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revising models

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mhjensen
2018-05-09 16:44:52 -04:00
parent 239a7c172c
commit c4865d0ada
13 changed files with 6021 additions and 983 deletions
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doc/pub/How2ReadData/html/How2ReadData.html
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@@ -65,9 +65,9 @@ div { text-align: justify; text-justify: inter-word; }
<!-- tocinfo
{'highest level': 2,
'sections': [('What is Machine Learning?', 2, None, '___sec0'),
('Types of Machine Learning', 2, None, '___sec1'),
('Different algorithms', 2, None, '___sec2'),
'sections': [('Introduction', 2, None, '___sec0'),
('Learning outcomes', 2, None, '___sec1'),
('Types of Machine Learning', 2, None, '___sec2'),
('Software and needed installations', 2, None, '___sec3'),
('Python installers', 2, None, '___sec4'),
('Installing R, C++, cython or Julia', 2, None, '___sec5'),
@@ -77,33 +77,61 @@ div { text-align: justify; text-justify: inter-word; }
None,
'___sec7'),
('Representing data, more examples', 2, None, '___sec8'),
('Predator-Prey model from ecology', 2, None, '___sec9'),
('Case study from Hudson bay', 2, None, '___sec10'),
('Hudson bay data', 2, None, '___sec11'),
('Plotting the data', 2, None, '___sec12'),
('Simple regression model', 2, None, '___sec9'),
('Simple regression model, now using _scikit-learn_',
2,
None,
'___sec10'),
('Simple regression model with gradient descent',
2,
None,
'___sec11'),
('Simple regression model with stochastic gradient descent',
2,
None,
'___sec12'),
('Polynomial regression', 2, None, '___sec13'),
('Predator-Prey model from ecology', 2, None, '___sec14'),
('Case study from Hudson bay', 2, None, '___sec15'),
('Hudson bay data', 2, None, '___sec16'),
('Plotting the data', 2, None, '___sec17'),
('Hares and lynx in Hudson bay from 1900 to 1920',
2,
None,
'___sec13'),
'___sec18'),
('Why now create a computer model for the hare and lynx '
'populations?',
2,
None,
'___sec14'),
('The traditional (top-down) approach', 2, None, '___sec15'),
('Basic mathematics notation', 2, None, '___sec16'),
'___sec19'),
('The traditional (top-down) approach', 2, None, '___sec20'),
('Basic mathematics notation', 2, None, '___sec21'),
('Basic dynamics of the population of hares',
2,
None,
'___sec17'),
('Basic dynamics of the population of lynx', 2, None, '___sec18'),
('Evolution equations', 2, None, '___sec19'),
('Adapt the model to the Hudson Bay case', 2, None, '___sec20'),
('The program', 2, None, '___sec21'),
('The plot', 2, None, '___sec22'),
('Linear regression in Python', 2, None, '___sec23'),
('Linear Least squares in R', 2, None, '___sec24'),
('Non-Linear Least squares in R', 2, None, '___sec25')]}
'___sec22'),
('Basic dynamics of the population of lynx', 2, None, '___sec23'),
('Evolution equations', 2, None, '___sec24'),
('Adapt the model to the Hudson Bay case', 2, None, '___sec25'),
('The program', 2, None, '___sec26'),
('The plot', 2, None, '___sec27'),
('Linear regression in Python', 2, None, '___sec28'),
('Linear Least squares in R', 2, None, '___sec29'),
('Non-Linear Least squares in R', 2, None, '___sec30'),
('Example: ecoli lab experiment', 2, None, '___sec31'),
('The program', 2, None, '___sec32'),
('The output', 2, None, '___sec33'),
('Parameter estimation', 2, None, '___sec34'),
('A program relevant for the biological problem',
2,
None,
'___sec35'),
('Simulating financial transcations', 2, None, '___sec36'),
('Project 4a): Simulation of Transactions', 3, None, '___sec37'),
('Project 4b): Recognizing the distribution',
3,
None,
'___sec38')]}
end of tocinfo -->
<body>
@@ -145,33 +173,131 @@ MathJax.Hub.Config({
<center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
<br>
<p>
<center><h4>Dec 10, 2017</h4></center> <!-- date -->
<center><h4>May 9, 2018</h4></center> <!-- date -->
<br>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec0">What is Machine Learning? </h2>
<h2 id="___sec0">Introduction </h2>
<p>
Machine learning is 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.
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 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. 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. It has also, especially in later
years, found applications in a wide variety of other areas, including
bioinformatics, economy, physics, finance and marketing.
processing and speech recognition.
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.
<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="_blank">Scikit-learn</a>, <a href="https://www.tensorflow.org/" target="_blank">Tensorflow</a>,
<a href="http://pytorch.org/" target="_blank">PyTorch</a> and <a href="https://keras.io/" target="_blank">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.
<p>
<!-- !split -->
<h2 id="___sec1">Learning outcomes </h2>
<p>
These 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 exercies, 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 large 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 a
statistical data analysis and its basic concepts such expectation
values, variance, covariance, correlation functions and errors, via
well-known probability distribution functions like 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 fit functions such Cubic splines and gradient
methods for data optimization 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, bootstrapping
and jackknife methods.
