Morten Hjorth-Jensen
3591 lines
228 KiB
HTML
3591 lines
228 KiB
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<!-- ------------------- main content ---------------------- -->
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<center><h1>Week 35: Introduction to the course, Logistics and Practicalities</h1></center> <!-- document title -->
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<p>
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<!-- author(s): Morten Hjorth-Jensen -->
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<center>
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<b>Morten Hjorth-Jensen</b> [1, 2]
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</center>
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<p>
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<!-- institution(s) -->
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<center>[1] <b>Department of Physics, University of Oslo</b></center>
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<center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
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<br>
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<p>
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<center><h4>Aug 28, 2021</h4></center> <!-- date -->
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<br>
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<p>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="overview-of-first-week">Overview of first week </h2>
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<p>
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<div class="alert alert-block alert-block alert-text-normal">
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<b></b>
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<p>
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<ul>
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<li> Wednesday August 25: Introduction to software and repetition of Python Programming</li>
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<li> Thursday August 26: First lecture: Presentation of the course, aims and content</li>
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<li> Thursday: Second Lecture: Start with simple linear regression and repetition of linear algebra and elements of statistics</li>
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<li> Friday August 27: Linear regression</li>
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<li> Computer lab: Wednesdays, 8am-6pm. First time: Wednesday August 25.</li>
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</ul>
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</div>
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<p>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="reading-recommendations">Reading Recommendations </h2>
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<p>
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<div class="alert alert-block alert-block alert-text-normal">
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<b></b>
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<p>
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For the reading assignments we use the following abbreviations:
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<ul>
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<li> GBC: Goodfellow, Bengio, and Courville, Deep Learning</li>
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<li> CMB: Christopher M. Bishop, Pattern Recognition and Machine Learning</li>
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<li> HTF: Hastie, Tibshirani, and Friedman, The Elements of Statistical Learning</li>
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<li> AG: Aurelien Geron, Hands‑On Machine Learning with Scikit‑Learn and TensorFlow</li>
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</ul>
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Reading recommendations this week: Refresh linear algebra, GBC chapters 1 and 2. CMB sections 1.1 and 3.1. HTF chapters 2 and 3. Install scikit-learn. See lecture notes for week 35 at <a href="https://compphysics.github.io/MachineLearning/doc/web/course.html" target="_blank"><tt>https://compphysics.github.io/MachineLearning/doc/web/course.html</tt></a>
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</div>
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<p>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="thursday-august-26">Thursday August 26 </h2>
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<p>
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The lectures will be recorded and updated videos will be posted after the lectures.
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<p>
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"Video of Lecture August 26, 2021":"https://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/h21/forelesningsvideoer/LectureThursdayAugust26.mp4?vrtx=view-as-webpage
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<p>
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<b>Zoom link for lectures</b>: <a href="https://msu.zoom.us/j/93311529525?pwd=a1VXSzY4aTFWVy9Rb05mNDJTZ09lZz09" target="_blank"><tt>https://msu.zoom.us/j/93311529525?pwd=a1VXSzY4aTFWVy9Rb05mNDJTZ09lZz09</tt></a>
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<ul>
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<li> Meeting ID: 933 1152 9525</li>
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<li> Passcode: 646102</li>
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</ul>
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<a href="https://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/h20/forelesningsvideoer/zoom_0.mp4?vrtx=view-as-webpage" target="_blank">Video of Lecture from Fall Semester 2020</a>.
