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<center><h1>Introduction to Applied Data Analysis and Machine Learning</h1></center> <!-- document title -->
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<!-- author(s): Morten Hjorth-Jensen -->
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<b>Morten Hjorth-Jensen</b> [1, 2]
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<!-- institution(s) -->
<center>[1] <b>Department of Physics, University of Oslo</b></center>
<center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
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<center><h4>Sep 19, 2020</h4></center> <!-- date -->
<br>
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<h2 id="___sec0" class="anchor">Introduction </h2>
<p>
During the last two decades there has been a swift and amazing
development of Machine Learning techniques and algorithms that impact
many areas in not only Science and Technology but also the Humanities,
Social Sciences, Medicine, Law, indeed, almost all possible
disciplines. The applications are incredibly many, from self-driving
cars to solving high-dimensional differential equations or complicated
quantum mechanical many-body problems. Machine Learning is perceived
by many as one of the main disruptive techniques nowadays.
<p>
Statistics, Data science and Machine Learning form important
fields of research in modern science. They describe how to learn and
make predictions from data, as well as allowing us to extract
important correlations about physical process and the underlying laws
of motion in large data sets. The latter, big data sets, appear
frequently in essentially all disciplines, from the traditional
Science, Technology, Mathematics and Engineering fields to Life
Science, Law, education research, the Humanities and the Social
Sciences.
<p>
It has become more
and more common to see research projects on big data in for example
the Social Sciences where extracting patterns from complicated survey
data is one of many research directions. Having a solid grasp of data
analysis and machine learning is thus becoming central to scientific
computing in many fields, and competences and skills within the fields
of machine learning and scientific computing are nowadays strongly
requested by many potential employers. The latter cannot be
overstated, familiarity with machine learning has almost become a
prerequisite for many of the most exciting employment opportunities,
whether they are in bioinformatics, life science, physics or finance,
in the private or the public sector. This author has had several
students or met students who have been hired recently based on their
skills and competences in scientific computing and data science, often
with marginal knowledge of machine learning.
<p>
Machine learning is a subfield of computer science, and is closely
related to computational statistics. It evolved from the study of
pattern recognition in artificial intelligence (AI) research, and has
made contributions to AI tasks like computer vision, natural language
processing and speech recognition. Many of the methods we will study are also
strongly rooted in basic mathematics and physics research.
<p>
Ideally, machine learning represents the science of giving computers
the ability to learn without being explicitly programmed. The idea is
that there exist generic algorithms which can be used to find patterns
in a broad class of data sets without having to write code
specifically for each problem. The algorithm will build its own logic
based on the data. You should however always keep in mind that
machines and algorithms are to a large extent developed by humans. The
insights and knowledge we have about a specific system, play a central
role when we develop a specific machine learning algorithm.
<p>
Machine learning is an extremely rich field, in spite of its young
age. The increases we have seen during the last three decades in
computational capabilities have been followed by developments of
methods and techniques for analyzing and handling large date sets,
relying heavily on statistics, computer science and mathematics. The
field is rather new and developing rapidly. Popular software packages
written in Python for machine learning like
<a href="http://scikit-learn.org/stable/" target="_self">Scikit-learn</a>,
<a href="https://www.tensorflow.org/" target="_self">Tensorflow</a>,
<a href="http://pytorch.org/" target="_self">PyTorch</a> and <a href="https://keras.io/" target="_self">Keras</a>, all
freely available at their respective GitHub sites, encompass
communities of developers in the thousands or more. And the number of
code developers and contributors keeps increasing. Not all the
algorithms and methods can be given a rigorous mathematical
justification, opening up thereby large rooms for experimenting and
trial and error and thereby exciting new developments. However, a
solid command of linear algebra, multivariate theory, probability
theory, statistical data analysis, understanding errors and Monte
Carlo methods are central elements in a proper understanding of many
of algorithms and methods we will discuss.
