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<title>1. Linear Regression, basic Elements &#8212; Applied Data Analysis and Machine Learning</title>
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Applied Data Analysis and Machine Learning
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About the course
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From Regression to Support Vector Machines
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1. Linear Regression, basic Elements
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2. Resampling Methods
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3. Ridge and Lasso Regression
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4. Logistic Regression
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<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#introduction">
1.1. Introduction
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#what-is-machine-learning">
1.2. What is Machine Learning?
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#software-and-needed-installations">
1.3. Software and needed installations
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#python-installers">
1.4. Python installers
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#useful-python-libraries">
1.5. Useful Python libraries
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#installing-r-c-cython-or-julia">
1.6. Installing R, C++, cython or Julia
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#installing-r-c-cython-numba-etc">
1.7. Installing R, C++, cython, Numba etc
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#numpy-examples-and-important-matrix-and-vector-handling-packages">
1.8. Numpy examples and Important Matrix and vector handling packages
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#basic-matrix-features">
1.9. Basic Matrix Features
</a>
<ul class="nav section-nav flex-column">
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#some-famous-matrices">
1.9.1. Some famous Matrices
</a>
</li>
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#more-basic-matrix-features">
1.9.2. More Basic Matrix Features
</a>
</li>
</ul>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#numpy-and-arrays">
1.10. Numpy and arrays
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#matrices-in-python">
1.11. Matrices in Python
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#meet-the-pandas">
1.12. Meet the Pandas
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#simple-linear-regression-model-using-scikit-learn">
1.13. Simple linear regression model using
<strong>
scikit-learn
</strong>
</a>
<ul class="nav section-nav flex-column">
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#organizing-our-data">
1.13.1. Organizing our data
</a>
</li>
</ul>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#id1">
1.14. Linear Regression, basic elements
</a>
<ul class="nav section-nav flex-column">
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#the-chi-2-function">
1.14.1. The
<span class="math notranslate nohighlight">
\(\chi^2\)
</span>
function
</a>
</li>
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#fitting-an-equation-of-state-for-dense-nuclear-matter">
1.14.2. Fitting an Equation of State for Dense Nuclear Matter
</a>
</li>
</ul>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#splitting-our-data-in-training-and-test-data">
1.15. Splitting our Data in Training and Test data
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#the-boston-housing-data-example">
1.16. The Boston housing data example
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#housing-data-the-code">
1.17. Housing data, the code
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#reducing-the-number-of-degrees-of-freedom-overarching-view">
1.18. Reducing the number of degrees of freedom, overarching view
</a>
<ul class="nav section-nav flex-column">
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#simple-preprocessing-examples-franke-function-and-regression">
1.18.1. Simple preprocessing examples, Franke function and regression
</a>
</li>
</ul>
</li>
</ul>
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<div>
<div class="section" id="linear-regression-basic-elements">
<h1><span class="section-number">1. </span>Linear Regression, basic Elements<a class="headerlink" href="#linear-regression-basic-elements" title="Permalink to this headline"></a></h1>
<p><a class="reference external" href="https://www.uio.no/studier/emner/matnat/fys/FYS-STK3155/h20/forelesningsvideoer/LectureAug21.mp4?vrtx=view-as-webpage">Video of Lecture</a></p>
<div class="section" id="introduction">
<h2><span class="section-number">1.1. </span>Introduction<a class="headerlink" href="#introduction" title="Permalink to this headline"></a></h2>
<p>Our emphasis throughout this series of lectures<br />
is on understanding the mathematical aspects of
different algorithms used in the fields of data analysis and machine learning.</p>
<p>However, where possible we will emphasize the
importance of using available software. We start thus with a hands-on
and top-down approach to machine learning. The aim is thus to start with
relevant data or data we have produced
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 in the beginning, start with simple
polynomials with random noise added. We will use the Python
software package <a class="reference external" href="http://scikit-learn.org/stable/">Scikit-Learn</a> and
introduce various machine learning algorithms to make fits of
the data and predictions. We move thereafter to more interesting
cases such as data from say experiments (below we will look at experimental nuclear binding energies as an example).
These are examples where we can easily set up the data and
then use machine learning algorithms included in for example
<strong>Scikit-Learn</strong>.</p>
<p>These examples will serve us the purpose of getting
started. Furthermore, they allow us to catch more than two birds with
a stone. They will allow us to bring in some programming specific
topics and tools as well as showing the power of various Python
libraries for machine learning and statistical data analysis.</p>
<p>Here, we will mainly focus on two
specific Python packages for Machine Learning, Scikit-Learn and
Tensorflow (see below for links etc). Moreover, the examples we
introduce will serve as inputs to many of our discussions later, as
well as allowing you to set up models and produce your own data and
get started with programming.</p>
</div>
<div class="section" id="what-is-machine-learning">
<h2><span class="section-number">1.2. </span>What is Machine Learning?<a class="headerlink" href="#what-is-machine-learning" title="Permalink to this headline"></a></h2>
<p>Statistics, data science and machine learning form important fields of
research in modern science. They describe how to learn and make
predictions from data, as well as allowing us to extract important
correlations about physical process and the underlying laws of motion
in large data sets. The latter, big data sets, appear frequently in
essentially all disciplines, from the traditional Science, Technology,
Mathematics and Engineering fields to Life Science, Law, education
research, the Humanities and the Social Sciences.</p>
<p>It has become more
and more common to see research projects on big data in for example
the Social Sciences where extracting patterns from complicated survey
data is one of many research directions. Having a solid grasp of data
analysis and machine learning is thus becoming central to scientific
computing in many fields, and competences and skills within the fields
of machine learning and scientific computing are nowadays strongly
requested by many potential employers. The latter cannot be
overstated, familiarity with machine learning has almost become a
prerequisite for many of the most exciting employment opportunities,
whether they are in bioinformatics, life science, physics or finance,
in the private or the public sector. This author has had several
students or met students who have been hired recently based on their
skills and competences in scientific computing and data science, often
with marginal knowledge of machine learning.</p>
<p>Machine learning is a subfield of computer science, and is closely
related to computational statistics. It evolved from the study of
pattern recognition in artificial intelligence (AI) research, and has
made contributions to AI tasks like computer vision, natural language
processing and speech recognition. Many of the methods we will study are also
strongly rooted in basic mathematics and physics research.</p>
<p>Ideally, machine learning represents the science of giving computers
the ability to learn without being explicitly programmed. The idea is
that there exist generic algorithms which can be used to find patterns
in a broad class of data sets without having to write code
specifically for each problem. The algorithm will build its own logic
based on the data. You should however always keep in mind that
machines and algorithms are to a large extent developed by humans. The
insights and knowledge we have about a specific system, play a central
role when we develop a specific machine learning algorithm.</p>
<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 class="reference external" href="http://scikit-learn.org/stable/">Scikit-learn</a>,
<a class="reference external" href="https://www.tensorflow.org/">Tensorflow</a>,
<a class="reference external" href="http://pytorch.org/">PyTorch</a> and <a class="reference external" href="https://keras.io/">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>
<p>The approaches to machine learning are many, but are often split into
two main categories. In <em>supervised learning</em> we know the answer to a
problem, and let the computer deduce the logic behind it. On the other
hand, <em>unsupervised learning</em> is a method for finding patterns and
relationship in data sets without any prior knowledge of the system.
Some authours also operate with a third category, namely
<em>reinforcement learning</em>. This is a paradigm of learning inspired by
behavioral psychology, where learning is achieved by trial-and-error,
solely from rewards and punishment.</p>
<p>Another way to categorize machine learning tasks is to consider the
desired output of a system. Some of the most common tasks are:</p>
<ul class="simple">
<li><p>Classification: Outputs are divided into two or more classes. The goal is to produce a model that assigns inputs into one of these classes. An example is to identify digits based on pictures of hand-written ones. Classification is typically supervised learning.</p></li>
<li><p>Regression: Finding a functional relationship between an input data set and a reference data set. The goal is to construct a function that maps input data to continuous output values.</p></li>
<li><p>Clustering: Data are divided into groups with certain common traits, without knowing the different groups beforehand. It is thus a form of unsupervised learning.</p></li>
</ul>
<p>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 (which can be subdivided into
training and test data), the second item is a model which is normally a
function of some parameters. The model reflects our knowledge of the system (or lack thereof). As an example, if we know that our data show a behavior similar to what would be predicted by a polynomial, fitting our data to a polynomial of some degree would then determin our model.</p>
<p>The last ingredient is a so-called <strong>cost</strong>
function which allows us to present an estimate on how good our model
is in reproducing the data it is supposed to train.<br />
At the heart of basically all ML algorithms there are so-called minimization algorithms, often we end up with various variants of <strong>gradient</strong> methods.</p>
</div>
<div class="section" id="software-and-needed-installations">
<h2><span class="section-number">1.3. </span>Software and needed installations<a class="headerlink" href="#software-and-needed-installations" title="Permalink to this headline"></a></h2>
<p>We will make extensive use of Python as programming language and its
myriad of available libraries. You will find
Jupyter notebooks invaluable in your work. You can run <strong>R</strong>
codes in the Jupyter/IPython notebooks, with the immediate benefit of
visualizing your data. You can also use compiled languages like C++,
Rust, Julia, Fortran etc if you prefer. The focus in these lectures will be
on Python.</p>
<p>If you have Python installed (we strongly recommend Python3) and you feel
pretty familiar with installing different packages, we recommend that
you install the following Python packages via <strong>pip</strong> as</p>
<ol class="simple">
<li><p>pip install numpy scipy matplotlib ipython scikit-learn mglearn sympy pandas pillow</p></li>
</ol>
<p>For Python3, replace <strong>pip</strong> with <strong>pip3</strong>.</p>
<p>For OSX users we recommend, after having installed Xcode, to
install <strong>brew</strong>. Brew allows for a seamless installation of additional
software via for example</p>
<ol class="simple">
<li><p>brew install python3</p></li>
</ol>
<p>For Linux users, with its variety of distributions like for example the widely popular Ubuntu distribution,
you can use <strong>pip</strong> as well and simply install Python as</p>
<ol class="simple">
<li><p>sudo apt-get install python3 (or python for pyhton2.7)</p></li>
</ol>
<p>etc etc.</p>
</div>
<div class="section" id="python-installers">
<h2><span class="section-number">1.4. </span>Python installers<a class="headerlink" href="#python-installers" title="Permalink to this headline"></a></h2>
<p>If you don’t want to perform these operations separately and venture
into the hassle of exploring how to set up dependencies and paths, we
recommend two widely used distrubutions which set up all relevant
dependencies for Python, namely</p>
<ul class="simple">
<li><p><a class="reference external" href="https://docs.anaconda.com/">Anaconda</a>,</p></li>
</ul>
<p>which is an open source
distribution of the Python and R programming languages for large-scale
data processing, predictive analytics, and scientific computing, that
aims to simplify package management and deployment. Package versions
are managed by the package management system <strong>conda</strong>.</p>
<ul class="simple">
<li><p><a class="reference external" href="https://www.enthought.com/product/canopy/">Enthought canopy</a></p></li>
</ul>
<p>is a Python
distribution for scientific and analytic computing distribution and
analysis environment, available for free and under a commercial
license.</p>
<p>Furthermore, <a class="reference external" href="https://colab.research.google.com/notebooks/welcome.ipynb">Google’s Colab</a> is a free Jupyter notebook environment that requires
no setup and runs entirely in the cloud. Try it out!</p>
</div>
<div class="section" id="useful-python-libraries">
<h2><span class="section-number">1.5. </span>Useful Python libraries<a class="headerlink" href="#useful-python-libraries" title="Permalink to this headline"></a></h2>
<p>Here we list several useful Python libraries we strongly recommend (if you use anaconda many of these are already there)</p>
<ul class="simple">
<li><p><a class="reference external" href="https://www.numpy.org/">NumPy</a> is a highly popular library for large, multi-dimensional arrays and matrices, along with a large collection of high-level mathematical functions to operate on these arrays</p></li>
<li><p><a class="reference external" href="https://pandas.pydata.org/">The pandas</a> library provides high-performance, easy-to-use data structures and data analysis tools</p></li>
<li><p><a class="reference external" href="http://xarray.pydata.org/en/stable/">Xarray</a> is a Python package that makes working with labelled multi-dimensional arrays simple, efficient, and fun!</p></li>
<li><p><a class="reference external" href="https://www.scipy.org/">Scipy</a> (pronounced “Sigh Pie”) is a Python-based ecosystem of open-source software for mathematics, science, and engineering.</p></li>
<li><p><a class="reference external" href="https://matplotlib.org/">Matplotlib</a> is a Python 2D plotting library which produces publication quality figures in a variety of hardcopy formats and interactive environments across platforms.</p></li>
<li><p><a class="reference external" href="https://github.com/HIPS/autograd">Autograd</a> can automatically differentiate native Python and Numpy code. It can handle a large subset of Python’s features, including loops, ifs, recursion and closures, and it can even take derivatives of derivatives of derivatives</p></li>
<li><p><a class="reference external" href="https://www.sympy.org/en/index.html">SymPy</a> is a Python library for symbolic mathematics.</p></li>
<li><p><a class="reference external" href="https://scikit-learn.org/stable/">scikit-learn</a> has simple and efficient tools for machine learning, data mining and data analysis</p></li>
<li><p><a class="reference external" href="https://www.tensorflow.org/">TensorFlow</a> is a Python library for fast numerical computing created and released by Google</p></li>
<li><p><a class="reference external" href="https://keras.io/">Keras</a> is a high-level neural networks API, written in Python and capable of running on top of TensorFlow, CNTK, or Theano</p></li>
<li><p>And many more such as <a class="reference external" href="https://pytorch.org/">pytorch</a>, <a class="reference external" href="https://pypi.org/project/Theano/">Theano</a> etc</p></li>
</ul>
</div>
<div class="section" id="installing-r-c-cython-or-julia">
<h2><span class="section-number">1.6. </span>Installing R, C++, cython or Julia<a class="headerlink" href="#installing-r-c-cython-or-julia" title="Permalink to this headline"></a></h2>
<p>You will also find it convenient to utilize <strong>R</strong>. We will mainly
use Python during our lectures and in various projects and exercises.
Those of you
already familiar with <strong>R</strong> should feel free to continue using <strong>R</strong>, keeping
however an eye on the parallel Python set ups. Similarly, if you are a
Python afecionado, feel free to explore <strong>R</strong> as well. Jupyter/Ipython
notebook allows you to run <strong>R</strong> codes interactively in your
browser. The software library <strong>R</strong> is really tailored for statistical data analysis
and allows for an easy usage of the tools and algorithms we will discuss in these
lectures.</p>
<p>To install <strong>R</strong> with Jupyter notebook
<a class="reference external" href="https://mpacer.org/maths/r-kernel-for-ipython-notebook">follow the link here</a></p>
</div>
<div class="section" id="installing-r-c-cython-numba-etc">
<h2><span class="section-number">1.7. </span>Installing R, C++, cython, Numba etc<a class="headerlink" href="#installing-r-c-cython-numba-etc" title="Permalink to this headline"></a></h2>
<p>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++ (or Fortran or other compiled languages) as programming
languages.</p>
<p>To add more entropy, <strong>cython</strong> can also be used when running your
notebooks. It means that Python with the jupyter notebook
setup allows you to integrate widely popular softwares and tools for
scientific computing. Similarly, the
<a class="reference external" href="https://numba.pydata.org/">Numba Python package</a> delivers increased performance
capabilities with minimal rewrites of your codes. With its
versatility, including symbolic operations, Python offers a unique
computational environment. Your jupyter notebook can easily be
converted into a nicely rendered <strong>PDF</strong> file or a Latex file for
further processing. For example, convert to latex as</p>
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span> pycod jupyter nbconvert filename.ipynb --to latex
</pre></div>
</div>
<p>And to add more versatility, the Python package <a class="reference external" href="http://www.sympy.org/en/index.html">SymPy</a> is a Python library for symbolic mathematics. It aims to become a full-featured computer algebra system (CAS) and is entirely written in Python.</p>
<p>Finally, if you wish to use the light mark-up language
<a class="reference external" href="https://github.com/hplgit/doconce">doconce</a> you can convert a standard ascii text file into various HTML
formats, ipython notebooks, latex files, pdf files etc with minimal edits. These lectures were generated using <strong>doconce</strong>.</p>
</div>
<div class="section" id="numpy-examples-and-important-matrix-and-vector-handling-packages">
<h2><span class="section-number">1.8. </span>Numpy examples and Important Matrix and vector handling packages<a class="headerlink" href="#numpy-examples-and-important-matrix-and-vector-handling-packages" title="Permalink to this headline"></a></h2>
<p>There are several central software libraries for linear algebra and eigenvalue problems. Several of the more
popular ones have been wrapped into ofter software packages like those from the widely used text <strong>Numerical Recipes</strong>. The original source codes in many of the available packages are often taken from the widely used
software package LAPACK, which follows two other popular packages
developed in the 1970s, namely EISPACK and LINPACK. We describe them shortly here.</p>
<ul class="simple">
<li><p>LINPACK: package for linear equations and least square problems.</p></li>
<li><p>LAPACK:package for solving symmetric, unsymmetric and generalized eigenvalue problems. From LAPACK’s website <a class="reference external" href="http://www.netlib.org">http://www.netlib.org</a> it is possible to download for free all source codes from this library. Both C/C++ and Fortran versions are available.</p></li>
<li><p>BLAS (I, II and III): (Basic Linear Algebra Subprograms) are routines that provide standard building blocks for performing basic vector and matrix operations. Blas I is vector operations, II vector-matrix operations and III matrix-matrix operations. Highly parallelized and efficient codes, all available for download from <a class="reference external" href="http://www.netlib.org">http://www.netlib.org</a>.</p></li>
</ul>
</div>
<div class="section" id="basic-matrix-features">
<h2><span class="section-number">1.9. </span>Basic Matrix Features<a class="headerlink" href="#basic-matrix-features" title="Permalink to this headline"></a></h2>
<p>Matrix properties reminder</p>
<div class="math notranslate nohighlight">
\[\begin{split}
\mathbf{A} =
\begin{bmatrix} a_{11} &amp; a_{12} &amp; a_{13} &amp; a_{14} \\
a_{21} &amp; a_{22} &amp; a_{23} &amp; a_{24} \\
a_{31} &amp; a_{32} &amp; a_{33} &amp; a_{34} \\
a_{41} &amp; a_{42} &amp; a_{43} &amp; a_{44}
\end{bmatrix}\qquad
\mathbf{I} =
\begin{bmatrix} 1 &amp; 0 &amp; 0 &amp; 0 \\
0 &amp; 1 &amp; 0 &amp; 0 \\
0 &amp; 0 &amp; 1 &amp; 0 \\
0 &amp; 0 &amp; 0 &amp; 1
\end{bmatrix}
\end{split}\]</div>
<p>The inverse of a matrix is defined by</p>
<div class="math notranslate nohighlight">
\[
\mathbf{A}^{-1} \cdot \mathbf{A} = I
\]</div>
<table border="1">
<thead>
<tr><th align="center"> Relations </th> <th align="center"> Name </th> <th align="center"> matrix elements </th> </tr>
</thead>
<tbody>
<tr><td align="center"> $A = A^{T}$ </td> <td align="center"> symmetric </td> <td align="center"> $a_{ij} = a_{ji}$ </td> </tr>
<tr><td align="center"> $A = \left (A^{T} \right )^{-1}$ </td> <td align="center"> real orthogonal </td> <td align="center"> $\sum_k a_{ik} a_{jk} = \sum_k a_{ki} a_{kj} = \delta_{ij}$ </td> </tr>
<tr><td align="center"> $A = A^{ * }$ </td> <td align="center"> real matrix </td> <td align="center"> $a_{ij} = a_{ij}^{ * }$ </td> </tr>
<tr><td align="center"> $A = A^{\dagger}$ </td> <td align="center"> hermitian </td> <td align="center"> $a_{ij} = a_{ji}^{ * }$ </td> </tr>
<tr><td align="center"> $A = \left (A^{\dagger} \right )^{-1}$ </td> <td align="center"> unitary </td> <td align="center"> $\sum_k a_{ik} a_{jk}^{ * } = \sum_k a_{ki}^{ * } a_{kj} = \delta_{ij}$ </td> </tr>
</tbody>
</table>
<div class="section" id="some-famous-matrices">
<h3><span class="section-number">1.9.1. </span>Some famous Matrices<a class="headerlink" href="#some-famous-matrices" title="Permalink to this headline"></a></h3>
<ul class="simple">
<li><p>Diagonal if <span class="math notranslate nohighlight">\(a_{ij}=0\)</span> for <span class="math notranslate nohighlight">\(i\ne j\)</span></p></li>
<li><p>Upper triangular if <span class="math notranslate nohighlight">\(a_{ij}=0\)</span> for <span class="math notranslate nohighlight">\(i &gt; j\)</span></p></li>
<li><p>Lower triangular if <span class="math notranslate nohighlight">\(a_{ij}=0\)</span> for <span class="math notranslate nohighlight">\(i &lt; j\)</span></p></li>
<li><p>Upper Hessenberg if <span class="math notranslate nohighlight">\(a_{ij}=0\)</span> for <span class="math notranslate nohighlight">\(i &gt; j+1\)</span></p></li>
<li><p>Lower Hessenberg if <span class="math notranslate nohighlight">\(a_{ij}=0\)</span> for <span class="math notranslate nohighlight">\(i &lt; j+1\)</span></p></li>
<li><p>Tridiagonal if <span class="math notranslate nohighlight">\(a_{ij}=0\)</span> for <span class="math notranslate nohighlight">\(|i -j| &gt; 1\)</span></p></li>
<li><p>Lower banded with bandwidth <span class="math notranslate nohighlight">\(p\)</span>: <span class="math notranslate nohighlight">\(a_{ij}=0\)</span> for <span class="math notranslate nohighlight">\(i &gt; j+p\)</span></p></li>
<li><p>Upper banded with bandwidth <span class="math notranslate nohighlight">\(p\)</span>: <span class="math notranslate nohighlight">\(a_{ij}=0\)</span> for <span class="math notranslate nohighlight">\(i &lt; j+p\)</span></p></li>
<li><p>Banded, block upper triangular, block lower triangular….</p></li>
</ul>
</div>
<div class="section" id="more-basic-matrix-features">
<h3><span class="section-number">1.9.2. </span>More Basic Matrix Features<a class="headerlink" href="#more-basic-matrix-features" title="Permalink to this headline"></a></h3>
<p>Some Equivalent Statements
For an <span class="math notranslate nohighlight">\(N\times N\)</span> matrix <span class="math notranslate nohighlight">\(\mathbf{A}\)</span> the following properties are all equivalent</p>
<ul class="simple">
<li><p>If the inverse of <span class="math notranslate nohighlight">\(\mathbf{A}\)</span> exists, <span class="math notranslate nohighlight">\(\mathbf{A}\)</span> is nonsingular.</p></li>
<li><p>The equation <span class="math notranslate nohighlight">\(\mathbf{Ax}=0\)</span> implies <span class="math notranslate nohighlight">\(\mathbf{x}=0\)</span>.</p></li>
<li><p>The rows of <span class="math notranslate nohighlight">\(\mathbf{A}\)</span> form a basis of <span class="math notranslate nohighlight">\(R^N\)</span>.</p></li>
<li><p>The columns of <span class="math notranslate nohighlight">\(\mathbf{A}\)</span> form a basis of <span class="math notranslate nohighlight">\(R^N\)</span>.</p></li>
<li><p><span class="math notranslate nohighlight">\(\mathbf{A}\)</span> is a product of elementary matrices.</p></li>
<li><p><span class="math notranslate nohighlight">\(0\)</span> is not eigenvalue of <span class="math notranslate nohighlight">\(\mathbf{A}\)</span>.</p></li>
</ul>
</div>
</div>
<div class="section" id="numpy-and-arrays">
<h2><span class="section-number">1.10. </span>Numpy and arrays<a class="headerlink" href="#numpy-and-arrays" title="Permalink to this headline"></a></h2>
<p><a class="reference external" href="http://www.numpy.org/">Numpy</a> provides an easy way to handle arrays in Python. The standard way to import this library is as</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
</pre></div>
</div>
</div>
</div>
<p>Here follows a simple example where we set up an array of ten elements, all determined by random numbers drawn according to the normal distribution,</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">n</span> <span class="o">=</span> <span class="mi">10</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="n">n</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[ 1.01043351 0.90745348 0.55248701 -0.3588324 0.26881845 0.56326636
0.10659197 -0.97644907 0.16328353 -2.30075596]
</pre></div>
</div>
</div>
</div>
<p>We defined a vector <span class="math notranslate nohighlight">\(x\)</span> with <span class="math notranslate nohighlight">\(n=10\)</span> elements with its values given by the Normal distribution <span class="math notranslate nohighlight">\(N(0,1)\)</span>.
