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project 2 ready, typos may be corrected later

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@@ -45,9 +45,13 @@ Automatically generated HTML file from DocOnce source
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@@ -106,14 +119,16 @@ MathJax.Hub.Config({
<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
<ul class="dropdown-menu">
<!-- navigation toc: --> <li><a href="#___sec0" style="font-size: 80%;"><b>Classification and Regression, from linear and logistic regression to neural networks</b></a></li>
<!-- navigation toc: --> <li><a href="#___sec1" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Part a): Producing the data</a></li>
<!-- navigation toc: --> <li><a href="#___sec2" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Part b): Estimating the coupling constant of the one-dimensional Ising model</a></li>
<!-- navigation toc: --> <li><a href="#___sec1" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Part a): Producing the data for the one-dimensional Ising model</a></li>
<!-- navigation toc: --> <li><a href="#___sec2" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Part b): Estimating the coupling constant of the one-dimensional Ising model using linear regression</a></li>
<!-- navigation toc: --> <li><a href="#___sec3" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Part c): Determine the phase of the two-dimensional Ising model</a></li>
<!-- navigation toc: --> <li><a href="#___sec4" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Part d): Classifying the Ising model phase using neural networks</a></li>
<!-- navigation toc: --> <li><a href="#___sec5" style="font-size: 80%;"><b>Background literature</b></a></li>
<!-- navigation toc: --> <li><a href="#___sec6" style="font-size: 80%;"><b>Introduction to numerical projects</b></a></li>
<!-- navigation toc: --> <li><a href="#___sec7" style="font-size: 80%;"><b>Format for electronic delivery of report and programs</b></a></li>
<!-- navigation toc: --> <li><a href="#___sec8" style="font-size: 80%;"><b>Software and needed installations</b></a></li>
<!-- navigation toc: --> <li><a href="#___sec4" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Part d): Regression analysis of the one-dimensional Ising model using neural networks</a></li>
<!-- navigation toc: --> <li><a href="#___sec5" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Part e): Classifying the Ising model phase using neural networks</a></li>
<!-- navigation toc: --> <li><a href="#___sec6" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Part f) Critical evaluation of the various algorithms</a></li>
<!-- navigation toc: --> <li><a href="#___sec7" style="font-size: 80%;"><b>Background literature</b></a></li>
<!-- navigation toc: --> <li><a href="#___sec8" style="font-size: 80%;"><b>Introduction to numerical projects</b></a></li>
<!-- navigation toc: --> <li><a href="#___sec9" style="font-size: 80%;"><b>Format for electronic delivery of report and programs</b></a></li>
<!-- navigation toc: --> <li><a href="#___sec10" style="font-size: 80%;"><b>Software and needed installations</b></a></li>
</ul>
</li>
@@ -147,24 +162,16 @@ MathJax.Hub.Config({
<center><b>Department of Physics, University of Oslo, Norway</b></center>
<br>
<p>
<center><h4>Oct 8, 2018</h4></center> <!-- date -->
<center><h4>Oct 9, 2018</h4></center> <!-- date -->
<br>
<ol>
<li> add about ising model</li>
<li> link to where we have the data</li>
<li> explain how to fit the model</li>
<li> link to mehta's article</li>
</ol>
<p>
</div> <!-- end jumbotron -->
<h2 id="___sec0" class="anchor">Classification and Regression, from linear and logistic regression to neural networks </h2>
<p>
The main aim of this project is to study both classification and
regression problems, starting with the regression algortihms studied
regression problems, starting with the regression algorithms studied
in project 1. We will include logistic regresion for classification
problems and write our own multilayer perceptron code for studying
both regression and classification problems. The codes developed in
@@ -174,19 +181,19 @@ also be utilized (and included in logistic regression and the neural
network codes) in the present analysis.
<p>
We will use the Ising model to generate our training data and will
focus mainly on supervised training. We will follow closely the recent
We will use the so-called Ising model for our training data and will
focus on supervised training. We will follow closely the recent
article of <a href="https://arxiv.org/abs/1803.08823" target="_self">Mehta et al, arXiv
1803.08823</a>. This article stands
out as an excellent review on machine learning (ML) algorithms applied
to typical physics problems. The added benefit is that each figure and
out as an excellent review on machine learning (ML) algorithms.
The added benefit is that each figure and
model presented in <a href="https://physics.bu.edu/~pankajm/MLnotebooks.html" target="_self">this article is accompanied by its jupyter
notebook</a>. This
means that we can start using these and compare with our own
results. In case you wish to use their data for the Ising model, their
data can be downloaded from the same link which lists to the jupyter
notebooks. See also at the end of the project description for more
information on how to install various Python packages.
means that we can start using these and compare with our own results.
They provide also the data set for the regression and classification
analysis that we will explore. In this sense, with their available
notebooks, it makes life easier since we can compare our own codes
with their codes.
<p>
With the abovementioned configurations we will determine, using first
@@ -202,75 +209,268 @@ that is the two-dimensional Ising model, will be studied using
logistic regression and deep neural networks. The aim is to develop
your own logistic regression code for the classification of the phases
(this is a binary model) and your multilayer perceptron code for the
classification and regression case.
classification and regression case. You can compare your own results with those obtained
using <b>scikit-learn</b> or <b>tensorflow</b> or other Python packages such as <b>keras</b> or other.
<p>
Feel free to use the notebooks to benchmark your code. If you wish to
write your own C++ or Fortran program for say a simple neural network
model and a logistic regression model, please feel free to do so. You can then benchmark your results
against the above jupyter notebooks.
write your own C++ or Fortran program for say a multilayer neural network
model and a logistic regression model, please feel free to do so.
<h3 id="___sec1" class="anchor">Part a): Producing the data </h3>
<h3 id="___sec1" class="anchor">Part a): Producing the data for the one-dimensional Ising model </h3>
<p>
You can use the Ising model data from the article of Mehta <em>et al.</em>,
or generate your own data. If you opt for using your own Ising model
code, you need to generate \( 10000 \) energy configurations with their
spin orientations after the system has reached its most likely
state. These energies and their corresponding spin orientations
represent then your data. We will use a fixed lattice of \( L\times L =
40 \times 40 \) spins in two dimensions and \( L=40 \) spins in one
dimension. Make sure the calculations have been equilibrated. For the
two-dimensional system, compute the configurations for three values of
the temperature, namely \( T=0.75 \) (ordered phase), \( T=2.3 \) (near the
critical point) and \( T=4.0 \) (disordered phase). For the
one-dimensional system it suffices to compute the various
configurations for one temperature only, say \( T=2.0 \). These are the
data you will use to study different ML algorithms. We generate our
data with \( J=1 \).
The model we will employ in our studies is the so-called <a href="https://en.wikipedia.org/wiki/Ising_model" target="_self">Ising
model</a>. Together with
models like the <a href="https://en.wikipedia.org/wiki/Potts_model" target="_self">Potts
model</a> and similar
so-called lattice models, the Ising model has been widely studied in
mathematics (in statistics in particular), physics, <a href="https://journals.aps.org/pre/abstract/10.1103/PhysRevE.93.062402" target="_self">life
science</a>,
chemistry and even in the <a href="https://www.springer.com/gp/book/9781461420316" target="_self">social sciences in order to model social
behavior</a>. It is a
simple binary value system where the variables of the model (spins often in
physics) can take two values only, for example \( \pm 1 \) or \( 0 \) and \( 1 \).
The system exhibits a phase transition in two or higher dimensions and
the first person to find the analytical expressions for various
expectation values was the Norwegian chemist <a href="https://en.wikipedia.org/wiki/Lars_Onsager" target="_self">Lars
Onsager</a> (Nobel prize in
chemistry) after a tour de force mathematics exercise.
<h3 id="___sec2" class="anchor">Part b): Estimating the coupling constant of the one-dimensional Ising model </h3>
<p>
In our discussions here we will stay with a physicist's approach and
call the variables for spin. You could replace this with any other
type of binary variables, ranging from a two political parties to blue
and red spheres. In its simplest form we define the energy of the
system as
$$
\begin{equation*}
E=-J\sum_{< kl>}^{N}s_ks_l,
\end{equation*}
$$
with \( s_k=\pm 1 \), \( N \) is the total number of spins,
\( J \) is a coupling constant expressing the strength of the interaction
between neighboring spins.
<p>
The symbol \( < kl> \) indicates that we sum over nearest
neighbors only.
Notice that for \( J>0 \) it is energetically favorable for neighboring spins
to be aligned. This feature leads to, at low enough temperatures,
a cooperative phenomenon called spontaneous magnetization. That is,
through interactions between nearest neighbors, a given magnetic
moment can influence the alignment of spins that are separated
from the given spin by a macroscopic distance. These long range correlations
between spins are associated with a long-range order in which
the lattice has a net magnetization in the absence of a magnetic field.
<p>
We start by considering the one-dimensional Ising model with nearest neighbor interactions. This model does not exhibit any phase transition.
<p>
Consider the 1D Ising model with nearest-neighbor interactions
$$
\begin{equation*}
E[\hat{s}]=-J\sum_{j=1}^{N}s_{j}s_{j+1},
\end{equation*}
$$
<p>
on a chain of length \( N \) with so-called periodic boundary conditions and \( S_j=\pm 1 \) Ising spin variables.
In one dimension, this model has no phase transition at finite temperature.
<p>
In the Python code below we generate, with a coupling coefficient set to \( J=1 \), a large number of spin configurations say \( 10000 \) as shown in the code below.
It means that our data will be a set of \( i=1\ldots n \) points of the form
\( \{(E[\boldsymbol{s}^i],\boldsymbol{s}^i)\} \).
Our task is to find the value of \( J \) from the data set using linear regression.
<p>
Here is the Python code you need to generate the training data, see
also the <a href="https://physics.bu.edu/~pankajm/ML-Notebooks/HTML/NB_CVI-linreg_ising.html" target="_self">notebook of Mehta et
al</a>.