<p>
The second part of the material covers several algorithms used in
machine learning.
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec1">Types of Machine Learning </h2>
<h2 id="___sec2">Types of Machine Learning </h2>
<p>
The approaches to machine learning are many, but are often split into two main categories.
@@ -179,7 +305,7 @@ 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 behavioural psychology, where learning is achieved by trial-and-error,
of learning inspired by behavioral psychology, where learning is achieved by trial-and-error,
solely from rewards and punishment.
<p>
@@ -192,32 +318,57 @@ Some of the most common tasks are:
<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>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec2">Different algorithms </h2>
In this course we will build our machine learning approach on a statistical foundation, with elements
from data analysis, stochastic processes etc before we proceed with the following machine learning algorithms
<ol>
<li> Linear regression and its variants</li>
<li> Decision tree algorithms, from simpler to more complex ones</li>
<li> Nearest neighbors models</li>
<li> Bayesian statistics</li>
<li> Support vector machines and finally various variants of</li>
<li> Artifical neural networks</li>
</ol>
Before we proceed however, there are several practicalities with data analysis and software tools we would
like to present. These tools will help us in our understanding of various machine learning algorithms.
The methods we cover have three main topics in common, irrespective of
whether we deal with supervised or unsupervised learning. The first
ingredient is normally our data set, the second is a model which is
normally a function of some parameters. The last ingredient is a
so-called <b>cost</b> function which allows us to present an estimate on
how good our model is in reproducing the data it is supposed to train.
<p>
Our emphasis here is on understanding the mathematical aspects of different algorithms, however, where possible
we will emphasize the importance of using available software.
Here we will build our machine learning approach on elements of the
statistical foundation discussed above, with elements from data
analysis, stochastic processes etc. We will discuss the following
machine learning algorithms
<ol>
<li> Linear regression and its variants, in essence polynomial regression</li>
<li> Decision tree algorithms, from simpler to more complex ones</li>
<li> Nearest neighbors models</li>
<li> Bayesian statistics and regression</li>
<li> Support vector machines and finally various variants of</li>
<li> Artifical neural networks and deep learning</li>
</ol>
Before we proceed however, there are several practicalities with data
analysis and software tools we would like to present. These tools will
help us in our understanding of various machine learning algorithms.
<p>
Our emphasis here is on understanding the mathematical aspects of
different algorithms, however, where possible we will emphasize the
importance of using available software. We start thus with a hands-on
and top-down approach machine learning. The aim is thus to start with
relevant data and use these to introduce statistical data analysis
concepts and machine learning algorithms before we delve into the
algorithms themselves. The examples we will use start with a simple
third-order polynomial with random noise added, and using the Python
software package <a href="http://scikit-learn.org/stable/" target="_blank">Scikit-learn</a> we
will introduce various machine learning algorithm s to make fits of
the data data and predictions. We move thereafter to more interesting
cases such as the simulation of financial transactions or disease
models. These are examples where we can easily set up the data and
then use machine learning algorithms using included in for example <b>scikit-learn</b>. Another model we
will consider is the so-called Ising model. Here we will use this
model to produce data for selected spin configurations and attempt to classify the data.
Finally, our last example consists of economic data from the OECD.
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec3">Software and needed installations </h2>
<p>
We will make intensive use of python as programming language and the myriad of available libraries.
Furthermore, you will find IPython/Jupyter notebooks invaluable in your work.
You can run <b>R</b> codes in the Jupyter/IPython notebooks, with the immediate benefit of visualizing your data.
@@ -279,7 +430,7 @@ To install <b>R</b> with Jupyter notebook <a href="https://mpacer.org/maths/r-ke
<h2 id="___sec6">Installing R, C++, cython or Julia </h2>
<p>
For the C++ affecianodas, Jupyter/IPython notebook allows you also to install C++ and run codes written in this language
For the C++ aficionados, Jupyter/IPython notebook allows you also to install C++ and run codes written in this language
interactively in the browser. Since we will emphasize writing many of the algorithms yourself, you can thus opt for
either Python or C++ as programming languages.