Another alternative is to declare a vector as follows</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>
<span class="nb">print</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[1 2 3]
</pre></div>
</div>
</div>
</div>
<p>Here we have defined a vector with three elements, with <span class="math notranslate nohighlight">\(x_0=1\)</span>, <span class="math notranslate nohighlight">\(x_1=2\)</span> and <span class="math notranslate nohighlight">\(x_2=3\)</span>. Note that both Python and C++
start numbering array elements from <span class="math notranslate nohighlight">\(0\)</span> and on. This means that a vector with <span class="math notranslate nohighlight">\(n\)</span> elements has a sequence of entities <span class="math notranslate nohighlight">\(x_0, x_1, x_2, \dots, x_{n-1}\)</span>. We could also let (recommended) Numpy to compute the logarithms of a specific array as</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">4</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">]))</span>
<span class="nb">print</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[1.38629436 1.94591015 2.07944154]
</pre></div>
</div>
</div>
</div>
<p>In the last example we used Numpy’s unary function <span class="math notranslate nohighlight">\(np.log\)</span>. This function is
highly tuned to compute array elements since the code is vectorized
and does not require looping. We normaly recommend that you use the
Numpy intrinsic functions instead of the corresponding <strong>log</strong> function
from Python’s <strong>math</strong> module. The looping is done explicitely by the
<strong>np.log</strong> function. The alternative, and slower way to compute the
logarithms of a vector would be to write</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">from</span> <span class="nn">math</span> <span class="kn">import</span> <span class="n">log</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">4</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">])</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">)):</span>
<span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
<span class="nb">print</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[1 1 2]
</pre></div>
</div>
</div>
</div>
<p>We note that our code is much longer already and we need to import the <strong>log</strong> function from the <strong>math</strong> module.
The attentive reader will also notice that the output is <span class="math notranslate nohighlight">\([1, 1, 2]\)</span>. Python interprets automagically our numbers as integers (like the <strong>automatic</strong> keyword in C++). To change this we could define our array elements to be double precision numbers as</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">4</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">],</span> <span class="n">dtype</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">float64</span><span class="p">))</span>
<span class="nb">print</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[1.38629436 1.94591015 2.07944154]
</pre></div>
</div>
</div>
</div>
<p>or simply write them as double precision numbers (Python uses 64 bits as default for floating point type variables), that is</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">4.0</span><span class="p">,</span> <span class="mf">7.0</span><span class="p">,</span> <span class="mf">8.0</span><span class="p">])</span>
<span class="nb">print</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output traceback highlight-ipythontb notranslate"><div class="highlight"><pre><span></span><span class="gt"> File</span><span class="nn"> &quot;&lt;ipython-input-7-f6d7a289d493&gt;&quot;</span><span class="gt">, line </span><span class="mi">3</span>
<span class="nb">print</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="o">^</span>
<span class="ne">SyntaxError</span>: invalid syntax
</pre></div>
</div>
</div>
</div>
<p>To check the number of bytes (remember that one byte contains eight bits for double precision variables), you can use simple use the <strong>itemsize</strong> functionality (the array <span class="math notranslate nohighlight">\(x\)</span> is actually an object which inherits the functionalities defined in Numpy) as</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">4.0</span><span class="p">,</span> <span class="mf">7.0</span><span class="p">,</span> <span class="mf">8.0</span><span class="p">])</span>
<span class="nb">print</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">itemsize</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="matrices-in-python">
<h2><span class="section-number">1.11. </span>Matrices in Python<a class="headerlink" href="#matrices-in-python" title="Permalink to this headline"></a></h2>
<p>Having defined vectors, we are now ready to try out matrices. We can
define a <span class="math notranslate nohighlight">\(3 \times 3 \)</span> real matrix <span class="math notranslate nohighlight">\(\hat{A}\)</span> as (recall that we user
lowercase letters for vectors and uppercase letters for matrices)</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="n">A</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span> <span class="p">[</span><span class="mf">4.0</span><span class="p">,</span> <span class="mf">7.0</span><span class="p">,</span> <span class="mf">8.0</span><span class="p">],</span> <span class="p">[</span><span class="mf">3.0</span><span class="p">,</span> <span class="mf">10.0</span><span class="p">,</span> <span class="mf">11.0</span><span class="p">],</span> <span class="p">[</span><span class="mf">4.0</span><span class="p">,</span> <span class="mf">5.0</span><span class="p">,</span> <span class="mf">7.0</span><span class="p">]</span> <span class="p">]))</span>
<span class="nb">print</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
<p>If we use the <strong>shape</strong> function we would get <span class="math notranslate nohighlight">\((3, 3)\)</span> as output, that is verifying that our matrix is a <span class="math notranslate nohighlight">\(3\times 3\)</span> matrix. We can slice the matrix and print for example the first column (Python organized matrix elements in a row-major order, see below) as</p>
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<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="n">A</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span> <span class="p">[</span><span class="mf">4.0</span><span class="p">,</span> <span class="mf">7.0</span><span class="p">,</span> <span class="mf">8.0</span><span class="p">],</span> <span class="p">[</span><span class="mf">3.0</span><span class="p">,</span> <span class="mf">10.0</span><span class="p">,</span> <span class="mf">11.0</span><span class="p">],</span> <span class="p">[</span><span class="mf">4.0</span><span class="p">,</span> <span class="mf">5.0</span><span class="p">,</span> <span class="mf">7.0</span><span class="p">]</span> <span class="p">]))</span>
<span class="c1"># print the first column, row-major order and elements start with 0</span>
<span class="nb">print</span><span class="p">(</span><span class="n">A</span><span class="p">[:,</span><span class="mi">0</span><span class="p">])</span>
</pre></div>
</div>
</div>
</div>
<p>We can continue this was by printing out other columns or rows. The example here prints out the second column</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="n">A</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span> <span class="p">[</span><span class="mf">4.0</span><span class="p">,</span> <span class="mf">7.0</span><span class="p">,</span> <span class="mf">8.0</span><span class="p">],</span> <span class="p">[</span><span class="mf">3.0</span><span class="p">,</span> <span class="mf">10.0</span><span class="p">,</span> <span class="mf">11.0</span><span class="p">],</span> <span class="p">[</span><span class="mf">4.0</span><span class="p">,</span> <span class="mf">5.0</span><span class="p">,</span> <span class="mf">7.0</span><span class="p">]</span> <span class="p">]))</span>
<span class="c1"># print the first column, row-major order and elements start with 0</span>
<span class="nb">print</span><span class="p">(</span><span class="n">A</span><span class="p">[</span><span class="mi">1</span><span class="p">,:])</span>
</pre></div>
</div>
</div>
</div>
<p>Numpy contains many other functionalities that allow us to slice, subdivide etc etc arrays. We strongly recommend that you look up the <a class="reference external" href="http://www.numpy.org/">Numpy website for more details</a>. Useful functions when defining a matrix are the <strong>np.zeros</strong> function which declares a matrix of a given dimension and sets all elements to zero</p>
<div class="cell docutils container">
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="n">n</span> <span class="o">=</span> <span class="mi">10</span>
<span class="c1"># define a matrix of dimension 10 x 10 and set all elements to zero</span>
<span class="n">A</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
<p>or initializing all elements to</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="n">n</span> <span class="o">=</span> <span class="mi">10</span>
<span class="c1"># define a matrix of dimension 10 x 10 and set all elements to one</span>
<span class="n">A</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
<p>or as unitarily distributed random numbers (see the material on random number generators in the statistics part)</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="n">n</span> <span class="o">=</span> <span class="mi">10</span>
<span class="c1"># define a matrix of dimension 10 x 10 and set all elements to random numbers with x \in [0, 1]</span>
<span class="n">A</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
<p>As we will see throughout these lectures, there are several extremely useful functionalities in Numpy.
As an example, consider the discussion of the covariance matrix. Suppose we have defined three vectors
<span class="math notranslate nohighlight">\(\hat{x}, \hat{y}, \hat{z}\)</span> with <span class="math notranslate nohighlight">\(n\)</span> elements each. The covariance matrix is defined as</p>
<div class="math notranslate nohighlight">
\[\begin{split}
\hat{\Sigma} = \begin{bmatrix} \sigma_{xx} &amp; \sigma_{xy} &amp; \sigma_{xz} \\
\sigma_{yx} &amp; \sigma_{yy} &amp; \sigma_{yz} \\
\sigma_{zx} &amp; \sigma_{zy} &amp; \sigma_{zz}
\end{bmatrix},
\end{split}\]</div>
<p>where for example</p>
<div class="math notranslate nohighlight">
\[
\sigma_{xy} =\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})(y_i- \overline{y}).
\]</div>
<p>The Numpy function <strong>np.cov</strong> calculates the covariance elements using the factor <span class="math notranslate nohighlight">\(1/(n-1)\)</span> instead of <span class="math notranslate nohighlight">\(1/n\)</span> since it assumes we do not have the exact mean values.
The following simple function uses the <strong>np.vstack</strong> function which takes each vector of dimension <span class="math notranslate nohighlight">\(1\times n\)</span> and produces a <span class="math notranslate nohighlight">\(3\times n\)</span> matrix <span class="math notranslate nohighlight">\(\hat{W}\)</span></p>
<div class="math notranslate nohighlight">
\[\begin{split}
\hat{W} = \begin{bmatrix} x_0 &amp; y_0 &amp; z_0 \\
x_1 &amp; y_1 &amp; z_1 \\
x_2 &amp; y_2 &amp; z_2 \\
\dots &amp; \dots &amp; \dots \\
x_{n-2} &amp; y_{n-2} &amp; z_{n-2} \\
x_{n-1} &amp; y_{n-1} &amp; z_{n-1}
\end{bmatrix},
\end{split}\]</div>
<p>which in turn is converted into into the <span class="math notranslate nohighlight">\(3\times 3\)</span> covariance matrix
<span class="math notranslate nohighlight">\(\hat{\Sigma}\)</span> via the Numpy function <strong>np.cov()</strong>. We note that we can also calculate
the mean value of each set of samples <span class="math notranslate nohighlight">\(\hat{x}\)</span> etc using the Numpy
function <strong>np.mean(x)</strong>. We can also extract the eigenvalues of the
covariance matrix through the <strong>np.linalg.eig()</strong> function.</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Importing various packages</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="n">n</span> <span class="o">=</span> <span class="mi">100</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="n">n</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
<span class="n">y</span> <span class="o">=</span> <span class="mi">4</span><span class="o">+</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="o">+</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="n">n</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">y</span><span class="p">))</span>
<span class="n">z</span> <span class="o">=</span> <span class="n">x</span><span class="o">**</span><span class="mi">3</span><span class="o">+</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="n">n</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">z</span><span class="p">))</span>
<span class="n">W</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">vstack</span><span class="p">((</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span><span class="p">))</span>
<span class="n">Sigma</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">cov</span><span class="p">(</span><span class="n">W</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">Sigma</span><span class="p">)</span>
<span class="n">Eigvals</span><span class="p">,</span> <span class="n">Eigvecs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">eig</span><span class="p">(</span><span class="n">Sigma</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">Eigvals</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="o">%</span><span class="k">matplotlib</span> inline
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">from</span> <span class="nn">scipy</span> <span class="kn">import</span> <span class="n">sparse</span>
<span class="n">eye</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">eye</span><span class="p">)</span>
<span class="n">sparse_mtx</span> <span class="o">=</span> <span class="n">sparse</span><span class="o">.</span><span class="n">csr_matrix</span><span class="p">(</span><span class="n">eye</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">sparse_mtx</span><span class="p">)</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">,</span><span class="mi">100</span><span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span><span class="n">marker</span><span class="o">=</span><span class="s1">&#39;x&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="meet-the-pandas">
<h2><span class="section-number">1.12. </span>Meet the Pandas<a class="headerlink" href="#meet-the-pandas" title="Permalink to this headline"></a></h2>
<!-- FIGURE: [fig/pandas.jpg, width=600 frac=0.8] -->
<p>Another useful Python package is
<a class="reference external" href="https://pandas.pydata.org/">pandas</a>, which is an open source library
providing high-performance, easy-to-use data structures and data
analysis tools for Python. <strong>pandas</strong> stands for panel data, a term borrowed from econometrics and is an efficient library for data analysis with an emphasis on tabular data.
<strong>pandas</strong> has two major classes, the <strong>DataFrame</strong> class with two-dimensional data objects and tabular data organized in columns and the class <strong>Series</strong> with a focus on one-dimensional data objects. Both classes allow you to index data easily as we will see in the examples below.
<strong>pandas</strong> allows you also to perform mathematical operations on the data, spanning from simple reshapings of vectors and matrices to statistical operations.</p>
<p>The following simple example shows how we can, in an easy way make tables of our data. Here we define a data set which includes names, place of birth and date of birth, and displays the data in an easy to read way. We will see repeated use of <strong>pandas</strong>, in particular in connection with classification of data.</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
<span class="kn">from</span> <span class="nn">IPython.display</span> <span class="kn">import</span> <span class="n">display</span>
<span class="n">data</span> <span class="o">=</span> <span class="p">{</span><span class="s1">&#39;First Name&#39;</span><span class="p">:</span> <span class="p">[</span><span class="s2">&quot;Frodo&quot;</span><span class="p">,</span> <span class="s2">&quot;Bilbo&quot;</span><span class="p">,</span> <span class="s2">&quot;Aragorn II&quot;</span><span class="p">,</span> <span class="s2">&quot;Samwise&quot;</span><span class="p">],</span>
<span class="s1">&#39;Last Name&#39;</span><span class="p">:</span> <span class="p">[</span><span class="s2">&quot;Baggins&quot;</span><span class="p">,</span> <span class="s2">&quot;Baggins&quot;</span><span class="p">,</span><span class="s2">&quot;Elessar&quot;</span><span class="p">,</span><span class="s2">&quot;Gamgee&quot;</span><span class="p">],</span>
<span class="s1">&#39;Place of birth&#39;</span><span class="p">:</span> <span class="p">[</span><span class="s2">&quot;Shire&quot;</span><span class="p">,</span> <span class="s2">&quot;Shire&quot;</span><span class="p">,</span> <span class="s2">&quot;Eriador&quot;</span><span class="p">,</span> <span class="s2">&quot;Shire&quot;</span><span class="p">],</span>
<span class="s1">&#39;Date of Birth T.A.&#39;</span><span class="p">:</span> <span class="p">[</span><span class="mi">2968</span><span class="p">,</span> <span class="mi">2890</span><span class="p">,</span> <span class="mi">2931</span><span class="p">,</span> <span class="mi">2980</span><span class="p">]</span>
<span class="p">}</span>
<span class="n">data_pandas</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">data</span><span class="p">)</span>
<span class="n">display</span><span class="p">(</span><span class="n">data_pandas</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
<p>In the above we have imported <strong>pandas</strong> with the shorthand <strong>pd</strong>, the latter has become the standard way we import <strong>pandas</strong>. We make then a list of various variables
and reorganize the aboves lists into a <strong>DataFrame</strong> and then print out a neat table with specific column labels as <em>Name</em>, <em>place of birth</em> and <em>date of birth</em>.
Displaying these results, we see that the indices are given by the default numbers from zero to three.
<strong>pandas</strong> is extremely flexible and we can easily change the above indices by defining a new type of indexing as</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">data_pandas</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">data</span><span class="p">,</span><span class="n">index</span><span class="o">=</span><span class="p">[</span><span class="s1">&#39;Frodo&#39;</span><span class="p">,</span><span class="s1">&#39;Bilbo&#39;</span><span class="p">,</span><span class="s1">&#39;Aragorn&#39;</span><span class="p">,</span><span class="s1">&#39;Sam&#39;</span><span class="p">])</span>
<span class="n">display</span><span class="p">(</span><span class="n">data_pandas</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
<p>Thereafter we display the content of the row which begins with the index <strong>Aragorn</strong></p>
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<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">display</span><span class="p">(</span><span class="n">data_pandas</span><span class="o">.</span><span class="n">loc</span><span class="p">[</span><span class="s1">&#39;Aragorn&#39;</span><span class="p">])</span>
</pre></div>
</div>
</div>
</div>
<p>We can easily append data to this, for example</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">new_hobbit</span> <span class="o">=</span> <span class="p">{</span><span class="s1">&#39;First Name&#39;</span><span class="p">:</span> <span class="p">[</span><span class="s2">&quot;Peregrin&quot;</span><span class="p">],</span>
<span class="s1">&#39;Last Name&#39;</span><span class="p">:</span> <span class="p">[</span><span class="s2">&quot;Took&quot;</span><span class="p">],</span>
<span class="s1">&#39;Place of birth&#39;</span><span class="p">:</span> <span class="p">[</span><span class="s2">&quot;Shire&quot;</span><span class="p">],</span>
<span class="s1">&#39;Date of Birth T.A.&#39;</span><span class="p">:</span> <span class="p">[</span><span class="mi">2990</span><span class="p">]</span>
<span class="p">}</span>
<span class="n">data_pandas</span><span class="o">=</span><span class="n">data_pandas</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">new_hobbit</span><span class="p">,</span> <span class="n">index</span><span class="o">=</span><span class="p">[</span><span class="s1">&#39;Pippin&#39;</span><span class="p">]))</span>
<span class="n">display</span><span class="p">(</span><span class="n">data_pandas</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
<p>Here are other examples where we use the <strong>DataFrame</strong> functionality to handle arrays, now with more interesting features for us, namely numbers. We set up a matrix
of dimensionality <span class="math notranslate nohighlight">\(10\times 5\)</span> and compute the mean value and standard deviation of each column. Similarly, we can perform mathematial operations like squaring the matrix elements and many other operations.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
<span class="kn">from</span> <span class="nn">IPython.display</span> <span class="kn">import</span> <span class="n">display</span>
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">100</span><span class="p">)</span>
<span class="c1"># setting up a 10 x 5 matrix</span>
<span class="n">rows</span> <span class="o">=</span> <span class="mi">10</span>
<span class="n">cols</span> <span class="o">=</span> <span class="mi">5</span>
<span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">rows</span><span class="p">,</span><span class="n">cols</span><span class="p">)</span>
<span class="n">df</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="n">display</span><span class="p">(</span><span class="n">df</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">mean</span><span class="p">())</span>
<span class="nb">print</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">std</span><span class="p">())</span>
<span class="n">display</span><span class="p">(</span><span class="n">df</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
<p>Thereafter we can select specific columns only and plot final results</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">df</span><span class="o">.</span><span class="n">columns</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;First&#39;</span><span class="p">,</span> <span class="s1">&#39;Second&#39;</span><span class="p">,</span> <span class="s1">&#39;Third&#39;</span><span class="p">,</span> <span class="s1">&#39;Fourth&#39;</span><span class="p">,</span> <span class="s1">&#39;Fifth&#39;</span><span class="p">]</span>
<span class="n">df</span><span class="o">.</span><span class="n">index</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
<span class="n">display</span><span class="p">(</span><span class="n">df</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="s1">&#39;Second&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span> <span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">info</span><span class="p">())</span>
<span class="nb">print</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">describe</span><span class="p">())</span>
<span class="kn">from</span> <span class="nn">pylab</span> <span class="kn">import</span> <span class="n">plt</span><span class="p">,</span> <span class="n">mpl</span>
<span class="n">plt</span><span class="o">.</span><span class="n">style</span><span class="o">.</span><span class="n">use</span><span class="p">(</span><span class="s1">&#39;seaborn&#39;</span><span class="p">)</span>
<span class="n">mpl</span><span class="o">.</span><span class="n">rcParams</span><span class="p">[</span><span class="s1">&#39;font.family&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="s1">&#39;serif&#39;</span>
<span class="n">df</span><span class="o">.</span><span class="n">cumsum</span><span class="p">()</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">lw</span><span class="o">=</span><span class="mf">2.0</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">6</span><span class="p">))</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
<span class="n">df</span><span class="o">.</span><span class="n">plot</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">6</span><span class="p">),</span> <span class="n">rot</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
</div>
</div>
<p>We can produce a <span class="math notranslate nohighlight">\(4\times 4\)</span> matrix</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">b</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">16</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">((</span><span class="mi">4</span><span class="p">,</span><span class="mi">4</span><span class="p">))</span>
<span class="nb">print</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
<span class="n">df1</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">df1</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
<p>and many other operations.</p>
<p>The <strong>Series</strong> class is another important class included in
<strong>pandas</strong>. You can view it as a specialization of <strong>DataFrame</strong> but where
we have just a single column of data. It shares many of the same features as _DataFrame. As with <strong>DataFrame</strong>,
most operations are vectorized, achieving thereby a high performance when dealing with computations of arrays, in particular labeled arrays.