<h2 id="___sec1" class="anchor">Learning outcomes </h2>
<p>
These sets of lectures aim at giving you an overview of central aspects of
statistical data analysis as well as some of the central algorithms
used in machine learning. We will introduce a variety of central
algorithms and methods essential for studies of data analysis and
machine learning.
<p>
Hands-on projects and experimenting with data and algorithms plays a central role in
these lectures, and our hope is, through the various
projects and exercises, to expose you to fundamental
research problems in these fields, with the aim to reproduce state of
the art scientific results. You will learn to develop and
structure codes for studying these systems, get acquainted with
computing facilities and learn to handle large scientific projects. A
good scientific and ethical conduct is emphasized throughout the
course. More specifically, you will
<ol>
<li> Learn about basic data analysis, Bayesian statistics, Monte Carlo methods, data optimization and machine learning;</li>
<li> Be capable of extending the acquired knowledge to other systems and cases;</li>
<li> Have an understanding of central algorithms used in data analysis and machine learning;</li>
<li> Gain knowledge of central aspects of Monte Carlo methods, Markov chains, Gibbs samplers and their possible applications, from numerical integration to simulation of stock markets;</li>
<li> Understand methods for regression and classification;</li>
<li> Learn about neural network, genetic algorithms and Boltzmann machines;</li>
<li> Work on numerical projects to illustrate the theory. The projects play a central role and you are expected to know modern programming languages like Python or C++, in addition to a basic knowledge of linear algebra (typically taught during the first one or two years of undergraduate studies).</li>
</ol>
There are several topics we will cover here, spanning from
statistical data analysis and its basic concepts such as expectation
values, variance, covariance, correlation functions and errors, via
well-known probability distribution functions like the uniform
distribution, the binomial distribution, the Poisson distribution and
simple and multivariate normal distributions to central elements of
Bayesian statistics and modeling. We will also remind the reader about
central elements from linear algebra and standard methods based on
linear algebra used to optimize (minimize) functions (the family of gradient descent methods)
and the Singular-value decomposition and
least square methods for parameterizing data.
<p>
We will also cover Monte Carlo methods, Markov chains, well-known
algorithms for sampling stochastic events like the Metropolis-Hastings
and Gibbs sampling methods. An important aspect of all our
calculations is a proper estimation of errors. Here we will also
discuss famous resampling techniques like the blocking, the bootstrapping
and the jackknife methods and the infamous bias-variance tradeoff.
<p>
The second part of the material covers several algorithms used in
machine learning.
<h2 id="___sec2" class="anchor">Machine Learning, a small (and probably biased) introduction </h2>
<p>
Ideally, machine learning represents the science of giving computers
the ability to learn without being explicitly programmed. The idea is
that there exist generic algorithms which can be used to find patterns
in a broad class of data sets without having to write code
specifically for each problem. The algorithm will build its own logic
based on the data. You should however always keep in mind that
machines and algorithms are to a large extent developed by humans. The
insights and knowledge we have about a specific system, play a central
role when we develop a specific machine learning algorithm.
<h2 id="___sec3" class="anchor">Machine Learning, an extremely rich field </h2>
<p>
Machine learning is an extremely rich field, in spite of its young
age. The increases we have seen during the last decades in
computational capabilities have been followed by developments of
methods and techniques for analyzing and handling large date sets,
relying heavily on statistics, computer science and mathematics. The
field is rather new and developing rapidly. Popular software libraries
written in Python for machine learning like
<a href="http://scikit-learn.org/stable/" target="_self">Scikit-learn</a>,
<a href="https://www.tensorflow.org/" target="_self">Tensorflow</a>,
<a href="http://pytorch.org/" target="_self">PyTorch</a> and <a href="https://keras.io/" target="_self">Keras</a>, all
freely available at their respective GitHub sites, encompass
communities of developers in the thousands or more. And the number of
code developers and contributors keeps increasing.