<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: #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">scipy.sparse</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">sp</span>
np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">12</span>)
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">warnings</span>
<span style="color: #408080; font-style: italic">#Comment this to turn on warnings</span>
warnings<span style="color: #666666">.</span>filterwarnings(<span style="color: #BA2121">&#39;ignore&#39;</span>)
<span style="color: #408080; font-style: italic">### define Ising model aprams</span>
<span style="color: #408080; font-style: italic"># system size</span>
L<span style="color: #666666">=40</span>
<span style="color: #408080; font-style: italic"># create 10000 random Ising states</span>
states<span style="color: #666666">=</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>choice([<span style="color: #666666">-1</span>, <span style="color: #666666">1</span>], size<span style="color: #666666">=</span>(<span style="color: #666666">10000</span>,L))
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">ising_energies</span>(states,L):
<span style="color: #BA2121; font-style: italic">&quot;&quot;&quot;</span>
<span style="color: #BA2121; font-style: italic"> This function calculates the energies of the states in the nn Ising Hamiltonian</span>
<span style="color: #BA2121; font-style: italic"> &quot;&quot;&quot;</span>
J<span style="color: #666666">=</span>np<span style="color: #666666">.</span>zeros((L,L),)
<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(L):
J[i,(i<span style="color: #666666">+1</span>)<span style="color: #666666">%</span>L]<span style="color: #666666">-=1.0</span>
<span style="color: #408080; font-style: italic"># compute energies</span>
E <span style="color: #666666">=</span> np<span style="color: #666666">.</span>einsum(<span style="color: #BA2121">&#39;...i,ij,...j-&gt;...&#39;</span>,states,J,states)
<span style="color: #008000; font-weight: bold">return</span> E
<span style="color: #408080; font-style: italic"># calculate Ising energies</span>
energies<span style="color: #666666">=</span>ising_energies(states,L)
</pre></div>
<p>
We can now recast the problem as a linear regression model using our codes from project 1.
The way we are going to build our model mimicks the way we could think of finding say the gravitional constant for the graviational force between two planets.
In the absence of any prior knowledge, one sensible choice is the all-to-all Ising model
$$
E_\mathrm{model}[\boldsymbol{s}^i] = - \sum_{j=1}^N \sum_{k=1}^N J_{j,k}s_{j}^is_{k}^i.
$$
<p>
Here \( i \) represents a particular spin configuration (one of the possible \( n \) configurations we generated with the code above).
<p>
This model is uniquely defined by the non-local coupling strengths \( J_{jk} \) which we want to learn.
The model is linear in \( \mathbf{J} \) which makes it possible to use linear regression.
<p>
To apply linear regression, we recast this model in the form
$$
E_\mathrm{model}^i \equiv \mathbf{X}^i \cdot \mathbf{J},
$$
<p>
where the vectors \( \mathbf{X}^i \) represent all two-body interactions
\( \{s_{j}^is_{k}^i \}_{j,k=1}^N \), and the index \( i \) runs over the
samples in the data set. To make the analogy complete, we can also
represent the dot product by a single index \( p = \{j,k\} \),
i.e. \( \mathbf{X}^i \cdot \mathbf{J}=X^i_pJ_p \). Note that the
regression model does not include the minus sign, so we expect to
learn negative \( J \)'s.
<p>
With these preliminaries, we are now ready to reutilize our codes from project 1.
<h3 id="___sec2" class="anchor">Part b): Estimating the coupling constant of the one-dimensional Ising model using linear regression </h3>
<p>
We start with the one-dimensional Ising model and use the data we have
generated with \( J=1 \). Use linear regression, Lasso and Ridge
regression as described section 6 and in Notebook 4 of <a href="https://physics.bu.edu/~pankajm/ML-Notebooks/HTML/NB_CVI-linreg_ising.html" target="_self">Mehta *et
al.*</a>. Discuss
the methods and how they perform in computing the coupling constant
\( J \). Give a critical analysis and discuss how to evaluate the <em>cost
function</em>. You should feel free to write your own code, see also the
lecture notes of
<a href="https://compphysics.github.io/MachineLearning/doc/web/course.html" target="_self">FYS-STK4155</a>,
in particular te material on least square methods. You can use
scikit-learn to perform these analyses. See below for instruction on
how to install scikit-learn.
generated with \( J=1 \) in the previous point. Use linear regression,
Lasso and Ridge regression as done in project 1. You can compare your
results with those of <a href="https://physics.bu.edu/~pankajm/ML-Notebooks/HTML/NB_CVI-linreg_ising.html" target="_self">Mehta
et al.</a>.
Make sure it is the 1D data which is used.
<p>
Discuss the methods and how they perform in computing the coupling
constant \( J \) and include a bias-variance analysis using either
cross-validation or bootstrap. Discuss also the mean squared error and
the \( R2 \) score as measures to assess your model.
<p>
Give a critical analysis of your results.
<h3 id="___sec3" class="anchor">Part c): Determine the phase of the two-dimensional Ising model </h3>
<p>
We switch now to binary classification methods and use logistic
regression to define the phases of the Ising model. Use described
section 7 and in Notebook 6 of <a href="https://physics.bu.edu/~pankajm/ML-Notebooks/HTML/NB_CVII-logreg_ising.html" target="_self">Mehta *et
al.*</a>. Discuss
the methods and how they perform. Give a critical analysis and discuss
how to evaluate the <em>cost function</em>. You should feel free to write
your own code.
<h3 id="___sec4" class="anchor">Part d): Classifying the Ising model phase using neural networks </h3>
regression to define the phases of the Ising model. This means that we switch to the two-dimensional Ising model
and use the data sets generated by <a href="https://physics.bu.edu/~pankajm/ML-Review-Datasets/isingMC/" target="_self">Mehta et al</a>
These energies and their corresponding spin orientation configurations
represent then your data. We will use a fixed lattice of \( L\times L =
40 \times 40 \) spins in two dimensions. The link above contains data for several temperatures.
The theoretical critical temperature for a phase transition is \( T_C\approx 2.269 \) in units of energy.
However, for a finite lattice the results representing the critical temperature are slightly higher (\( T_C \approx 2.3 \)).
<p>
We end the classification problem of the phases of the Ising model by
employing the algorithm for so-called feed-forward deep neural
networks (see section 9 of Mehta <em>et al.</em>). The method is described in
<a href="https://physics.bu.edu/~pankajm/ML-Notebooks/HTML/NB_CIX-DNN_ising_TFlow.html" target="_self">notebook
12</a>.
Our goal here, using logistic regression, is to train our model to
predict the phase of a sample given the spin configuration, whether it
represents a state above the critical temperature or below. The
configurations representing states below the critical temperature are
called ordered states (the spins tend to point in one direction,
resulting in a net magnetic moment) while those above the critical
temperature are called disordered. Since a finite lattice like this
does not exhibit a clear sign of a phase transition we will mainly
stay with either orderer or disoredered phases. You could include the
critical phase if you want.
<p>
You can use tensorflow to perform these analyses. See below for instruction on how to install tensorflow.
Your aim here is thus to read in these data (use the examples from
<a href="https://physics.bu.edu/~pankajm/ML-Notebooks/HTML/NB_CVII-logreg_ising.html" target="_self">Mehta et
al</a>)
and write your own code for doing logistic regression, see the lecture
notes on <a href="https://compphysics.github.io/MachineLearning/doc/pub/LogReg/html/LogReg-bs.html" target="_self">logistic
regression</a>.
<h2 id="___sec5" class="anchor">Background literature </h2>
<p>
You should include either bootstrap or cross-validation in setting up
your model (as done in project 1) and compute the \( R2 \) score and the
mean-square error.
We will use both ordered and disordered states to train
the logistic regressor and, once the supervised training procedure is
complete, you should evaluate the performance of your classification model on
unseen ordered, disordered and possibly critical states.
<p>
In order to find the optimal parameters of your logistic regressor you should
include a gradient descent solver, as discussed in the <a href="https://compphysics.github.io/MachineLearning/doc/pub/Splines/html/Splines-bs.html" target="_self">gradient
descent
lectures</a>.
Since we don't have so many data points, you may just code the
standard gradient descent with a given learning rate, or even attempt
to use the Newton-Raphson method. Alternatively, it may be useful for
the next part on neural networks to implement a stochastic gradient
descent. For all gradient methods, you can use <b>scikit-learn</b>'s toolbox for
optimization methods instead of writing your own code.
<p>
The notebook of <a href="https://physics.bu.edu/~pankajm/ML-Notebooks/HTML/NB_CVII-logreg_ising.html" target="_self">Mehta et al</a> is highly recommended in order to benchmark your code and results.
<h3 id="___sec4" class="anchor">Part d): Regression analysis of the one-dimensional Ising model using neural networks </h3>
<p>
Your aim now, and this is the central part of this project, is to
write to your own multilayer perceptron model implementing the back
propagation algorithm discussed in the <a href="https://compphysics.github.io/MachineLearning/doc/pub/NeuralNet/html/NeuralNet-bs.html" target="_self">lecture
slides</a>. We
start with the regression case discussed in parts a) and b) but train
now the network to find the optimal weights and biases. You are free
to use the codes in the above lecture slides as starting points.
<p>
Train your network and compare the results with those from your linear regression code.
You can test your results against a similar code using _scikit_learn_ (see the examples in teh above lecture notes) or <b>tensorflow/keras</b>.
<p>
You should have the same elements as in the regression examples, including the \( R2 \) score, the MSE, and bootstrap or cross-validation.
<p>
A useful reference on the back progagation algorithm is <a href="http://neuralnetworksanddeeplearning.com/" target="_self">Nielsen's book</a>. It is an excellent read.
<h3 id="___sec5" class="anchor">Part e): Classifying the Ising model phase using neural networks </h3>
<p>
Finally, change now your cost function to the \( log \) cross-entropy classification cost function for the case discussed in part c). Train your network again and
compare the results with those from your logistic regression code i c).
Here again you can compare your results with those of <a href="https://physics.bu.edu/~pankajm/ML-Notebooks/HTML/NB_CIX-DNN_ising_TFlow.html" target="_self">Mehta et al</a>. There they used <b>tensorflow</b> to classify the phases.