@@ -354,7 +505,7 @@ line <span style="color: #666666">=</span> np<span style="color: #666666">.</spa
reg <span style="color: #666666">=</span> DecisionTreeRegressor(min_samples_split<span style="color: #666666">=3</span>)<span style="color: #666666">.</span>fit(x,y)
plt<span style="color: #666666">.</span>plot(line, reg<span style="color: #666666">.</span>predict(line), label<span style="color: #666666">=</span><span style="color: #BA2121">&quot;decision tree&quot;</span>)
regline <span style="color: #666666">=</span> LinearRegression()<span style="color: #666666">.</span>fit(x,y)
plt<span style="color: #666666">.</span>plot(line, regline<span style="color: #666666">.</span>predict(line), label<span style="color: #666666">=</span> <span style="color: #BA2121">&quot;Linear Rgression&quot;</span>)
plt<span style="color: #666666">.</span>plot(line, regline<span style="color: #666666">.</span>predict(line), label<span style="color: #666666">=</span> <span style="color: #BA2121">&quot;Linear Regression&quot;</span>)
plt<span style="color: #666666">.</span>show()
</pre></div>
@@ -364,7 +515,144 @@ plt<span style="color: #666666">.</span>show()
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec9">Predator-Prey model from ecology </h2>
<h2 id="___sec9">Simple regression model </h2>
Add info about the equations
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #408080; font-style: italic"># Importing various packages</span>
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">random</span> <span style="color: #008000; font-weight: bold">import</span> random, seed
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
x <span style="color: #666666">=</span> <span style="color: #666666">2*</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(<span style="color: #666666">100</span>,<span style="color: #666666">1</span>)
y <span style="color: #666666">=</span> <span style="color: #666666">4+3*</span>x<span style="color: #666666">+</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(<span style="color: #666666">100</span>,<span style="color: #666666">1</span>)
xb <span style="color: #666666">=</span> np<span style="color: #666666">.</span>c_[np<span style="color: #666666">.</span>ones((<span style="color: #666666">100</span>,<span style="color: #666666">1</span>)), x]
theta <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>inv(xb<span style="color: #666666">.</span>T<span style="color: #666666">.</span>dot(xb))<span style="color: #666666">.</span>dot(xb<span style="color: #666666">.</span>T)<span style="color: #666666">.</span>dot(y)
xnew <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([[<span style="color: #666666">0</span>],[<span style="color: #666666">2</span>]])
xbnew <span style="color: #666666">=</span> np<span style="color: #666666">.</span>c_[np<span style="color: #666666">.</span>ones((<span style="color: #666666">2</span>,<span style="color: #666666">1</span>)), xnew]
ypredict <span style="color: #666666">=</span> xbnew<span style="color: #666666">.</span>dot(theta)
plt<span style="color: #666666">.</span>plot(xnew, ypredict, <span style="color: #BA2121">&quot;r-&quot;</span>)
plt<span style="color: #666666">.</span>plot(x, y ,<span style="color: #BA2121">&#39;ro&#39;</span>)
plt<span style="color: #666666">.</span>axis([<span style="color: #666666">0</span>,<span style="color: #666666">2.0</span>,<span style="color: #666666">0</span>, <span style="color: #666666">15.0</span>])
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">r&#39;$x$&#39;</span>)
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">r&#39;$y$&#39;</span>)
plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">r&#39;Linear Regression&#39;</span>)
plt<span style="color: #666666">.</span>show()
</pre></div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec10">Simple regression model, now using <b>scikit-learn</b> </h2>
Add info about the equations
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #408080; font-style: italic"># Importing various packages</span>
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">random</span> <span style="color: #008000; font-weight: bold">import</span> random, seed
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> LinearRegression
x <span style="color: #666666">=</span> <span style="color: #666666">2*</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(<span style="color: #666666">100</span>,<span style="color: #666666">1</span>)
y <span style="color: #666666">=</span> <span style="color: #666666">4+3*</span>x<span style="color: #666666">+</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(<span style="color: #666666">100</span>,<span style="color: #666666">1</span>)
linreg <span style="color: #666666">=</span> LinearRegression()
linreg<span style="color: #666666">.</span>fit(x,y)
xnew <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([[<span style="color: #666666">0</span>],[<span style="color: #666666">2</span>]])
ypredict <span style="color: #666666">=</span> linreg<span style="color: #666666">.</span>predict(xnew)
plt<span style="color: #666666">.</span>plot(xnew, ypredict, <span style="color: #BA2121">&quot;r-&quot;</span>)
plt<span style="color: #666666">.</span>plot(x, y ,<span style="color: #BA2121">&#39;ro&#39;</span>)
plt<span style="color: #666666">.</span>axis([<span style="color: #666666">0</span>,<span style="color: #666666">2.0</span>,<span style="color: #666666">0</span>, <span style="color: #666666">15.0</span>])
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">r&#39;$x$&#39;</span>)
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">r&#39;$y$&#39;</span>)
plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">r&#39;Random numbers &#39;</span>)
plt<span style="color: #666666">.</span>show()
</pre></div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec11">Simple regression model with gradient descent </h2>
Add info about the equations, play around with different learning rates
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #408080; font-style: italic"># Importing various packages</span>
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">math</span> <span style="color: #008000; font-weight: bold">import</span> exp, sqrt
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">random</span> <span style="color: #008000; font-weight: bold">import</span> random, seed
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
x <span style="color: #666666">=</span> <span style="color: #666666">2*</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(<span style="color: #666666">100</span>,<span style="color: #666666">1</span>)
y <span style="color: #666666">=</span> <span style="color: #666666">4+3*</span>x<span style="color: #666666">+</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(<span style="color: #666666">100</span>,<span style="color: #666666">1</span>)
xb <span style="color: #666666">=</span> np<span style="color: #666666">.</span>c_[np<span style="color: #666666">.</span>ones((<span style="color: #666666">100</span>,<span style="color: #666666">1</span>)), x]