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<p>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="lectures-and-computerlab">Lectures and ComputerLab </h2>
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<p>
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<div class="alert alert-block alert-block alert-text-normal">
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<b></b>
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<p>
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<ul>
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<li> Lectures: Thursday (12.15pm-2pm and Friday (12.15pm-2pm).</li>
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<li> Weekly reading assignments and videos needed to solve projects and exercises.</li>
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<li> Weekly exercises when not working on projects. You can hand in exercises if you want.</li>
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<li> Detailed lecture notes, exercises, all programs presented, projects etc can be found at the homepage of the course.</li>
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<li> Weekly plans and all other information are on the official webpage.</li>
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<li> No final exam, three projects that are graded and have to be approved.</li>
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</ul>
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</div>
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<p>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="announcement">Announcement </h2>
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<p>
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<b>NORA AI competetion:</b> See the link here <a href="https://www.nora.ai/Competition/image-segmentation.html" target="_blank"><tt>https://www.nora.ai/Competition/image-segmentation.html</tt></a>
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<p>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="communication-channels">Communication channels </h2>
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<ul>
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<li> Chat and communications via <a href="canvas.uio.no" target="_blank"><tt>canvas.uio.no</tt></a>, GDPR safe</li>
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<li> <b>Slack</b> channel: machinelearninguio.slack.com</li>
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<li> <b>Piazza</b> : enlist at <a href="https:piazza.com/uio.no/fall2021/fysstk4155" target="_blank"><tt>https:piazza.com/uio.no/fall2021/fysstk4155</tt></a></li>
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</ul>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="course-format">Course Format </h2>
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<p>
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<div class="alert alert-block alert-block alert-text-normal">
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<b></b>
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<p>
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<ul>
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<li> Three compulsory projects. Electronic reports only using <a href="https://www.uio.no/english/services/it/education/canvas/" target="_blank">Canvas</a> to hand in projects and <a href="https://git-scm.com/" target="_blank">git</a> as version control software and <a href="https://github.com/" target="_blank">GitHub</a> for repository (or <a href="https://about.gitlab.com/" target="_blank">GitLab</a>) of all your material.</li>
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<li> Evaluation and grading: The three projects are graded and each counts 1/3 of the final mark. No final written or oral exam.
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<ol type="a"></li>
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<li> For the last project each group/participant submits a proposal or works with suggested (by us) proposals for the project.</li>
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<li> If possible, we would like to organize the last project as a workshop where each group makes a poster and presents this to all other participants of the course</li>
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<li> Poster session where all participants can study and discuss the other proposals.</li>
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<li> Based on feedback etc, each group finalizes the report and submits for grading.</li>
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</ol>
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<li> Python is the default programming language, but feel free to use C/C++ and/or Fortran or other programming languages. All source codes discussed during the lectures can be found at the webpage and <a href="https://github.com/CompPhysics/MachineLearning/tree/master/doc/Programs" target="_blank">github address</a> of the course.</li>
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</ul>
|
|
</div>
|
|
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="teachers">Teachers </h2>
|
|
|
|
<p>
|
|
<div class="alert alert-block alert-block alert-text-normal">
|
|
<b></b>
|
|
<p>
|
|
|
|
<p>
|
|
<b>Teachers :</b>
|
|
|
|
<ul>
|
|
<li> Morten Hjorth-Jensen, morten.hjorth-jensen@fys.uio.no</li>
|
|
|
|
<ul>
|
|
<li> <b>Phone</b>: +47-48257387</li>
|
|
<li> <b>Office</b>: Department of Physics, University of Oslo, Eastern wing, room FØ470</li>
|
|
<li> <b>Office hours</b>: <em>Anytime</em>! Individual or group office hours can be arranged either in person or via zoom. Feel free to send an email for planning.</li>
|
|
</ul>
|
|
|
|
<li> Øyvind Sigmundson Schøyen, oyvinssc@student.matnat.uio.no</li>
|
|
|
|
<ul>
|
|
<li> <b>Office</b>: Department of Physics, University of Oslo, Eastern wing, room FØ452</li>
|
|
</ul>
|
|
|
|
<li> Stian Dysthe Bilek stian.bilek@fys.uio.no</li>
|
|
|
|
<ul>
|
|
<li> <b>Office</b>: Department of Physics, University of Oslo, Eastern wing, room FØ450</li>
|
|
</ul>
|
|
|
|
<li> Linus Ekstrøm, linueks@gmail.com, linus.ekstrom@fys.uio.no</li>
|
|
<li> Nicholas Karlsen, nicholaskarlsen1102@gmail.com, nicholas.karlsen@fys.uio.no</li>
|
|
<li> Bendik Steinsvåg Dalen, b.s.dalen@fys.uio.no</li>
|
|
<li> Philip Karim Sørli Niane, p.k.s.niane@fys.uio.no</li>
|
|
</ul>
|
|
</div>
|
|
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="deadlines-for-projects-tentative">Deadlines for projects (tentative) </h2>
|
|
|
|
<p>
|
|
<div class="alert alert-block alert-block alert-text-normal">
|
|
<b></b>
|
|
<p>
|
|
|
|
<ol>
|
|
<li> Project 1: October 4 (available September 10) graded with feedback)</li>
|
|
<li> Project 2: November 8 (available October 8, graded with feedback)</li>
|
|
<li> Project 3: December 13 (available November 12, graded with feedback)</li>
|
|
</ol>
|
|
|
|
Projects are handed in using <b>Canvas</b>. We use Github as repository for codes, benchmark calculations etc. Comments and feedback on projects only via <b>Canvas</b>.
