the
我想首先开始讨论

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problem formulation for PCA,
PCA问题的公式描述

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in other words let us try
也就是说

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to formulate precisely exactly what
我们用公式准确地精确地描述

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we would like PCA to do.
我们想让 PCA 来做什么

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Let's say we have a dataset like
假设 我们有这样的一个数据集

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this, so this is a dataset
这个数据集含有

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of example X in R2,
二维实数R^2空间内的样本X

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and let's say I want
假设我想

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to reduce the dimension of the
对数据进行降维

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data from two dimensional to one dimensional.
从二维降到一维

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In other words I would like to find
也就是说，我想找到

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a line onto which to project the data.
一条直线，将数据投影到这条直线上

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So what seems like a good line onto which to project the data?
那怎么找到一条好的直线来投影这些数据呢？

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A line like this might be a pretty good choice.
这样的一条直线也许是个不错的选择

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And the reason you think
你认为

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this might be a good choice is
这是一个不错的选择的原因是

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that if you look at
如果你观察

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where the projected versions of the points goes.
投影到直线上的点的位置

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I'm gonna take this point and project it down here and get that.
我将这个点 投影到直线上 得到这个点

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This point gets projected here, to
这点被投影到这里

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here, to here, to here
这里 这里 以及这里

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what we find is that the
我们发现

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distance between each point
每个点到它们对应的

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and the projected version is pretty small.
投影到直线上的点之间的距离非常小

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That is, these blue
也就是说 这些蓝色的

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line segments are pretty short.
线段非常的短

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So what PCA
所以 正式的说

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does, formally, is it tries
PCA所作的就是

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to find a lower-dimensional surface
寻找一个低维的面

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really a line in
在这个例子中

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this case, onto which to
其实是一条直线

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project the data, so that
数据投射在上面  使得

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the sum of squares of these
这些蓝色小线段的平方和

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little blue line segments is minimized.
达到最小化

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The length of those blue line
这些蓝色线段的长度

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segments, that's sometimes also called
时常被叫做

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the projection error, and so what PCA
投影误差

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does is it tries to find
所以PCA所做的就是寻找

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the surface onto which to
一个投影平面

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project the data so as to
对数据进行投影

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minimize that. As an a
使得这个能够最小化

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side, before applying PCA
另外，在应用PCA之前

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it's standard practice to
常规的做法是

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first perform mean normalization and
先进行均值归一化和

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feature scaling so that the
特征规范化，使得

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features, X1 and X2,
特征X1和X2

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should have zero mean and
均值为0

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should have comparable ranges of values.
数值在可比较的范围之内

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I've already done this for this
在这个例子里，我已经这么做了

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example, but I'll come
但是

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back to this later and talk more
在后面，我将回过来，讨论更多的

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about feature scaling and mean normalization in the context of PCA later.
PCA背景下的特征规范化和均值归一化问题

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Coming back to this example,
回到这个例子

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in contrast to the red
对比

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lines that I just drew here's
我刚画好的红线

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a different line onto which I could project my data.
这是我对数据进行投影的一条不同的直线

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This magenta line.
这条品红色的线

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And as you can see, you
如你所见

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know, this magenta line is a
你知道，这条品红色直线

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much worse direction onto which to project my data, right?
是一个非常糟糕的方向来投影我的数据，对吧？

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So if I were to project my
所以，如果我将数据

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data onto the magenta
投影到这条品红色的直线上

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line, like the other set of points like that.
像我们刚才做的那样

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And the projection errors, that
这样，投影误差

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is these blue line segments would be huge.
就是这些蓝色的线段，将会很大

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So these points have to
所以，这些点将会

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move a huge distance in
移动很长一段距离

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order to get onto
才能投影到

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the, in order to
才能

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get projected onto the magenta line.
投影到这条品红色直线上

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And so that's why PCA
因此，这就是为什么PCA

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principle component analysis would choose
主成分分析方法会选择

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something like the red line
红色的这条直线

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rather than like the magenta line down here.
而不是品红色的这条直线

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Let's write out the PCA problem a little more formally.
我们正式一点地写出PCA问题

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The goal of PCA if we
PCA的目标是

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want to reduce data from two-
将数据从二维

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dimensional to one-dimensional is
降到一维

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we're going to try to find
我们将试着寻找

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a vector, that is
一个向量

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a vector Ui
向量Ui

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which is going to be in Rn,
属于Rn空间中的向量

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so that would be in R2 in this case,
在这个例子中是R2空间中的

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going to find direction onto
寻找一个对数据进行投影的方向

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which to project the data so as to minimize the projection error.
使得投影误差能够最小

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So in this example I'm
在这个例子里

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hoping that PCA will find this
希望PCA寻找到