As we will see below it leads also to a very concice code close to the mathematical operations we may be interested in.
For multidimensional arrays, we recommend strongly <a class="reference external" href="http://xarray.pydata.org/en/stable/">xarray</a>. <strong>xarray</strong> has much of the same flexibility as <strong>pandas</strong>, but allows for the extension to higher dimensions than two. We will see examples later of the usage of both <strong>pandas</strong> and <strong>xarray</strong>.</p>
<p>In order to study various Machine Learning algorithms, we need to
access data. Acccessing data is an essential step in all machine
learning algorithms. In particular, setting up the so-called <strong>design
matrix</strong> (to be defined below) is often the first element we need in
order to perform our calculations. To set up the design matrix means
reading (and later, when the calculations are done, writing) data
in various formats, The formats span from reading files from disk,
loading data from databases and interacting with online sources
like web application programming interfaces (APIs).</p>
<p>In handling various input formats, as discussed above, we will mainly stay with <strong>pandas</strong>,
a Python package which allows us, in a seamless and painless way, to
deal with a multitude of formats, from standard <strong>csv</strong> (comma separated
values) files, via <strong>excel</strong>, <strong>html</strong> to <strong>hdf5</strong> formats. With <strong>pandas</strong>
and the <strong>DataFrame</strong> and <strong>Series</strong> functionalities we are able to convert text data
into the calculational formats we need for a specific algorithm. And our code is going to be
pretty close the basic mathematical expressions.</p>
<p>Our first data set is going to be a classic from nuclear physics, namely all
available data on binding energies. Don’t be intimidated if you are not familiar with nuclear physics. It serves simply as an example here of a data set.</p>
<p>We will show some of the
strengths of packages like <strong>Scikit-Learn</strong> in fitting nuclear binding energies to
specific functions using linear regression first. Then, as a teaser, we will show you how
you can easily implement other algorithms like decision trees and random forests and neural networks.</p>
<p>But before we really start with nuclear physics data, let’s just look at some simpler polynomial fitting cases, such as,
(don’t be offended) fitting straight lines!</p>
</div>
<div class="section" id="simple-linear-regression-model-using-scikit-learn">
<h2><span class="section-number">1.13. </span>Simple linear regression model using <strong>scikit-learn</strong><a class="headerlink" href="#simple-linear-regression-model-using-scikit-learn" title="Permalink to this headline"></a></h2>
<p>We start with perhaps our simplest possible example, using <strong>Scikit-Learn</strong> to perform linear regression analysis on a data set produced by us.</p>
<p>What follows is a simple Python code where we have defined a function
<span class="math notranslate nohighlight">\(y\)</span> in terms of the variable <span class="math notranslate nohighlight">\(x\)</span>. Both are defined as vectors with <span class="math notranslate nohighlight">\(100\)</span> entries.
The numbers in the vector <span class="math notranslate nohighlight">\(\hat{x}\)</span> are given
by random numbers generated with a uniform distribution with entries
<span class="math notranslate nohighlight">\(x_i \in [0,1]\)</span> (more about probability distribution functions
later). These values are then used to define a function <span class="math notranslate nohighlight">\(y(x)\)</span>
(tabulated again as a vector) with a linear dependence on <span class="math notranslate nohighlight">\(x\)</span> plus a
random noise added via the normal distribution.</p>
<p>The Numpy functions are imported used the <strong>import numpy as np</strong>
statement and the random number generator for the uniform distribution
is called using the function <strong>np.random.rand()</strong>, where we specificy
that we want <span class="math notranslate nohighlight">\(100\)</span> random variables. Using Numpy we define
automatically an array with the specified number of elements, <span class="math notranslate nohighlight">\(100\)</span> in
our case. With the Numpy function <strong>randn()</strong> we can compute random
numbers with the normal distribution (mean value <span class="math notranslate nohighlight">\(\mu\)</span> equal to zero and
variance <span class="math notranslate nohighlight">\(\sigma^2\)</span> set to one) and produce the values of <span class="math notranslate nohighlight">\(y\)</span> assuming a linear
dependence as function of <span class="math notranslate nohighlight">\(x\)</span></p>
<div class="math notranslate nohighlight">
\[
y = 2x+N(0,1),
\]</div>
<p>where <span class="math notranslate nohighlight">\(N(0,1)\)</span> represents random numbers generated by the normal
distribution. From <strong>Scikit-Learn</strong> we import then the
<strong>LinearRegression</strong> functionality and make a prediction <span class="math notranslate nohighlight">\(\tilde{y} =
\alpha + \beta x\)</span> using the function <strong>fit(x,y)</strong>. We call the set of
data <span class="math notranslate nohighlight">\((\hat{x},\hat{y})\)</span> for our training data. The Python package
<strong>scikit-learn</strong> has also a functionality which extracts the above
fitting parameters <span class="math notranslate nohighlight">\(\alpha\)</span> and <span class="math notranslate nohighlight">\(\beta\)</span> (see below). Later we will
distinguish between training data and test data.</p>
<p>For plotting we use the Python package
<a class="reference external" href="https://matplotlib.org/">matplotlib</a> which produces publication
quality figures. Feel free to explore the extensive
<a class="reference external" href="https://matplotlib.org/gallery/index.html">gallery</a> of examples. In
this example we plot our original values of <span class="math notranslate nohighlight">\(x\)</span> and <span class="math notranslate nohighlight">\(y\)</span> as well as the
prediction <strong>ypredict</strong> (<span class="math notranslate nohighlight">\(\tilde{y}\)</span>), which attempts at fitting our
data with a straight line.</p>
<p>The Python code follows here.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Importing various packages</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <span class="n">LinearRegression</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">100</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">+</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">100</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="n">linreg</span> <span class="o">=</span> <span class="n">LinearRegression</span><span class="p">()</span>
<span class="n">linreg</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">)</span>
<span class="n">xnew</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">0</span><span class="p">],[</span><span class="mi">1</span><span class="p">]])</span>
<span class="n">ypredict</span> <span class="o">=</span> <span class="n">linreg</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">xnew</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xnew</span><span class="p">,</span> <span class="n">ypredict</span><span class="p">,</span> <span class="s2">&quot;r-&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="p">,</span><span class="s1">&#39;ro&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mf">1.0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span> <span class="mf">5.0</span><span class="p">])</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">&#39;$x$&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">&#39;$y$&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="sa">r</span><span class="s1">&#39;Simple Linear Regression&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
</div>
</div>
<p>This example serves several aims. It allows us to demonstrate several
aspects of data analysis and later machine learning algorithms. The
immediate visualization shows that our linear fit is not
impressive. It goes through the data points, but there are many
outliers which are not reproduced by our linear regression. We could
now play around with this small program and change for example the
factor in front of <span class="math notranslate nohighlight">\(x\)</span> and the normal distribution. Try to change the
function <span class="math notranslate nohighlight">\(y\)</span> to</p>
<div class="math notranslate nohighlight">
\[
y = 10x+0.01 \times N(0,1),
\]</div>
<p>where <span class="math notranslate nohighlight">\(x\)</span> is defined as before. Does the fit look better? Indeed, by
reducing the role of the noise given by the normal distribution we see immediately that
our linear prediction seemingly reproduces better the training
set. However, this testing ‘by the eye’ is obviouly not satisfactory in the
long run. Here we have only defined the training data and our model, and
have not discussed a more rigorous approach to the <strong>cost</strong> function.</p>
<p>We need more rigorous criteria in defining whether we have succeeded or
not in modeling our training data. You will be surprised to see that
many scientists seldomly venture beyond this ‘by the eye’ approach. A
standard approach for the <em>cost</em> function is the so-called <span class="math notranslate nohighlight">\(\chi^2\)</span>
function (a variant of the mean-squared error (MSE))</p>
<div class="math notranslate nohighlight">
\[
\chi^2 = \frac{1}{n}
\sum_{i=0}^{n-1}\frac{(y_i-\tilde{y}_i)^2}{\sigma_i^2},
\]</div>
<p>where <span class="math notranslate nohighlight">\(\sigma_i^2\)</span> is the variance (to be defined later) of the entry
<span class="math notranslate nohighlight">\(y_i\)</span>. We may not know the explicit value of <span class="math notranslate nohighlight">\(\sigma_i^2\)</span>, it serves
however the aim of scaling the equations and make the cost function
dimensionless.</p>
<p>Minimizing the cost function is a central aspect of
our discussions to come. Finding its minima as function of the model
parameters (<span class="math notranslate nohighlight">\(\alpha\)</span> and <span class="math notranslate nohighlight">\(\beta\)</span> in our case) will be a recurring
theme in these series of lectures. Essentially all machine learning
algorithms we will discuss center around the minimization of the
chosen cost function. This depends in turn on our specific
model for describing the data, a typical situation in supervised
learning. Automatizing the search for the minima of the cost function is a
central ingredient in all algorithms. Typical methods which are
employed are various variants of <strong>gradient</strong> methods. These will be
discussed in more detail later. Again, you’ll be surprised to hear that
many practitioners minimize the above function ‘’by the eye’, popularly dubbed as
‘chi by the eye’. That is, change a parameter and see (visually and numerically) that
the <span class="math notranslate nohighlight">\(\chi^2\)</span> function becomes smaller.</p>
<p>There are many ways to define the cost function. A simpler approach is to look at the relative difference between the training data and the predicted data, that is we define
the relative error (why would we prefer the MSE instead of the relative error?) as</p>
<div class="math notranslate nohighlight">
\[
\epsilon_{\mathrm{relative}}= \frac{\vert \hat{y} -\hat{\tilde{y}}\vert}{\vert \hat{y}\vert}.
\]</div>
<p>The squared cost function results in an arithmetic mean-unbiased
estimator, and the absolute-value cost function results in a
median-unbiased estimator (in the one-dimensional case, and a
geometric median-unbiased estimator for the multi-dimensional
case). The squared cost function has the disadvantage that it has the tendency
to be dominated by outliers.</p>
<p>We can modify easily the above Python code and plot the relative error instead</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <span class="n">LinearRegression</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">100</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <span class="mi">5</span><span class="o">*</span><span class="n">x</span><span class="o">+</span><span class="mf">0.01</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">100</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="n">linreg</span> <span class="o">=</span> <span class="n">LinearRegression</span><span class="p">()</span>
<span class="n">linreg</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">)</span>
<span class="n">ypredict</span> <span class="o">=</span> <span class="n">linreg</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">ypredict</span><span class="o">-</span><span class="n">y</span><span class="p">)</span><span class="o">/</span><span class="nb">abs</span><span class="p">(</span><span class="n">y</span><span class="p">),</span> <span class="s2">&quot;ro&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mf">1.0</span><span class="p">,</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">])</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">&#39;$x$&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">&#39;$\epsilon_{\mathrm</span><span class="si">{relative}</span><span class="s1">}$&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="sa">r</span><span class="s1">&#39;Relative error&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
</div>
</div>
<p>Depending on the parameter in front of the normal distribution, we may
have a small or larger relative error. Try to play around with
different training data sets and study (graphically) the value of the
relative error.</p>
<p>As mentioned above, <strong>Scikit-Learn</strong> has an impressive functionality.
We can for example extract the values of <span class="math notranslate nohighlight">\(\alpha\)</span> and <span class="math notranslate nohighlight">\(\beta\)</span> and
their error estimates, or the variance and standard deviation and many
other properties from the statistical data analysis.</p>
<p>Here we show an
example of the functionality of <strong>Scikit-Learn</strong>.</p>
<div class="cell docutils container">
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <span class="n">LinearRegression</span>
<span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">mean_squared_error</span><span class="p">,</span> <span class="n">r2_score</span><span class="p">,</span> <span class="n">mean_squared_log_error</span><span class="p">,</span> <span class="n">mean_absolute_error</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">100</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <span class="mf">2.0</span><span class="o">+</span> <span class="mi">5</span><span class="o">*</span><span class="n">x</span><span class="o">+</span><span class="mf">0.5</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">100</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="n">linreg</span> <span class="o">=</span> <span class="n">LinearRegression</span><span class="p">()</span>
<span class="n">linreg</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">)</span>
<span class="n">ypredict</span> <span class="o">=</span> <span class="n">linreg</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;The intercept alpha: </span><span class="se">\n</span><span class="s1">&#39;</span><span class="p">,</span> <span class="n">linreg</span><span class="o">.</span><span class="n">intercept_</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Coefficient beta : </span><span class="se">\n</span><span class="s1">&#39;</span><span class="p">,</span> <span class="n">linreg</span><span class="o">.</span><span class="n">coef_</span><span class="p">)</span>
<span class="c1"># The mean squared error </span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Mean squared error: </span><span class="si">%.2f</span><span class="s2">&quot;</span> <span class="o">%</span> <span class="n">mean_squared_error</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">ypredict</span><span class="p">))</span>
<span class="c1"># Explained variance score: 1 is perfect prediction </span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Variance score: </span><span class="si">%.2f</span><span class="s1">&#39;</span> <span class="o">%</span> <span class="n">r2_score</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">ypredict</span><span class="p">))</span>
<span class="c1"># Mean squared log error </span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Mean squared log error: </span><span class="si">%.2f</span><span class="s1">&#39;</span> <span class="o">%</span> <span class="n">mean_squared_log_error</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">ypredict</span><span class="p">)</span> <span class="p">)</span>
<span class="c1"># Mean absolute error </span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Mean absolute error: </span><span class="si">%.2f</span><span class="s1">&#39;</span> <span class="o">%</span> <span class="n">mean_absolute_error</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">ypredict</span><span class="p">))</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">ypredict</span><span class="p">,</span> <span class="s2">&quot;r-&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="p">,</span><span class="s1">&#39;ro&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">([</span><span class="mf">0.0</span><span class="p">,</span><span class="mf">1.0</span><span class="p">,</span><span class="mf">1.5</span><span class="p">,</span> <span class="mf">7.0</span><span class="p">])</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">&#39;$x$&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">&#39;$y$&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="sa">r</span><span class="s1">&#39;Linear Regression fit &#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
</div>
</div>
<p>The function <strong>coef</strong> gives us the parameter <span class="math notranslate nohighlight">\(\beta\)</span> of our fit while <strong>intercept</strong> yields
<span class="math notranslate nohighlight">\(\alpha\)</span>. Depending on the constant in front of the normal distribution, we get values near or far from <span class="math notranslate nohighlight">\(alpha =2\)</span> and <span class="math notranslate nohighlight">\(\beta =5\)</span>. Try to play around with different parameters in front of the normal distribution. The function <strong>meansquarederror</strong> gives us the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error or loss defined as</p>
<div class="math notranslate nohighlight">
\[
MSE(\hat{y},\hat{\tilde{y}}) = \frac{1}{n}
\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2,
\]</div>
<p>The smaller the value, the better the fit. Ideally we would like to
have an MSE equal zero. The attentive reader has probably recognized
this function as being similar to the <span class="math notranslate nohighlight">\(\chi^2\)</span> function defined above.</p>
<p>The <strong>r2score</strong> function computes <span class="math notranslate nohighlight">\(R^2\)</span>, the coefficient of
determination. It provides a measure of how well future samples are
likely to be predicted by the model. Best possible score is 1.0 and it
can be negative (because the model can be arbitrarily worse). A
constant model that always predicts the expected value of <span class="math notranslate nohighlight">\(\hat{y}\)</span>,
disregarding the input features, would get a <span class="math notranslate nohighlight">\(R^2\)</span> score of <span class="math notranslate nohighlight">\(0.0\)</span>.</p>
<p>If <span class="math notranslate nohighlight">\(\tilde{\hat{y}}_i\)</span> is the predicted value of the <span class="math notranslate nohighlight">\(i-th\)</span> sample and <span class="math notranslate nohighlight">\(y_i\)</span> is the corresponding true value, then the score <span class="math notranslate nohighlight">\(R^2\)</span> is defined as</p>
<div class="math notranslate nohighlight">
\[
R^2(\hat{y}, \tilde{\hat{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2},
\]</div>
<p>where we have defined the mean value of <span class="math notranslate nohighlight">\(\hat{y}\)</span> as</p>
<div class="math notranslate nohighlight">
\[
\bar{y} = \frac{1}{n} \sum_{i=0}^{n - 1} y_i.
\]</div>
<p>Another quantity taht we will meet again in our discussions of regression analysis is
the mean absolute error (MAE), a risk metric corresponding to the expected value of the absolute error loss or what we call the <span class="math notranslate nohighlight">\(l1\)</span>-norm loss. In our discussion above we presented the relative error.
The MAE is defined as follows</p>
<div class="math notranslate nohighlight">
\[
\text{MAE}(\hat{y}, \hat{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n-1} \left| y_i - \tilde{y}_i \right|.
\]</div>
<p>We present the
squared logarithmic (quadratic) error</p>
<div class="math notranslate nohighlight">
\[
\text{MSLE}(\hat{y}, \hat{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n - 1} (\log_e (1 + y_i) - \log_e (1 + \tilde{y}_i) )^2,
\]</div>
<p>where <span class="math notranslate nohighlight">\(\log_e (x)\)</span> stands for the natural logarithm of <span class="math notranslate nohighlight">\(x\)</span>. This error
estimate is best to use when targets having exponential growth, such
as population counts, average sales of a commodity over a span of
years etc.</p>
<p>Finally, another cost function is the Huber cost function used in robust regression.</p>
<p>The rationale behind this possible cost function is its reduced
sensitivity to outliers in the data set. In our discussions on
dimensionality reduction and normalization of data we will meet other
ways of dealing with outliers.</p>
<p>The Huber cost function is defined as</p>
<div class="math notranslate nohighlight">
\[\begin{split}
H_{\delta}(a)={\begin{cases}{\frac {1}{2}}{a^{2}}&amp;{\text{for }}|a|\leq \delta ,\\\delta (|a|-{\frac {1}{2}}\delta ),&amp;{\text{otherwise.}}\end{cases}}}.
\end{split}\]</div>
<p>Here <span class="math notranslate nohighlight">\(a=\boldsymbol{y} - \boldsymbol{\tilde{y}}\)</span>.
We will discuss in more
detail these and other functions in the various lectures. We conclude this part with another example. Instead of
a linear <span class="math notranslate nohighlight">\(x\)</span>-dependence we study now a cubic polynomial and use the polynomial regression analysis tools of scikit-learn.</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">random</span>
<span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <span class="n">Ridge</span>
<span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">import</span> <span class="n">PolynomialFeatures</span>
<span class="kn">from</span> <span class="nn">sklearn.pipeline</span> <span class="kn">import</span> <span class="n">make_pipeline</span>
<span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <span class="n">LinearRegression</span>
<span class="n">x</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mf">0.02</span><span class="p">,</span><span class="mf">0.98</span><span class="p">,</span><span class="mi">200</span><span class="p">)</span>
<span class="n">noise</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="n">sample</span><span class="p">((</span><span class="nb">range</span><span class="p">(</span><span class="mi">200</span><span class="p">)),</span><span class="mi">200</span><span class="p">))</span>
<span class="n">y</span><span class="o">=</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span><span class="o">*</span><span class="n">noise</span>
<span class="n">yn</span><span class="o">=</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span><span class="o">*</span><span class="mi">100</span>
<span class="n">poly3</span> <span class="o">=</span> <span class="n">PolynomialFeatures</span><span class="p">(</span><span class="n">degree</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">poly3</span><span class="o">.</span><span class="n">fit_transform</span><span class="p">(</span><span class="n">x</span><span class="p">[:,</span><span class="n">np</span><span class="o">.</span><span class="n">newaxis</span><span class="p">])</span>
<span class="n">clf3</span> <span class="o">=</span> <span class="n">LinearRegression</span><span class="p">()</span>
<span class="n">clf3</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">y</span><span class="p">)</span>
<span class="n">Xplot</span><span class="o">=</span><span class="n">poly3</span><span class="o">.</span><span class="n">fit_transform</span><span class="p">(</span><span class="n">x</span><span class="p">[:,</span><span class="n">np</span><span class="o">.</span><span class="n">newaxis</span><span class="p">])</span>
<span class="n">poly3_plot</span><span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">clf3</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">Xplot</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Cubic Fit&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">yn</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;red&#39;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;True Cubic&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Data&#39;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;orange&#39;</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
<span class="k">def</span> <span class="nf">error</span><span class="p">(</span><span class="n">a</span><span class="p">):</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">y</span><span class="p">:</span>
<span class="n">err</span><span class="o">=</span><span class="p">(</span><span class="n">y</span><span class="o">-</span><span class="n">yn</span><span class="p">)</span><span class="o">/</span><span class="n">yn</span>
<span class="k">return</span> <span class="nb">abs</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">err</span><span class="p">))</span><span class="o">/</span><span class="nb">len</span><span class="p">(</span><span class="n">err</span><span class="p">)</span>
<span class="nb">print</span> <span class="p">(</span><span class="n">error</span><span class="p">(</span><span class="n">y</span><span class="p">))</span>
</pre></div>
</div>
</div>
</div>
<p>Let us now dive into nuclear physics and remind ourselves briefly about some basic features about binding
energies. A basic quantity which can be measured for the ground
states of nuclei is the atomic mass <span class="math notranslate nohighlight">\(M(N, Z)\)</span> of the neutral atom with
atomic mass number <span class="math notranslate nohighlight">\(A\)</span> and charge <span class="math notranslate nohighlight">\(Z\)</span>. The number of neutrons is <span class="math notranslate nohighlight">\(N\)</span>. There are indeed several sophisticated experiments worldwide which allow us to measure this quantity to high precision (parts per million even).</p>
<p>Atomic masses are usually tabulated in terms of the mass excess defined by</p>
<div class="math notranslate nohighlight">
\[
\Delta M(N, Z) = M(N, Z) - uA,
\]</div>
<p>where <span class="math notranslate nohighlight">\(u\)</span> is the Atomic Mass Unit</p>
<div class="math notranslate nohighlight">
\[
u = M(^{12}\mathrm{C})/12 = 931.4940954(57) \hspace{0.1cm} \mathrm{MeV}/c^2.