|
|
|
|
|
|
</div>
|
|
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="recommended-textbooks">Recommended textbooks </h2>
|
|
|
|
<ol>
|
|
<li> The lecture notes are collected as a jupyter-book at <a href="https://compphysics.github.io/MachineLearning/doc/LectureNotes/_build/html/intro.html." target="_blank"><tt>https://compphysics.github.io/MachineLearning/doc/LectureNotes/_build/html/intro.html.</tt></a></li>
|
|
</ol>
|
|
|
|
In addition to the lecture notes, we recommend the books of Bishop and Goodfellow et al. We will follow these texts closely and the weekly reading assignments refer to these two texts. The text by Hastie et al is also widely used in the Machine Learning community. Finally, we also recommend the hands-on text by Geron, see below.
|
|
|
|
<ol>
|
|
<li> Christopher M. Bishop, Pattern Recognition and Machine Learning, Springer, <a href="https://www.springer.com/gp/book/9780387310732." target="_blank"><tt>https://www.springer.com/gp/book/9780387310732.</tt></a></li>
|
|
<li> Ian Goodfellow, Yoshua Bengio, and Aaron Courville. The different chapters are available for free at <a href="https://www.deeplearningbook.org/." target="_blank"><tt>https://www.deeplearningbook.org/.</tt></a> Chapters 2-14 are highly recommended. The lectures follow to a larg extent this text. The weekly plans will include reading suggestions from these two textbooks.</li>
|
|
</ol>
|
|
|
|
Additional textbooks:
|
|
|
|
<ol>
|
|
<li> Trevor Hastie, Robert Tibshirani, Jerome H. Friedman, The Elements of Statistical Learning, Springer, <a href="https://www.springer.com/gp/book/9780387848570." target="_blank"><tt>https://www.springer.com/gp/book/9780387848570.</tt></a> This is a well-known text and serves as additional literature.</li>
|
|
<li> Aurelien Geron, Hands‑On Machine Learning with Scikit‑Learn and TensorFlow, O'Reilly, <a href="https://www.oreilly.com/library/view/hands-on-machine-learning/9781492032632/." target="_blank"><tt>https://www.oreilly.com/library/view/hands-on-machine-learning/9781492032632/.</tt></a> This text is very useful since it contains many code examples and hands-on applications of all algorithms discussed in this course.</li>
|
|
</ol>
|
|
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="prerequisites">Prerequisites </h2>
|
|
|
|
<p>
|
|
Basic knowledge in programming and mathematics, with an emphasis on
|
|
linear algebra. Knowledge of Python or/and C++ as programming
|
|
languages is strongly recommended and experience with Jupiter notebook
|
|
is recommended. Required courses are the equivalents to the University
|
|
of Oslo mathematics courses MAT1100, MAT1110, MAT1120 and at least one
|
|
of the corresponding computing and programming courses INF1000/INF1110
|
|
or MAT-INF1100/MAT-INF1100L/BIOS1100/KJM-INF1100. Most universities
|
|
offer nowadays a basic programming course (often compulsory) where
|
|
Python is the recurring programming language.
