I would like to give a brief overview of
the course. This course is called
Computational Methods for Data Analysis.
And it tries to take a very broad
viewpoint about data and how you would use
this data in informing whatever system or
application you are interested in. To
start with there is a little bit of a
review of statistics. It's included in the
notes of the book. I am expecting that you
would know at least a very strong, have a
very good background in linear algebra, a
good background in some computing. So that
we're going to basically use the computing
which is going to be Matlab based, in
addition to some statistical methods,
which I will introduce as we go along. And
in addition to these statistical methods,
we're going to start building upon this
towards trying to figure out how to use
this data to inform your system. The
starting point for all this is time
frequency analyses. How you would maybe
take signals that are noisy in time. And
how you might think about, representing
them both in time, in frequency in making
use of this in filtering to make very nice
predictions about what your data is
telling you in such systems. Time
frequency analysis is a very important
aspect of many applications, and including
data and image processing. In fact, that's
eventually where this time frequency
processing will lead, is this idea of
image processing. In fact, many of the
applications in the course will be brought
in, in the sense of in will be
contextualized in an image processing way.
Because this is a very natural way for us
to actually see what our data analysis
techniques and reductions can do for us.
So, we will pursue that line of at the
beginning of the course, start along that
line. And as we move along, we will start
engaging in this idea of image processing
in this time frequency analysis. Both
image cleanup, image de-noising. How do
you use filtering and spectral analysis?
Wavelet analysis towards understanding how
you might be able to manipulate your data,
clean date up as it were in a meaningful w
ay removing noise, doing that kind of
processing that becomes really quite
important across a wide, wide variety of,
variety of applications. Once we have some
of this down, we're going to start off and
move a little bit more towards the
computer science aspects and introduce the
concept of machine learning. And in this
context, we're integrating essentially
some ideas of computer science. Some
statistics together with the idea that we
could train the computer to recognize
patterns in data and then exploit those
patterns. Make statistical decisions based
upon underlying features of your data.
Underlying features of data is actually
ultimately what everything in the course
is going to revolve towards. this first
simple example of even learning how to
identify dogs from cats. How would you
write a routine that would help you in
fact do this in a systematic way. And most
of this is done by doing this machine
learning architecture, which looks for
features in your data. One of the most
common ways to look for features in the
data is with, through what's called a
Singular Value Decomposition. This is also
known as principal component analysis or
proper thought, you know, mode
decomposition. In fact, there are quite a
number of different names by which this
goes by. And it's a extremely powerful
technique for essentially moving your data
into a space where you can look for
clusters, meaningful clusters of data that
you would not be able to pick out with the
naked eye. And it's this kind of
clustering of data or looking at these
load dimensional structures in your data
that allows us to actually build efficient
codes for understanding how our data
behaves in the first place. so, we're
going to pursue this quite a bit. And
after this example of sort of moral
traditional computer science type
application, we're going to move into what
I called complex dynamical systems. So,
systems in which you can think about
either whether climate modeling, weather
prediction or even any kind of neuro,
neuro system where you have lots of
dynamics going on. And your ability to
actually write down some kind of governing
equations for this is fairly limited or
perhaps, imprecise. Yet, at the same time,
you've collected significant amounts of
data on your system. And what you would
like to be able to do is maybe construct
or reproduce, or predict how the future
state of the system will evolve. So, we're
going to start placing some of these data
analysis techniques in the context of what
we might be considered more scientific
applications that come from traditional
STEM fields. So, for instance, fluid flow.
and so, what we'd like to do is think
about looking for structures in our data
and our data analysis dynamic. Our data
keeps being recorded, we keep gathering
information from our data. And how can we
use this information to either inform a
model, or made predictive statements about
the future state of that system. The work
here will involve, again, either principal
components, if you would like to call them
that. Or proper orthogonally composition.
In either case, the idea here is to
project your data onto some low
dimensional, underlying manifold on which
all the dynamics happens, okay? So, this
is often called dimensionality reduction.
You take this very large complex system
which either will take you quite a bit of
long time to solve, or you maybe don't
have governing equations for your siftem,
system. However, you have a lot of
measurements that may be even very
precise, and how do you use the data alone
to inform the future state of your system.
This is going to be the focus of the
second part of the course, which is, let's
take a system that is related to some
physical system we would like to quantify,
how do we use the data and integrate it
with our more traditional progistic
scientific computing. In this, we're going
to introduce two methods that are even
what might be called equation free. One
is, in fact, called equation free
modeling. And this doesn't mean you don't
have equations. It simply means that you
might actually have, not h ave the
governing equations you need to understand
the evolution of your system. Maybe it's
too complex, maybe you don't have a handle
on how to actually write down governing
equations. However, you have data
measurements. And can you use those data
measurements alone to inform the future
state of your system. So, this is called
equation-free modeling. another system for
trying to do such modeling procedures is
called dynamic mode decomposition in which
you project your data onto some
low-dimensional manifold to try to predict
the future state of the system by taking
snapshots of data and try and understand
from my low-dimensional features, can I
actually predict perhaps my state of the
system much further out? An so by using
this, you can start thinking about framing
control problems framing accurate ways to
predict future states of the system an so
forth. So these are kind of the
highlighted objectives. Which is, we are
after understanding data in the context of
physical systems, physical biological
systems. We would like to integrate these
two together. An some effort will be made
here, or a lot of effort will be made
here, especially after the first part of
the book towards bringing this information
together in such a way that it gives you
very powerful tools in your own
applications, or own research that you
might have going on, or, or even in your
own company. and so, the idea would be to,
to try to do this within the context of
smaller toy models to demonstrate the
concepts. Not only that, most of it will
be done in the context of using Matlab.
Matlab is a high level language which
allows you to sort of testbed these ideas
in a very generic way and to build forward
you know, sort of excellent ways to sort
of see if the method can work, or how it
fails or what the limitations are. So,
with all that combined, you have this idea
of bringing statistics and computer
science ideas in with traditional applied
math complex dynamical systems idea, and
bringing this all together in a very nice
way which you can int egrate all these
extremely powerful tools into one
overarching method in which you can make
maximum use of these schemes towards
understanding your problems. So that will
be the flow of, of the course. I think
there's a lot of very interesting
applications, as well as examples that I
give in the course, that you, you can see
where this can be useful. and I my hope
is, of course, that these methods will
transfer from using them here in this
course towards using them in a much
broader context both in perhaps, your own
work or your own personal research.