\]</div>
<p>The nucleon masses are</p>
<div class="math notranslate nohighlight">
\[
m_p = 1.00727646693(9)u,
\]</div>
<p>and</p>
<div class="math notranslate nohighlight">
\[
m_n = 939.56536(8)\hspace{0.1cm} \mathrm{MeV}/c^2 = 1.0086649156(6)u.
\]</div>
<p>In the <a class="reference external" href="http://nuclearmasses.org/resources_folder/Wang_2017_Chinese_Phys_C_41_030003.pdf">2016 mass evaluation of by W.J.Huang, G.Audi, M.Wang, F.G.Kondev, S.Naimi and X.Xu</a>
there are data on masses and decays of 3437 nuclei.</p>
<p>The nuclear binding energy is defined as the energy required to break
up a given nucleus into its constituent parts of <span class="math notranslate nohighlight">\(N\)</span> neutrons and <span class="math notranslate nohighlight">\(Z\)</span>
protons. In terms of the atomic masses <span class="math notranslate nohighlight">\(M(N, Z)\)</span> the binding energy is
defined by</p>
<div class="math notranslate nohighlight">
\[
BE(N, Z) = ZM_H c^2 + Nm_n c^2 - M(N, Z)c^2 ,
\]</div>
<p>where <span class="math notranslate nohighlight">\(M_H\)</span> is the mass of the hydrogen atom and <span class="math notranslate nohighlight">\(m_n\)</span> is the mass of the neutron.
In terms of the mass excess the binding energy is given by</p>
<div class="math notranslate nohighlight">
\[
BE(N, Z) = Z\Delta_H c^2 + N\Delta_n c^2 -\Delta(N, Z)c^2 ,
\]</div>
<p>where <span class="math notranslate nohighlight">\(\Delta_H c^2 = 7.2890\)</span> MeV and <span class="math notranslate nohighlight">\(\Delta_n c^2 = 8.0713\)</span> MeV.</p>
<p>A popular and physically intuitive model which can be used to parametrize
the experimental binding energies as function of <span class="math notranslate nohighlight">\(A\)</span>, is the so-called
<strong>liquid drop model</strong>. The ansatz is based on the following expression</p>
<div class="math notranslate nohighlight">
\[
BE(N,Z) = a_1A-a_2A^{2/3}-a_3\frac{Z^2}{A^{1/3}}-a_4\frac{(N-Z)^2}{A},
\]</div>
<p>where <span class="math notranslate nohighlight">\(A\)</span> stands for the number of nucleons and the <span class="math notranslate nohighlight">\(a_i\)</span>s are parameters which are determined by a fit
to the experimental data.</p>
<p>To arrive at the above expression we have assumed that we can make the following assumptions:</p>
<ul class="simple">
<li><p>There is a volume term <span class="math notranslate nohighlight">\(a_1A\)</span> proportional with the number of nucleons (the energy is also an extensive quantity). When an assembly of nucleons of the same size is packed together into the smallest volume, each interior nucleon has a certain number of other nucleons in contact with it. This contribution is proportional to the volume.</p></li>
<li><p>There is a surface energy term <span class="math notranslate nohighlight">\(a_2A^{2/3}\)</span>. The assumption here is that a nucleon at the surface of a nucleus interacts with fewer other nucleons than one in the interior of the nucleus and hence its binding energy is less. This surface energy term takes that into account and is therefore negative and is proportional to the surface area.</p></li>
<li><p>There is a Coulomb energy term <span class="math notranslate nohighlight">\(a_3\frac{Z^2}{A^{1/3}}\)</span>. The electric repulsion between each pair of protons in a nucleus yields less binding.</p></li>
<li><p>There is an asymmetry term <span class="math notranslate nohighlight">\(a_4\frac{(N-Z)^2}{A}\)</span>. This term is associated with the Pauli exclusion principle and reflects the fact that the proton-neutron interaction is more attractive on the average than the neutron-neutron and proton-proton interactions.</p></li>
</ul>
<p>We could also add a so-called pairing term, which is a correction term that
arises from the tendency of proton pairs and neutron pairs to
occur. An even number of particles is more stable than an odd number.</p>
<div class="section" id="organizing-our-data">
<h3><span class="section-number">1.13.1. </span>Organizing our data<a class="headerlink" href="#organizing-our-data" title="Permalink to this headline"></a></h3>
<p>Let us start with reading and organizing our data.
We start with the compilation of masses and binding energies from 2016.
After having downloaded this file to our own computer, we are now ready to read the file and start structuring our data.</p>
<p>We start with preparing folders for storing our calculations and the data file over masses and binding energies. We import also various modules that we will find useful in order to present various Machine Learning methods. Here we focus mainly on the functionality of <strong>scikit-learn</strong>.</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Common imports</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">import</span> <span class="nn">sklearn.linear_model</span> <span class="k">as</span> <span class="nn">skl</span>
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
<span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">mean_squared_error</span><span class="p">,</span> <span class="n">r2_score</span><span class="p">,</span> <span class="n">mean_absolute_error</span>
<span class="kn">import</span> <span class="nn">os</span>
<span class="c1"># Where to save the figures and data files</span>
<span class="n">PROJECT_ROOT_DIR</span> <span class="o">=</span> <span class="s2">&quot;Results&quot;</span>
<span class="n">FIGURE_ID</span> <span class="o">=</span> <span class="s2">&quot;Results/FigureFiles&quot;</span>
<span class="n">DATA_ID</span> <span class="o">=</span> <span class="s2">&quot;DataFiles/&quot;</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="n">PROJECT_ROOT_DIR</span><span class="p">):</span>
<span class="n">os</span><span class="o">.</span><span class="n">mkdir</span><span class="p">(</span><span class="n">PROJECT_ROOT_DIR</span><span class="p">)</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="n">FIGURE_ID</span><span class="p">):</span>
<span class="n">os</span><span class="o">.</span><span class="n">makedirs</span><span class="p">(</span><span class="n">FIGURE_ID</span><span class="p">)</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="n">DATA_ID</span><span class="p">):</span>
<span class="n">os</span><span class="o">.</span><span class="n">makedirs</span><span class="p">(</span><span class="n">DATA_ID</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">image_path</span><span class="p">(</span><span class="n">fig_id</span><span class="p">):</span>
<span class="k">return</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">FIGURE_ID</span><span class="p">,</span> <span class="n">fig_id</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">data_path</span><span class="p">(</span><span class="n">dat_id</span><span class="p">):</span>
<span class="k">return</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">DATA_ID</span><span class="p">,</span> <span class="n">dat_id</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">save_fig</span><span class="p">(</span><span class="n">fig_id</span><span class="p">):</span>
<span class="n">plt</span><span class="o">.</span><span class="n">savefig</span><span class="p">(</span><span class="n">image_path</span><span class="p">(</span><span class="n">fig_id</span><span class="p">)</span> <span class="o">+</span> <span class="s2">&quot;.png&quot;</span><span class="p">,</span> <span class="nb">format</span><span class="o">=</span><span class="s1">&#39;png&#39;</span><span class="p">)</span>
<span class="n">infile</span> <span class="o">=</span> <span class="nb">open</span><span class="p">(</span><span class="n">data_path</span><span class="p">(</span><span class="s2">&quot;MassEval2016.dat&quot;</span><span class="p">),</span><span class="s1">&#39;r&#39;</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
<p>Before we proceed, we define also a function for making our plots. You can obviously avoid this and simply set up various <strong>matplotlib</strong> commands every time you need them. You may however find it convenient to collect all such commands in one function and simply call this function.</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">pylab</span> <span class="kn">import</span> <span class="n">plt</span><span class="p">,</span> <span class="n">mpl</span>
<span class="n">plt</span><span class="o">.</span><span class="n">style</span><span class="o">.</span><span class="n">use</span><span class="p">(</span><span class="s1">&#39;seaborn&#39;</span><span class="p">)</span>
<span class="n">mpl</span><span class="o">.</span><span class="n">rcParams</span><span class="p">[</span><span class="s1">&#39;font.family&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="s1">&#39;serif&#39;</span>
<span class="k">def</span> <span class="nf">MakePlot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span> <span class="n">styles</span><span class="p">,</span> <span class="n">labels</span><span class="p">,</span> <span class="n">axlabels</span><span class="p">):</span>
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">6</span><span class="p">))</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">)):</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">y</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">styles</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">label</span> <span class="o">=</span> <span class="n">labels</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="n">axlabels</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="n">axlabels</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
<p>Our next step is to read the data on experimental binding energies and
reorganize them as functions of the mass number <span class="math notranslate nohighlight">\(A\)</span>, the number of
protons <span class="math notranslate nohighlight">\(Z\)</span> and neutrons <span class="math notranslate nohighlight">\(N\)</span> using <strong>pandas</strong>. Before we do this it is
always useful (unless you have a binary file or other types of compressed
data) to actually open the file and simply take a look at it!</p>
<p>In particular, the program that outputs the final nuclear masses is written in Fortran with a specific format. It means that we need to figure out the format and which columns contain the data we are interested in. Pandas comes with a function that reads formatted output. After having admired the file, we are now ready to start massaging it with <strong>pandas</strong>. The file begins with some basic format information.</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="sd">&quot;&quot;&quot; </span>
<span class="sd">This is taken from the data file of the mass 2016 evaluation. </span>
<span class="sd">All files are 3436 lines long with 124 character per line. </span>
<span class="sd"> Headers are 39 lines long. </span>
<span class="sd"> col 1 : Fortran character control: 1 = page feed 0 = line feed </span>
<span class="sd"> format : a1,i3,i5,i5,i5,1x,a3,a4,1x,f13.5,f11.5,f11.3,f9.3,1x,a2,f11.3,f9.3,1x,i3,1x,f12.5,f11.5 </span>
<span class="sd"> These formats are reflected in the pandas widths variable below, see the statement </span>
<span class="sd"> widths=(1,3,5,5,5,1,3,4,1,13,11,11,9,1,2,11,9,1,3,1,12,11,1), </span>
<span class="sd"> Pandas has also a variable header, with length 39 in this case. </span>
<span class="sd">&quot;&quot;&quot;</span>
</pre></div>
</div>
</div>
</div>
<p>The data we are interested in are in columns 2, 3, 4 and 11, giving us
the number of neutrons, protons, mass numbers and binding energies,
respectively. We add also for the sake of completeness the element name. The data are in fixed-width formatted lines and we will
covert them into the <strong>pandas</strong> DataFrame structure.</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Read the experimental data with Pandas</span>
<span class="n">Masses</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_fwf</span><span class="p">(</span><span class="n">infile</span><span class="p">,</span> <span class="n">usecols</span><span class="o">=</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">11</span><span class="p">),</span>
<span class="n">names</span><span class="o">=</span><span class="p">(</span><span class="s1">&#39;N&#39;</span><span class="p">,</span> <span class="s1">&#39;Z&#39;</span><span class="p">,</span> <span class="s1">&#39;A&#39;</span><span class="p">,</span> <span class="s1">&#39;Element&#39;</span><span class="p">,</span> <span class="s1">&#39;Ebinding&#39;</span><span class="p">),</span>
<span class="n">widths</span><span class="o">=</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">13</span><span class="p">,</span><span class="mi">11</span><span class="p">,</span><span class="mi">11</span><span class="p">,</span><span class="mi">9</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">11</span><span class="p">,</span><span class="mi">9</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">12</span><span class="p">,</span><span class="mi">11</span><span class="p">,</span><span class="mi">1</span><span class="p">),</span>
<span class="n">header</span><span class="o">=</span><span class="mi">39</span><span class="p">,</span>
<span class="n">index_col</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
<span class="c1"># Extrapolated values are indicated by &#39;#&#39; in place of the decimal place, so</span>
<span class="c1"># the Ebinding column won&#39;t be numeric. Coerce to float and drop these entries.</span>
<span class="n">Masses</span><span class="p">[</span><span class="s1">&#39;Ebinding&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">to_numeric</span><span class="p">(</span><span class="n">Masses</span><span class="p">[</span><span class="s1">&#39;Ebinding&#39;</span><span class="p">],</span> <span class="n">errors</span><span class="o">=</span><span class="s1">&#39;coerce&#39;</span><span class="p">)</span>
<span class="n">Masses</span> <span class="o">=</span> <span class="n">Masses</span><span class="o">.</span><span class="n">dropna</span><span class="p">()</span>
<span class="c1"># Convert from keV to MeV.</span>
<span class="n">Masses</span><span class="p">[</span><span class="s1">&#39;Ebinding&#39;</span><span class="p">]</span> <span class="o">/=</span> <span class="mi">1000</span>
<span class="c1"># Group the DataFrame by nucleon number, A.</span>
<span class="n">Masses</span> <span class="o">=</span> <span class="n">Masses</span><span class="o">.</span><span class="n">groupby</span><span class="p">(</span><span class="s1">&#39;A&#39;</span><span class="p">)</span>
<span class="c1"># Find the rows of the grouped DataFrame with the maximum binding energy.</span>
<span class="n">Masses</span> <span class="o">=</span> <span class="n">Masses</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">t</span><span class="p">[</span><span class="n">t</span><span class="o">.</span><span class="n">Ebinding</span><span class="o">==</span><span class="n">t</span><span class="o">.</span><span class="n">Ebinding</span><span class="o">.</span><span class="n">max</span><span class="p">()])</span>
</pre></div>
</div>
</div>
</div>
<p>We have now read in the data, grouped them according to the variables we are interested in.
We see how easy it is to reorganize the data using <strong>pandas</strong>. If we
were to do these operations in C/C++ or Fortran, we would have had to
write various functions/subroutines which perform the above
reorganizations for us. Having reorganized the data, we can now start
to make some simple fits using both the functionalities in <strong>numpy</strong> and
<strong>Scikit-Learn</strong> afterwards.</p>
<p>Now we define five variables which contain
the number of nucleons <span class="math notranslate nohighlight">\(A\)</span>, the number of protons <span class="math notranslate nohighlight">\(Z\)</span> and the number of neutrons <span class="math notranslate nohighlight">\(N\)</span>, the element name and finally the energies themselves.</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">A</span> <span class="o">=</span> <span class="n">Masses</span><span class="p">[</span><span class="s1">&#39;A&#39;</span><span class="p">]</span>
<span class="n">Z</span> <span class="o">=</span> <span class="n">Masses</span><span class="p">[</span><span class="s1">&#39;Z&#39;</span><span class="p">]</span>
<span class="n">N</span> <span class="o">=</span> <span class="n">Masses</span><span class="p">[</span><span class="s1">&#39;N&#39;</span><span class="p">]</span>
<span class="n">Element</span> <span class="o">=</span> <span class="n">Masses</span><span class="p">[</span><span class="s1">&#39;Element&#39;</span><span class="p">]</span>
<span class="n">Energies</span> <span class="o">=</span> <span class="n">Masses</span><span class="p">[</span><span class="s1">&#39;Ebinding&#39;</span><span class="p">]</span>
<span class="nb">print</span><span class="p">(</span><span class="n">Masses</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
<p>The next step, and we will define this mathematically later, is to set up the so-called <strong>design matrix</strong>. We will throughout call this matrix <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span>.
It has dimensionality <span class="math notranslate nohighlight">\(p\times n\)</span>, where <span class="math notranslate nohighlight">\(n\)</span> is the number of data points and <span class="math notranslate nohighlight">\(p\)</span> are the so-called predictors. In our case here they are given by the number of polynomials in <span class="math notranslate nohighlight">\(A\)</span> we wish to include in the fit.</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Now we set up the design matrix X</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">A</span><span class="p">),</span><span class="mi">5</span><span class="p">))</span>
<span class="n">X</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
<span class="n">X</span><span class="p">[:,</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">A</span>
<span class="n">X</span><span class="p">[:,</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="n">A</span><span class="o">**</span><span class="p">(</span><span class="mf">2.0</span><span class="o">/</span><span class="mf">3.0</span><span class="p">)</span>
<span class="n">X</span><span class="p">[:,</span><span class="mi">3</span><span class="p">]</span> <span class="o">=</span> <span class="n">A</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="mf">1.0</span><span class="o">/</span><span class="mf">3.0</span><span class="p">)</span>
<span class="n">X</span><span class="p">[:,</span><span class="mi">4</span><span class="p">]</span> <span class="o">=</span> <span class="n">A</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="mf">1.0</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
<p>With <strong>scikitlearn</strong> we are now ready to use linear regression and fit our data.</p>
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<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">clf</span> <span class="o">=</span> <span class="n">skl</span><span class="o">.</span><span class="n">LinearRegression</span><span class="p">()</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Energies</span><span class="p">)</span>
<span class="n">fity</span> <span class="o">=</span> <span class="n">clf</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
<p>Pretty simple!<br />
Now we can print measures of how our fit is doing, the coefficients from the fits and plot the final fit together with our data.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># The mean squared error </span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Mean squared error: </span><span class="si">%.2f</span><span class="s2">&quot;</span> <span class="o">%</span> <span class="n">mean_squared_error</span><span class="p">(</span><span class="n">Energies</span><span class="p">,</span> <span class="n">fity</span><span class="p">))</span>
<span class="c1"># Explained variance score: 1 is perfect prediction </span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Variance score: </span><span class="si">%.2f</span><span class="s1">&#39;</span> <span class="o">%</span> <span class="n">r2_score</span><span class="p">(</span><span class="n">Energies</span><span class="p">,</span> <span class="n">fity</span><span class="p">))</span>
<span class="c1"># Mean absolute error </span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Mean absolute error: </span><span class="si">%.2f</span><span class="s1">&#39;</span> <span class="o">%</span> <span class="n">mean_absolute_error</span><span class="p">(</span><span class="n">Energies</span><span class="p">,</span> <span class="n">fity</span><span class="p">))</span>
<span class="nb">print</span><span class="p">(</span><span class="n">clf</span><span class="o">.</span><span class="n">coef_</span><span class="p">,</span> <span class="n">clf</span><span class="o">.</span><span class="n">intercept_</span><span class="p">)</span>
<span class="n">Masses</span><span class="p">[</span><span class="s1">&#39;Eapprox&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">fity</span>
<span class="c1"># Generate a plot comparing the experimental with the fitted values values.</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">()</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">&#39;$A = N + Z$&#39;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">&#39;$E_\mathrm</span><span class="si">{bind}</span><span class="s1">\,/\mathrm</span><span class="si">{MeV}</span><span class="s1">$&#39;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">Masses</span><span class="p">[</span><span class="s1">&#39;A&#39;</span><span class="p">],</span> <span class="n">Masses</span><span class="p">[</span><span class="s1">&#39;Ebinding&#39;</span><span class="p">],</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.7</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span>
<span class="n">label</span><span class="o">=</span><span class="s1">&#39;Ame2016&#39;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">Masses</span><span class="p">[</span><span class="s1">&#39;A&#39;</span><span class="p">],</span> <span class="n">Masses</span><span class="p">[</span><span class="s1">&#39;Eapprox&#39;</span><span class="p">],</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.7</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">&#39;m&#39;</span><span class="p">,</span>
<span class="n">label</span><span class="o">=</span><span class="s1">&#39;Fit&#39;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
<span class="n">save_fig</span><span class="p">(</span><span class="s2">&quot;Masses2016&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
</div>
</div>
<p>As a teaser, let us now see how we can do this with decision trees using <strong>scikit-learn</strong>. Later we will switch to so-called <strong>random forests</strong>!</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#Decision Tree Regression</span>
<span class="kn">from</span> <span class="nn">sklearn.tree</span> <span class="kn">import</span> <span class="n">DecisionTreeRegressor</span>
<span class="n">regr_1</span><span class="o">=</span><span class="n">DecisionTreeRegressor</span><span class="p">(</span><span class="n">max_depth</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
<span class="n">regr_2</span><span class="o">=</span><span class="n">DecisionTreeRegressor</span><span class="p">(</span><span class="n">max_depth</span><span class="o">=</span><span class="mi">7</span><span class="p">)</span>
<span class="n">regr_3</span><span class="o">=</span><span class="n">DecisionTreeRegressor</span><span class="p">(</span><span class="n">max_depth</span><span class="o">=</span><span class="mi">9</span><span class="p">)</span>
<span class="n">regr_1</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Energies</span><span class="p">)</span>
<span class="n">regr_2</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Energies</span><span class="p">)</span>
<span class="n">regr_3</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Energies</span><span class="p">)</span>
<span class="n">y_1</span> <span class="o">=</span> <span class="n">regr_1</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="n">y_2</span> <span class="o">=</span> <span class="n">regr_2</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="n">y_3</span><span class="o">=</span><span class="n">regr_3</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="n">Masses</span><span class="p">[</span><span class="s1">&#39;Eapprox&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">y_3</span>
<span class="c1"># Plot the results</span>
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">Energies</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;blue&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Data&quot;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">y_1</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;red&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;max_depth=5&quot;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">y_2</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;green&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;max_depth=7&quot;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">y_3</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;m&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;max_depth=9&quot;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;$A$&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;$E$[MeV]&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Decision Tree Regression&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
<span class="n">save_fig</span><span class="p">(</span><span class="s2">&quot;Masses2016Trees&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
<span class="nb">print</span><span class="p">(</span><span class="n">Masses</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span> <span class="p">(</span><span class="n">Energies</span><span class="o">-</span><span class="n">y_1</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
</pre></div>
</div>
</div>
</div>
<p>The <strong>seaborn</strong> package allows us to visualize data in an efficient way. Note that we use <strong>scikit-learn</strong>’s multi-layer perceptron (or feed forward neural network)
functionality.</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.neural_network</span> <span class="kn">import</span> <span class="n">MLPRegressor</span>
<span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">accuracy_score</span>
<span class="kn">import</span> <span class="nn">seaborn</span> <span class="k">as</span> <span class="nn">sns</span>
<span class="n">X_train</span> <span class="o">=</span> <span class="n">X</span>
<span class="n">Y_train</span> <span class="o">=</span> <span class="n">Energies</span>
<span class="n">n_hidden_neurons</span> <span class="o">=</span> <span class="mi">100</span>
<span class="n">epochs</span> <span class="o">=</span> <span class="mi">100</span>
<span class="c1"># store models for later use</span>
<span class="n">eta_vals</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">logspace</span><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">7</span><span class="p">)</span>
<span class="n">lmbd_vals</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">logspace</span><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">7</span><span class="p">)</span>
<span class="c1"># store the models for later use</span>
<span class="n">DNN_scikit</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">eta_vals</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">lmbd_vals</span><span class="p">)),</span> <span class="n">dtype</span><span class="o">=</span><span class="nb">object</span><span class="p">)</span>
<span class="n">train_accuracy</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">eta_vals</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">lmbd_vals</span><span class="p">)))</span>
<span class="n">sns</span><span class="o">.</span><span class="n">set</span><span class="p">()</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">eta</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">eta_vals</span><span class="p">):</span>
<span class="k">for</span> <span class="n">j</span><span class="p">,</span> <span class="n">lmbd</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">lmbd_vals</span><span class="p">):</span>
<span class="n">dnn</span> <span class="o">=</span> <span class="n">MLPRegressor</span><span class="p">(</span><span class="n">hidden_layer_sizes</span><span class="o">=</span><span class="p">(</span><span class="n">n_hidden_neurons</span><span class="p">),</span> <span class="n">activation</span><span class="o">=</span><span class="s1">&#39;logistic&#39;</span><span class="p">,</span>
<span class="n">alpha</span><span class="o">=</span><span class="n">lmbd</span><span class="p">,</span> <span class="n">learning_rate_init</span><span class="o">=</span><span class="n">eta</span><span class="p">,</span> <span class="n">max_iter</span><span class="o">=</span><span class="n">epochs</span><span class="p">)</span>
<span class="n">dnn</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">Y_train</span><span class="p">)</span>
<span class="n">DNN_scikit</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">dnn</span>
<span class="n">train_accuracy</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">dnn</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">Y_train</span><span class="p">)</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span> <span class="o">=</span> <span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">10</span><span class="p">))</span>
<span class="n">sns</span><span class="o">.</span><span class="n">heatmap</span><span class="p">(</span><span class="n">train_accuracy</span><span class="p">,</span> <span class="n">annot</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s2">&quot;viridis&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">&quot;Training Accuracy&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">&quot;$\eta$&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">&quot;$\lambda$&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
</div>
</div>
</div>
</div>
<div class="section" id="id1">
<h2><span class="section-number">1.14. </span>Linear Regression, basic elements<a class="headerlink" href="#id1" title="Permalink to this headline"></a></h2>
<p><a class="reference external" href="https://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/h20/forelesningsvideoer/LectureAug27.mp4?vrtx=view-as-webpage">Video of Lecture</a>.</p>
<p>Fitting a continuous function with linear parameterization in terms of the parameters <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span>.</p>
<ul class="simple">
<li><p>Method of choice for fitting a continuous function!</p></li>
<li><p>Gives an excellent introduction to central Machine Learning features with <strong>understandable pedagogical</strong> links to other methods like <strong>Neural Networks</strong>, <strong>Support Vector Machines</strong> etc</p></li>
<li><p>Analytical expression for the fitting parameters <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span></p></li>
<li><p>Analytical expressions for statistical propertiers like mean values, variances, confidence intervals and more</p></li>
<li><p>Analytical relation with probabilistic interpretations</p></li>
<li><p>Easy to introduce basic concepts like bias-variance tradeoff, cross-validation, resampling and regularization techniques and many other ML topics</p></li>
<li><p>Easy to code! And links well with classification problems and logistic regression and neural networks</p></li>
<li><p>Allows for <strong>easy</strong> hands-on understanding of gradient descent methods</p></li>
<li><p>and many more features</p></li>
</ul>
<p>For more discussions of Ridge and Lasso regression, <a class="reference external" href="https://arxiv.org/abs/1509.09169">Wessel van Wieringen’s</a> article is highly recommended.