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="learning-outcomes">Learning outcomes </h2>
|
|
|
|
<p>
|
|
<div class="alert alert-block alert-block alert-text-normal">
|
|
<b></b>
|
|
<p>
|
|
|
|
<p>
|
|
This course aims at giving you insights and knowledge about many of
|
|
the central algorithms used in Data Analysis and Machine Learning.
|
|
The course is project based and through various numerical projects,
|
|
normally three, you will be exposed to fundamental research problems
|
|
in these fields, with the aim to reproduce state of the art scientific
|
|
results. Both supervised and unsupervised methods will be covered. The
|
|
emphasis is on a frequentist approach, although we will try to link it
|
|
with a Bayesian approach as well. You will learn to develop and
|
|
structure large codes for studying different cases where Machine
|
|
Learning is applied to, get acquainted with computing facilities and
|
|
learn to handle large scientific projects. A good scientific and
|
|
ethical conduct is emphasized throughout the course. More
|
|
specifically, after this course you will
|
|
|
|
<ul>
|
|
<li> Learn about basic data analysis, statistical analysis, Bayesian statistics, Monte Carlo sampling, data optimization and machine learning;</li>
|
|
<li> Be capable of extending the acquired knowledge to other systems and cases;</li>
|
|
<li> Have an understanding of central algorithms used in data analysis and machine learning;</li>
|
|
<li> Understand linear methods for regression and classification, from ordinary least squares, via Lasso and Ridge to Logistic regression;</li>
|
|
<li> Learn about neural networks and deep learning methods for supervised and unsupervised learning. Emphasis on feed forward neural networks, convolutional and recurrent neural networks;</li>
|
|
<li> Learn about about decision trees, random forests, bagging and boosting methods;</li>
|
|
<li> Learn about support vector machines and kernel transformations;</li>
|
|
<li> Reduction of data sets, from PCA to clustering;</li>
|
|
<li> Autoencoders and Reinforcement Learning;</li>
|
|
<li> Work on numerical projects to illustrate the theory. The projects play a central role and you are expected to know modern programming languages like Python or C++ and/or Fortran (Fortran2003 or later) or Julia or other.</li>
|
|
</ul>
|
|
</div>
|
|
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="topics-covered-in-this-course-statistical-analysis-and-optimization-of-data">Topics covered in this course: Statistical analysis and optimization of data </h2>
|
|
|
|
<p>
|
|
The course has two central parts
|
|
|
|
<ol>
|
|
<li> Statistical analysis and optimization of data</li>
|
|
<li> Machine learning</li>
|
|
</ol>
|
|
|
|
These topics will be scattered thorughout the course and may not necessarily be taught separately. Rather, we will often take an approach (during the lectures and project/exercise sessions) where say elements from statistical data analysis are mixed with specific Machine Learning algorithms
|
|
|
|
<p>
|
|
<div class="alert alert-block alert-block alert-text-normal">
|
|
<b>Statistical analysis and optimization of data</b>
|
|
<p>
|
|
|
|
<p>
|
|
We plan to cover the following topics:
|
|
|
|
<ul>
|
|
<li> Basic concepts, expectation values, variance, covariance, correlation functions and errors;</li>
|
|
<li> Simpler models, binomial distribution, the Poisson distribution, simple and multivariate normal distributions;</li>
|
|
<li> Central elements of Bayesian statistics and modeling;</li>
|
|
<li> Gradient methods for data optimization,</li>
|
|
<li> Monte Carlo methods, Markov chains, Gibbs sampling and Metropolis-Hastings sampling;</li>
|
|
<li> Estimation of errors and resampling techniques such as the cross-validation, blocking, bootstrapping and jackknife methods;</li>
|
|
<li> Principal Component Analysis (PCA) and its mathematical foundation</li>
|
|
</ul>
|
|
</div>
|
|
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="topics-covered-in-this-course-machine-learning">Topics covered in this course: Machine Learning </h2>
|
|
|
|
<p>
|
|
<div class="alert alert-block alert-block alert-text-normal">
|
|
<b></b>
|
|
<p>
|
|
The following topics will be covered
|
|
|
|
<ul>
|
|
<li> Linear Regression and Logistic Regression;</li>
|
|
<li> Neural networks and deep learning, including convolutional and recurrent neural networks</li>
|
|
<li> Decisions trees, Random Forests, Bagging and Boosting</li>
|
|
<li> Support vector machines</li>
|
|
<li> Bayesian linear and logistic regression</li>
|
|
<li> Boltzmann Machines</li>
|
|
<li> Unsupervised learning Dimensionality reduction, from PCA to clustering</li>
|
|
</ul>
|
|
|
|
Hands-on demonstrations, exercises and projects aim at deepening your understanding of these topics.