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vector, which I'm going
向量，我将它叫做

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to call U1, so that
U1，所以

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when I project the data onto
当我把数据投影到

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the line that I defined
我定义的这条直线

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by extending out this vector,
通过延长这个向量得到的直线

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I end up with pretty small
最后我得到非常小的

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reconstruction errors and reference
重建误差

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data looks like this.
参考数据看上去是这样的

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And by the way,
此外

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I should mention that whether PCA
我应该指出的是，无论PCA

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gives me U1 or negative U1, it doesn't matter.
给出的是这个U1，还是负的U1，都没关系

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So if it gives me a positive
如果它给出的是正的向量

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vector in this direction that's fine, if it gives me,
在这个方向上，这没问题

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sort of the opposite vector facing
如果给出的是相反的向量

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in the opposite direction, so that
在相反的方向上

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would be -U1, draw that in
也就是-U1，用蓝色来画

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blue instead, whether it gives me positive
无论给的是正的

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U1 negative U1,
还是负的U1

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it doesn't matter, because each of
都没关系，因为

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these vectors defines the
每一个方向都定义了

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same red line onto which
相同的红色直线

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I'm projecting my data. So this
也就是我将投影的方向

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is a case of reducing data
这就是将

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from 2 dimensional to 1 dimensional.
2维数据降到1维的例子

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In the more general case we
更一般的情况是

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have N dimensional data and
我们有N维的数据

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we want to reduce it K dimensions.
想降到K维

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In that case, we want to
在这种情况下

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find not just a single vector
我们不仅仅只寻找单个的向量

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onto which to project the data
来对数据进行投影

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but we want to find K dimensions
我们想寻找K个方向

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onto which to project the data.
来对数据进行投影

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So as to minimize this projection error.
为了最小化投影误差

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So here's an example.
这是一个例子

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If I have a 3D point
如果我有3维数据点

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cloud like this then maybe
比如说像这样的

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what I want to do is find
我想要做的是

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vectors, sorry find a pair of vectors,
寻找向量，对不起，是寻找两个向量

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and I'm gonna call these vectors
我将这些向量叫做...

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lets draw these in red, I'm going
我们用红线画出来

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to find a pair of vectors,
我要寻找两个向量

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extending for the origin here's
从原点延伸出来

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U1, and here's
这是U1

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my second vector U2,
这是第二个向量U2

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and together these two
这两个向量一起

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vectors define a plane,
定义了一个平面

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or they define a 2D surface,
或者说，定义了一个2维平面

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kind of like this, sort of,
类似于这样的

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2D surface, onto which I'm going to project my data.
2维平面，我将把数据投影到上面

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For those of you that are
对于你们其中

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familiar with linear algebra, for
熟悉线性代数的人来说

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those of you that are really experts
对于你们其中真的

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in linear algebra, the formal
精通线性代数的人来说，

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definition of this is that
对这个正式的定义是

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we're going to find a set of
我们将寻找一组向量

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vectors, U1 U2 maybe up
U1，U2，也许

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to Uk, and what
一直到Uk

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we're going to do is project
我们将要做的是

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the data onto the linear
将数据投影到

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subspace spanned by this set of k vectors.
这k个向量张开的线性子空间上

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But if you are not familiar
但是如果你不熟悉

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with linear algebra, just think
线性代数，那就想成是

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of it as finding k directions
寻找k个方向

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instead of just one direction, onto which to project the data.
而不是只寻找一个方向，对数据进行投影

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So, finding a k-dimensional surface,
所以，寻找一个k维的平面

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really finding a 2D plane
在这里是寻找2维的平面

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in this case, shown in this
如图所示

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figure, where we can
这里我们用

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define the position of the points in the plane using k directions.
k个方向来定义平面中这些点的位置

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That's why for PCA, we
这就是为什么，对于PCA

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want to find k vectors onto which to project the data.
我们要寻找k个向量来对数据进行投影

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And so, more formally, in
因此，更正式一点的说

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PCA what we want
在PCA中，我们想做的就是

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to do is find this way
寻找到这种方式

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to project the data so as
对数据进行投影

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to minimize the, sort of,
进而最小化投影距离

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projection distance, which is distance between points and projections.
也就是数据点和投影后的点之间的距离

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In this so 3D example,
在这个3维的例子里

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too, given a point we
给定一个点

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would take the point and project
我们想将这个点

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it onto this 2D surface.
投影到2维平面上

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When you're done with that,
当你完成了那个

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and so the projection error would
因此投影误差就是

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be, you know, the distance between the
也就是

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point and where it gets projected down.
这点与投影到

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to my 2D surface,
2维平面之后的点之间的距离

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and so what PCA does is it'll
因此PCA做的是