Similarly, <a class="reference external" href="https://arxiv.org/abs/1803.08823">Mehta et al’s article</a> is also recommended.</p>
<p>Regression modeling deals with the description of the sampling distribution of a given random variable <span class="math notranslate nohighlight">\(y\)</span> and how it varies as function of another variable or a set of such variables <span class="math notranslate nohighlight">\(\boldsymbol{x} =[x_0, x_1,\dots, x_{n-1}]^T\)</span>.
The first variable is called the <strong>dependent</strong>, the <strong>outcome</strong> or the <strong>response</strong> variable while the set of variables <span class="math notranslate nohighlight">\(\boldsymbol{x}\)</span> is called the independent variable, or the predictor variable or the explanatory variable.</p>
<p>A regression model aims at finding a likelihood function <span class="math notranslate nohighlight">\(p(\boldsymbol{y}\vert \boldsymbol{x})\)</span>, that is the conditional distribution for <span class="math notranslate nohighlight">\(\boldsymbol{y}\)</span> with a given <span class="math notranslate nohighlight">\(\boldsymbol{x}\)</span>. The estimation of <span class="math notranslate nohighlight">\(p(\boldsymbol{y}\vert \boldsymbol{x})\)</span> is made using a data set with</p>
<ul class="simple">
<li><p><span class="math notranslate nohighlight">\(n\)</span> cases <span class="math notranslate nohighlight">\(i = 0, 1, 2, \dots, n-1\)</span></p></li>
<li><p>Response (target, dependent or outcome) variable <span class="math notranslate nohighlight">\(y_i\)</span> with <span class="math notranslate nohighlight">\(i = 0, 1, 2, \dots, n-1\)</span></p></li>
<li><p><span class="math notranslate nohighlight">\(p\)</span> so-called explanatory (independent or predictor) variables <span class="math notranslate nohighlight">\(\boldsymbol{x}_i=[x_{i0}, x_{i1}, \dots, x_{ip-1}]\)</span> with <span class="math notranslate nohighlight">\(i = 0, 1, 2, \dots, n-1\)</span> and explanatory variables running from <span class="math notranslate nohighlight">\(0\)</span> to <span class="math notranslate nohighlight">\(p-1\)</span>. See below for more explicit examples.</p></li>
</ul>
<p>The goal of the regression analysis is to extract/exploit relationship between <span class="math notranslate nohighlight">\(\boldsymbol{y}\)</span> and <span class="math notranslate nohighlight">\(\boldsymbol{x}\)</span> in or to infer causal dependencies, approximations to the likelihood functions, functional relationships and to make predictions, making fits and many other things.</p>
<p>Consider an experiment in which <span class="math notranslate nohighlight">\(p\)</span> characteristics of <span class="math notranslate nohighlight">\(n\)</span> samples are
measured. The data from this experiment, for various explanatory variables <span class="math notranslate nohighlight">\(p\)</span> are normally represented by a matrix<br />
<span class="math notranslate nohighlight">\(\mathbf{X}\)</span>.</p>
<p>The matrix <span class="math notranslate nohighlight">\(\mathbf{X}\)</span> is called the <em>design
matrix</em>. Additional information of the samples is available in the
form of <span class="math notranslate nohighlight">\(\boldsymbol{y}\)</span> (also as above). The variable <span class="math notranslate nohighlight">\(\boldsymbol{y}\)</span> is
generally referred to as the <em>response variable</em>. The aim of
regression analysis is to explain <span class="math notranslate nohighlight">\(\boldsymbol{y}\)</span> in terms of
<span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span> through a functional relationship like <span class="math notranslate nohighlight">\(y_i =
f(\mathbf{X}_{i,\ast})\)</span>. When no prior knowledge on the form of
<span class="math notranslate nohighlight">\(f(\cdot)\)</span> is available, it is common to assume a linear relationship
between <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span> and <span class="math notranslate nohighlight">\(\boldsymbol{y}\)</span>. This assumption gives rise to
the <em>linear regression model</em> where <span class="math notranslate nohighlight">\(\boldsymbol{\beta} = [\beta_0, \ldots,
\beta_{p-1}]^{T}\)</span> are the <em>regression parameters</em>.</p>
<p>Linear regression gives us a set of analytical equations for the parameters <span class="math notranslate nohighlight">\(\beta_j\)</span>.</p>
<p>In order to understand the relation among the predictors <span class="math notranslate nohighlight">\(p\)</span>, the set of data <span class="math notranslate nohighlight">\(n\)</span> and the target (outcome, output etc) <span class="math notranslate nohighlight">\(\boldsymbol{y}\)</span>,
consider the model we discussed for describing nuclear binding energies.</p>
<p>There we assumed that we could parametrize the data using a polynomial approximation based on the liquid drop model.
Assuming</p>
<div class="math notranslate nohighlight">
\[
BE(A) = a_0+a_1A+a_2A^{2/3}+a_3A^{-1/3}+a_4A^{-1},
\]</div>
<p>we have five predictors, that is the intercept, the <span class="math notranslate nohighlight">\(A\)</span> dependent term, the <span class="math notranslate nohighlight">\(A^{2/3}\)</span> term and the <span class="math notranslate nohighlight">\(A^{-1/3}\)</span> and <span class="math notranslate nohighlight">\(A^{-1}\)</span> terms.
This gives <span class="math notranslate nohighlight">\(p=0,1,2,3,4\)</span>. Furthermore we have <span class="math notranslate nohighlight">\(n\)</span> entries for each predictor. It means that our design matrix is a
<span class="math notranslate nohighlight">\(p\times n\)</span> matrix <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span>.</p>
<p>Here the predictors are based on a model we have made. A popular data set which is widely encountered in ML applications is the
so-called <a class="reference external" href="https://www.sciencedirect.com/science/article/pii/S0957417407006719?via%3Dihub">credit card default data from Taiwan</a>. The data set contains data on <span class="math notranslate nohighlight">\(n=30000\)</span> credit card holders with predictors like gender, marital status, age, profession, education, etc. In total there are <span class="math notranslate nohighlight">\(24\)</span> such predictors or attributes leading to a design matrix of dimensionality <span class="math notranslate nohighlight">\(24 \times 30000\)</span>. This is however a classification problem and we will come back to it when we discuss Logistic Regression.</p>
<p>Before we proceed let us study a case from linear algebra where we aim at fitting a set of data <span class="math notranslate nohighlight">\(\boldsymbol{y}=[y_0,y_1,\dots,y_{n-1}]\)</span>. We could think of these data as a result of an experiment or a complicated numerical experiment. These data are functions of a series of variables <span class="math notranslate nohighlight">\(\boldsymbol{x}=[x_0,x_1,\dots,x_{n-1}]\)</span>, that is <span class="math notranslate nohighlight">\(y_i = y(x_i)\)</span> with <span class="math notranslate nohighlight">\(i=0,1,2,\dots,n-1\)</span>. The variables <span class="math notranslate nohighlight">\(x_i\)</span> could represent physical quantities like time, temperature, position etc. We assume that <span class="math notranslate nohighlight">\(y(x)\)</span> is a smooth function.</p>
<p>Since obtaining these data points may not be trivial, we want to use these data to fit a function which can allow us to make predictions for values of <span class="math notranslate nohighlight">\(y\)</span> which are not in the present set. The perhaps simplest approach is to assume we can parametrize our function in terms of a polynomial of degree <span class="math notranslate nohighlight">\(n-1\)</span> with <span class="math notranslate nohighlight">\(n\)</span> points, that is</p>
<div class="math notranslate nohighlight">
\[
y=y(x) \rightarrow y(x_i)=\tilde{y}_i+\epsilon_i=\sum_{j=0}^{n-1} \beta_j x_i^j+\epsilon_i,
\]</div>
<p>where <span class="math notranslate nohighlight">\(\epsilon_i\)</span> is the error in our approximation.</p>
<p>For every set of values <span class="math notranslate nohighlight">\(y_i,x_i\)</span> we have thus the corresponding set of equations</p>
<div class="math notranslate nohighlight">
\[\begin{split}
\begin{align*}
y_0&amp;=\beta_0+\beta_1x_0^1+\beta_2x_0^2+\dots+\beta_{n-1}x_0^{n-1}+\epsilon_0\\
y_1&amp;=\beta_0+\beta_1x_1^1+\beta_2x_1^2+\dots+\beta_{n-1}x_1^{n-1}+\epsilon_1\\
y_2&amp;=\beta_0+\beta_1x_2^1+\beta_2x_2^2+\dots+\beta_{n-1}x_2^{n-1}+\epsilon_2\\
\dots &amp; \dots \\
y_{n-1}&amp;=\beta_0+\beta_1x_{n-1}^1+\beta_2x_{n-1}^2+\dots+\beta_{n-1}x_{n-1}^{n-1}+\epsilon_{n-1}.\\
\end{align*}
\end{split}\]</div>
<p>Defining the vectors</p>
<div class="math notranslate nohighlight">
\[
\boldsymbol{y} = [y_0,y_1, y_2,\dots, y_{n-1}]^T,
\]</div>
<p>and</p>
<div class="math notranslate nohighlight">
\[
\boldsymbol{\beta} = [\beta_0,\beta_1, \beta_2,\dots, \beta_{n-1}]^T,
\]</div>
<p>and</p>
<div class="math notranslate nohighlight">
\[
\boldsymbol{\epsilon} = [\epsilon_0,\epsilon_1, \epsilon_2,\dots, \epsilon_{n-1}]^T,
\]</div>
<p>and the design matrix</p>
<div class="math notranslate nohighlight">
\[\begin{split}
\boldsymbol{X}=
\begin{bmatrix}
1&amp; x_{0}^1 &amp;x_{0}^2&amp; \dots &amp; \dots &amp;x_{0}^{n-1}\\
1&amp; x_{1}^1 &amp;x_{1}^2&amp; \dots &amp; \dots &amp;x_{1}^{n-1}\\
1&amp; x_{2}^1 &amp;x_{2}^2&amp; \dots &amp; \dots &amp;x_{2}^{n-1}\\
\dots&amp; \dots &amp;\dots&amp; \dots &amp; \dots &amp;\dots\\
1&amp; x_{n-1}^1 &amp;x_{n-1}^2&amp; \dots &amp; \dots &amp;x_{n-1}^{n-1}\\
\end{bmatrix}
\end{split}\]</div>
<p>we can rewrite our equations as</p>
<div class="math notranslate nohighlight">
\[
\boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta}+\boldsymbol{\epsilon}.
\]</div>
<p>The above design matrix is called a <a class="reference external" href="https://en.wikipedia.org/wiki/Vandermonde_matrix">Vandermonde matrix</a>.</p>
<p>We are obviously not limited to the above polynomial expansions. We
could replace the various powers of <span class="math notranslate nohighlight">\(x\)</span> with elements of Fourier
series or instead of <span class="math notranslate nohighlight">\(x_i^j\)</span> we could have <span class="math notranslate nohighlight">\(\cos{(j x_i)}\)</span> or <span class="math notranslate nohighlight">\(\sin{(j
x_i)}\)</span>, or time series or other orthogonal functions. For every set
of values <span class="math notranslate nohighlight">\(y_i,x_i\)</span> we can then generalize the equations to</p>
<div class="math notranslate nohighlight">
\[\begin{split}
\begin{align*}
y_0&amp;=\beta_0x_{00}+\beta_1x_{01}+\beta_2x_{02}+\dots+\beta_{n-1}x_{0n-1}+\epsilon_0\\
y_1&amp;=\beta_0x_{10}+\beta_1x_{11}+\beta_2x_{12}+\dots+\beta_{n-1}x_{1n-1}+\epsilon_1\\
y_2&amp;=\beta_0x_{20}+\beta_1x_{21}+\beta_2x_{22}+\dots+\beta_{n-1}x_{2n-1}+\epsilon_2\\
\dots &amp; \dots \\
y_{i}&amp;=\beta_0x_{i0}+\beta_1x_{i1}+\beta_2x_{i2}+\dots+\beta_{n-1}x_{in-1}+\epsilon_i\\
\dots &amp; \dots \\
y_{n-1}&amp;=\beta_0x_{n-1,0}+\beta_1x_{n-1,2}+\beta_2x_{n-1,2}+\dots+\beta_{n-1}x_{n-1,n-1}+\epsilon_{n-1}.\\
\end{align*}
\end{split}\]</div>
<p><strong>Note that we have <span class="math notranslate nohighlight">\(p=n\)</span> here. The matrix is symmetric. This is generally not the case!</strong></p>
<p>We redefine in turn the matrix <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span> as</p>
<div class="math notranslate nohighlight">
\[\begin{split}
\boldsymbol{X}=
\begin{bmatrix}
x_{00}&amp; x_{01} &amp;x_{02}&amp; \dots &amp; \dots &amp;x_{0,n-1}\\
x_{10}&amp; x_{11} &amp;x_{12}&amp; \dots &amp; \dots &amp;x_{1,n-1}\\
x_{20}&amp; x_{21} &amp;x_{22}&amp; \dots &amp; \dots &amp;x_{2,n-1}\\
\dots&amp; \dots &amp;\dots&amp; \dots &amp; \dots &amp;\dots\\
x_{n-1,0}&amp; x_{n-1,1} &amp;x_{n-1,2}&amp; \dots &amp; \dots &amp;x_{n-1,n-1}\\
\end{bmatrix}
\end{split}\]</div>
<p>and without loss of generality we rewrite again our equations as</p>
<div class="math notranslate nohighlight">
\[
\boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta}+\boldsymbol{\epsilon}.