|
|
|
|
|
|
</div>
|
|
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="extremely-useful-tools-strongly-recommended">Extremely useful tools, strongly recommended </h2>
|
|
|
|
<p>
|
|
<div class="alert alert-block alert-block alert-text-normal">
|
|
<b>and discussed at the lab sessions</b>
|
|
<p>
|
|
|
|
<ul>
|
|
<li> GIT for version control, and GitHub or GitLab as repositories, highly recommended. This will be discussed during the first exercise session</li>
|
|
<li> Anaconda and other Python environments, see intro slides and links to programming resources at <a href="https://computationalscienceuio.github.io/RefreshProgrammingSkills/intro.html" target="_blank"><tt>https://computationalscienceuio.github.io/RefreshProgrammingSkills/intro.html</tt></a></li>
|
|
</ul>
|
|
</div>
|
|
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="other-courses-on-data-science-and-machine-learning-at-uio">Other courses on Data science and Machine Learning at UiO </h2>
|
|
|
|
<p>
|
|
The link here <a href="https://www.mn.uio.no/english/research/about/centre-focus/innovation/data-science/studies/" target="_blank"><tt>https://www.mn.uio.no/english/research/about/centre-focus/innovation/data-science/studies/</tt></a> gives an excellent overview of courses on Machine learning at UiO.
|
|
|
|
<ol>
|
|
<li> <a href="http://www.uio.no/studier/emner/matnat/math/STK2100/index-eng.html" target="_blank">STK2100 Machine learning and statistical methods for prediction and classification</a>.</li>
|
|
<li> <a href="https://www.uio.no/studier/emner/matnat/ifi/IN3050/index-eng.html" target="_blank">IN3050/4050 Introduction to Artificial Intelligence and Machine Learning</a>. Introductory course in machine learning and AI with an algorithmic approach.</li>
|
|
<li> <a href="http://www.uio.no/studier/emner/matnat/math/STK-INF3000/index-eng.html" target="_blank">STK-INF3000/4000 Selected Topics in Data Science</a>. The course provides insight into selected contemporary relevant topics within Data Science.</li>
|
|
<li> <a href="https://www.uio.no/studier/emner/matnat/ifi/IN4080/index.html" target="_blank">IN4080 Natural Language Processing</a>. Probabilistic and machine learning techniques applied to natural language processing.</li>
|
|
<li> <a href="https://www.uio.no/studier/emner/matnat/math/STK-IN4300/index-eng.html" target="_blank">STK-IN4300 Statistical learning methods in Data Science</a>. An advanced introduction to statistical and machine learning. For students with a good mathematics and statistics background.</li>
|
|
<li> <a href="http://www.uio.no/studier/emner/matnat/ifi/INF4490/" target="_blank">INF4490 Biologically Inspired Computing</a>. An introduction to self-adapting methods also called artificial intelligence or machine learning.</li>
|
|
<li> <a href="https://www.uio.no/studier/emner/matnat/ifi/IN-STK5000/index-eng.html" target="_blank">IN-STK5000 Adaptive Methods for Data-Based Decision Making</a>. Methods for adaptive collection and processing of data based on machine learning techniques.</li>
|
|
<li> <a href="https://www.uio.no/studier/emner/matnat/ifi/IN5400/" target="_blank">IN5400/INF5860 Machine Learning for Image Analysis</a>. An introduction to deep learning with particular emphasis on applications within Image analysis, but useful for other application areas too.</li>
|
|
<li> <a href="https://www.uio.no/studier/emner/matnat/its/TEK5040/" target="_blank">TEK5040 Deep learning for autonomous systems</a>. The course addresses advanced algorithms and architectures for deep learning with neural networks. The course provides an introduction to how deep-learning techniques can be used in the construction of key parts of advanced autonomous systems that exist in physical environments and cyber environments.</li>
|
|
<li> <a href="https://www.uio.no/studier/emner/matnat/math/STK4051/index-eng.html" target="_blank">STK4051 Computational Statistics</a></li>
|
|
<li> <a href="https://www.uio.no/studier/emner/matnat/math/STK4021/index-eng.html" target="_blank">STK4021 Applied Bayesian Analysis and Numerical Methods</a></li>
|
|
</ol>
|
|
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="introduction">Introduction </h2>
|
|
|
|
<p>
|
|
Our emphasis throughout this series of lectures
|
|
is on understanding the mathematical aspects of
|
|
different algorithms used in the fields of data analysis and machine learning.