<h3 id="___sec6" class="anchor">Part f) Critical evaluation of the various algorithms </h3>
<p>
After all these glorious calculations, you should now summarize the various algorithms and come with a critical evaluation of their pros and cons. Which algorithm works best for the regression case and which is best for the classification case. These codes will also be part of your final project 3, but now applied to other data sets.
<h2 id="___sec7" class="anchor">Background literature </h2>
<ol>
<li> The text of Michael Nielsen is highly recommended, see <a href="http://neuralnetworksanddeeplearning.com/" target="_self">Nielsen's book</a>. It is an excellent read.</li>
<li> The textbook of <a href="https://www.springer.com/gp/book/9780387848570" target="_self">Trevor Hastie, Robert Tibshirani, Jerome H. Friedman, The Elements of Statistical Learning, Springer</a>, chapters 3 and 7 are the most relevant ones for the analysis here.</li>
<li> <a href="https://arxiv.org/abs/1803.08823" target="_self">Mehta et al, arXiv 1803.08823</a>, <em>A high-bias, low-variance introduction to Machine Learning for physicists</em>, ArXiv:1803.08823.</li>
</ol>
@@ -283,7 +483,7 @@ If you wish to read more about the Ising model and statistical physics here are
<li> <a href="https://global.oup.com/academic/product/monte-carlo-methods-in-statistical-physics-9780198517979?cc=no&lang=en&" target="_self">M. E. J. Newman and T. Barkema</a>, <em>Monte Carlo Methods in Statistical Physics</em>, Oxford, see chapters 3 and 4.</li>
</ol>
<h2 id="___sec6" class="anchor">Introduction to numerical projects </h2>
<h2 id="___sec8" class="anchor">Introduction to numerical projects </h2>
<p>
Here follows a brief recipe and recommendation on how to write a report for each
@@ -301,7 +501,7 @@ project.
<li> Try to establish a practice where you log your work at the computerlab. You may find such a logbook very handy at later stages in your work, especially when you don't properly remember what a previous test version of your program did. Here you could also record the time spent on solving the exercise, various algorithms you may have tested or other topics which you feel worthy of mentioning.</li>
</ul>
<h2 id="___sec7" class="anchor">Format for electronic delivery of report and programs </h2>
<h2 id="___sec9" class="anchor">Format for electronic delivery of report and programs </h2>
<p>
The preferred format for the report is a PDF file. You can also use DOC or postscript formats or as an ipython notebook file. As programming language we prefer that you choose between C/C++, Fortran2008 or Python. The following prescription should be followed when preparing the report:
@@ -318,7 +518,7 @@ Finally,
we encourage you to collaborate. Optimal working groups consist of
2-3 students. You can then hand in a common report.
<h2 id="___sec8" class="anchor">Software and needed installations </h2>
<h2 id="___sec10" class="anchor">Software and needed installations </h2>
<p>
If you have Python installed (we recommend Python3) and you feel pretty familiar with installing different packages,
doc/Projects/2018/Project2/html/Project2.html
+275 -75
View File
@@ -44,9 +44,13 @@ div { text-align: justify; text-justify: inter-word; }
2,
None,
'___sec0'),
('Part a): Producing the data', 3, None, '___sec1'),
('Part a): Producing the data for the one-dimensional Ising '
'model',
3,
None,
'___sec1'),
('Part b): Estimating the coupling constant of the '
'one-dimensional Ising model',
'one-dimensional Ising model using linear regression',
3,
None,
'___sec2'),
@@ -55,18 +59,27 @@ div { text-align: justify; text-justify: inter-word; }
3,
None,
'___sec3'),
('Part d): Classifying the Ising model phase using neural '
'networks',
('Part d): Regression analysis of the one-dimensional Ising '
'model using neural networks',
3,
None,
'___sec4'),
('Background literature', 2, None, '___sec5'),
('Introduction to numerical projects', 2, None, '___sec6'),
('Part e): Classifying the Ising model phase using neural '
'networks',
3,
None,
'___sec5'),
('Part f) Critical evaluation of the various algorithms',
3,
None,
'___sec6'),
('Background literature', 2, None, '___sec7'),
('Introduction to numerical projects', 2, None, '___sec8'),
('Format for electronic delivery of report and programs',
2,
None,
'___sec7'),
('Software and needed installations', 2, None, '___sec8')]}
'___sec9'),
('Software and needed installations', 2, None, '___sec10')]}
end of tocinfo -->
<body>
@@ -107,20 +120,14 @@ MathJax.Hub.Config({
<center><b>Department of Physics, University of Oslo, Norway</b></center>
<br>
<p>
<center><h4>Oct 8, 2018</h4></center> <!-- date -->
<center><h4>Oct 9, 2018</h4></center> <!-- date -->
<br>
<ol>
<li> add about ising model</li>
<li> link to where we have the data</li>
<li> explain how to fit the model</li>
<li> link to mehta's article</li>
</ol>
<h2 id="___sec0">Classification and Regression, from linear and logistic regression to neural networks </h2>
<p>
The main aim of this project is to study both classification and
regression problems, starting with the regression algortihms studied
regression problems, starting with the regression algorithms studied
in project 1. We will include logistic regresion for classification
problems and write our own multilayer perceptron code for studying
both regression and classification problems. The codes developed in
@@ -130,19 +137,19 @@ also be utilized (and included in logistic regression and the neural
network codes) in the present analysis.
<p>
We will use the Ising model to generate our training data and will
focus mainly on supervised training. We will follow closely the recent
We will use the so-called Ising model for our training data and will
focus on supervised training. We will follow closely the recent
article of <a href="https://arxiv.org/abs/1803.08823" target="_blank">Mehta et al, arXiv
1803.08823</a>. This article stands
out as an excellent review on machine learning (ML) algorithms applied
to typical physics problems. The added benefit is that each figure and
out as an excellent review on machine learning (ML) algorithms.
The added benefit is that each figure and
model presented in <a href="https://physics.bu.edu/~pankajm/MLnotebooks.html" target="_blank">this article is accompanied by its jupyter
notebook</a>. This
means that we can start using these and compare with our own
results. In case you wish to use their data for the Ising model, their
data can be downloaded from the same link which lists to the jupyter
notebooks. See also at the end of the project description for more
information on how to install various Python packages.
means that we can start using these and compare with our own results.
They provide also the data set for the regression and classification
analysis that we will explore. In this sense, with their available
notebooks, it makes life easier since we can compare our own codes
with their codes.
<p>
With the abovementioned configurations we will determine, using first
@@ -158,75 +165,268 @@ that is the two-dimensional Ising model, will be studied using
logistic regression and deep neural networks. The aim is to develop
your own logistic regression code for the classification of the phases
(this is a binary model) and your multilayer perceptron code for the
classification and regression case.
classification and regression case. You can compare your own results with those obtained
using <b>scikit-learn</b> or <b>tensorflow</b> or other Python packages such as <b>keras</b> or other.
<p>
Feel free to use the notebooks to benchmark your code. If you wish to
write your own C++ or Fortran program for say a simple neural network
model and a logistic regression model, please feel free to do so. You can then benchmark your results
against the above jupyter notebooks.
write your own C++ or Fortran program for say a multilayer neural network
model and a logistic regression model, please feel free to do so.
<h3 id="___sec1">Part a): Producing the data </h3>
<h3 id="___sec1">Part a): Producing the data for the one-dimensional Ising model </h3>
<p>
You can use the Ising model data from the article of Mehta <em>et al.</em>,
or generate your own data. If you opt for using your own Ising model
code, you need to generate \( 10000 \) energy configurations with their
spin orientations after the system has reached its most likely
state. These energies and their corresponding spin orientations
represent then your data. We will use a fixed lattice of \( L\times L =
40 \times 40 \) spins in two dimensions and \( L=40 \) spins in one
dimension. Make sure the calculations have been equilibrated. For the
two-dimensional system, compute the configurations for three values of
the temperature, namely \( T=0.75 \) (ordered phase), \( T=2.3 \) (near the
critical point) and \( T=4.0 \) (disordered phase). For the
one-dimensional system it suffices to compute the various
configurations for one temperature only, say \( T=2.0 \). These are the
data you will use to study different ML algorithms. We generate our
data with \( J=1 \).
The model we will employ in our studies is the so-called <a href="https://en.wikipedia.org/wiki/Ising_model" target="_blank">Ising
model</a>. Together with
models like the <a href="https://en.wikipedia.org/wiki/Potts_model" target="_blank">Potts
model</a> and similar
so-called lattice models, the Ising model has been widely studied in
mathematics (in statistics in particular), physics, <a href="https://journals.aps.org/pre/abstract/10.1103/PhysRevE.93.062402" target="_blank">life
science</a>,
chemistry and even in the <a href="https://www.springer.com/gp/book/9781461420316" target="_blank">social sciences in order to model social
behavior</a>. It is a
simple binary value system where the variables of the model (spins often in
physics) can take two values only, for example \( \pm 1 \) or \( 0 \) and \( 1 \).
The system exhibits a phase transition in two or higher dimensions and
the first person to find the analytical expressions for various
expectation values was the Norwegian chemist <a href="https://en.wikipedia.org/wiki/Lars_Onsager" target="_blank">Lars
Onsager</a> (Nobel prize in
chemistry) after a tour de force mathematics exercise.
<h3 id="___sec2">Part b): Estimating the coupling constant of the one-dimensional Ising model </h3>
<p>
In our discussions here we will stay with a physicist's approach and
call the variables for spin. You could replace this with any other
type of binary variables, ranging from a two political parties to blue
and red spheres. In its simplest form we define the energy of the
system as
$$
\begin{equation*}
E=-J\sum_{< kl>}^{N}s_ks_l,
\end{equation*}
$$
with \( s_k=\pm 1 \), \( N \) is the total number of spins,
\( J \) is a coupling constant expressing the strength of the interaction
between neighboring spins.
<p>
The symbol \( < kl> \) indicates that we sum over nearest
neighbors only.