<h2 id="___sec4" class="anchor">A multidisciplinary approach </h2>
<p>
Not all the
algorithms and methods can be given a rigorous mathematical
justification (for example decision trees and random forests), opening up thereby large rooms for experimenting and
trial and error and thereby exciting new developments. However, a
solid command of linear algebra, multivariate theory, probability
theory, statistical data analysis, understanding errors and Monte
Carlo methods are central elements in a proper understanding of many
of the algorithms and methods we will discuss.
<h2 id="___sec5" class="anchor">Types of Machine Learning </h2>
<p>
The approaches to machine learning are many, but are often split into
two main categories. In <em>supervised learning</em> we know the answer to a
problem, and let the computer deduce the logic behind it. On the other
hand, <em>unsupervised learning</em> is a method for finding patterns and
relationship in data sets without any prior knowledge of the system.
Some authours also operate with a third category, namely
<em>reinforcement learning</em>. This is a paradigm of learning inspired by
behavioral psychology, where learning is achieved by trial-and-error,
solely from rewards and punishment.
<p>
Another way to categorize machine learning tasks is to consider the
desired output of a system. Some of the most common tasks are:
<p>
<!-- !bpop -->
<ul>
<li> Classification: Outputs are divided into two or more classes. The goal is to produce a model that assigns inputs into one of these classes. An example is to identify digits based on pictures of hand-written ones. Classification is often supervised learning.</li>
<li> Regression: Finding a functional relationship between an input data set and a reference data set. The goal is to construct a function that maps input data to continuous output values.</li>
<li> Clustering: Data are divided into groups with certain common traits, without knowing the different groups beforehand. It is thus a form of unsupervised learning.</li>
</ul>
<!-- !epop -->
<h2 id="___sec6" class="anchor">Essential elements of ML </h2>
<p>
The methods we cover have three main topics in common, irrespective of
whether we deal with supervised or unsupervised learning.
<!-- !bpop -->
<ul>
<li> The first ingredient is normally our data set (which can be subdivided into training, validation and test data). Many find the most difficult part of using Machine Learning to be the set up of your data in a meaningful way.</li>
<li> The second item is a model which is normally a function of some parameters. The model reflects our knowledge of the system (or lack thereof). As an example, if we know that our data show a behavior similar to what would be predicted by a polynomial, fitting our data to a polynomial of some degree would then determin our model.</li>
<li> The last ingredient is a so-called <b>cost/loss</b> function (or error function) which allows us to present an estimate on how good our model is in reproducing the data it is supposed to train.</li>
</ul>
<!-- !epop -->
<h2 id="___sec7" class="anchor">An optimization/minimization problem </h2>
<p>
At the heart of basically all Machine Learning algorithms we will encounter so-called minimization or optimization algorithms. A large family of such methods are so-called <b>gradient methods</b>.
<h2 id="___sec8" class="anchor">A Frequentist approach to data analysis </h2>
<p>
When you hear phrases like <b>predictions and estimations</b> and
<b>correlations and causations</b>, what do you think of? May be you think
of the difference between classifying new data points and generating
new data points.
Or perhaps you consider that correlations represent some kind of symmetric statements like
if \( A \) is correlated with \( B \), then \( B \) is correlated with
\( A \). Causation on the other hand is directional, that is if \( A \) causes \( B \), \( B \) does not
necessarily cause \( A \).
<p>
These concepts are in some sense the difference between machine
learning and statistics. In machine learning and prediction based
tasks, we are often interested in developing algorithms that are
capable of learning patterns from given data in an automated fashion,
and then using these learned patterns to make predictions or
assessments of newly given data. In many cases, our primary concern
is the quality of the predictions or assessments, and we are less
concerned about the underlying patterns that were learned in order
to make these predictions.
<p>
In machine learning we normally use <a href="https://en.wikipedia.org/wiki/Frequentist_inference" target="_self">a so-called frequentist approach</a>,
where the aim is to make predictions and find correlations. We focus
less on for example extracting a probability distribution function (PDF). The PDF can be
used in turn to make estimations and find causations such as given \( A \)
what is the likelihood of finding \( B \).