\]</div>
<p>The left-hand side of this equation is kwown. Our error vector <span class="math notranslate nohighlight">\(\boldsymbol{\epsilon}\)</span> and the parameter vector <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span> are our unknow quantities. How can we obtain the optimal set of <span class="math notranslate nohighlight">\(\beta_i\)</span> values?</p>
<p>We have defined the matrix <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span> via the equations</p>
<div class="math notranslate nohighlight">
\[\begin{split}
\begin{align*}
y_0&amp;=\beta_0x_{00}+\beta_1x_{01}+\beta_2x_{02}+\dots+\beta_{n-1}x_{0n-1}+\epsilon_0\\
y_1&amp;=\beta_0x_{10}+\beta_1x_{11}+\beta_2x_{12}+\dots+\beta_{n-1}x_{1n-1}+\epsilon_1\\
y_2&amp;=\beta_0x_{20}+\beta_1x_{21}+\beta_2x_{22}+\dots+\beta_{n-1}x_{2n-1}+\epsilon_1\\
\dots &amp; \dots \\
y_{i}&amp;=\beta_0x_{i0}+\beta_1x_{i1}+\beta_2x_{i2}+\dots+\beta_{n-1}x_{in-1}+\epsilon_1\\
\dots &amp; \dots \\
y_{n-1}&amp;=\beta_0x_{n-1,0}+\beta_1x_{n-1,2}+\beta_2x_{n-1,2}+\dots+\beta_{n-1}x_{n-1,n-1}+\epsilon_{n-1}.\\
\end{align*}
\end{split}\]</div>
<p>As we noted above, we stayed with a system with the design matrix
<span class="math notranslate nohighlight">\(\boldsymbol{X}\in {\mathbb{R}}^{n\times n}\)</span>, that is we have <span class="math notranslate nohighlight">\(p=n\)</span>. For reasons to come later (algorithmic arguments) we will hereafter define
our matrix as <span class="math notranslate nohighlight">\(\boldsymbol{X}\in {\mathbb{R}}^{n\times p}\)</span>, with the predictors refering to the column numbers and the entries <span class="math notranslate nohighlight">\(n\)</span> being the row elements.</p>
<p>In our <a class="reference external" href="https://compphysics.github.io/MachineLearning/doc/pub/How2ReadData/html/How2ReadData.html">introductory notes</a> we looked at the so-called <a class="reference external" href="https://en.wikipedia.org/wiki/Semi-empirical_mass_formula">liquid drop model</a>. Let us remind ourselves about what we did by looking at the code.</p>
<p>We restate the parts of the code we are most interested in.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Common imports</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">from</span> <span class="nn">IPython.display</span> <span class="kn">import</span> <span class="n">display</span>
<span class="kn">import</span> <span class="nn">os</span>
<span class="c1"># Where to save the figures and data files</span>
<span class="n">PROJECT_ROOT_DIR</span> <span class="o">=</span> <span class="s2">&quot;Results&quot;</span>
<span class="n">FIGURE_ID</span> <span class="o">=</span> <span class="s2">&quot;Results/FigureFiles&quot;</span>
<span class="n">DATA_ID</span> <span class="o">=</span> <span class="s2">&quot;DataFiles/&quot;</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="n">PROJECT_ROOT_DIR</span><span class="p">):</span>
<span class="n">os</span><span class="o">.</span><span class="n">mkdir</span><span class="p">(</span><span class="n">PROJECT_ROOT_DIR</span><span class="p">)</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="n">FIGURE_ID</span><span class="p">):</span>
<span class="n">os</span><span class="o">.</span><span class="n">makedirs</span><span class="p">(</span><span class="n">FIGURE_ID</span><span class="p">)</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="n">DATA_ID</span><span class="p">):</span>
<span class="n">os</span><span class="o">.</span><span class="n">makedirs</span><span class="p">(</span><span class="n">DATA_ID</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">image_path</span><span class="p">(</span><span class="n">fig_id</span><span class="p">):</span>
<span class="k">return</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">FIGURE_ID</span><span class="p">,</span> <span class="n">fig_id</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">data_path</span><span class="p">(</span><span class="n">dat_id</span><span class="p">):</span>
<span class="k">return</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">DATA_ID</span><span class="p">,</span> <span class="n">dat_id</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">save_fig</span><span class="p">(</span><span class="n">fig_id</span><span class="p">):</span>
<span class="n">plt</span><span class="o">.</span><span class="n">savefig</span><span class="p">(</span><span class="n">image_path</span><span class="p">(</span><span class="n">fig_id</span><span class="p">)</span> <span class="o">+</span> <span class="s2">&quot;.png&quot;</span><span class="p">,</span> <span class="nb">format</span><span class="o">=</span><span class="s1">&#39;png&#39;</span><span class="p">)</span>
<span class="n">infile</span> <span class="o">=</span> <span class="nb">open</span><span class="p">(</span><span class="n">data_path</span><span class="p">(</span><span class="s2">&quot;MassEval2016.dat&quot;</span><span class="p">),</span><span class="s1">&#39;r&#39;</span><span class="p">)</span>
<span class="c1"># Read the experimental data with Pandas</span>
<span class="n">Masses</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_fwf</span><span class="p">(</span><span class="n">infile</span><span class="p">,</span> <span class="n">usecols</span><span class="o">=</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">11</span><span class="p">),</span>
<span class="n">names</span><span class="o">=</span><span class="p">(</span><span class="s1">&#39;N&#39;</span><span class="p">,</span> <span class="s1">&#39;Z&#39;</span><span class="p">,</span> <span class="s1">&#39;A&#39;</span><span class="p">,</span> <span class="s1">&#39;Element&#39;</span><span class="p">,</span> <span class="s1">&#39;Ebinding&#39;</span><span class="p">),</span>
<span class="n">widths</span><span class="o">=</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">13</span><span class="p">,</span><span class="mi">11</span><span class="p">,</span><span class="mi">11</span><span class="p">,</span><span class="mi">9</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">11</span><span class="p">,</span><span class="mi">9</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">12</span><span class="p">,</span><span class="mi">11</span><span class="p">,</span><span class="mi">1</span><span class="p">),</span>
<span class="n">header</span><span class="o">=</span><span class="mi">39</span><span class="p">,</span>
<span class="n">index_col</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
<span class="c1"># Extrapolated values are indicated by &#39;#&#39; in place of the decimal place, so</span>
<span class="c1"># the Ebinding column won&#39;t be numeric. Coerce to float and drop these entries.</span>
<span class="n">Masses</span><span class="p">[</span><span class="s1">&#39;Ebinding&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">to_numeric</span><span class="p">(</span><span class="n">Masses</span><span class="p">[</span><span class="s1">&#39;Ebinding&#39;</span><span class="p">],</span> <span class="n">errors</span><span class="o">=</span><span class="s1">&#39;coerce&#39;</span><span class="p">)</span>
<span class="n">Masses</span> <span class="o">=</span> <span class="n">Masses</span><span class="o">.</span><span class="n">dropna</span><span class="p">()</span>
<span class="c1"># Convert from keV to MeV.</span>
<span class="n">Masses</span><span class="p">[</span><span class="s1">&#39;Ebinding&#39;</span><span class="p">]</span> <span class="o">/=</span> <span class="mi">1000</span>
<span class="c1"># Group the DataFrame by nucleon number, A.</span>
<span class="n">Masses</span> <span class="o">=</span> <span class="n">Masses</span><span class="o">.</span><span class="n">groupby</span><span class="p">(</span><span class="s1">&#39;A&#39;</span><span class="p">)</span>
<span class="c1"># Find the rows of the grouped DataFrame with the maximum binding energy.</span>
<span class="n">Masses</span> <span class="o">=</span> <span class="n">Masses</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">t</span><span class="p">[</span><span class="n">t</span><span class="o">.</span><span class="n">Ebinding</span><span class="o">==</span><span class="n">t</span><span class="o">.</span><span class="n">Ebinding</span><span class="o">.</span><span class="n">max</span><span class="p">()])</span>
<span class="n">A</span> <span class="o">=</span> <span class="n">Masses</span><span class="p">[</span><span class="s1">&#39;A&#39;</span><span class="p">]</span>
<span class="n">Z</span> <span class="o">=</span> <span class="n">Masses</span><span class="p">[</span><span class="s1">&#39;Z&#39;</span><span class="p">]</span>
<span class="n">N</span> <span class="o">=</span> <span class="n">Masses</span><span class="p">[</span><span class="s1">&#39;N&#39;</span><span class="p">]</span>
<span class="n">Element</span> <span class="o">=</span> <span class="n">Masses</span><span class="p">[</span><span class="s1">&#39;Element&#39;</span><span class="p">]</span>
<span class="n">Energies</span> <span class="o">=</span> <span class="n">Masses</span><span class="p">[</span><span class="s1">&#39;Ebinding&#39;</span><span class="p">]</span>
<span class="c1"># Now we set up the design matrix X</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">A</span><span class="p">),</span><span class="mi">5</span><span class="p">))</span>
<span class="n">X</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
<span class="n">X</span><span class="p">[:,</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">A</span>
<span class="n">X</span><span class="p">[:,</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="n">A</span><span class="o">**</span><span class="p">(</span><span class="mf">2.0</span><span class="o">/</span><span class="mf">3.0</span><span class="p">)</span>
<span class="n">X</span><span class="p">[:,</span><span class="mi">3</span><span class="p">]</span> <span class="o">=</span> <span class="n">A</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="mf">1.0</span><span class="o">/</span><span class="mf">3.0</span><span class="p">)</span>
<span class="n">X</span><span class="p">[:,</span><span class="mi">4</span><span class="p">]</span> <span class="o">=</span> <span class="n">A</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="mf">1.0</span><span class="p">)</span>
<span class="c1"># Then nice printout using pandas</span>
<span class="n">DesignMatrix</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="n">DesignMatrix</span><span class="o">.</span><span class="n">index</span> <span class="o">=</span> <span class="n">A</span>
<span class="n">DesignMatrix</span><span class="o">.</span><span class="n">columns</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;1&#39;</span><span class="p">,</span> <span class="s1">&#39;A&#39;</span><span class="p">,</span> <span class="s1">&#39;A^(2/3)&#39;</span><span class="p">,</span> <span class="s1">&#39;A^(-1/3)&#39;</span><span class="p">,</span> <span class="s1">&#39;1/A&#39;</span><span class="p">]</span>
<span class="n">display</span><span class="p">(</span><span class="n">DesignMatrix</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
<p>With <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\in {\mathbb{R}}^{p\times 1}\)</span>, it means that we will hereafter write our equations for the approximation as</p>
<div class="math notranslate nohighlight">
\[
\boldsymbol{\tilde{y}}= \boldsymbol{X}\boldsymbol{\beta},
\]</div>
<p>throughout these lectures.</p>
<p>With the above we use the design matrix to define the approximation <span class="math notranslate nohighlight">\(\boldsymbol{\tilde{y}}\)</span> via the unknown quantity <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span> as</p>
<div class="math notranslate nohighlight">
\[
\boldsymbol{\tilde{y}}= \boldsymbol{X}\boldsymbol{\beta},
\]</div>
<p>and in order to find the optimal parameters <span class="math notranslate nohighlight">\(\beta_i\)</span> instead of solving the above linear algebra problem, we define a function which gives a measure of the spread between the values <span class="math notranslate nohighlight">\(y_i\)</span> (which represent hopefully the exact values) and the parameterized values <span class="math notranslate nohighlight">\(\tilde{y}_i\)</span>, namely</p>
<div class="math notranslate nohighlight">
\[
C(\boldsymbol{\beta})=\frac{1}{n}\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2=\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{\tilde{y}}\right)^T\left(\boldsymbol{y}-\boldsymbol{\tilde{y}}\right)\right\},
\]</div>
<p>or using the matrix <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span> and in a more compact matrix-vector notation as</p>
<div class="math notranslate nohighlight">
\[
C(\boldsymbol{\beta})=\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right\}.
\]</div>
<p>This function is one possible way to define the so-called cost function.</p>
<p>It is also common to define
the function <span class="math notranslate nohighlight">\(C\)</span> as</p>
<div class="math notranslate nohighlight">
\[
C(\boldsymbol{\beta})=\frac{1}{2n}\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2,
\]</div>
<p>since when taking the first derivative with respect to the unknown parameters <span class="math notranslate nohighlight">\(\beta\)</span>, the factor of <span class="math notranslate nohighlight">\(2\)</span> cancels out.</p>
<p>The function</p>
<div class="math notranslate nohighlight">
\[
C(\boldsymbol{\beta})=\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right\},
\]</div>
<p>can be linked to the variance of the quantity <span class="math notranslate nohighlight">\(y_i\)</span> if we interpret the latter as the mean value.
When linking (see the discussion below) with the maximum likelihood approach below, we will indeed interpret <span class="math notranslate nohighlight">\(y_i\)</span> as a mean value</p>
<div class="math notranslate nohighlight">
\[
y_{i}=\langle y_i \rangle = \beta_0x_{i,0}+\beta_1x_{i,1}+\beta_2x_{i,2}+\dots+\beta_{n-1}x_{i,n-1}+\epsilon_i,
\]</div>
<p>where <span class="math notranslate nohighlight">\(\langle y_i \rangle\)</span> is the mean value. Keep in mind also that
till now we have treated <span class="math notranslate nohighlight">\(y_i\)</span> as the exact value. Normally, the
response (dependent or outcome) variable <span class="math notranslate nohighlight">\(y_i\)</span> the outcome of a
numerical experiment or another type of experiment and is thus only an
approximation to the true value. It is then always accompanied by an
error estimate, often limited to a statistical error estimate given by
the standard deviation discussed earlier. In the discussion here we
will treat <span class="math notranslate nohighlight">\(y_i\)</span> as our exact value for the response variable.</p>
<p>In order to find the parameters <span class="math notranslate nohighlight">\(\beta_i\)</span> we will then minimize the spread of <span class="math notranslate nohighlight">\(C(\boldsymbol{\beta})\)</span>, that is we are going to solve the problem</p>
<div class="math notranslate nohighlight">
\[
{\displaystyle \min_{\boldsymbol{\beta}\in
{\mathbb{R}}^{p}}}\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right\}.
\]</div>
<p>In practical terms it means we will require</p>
<div class="math notranslate nohighlight">
\[
\frac{\partial C(\boldsymbol{\beta})}{\partial \beta_j} = \frac{\partial }{\partial \beta_j}\left[ \frac{1}{n}\sum_{i=0}^{n-1}\left(y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}\right)^2\right]=0,
\]</div>
<p>which results in</p>
<div class="math notranslate nohighlight">
\[
\frac{\partial C(\boldsymbol{\beta})}{\partial \beta_j} = -\frac{2}{n}\left[ \sum_{i=0}^{n-1}x_{ij}\left(y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}\right)\right]=0,
\]</div>
<p>or in a matrix-vector form as</p>
<div class="math notranslate nohighlight">
\[
\frac{\partial C(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = 0 = \boldsymbol{X}^T\left( \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right).
\]</div>
<p>We can rewrite</p>
<div class="math notranslate nohighlight">
\[
\frac{\partial C(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = 0 = \boldsymbol{X}^T\left( \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right),
\]</div>
<p>as</p>
<div class="math notranslate nohighlight">
\[
\boldsymbol{X}^T\boldsymbol{y} = \boldsymbol{X}^T\boldsymbol{X}\boldsymbol{\beta},
\]</div>
<p>and if the matrix <span class="math notranslate nohighlight">\(\boldsymbol{X}^T\boldsymbol{X}\)</span> is invertible we have the solution</p>
<div class="math notranslate nohighlight">
\[
\boldsymbol{\beta} =\left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}.
\]</div>
<p>We note also that since our design matrix is defined as <span class="math notranslate nohighlight">\(\boldsymbol{X}\in
{\mathbb{R}}^{n\times p}\)</span>, the product <span class="math notranslate nohighlight">\(\boldsymbol{X}^T\boldsymbol{X} \in
{\mathbb{R}}^{p\times p}\)</span>. In the above case we have that <span class="math notranslate nohighlight">\(p \ll n\)</span>,
in our case <span class="math notranslate nohighlight">\(p=5\)</span> meaning that we end up with inverting a small
<span class="math notranslate nohighlight">\(5\times 5\)</span> matrix. This is a rather common situation, in many cases we end up with low-dimensional
matrices to invert. The methods discussed here and for many other
supervised learning algorithms like classification with logistic
regression or support vector machines, exhibit dimensionalities which
allow for the usage of direct linear algebra methods such as <strong>LU</strong> decomposition or <strong>Singular Value Decomposition</strong> (SVD) for finding the inverse of the matrix
<span class="math notranslate nohighlight">\(\boldsymbol{X}^T\boldsymbol{X}\)</span>.</p>
<p><strong>Small question</strong>: Do you think the example we have at hand here (the nuclear binding energies) can lead to problems in inverting the matrix <span class="math notranslate nohighlight">\(\boldsymbol{X}^T\boldsymbol{X}\)</span>? What kind of problems can we expect?</p>
<p>The following matrix and vector relation will be useful here and for the rest of the course. Vectors are always written as boldfaced lower case letters and
matrices as upper case boldfaced letters.</p>
<p>4
8</p>
<p>&lt;
&lt;
&lt;
!
!
M
A
T
H
_
B
L
O
C
K</p>
<p>4
9</p>
<p>&lt;
&lt;
&lt;
!
!
M
A
T
H
_
B
L
O
C
K</p>
<p>5
0</p>
<p>&lt;
&lt;
&lt;
!
!
M
A
T
H
_
B
L
O
C
K</p>
<div class="math notranslate nohighlight">
\[
\frac{\partial \log{\vert\boldsymbol{A}\vert}}{\partial \boldsymbol{A}} = (\boldsymbol{A}^{-1})^T.
\]</div>
<p>The residuals <span class="math notranslate nohighlight">\(\boldsymbol{\epsilon}\)</span> are in turn given by</p>
<div class="math notranslate nohighlight">
\[
\boldsymbol{\epsilon} = \boldsymbol{y}-\boldsymbol{\tilde{y}} = \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta},
\]</div>
<p>and with</p>
<div class="math notranslate nohighlight">
\[
\boldsymbol{X}^T\left( \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)= 0,
\]</div>
<p>we have</p>
<div class="math notranslate nohighlight">
\[
\boldsymbol{X}^T\boldsymbol{\epsilon}=\boldsymbol{X}^T\left( \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)= 0,
\]</div>
<p>meaning that the solution for <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span> is the one which minimizes the residuals. Later we will link this with the maximum likelihood approach.</p>
<p>Let us now return to our nuclear binding energies and simply code the above equations.</p>
<p>It is rather straightforward to implement the matrix inversion and obtain the parameters <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span>. After having defined the matrix <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span> we simply need to
write</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># matrix inversion to find beta</span>
<span class="n">beta</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">inv</span><span class="p">(</span><span class="n">X</span><span class="o">.</span><span class="n">T</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">X</span><span class="p">))</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">X</span><span class="o">.</span><span class="n">T</span><span class="p">)</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">Energies</span><span class="p">)</span>
<span class="c1"># and then make the prediction</span>
<span class="n">ytilde</span> <span class="o">=</span> <span class="n">X</span> <span class="o">@</span> <span class="n">beta</span>
</pre></div>
</div>
</div>
</div>
<p>Alternatively, you can use the least squares functionality in <strong>Numpy</strong> as</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">fit</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">lstsq</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Energies</span><span class="p">,</span> <span class="n">rcond</span> <span class="o">=</span><span class="kc">None</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
<span class="n">ytildenp</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">fit</span><span class="p">,</span><span class="n">X</span><span class="o">.</span><span class="n">T</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
<p>And finally we plot our fit with and compare with data</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">Masses</span><span class="p">[</span><span class="s1">&#39;Eapprox&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">ytilde</span>
<span class="c1"># Generate a plot comparing the experimental with the fitted values values.</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">()</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">&#39;$A = N + Z$&#39;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">&#39;$E_\mathrm</span><span class="si">{bind}</span><span class="s1">\,/\mathrm</span><span class="si">{MeV}</span><span class="s1">$&#39;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">Masses</span><span class="p">[</span><span class="s1">&#39;A&#39;</span><span class="p">],</span> <span class="n">Masses</span><span class="p">[</span><span class="s1">&#39;Ebinding&#39;</span><span class="p">],</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.7</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span>
<span class="n">label</span><span class="o">=</span><span class="s1">&#39;Ame2016&#39;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">Masses</span><span class="p">[</span><span class="s1">&#39;A&#39;</span><span class="p">],</span> <span class="n">Masses</span><span class="p">[</span><span class="s1">&#39;Eapprox&#39;</span><span class="p">],</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.7</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">&#39;m&#39;</span><span class="p">,</span>
<span class="n">label</span><span class="o">=</span><span class="s1">&#39;Fit&#39;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
<span class="n">save_fig</span><span class="p">(</span><span class="s2">&quot;Masses2016OLS&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
</div>
</div>
<p>We can easily test our fit by computing the <span class="math notranslate nohighlight">\(R2\)</span> score that we discussed in connection with the functionality of <strong>Scikit-Learn</strong> in the introductory slides.
Since we are not using <strong>Scikit-Learn</strong> here we can define our own <span class="math notranslate nohighlight">\(R2\)</span> function as</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">R2</span><span class="p">(</span><span class="n">y_data</span><span class="p">,</span> <span class="n">y_model</span><span class="p">):</span>
<span class="k">return</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">((</span><span class="n">y_data</span> <span class="o">-</span> <span class="n">y_model</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">((</span><span class="n">y_data</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">y_data</span><span class="p">))</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
<p>and we would be using it as</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="n">R2</span><span class="p">(</span><span class="n">Energies</span><span class="p">,</span><span class="n">ytilde</span><span class="p">))</span>
</pre></div>
</div>
</div>
</div>
<p>We can easily add our <strong>MSE</strong> score as</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">MSE</span><span class="p">(</span><span class="n">y_data</span><span class="p">,</span><span class="n">y_model</span><span class="p">):</span>
<span class="n">n</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="n">y_model</span><span class="p">)</span>
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">((</span><span class="n">y_data</span><span class="o">-</span><span class="n">y_model</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="n">n</span>
<span class="nb">print</span><span class="p">(</span><span class="n">MSE</span><span class="p">(</span><span class="n">Energies</span><span class="p">,</span><span class="n">ytilde</span><span class="p">))</span>
</pre></div>
</div>
</div>
</div>
<p>and finally the relative error as</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">RelativeError</span><span class="p">(</span><span class="n">y_data</span><span class="p">,</span><span class="n">y_model</span><span class="p">):</span>
<span class="k">return</span> <span class="nb">abs</span><span class="p">((</span><span class="n">y_data</span><span class="o">-</span><span class="n">y_model</span><span class="p">)</span><span class="o">/</span><span class="n">y_data</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">RelativeError</span><span class="p">(</span><span class="n">Energies</span><span class="p">,</span> <span class="n">ytilde</span><span class="p">))</span>
</pre></div>
</div>
</div>
</div>
<div class="section" id="the-chi-2-function">
<h3><span class="section-number">1.14.1. </span>The <span class="math notranslate nohighlight">\(\chi^2\)</span> function<a class="headerlink" href="#the-chi-2-function" title="Permalink to this headline"></a></h3>
<p>Normally, the response (dependent or outcome) variable <span class="math notranslate nohighlight">\(y_i\)</span> is the
outcome of a numerical experiment or another type of experiment and is
thus only an approximation to the true value. It is then always
accompanied by an error estimate, often limited to a statistical error
estimate given by the standard deviation discussed earlier. In the
discussion here we will treat <span class="math notranslate nohighlight">\(y_i\)</span> as our exact value for the
response variable.</p>
<p>Introducing the standard deviation <span class="math notranslate nohighlight">\(\sigma_i\)</span> for each measurement
<span class="math notranslate nohighlight">\(y_i\)</span>, we define now the <span class="math notranslate nohighlight">\(\chi^2\)</span> function (omitting the <span class="math notranslate nohighlight">\(1/n\)</span> term)
as</p>
<div class="math notranslate nohighlight">
\[
\chi^2(\boldsymbol{\beta})=\frac{1}{n}\sum_{i=0}^{n-1}\frac{\left(y_i-\tilde{y}_i\right)^2}{\sigma_i^2}=\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{\tilde{y}}\right)^T\frac{1}{\boldsymbol{\Sigma^2}}\left(\boldsymbol{y}-\boldsymbol{\tilde{y}}\right)\right\},
\]</div>
<p>where the matrix <span class="math notranslate nohighlight">\(\boldsymbol{\Sigma}\)</span> is a diagonal matrix with <span class="math notranslate nohighlight">\(\sigma_i\)</span> as matrix elements.</p>
<p>In order to find the parameters <span class="math notranslate nohighlight">\(\beta_i\)</span> we will then minimize the spread of <span class="math notranslate nohighlight">\(\chi^2(\boldsymbol{\beta})\)</span> by requiring</p>
<div class="math notranslate nohighlight">
\[
\frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \beta_j} = \frac{\partial }{\partial \beta_j}\left[ \frac{1}{n}\sum_{i=0}^{n-1}\left(\frac{y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}}{\sigma_i}\right)^2\right]=0,
\]</div>
<p>which results in</p>
<div class="math notranslate nohighlight">
\[
\frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \beta_j} = -\frac{2}{n}\left[ \sum_{i=0}^{n-1}\frac{x_{ij}}{\sigma_i}\left(\frac{y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}}{\sigma_i}\right)\right]=0,
\]</div>
<p>or in a matrix-vector form as</p>
<div class="math notranslate nohighlight">
\[
\frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = 0 = \boldsymbol{A}^T\left( \boldsymbol{b}-\boldsymbol{A}\boldsymbol{\beta}\right).