Notice that for \( J>0 \) it is energetically favorable for neighboring spins
to be aligned. This feature leads to, at low enough temperatures,
a cooperative phenomenon called spontaneous magnetization. That is,
through interactions between nearest neighbors, a given magnetic
moment can influence the alignment of spins that are separated
from the given spin by a macroscopic distance. These long range correlations
between spins are associated with a long-range order in which
the lattice has a net magnetization in the absence of a magnetic field.
<p>
We start by considering the one-dimensional Ising model with nearest neighbor interactions. This model does not exhibit any phase transition.
<p>
Consider the 1D Ising model with nearest-neighbor interactions
$$
\begin{equation*}
E[\hat{s}]=-J\sum_{j=1}^{N}s_{j}s_{j+1},
\end{equation*}
$$
<p>
on a chain of length \( N \) with so-called periodic boundary conditions and \( S_j=\pm 1 \) Ising spin variables.
In one dimension, this model has no phase transition at finite temperature.
<p>
In the Python code below we generate, with a coupling coefficient set to \( J=1 \), a large number of spin configurations say \( 10000 \) as shown in the code below.
It means that our data will be a set of \( i=1\ldots n \) points of the form
\( \{(E[\boldsymbol{s}^i],\boldsymbol{s}^i)\} \).
Our task is to find the value of \( J \) from the data set using linear regression.
<p>
Here is the Python code you need to generate the training data, see
also the <a href="https://physics.bu.edu/~pankajm/ML-Notebooks/HTML/NB_CVI-linreg_ising.html" target="_blank">notebook of Mehta et
al</a>.
<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: #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">scipy.sparse</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">sp</span>
np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">12</span>)
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">warnings</span>
<span style="color: #408080; font-style: italic">#Comment this to turn on warnings</span>
warnings<span style="color: #666666">.</span>filterwarnings(<span style="color: #BA2121">&#39;ignore&#39;</span>)
<span style="color: #408080; font-style: italic">### define Ising model aprams</span>
<span style="color: #408080; font-style: italic"># system size</span>
L<span style="color: #666666">=40</span>
<span style="color: #408080; font-style: italic"># create 10000 random Ising states</span>
states<span style="color: #666666">=</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>choice([<span style="color: #666666">-1</span>, <span style="color: #666666">1</span>], size<span style="color: #666666">=</span>(<span style="color: #666666">10000</span>,L))
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">ising_energies</span>(states,L):
<span style="color: #BA2121; font-style: italic">&quot;&quot;&quot;</span>
<span style="color: #BA2121; font-style: italic"> This function calculates the energies of the states in the nn Ising Hamiltonian</span>
<span style="color: #BA2121; font-style: italic"> &quot;&quot;&quot;</span>
J<span style="color: #666666">=</span>np<span style="color: #666666">.</span>zeros((L,L),)
<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(L):
J[i,(i<span style="color: #666666">+1</span>)<span style="color: #666666">%</span>L]<span style="color: #666666">-=1.0</span>
<span style="color: #408080; font-style: italic"># compute energies</span>
E <span style="color: #666666">=</span> np<span style="color: #666666">.</span>einsum(<span style="color: #BA2121">&#39;...i,ij,...j-&gt;...&#39;</span>,states,J,states)
<span style="color: #008000; font-weight: bold">return</span> E
<span style="color: #408080; font-style: italic"># calculate Ising energies</span>
energies<span style="color: #666666">=</span>ising_energies(states,L)
</pre></div>
<p>
We can now recast the problem as a linear regression model using our codes from project 1.
The way we are going to build our model mimicks the way we could think of finding say the gravitional constant for the graviational force between two planets.
In the absence of any prior knowledge, one sensible choice is the all-to-all Ising model
$$
E_\mathrm{model}[\boldsymbol{s}^i] = - \sum_{j=1}^N \sum_{k=1}^N J_{j,k}s_{j}^is_{k}^i.
$$
<p>
Here \( i \) represents a particular spin configuration (one of the possible \( n \) configurations we generated with the code above).
<p>
This model is uniquely defined by the non-local coupling strengths \( J_{jk} \) which we want to learn.
The model is linear in \( \mathbf{J} \) which makes it possible to use linear regression.
<p>
To apply linear regression, we recast this model in the form
$$
E_\mathrm{model}^i \equiv \mathbf{X}^i \cdot \mathbf{J},
$$
<p>
where the vectors \( \mathbf{X}^i \) represent all two-body interactions
\( \{s_{j}^is_{k}^i \}_{j,k=1}^N \), and the index \( i \) runs over the
samples in the data set. To make the analogy complete, we can also
represent the dot product by a single index \( p = \{j,k\} \),
i.e. \( \mathbf{X}^i \cdot \mathbf{J}=X^i_pJ_p \). Note that the
regression model does not include the minus sign, so we expect to
learn negative \( J \)'s.
<p>
With these preliminaries, we are now ready to reutilize our codes from project 1.
<h3 id="___sec2">Part b): Estimating the coupling constant of the one-dimensional Ising model using linear regression </h3>
<p>
We start with the one-dimensional Ising model and use the data we have
generated with \( J=1 \). Use linear regression, Lasso and Ridge
regression as described section 6 and in Notebook 4 of <a href="https://physics.bu.edu/~pankajm/ML-Notebooks/HTML/NB_CVI-linreg_ising.html" target="_blank">Mehta *et
al.*</a>. Discuss
the methods and how they perform in computing the coupling constant
\( J \). Give a critical analysis and discuss how to evaluate the <em>cost
function</em>. You should feel free to write your own code, see also the
lecture notes of
<a href="https://compphysics.github.io/MachineLearning/doc/web/course.html" target="_blank">FYS-STK4155</a>,
in particular te material on least square methods. You can use
scikit-learn to perform these analyses. See below for instruction on
how to install scikit-learn.
generated with \( J=1 \) in the previous point. Use linear regression,
Lasso and Ridge regression as done in project 1. You can compare your
results with those of <a href="https://physics.bu.edu/~pankajm/ML-Notebooks/HTML/NB_CVI-linreg_ising.html" target="_blank">Mehta
et al.</a>.
Make sure it is the 1D data which is used.
<p>
Discuss the methods and how they perform in computing the coupling
constant \( J \) and include a bias-variance analysis using either
cross-validation or bootstrap. Discuss also the mean squared error and
the \( R2 \) score as measures to assess your model.
<p>
Give a critical analysis of your results.
<h3 id="___sec3">Part c): Determine the phase of the two-dimensional Ising model </h3>
<p>
We switch now to binary classification methods and use logistic
regression to define the phases of the Ising model. Use described
section 7 and in Notebook 6 of <a href="https://physics.bu.edu/~pankajm/ML-Notebooks/HTML/NB_CVII-logreg_ising.html" target="_blank">Mehta *et
al.*</a>. Discuss
the methods and how they perform. Give a critical analysis and discuss
how to evaluate the <em>cost function</em>. You should feel free to write
your own code.
<h3 id="___sec4">Part d): Classifying the Ising model phase using neural networks </h3>
regression to define the phases of the Ising model. This means that we switch to the two-dimensional Ising model
and use the data sets generated by <a href="https://physics.bu.edu/~pankajm/ML-Review-Datasets/isingMC/" target="_blank">Mehta et al</a>
These energies and their corresponding spin orientation configurations
represent then your data. We will use a fixed lattice of \( L\times L =
40 \times 40 \) spins in two dimensions. The link above contains data for several temperatures.
The theoretical critical temperature for a phase transition is \( T_C\approx 2.269 \) in units of energy.
However, for a finite lattice the results representing the critical temperature are slightly higher (\( T_C \approx 2.3 \)).
<p>
We end the classification problem of the phases of the Ising model by
employing the algorithm for so-called feed-forward deep neural
networks (see section 9 of Mehta <em>et al.</em>). The method is described in
<a href="https://physics.bu.edu/~pankajm/ML-Notebooks/HTML/NB_CIX-DNN_ising_TFlow.html" target="_blank">notebook
12</a>.
Our goal here, using logistic regression, is to train our model to
predict the phase of a sample given the spin configuration, whether it
represents a state above the critical temperature or below. The
configurations representing states below the critical temperature are
called ordered states (the spins tend to point in one direction,
resulting in a net magnetic moment) while those above the critical
temperature are called disordered. Since a finite lattice like this
does not exhibit a clear sign of a phase transition we will mainly
stay with either orderer or disoredered phases. You could include the
critical phase if you want.
<p>
You can use tensorflow to perform these analyses. See below for instruction on how to install tensorflow.
Your aim here is thus to read in these data (use the examples from
<a href="https://physics.bu.edu/~pankajm/ML-Notebooks/HTML/NB_CVII-logreg_ising.html" target="_blank">Mehta et
al</a>)
and write your own code for doing logistic regression, see the lecture
notes on <a href="https://compphysics.github.io/MachineLearning/doc/pub/LogReg/html/LogReg-bs.html" target="_blank">logistic
regression</a>.
<h2 id="___sec5">Background literature </h2>
<p>
You should include either bootstrap or cross-validation in setting up
your model (as done in project 1) and compute the \( R2 \) score and the
mean-square error.
We will use both ordered and disordered states to train
the logistic regressor and, once the supervised training procedure is
complete, you should evaluate the performance of your classification model on
unseen ordered, disordered and possibly critical states.
<p>
In order to find the optimal parameters of your logistic regressor you should
include a gradient descent solver, as discussed in the <a href="https://compphysics.github.io/MachineLearning/doc/pub/Splines/html/Splines-bs.html" target="_blank">gradient
descent
lectures</a>.
Since we don't have so many data points, you may just code the
standard gradient descent with a given learning rate, or even attempt
to use the Newton-Raphson method. Alternatively, it may be useful for
the next part on neural networks to implement a stochastic gradient
descent. For all gradient methods, you can use <b>scikit-learn</b>'s toolbox for
optimization methods instead of writing your own code.
<p>
The notebook of <a href="https://physics.bu.edu/~pankajm/ML-Notebooks/HTML/NB_CVII-logreg_ising.html" target="_blank">Mehta et al</a> is highly recommended in order to benchmark your code and results.
