Name: データ分析6：主成分分析（PCA） - コンピュータマニア (Data Analysis 6: Principal Component Analysis (PCA) - Computerphile)
Uploaded: 2021-01-14T10:13:28.000Z
Duration: 20 min 9 s
Description: VoiceTubeの動画で発音を聞きながら英語表現を覚えよう！学べる英語：

様態の副詞

Principal component analysis is perhaps the most widely used data reduction technique on the planet

Everyone uses it but here's the thing. It doesn't actually do data reduction

Principal component analysis is the idea of trying to find a different view for our data in which we can separate it better

And the idea is that what we want to try and do is reframe our data

Maybe move it around so that we can better separate things out better cluster things perhaps it's better for machine learning

Axes by the most to least useful in some sense. So then we can perform a separate data reduction

Technique later by taking the slightly less useful axes away by or in this case

Dimensions or attributes of our data PCA is commonly pitched as a data reduction technique. Actually. It's a data transformation technique

It just makes our data and meaning able to production later

So let's imagine we have some attributes and we we know that some are correlated some are not correlated

The problem is that maybe we don't want to just delete some of the attributes may be

But it's not it doesn't mean we definitely want to delete attribute two and keep only attribute one on the other hand maybe

We do need to reduce some of the number of dimensions we've got or maybe we just want to try and make our data

More amenable to things like clustering. So let's look at a quick example

Typically PCA is done over many dimensions when we've got lots and lots of attributes

I'm just going to show two because obviously I starts to break down when I try and draw that many on the page

And what we want to try and do is work out what the contribution of each of these is to our data set

Which of these is useful which of these is not useful and now obviously if we had many dimensions, you know

Seven hundred ten thousand we can still apply the same technique. So maybe we have some data that's like this

We have some datasets over here and perhaps we have a little gap and maybe some data over here and in general

Our data is kind of increasing like this. So this means that attribute one and attribute two are positively correlated to some extent

but maybe the correlation is not so strong that we just wanted elite attribute to what we want to try and do is

Transform our data into a way where these are more useful imagine that you've got some data but looks a bit like this

We take a different view we can see there's actually two

objects and then we can separate them out and maybe if you were to take them again you could see there was four objects and

So on this is the idea what PCA is going to do is find new axes for this data

That separate it better for PCA to work. What we would start by doing is standardizing our data

So all of our dimensions attribute one attribute to all of the attributes are going to be centered around zero

And they're going to have a standard deviation of one

PCA will not work really at all. If you have widely different scales for your data

So what we want to try and do is find a direction or an axis through these two attributes

That separates out our data better than individual attributes

Do let's see how this data looks just from attribute one

you can see that it's sort of got this amount of spread in actually Bute one and they kind of

Dotted around like this and they sort of should go all the way along like this so you can't really see anything on here

Meaningful about these two groups, right?

And of course the more dimensions you have the more this could be a problem

Similarly about tribute to if we trace along here it goes from this range to this range

This is the variance of attribute - like the range and we can see that roughly speaking

The data is as spread out in attribute one as it is in attribute to that spread is about the same and both of them

are kind of useful for looking of a data but not really because again,

We have an equal distribution of point all the way along here. So that's not hugely useful

All right. So if we look at just attribute one, that's not hugely helpful

If we look at just attribute two, that's not hugely helpful either. So what can we do?

what we want to try and do is find a new axis like some new attribute that fits through this data like this and can

Really separate everything out because the spread of this data is actually diagonally in some sense not this way or this way

So what principal component analysis is going to do is find this principal component miss axis through our data like this

Such that when we look at the spread of a data, it's maximized, right?

So the data is as spread out as we can find it

And this is going to happen over any number of attributes

attribute to attribute three attribute for all the way to attribute n when we've got maybe 700 or 800 or

$1000 so at the moment which is fitting one principal component, this is one line through our two-dimensional data

There's going to be more principal components later, right?

But what we want to do is we want to pick the direction through this data

However, many attributes it has that has the most spread. So how do we measure this?

There's really two goals which were exactly the same one is to maximize the variance

So we find a direction for this line such that these points at the very edge are farthest apart

The other one is that we minimize the error

so we take this error from here this distance this distance from all these points to our new axes and we minimize it so

You can imagine if we do this for all our points we can get the sum of the squared

distances from these points to this line and then as we move this line around

Sometimes it's not if we have a line that goes like this

Some of these lines are going to be very large like this and that's going to be a higher amount of error

So what we'll find is that if we do this our first principal component will sit through whichever direction in the data

minimizes these distances and by definition

Maximizes this spread which makes this axis super useful if we use this axis now as our new X and we rotate this whole page

All our data is lovely and separated. And actually we have two distinct clusters in this data set, right?

Now as I mentioned PCA doesn't typically reduce the number of attributes from two to one just like that

We're going to have another principal component which represents the second amount of most variance orthogonal e so at ninety degrees

We find the first principal component which maximizes variance and then we find the next one along that maximizes events in the next direction

Now if there were multiple dimensions we'd keep applying this process

We keep finding new axes for our data that systematically show more and more of a spread of our data

But we're crucially we're ordering this by the amount of variance that they represent

So this is PC one or principal component one. This is principal component two and

Principal component one is always going to have the most

Varied data in it principal component to the next most three the next most all the way to the end with the least

Side-effect of this process is that we're going to have new axes through our data

Which and we're going to have a same number of axes as there are original dimensions in our data

But they're going to get less and less useful in terms of the variance of our data as we go forward

So PC one is going to be the most important

Pc-1 pc2 a little bit less spread out PC three a little bit less still all the way down to PC n all the way

Down here if you wanted to perform dimensionality reduction because you felt you had too many dimensions to your data

you could just for example keep the first 10 principal components project your data into that space and

we won't go into the mathematics of how to calculate these principal components because you can find out very easily online and all has a

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データ分析6：主成分分析（PCA） - コンピュータマニア (Data Analysis 6: Principal Component Analysis (PCA) - Computerphile)

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