Principal Component Analysis Problem Formulation

For the problem of dimensionality reduction, by far the most popular,

by far the most commonly used algorithm is something called

principle components analysis, or PCA.

In this video, I'd like to start talking about the problem formulation for PCA.

In other words, let's try to formulate,

precisely, exactly what we would like PCA to do.

Let's say we have a data set like this.

So, this is a data set of examples x and R2 and let's say I want to

reduce the dimension of the data from two-dimensional to one-dimensional.

In other words, I would like to find a line onto which to project the data.

So what seems like a good line onto which to project the data,

it's a line like this, might be a pretty good choice.

And the reason we think this might be a good choice is that if you look at where

the projected versions of the point scales, so I take this point and

project it down here.

Get that, this point gets projected here, to here, to here, to here.

What we find is that the distance between each point and

the projected version is pretty small.

That is, these blue line segments are pretty short.

So what PCA does formally is it tries to find a lower dimensional surface,

really a line in this case, onto which to project the data so

that the sum of squares of these little blue line segments is minimized.

The length of those blue line segments,

that's sometimes also called the projection error.

And so what PCA does is it tries to find a surface onto which to project the data so

as to minimize that.

As an aside, before applying PCA, it's standard practice to first

perform mean normalization at feature scaling so that the features x1 and

x2 should have zero mean, and should have comparable ranges of values.

I've already done this for this example, but I'll come back to this later and

talk more about feature scaling and the normalization in the context of PCA later.

But coming back to this example, in contrast to the red line that I just drew,

here's a different line onto which I could project my data,

which is this magenta line.

And, as we'll see, this magenta line is a much worse direction

onto which to project my data, right?

So if I were to project my data onto the magenta line,

we'd get a set of points like that.

And the projection errors, that is these blue line segments, will be huge.

So these points have to move a huge distance in order to

get projected onto the magenta line.

And so that's why PCA, principal components analysis,

will choose something like the red line rather than the magenta line down here.

Let's write out the PCA problem a little more formally.

The goal of PCA, if we want to reduce data from two-dimensional to

one-dimensional is, we're going to try find a vector that is a vector u1,

which is going to be an Rn, so that would be an R2 in this case.

I'm gonna find the direction onto which to project the data, so

it's to minimize the projection error.

So, in this example I'm hoping that PCA will find this vector,

which l wanna call u(1), so that when I project the data onto

the line that I define by extending out this vector,

I end up with pretty small reconstruction errors.

And that reference of data that looks like this.

And by the way, I should mention that where the PCA gives me u(1) or

-u(1), doesn't matter.

So if it gives me a positive vector in this direction, that's fine.

If it gives me the opposite vector facing in the opposite direction, so

that would be like minus u(1).

Let's draw that in blue instead, right?

But it gives a positive u(1) or negative u(1), it doesn't matter because

each of these vectors defines the same red line onto which I'm projecting my data.

So this is a case of reducing data from two-dimensional to one-dimensional.

In the more general case we have n-dimensional data and

we'll want to reduce it to k-dimensions.

In that case we want to find not just a single vector onto which to project

the data but we want to find k-dimensions onto which to project the data.

So as to minimize this projection error.

So here's the example.

If I have a 3D point cloud like this, then maybe what I want to do is find vectors.

So find a pair of vectors.

And I'm gonna call these vectors.

Let's draw these in red.

I'm going to find a pair of vectors, sustained from the origin.

Here's u(1), and here's my second vector, u(2).

And together, these two vectors define a plane, or they define a 2D surface, right?

Like this with a 2D surface onto which I am going to project my data.

For those of you that are familiar with linear algebra, for

this year they're really experts in linear algebra, the formal definition of this is

that we are going to find the set of vectors u(1), u(2), maybe up to u(k).

And what we're going to do is project the data

onto the linear subspace spanned by this set of k vectors.

But if you're not familiar with linear algebra, just think of it as

finding k directions instead of just one direction onto which to project the data.

So finding a k-dimensional surface is really finding a 2D plane in this case,

shown in this figure,

where we can define the position of the points in a plane using k directions.

And that's why for PCA we want to find k vectors onto which to project the data.

And so more formally in PCA, what we want to do is find this way to project the data

so as to minimize the sort of projection distance,

which is the distance between the points and the projections.

And so in this 3D example too.

Given a point we would take the point and project it onto this 2D surface.

We are done with that.

And so the projection error would be, the distance between the point and

where it gets projected down to my 2D surface.

And so what PCA does is I try to find the line, or a plane, or whatever,

onto which to project the data, to try to minimize that square projection,

that 90 degree or that orthogonal projection error.

Finally, one question I sometimes get asked is how does PCA relate to

linear regression?

Because when explaining PCA, I sometimes end up drawing diagrams like these and

that looks a little bit like linear regression.

It turns out PCA is not linear regression, and

despite some cosmetic similarity, these are actually totally different algorithms.

If we were doing linear regression, what we would do would be, on the left we would

be trying to predict the value of some variable y given some info features x.

And so linear regression, what we're doing is we're fitting a straight line so

as to minimize the square error between point and this straight line.

And so what we're minimizing would be the squared magnitude of these blue lines.

And notice that I'm drawing these blue lines vertically.

That these blue lines are the vertical distance between the point and

the value predicted by the hypothesis.

Whereas in contrast, in PCA, what it does is it tries to minimize

the magnitude of these blue lines, which are drawn at an angle.

These are really the shortest orthogonal distances.

The shortest distance between the point x and this red line.

And this gives very different effects depending on the dataset.

And more generally, when you're doing linear regression,

there is this distinguished variable y they we're trying to predict.

All that linear regression as well as taking all the values of x and

try to use that to predict y.

Whereas in PCA, there is no distinguish, or

there is no special variable y that we're trying to predict.

And instead, we have a list of features, x1, x2, and so on, up to xn, and

all of these features are treated equally, so no one of them is special.

As one last example, if I have three-dimensional data and

I want to reduce data from 3D to 2D, so maybe I wanna find two directions,

u(1) and u(2), onto which to project my data.

Then what I have is I have three features, x1, x2, x3, and

all of these are treated alike.

All of these are treated symmetrically and

there's no special variable y that I'm trying to predict.

And so PCA is not a linear regression, and

even though at some cosmetic level they might look related,

these are actually very different algorithms.

So hopefully you now understand what PCA is doing.

It's trying to find a lower dimensional surface onto which to project the data, so

as to minimize this squared projection error.

To minimize the square distance between each point and

the location of where it gets projected.

In the next video, we'll start to talk about how to

actually find this lower dimensional surface onto which to project the data.

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