\]</div>
<p>where we have defined the matrix <span class="math notranslate nohighlight">\(\boldsymbol{A} =\boldsymbol{X}/\boldsymbol{\Sigma}\)</span> with matrix elements <span class="math notranslate nohighlight">\(a_{ij} = x_{ij}/\sigma_i\)</span> and the vector <span class="math notranslate nohighlight">\(\boldsymbol{b}\)</span> with elements <span class="math notranslate nohighlight">\(b_i = y_i/\sigma_i\)</span>.</p>
<p>We can rewrite</p>
<div class="math notranslate nohighlight">
\[
\frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = 0 = \boldsymbol{A}^T\left( \boldsymbol{b}-\boldsymbol{A}\boldsymbol{\beta}\right),
\]</div>
<p>as</p>
<div class="math notranslate nohighlight">
\[
\boldsymbol{A}^T\boldsymbol{b} = \boldsymbol{A}^T\boldsymbol{A}\boldsymbol{\beta},
\]</div>
<p>and if the matrix <span class="math notranslate nohighlight">\(\boldsymbol{A}^T\boldsymbol{A}\)</span> is invertible we have the solution</p>
<div class="math notranslate nohighlight">
\[
\boldsymbol{\beta} =\left(\boldsymbol{A}^T\boldsymbol{A}\right)^{-1}\boldsymbol{A}^T\boldsymbol{b}.
\]</div>
<p>If we then introduce the matrix</p>
<div class="math notranslate nohighlight">
\[
\boldsymbol{H} = \left(\boldsymbol{A}^T\boldsymbol{A}\right)^{-1},
\]</div>
<p>we have then the following expression for the parameters <span class="math notranslate nohighlight">\(\beta_j\)</span> (the matrix elements of <span class="math notranslate nohighlight">\(\boldsymbol{H}\)</span> are <span class="math notranslate nohighlight">\(h_{ij}\)</span>)</p>
<div class="math notranslate nohighlight">
\[
\beta_j = \sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}\frac{y_i}{\sigma_i}\frac{x_{ik}}{\sigma_i} = \sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}b_ia_{ik}
\]</div>
<p>We state without proof the expression for the uncertainty in the parameters <span class="math notranslate nohighlight">\(\beta_j\)</span> as (we leave this as an exercise)</p>
<div class="math notranslate nohighlight">
\[
\sigma^2(\beta_j) = \sum_{i=0}^{n-1}\sigma_i^2\left( \frac{\partial \beta_j}{\partial y_i}\right)^2,
\]</div>
<p>resulting in</p>
<div class="math notranslate nohighlight">
\[
\sigma^2(\beta_j) = \left(\sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}a_{ik}\right)\left(\sum_{l=0}^{p-1}h_{jl}\sum_{m=0}^{n-1}a_{ml}\right) = h_{jj}!
\]</div>
<p>The first step here is to approximate the function <span class="math notranslate nohighlight">\(y\)</span> with a first-order polynomial, that is we write</p>
<div class="math notranslate nohighlight">
\[
y=y(x) \rightarrow y(x_i) \approx \beta_0+\beta_1 x_i.
\]</div>
<p>By computing the derivatives of <span class="math notranslate nohighlight">\(\chi^2\)</span> with respect to <span class="math notranslate nohighlight">\(\beta_0\)</span> and <span class="math notranslate nohighlight">\(\beta_1\)</span> show that these are given by</p>
<div class="math notranslate nohighlight">
\[
\frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \beta_0} = -2\left[ \frac{1}{n}\sum_{i=0}^{n-1}\left(\frac{y_i-\beta_0-\beta_1x_{i}}{\sigma_i^2}\right)\right]=0,
\]</div>
<p>and</p>
<div class="math notranslate nohighlight">
\[
\frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \beta_1} = -\frac{2}{n}\left[ \sum_{i=0}^{n-1}x_i\left(\frac{y_i-\beta_0-\beta_1x_{i}}{\sigma_i^2}\right)\right]=0.
\]</div>
<p>For a linear fit (a first-order polynomial) we don’t need to invert a matrix!!<br />
Defining</p>
<div class="math notranslate nohighlight">
\[
\gamma = \sum_{i=0}^{n-1}\frac{1}{\sigma_i^2},
\]</div>
<div class="math notranslate nohighlight">
\[
\gamma_x = \sum_{i=0}^{n-1}\frac{x_{i}}{\sigma_i^2},
\]</div>
<div class="math notranslate nohighlight">
\[
\gamma_y = \sum_{i=0}^{n-1}\left(\frac{y_i}{\sigma_i^2}\right),
\]</div>
<div class="math notranslate nohighlight">
\[
\gamma_{xx} = \sum_{i=0}^{n-1}\frac{x_ix_{i}}{\sigma_i^2},
\]</div>
<div class="math notranslate nohighlight">
\[
\gamma_{xy} = \sum_{i=0}^{n-1}\frac{y_ix_{i}}{\sigma_i^2},
\]</div>
<p>we obtain</p>
<div class="math notranslate nohighlight">
\[
\beta_0 = \frac{\gamma_{xx}\gamma_y-\gamma_x\gamma_y}{\gamma\gamma_{xx}-\gamma_x^2},
\]</div>
<div class="math notranslate nohighlight">
\[
\beta_1 = \frac{\gamma_{xy}\gamma-\gamma_x\gamma_y}{\gamma\gamma_{xx}-\gamma_x^2}.
\]</div>
<p>This approach (different linear and non-linear regression) suffers
often from both being underdetermined and overdetermined in the
unknown coefficients <span class="math notranslate nohighlight">\(\beta_i\)</span>. A better approach is to use the
Singular Value Decomposition (SVD) method discussed below. Or using
Lasso and Ridge regression. See below.</p>
</div>
<div class="section" id="fitting-an-equation-of-state-for-dense-nuclear-matter">
<h3><span class="section-number">1.14.2. </span>Fitting an Equation of State for Dense Nuclear Matter<a class="headerlink" href="#fitting-an-equation-of-state-for-dense-nuclear-matter" title="Permalink to this headline"></a></h3>
<p>Before we continue, let us introduce yet another example. We are going to fit the
nuclear equation of state using results from many-body calculations.
The equation of state we have made available here, as function of
density, has been derived using modern nucleon-nucleon potentials with
<a class="reference external" href="https://www.sciencedirect.com/science/article/pii/S0370157399001106">the addition of three-body
forces</a>. This
time the file is presented as a standard <strong>csv</strong> file.</p>
<p>The beginning of the Python code here is similar to what you have seen
before, with the same initializations and declarations. We use also
<strong>pandas</strong> again, rather extensively in order to organize our data.</p>
<p>The difference now is that we use <strong>Scikit-Learn’s</strong> regression tools
instead of our own matrix inversion implementation. Furthermore, we
sneak in <strong>Ridge</strong> regression (to be discussed below) which includes a
hyperparameter <span class="math notranslate nohighlight">\(\lambda\)</span>, also to be explained below.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Common imports</span>
<span class="kn">import</span> <span class="nn">os</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">import</span> <span class="nn">sklearn.linear_model</span> <span class="k">as</span> <span class="nn">skl</span>
<span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">mean_squared_error</span><span class="p">,</span> <span class="n">r2_score</span><span class="p">,</span> <span class="n">mean_absolute_error</span>
<span class="c1"># Where to save the figures and data files</span>
<span class="n">PROJECT_ROOT_DIR</span> <span class="o">=</span> <span class="s2">&quot;Results&quot;</span>
<span class="n">FIGURE_ID</span> <span class="o">=</span> <span class="s2">&quot;Results/FigureFiles&quot;</span>
<span class="n">DATA_ID</span> <span class="o">=</span> <span class="s2">&quot;DataFiles/&quot;</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="n">PROJECT_ROOT_DIR</span><span class="p">):</span>
<span class="n">os</span><span class="o">.</span><span class="n">mkdir</span><span class="p">(</span><span class="n">PROJECT_ROOT_DIR</span><span class="p">)</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="n">FIGURE_ID</span><span class="p">):</span>
<span class="n">os</span><span class="o">.</span><span class="n">makedirs</span><span class="p">(</span><span class="n">FIGURE_ID</span><span class="p">)</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="n">DATA_ID</span><span class="p">):</span>
<span class="n">os</span><span class="o">.</span><span class="n">makedirs</span><span class="p">(</span><span class="n">DATA_ID</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">image_path</span><span class="p">(</span><span class="n">fig_id</span><span class="p">):</span>
<span class="k">return</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">FIGURE_ID</span><span class="p">,</span> <span class="n">fig_id</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">data_path</span><span class="p">(</span><span class="n">dat_id</span><span class="p">):</span>
<span class="k">return</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">DATA_ID</span><span class="p">,</span> <span class="n">dat_id</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">save_fig</span><span class="p">(</span><span class="n">fig_id</span><span class="p">):</span>
<span class="n">plt</span><span class="o">.</span><span class="n">savefig</span><span class="p">(</span><span class="n">image_path</span><span class="p">(</span><span class="n">fig_id</span><span class="p">)</span> <span class="o">+</span> <span class="s2">&quot;.png&quot;</span><span class="p">,</span> <span class="nb">format</span><span class="o">=</span><span class="s1">&#39;png&#39;</span><span class="p">)</span>
<span class="n">infile</span> <span class="o">=</span> <span class="nb">open</span><span class="p">(</span><span class="n">data_path</span><span class="p">(</span><span class="s2">&quot;EoS.csv&quot;</span><span class="p">),</span><span class="s1">&#39;r&#39;</span><span class="p">)</span>
<span class="c1"># Read the EoS data as csv file and organize the data into two arrays with density and energies</span>
<span class="n">EoS</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_csv</span><span class="p">(</span><span class="n">infile</span><span class="p">,</span> <span class="n">names</span><span class="o">=</span><span class="p">(</span><span class="s1">&#39;Density&#39;</span><span class="p">,</span> <span class="s1">&#39;Energy&#39;</span><span class="p">))</span>
<span class="n">EoS</span><span class="p">[</span><span class="s1">&#39;Energy&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">to_numeric</span><span class="p">(</span><span class="n">EoS</span><span class="p">[</span><span class="s1">&#39;Energy&#39;</span><span class="p">],</span> <span class="n">errors</span><span class="o">=</span><span class="s1">&#39;coerce&#39;</span><span class="p">)</span>
<span class="n">EoS</span> <span class="o">=</span> <span class="n">EoS</span><span class="o">.</span><span class="n">dropna</span><span class="p">()</span>
<span class="n">Energies</span> <span class="o">=</span> <span class="n">EoS</span><span class="p">[</span><span class="s1">&#39;Energy&#39;</span><span class="p">]</span>
<span class="n">Density</span> <span class="o">=</span> <span class="n">EoS</span><span class="p">[</span><span class="s1">&#39;Density&#39;</span><span class="p">]</span>
<span class="c1"># The design matrix now as function of various polytrops</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">Density</span><span class="p">),</span><span class="mi">4</span><span class="p">))</span>
<span class="n">X</span><span class="p">[:,</span><span class="mi">3</span><span class="p">]</span> <span class="o">=</span> <span class="n">Density</span><span class="o">**</span><span class="p">(</span><span class="mf">4.0</span><span class="o">/</span><span class="mf">3.0</span><span class="p">)</span>
<span class="n">X</span><span class="p">[:,</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="n">Density</span>
<span class="n">X</span><span class="p">[:,</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">Density</span><span class="o">**</span><span class="p">(</span><span class="mf">2.0</span><span class="o">/</span><span class="mf">3.0</span><span class="p">)</span>
<span class="n">X</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
<span class="c1"># We use now Scikit-Learn&#39;s linear regressor and ridge regressor</span>
<span class="c1"># OLS part</span>
<span class="n">clf</span> <span class="o">=</span> <span class="n">skl</span><span class="o">.</span><span class="n">LinearRegression</span><span class="p">()</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Energies</span><span class="p">)</span>
<span class="n">ytilde</span> <span class="o">=</span> <span class="n">clf</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="n">EoS</span><span class="p">[</span><span class="s1">&#39;Eols&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">ytilde</span>
<span class="c1"># The mean squared error </span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Mean squared error: </span><span class="si">%.2f</span><span class="s2">&quot;</span> <span class="o">%</span> <span class="n">mean_squared_error</span><span class="p">(</span><span class="n">Energies</span><span class="p">,</span> <span class="n">ytilde</span><span class="p">))</span>
<span class="c1"># Explained variance score: 1 is perfect prediction </span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Variance score: </span><span class="si">%.2f</span><span class="s1">&#39;</span> <span class="o">%</span> <span class="n">r2_score</span><span class="p">(</span><span class="n">Energies</span><span class="p">,</span> <span class="n">ytilde</span><span class="p">))</span>
<span class="c1"># Mean absolute error </span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Mean absolute error: </span><span class="si">%.2f</span><span class="s1">&#39;</span> <span class="o">%</span> <span class="n">mean_absolute_error</span><span class="p">(</span><span class="n">Energies</span><span class="p">,</span> <span class="n">ytilde</span><span class="p">))</span>
<span class="nb">print</span><span class="p">(</span><span class="n">clf</span><span class="o">.</span><span class="n">coef_</span><span class="p">,</span> <span class="n">clf</span><span class="o">.</span><span class="n">intercept_</span><span class="p">)</span>
<span class="c1"># The Ridge regression with a hyperparameter lambda = 0.1</span>
<span class="n">_lambda</span> <span class="o">=</span> <span class="mf">0.1</span>
<span class="n">clf_ridge</span> <span class="o">=</span> <span class="n">skl</span><span class="o">.</span><span class="n">Ridge</span><span class="p">(</span><span class="n">alpha</span><span class="o">=</span><span class="n">_lambda</span><span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Energies</span><span class="p">)</span>
<span class="n">yridge</span> <span class="o">=</span> <span class="n">clf_ridge</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="n">EoS</span><span class="p">[</span><span class="s1">&#39;Eridge&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">yridge</span>
<span class="c1"># The mean squared error </span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Mean squared error: </span><span class="si">%.2f</span><span class="s2">&quot;</span> <span class="o">%</span> <span class="n">mean_squared_error</span><span class="p">(</span><span class="n">Energies</span><span class="p">,</span> <span class="n">yridge</span><span class="p">))</span>
<span class="c1"># Explained variance score: 1 is perfect prediction </span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Variance score: </span><span class="si">%.2f</span><span class="s1">&#39;</span> <span class="o">%</span> <span class="n">r2_score</span><span class="p">(</span><span class="n">Energies</span><span class="p">,</span> <span class="n">yridge</span><span class="p">))</span>
<span class="c1"># Mean absolute error </span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Mean absolute error: </span><span class="si">%.2f</span><span class="s1">&#39;</span> <span class="o">%</span> <span class="n">mean_absolute_error</span><span class="p">(</span><span class="n">Energies</span><span class="p">,</span> <span class="n">yridge</span><span class="p">))</span>
<span class="nb">print</span><span class="p">(</span><span class="n">clf_ridge</span><span class="o">.</span><span class="n">coef_</span><span class="p">,</span> <span class="n">clf_ridge</span><span class="o">.</span><span class="n">intercept_</span><span class="p">)</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">()</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">&#39;$\rho[\mathrm</span><span class="si">{fm}</span><span class="s1">^{-3}]$&#39;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">&#39;Energy per particle&#39;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">EoS</span><span class="p">[</span><span class="s1">&#39;Density&#39;</span><span class="p">],</span> <span class="n">EoS</span><span class="p">[</span><span class="s1">&#39;Energy&#39;</span><span class="p">],</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.7</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span>
<span class="n">label</span><span class="o">=</span><span class="s1">&#39;Theoretical data&#39;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">EoS</span><span class="p">[</span><span class="s1">&#39;Density&#39;</span><span class="p">],</span> <span class="n">EoS</span><span class="p">[</span><span class="s1">&#39;Eols&#39;</span><span class="p">],</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.7</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">&#39;m&#39;</span><span class="p">,</span>
<span class="n">label</span><span class="o">=</span><span class="s1">&#39;OLS&#39;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">EoS</span><span class="p">[</span><span class="s1">&#39;Density&#39;</span><span class="p">],</span> <span class="n">EoS</span><span class="p">[</span><span class="s1">&#39;Eridge&#39;</span><span class="p">],</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.7</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s1">&#39;g&#39;</span><span class="p">,</span>
<span class="n">label</span><span class="o">=</span><span class="s1">&#39;Ridge $\lambda = 0.1$&#39;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
<span class="n">save_fig</span><span class="p">(</span><span class="s2">&quot;EoSfitting&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
</div>
</div>
<p>The above simple polynomial in density <span class="math notranslate nohighlight">\(\rho\)</span> gives an excellent fit
to the data.</p>
<p>We note also that there is a small deviation between the
standard OLS and the Ridge regression at higher densities. We discuss this in more detail
below.</p>
</div>
</div>
<div class="section" id="splitting-our-data-in-training-and-test-data">
<h2><span class="section-number">1.15. </span>Splitting our Data in Training and Test data<a class="headerlink" href="#splitting-our-data-in-training-and-test-data" title="Permalink to this headline"></a></h2>
<p>It is normal in essentially all Machine Learning studies to split the
data in a training set and a test set (sometimes also an additional
validation set). <strong>Scikit-Learn</strong> has an own function for this. There
is no explicit recipe for how much data should be included as training
data and say test data. An accepted rule of thumb is to use
approximately <span class="math notranslate nohighlight">\(2/3\)</span> to <span class="math notranslate nohighlight">\(4/5\)</span> of the data as training data. We will
postpone a discussion of this splitting to the end of these notes and
our discussion of the so-called <strong>bias-variance</strong> tradeoff. Here we
limit ourselves to repeat the above equation of state fitting example
but now splitting the data into a training set and a test set.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">os</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
<span class="c1"># Where to save the figures and data files</span>
<span class="n">PROJECT_ROOT_DIR</span> <span class="o">=</span> <span class="s2">&quot;Results&quot;</span>
<span class="n">FIGURE_ID</span> <span class="o">=</span> <span class="s2">&quot;Results/FigureFiles&quot;</span>
<span class="n">DATA_ID</span> <span class="o">=</span> <span class="s2">&quot;DataFiles/&quot;</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="n">PROJECT_ROOT_DIR</span><span class="p">):</span>
<span class="n">os</span><span class="o">.</span><span class="n">mkdir</span><span class="p">(</span><span class="n">PROJECT_ROOT_DIR</span><span class="p">)</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="n">FIGURE_ID</span><span class="p">):</span>
<span class="n">os</span><span class="o">.</span><span class="n">makedirs</span><span class="p">(</span><span class="n">FIGURE_ID</span><span class="p">)</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="n">DATA_ID</span><span class="p">):</span>
<span class="n">os</span><span class="o">.</span><span class="n">makedirs</span><span class="p">(</span><span class="n">DATA_ID</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">image_path</span><span class="p">(</span><span class="n">fig_id</span><span class="p">):</span>
<span class="k">return</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">FIGURE_ID</span><span class="p">,</span> <span class="n">fig_id</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">data_path</span><span class="p">(</span><span class="n">dat_id</span><span class="p">):</span>
<span class="k">return</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">DATA_ID</span><span class="p">,</span> <span class="n">dat_id</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">save_fig</span><span class="p">(</span><span class="n">fig_id</span><span class="p">):</span>
<span class="n">plt</span><span class="o">.</span><span class="n">savefig</span><span class="p">(</span><span class="n">image_path</span><span class="p">(</span><span class="n">fig_id</span><span class="p">)</span> <span class="o">+</span> <span class="s2">&quot;.png&quot;</span><span class="p">,</span> <span class="nb">format</span><span class="o">=</span><span class="s1">&#39;png&#39;</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">R2</span><span class="p">(</span><span class="n">y_data</span><span class="p">,</span> <span class="n">y_model</span><span class="p">):</span>
<span class="k">return</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">((</span><span class="n">y_data</span> <span class="o">-</span> <span class="n">y_model</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">((</span><span class="n">y_data</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">y_data</span><span class="p">))</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">MSE</span><span class="p">(</span><span class="n">y_data</span><span class="p">,</span><span class="n">y_model</span><span class="p">):</span>
<span class="n">n</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="n">y_model</span><span class="p">)</span>
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">((</span><span class="n">y_data</span><span class="o">-</span><span class="n">y_model</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="n">n</span>
<span class="n">infile</span> <span class="o">=</span> <span class="nb">open</span><span class="p">(</span><span class="n">data_path</span><span class="p">(</span><span class="s2">&quot;EoS.csv&quot;</span><span class="p">),</span><span class="s1">&#39;r&#39;</span><span class="p">)</span>
<span class="c1"># Read the EoS data as csv file and organized into two arrays with density and energies</span>
<span class="n">EoS</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_csv</span><span class="p">(</span><span class="n">infile</span><span class="p">,</span> <span class="n">names</span><span class="o">=</span><span class="p">(</span><span class="s1">&#39;Density&#39;</span><span class="p">,</span> <span class="s1">&#39;Energy&#39;</span><span class="p">))</span>
<span class="n">EoS</span><span class="p">[</span><span class="s1">&#39;Energy&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">to_numeric</span><span class="p">(</span><span class="n">EoS</span><span class="p">[</span><span class="s1">&#39;Energy&#39;</span><span class="p">],</span> <span class="n">errors</span><span class="o">=</span><span class="s1">&#39;coerce&#39;</span><span class="p">)</span>
<span class="n">EoS</span> <span class="o">=</span> <span class="n">EoS</span><span class="o">.</span><span class="n">dropna</span><span class="p">()</span>
<span class="n">Energies</span> <span class="o">=</span> <span class="n">EoS</span><span class="p">[</span><span class="s1">&#39;Energy&#39;</span><span class="p">]</span>
<span class="n">Density</span> <span class="o">=</span> <span class="n">EoS</span><span class="p">[</span><span class="s1">&#39;Density&#39;</span><span class="p">]</span>
<span class="c1"># The design matrix now as function of various polytrops</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">Density</span><span class="p">),</span><span class="mi">5</span><span class="p">))</span>
<span class="n">X</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
<span class="n">X</span><span class="p">[:,</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">Density</span><span class="o">**</span><span class="p">(</span><span class="mf">2.0</span><span class="o">/</span><span class="mf">3.0</span><span class="p">)</span>
<span class="n">X</span><span class="p">[:,</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="n">Density</span>
<span class="n">X</span><span class="p">[:,</span><span class="mi">3</span><span class="p">]</span> <span class="o">=</span> <span class="n">Density</span><span class="o">**</span><span class="p">(</span><span class="mf">4.0</span><span class="o">/</span><span class="mf">3.0</span><span class="p">)</span>
<span class="n">X</span><span class="p">[:,</span><span class="mi">4</span><span class="p">]</span> <span class="o">=</span> <span class="n">Density</span><span class="o">**</span><span class="p">(</span><span class="mf">5.0</span><span class="o">/</span><span class="mf">3.0</span><span class="p">)</span>
<span class="c1"># We split the data in test and training data</span>
<span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Energies</span><span class="p">,</span> <span class="n">test_size</span><span class="o">=</span><span class="mf">0.2</span><span class="p">)</span>
<span class="c1"># matrix inversion to find beta</span>
<span class="n">beta</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">inv</span><span class="p">(</span><span class="n">X_train</span><span class="o">.</span><span class="n">T</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">X_train</span><span class="p">))</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">X_train</span><span class="o">.</span><span class="n">T</span><span class="p">)</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">y_train</span><span class="p">)</span>
<span class="c1"># and then make the prediction</span>
<span class="n">ytilde</span> <span class="o">=</span> <span class="n">X_train</span> <span class="o">@</span> <span class="n">beta</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Training R2&quot;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">R2</span><span class="p">(</span><span class="n">y_train</span><span class="p">,</span><span class="n">ytilde</span><span class="p">))</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Training MSE&quot;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">MSE</span><span class="p">(</span><span class="n">y_train</span><span class="p">,</span><span class="n">ytilde</span><span class="p">))</span>
<span class="n">ypredict</span> <span class="o">=</span> <span class="n">X_test</span> <span class="o">@</span> <span class="n">beta</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Test R2&quot;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">R2</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span><span class="n">ypredict</span><span class="p">))</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Test MSE&quot;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">MSE</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span><span class="n">ypredict</span><span class="p">))</span>
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="the-boston-housing-data-example">
<h2><span class="section-number">1.16. </span>The Boston housing data example<a class="headerlink" href="#the-boston-housing-data-example" title="Permalink to this headline"></a></h2>