theta_linreg <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>inv(xb<span style="color: #666666">.</span>T<span style="color: #666666">.</span>dot(xb))<span style="color: #666666">.</span>dot(xb<span style="color: #666666">.</span>T)<span style="color: #666666">.</span>dot(y)
<span style="color: #008000; font-weight: bold">print</span>(theta_linreg)
theta <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(<span style="color: #666666">2</span>,<span style="color: #666666">1</span>)
eta <span style="color: #666666">=</span> <span style="color: #666666">0.1</span>
Niterations <span style="color: #666666">=</span> <span style="color: #666666">1000</span>
m <span style="color: #666666">=</span> <span style="color: #666666">100</span>
<span style="color: #008000; font-weight: bold">for</span> <span style="color: #008000">iter</span> <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(Niterations):
gradients <span style="color: #666666">=</span> <span style="color: #666666">2.0/</span>m<span style="color: #666666">*</span>xb<span style="color: #666666">.</span>T<span style="color: #666666">.</span>dot(xb<span style="color: #666666">.</span>dot(theta)<span style="color: #666666">-</span>y)
theta <span style="color: #666666">-=</span> eta<span style="color: #666666">*</span>gradients
<span style="color: #008000; font-weight: bold">print</span>(theta)
xnew <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([[<span style="color: #666666">0</span>],[<span style="color: #666666">2</span>]])
xbnew <span style="color: #666666">=</span> np<span style="color: #666666">.</span>c_[np<span style="color: #666666">.</span>ones((<span style="color: #666666">2</span>,<span style="color: #666666">1</span>)), xnew]
ypredict <span style="color: #666666">=</span> xbnew<span style="color: #666666">.</span>dot(theta)
ypredict2 <span style="color: #666666">=</span> xbnew<span style="color: #666666">.</span>dot(theta_linreg)
plt<span style="color: #666666">.</span>plot(xnew, ypredict, <span style="color: #BA2121">&quot;r-&quot;</span>)
plt<span style="color: #666666">.</span>plot(xnew, ypredict2, <span style="color: #BA2121">&quot;b-&quot;</span>)
plt<span style="color: #666666">.</span>plot(x, y ,<span style="color: #BA2121">&#39;ro&#39;</span>)
plt<span style="color: #666666">.</span>axis([<span style="color: #666666">0</span>,<span style="color: #666666">2.0</span>,<span style="color: #666666">0</span>, <span style="color: #666666">15.0</span>])
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">r&#39;$x$&#39;</span>)
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">r&#39;$y$&#39;</span>)
plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">r&#39;Random numbers &#39;</span>)
plt<span style="color: #666666">.</span>show()
</pre></div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec12">Simple regression model with stochastic gradient descent </h2>
Add info about the equations, play around with different learning rates
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #408080; font-style: italic"># Importing various packages</span>
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">math</span> <span style="color: #008000; font-weight: bold">import</span> exp, sqrt
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">random</span> <span style="color: #008000; font-weight: bold">import</span> random, seed
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> SGDRegressor
x <span style="color: #666666">=</span> <span style="color: #666666">2*</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(<span style="color: #666666">100</span>,<span style="color: #666666">1</span>)
y <span style="color: #666666">=</span> <span style="color: #666666">4+3*</span>x<span style="color: #666666">+</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(<span style="color: #666666">100</span>,<span style="color: #666666">1</span>)
xb <span style="color: #666666">=</span> np<span style="color: #666666">.</span>c_[np<span style="color: #666666">.</span>ones((<span style="color: #666666">100</span>,<span style="color: #666666">1</span>)), x]
theta_linreg <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>inv(xb<span style="color: #666666">.</span>T<span style="color: #666666">.</span>dot(xb))<span style="color: #666666">.</span>dot(xb<span style="color: #666666">.</span>T)<span style="color: #666666">.</span>dot(y)
<span style="color: #008000; font-weight: bold">print</span>(theta_linreg)
sgdreg <span style="color: #666666">=</span> SGDRegressor(n_iter <span style="color: #666666">=</span> <span style="color: #666666">50</span>, penalty<span style="color: #666666">=</span><span style="color: #008000">None</span>, eta0<span style="color: #666666">=0.1</span>)
sgdreg<span style="color: #666666">.</span>fit(x,y<span style="color: #666666">.</span>ravel())
<span style="color: #008000; font-weight: bold">print</span>(sgdreg<span style="color: #666666">.</span>intercept_, sgdreg<span style="color: #666666">.</span>coef_)
</pre></div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec13">Polynomial regression </h2>
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>
</pre></div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec14">Predator-Prey model from ecology </h2>
<p>
<div class="alert alert-block alert-block alert-text-normal">
@@ -388,7 +676,7 @@ scientific method:
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec10">Case study from Hudson bay </h2>
<h2 id="___sec15">Case study from Hudson bay </h2>
<p>
<div class="alert alert-block alert-block alert-text-normal">
@@ -409,7 +697,7 @@ Here we start by
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec11">Hudson bay data </h2>
<h2 id="___sec16">Hudson bay data </h2>
<p>
<div class="alert alert-block alert-block alert-text-normal">
@@ -458,7 +746,7 @@ One reason that this particular system has been so extensively studied is that t
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec12">Plotting the data </h2>
<h2 id="___sec17">Plotting the data </h2>
<p>
<div class="alert alert-block alert-block alert-text-normal">
@@ -494,7 +782,7 @@ plt<span style="color: #666666">.</span>show()
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec13">Hares and lynx in Hudson bay from 1900 to 1920 </h2>
<h2 id="___sec18">Hares and lynx in Hudson bay from 1900 to 1920 </h2>
<p>
<br /><br /><center><p><img src="fig/Hudson_Bay_data.png" align="bottom" width=700></p></center><br /><br />
@@ -502,7 +790,7 @@ plt<span style="color: #666666">.</span>show()
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec14">Why now create a computer model for the hare and lynx populations? </h2>
<h2 id="___sec19">Why now create a computer model for the hare and lynx populations? </h2>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<p>
@@ -535,7 +823,7 @@ climate and other complicating factors. How significant are these?