<h3 id="___sec4">Part d): Regression analysis of the one-dimensional Ising model using neural networks </h3>
<p>
Your aim now, and this is the central part of this project, is to
write to your own multilayer perceptron model implementing the back
propagation algorithm discussed in the <a href="https://compphysics.github.io/MachineLearning/doc/pub/NeuralNet/html/NeuralNet-bs.html" target="_blank">lecture
slides</a>. We
start with the regression case discussed in parts a) and b) but train
now the network to find the optimal weights and biases. You are free
to use the codes in the above lecture slides as starting points.
<p>
Train your network and compare the results with those from your linear regression code.
You can test your results against a similar code using _scikit_learn_ (see the examples in teh above lecture notes) or <b>tensorflow/keras</b>.
<p>
You should have the same elements as in the regression examples, including the \( R2 \) score, the MSE, and bootstrap or cross-validation.
<p>
A useful reference on the back progagation algorithm is <a href="http://neuralnetworksanddeeplearning.com/" target="_blank">Nielsen's book</a>. It is an excellent read.
<h3 id="___sec5">Part e): Classifying the Ising model phase using neural networks </h3>
<p>
Finally, change now your cost function to the \( log \) cross-entropy classification cost function for the case discussed in part c). Train your network again and
compare the results with those from your logistic regression code i c).
Here again you can compare your results with those of <a href="https://physics.bu.edu/~pankajm/ML-Notebooks/HTML/NB_CIX-DNN_ising_TFlow.html" target="_blank">Mehta et al</a>. There they used <b>tensorflow</b> to classify the phases.
<h3 id="___sec6">Part f) Critical evaluation of the various algorithms </h3>
<p>
After all these glorious calculations, you should now summarize the various algorithms and come with a critical evaluation of their pros and cons. Which algorithm works best for the regression case and which is best for the classification case. These codes will also be part of your final project 3, but now applied to other data sets.
<h2 id="___sec7">Background literature </h2>
<ol>
<li> The text of Michael Nielsen is highly recommended, see <a href="http://neuralnetworksanddeeplearning.com/" target="_blank">Nielsen's book</a>. It is an excellent read.</li>
<li> The textbook of <a href="https://www.springer.com/gp/book/9780387848570" target="_blank">Trevor Hastie, Robert Tibshirani, Jerome H. Friedman, The Elements of Statistical Learning, Springer</a>, chapters 3 and 7 are the most relevant ones for the analysis here.</li>
<li> <a href="https://arxiv.org/abs/1803.08823" target="_blank">Mehta et al, arXiv 1803.08823</a>, <em>A high-bias, low-variance introduction to Machine Learning for physicists</em>, ArXiv:1803.08823.</li>
</ol>
@@ -239,7 +439,7 @@ If you wish to read more about the Ising model and statistical physics here are
<li> <a href="https://global.oup.com/academic/product/monte-carlo-methods-in-statistical-physics-9780198517979?cc=no&lang=en&" target="_blank">M. E. J. Newman and T. Barkema</a>, <em>Monte Carlo Methods in Statistical Physics</em>, Oxford, see chapters 3 and 4.</li>
</ol>
<h2 id="___sec6">Introduction to numerical projects </h2>
<h2 id="___sec8">Introduction to numerical projects </h2>
<p>
Here follows a brief recipe and recommendation on how to write a report for each
@@ -257,7 +457,7 @@ project.
<li> Try to establish a practice where you log your work at the computerlab. You may find such a logbook very handy at later stages in your work, especially when you don't properly remember what a previous test version of your program did. Here you could also record the time spent on solving the exercise, various algorithms you may have tested or other topics which you feel worthy of mentioning.</li>
</ul>
<h2 id="___sec7">Format for electronic delivery of report and programs </h2>
<h2 id="___sec9">Format for electronic delivery of report and programs </h2>
<p>
The preferred format for the report is a PDF file. You can also use DOC or postscript formats or as an ipython notebook file. As programming language we prefer that you choose between C/C++, Fortran2008 or Python. The following prescription should be followed when preparing the report:
@@ -274,7 +474,7 @@ Finally,
we encourage you to collaborate. Optimal working groups consist of
2-3 students. You can then hand in a common report.
<h2 id="___sec8">Software and needed installations </h2>
<h2 id="___sec10">Software and needed installations </h2>
<p>
If you have Python installed (we recommend Python3) and you feel pretty familiar with installing different packages,
doc/Projects/2018/Project2/ipynb/ipynb-Project2-src.tar.gz
BIN
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Binary file not shown.
doc/Projects/2018/Project2/pdf/Project2.p.tex
+238 -68
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@@ -45,6 +45,12 @@ final, % draft: marks overfull hboxes, figures with paths
\usepackage[pdftex]{graphicx}
\usepackage{ptex2tex}
% #ifdef MINTED
\usepackage{minted}
\usemintedstyle{default}
% #endif
\usepackage[T1]{fontenc}
%\usepackage[latin1]{inputenc}
\usepackage{ucs}
@@ -149,28 +155,17 @@ Project 2 on Machine Learning, deadline November 5
% --- begin date ---
\begin{center}
Oct 8, 2018
Oct 9, 2018
\end{center}
% --- end date ---
\vspace{1cm}
\begin{enumerate}
\item add about ising model
\item link to where we have the data
\item explain how to fit the model
\item link to mehta's article
\end{enumerate}
\noindent
\subsection{Classification and Regression, from linear and logistic regression to neural networks}
The main aim of this project is to study both classification and
regression problems, starting with the regression algortihms studied
regression problems, starting with the regression algorithms studied
in project 1. We will include logistic regresion for classification
problems and write our own multilayer perceptron code for studying
both regression and classification problems. The codes developed in
@@ -179,19 +174,20 @@ computation of the mean-squared error and the R2 score function can
also be utilized (and included in logistic regression and the neural
network codes) in the present analysis.
We will use the Ising model to generate our training data and will
focus mainly on supervised training. We will follow closely the recent
We will use the so-called Ising model for our training data and will
focus on supervised training. We will follow closely the recent
article of \href{{https://arxiv.org/abs/1803.08823}}{Mehta et al, arXiv
1803.08823}. This article stands
out as an excellent review on machine learning (ML) algorithms applied
to typical physics problems. The added benefit is that each figure and
out as an excellent review on machine learning (ML) algorithms.
The added benefit is that each figure and
model presented in \href{{https://physics.bu.edu/~pankajm/MLnotebooks.html}}{this article is accompanied by its jupyter
notebook}. This
means that we can start using these and compare with our own
results. In case you wish to use their data for the Ising model, their
data can be downloaded from the same link which lists to the jupyter
notebooks. See also at the end of the project description for more
information on how to install various Python packages.
means that we can start using these and compare with our own results.
They provide also the data set for the regression and classification
analysis that we will explore. In this sense, with their available
notebooks, it makes life easier since we can compare our own codes
with their codes.
@@ -208,69 +204,243 @@ that is the two-dimensional Ising model, will be studied using
logistic regression and deep neural networks. The aim is to develop
your own logistic regression code for the classification of the phases
(this is a binary model) and your multilayer perceptron code for the
classification and regression case.
classification and regression case. You can compare your own results with those obtained
using \textbf{scikit-learn} or \textbf{tensorflow} or other Python packages such as \textbf{keras} or other.
Feel free to use the notebooks to benchmark your code. If you wish to
write your own C++ or Fortran program for say a simple neural network
model and a logistic regression model, please feel free to do so. You can then benchmark your results
against the above jupyter notebooks.
write your own C++ or Fortran program for say a multilayer neural network
model and a logistic regression model, please feel free to do so.
\paragraph{Part a): Producing the data.}
You can use the Ising model data from the article of Mehta \emph{et al.},
or generate your own data. If you opt for using your own Ising model
code, you need to generate $10000$ energy configurations with their
spin orientations after the system has reached its most likely
state. These energies and their corresponding spin orientations
represent then your data. We will use a fixed lattice of $L\times L =
40 \times 40$ spins in two dimensions and $L=40$ spins in one
dimension. Make sure the calculations have been equilibrated. For the
two-dimensional system, compute the configurations for three values of
the temperature, namely $T=0.75$ (ordered phase), $T=2.3$ (near the
critical point) and $T=4.0$ (disordered phase). For the
one-dimensional system it suffices to compute the various
configurations for one temperature only, say $T=2.0$. These are the
data you will use to study different ML algorithms. We generate our
data with $J=1$.
\paragraph{Part a): Producing the data for the one-dimensional Ising model.}
The model we will employ in our studies is the so-called \href{{https://en.wikipedia.org/wiki/Ising_model}}{Ising
model}. Together with
models like the \href{{https://en.wikipedia.org/wiki/Potts_model}}{Potts
model} and similar
so-called lattice models, the Ising model has been widely studied in
mathematics (in statistics in particular), physics, \href{{https://journals.aps.org/pre/abstract/10.1103/PhysRevE.93.062402}}{life
science},
chemistry and even in the \href{{https://www.springer.com/gp/book/9781461420316}}{social sciences in order to model social
behavior}. It is a
simple binary value system where the variables of the model (spins often in
physics) can take two values only, for example $\pm 1$ or $0$ and $1$.
The system exhibits a phase transition in two or higher dimensions and
the first person to find the analytical expressions for various
expectation values was the Norwegian chemist \href{{https://en.wikipedia.org/wiki/Lars_Onsager}}{Lars
Onsager} (Nobel prize in
chemistry) after a tour de force mathematics exercise.
\paragraph{Part b): Estimating the coupling constant of the one-dimensional Ising model.}
In our discussions here we will stay with a physicist's approach and
call the variables for spin. You could replace this with any other
type of binary variables, ranging from a two political parties to blue
and red spheres. In its simplest form we define the energy of the
system as
\begin{equation*}
E=-J\sum_{<kl>}^{N}s_ks_l,
\end{equation*}
with $s_k=\pm 1$, $N$ is the total number of spins,
$J$ is a coupling constant expressing the strength of the interaction
between neighboring spins.