<p>The Boston housing<br />
data set was originally a part of UCI Machine Learning Repository
and has been removed now. The data set is now included in <strong>Scikit-Learn</strong>’s
library. There are 506 samples and 13 feature (predictor) variables
in this data set. The objective is to predict the value of prices of
the house using the features (predictors) listed here.</p>
<p>The features/predictors are</p>
<ol class="simple">
<li><p>CRIM: Per capita crime rate by town</p></li>
<li><p>ZN: Proportion of residential land zoned for lots over 25000 square feet</p></li>
<li><p>INDUS: Proportion of non-retail business acres per town</p></li>
<li><p>CHAS: Charles River dummy variable (= 1 if tract bounds river; 0 otherwise)</p></li>
<li><p>NOX: Nitric oxide concentration (parts per 10 million)</p></li>
<li><p>RM: Average number of rooms per dwelling</p></li>
<li><p>AGE: Proportion of owner-occupied units built prior to 1940</p></li>
<li><p>DIS: Weighted distances to five Boston employment centers</p></li>
<li><p>RAD: Index of accessibility to radial highways</p></li>
<li><p>TAX: Full-value property tax rate per USD10000</p></li>
<li><p>B: <span class="math notranslate nohighlight">\(1000(Bk - 0.63)^2\)</span>, where <span class="math notranslate nohighlight">\(Bk\)</span> is the proportion of [people of African American descent] by town</p></li>
<li><p>LSTAT: Percentage of lower status of the population</p></li>
<li><p>MEDV: Median value of owner-occupied homes in USD 1000s</p></li>
</ol>
</div>
<div class="section" id="housing-data-the-code">
<h2><span class="section-number">1.17. </span>Housing data, the code<a class="headerlink" href="#housing-data-the-code" title="Permalink to this headline"></a></h2>
<p>We start by importing the libraries</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
<span class="kn">import</span> <span class="nn">seaborn</span> <span class="k">as</span> <span class="nn">sns</span>
</pre></div>
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</div>
</div>
<p>and load the Boston Housing DataSet from <strong>Scikit-Learn</strong></p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">load_boston</span>
<span class="n">boston_dataset</span> <span class="o">=</span> <span class="n">load_boston</span><span class="p">()</span>
<span class="c1"># boston_dataset is a dictionary</span>
<span class="c1"># let&#39;s check what it contains</span>
<span class="n">boston_dataset</span><span class="o">.</span><span class="n">keys</span><span class="p">()</span>
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</div>
</div>
<p>Then we invoke Pandas</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">boston</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">boston_dataset</span><span class="o">.</span><span class="n">data</span><span class="p">,</span> <span class="n">columns</span><span class="o">=</span><span class="n">boston_dataset</span><span class="o">.</span><span class="n">feature_names</span><span class="p">)</span>
<span class="n">boston</span><span class="o">.</span><span class="n">head</span><span class="p">()</span>
<span class="n">boston</span><span class="p">[</span><span class="s1">&#39;MEDV&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">boston_dataset</span><span class="o">.</span><span class="n">target</span>
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</div>
<p>and preprocess the data</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># check for missing values in all the columns</span>
<span class="n">boston</span><span class="o">.</span><span class="n">isnull</span><span class="p">()</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span>
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</div>
<p>We can then visualize the data</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># set the size of the figure</span>
<span class="n">sns</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="n">rc</span><span class="o">=</span><span class="p">{</span><span class="s1">&#39;figure.figsize&#39;</span><span class="p">:(</span><span class="mf">11.7</span><span class="p">,</span><span class="mf">8.27</span><span class="p">)})</span>
<span class="c1"># plot a histogram showing the distribution of the target values</span>
<span class="n">sns</span><span class="o">.</span><span class="n">distplot</span><span class="p">(</span><span class="n">boston</span><span class="p">[</span><span class="s1">&#39;MEDV&#39;</span><span class="p">],</span> <span class="n">bins</span><span class="o">=</span><span class="mi">30</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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</div>
</div>
</div>
<p>It is now useful to look at the correlation matrix</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># compute the pair wise correlation for all columns </span>
<span class="n">correlation_matrix</span> <span class="o">=</span> <span class="n">boston</span><span class="o">.</span><span class="n">corr</span><span class="p">()</span><span class="o">.</span><span class="n">round</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
<span class="c1"># use the heatmap function from seaborn to plot the correlation matrix</span>
<span class="c1"># annot = True to print the values inside the square</span>
<span class="n">sns</span><span class="o">.</span><span class="n">heatmap</span><span class="p">(</span><span class="n">data</span><span class="o">=</span><span class="n">correlation_matrix</span><span class="p">,</span> <span class="n">annot</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
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</div>
</div>
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<p>From the above coorelation plot we can see that <strong>MEDV</strong> is strongly correlated to <strong>LSTAT</strong> and <strong>RM</strong>. We see also that <strong>RAD</strong> and <strong>TAX</strong> are stronly correlated, but we don’t include this in our features together to avoid multi-colinearity</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">20</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span>
<span class="n">features</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;LSTAT&#39;</span><span class="p">,</span> <span class="s1">&#39;RM&#39;</span><span class="p">]</span>
<span class="n">target</span> <span class="o">=</span> <span class="n">boston</span><span class="p">[</span><span class="s1">&#39;MEDV&#39;</span><span class="p">]</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">col</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">features</span><span class="p">):</span>
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">features</span><span class="p">)</span> <span class="p">,</span> <span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">boston</span><span class="p">[</span><span class="n">col</span><span class="p">]</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">target</span>
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">marker</span><span class="o">=</span><span class="s1">&#39;o&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="n">col</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="n">col</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;MEDV&#39;</span><span class="p">)</span>
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</div>
</div>
</div>
<p>Now we start training our model</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">X</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">c_</span><span class="p">[</span><span class="n">boston</span><span class="p">[</span><span class="s1">&#39;LSTAT&#39;</span><span class="p">],</span> <span class="n">boston</span><span class="p">[</span><span class="s1">&#39;RM&#39;</span><span class="p">]],</span> <span class="n">columns</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;LSTAT&#39;</span><span class="p">,</span><span class="s1">&#39;RM&#39;</span><span class="p">])</span>
<span class="n">Y</span> <span class="o">=</span> <span class="n">boston</span><span class="p">[</span><span class="s1">&#39;MEDV&#39;</span><span class="p">]</span>
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</div>
</div>
<p>We split the data into training and test sets</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
<span class="c1"># splits the training and test data set in 80% : 20%</span>
<span class="c1"># assign random_state to any value.This ensures consistency.</span>
<span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">Y_train</span><span class="p">,</span> <span class="n">Y_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">test_size</span> <span class="o">=</span> <span class="mf">0.2</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">X_train</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">X_test</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">Y_train</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">Y_test</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
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</div>
</div>
<p>Then we use the linear regression functionality from <strong>Scikit-Learn</strong></p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <span class="n">LinearRegression</span>
<span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">mean_squared_error</span><span class="p">,</span> <span class="n">r2_score</span>
<span class="n">lin_model</span> <span class="o">=</span> <span class="n">LinearRegression</span><span class="p">()</span>
<span class="n">lin_model</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">Y_train</span><span class="p">)</span>
<span class="c1"># model evaluation for training set</span>
<span class="n">y_train_predict</span> <span class="o">=</span> <span class="n">lin_model</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
<span class="n">rmse</span> <span class="o">=</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">mean_squared_error</span><span class="p">(</span><span class="n">Y_train</span><span class="p">,</span> <span class="n">y_train_predict</span><span class="p">)))</span>
<span class="n">r2</span> <span class="o">=</span> <span class="n">r2_score</span><span class="p">(</span><span class="n">Y_train</span><span class="p">,</span> <span class="n">y_train_predict</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;The model performance for training set&quot;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;--------------------------------------&quot;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;RMSE is </span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">rmse</span><span class="p">))</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;R2 score is </span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">r2</span><span class="p">))</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;</span><span class="se">\n</span><span class="s2">&quot;</span><span class="p">)</span>
<span class="c1"># model evaluation for testing set</span>
<span class="n">y_test_predict</span> <span class="o">=</span> <span class="n">lin_model</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
<span class="c1"># root mean square error of the model</span>
<span class="n">rmse</span> <span class="o">=</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">mean_squared_error</span><span class="p">(</span><span class="n">Y_test</span><span class="p">,</span> <span class="n">y_test_predict</span><span class="p">)))</span>
<span class="c1"># r-squared score of the model</span>
<span class="n">r2</span> <span class="o">=</span> <span class="n">r2_score</span><span class="p">(</span><span class="n">Y_test</span><span class="p">,</span> <span class="n">y_test_predict</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;The model performance for testing set&quot;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;--------------------------------------&quot;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;RMSE is </span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">rmse</span><span class="p">))</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;R2 score is </span><span class="si">{}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">r2</span><span class="p">))</span>
</pre></div>
</div>
</div>
</div>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># plotting the y_test vs y_pred</span>
<span class="c1"># ideally should have been a straight line</span>
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">Y_test</span><span class="p">,</span> <span class="n">y_test_predict</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="reducing-the-number-of-degrees-of-freedom-overarching-view">
<h2><span class="section-number">1.18. </span>Reducing the number of degrees of freedom, overarching view<a class="headerlink" href="#reducing-the-number-of-degrees-of-freedom-overarching-view" title="Permalink to this headline"></a></h2>
<p>Many Machine Learning problems involve thousands or even millions of
features for each training instance. Not only does this make training
extremely slow, it can also make it much harder to find a good
solution, as we will see. This problem is often referred to as the
curse of dimensionality. Fortunately, in real-world problems, it is
often possible to reduce the number of features considerably, turning
an intractable problem into a tractable one.</p>
<p>Later we will discuss some of the most popular dimensionality reduction
techniques: the principal component analysis (PCA), Kernel PCA, and
Locally Linear Embedding (LLE).</p>
<p>Principal component analysis and its various variants deal with the
problem of fitting a low-dimensional <a class="reference external" href="https://en.wikipedia.org/wiki/Affine_space">affine
subspace</a> to a set of of
data points in a high-dimensional space. With its family of methods it
is one of the most used tools in data modeling, compression and
visualization.</p>
<p>Before we proceed however, we will discuss how to preprocess our
data. Till now and in connection with our previous examples we have
not met so many cases where we are too sensitive to the scaling of our
data. Normally the data may need a rescaling and/or may be sensitive
to extreme values. Scaling the data renders our inputs much more
suitable for the algorithms we want to employ.</p>
<p><strong>Scikit-Learn</strong> has several functions which allow us to rescale the
data, normally resulting in much better results in terms of various
accuracy scores. The <strong>StandardScaler</strong> function in <strong>Scikit-Learn</strong>
ensures that for each feature/predictor we study the mean value is
zero and the variance is one (every column in the design/feature
matrix). This scaling has the drawback that it does not ensure that
we have a particular maximum or minimum in our data set. Another
function included in <strong>Scikit-Learn</strong> is the <strong>MinMaxScaler</strong> which
ensures that all features are exactly between <span class="math notranslate nohighlight">\(0\)</span> and <span class="math notranslate nohighlight">\(1\)</span>. The</p>
<p>The <strong>Normalizer</strong> scales each data
point such that the feature vector has a euclidean length of one. In other words, it
projects a data point on the circle (or sphere in the case of higher dimensions) with a
radius of 1. This means every data point is scaled by a different number (by the
inverse of it’s length).
This normalization is often used when only the direction (or angle) of the data matters,
not the length of the feature vector.</p>
<p>The <strong>RobustScaler</strong> works similarly to the StandardScaler in that it
ensures statistical properties for each feature that guarantee that
they are on the same scale. However, the RobustScaler uses the median
and quartiles, instead of mean and variance. This makes the
RobustScaler ignore data points that are very different from the rest
(like measurement errors). These odd data points are also called
outliers, and might often lead to trouble for other scaling
techniques.</p>
<div class="section" id="simple-preprocessing-examples-franke-function-and-regression">
<h3><span class="section-number">1.18.1. </span>Simple preprocessing examples, Franke function and regression<a class="headerlink" href="#simple-preprocessing-examples-franke-function-and-regression" title="Permalink to this headline"></a></h3>
<div class="cell docutils container">
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Common imports</span>
<span class="kn">import</span> <span class="nn">os</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">import</span> <span class="nn">sklearn.linear_model</span> <span class="k">as</span> <span class="nn">skl</span>
<span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">mean_squared_error</span>
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
<span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">import</span> <span class="n">MinMaxScaler</span><span class="p">,</span> <span class="n">StandardScaler</span><span class="p">,</span> <span class="n">Normalizer</span>
<span class="c1"># Where to save the figures and data files</span>
<span class="n">PROJECT_ROOT_DIR</span> <span class="o">=</span> <span class="s2">&quot;Results&quot;</span>
<span class="n">FIGURE_ID</span> <span class="o">=</span> <span class="s2">&quot;Results/FigureFiles&quot;</span>
<span class="n">DATA_ID</span> <span class="o">=</span> <span class="s2">&quot;DataFiles/&quot;</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="n">PROJECT_ROOT_DIR</span><span class="p">):</span>
<span class="n">os</span><span class="o">.</span><span class="n">mkdir</span><span class="p">(</span><span class="n">PROJECT_ROOT_DIR</span><span class="p">)</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="n">FIGURE_ID</span><span class="p">):</span>
<span class="n">os</span><span class="o">.</span><span class="n">makedirs</span><span class="p">(</span><span class="n">FIGURE_ID</span><span class="p">)</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="n">DATA_ID</span><span class="p">):</span>
<span class="n">os</span><span class="o">.</span><span class="n">makedirs</span><span class="p">(</span><span class="n">DATA_ID</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">image_path</span><span class="p">(</span><span class="n">fig_id</span><span class="p">):</span>
<span class="k">return</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">FIGURE_ID</span><span class="p">,</span> <span class="n">fig_id</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">data_path</span><span class="p">(</span><span class="n">dat_id</span><span class="p">):</span>
<span class="k">return</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">DATA_ID</span><span class="p">,</span> <span class="n">dat_id</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">save_fig</span><span class="p">(</span><span class="n">fig_id</span><span class="p">):</span>
<span class="n">plt</span><span class="o">.</span><span class="n">savefig</span><span class="p">(</span><span class="n">image_path</span><span class="p">(</span><span class="n">fig_id</span><span class="p">)</span> <span class="o">+</span> <span class="s2">&quot;.png&quot;</span><span class="p">,</span> <span class="nb">format</span><span class="o">=</span><span class="s1">&#39;png&#39;</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">FrankeFunction</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">):</span>
<span class="n">term1</span> <span class="o">=</span> <span class="mf">0.75</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="p">(</span><span class="mf">0.25</span><span class="o">*</span><span class="p">(</span><span class="mi">9</span><span class="o">*</span><span class="n">x</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span> <span class="o">-</span> <span class="mf">0.25</span><span class="o">*</span><span class="p">((</span><span class="mi">9</span><span class="o">*</span><span class="n">y</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
<span class="n">term2</span> <span class="o">=</span> <span class="mf">0.75</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="p">((</span><span class="mi">9</span><span class="o">*</span><span class="n">x</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="mf">49.0</span> <span class="o">-</span> <span class="mf">0.1</span><span class="o">*</span><span class="p">(</span><span class="mi">9</span><span class="o">*</span><span class="n">y</span><span class="o">+</span><span class="mi">1</span><span class="p">))</span>
<span class="n">term3</span> <span class="o">=</span> <span class="mf">0.5</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="p">(</span><span class="mi">9</span><span class="o">*</span><span class="n">x</span><span class="o">-</span><span class="mi">7</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="mf">4.0</span> <span class="o">-</span> <span class="mf">0.25</span><span class="o">*</span><span class="p">((</span><span class="mi">9</span><span class="o">*</span><span class="n">y</span><span class="o">-</span><span class="mi">3</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
<span class="n">term4</span> <span class="o">=</span> <span class="o">-</span><span class="mf">0.2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="p">(</span><span class="mi">9</span><span class="o">*</span><span class="n">x</span><span class="o">-</span><span class="mi">4</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="p">(</span><span class="mi">9</span><span class="o">*</span><span class="n">y</span><span class="o">-</span><span class="mi">7</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
<span class="k">return</span> <span class="n">term1</span> <span class="o">+</span> <span class="n">term2</span> <span class="o">+</span> <span class="n">term3</span> <span class="o">+</span> <span class="n">term4</span>
<span class="k">def</span> <span class="nf">create_X</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">n</span> <span class="p">):</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">ravel</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">ravel</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
<span class="n">N</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="n">l</span> <span class="o">=</span> <span class="nb">int</span><span class="p">((</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">n</span><span class="o">+</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span> <span class="c1"># Number of elements in beta</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="n">N</span><span class="p">,</span><span class="n">l</span><span class="p">))</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">):</span>
<span class="n">q</span> <span class="o">=</span> <span class="nb">int</span><span class="p">((</span><span class="n">i</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
<span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">):</span>
<span class="n">X</span><span class="p">[:,</span><span class="n">q</span><span class="o">+</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="n">i</span><span class="o">-</span><span class="n">k</span><span class="p">))</span><span class="o">*</span><span class="p">(</span><span class="n">y</span><span class="o">**</span><span class="n">k</span><span class="p">)</span>
<span class="k">return</span> <span class="n">X</span>
<span class="c1"># Making meshgrid of datapoints and compute Franke&#39;s function</span>
<span class="n">n</span> <span class="o">=</span> <span class="mi">5</span>
<span class="n">N</span> <span class="o">=</span> <span class="mi">1000</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">N</span><span class="p">))</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">N</span><span class="p">))</span>
<span class="n">z</span> <span class="o">=</span> <span class="n">FrankeFunction</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">create_X</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="n">n</span><span class="p">)</span>
<span class="c1"># split in training and test data</span>
<span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">z</span><span class="p">,</span><span class="n">test_size</span><span class="o">=</span><span class="mf">0.2</span><span class="p">)</span>
<span class="n">clf</span> <span class="o">=</span> <span class="n">skl</span><span class="o">.</span><span class="n">LinearRegression</span><span class="p">()</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
<span class="c1"># The mean squared error and R2 score</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;MSE before scaling: </span><span class="si">{:.2f}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">mean_squared_error</span><span class="p">(</span><span class="n">clf</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">),</span> <span class="n">y_test</span><span class="p">)))</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;R2 score before scaling </span><span class="si">{:.2f}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">clf</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span><span class="n">y_test</span><span class="p">)))</span>
<span class="n">scaler</span> <span class="o">=</span> <span class="n">StandardScaler</span><span class="p">()</span>
<span class="n">scaler</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
<span class="n">X_train_scaled</span> <span class="o">=</span> <span class="n">scaler</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
<span class="n">X_test_scaled</span> <span class="o">=</span> <span class="n">scaler</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Feature min values before scaling:</span><span class="se">\n</span><span class="s2"> </span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">X_train</span><span class="o">.</span><span class="n">min</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)))</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Feature max values before scaling:</span><span class="se">\n</span><span class="s2"> </span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">X_train</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)))</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Feature min values after scaling:</span><span class="se">\n</span><span class="s2"> </span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">X_train_scaled</span><span class="o">.</span><span class="n">min</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)))</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Feature max values after scaling:</span><span class="se">\n</span><span class="s2"> </span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">X_train_scaled</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)))</span>
<span class="n">clf</span> <span class="o">=</span> <span class="n">skl</span><span class="o">.</span><span class="n">LinearRegression</span><span class="p">()</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train_scaled</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;MSE after scaling: </span><span class="si">{:.2f}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">mean_squared_error</span><span class="p">(</span><span class="n">clf</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test_scaled</span><span class="p">),</span> <span class="n">y_test</span><span class="p">)))</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;R2 score for scaled data: </span><span class="si">{:.2f}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">clf</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_test_scaled</span><span class="p">,</span><span class="n">y_test</span><span class="p">)))</span>
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