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec15">The traditional (top-down) approach </h2>
<h2 id="___sec20">The traditional (top-down) approach </h2>
<p>
<div class="alert alert-block alert-block alert-text-normal">
@@ -568,7 +856,7 @@ ODEs</em> (which cannot be solved)
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec16">Basic mathematics notation </h2>
<h2 id="___sec21">Basic mathematics notation </h2>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<p>
@@ -588,7 +876,7 @@ ODEs</em> (which cannot be solved)
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec17">Basic dynamics of the population of hares </h2>
<h2 id="___sec22">Basic dynamics of the population of hares </h2>
<p>
<div class="alert alert-block alert-block alert-text-normal">
@@ -616,7 +904,7 @@ $$ \Delta H = a\Delta t H^n - b \Delta t H^nL^n$$
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec18">Basic dynamics of the population of lynx </h2>
<h2 id="___sec23">Basic dynamics of the population of lynx </h2>
<p>
<div class="alert alert-block alert-block alert-text-normal">
@@ -645,7 +933,7 @@ $$ \Delta L = d\Delta t H^nL^n - c\Delta t L^n$$
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec19">Evolution equations </h2>
<h2 id="___sec24">Evolution equations </h2>
<p>
<div class="alert alert-block alert-block alert-text-normal">
@@ -674,7 +962,7 @@ Note:
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec20">Adapt the model to the Hudson Bay case </h2>
<h2 id="___sec25">Adapt the model to the Hudson Bay case </h2>
<p>
<div class="alert alert-block alert-block alert-text-normal">
@@ -697,7 +985,7 @@ Note:
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec21">The program </h2>
<h2 id="___sec26">The program </h2>
<p>
<div class="alert alert-block alert-block alert-text-normal">
@@ -757,7 +1045,7 @@ plt<span style="color: #666666">.</span>show()
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec22">The plot </h2>
<h2 id="___sec27">The plot </h2>
<p>
<br /><br /><center><p><img src="fig/Hudson_Bay_sim.png" align="bottom" width=700></p></center><br /><br />
@@ -768,7 +1056,7 @@ If we perform a least-square fitting, we can find optimal values for the paramet
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec23">Linear regression in Python </h2>
<h2 id="___sec28">Linear regression in Python </h2>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<p>
@@ -801,7 +1089,7 @@ plt<span style="color: #666666">.</span>show()
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec24">Linear Least squares in R </h2>
<h2 id="___sec29">Linear Least squares in R </h2>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<p>
@@ -835,7 +1123,7 @@ predict(linearMod,<span style="color: #B00040">data.frame</span>(Year<span style
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec25">Non-Linear Least squares in R </h2>
<h2 id="___sec30">Non-Linear Least squares in R </h2>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<p>
@@ -867,12 +1155,360 @@ text(<span style="color: #666666">0</span>, <span style="color: #666666">0.5</sp
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec31">Example: ecoli lab experiment </h2>
<p>
<div class="alert alert-block alert-notice alert-text-normal">
<b>Typical pattern:</b>
<p>
The population grows faster and faster. <a href="http://www.zo.utexas.edu/courses/Thoc/PopGrowth.html" target="_blank">Why? Is there an underlying (general) mechanism</a>?