<h2 id="___sec9" class="anchor">What is a good model? </h2>
<p>
In science and engineering we often end up in situations where we want to infer (or learn) a
quantitative model \( M \) for a given set of sample points \( \boldsymbol{X} \in [x_1, x_2,\dots x_N] \).
<p>
As we will see repeatedely in these lectures, we could try to fit these data points to a model given by a
straight line, or if we wish to be more sophisticated to a more complex
function.
<p>
The reason for inferring such a model is that it
serves many useful purposes. On the one hand, the model can reveal information
encoded in the data or underlying mechanisms from which the data were generated. For instance, we could discover important
corelations that relate interesting physics interpretations.
<p>
In addition, it can simplify the representation of the given data set and help
us in making predictions about future data samples.
<p>
A first important consideration to keep in mind is that inferring the <em>correct</em> model
for a given data set is an elusive, if not impossible, task. The fundamental difficulty
is that if we are not specific about what we mean by a <em>correct</em> model, there
could easily be many different models that fit the given data set <em>equally well</em>.
<h2 id="___sec10" class="anchor">What is a good model? Can we define it? </h2>
<p>
The central question is this: what leads us to say that a model is correct or
optimal for a given data set? To make the model inference problem well posed, i.e.,
to guarantee that there is a unique optimal model for the given data, we need to
impose additional assumptions or restrictions on the class of models considered. To
this end, we should not be looking for just any model that can describe the data.
Instead, we should look for a <b>model</b> \( M \) that is the best among a restricted class
of models. In addition, to make the model inference problem computationally
tractable, we need to specify how restricted the class of models needs to be. A
common strategy is to start
with the simplest possible class of models that is just necessary to describe the data
or solve the problem at hand. More precisely, the model class should be rich enough
to contain at least one model that can fit the data to a desired accuracy and yet be
restricted enough that it is relatively simple to find the best model for the given data.
<p>
Thus, the most popular strategy is to start from the
simplest class of models and increase the complexity of the models only when the
simpler models become inadequate. For instance, if we work with a regression problem to fit a set of sample points, one
may first try the simplest class of models, namely linear models, followed obviously by more complex models.
<p>
How to evaluate which model fits best the data is something we will come back to over and over again in these set of lectures.
<h2 id="___sec11" class="anchor">Choice of Programming Language </h2>
<p>
Python plays nowadays a central role in the development of machine
learning techniques and tools for data analysis. In particular, seen
the wealth of machine learning and data analysis libraries written in
Python, easy to use libraries with immediate visualization(and not the
least impressive galleries of existing examples), the popularity of the
Jupyter notebook framework with the possibility to run <b>R</b> codes or
compiled programs written in C++, and much more made our choice of
programming language for this series of lectures easy. However,
since the focus here is not only on using existing Python libraries such
as <b>Scikit-Learn</b>, <b>Tensorflow</b> and <b>Pytorch</b>, but also on developing your own
algorithms and codes, we will as far as possible present many of these
algorithms either as a Python codes or C++ or Fortran (or other languages) codes.
<h2 id="___sec12" class="anchor">Data handling, machine learning and ethical aspects </h2>
<p>
In most of the cases we will study, we will either generate the data
to analyze ourselves (both for supervised learning and unsupervised
learning) or we will recur again and again to data present in say
<b>Scikit-Learn</b> or <b>Tensorflow</b>. Many of the examples we end up
dealing with are from a privacy and data protection point of view,
rather inoccuous and boring results of numerical
calculations. However, this does not hinder us from developing a sound
ethical attitude to the data we use, how we analyze the data and how
we handle the data.