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try to find a line or
寻找一条直线

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plane or whatever onto which
或者平面，诸如此类等等

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to project the data, to try
对数据进行投影

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to minimize that squared projection,
来最小化平方投影

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that 90 degree, or that orthogonal projection error.
90度的或者正交的投影误差

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Finally, one question that I
最后

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sometimes get asked is how
一个我有时会被问到的问题是

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does PCA relate to
PCA和线性回归有怎么样的关系？

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linear regression, because when explaining
因为当我解释PCA的时候

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PCA I sometimes end up
我有时候会以

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drawing diagrams like these and that looks a little bit like linear regression.
画这样的图表来结束，看上去有点像线性回归

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It turns out PCA is not
但是，事实是

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linear regression, and despite
PCA不是线性回国

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some cosmetic similarity these are actually totally different algorithms.
尽管看上去有一些相似性，但是它们确实是两种不同的算法

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If we were doing linear regression
如果我们做线性回国

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what we would do would be, on
我们 做的是

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the left we would be trying
看左边，我们想要

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to predict the value of some
在给定某个输入特征x的情况下

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variable y given some input
预测某个变量y的数值

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features x. And so linear
因此，对于线性回归

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regression, what we're doing
我们想做的是

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is we're fitting a straight line
拟合一条直线

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so as to minimize the squared
来最小化

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error between a point and the straight line.
点和直线之间的平方误差

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And so what we'd be minimizing
所以我们要最小化的是

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would be the squared magnitude of these blue lines.
这些蓝线幅值的平方

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And notice I'm drawing these
注意我画的这些

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blue lines vertically, that they
蓝色的垂直线

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are the vertical distance between
这是垂直距离

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a point and the value
它是某个点

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predicted by the hypothesis, whereas in
与通过假设的得到的其预测值之间的距离

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contrast, in PCA, what it
与此想反

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does is it tries to
PCA要做的是

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minimize the magnitude of these blue lines,
最小化这些蓝色直线的幅值

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which are drawn at an angle,
倾斜地画出来的

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these are really the shortest orthogonal
这实际上是最短的

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distances, the shortest distance between
直角距离

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the point X and this
点X

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red line, and this
跟红色直线之间的最短距离

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gives very different effects, depending
这是一种非常不同的效果

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on the data set.
取决于数据集

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And more generally generally and
更更更一般的是

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more generally when you're doing
当你做

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linear regression there is this
线性回归的时候，有一个

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distinguished variable y that
特别的变量y

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we're trying to predict, all that
是我们将要预测的

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linear regression is about is taking all the values
线性回归所要做的就是

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of X and try to use
用X的所有的值来

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that to predict Y. Whereas
预测y

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in PCA, there is no
然而在PCA中

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distinguished or there is
没有这么一个特别的或者

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no special variable Y that
特殊的变量y

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we're trying to predict, and instead
是我们要预测的

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we have a list of features
我们所拥有的是

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x1, x2, and so on
特征x1，x2，等

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up to xN, and all of
一直到xN

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these features are treated equally.
所有的这些特征都是被同样地对待

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So, no one of them is special.
因此，它们中没有一个是特殊的

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As one last example, if I
最后一个例子中

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have three-dimensional data, and
如果我有3维数据

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I want to reduce data from
我要将这些数据

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3D to 2D, so maybe
从3维降到2维

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I want to find two directions,
我就要找到2个方向

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you know, u1, and u2,
也就是u1和u2

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onto which to project my data,
将数据投影到它们上面

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then what I have is I
然后我得到的是

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have three features, x1, x2,
我有3个特征，x1，x2

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x3, and all of these are treated alike.
x3，所有的这些都是被同样地对待

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All of these are treated symmetrically
这些都是被均等地对待

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and there is no special variable
没有特殊的变量

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y that I'm trying to predict.
y，需要被预测

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And so PCA is not
因此，PCA

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linear regression, and even
不是线性回归

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though at some cosmetic level they
尽管一定程度的相似性

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might look related, these are
使得它们看上去是有关联的

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actually very different algorithms. So,
但它们实际上非常不同的算法

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hopefully you now understand what
因此，希望你们能理解

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PCA is doing: it's trying
PCA是做什么的

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to find a lower dimensional
它是寻找到一个低维的平面

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surface onto which to project
对数据进行投影

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the data so as to
以便

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minimize this squared projection error,
最小化投影误差的平方

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to minimize the squared distance between
最小化每个点

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each point and the location of where it gets projected.
与投影后的对应点之间的距离的平方值

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In the next video we'll start
在下一段视频中

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to talk about how to actually
我们将开始讨论

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find this lower dimensional surface
如果真正地找到这个低维平面

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onto which to project the data.
来对数据进行投影