The symbol $<kl>$ indicates that we sum over nearest
neighbors only.
Notice that for $J>0$ it is energetically favorable for neighboring spins
to be aligned. This feature leads to, at low enough temperatures,
a cooperative phenomenon called spontaneous magnetization. That is,
through interactions between nearest neighbors, a given magnetic
moment can influence the alignment of spins that are separated
from the given spin by a macroscopic distance. These long range correlations
between spins are associated with a long-range order in which
the lattice has a net magnetization in the absence of a magnetic field.
We start by considering the one-dimensional Ising model with nearest neighbor interactions. This model does not exhibit any phase transition.
Consider the 1D Ising model with nearest-neighbor interactions
\begin{equation*}
E[\hat{s}]=-J\sum_{j=1}^{N}s_{j}s_{j+1},
\end{equation*}
on a chain of length $N$ with so-called periodic boundary conditions and $S_j=\pm 1$ Ising spin variables.
In one dimension, this model has no phase transition at finite temperature.
In the Python code below we generate, with a coupling coefficient set to $J=1$, a large number of spin configurations say $10000$ as shown in the code below.
It means that our data will be a set of $i=1\ldots n$ points of the form
$\{(E[\boldsymbol{s}^i],\boldsymbol{s}^i)\}$.
Our task is to find the value of $J$ from the data set using linear regression.
Here is the Python code you need to generate the training data, see
also the \href{{https://physics.bu.edu/~pankajm/ML-Notebooks/HTML/NB_CVI-linreg_ising.html}}{notebook of Mehta et
al}.
\bpycod
import numpy as np
import scipy.sparse as sp
np.random.seed(12)
import warnings
#Comment this to turn on warnings
warnings.filterwarnings('ignore')
### define Ising model aprams
# system size
L=40
# create 10000 random Ising states
states=np.random.choice([-1, 1], size=(10000,L))
def ising_energies(states,L):
"""
This function calculates the energies of the states in the nn Ising Hamiltonian
"""
J=np.zeros((L,L),)
for i in range(L):
J[i,(i+1)%L]-=1.0
# compute energies
E = np.einsum('...i,ij,...j->...',states,J,states)
return E
# calculate Ising energies
energies=ising_energies(states,L)
\epycod
We can now recast the problem as a linear regression model using our codes from project 1.
The way we are going to build our model mimicks the way we could think of finding say the gravitional constant for the graviational force between two planets.
In the absence of any prior knowledge, one sensible choice is the all-to-all Ising model
\[
E_\mathrm{model}[\boldsymbol{s}^i] = - \sum_{j=1}^N \sum_{k=1}^N J_{j,k}s_{j}^is_{k}^i.
\]
Here $i$ represents a particular spin configuration (one of the possible $n$ configurations we generated with the code above).
This model is uniquely defined by the non-local coupling strengths $J_{jk}$ which we want to learn.
The model is linear in $\mathbf{J}$ which makes it possible to use linear regression.
To apply linear regression, we recast this model in the form
\[
E_\mathrm{model}^i \equiv \mathbf{X}^i \cdot \mathbf{J},
\]
where the vectors $\mathbf{X}^i$ represent all two-body interactions
$\{s_{j}^is_{k}^i \}_{j,k=1}^N$, and the index $i$ runs over the
samples in the data set. To make the analogy complete, we can also
represent the dot product by a single index $p = \{j,k\}$,
i.e.~$\mathbf{X}^i \cdot \mathbf{J}=X^i_pJ_p$. Note that the
regression model does not include the minus sign, so we expect to
learn negative $J$'s.
With these preliminaries, we are now ready to reutilize our codes from project 1.
\paragraph{Part b): Estimating the coupling constant of the one-dimensional Ising model using linear regression.}
We start with the one-dimensional Ising model and use the data we have
generated with $J=1$. Use linear regression, Lasso and Ridge
regression as described section 6 and in Notebook 4 of \href{{https://physics.bu.edu/~pankajm/ML-Notebooks/HTML/NB_CVI-linreg_ising.html}}{Mehta *et
al.*}. Discuss
the methods and how they perform in computing the coupling constant
$J$. Give a critical analysis and discuss how to evaluate the \emph{cost
function}. You should feel free to write your own code, see also the
lecture notes of
\href{{https://compphysics.github.io/MachineLearning/doc/web/course.html}}{FYS-STK4155},
in particular te material on least square methods. You can use
scikit-learn to perform these analyses. See below for instruction on
how to install scikit-learn.
generated with $J=1$ in the previous point. Use linear regression,
Lasso and Ridge regression as done in project 1. You can compare your
results with those of \href{{https://physics.bu.edu/~pankajm/ML-Notebooks/HTML/NB_CVI-linreg_ising.html}}{Mehta
et al.}.
Make sure it is the 1D data which is used.
Discuss the methods and how they perform in computing the coupling
constant $J$ and include a bias-variance analysis using either
cross-validation or bootstrap. Discuss also the mean squared error and
the $R2$ score as measures to assess your model.
Give a critical analysis of your results.
\paragraph{Part c): Determine the phase of the two-dimensional Ising model.}
We switch now to binary classification methods and use logistic
regression to define the phases of the Ising model. Use described
section 7 and in Notebook 6 of \href{{https://physics.bu.edu/~pankajm/ML-Notebooks/HTML/NB_CVII-logreg_ising.html}}{Mehta *et
al.*}. Discuss
the methods and how they perform. Give a critical analysis and discuss
how to evaluate the \emph{cost function}. You should feel free to write
your own code.
regression to define the phases of the Ising model. This means that we switch to the two-dimensional Ising model
and use the data sets generated by \href{{https://physics.bu.edu/~pankajm/ML-Review-Datasets/isingMC/}}{Mehta et al}
These energies and their corresponding spin orientation configurations
represent then your data. We will use a fixed lattice of $L\times L =
40 \times 40$ spins in two dimensions. The link above contains data for several temperatures.
The theoretical critical temperature for a phase transition is $T_C\approx 2.269$ in units of energy.
However, for a finite lattice the results representing the critical temperature are slightly higher ($T_C \approx 2.3$).
Our goal here, using logistic regression, is to train our model to
predict the phase of a sample given the spin configuration, whether it
represents a state above the critical temperature or below. The
configurations representing states below the critical temperature are
called ordered states (the spins tend to point in one direction,
resulting in a net magnetic moment) while those above the critical
temperature are called disordered. Since a finite lattice like this
does not exhibit a clear sign of a phase transition we will mainly
stay with either orderer or disoredered phases. You could include the
critical phase if you want.
Your aim here is thus to read in these data (use the examples from
\href{{https://physics.bu.edu/~pankajm/ML-Notebooks/HTML/NB_CVII-logreg_ising.html}}{Mehta et
al})
and write your own code for doing logistic regression, see the lecture
notes on \href{{https://compphysics.github.io/MachineLearning/doc/pub/LogReg/html/LogReg-bs.html}}{logistic
regression}.
You should include either bootstrap or cross-validation in setting up
your model (as done in project 1) and compute the $R2$ score and the
mean-square error.
We will use both ordered and disordered states to train
the logistic regressor and, once the supervised training procedure is
complete, you should evaluate the performance of your classification model on
unseen ordered, disordered and possibly critical states.
In order to find the optimal parameters of your logistic regressor you should
include a gradient descent solver, as discussed in the \href{{https://compphysics.github.io/MachineLearning/doc/pub/Splines/html/Splines-bs.html}}{gradient
descent
lectures}.
Since we don't have so many data points, you may just code the
standard gradient descent with a given learning rate, or even attempt
to use the Newton-Raphson method. Alternatively, it may be useful for
the next part on neural networks to implement a stochastic gradient
descent. For all gradient methods, you can use \textbf{scikit-learn}'s toolbox for
optimization methods instead of writing your own code.
The notebook of \href{{https://physics.bu.edu/~pankajm/ML-Notebooks/HTML/NB_CVII-logreg_ising.html}}{Mehta et al} is highly recommended in order to benchmark your code and results.
\paragraph{Part d): Regression analysis of the one-dimensional Ising model using neural networks.}
Your aim now, and this is the central part of this project, is to
write to your own multilayer perceptron model implementing the back
propagation algorithm discussed in the \href{{https://compphysics.github.io/MachineLearning/doc/pub/NeuralNet/html/NeuralNet-bs.html}}{lecture
slides}. We
start with the regression case discussed in parts a) and b) but train
now the network to find the optimal weights and biases. You are free
to use the codes in the above lecture slides as starting points.
Train your network and compare the results with those from your linear regression code.
You can test your results against a similar code using _scikit_learn_ (see the examples in teh above lecture notes) or \textbf{tensorflow/keras}.
You should have the same elements as in the regression examples, including the $R2$ score, the MSE, and bootstrap or cross-validation.
A useful reference on the back progagation algorithm is \href{{http://neuralnetworksanddeeplearning.com/}}{Nielsen's book}. It is an excellent read.
\paragraph{Part e): Classifying the Ising model phase using neural networks.}
Finally, change now your cost function to the $log$ cross-entropy classification cost function for the case discussed in part c). Train your network again and
compare the results with those from your logistic regression code i c).
Here again you can compare your results with those of \href{{https://physics.bu.edu/~pankajm/ML-Notebooks/HTML/NB_CIX-DNN_ising_TFlow.html}}{Mehta et al}. There they used \textbf{tensorflow} to classify the phases.
\paragraph{Part f) Critical evaluation of the various algorithms.}
After all these glorious calculations, you should now summarize the various algorithms and come with a critical evaluation of their pros and cons. Which algorithm works best for the regression case and which is best for the classification case. These codes will also be part of your final project 3, but now applied to other data sets.
\paragraph{Part d): Classifying the Ising model phase using neural networks.}
We end the classification problem of the phases of the Ising model by
employing the algorithm for so-called feed-forward deep neural
networks (see section 9 of Mehta \emph{et al.}). The method is described in
\href{{https://physics.bu.edu/~pankajm/ML-Notebooks/HTML/NB_CIX-DNN_ising_TFlow.html}}{notebook
12}.