</div>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<p>
<ol>
<li> Cells divide after \( T \) seconds on average (one generation)</li>
<li> \( 2N \) celles divide into twice as many new cells \( \Delta N \) in a time
interval \( \Delta t \) as \( N \) cells would: \( \Delta N \propto N \)</li>
<li> \( N \) cells result in twice as many new individuals \( \Delta N \) in
time \( 2\Delta t \) as in time \( \Delta t \): \( \Delta N \propto\Delta t \)</li>
<li> Same proportionality wrt death (repeat reasoning)</li>
<li> Proposed model: \( \Delta N = b\Delta t N - d\Delta tN \) for some unknown
constants \( b \) (births) and \( d \) (deaths)</li>
<li> Describe evolution in discrete time: \( t_n=n\Delta t \)</li>
<li> Program-friendly notation: \( N \) at \( t_n \) is \( N^n \)</li>
<li> Math model: \( N^{n+1} = N^n + r\Delta t\, N \) (with \( \ r=b-d \))</li>
<li> Program model: <code>N[n+1] = N[n] + r*dt*N[n]</code></li>
</ol>
</div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec32">The program </h2>
<p>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<p>
Let us solve the difference equation in as simple way as possible,
just to train some programming: \( r=1.5 \), \( N^0=1 \), \( \Delta t=0.5 \)
<p>
<!-- code=python (!bc pypro) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
t <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(<span style="color: #666666">0</span>, <span style="color: #666666">10</span>, <span style="color: #666666">21</span>) <span style="color: #408080; font-style: italic"># 20 intervals in [0, 10]</span>
dt <span style="color: #666666">=</span> t[<span style="color: #666666">1</span>] <span style="color: #666666">-</span> t[<span style="color: #666666">0</span>]
N <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(t<span style="color: #666666">.</span>size)
N[<span style="color: #666666">0</span>] <span style="color: #666666">=</span> <span style="color: #666666">1</span>
r <span style="color: #666666">=</span> <span style="color: #666666">0.5</span>
<span style="color: #008000; font-weight: bold">for</span> n <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #666666">0</span>, N<span style="color: #666666">.</span>size<span style="color: #666666">-1</span>, <span style="color: #666666">1</span>):
N[n<span style="color: #666666">+1</span>] <span style="color: #666666">=</span> N[n] <span style="color: #666666">+</span> r<span style="color: #666666">*</span>dt<span style="color: #666666">*</span>N[n]
<span style="color: #008000; font-weight: bold">print</span> <span style="color: #BA2121">&#39;N[</span><span style="color: #BB6688; font-weight: bold">%d</span><span style="color: #BA2121">]=</span><span style="color: #BB6688; font-weight: bold">%.1f</span><span style="color: #BA2121">&#39;</span> <span style="color: #666666">%</span> (n<span style="color: #666666">+1</span>, N[n<span style="color: #666666">+1</span>])
</pre></div>
</div>
<p>
% if FORMAT != 'ipynb':
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec33">The output </h2>
<p>
<!-- code=text typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>N[1]=1.2
N[2]=1.6
N[3]=2.0
N[4]=2.4
N[5]=3.1
N[6]=3.8
N[7]=4.8
N[8]=6.0
N[9]=7.5
N[10]=9.3
N[11]=11.6
N[12]=14.6
N[13]=18.2
N[14]=22.7
N[15]=28.4
N[16]=35.5
N[17]=44.4
N[18]=55.5
N[19]=69.4
N[20]=86.7
</pre></div>
<p>
% endif
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec34">Parameter estimation </h2>
<p>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<p>
<ul>
<li> We do not know \( r \)</li>
<li> How can we estimate \( r \) from data?</li>
</ul>
We can use the difference equation with the experimental data
$$ N^{n+1} = N^n + r\Delta t N^n$$
Say \( N^{n+1} \) and \( N^n \) are known from data, solve wrt \( r \):
$$ r = \frac{N^{n+1}-N^n}{N^n\Delta t} $$
<p>
Use experimental data in the fraction, say \( t_1=600 \), \( t_2=1200 \),
\( N^1=140 \), \( N^2=250 \): \( r=0.0013 \).
