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I would like to give a brief overview of
the course. This course is called

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Computational Methods for Data Analysis.
And it tries to take a very broad

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viewpoint about data and how you would use
this data in informing whatever system or

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application you are interested in. To
start with there is a little bit of a

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review of statistics. It's included in the
notes of the book. I am expecting that you

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would know at least a very strong, have a
very good background in linear algebra, a

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good background in some computing. So that
we're going to basically use the computing

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which is going to be Matlab based, in
addition to some statistical methods,

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which I will introduce as we go along. And
in addition to these statistical methods,

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we're going to start building upon this
towards trying to figure out how to use

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this data to inform your system. The
starting point for all this is time

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frequency analyses. How you would maybe
take signals that are noisy in time. And

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how you might think about, representing
them both in time, in frequency in making

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use of this in filtering to make very nice
predictions about what your data is

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telling you in such systems. Time
frequency analysis is a very important

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aspect of many applications, and including
data and image processing. In fact, that's

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eventually where this time frequency
processing will lead, is this idea of

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image processing. In fact, many of the
applications in the course will be brought

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in, in the sense of in will be
contextualized in an image processing way.

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Because this is a very natural way for us
to actually see what our data analysis

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techniques and reductions can do for us.
So, we will pursue that line of at the

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beginning of the course, start along that
line. And as we move along, we will start

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engaging in this idea of image processing
in this time frequency analysis. Both

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image cleanup, image de-noising. How do
you use filtering and spectral analysis?

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Wavelet analysis towards understanding how
you might be able to manipulate your data,

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clean date up as it were in a meaningful w
ay removing noise, doing that kind of

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processing that becomes really quite
important across a wide, wide variety of,

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variety of applications. Once we have some
of this down, we're going to start off and

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move a little bit more towards the
computer science aspects and introduce the

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concept of machine learning. And in this
context, we're integrating essentially

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some ideas of computer science. Some
statistics together with the idea that we

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could train the computer to recognize
patterns in data and then exploit those

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patterns. Make statistical decisions based
upon underlying features of your data.

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Underlying features of data is actually
ultimately what everything in the course

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is going to revolve towards. this first
simple example of even learning how to

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identify dogs from cats. How would you
write a routine that would help you in

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fact do this in a systematic way. And most
of this is done by doing this machine

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learning architecture, which looks for
features in your data. One of the most

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common ways to look for features in the
data is with, through what's called a

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Singular Value Decomposition. This is also
known as principal component analysis or

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proper thought, you know, mode
decomposition. In fact, there are quite a

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number of different names by which this
goes by. And it's a extremely powerful

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technique for essentially moving your data
into a space where you can look for

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clusters, meaningful clusters of data that
you would not be able to pick out with the

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naked eye. And it's this kind of
clustering of data or looking at these

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load dimensional structures in your data
that allows us to actually build efficient

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codes for understanding how our data
behaves in the first place. so, we're

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going to pursue this quite a bit. And
after this example of sort of moral

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traditional computer science type
application, we're going to move into what

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I called complex dynamical systems. So,
systems in which you can think about

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either whether climate modeling, weather
prediction or even any kind of neuro,

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neuro system where you have lots of
dynamics going on. And your ability to

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actually write down some kind of governing
equations for this is fairly limited or

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perhaps, imprecise. Yet, at the same time,
you've collected significant amounts of

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data on your system. And what you would
like to be able to do is maybe construct

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or reproduce, or predict how the future
state of the system will evolve. So, we're

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going to start placing some of these data
analysis techniques in the context of what

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we might be considered more scientific
applications that come from traditional

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STEM fields. So, for instance, fluid flow.
and so, what we'd like to do is think

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about looking for structures in our data
and our data analysis dynamic. Our data

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keeps being recorded, we keep gathering
information from our data. And how can we

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use this information to either inform a
model, or made predictive statements about

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the future state of that system. The work
here will involve, again, either principal

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components, if you would like to call them
that. Or proper orthogonally composition.

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In either case, the idea here is to
project your data onto some low

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dimensional, underlying manifold on which
all the dynamics happens, okay? So, this

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is often called dimensionality reduction.
You take this very large complex system

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which either will take you quite a bit of
long time to solve, or you maybe don't

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have governing equations for your siftem,
system. However, you have a lot of

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measurements that may be even very
precise, and how do you use the data alone

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to inform the future state of your system.
This is going to be the focus of the

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second part of the course, which is, let's
take a system that is related to some

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physical system we would like to quantify,
how do we use the data and integrate it

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with our more traditional progistic
scientific computing. In this, we're going

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to introduce two methods that are even
what might be called equation free. One

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is, in fact, called equation free
modeling. And this doesn't mean you don't

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have equations. It simply means that you
might actually have, not h ave the

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governing equations you need to understand
the evolution of your system. Maybe it's

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too complex, maybe you don't have a handle
on how to actually write down governing

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equations. However, you have data
measurements. And can you use those data

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measurements alone to inform the future
state of your system. So, this is called

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equation-free modeling. another system for
trying to do such modeling procedures is

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called dynamic mode decomposition in which
you project your data onto some

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low-dimensional manifold to try to predict
the future state of the system by taking

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snapshots of data and try and understand
from my low-dimensional features, can I

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actually predict perhaps my state of the
system much further out? An so by using

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this, you can start thinking about framing
control problems framing accurate ways to

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predict future states of the system an so
forth. So these are kind of the

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highlighted objectives. Which is, we are
after understanding data in the context of

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physical systems, physical biological
systems. We would like to integrate these

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two together. An some effort will be made
here, or a lot of effort will be made

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here, especially after the first part of
the book towards bringing this information

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together in such a way that it gives you
very powerful tools in your own

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applications, or own research that you
might have going on, or, or even in your

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own company. and so, the idea would be to,
to try to do this within the context of

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smaller toy models to demonstrate the
concepts. Not only that, most of it will

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be done in the context of using Matlab.
Matlab is a high level language which

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allows you to sort of testbed these ideas
in a very generic way and to build forward

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you know, sort of excellent ways to sort
of see if the method can work, or how it

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fails or what the limitations are. So,
with all that combined, you have this idea

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of bringing statistics and computer
science ideas in with traditional applied

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math complex dynamical systems idea, and
bringing this all together in a very nice

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way which you can int egrate all these
extremely powerful tools into one

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overarching method in which you can make
maximum use of these schemes towards

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understanding your problems. So that will
be the flow of, of the course. I think

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there's a lot of very interesting
applications, as well as examples that I

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give in the course, that you, you can see
where this can be useful. and I my hope

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is, of course, that these methods will
transfer from using them here in this

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course towards using them in a much
broader context both in perhaps, your own

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work or your own personal research.