You can use tensorflow to perform these analyses. See below for instruction on how to install tensorflow.
\subsection{Background literature}
\begin{enumerate}
\item The text of Michael Nielsen is highly recommended, see \href{{http://neuralnetworksanddeeplearning.com/}}{Nielsen's book}. It is an excellent read.
\item The textbook of \href{{https://www.springer.com/gp/book/9780387848570}}{Trevor Hastie, Robert Tibshirani, Jerome H. Friedman, The Elements of Statistical Learning, Springer}, chapters 3 and 7 are the most relevant ones for the analysis here.
\item \href{{https://arxiv.org/abs/1803.08823}}{Mehta et al, arXiv 1803.08823}, \emph{A high-bias, low-variance introduction to Machine Learning for physicists}, ArXiv:1803.08823.
doc/Projects/2018/Project2/pdf/Project2.pdf
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doc/Projects/2018/Project2/pdf/Project2.tex
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@@ -20,6 +20,8 @@ final, % draft: marks overfull hboxes, figures with paths
\usepackage[pdftex]{graphicx}
\usepackage{fancyvrb} % packages needed for verbatim environments
\usepackage[T1]{fontenc}
%\usepackage[latin1]{inputenc}
\usepackage{ucs}
@@ -123,28 +125,17 @@ Project 2 on Machine Learning, deadline November 5
% --- begin date ---
\begin{center}
Oct 8, 2018
Oct 9, 2018
\end{center}
% --- end date ---
\vspace{1cm}
\begin{enumerate}
\item add about ising model
\item link to where we have the data
\item explain how to fit the model
\item link to mehta's article
\end{enumerate}
\noindent
\subsection*{Classification and Regression, from linear and logistic regression to neural networks}
The main aim of this project is to study both classification and
regression problems, starting with the regression algortihms studied
regression problems, starting with the regression algorithms studied
in project 1. We will include logistic regresion for classification
problems and write our own multilayer perceptron code for studying
both regression and classification problems. The codes developed in
@@ -153,19 +144,20 @@ computation of the mean-squared error and the R2 score function can
also be utilized (and included in logistic regression and the neural
network codes) in the present analysis.
We will use the Ising model to generate our training data and will
focus mainly on supervised training. We will follow closely the recent
We will use the so-called Ising model for our training data and will
focus on supervised training. We will follow closely the recent
article of \href{{https://arxiv.org/abs/1803.08823}}{Mehta et al, arXiv
1803.08823}. This article stands
out as an excellent review on machine learning (ML) algorithms applied
to typical physics problems. The added benefit is that each figure and
out as an excellent review on machine learning (ML) algorithms.
The added benefit is that each figure and
model presented in \href{{https://physics.bu.edu/~pankajm/MLnotebooks.html}}{this article is accompanied by its jupyter
notebook}. This
means that we can start using these and compare with our own
results. In case you wish to use their data for the Ising model, their
data can be downloaded from the same link which lists to the jupyter
notebooks. See also at the end of the project description for more
information on how to install various Python packages.
means that we can start using these and compare with our own results.
They provide also the data set for the regression and classification
analysis that we will explore. In this sense, with their available
notebooks, it makes life easier since we can compare our own codes
with their codes.
@@ -182,69 +174,243 @@ that is the two-dimensional Ising model, will be studied using
logistic regression and deep neural networks. The aim is to develop
your own logistic regression code for the classification of the phases
(this is a binary model) and your multilayer perceptron code for the
classification and regression case.
classification and regression case. You can compare your own results with those obtained
using \textbf{scikit-learn} or \textbf{tensorflow} or other Python packages such as \textbf{keras} or other.
Feel free to use the notebooks to benchmark your code. If you wish to
write your own C++ or Fortran program for say a simple neural network
model and a logistic regression model, please feel free to do so. You can then benchmark your results
against the above jupyter notebooks.
write your own C++ or Fortran program for say a multilayer neural network
model and a logistic regression model, please feel free to do so.
\paragraph{Part a): Producing the data.}
You can use the Ising model data from the article of Mehta \emph{et al.},
or generate your own data. If you opt for using your own Ising model
code, you need to generate $10000$ energy configurations with their
spin orientations after the system has reached its most likely
state. These energies and their corresponding spin orientations
represent then your data. We will use a fixed lattice of $L\times L =
40 \times 40$ spins in two dimensions and $L=40$ spins in one
dimension. Make sure the calculations have been equilibrated. For the
two-dimensional system, compute the configurations for three values of
the temperature, namely $T=0.75$ (ordered phase), $T=2.3$ (near the
critical point) and $T=4.0$ (disordered phase). For the
one-dimensional system it suffices to compute the various
configurations for one temperature only, say $T=2.0$. These are the
data you will use to study different ML algorithms. We generate our
data with $J=1$.
\paragraph{Part a): Producing the data for the one-dimensional Ising model.}
The model we will employ in our studies is the so-called \href{{https://en.wikipedia.org/wiki/Ising_model}}{Ising
model}. Together with
models like the \href{{https://en.wikipedia.org/wiki/Potts_model}}{Potts
model} and similar
so-called lattice models, the Ising model has been widely studied in
mathematics (in statistics in particular), physics, \href{{https://journals.aps.org/pre/abstract/10.1103/PhysRevE.93.062402}}{life
science},
chemistry and even in the \href{{https://www.springer.com/gp/book/9781461420316}}{social sciences in order to model social
behavior}. It is a
simple binary value system where the variables of the model (spins often in
physics) can take two values only, for example $\pm 1$ or $0$ and $1$.
The system exhibits a phase transition in two or higher dimensions and
the first person to find the analytical expressions for various
expectation values was the Norwegian chemist \href{{https://en.wikipedia.org/wiki/Lars_Onsager}}{Lars
Onsager} (Nobel prize in
chemistry) after a tour de force mathematics exercise.
\paragraph{Part b): Estimating the coupling constant of the one-dimensional Ising model.}
In our discussions here we will stay with a physicist's approach and
call the variables for spin. You could replace this with any other
type of binary variables, ranging from a two political parties to blue
and red spheres. In its simplest form we define the energy of the
system as
\begin{equation*}
E=-J\sum_{<kl>}^{N}s_ks_l,
\end{equation*}
with $s_k=\pm 1$, $N$ is the total number of spins,
$J$ is a coupling constant expressing the strength of the interaction
between neighboring spins.
The symbol $<kl>$ indicates that we sum over nearest
neighbors only.
Notice that for $J>0$ it is energetically favorable for neighboring spins
to be aligned. This feature leads to, at low enough temperatures,
a cooperative phenomenon called spontaneous magnetization. That is,
through interactions between nearest neighbors, a given magnetic
moment can influence the alignment of spins that are separated
from the given spin by a macroscopic distance. These long range correlations
between spins are associated with a long-range order in which
the lattice has a net magnetization in the absence of a magnetic field.
We start by considering the one-dimensional Ising model with nearest neighbor interactions. This model does not exhibit any phase transition.
Consider the 1D Ising model with nearest-neighbor interactions
\begin{equation*}
E[\hat{s}]=-J\sum_{j=1}^{N}s_{j}s_{j+1},
\end{equation*}
on a chain of length $N$ with so-called periodic boundary conditions and $S_j=\pm 1$ Ising spin variables.
In one dimension, this model has no phase transition at finite temperature.
In the Python code below we generate, with a coupling coefficient set to $J=1$, a large number of spin configurations say $10000$ as shown in the code below.
It means that our data will be a set of $i=1\ldots n$ points of the form
$\{(E[\boldsymbol{s}^i],\boldsymbol{s}^i)\}$.
Our task is to find the value of $J$ from the data set using linear regression.
Here is the Python code you need to generate the training data, see
also the \href{{https://physics.bu.edu/~pankajm/ML-Notebooks/HTML/NB_CVI-linreg_ising.html}}{notebook of Mehta et
al}.
\begin{print}
import numpy as np
import scipy.sparse as sp
np.random.seed(12)
import warnings
#Comment this to turn on warnings
warnings.filterwarnings('ignore')
### define Ising model aprams
# system size
L=40
# create 10000 random Ising states
states=np.random.choice([-1, 1], size=(10000,L))
def ising_energies(states,L):
"""
This function calculates the energies of the states in the nn Ising Hamiltonian
"""
J=np.zeros((L,L),)
for i in range(L):
J[i,(i+1)%L]-=1.0
# compute energies
E = np.einsum('...i,ij,...j->...',states,J,states)
return E
# calculate Ising energies
energies=ising_energies(states,L)
\end{print}
We can now recast the problem as a linear regression model using our codes from project 1.
The way we are going to build our model mimicks the way we could think of finding say the gravitional constant for the graviational force between two planets.
In the absence of any prior knowledge, one sensible choice is the all-to-all Ising model
\[
E_\mathrm{model}[\boldsymbol{s}^i] = - \sum_{j=1}^N \sum_{k=1}^N J_{j,k}s_{j}^is_{k}^i.
\]
Here $i$ represents a particular spin configuration (one of the possible $n$ configurations we generated with the code above).
This model is uniquely defined by the non-local coupling strengths $J_{jk}$ which we want to learn.
The model is linear in $\mathbf{J}$ which makes it possible to use linear regression.
To apply linear regression, we recast this model in the form
\[
E_\mathrm{model}^i \equiv \mathbf{X}^i \cdot \mathbf{J},
\]
where the vectors $\mathbf{X}^i$ represent all two-body interactions
$\{s_{j}^is_{k}^i \}_{j,k=1}^N$, and the index $i$ runs over the
samples in the data set. To make the analogy complete, we can also
represent the dot product by a single index $p = \{j,k\}$,
i.e.~$\mathbf{X}^i \cdot \mathbf{J}=X^i_pJ_p$. Note that the
regression model does not include the minus sign, so we expect to
learn negative $J$'s.
With these preliminaries, we are now ready to reutilize our codes from project 1.