<p>
The most immediate and simplest possible ethical aspects deal with our
approach to the scientific process. Nowadays, with version control
software like <a href="https://git-scm.com/" target="_self">Git</a> and various online
repositories like <a href="https://github.com/" target="_self">Github</a>,
<a href="https://about.gitlab.com/" target="_self">Gitlab</a> etc, we can easily make our codes
and data sets we have used, freely and easily accessible to a wider
community. This helps us almost automagically in making our science
reproducible. The large open-source development communities involved
in say <a href="http://scikit-learn.org/stable/" target="_self">Scikit-Learn</a>,
<a href="https://www.tensorflow.org/" target="_self">Tensorflow</a>,
<a href="http://pytorch.org/" target="_self">PyTorch</a> and <a href="https://keras.io/" target="_self">Keras</a>, are
all excellent examples of this. The codes can be tested and improved
upon continuosly, helping thereby our scientific community at large in
developing data analysis and machine learning tools. It is much
easier today to gain traction and acceptance for making your science
reproducible. From a societal stand, this is an important element
since many of the developers are employees of large public institutions like
universities and research labs. Our fellow taxpayers do deserve to get
something back for their bucks.
<p>
However, this more mechanical aspect of the ethics of science (in
particular the reproducibility of scientific results) is something
which is obvious and everybody should do so as part of the dialectics of
science. The fact that many scientists are not willing to share their codes or
data is detrimental to the scientific discourse.
<p>
Before we proceed, we should add a disclaimer. Even though
we may dream of computers developing some kind of higher learning
capabilities, at the end (even if the artificial intelligence
community keeps touting our ears full of fancy futuristic avenues), it is we, yes you reading these lines,
who end up constructing and instructing, via various algorithms, the
machine learning approaches. Self-driving cars for example, rely on sofisticated
programs which take into account all possible situations a car can
encounter. In addition, extensive usage of training data from GPS
information, maps etc, are typically fed into the software for
self-driving cars. Adding to this various sensors and cameras that
feed information to the programs, there are zillions of ethical issues
which arise from this.
<p>
For self-driving cars, where basically many of the standard machine
learning algorithms discussed here enter into the codes, at a certain
stage we have to make choices. Yes, we , the lads and lasses who wrote
a program for a specific brand of a self-driving car. As an example,
all carmakers have as their utmost priority the security of the
driver and the accompanying passengers. A famous European carmaker, which is
one of the leaders in the market of self-driving cars, had <b>if</b>
statements of the following type: suppose there are two obstacles in
front of you and you cannot avoid to collide with one of them. One of
the obstacles is a monstertruck while the other one is a kindergarten
class trying to cross the road. The self-driving car algo would then
opt for the hitting the small folks instead of the monstertruck, since
the likelihood of surving a collision with our future citizens, is
much higher.
<p>
This leads to serious ethical aspects. Why should we opt for such an
option? Who decides and who is entitled to make such choices? Keep in
mind that many of the algorithms you will encounter in this series of
lectures or hear about later, are indeed based on simple programming
instructions. And you are very likely to be one of the people who may
end up writing such a code. Thus, developing a sound ethical attitude
to what we do, an approach well beyond the simple mechanistic one of
making our science available and reproducible, is much needed. The
example of the self-driving cars is just one of infinitely many cases
where we have to make choices. When you analyze data on economic
inequalities, who guarantees that you are not weighting some data in a
particular way, perhaps because you dearly want a specific conclusion
which may support your political views? Or what about the recent
claims that a famous IT company like Apple has a sexist bias on the
their recently <a href="https://qz.com/1748321/the-role-of-goldman-sachs-algorithms-in-the-apple-credit-card-scandal/" target="_self">launched credit card</a>?
<p>
We do not have the answers here, nor will we venture into a deeper
discussions of these aspects, but we want you think over these topics
in a more overarching way. A statistical data analysis with its dry
numbers and graphs meant to guide the eye, does not necessarily
reflect the truth, whatever that is. As a scientist, and after a
university education, you are supposedly a better citizen, with an
improved critical view and understanding of the scientific method, and
perhaps some deeper understanding of the ethics of science at
large. Use these insights. Be a critical citizen. You owe it to our
society.
<p>
<p>
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