In the last video, we talked
about dimensionality reduction for
the purpose of compressing the data.
In this video, I'd like
to tell you about a second application
of dimensionality reduction and that is to visualize the data.
For a lot of machine learning
applications, it really helps
us to develop effective learning
algorithms, if we can understand our data better.
If there is some way of visualizing
the data better, and so,
dimensionality reduction offers us,
often, another useful tool to do so.
Let's start with an example.
Let's say we've collected a large
data set of many statistics
and facts about different countries around the world.
So, maybe the first feature, X1
is the country's GDP, or the
Gross Domestic Product, and
X2 is a per capita, meaning
the per person GDP, X3
human development index, life
expectancy, X5, X6 and so on.
And we may have a huge data set
like this, where, you know,
maybe 50 features for
every country, and we have a huge set of countries.
So is there something
we can do to try to understand our data better?
I've given this huge table of numbers.
How do you visualize this data?
If you have 50 features, it's
very difficult to plot 50-dimensional
data.
What is a good way to examine this data?
Using dimensionality reduction, what
we can do is, instead of
having each country represented by
this featured vector, xi, which
is 50-dimensional, so instead
of, say, having a country
like Canada, instead of
having 50 numbers to represent the features
of Canada, let's say we
can come up with a different feature
representation that is these
z vectors, that is in R2.
If that's the case, if we
can have just a pair of
numbers, z1 and z2 that
somehow, summarizes my 50
numbers, maybe what we
can do  [xx] is to plot
these countries in R2 and
use that to try to
understand the space in
[xx] of features of different
countries [xx]  the better and
so, here, what you
can do is reduce the
data from 50
D, from 50 dimensions
to 2D, so you can
plot this as a 2
dimensional plot, and, when
you do that, it turns out
that, if you look at the
output of the Dimensionality Reduction algorithms,
It usually doesn't astride a
physical meaning to these new features you want [xx] to.
It's often up to us to figure out you know, roughly what these features means.
But, And if you plot those features, here is what you might find.
So, here, every country
is represented by a point
ZI, which is an R2
and so each of those.
Dots, and this figure
represents a country, and so,
here's Z1 and here's
Z2, and [xx] [xx] of these.
So, you might find,
for example, That the horizontial
axis the Z1 axis
corresponds roughly to the
overall country size, or
the overall economic activity of a country.
So the overall GDP, overall
economic size of a country.
Whereas the vertical axis in our
data might correspond to the
per person GDP.
Or the per person well being,
or the per person economic activity, and,
you might find that, given these
50 features, you know, these
are really the 2 main dimensions
of the deviation, and so, out
here you may have a country
like the U.S.A., which
is a relatively large GDP,
you know, is a very
large GDP and a relatively
high per-person GDP as well.
Whereas here you might have
a country like Singapore, which
actually has a very
high per person GDP as well,
but because Singapore is a much
smaller country the overall
economy size of Singapore
is much smaller than the US.
And, over here, you would
have countries where individuals
are unfortunately some are less
well off, maybe shorter life expectancy,
less health care, less economic
maturity that's why smaller
countries, whereas a point
like this will correspond to
a country that has a
fair, has a substantial amount of
economic activity, but where individuals
tend to be somewhat less well off.
So you might find that
the axes Z1 and Z2
can help you to most succinctly
capture really what are the
two main dimensions of the variations
amongst different countries.
Such as the overall economic
activity of the country projected
by the size of the
country's overall economy as well
as the per-person individual
well-being, measured by per-person
GDP, per-person healthcare, and things like that.
So that's how you can
use dimensionality reduction, in
order to reduce data from
50 dimensions or whatever, down
to two dimensions, or maybe
down to three dimensions, so that
you can plot it and understand your data better.
In the next video, we'll start
to develop a specific algorithm,
called PCA, or Principal Component
Analysis, which will allow
us to do this and also
do the earlier application I talked about of compressing the data.