\paragraph{Part b): Estimating the coupling constant of the one-dimensional Ising model using linear regression.}
We start with the one-dimensional Ising model and use the data we have
generated with $J=1$. Use linear regression, Lasso and Ridge
regression as described section 6 and in Notebook 4 of \href{{https://physics.bu.edu/~pankajm/ML-Notebooks/HTML/NB_CVI-linreg_ising.html}}{Mehta *et
al.*}. Discuss
the methods and how they perform in computing the coupling constant
$J$. Give a critical analysis and discuss how to evaluate the \emph{cost
function}. You should feel free to write your own code, see also the
lecture notes of
\href{{https://compphysics.github.io/MachineLearning/doc/web/course.html}}{FYS-STK4155},
in particular te material on least square methods. You can use
scikit-learn to perform these analyses. See below for instruction on
how to install scikit-learn.
generated with $J=1$ in the previous point. Use linear regression,
Lasso and Ridge regression as done in project 1. You can compare your
results with those of \href{{https://physics.bu.edu/~pankajm/ML-Notebooks/HTML/NB_CVI-linreg_ising.html}}{Mehta
et al.}.
Make sure it is the 1D data which is used.
Discuss the methods and how they perform in computing the coupling
constant $J$ and include a bias-variance analysis using either
cross-validation or bootstrap. Discuss also the mean squared error and
the $R2$ score as measures to assess your model.
Give a critical analysis of your results.
\paragraph{Part c): Determine the phase of the two-dimensional Ising model.}
We switch now to binary classification methods and use logistic
regression to define the phases of the Ising model. Use described
section 7 and in Notebook 6 of \href{{https://physics.bu.edu/~pankajm/ML-Notebooks/HTML/NB_CVII-logreg_ising.html}}{Mehta *et
al.*}. Discuss
the methods and how they perform. Give a critical analysis and discuss
how to evaluate the \emph{cost function}. You should feel free to write
your own code.
regression to define the phases of the Ising model. This means that we switch to the two-dimensional Ising model
and use the data sets generated by \href{{https://physics.bu.edu/~pankajm/ML-Review-Datasets/isingMC/}}{Mehta et al}
These energies and their corresponding spin orientation configurations
represent then your data. We will use a fixed lattice of $L\times L =
40 \times 40$ spins in two dimensions. The link above contains data for several temperatures.
The theoretical critical temperature for a phase transition is $T_C\approx 2.269$ in units of energy.
However, for a finite lattice the results representing the critical temperature are slightly higher ($T_C \approx 2.3$).
Our goal here, using logistic regression, is to train our model to
predict the phase of a sample given the spin configuration, whether it
represents a state above the critical temperature or below. The
configurations representing states below the critical temperature are
called ordered states (the spins tend to point in one direction,
resulting in a net magnetic moment) while those above the critical
temperature are called disordered. Since a finite lattice like this
does not exhibit a clear sign of a phase transition we will mainly
stay with either orderer or disoredered phases. You could include the
critical phase if you want.
Your aim here is thus to read in these data (use the examples from
\href{{https://physics.bu.edu/~pankajm/ML-Notebooks/HTML/NB_CVII-logreg_ising.html}}{Mehta et
al})
and write your own code for doing logistic regression, see the lecture
notes on \href{{https://compphysics.github.io/MachineLearning/doc/pub/LogReg/html/LogReg-bs.html}}{logistic
regression}.
You should include either bootstrap or cross-validation in setting up
your model (as done in project 1) and compute the $R2$ score and the
mean-square error.
We will use both ordered and disordered states to train
the logistic regressor and, once the supervised training procedure is
complete, you should evaluate the performance of your classification model on
unseen ordered, disordered and possibly critical states.
In order to find the optimal parameters of your logistic regressor you should
include a gradient descent solver, as discussed in the \href{{https://compphysics.github.io/MachineLearning/doc/pub/Splines/html/Splines-bs.html}}{gradient
descent
lectures}.
Since we don't have so many data points, you may just code the
standard gradient descent with a given learning rate, or even attempt
to use the Newton-Raphson method. Alternatively, it may be useful for
the next part on neural networks to implement a stochastic gradient
descent. For all gradient methods, you can use \textbf{scikit-learn}'s toolbox for
optimization methods instead of writing your own code.
The notebook of \href{{https://physics.bu.edu/~pankajm/ML-Notebooks/HTML/NB_CVII-logreg_ising.html}}{Mehta et al} is highly recommended in order to benchmark your code and results.
\paragraph{Part d): Regression analysis of the one-dimensional Ising model using neural networks.}
Your aim now, and this is the central part of this project, is to
write to your own multilayer perceptron model implementing the back
propagation algorithm discussed in the \href{{https://compphysics.github.io/MachineLearning/doc/pub/NeuralNet/html/NeuralNet-bs.html}}{lecture
slides}. We
start with the regression case discussed in parts a) and b) but train
now the network to find the optimal weights and biases. You are free
to use the codes in the above lecture slides as starting points.
Train your network and compare the results with those from your linear regression code.
You can test your results against a similar code using _scikit_learn_ (see the examples in teh above lecture notes) or \textbf{tensorflow/keras}.
You should have the same elements as in the regression examples, including the $R2$ score, the MSE, and bootstrap or cross-validation.
A useful reference on the back progagation algorithm is \href{{http://neuralnetworksanddeeplearning.com/}}{Nielsen's book}. It is an excellent read.
\paragraph{Part e): Classifying the Ising model phase using neural networks.}
Finally, change now your cost function to the $log$ cross-entropy classification cost function for the case discussed in part c). Train your network again and
compare the results with those from your logistic regression code i c).
Here again you can compare your results with those of \href{{https://physics.bu.edu/~pankajm/ML-Notebooks/HTML/NB_CIX-DNN_ising_TFlow.html}}{Mehta et al}. There they used \textbf{tensorflow} to classify the phases.
\paragraph{Part f) Critical evaluation of the various algorithms.}
After all these glorious calculations, you should now summarize the various algorithms and come with a critical evaluation of their pros and cons. Which algorithm works best for the regression case and which is best for the classification case. These codes will also be part of your final project 3, but now applied to other data sets.
\paragraph{Part d): Classifying the Ising model phase using neural networks.}
We end the classification problem of the phases of the Ising model by
employing the algorithm for so-called feed-forward deep neural
networks (see section 9 of Mehta \emph{et al.}). The method is described in
\href{{https://physics.bu.edu/~pankajm/ML-Notebooks/HTML/NB_CIX-DNN_ising_TFlow.html}}{notebook
12}.
You can use tensorflow to perform these analyses. See below for instruction on how to install tensorflow.
\subsection*{Background literature}
\begin{enumerate}
\item The text of Michael Nielsen is highly recommended, see \href{{http://neuralnetworksanddeeplearning.com/}}{Nielsen's book}. It is an excellent read.
\item The textbook of \href{{https://www.springer.com/gp/book/9780387848570}}{Trevor Hastie, Robert Tibshirani, Jerome H. Friedman, The Elements of Statistical Learning, Springer}, chapters 3 and 7 are the most relevant ones for the analysis here.
\item \href{{https://arxiv.org/abs/1803.08823}}{Mehta et al, arXiv 1803.08823}, \emph{A high-bias, low-variance introduction to Machine Learning for physicists}, ArXiv:1803.08823.
doc/src/Projects/2018/Project2/Project2.do.txt
+6 -12
View File
@@ -2,11 +2,6 @@ TITLE: Project 2 on Machine Learning, deadline November 5
AUTHOR: "Data Analysis and Machine Learning FYS-STK3155/FYS4155":"http://www.uio.no/studier/emner/matnat/fys/FYS3155/index-eng.html" {copyright, 1999-present|CC BY-NC} at Department of Physics, University of Oslo, Norway
DATE: today
o add note to nielsen's text
o add about ising model
o link to where we have the data
o explain how to fit the model
o link to mehta's article
===== Classification and Regression, from linear and logistic regression to neural networks =====
@@ -88,14 +83,12 @@ system as
!bt
\begin{equation*}
E=-J\sum_{<kl>}^{N}s_ks_l-{\cal B}\sum_k^Ns_k,
E=-J\sum_{<kl>}^{N}s_ks_l,
\end{equation*}
!et
with $s_k=\pm 1$, $N$ is the total number of spins,
$J$ is a coupling constant expressing the strength of the interaction
between neighboring spins and
${\cal B}$ is an external magnetic field interacting with the magnetic
moment set up by the spins. We will discard the magnetic field part.
between neighboring spins.
The symbol $<kl>$ indicates that we sum over nearest
neighbors only.
@@ -119,6 +112,7 @@ Consider the 1D Ising model with nearest-neighbor interactions
E[\hat{s}]=-J\sum_{j=1}^{N}s_{j}s_{j+1},
\end{equation*}
!et
on a chain of length $N$ with so-called periodic boundary conditions and $S_j=\pm 1$ Ising spin variables.
In one dimension, this model has no phase transition at finite temperature.
@@ -136,7 +130,6 @@ import numpy as np
import scipy.sparse as sp
np.random.seed(12)
import warnings
#Comment this to turn on warnings
warnings.filterwarnings('ignore')
@@ -172,17 +165,18 @@ In the absence of any prior knowledge, one sensible choice is the all-to-all Isi
E_\mathrm{model}[\boldsymbol{s}^i] = - \sum_{j=1}^N \sum_{k=1}^N J_{j,k}s_{j}^is_{k}^i.
\]
!et
Here $i$ represents a particular spin configuration (one of the possible $n$ configurations we generated with the code above).
This model is uniquely defined by the non-local coupling strengths $J_{jk}$ which we want to learn.
The model is linear in ${\mathbf J}$ which makes it possible to use linear regression.
The model is linear in $\mathbf{J}$ which makes it possible to use linear regression.
To apply linear regression, we recast this model in the form
!bt
\[
E_\mathrm{model}^i \equiv \mathbf{X}^i \cdot \mathbf{J},
\]
!ec
!et
where the vectors $\mathbf{X}^i$ represent all two-body interactions
$\{s_{j}^is_{k}^i \}_{j,k=1}^N$, and the index $i$ runs over the