</div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec35">A program relevant for the biological problem </h2>
<p>
<!-- exact r = 0.000694 -->
<p>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<p>
<p>
<!-- code=python (!bc pypro) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
<span style="color: #408080; font-style: italic"># Estimate r</span>
data <span style="color: #666666">=</span> np<span style="color: #666666">.</span>loadtxt(<span style="color: #BA2121">&#39;ecoli.csv&#39;</span>, delimiter<span style="color: #666666">=</span><span style="color: #BA2121">&#39;,&#39;</span>)
t_e <span style="color: #666666">=</span> data[:,<span style="color: #666666">0</span>]
N_e <span style="color: #666666">=</span> data[:,<span style="color: #666666">1</span>]
i <span style="color: #666666">=</span> <span style="color: #666666">2</span> <span style="color: #408080; font-style: italic"># Data point (i,i+1) used to estimate r</span>
r <span style="color: #666666">=</span> (N_e[i<span style="color: #666666">+1</span>] <span style="color: #666666">-</span> N_e[i])<span style="color: #666666">/</span>(N_e[i]<span style="color: #666666">*</span>(t_e[i<span style="color: #666666">+1</span>] <span style="color: #666666">-</span> t_e[i]))
<span style="color: #008000; font-weight: bold">print</span> <span style="color: #BA2121">&#39;Estimated r=</span><span style="color: #BB6688; font-weight: bold">%.5f</span><span style="color: #BA2121">&#39;</span> <span style="color: #666666">%</span> r
<span style="color: #408080; font-style: italic"># Can experiment with r values and see if the model can</span>
<span style="color: #408080; font-style: italic"># match the data better</span>
T <span style="color: #666666">=</span> <span style="color: #666666">1200</span> <span style="color: #408080; font-style: italic"># cell can divide after T sec</span>
t_max <span style="color: #666666">=</span> <span style="color: #666666">5*</span>T <span style="color: #408080; font-style: italic"># 5 generations in experiment</span>
t <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(<span style="color: #666666">0</span>, t_max, <span style="color: #666666">1000</span>)
dt <span style="color: #666666">=</span> t[<span style="color: #666666">1</span>] <span style="color: #666666">-</span> t[<span style="color: #666666">0</span>]
N <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(t<span style="color: #666666">.</span>size)
N[<span style="color: #666666">0</span>] <span style="color: #666666">=</span> <span style="color: #666666">100</span>
<span style="color: #008000; font-weight: bold">for</span> n <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #666666">0</span>, <span style="color: #008000">len</span>(t)<span style="color: #666666">-1</span>, <span style="color: #666666">1</span>):
N[n<span style="color: #666666">+1</span>] <span style="color: #666666">=</span> N[n] <span style="color: #666666">+</span> r<span style="color: #666666">*</span>dt<span style="color: #666666">*</span>N[n]
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
plt<span style="color: #666666">.</span>plot(t, N, <span style="color: #BA2121">&#39;r-&#39;</span>, t_e, N_e, <span style="color: #BA2121">&#39;bo&#39;</span>)
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">&#39;time [s]&#39;</span>); plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">&#39;N&#39;</span>)
plt<span style="color: #666666">.</span>legend([<span style="color: #BA2121">&#39;model&#39;</span>, <span style="color: #BA2121">&#39;experiment&#39;</span>], loc<span style="color: #666666">=</span><span style="color: #BA2121">&#39;upper left&#39;</span>)
plt<span style="color: #666666">.</span>show()
</pre></div>
<p>
Change <code>r</code> in the program and play around to make a better fit!
</div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec36">Simulating financial transcations </h2>
<p>
The aim of this project is to simulate financial transactions among financial agents
using Monte Carlo methods. The final goal is to extract a distribution of income as function
of the income \( m \). From Pareto's work (<a href="http://www.institutcoppet.org/2012/05/08/cours-deconomie-politique-1896-de-vilfredo-pareto" target="_blank">V.&nbsp;Pareto, 1897</a>) it is known from empirical studies
that the higher end of the distribution of money follows a distribution
$$
w_m\propto m^{-1-\alpha},
$$
with \( \alpha\in [1,2] \). We will here follow the analysis made by <a href="http://www.sciencedirect.com/science/article/pii/S0378437104004327" target="_blank">Patriarca and collaborators</a>.
<p>
Here we will study numerically the relation between the micro-dynamic relations among financial
agents and the resulting macroscopic money distribution.
<p>
We assume we have \( N \) agents that exchange money in pairs \( (i,j) \). We assume also that all agents
start with the same amount of money \( m_0 > 0 \). At a given 'time step', we choose randomly a pair
of agents \( (i,j) \) and let a transaction take place. This means that agent \( i \)'s money \( m_i \) changes
to \( m_i' \) and similarly we have \( m_j\rightarrow m_j' \).
Money is conserved during a transaction, meaning that
$$
\begin{equation}
m_i+m_j=m_i'+m_j'.
\label{eq:conserve}
\end{equation}
$$
The change is done via a random reassignement (a random number) \( \epsilon \), meaning that
$$
\begin{equation*}
m_i' = \epsilon(m_i+m_j),
\end{equation*}
$$
leading to
$$
\begin{equation*}
m_j'= (1-\epsilon)(m_i+m_j).
\end{equation*}
$$
The number \( \epsilon \) is extracted from a uniform distribution.
In this simple model, no agents are left with a debt, that is \( m\ge 0 \).
Due to the conservation law above, one can show that the system relaxes toward an equilibrium
state given by a Gibbs distribution
$$
\begin{equation*}
w_m=\beta \exp{(-\beta m)},
\end{equation*}
$$
with
$$
\begin{equation*}
\beta = \frac{1}{\langle m\rangle},
\end{equation*}
$$
and \( \langle m\rangle=\sum_i m_i/N=m_0 \), the average money.
It means that after equilibrium has been reached that the majority of agents is left with a small
number of money, while the number of richest agents, those with \( m \) larger than a specific value \( m' \),
exponentially decreases with \( m' \).
<p>
We assume that we have \( N=500 \) agents. In each simulation, we need a sufficiently large number of transactions, say \( 10^7 \). Our aim is find the final equilibrium distribution \( w_m \). In order to do that we would need
several runs of the above simulations, at least \( 10^3-10^4 \) runs (experiments).
<h3 id="___sec37">Project 4a): Simulation of Transactions </h3>
Your task is to first set up an algorithm which simulates the above transactions with an initial
amount \( m_0 \).
The challenge here is to figure out a Monte Carlo simulation based on the
above equations.
You will in particular need to make an algorithm which sets up a histogram as function of \( m \).
This histogram contains the number of times a value \( m \) is registered and represents
\( w_m\Delta m \). You will need to set up a value for the interval \( \Delta m \) (typically \( 0.01-0.05 \)).
That means you need to account for the number of times you register an income in the interval
\( m,m+\Delta m \). The number of times you register this income, represents the value that enters the histogram.
You will also need to find a criterion for when the equilibrium situation has been reached.
<h3 id="___sec38">Project 4b): Recognizing the distribution </h3>
Make thereafter a plot of \( \log{(w_m)} \) as function of \( m \)
and see if you get a straight line.
Comment the result.
<p>
We can then change our model to allow for a saving criterion, meaning that the agents save
a fraction \( \lambda \) of the money they have before the transaction is made. The final distribution will then no longer be given by Gibbs distribution. It could also include a taxation on financial transactions.
<p>
The conservation law of Eq. \eqref{eq:conserve} holds, but the money to be shared in a transaction between
agent \( i \) and agent \( j \) is now \( (1-\lambda)(m_i+m_j) \). This means that we have
$$
\begin{equation*}
m_i' = \lambda m_i+\epsilon(1-\lambda)(m_i+m_j),
\end{equation*}
$$
and
$$
\begin{equation*}
m_j' = \lambda m_j+(1-\epsilon)(1-\lambda)(m_i+m_j),
\end{equation*}
$$
which can be written as
$$
\begin{equation*}
m_i'=m_i+\delta m
\end{equation*}
$$
and
$$
\begin{equation*}
m_j'=m_j-\delta m,
\end{equation*}
$$
with
$$
\begin{equation*}
\delta m=(1-\lambda)(\epsilon m_j-(1-\epsilon)m_i),
\end{equation*}
$$
showing how money is conserved during a transaction.
Select values of \( \lambda =0.25,0.5 \) and \( \lambda=0.9 \) and try to extract the corresponding
equilibrium distributions and compare these with the Gibbs distribution. Comment your results.
Extract a parametrization of the above curves, see for example <a href="http://www.sciencedirect.com/science/article/pii/S0378437104004327" target="_blank">Patriarca and collaborators</a> and see if you can parametrize the high-end tails of the distributions in terms of power laws. Comment your results.
<p>
In the rest of this project we will follow the work of <a href="http://www.sciencedirect.com/science/article/pii/S0378437114006967" target="_blank">Goswami and Sen</a>.
In the studies above the agents were selected randomly, irrespective of whether we allowed for
saving or not during a transaction. What is often observed is that various agents tend to make preferences for for whom to interact with. We will now study the evolution of the distribution of wealth \( w_m \) by assuming that there is a likelihood
$$
p_{ij} \propto \vert m_i-m_j\vert^{-\alpha},
$$
for an interaction between agents \( i \) and \( j \) with respective wealths \( m_i \) and \( m_j \). The parameter \( \alpha > 0 \). For \( \alpha=0 \) we recover our model from part 5a).
Perform the same analysis as previously with \( N=500 \) as well as with \( N=1000 \) agents and study the distribution of wealth for \( \alpha =0.5 \), \( \alpha =1.0 \), \( \alpha =1.5 \) and \( \alpha =2.0 \).
You should try to reproduce Figure 1 of <a href="http://www.sciencedirect.com/science/article/pii/S0378437114006967" target="_blank">Goswami and Sen</a>.
Extract the tail of the distribution and see if it follows a Pareto distribution
$$
w_m\propto m^{-1-\alpha}.
$$
What happens if \( \alpha \gg 1 \)?
<p>
Perform the analysis with and without a saving \( \lambda \) on each transaction and comment your results.
We add to the previous probability the possibility that two agents who interact have performed similar transactions earlier. That is, in addition to being financially close, we assume that the likelihood for interacting increases if two agents have interacted earlier.
We add this feature by modifying the previous likelihood to
$$
p_{ij} \propto \vert m_i-m_j\vert^{-\alpha}\left(c_{ij}+1\right)^{\gamma},
$$
where \( c_{ij} \) represents the number of previous interactions that have taken place between \( i \) and \( j \). The factor \( 1 \) is added in order to ensure that if they have not interacted earlier they can still interact. Perform similar studies as above with \( N=1000 \), \( \alpha=1.0 \) and \( \alpha=2.0 \) using \( \gamma = 0.0, 1.0, 2.0, 3.0 \) and \( 4.0 \). Plot the wealth distributions for these cases and try to extract eventual power law tails with and without a saving \( \lambda \) in each transaction. Comment your results and compare them with figures 5 and 6 of <a href="http://www.sciencedirect.com/science/article/pii/S0378437114006967" target="_blank">Goswami and Sen</a>.
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