|
|
|
|
<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 href="http://scikit-learn.org/stable/" target="_blank">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
|
|
<b>Scikit-Learn</b>.
|
|
|
|
<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>
|
|
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>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="what-is-machine-learning">What is Machine Learning? </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>
|
|
It has become more
|
|
and more common to see research projects on big data in for example
|
|
the Social Sciences where extracting patterns from complicated survey
|
|
data is one of many research directions. Having a solid grasp of data
|
|
analysis and machine learning is thus becoming central to scientific
|
|
computing in many fields, and competences and skills within the fields
|
|
of machine learning and scientific computing are nowadays strongly
|
|
requested by many potential employers. The latter cannot be
|
|
overstated, familiarity with machine learning has almost become a
|
|
prerequisite for many of the most exciting employment opportunities,
|
|
whether they are in bioinformatics, life science, physics or finance,
|
|
in the private or the public sector. This author has had several
|
|
students or met students who have been hired recently based on their
|
|
skills and competences in scientific computing and data science, often
|
|
with marginal knowledge of machine learning.
|
|
|
|
<p>
|
|
Machine learning is a subfield of computer science, and is closely
|
|
related to computational statistics. It evolved from the study of
|
|
pattern recognition in artificial intelligence (AI) research, and has
|
|
made contributions to AI tasks like computer vision, natural language
|
|
processing and speech recognition. Many of the methods we will study are also
|
|
strongly rooted in basic mathematics and physics research.
|
|
|
|
<p>
|
|
Ideally, machine learning represents the science of giving computers
|
|
the ability to learn without being explicitly programmed. The idea is
|
|
that there exist generic algorithms which can be used to find patterns
|
|
in a broad class of data sets without having to write code
|
|
specifically for each problem. The algorithm will build its own logic
|
|
based on the data. You should however always keep in mind that
|
|
machines and algorithms are to a large extent developed by humans. The
|
|
insights and knowledge we have about a specific system, play a central
|
|
role when we develop a specific machine learning algorithm.
|
|
|
|
<p>
|
|
Machine learning is an extremely rich field, in spite of its young
|
|
age. The increases we have seen during the last three decades in
|
|
computational capabilities have been followed by developments of
|
|
methods and techniques for analyzing and handling large date sets,
|
|
relying heavily on statistics, computer science and mathematics. The
|
|
field is rather new and developing rapidly. Popular software packages
|
|
written in Python for machine learning like
|
|
<a href="http://scikit-learn.org/stable/" target="_blank">Scikit-learn</a>,
|
|
<a href="https://www.tensorflow.org/" target="_blank">Tensorflow</a>,
|
|
<a href="http://pytorch.org/" target="_blank">PyTorch</a> and <a href="https://keras.io/" target="_blank">Keras</a>, all
|
|
freely available at their respective GitHub sites, encompass
|
|
communities of developers in the thousands or more. And the number of
|
|
code developers and contributors keeps increasing. Not all the
|
|
algorithms and methods can be given a rigorous mathematical
|
|
justification, opening up thereby large rooms for experimenting and
|
|
trial and error and thereby exciting new developments. However, a
|
|
solid command of linear algebra, multivariate theory, probability
|
|
theory, statistical data analysis, understanding errors and Monte
|
|
Carlo methods are central elements in a proper understanding of many
|
|
of algorithms and methods we will discuss.
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="types-of-machine-learning">Types of Machine Learning </h2>
|
|
|
|
<p>
|
|
The approaches to machine learning are many, but are often split into
|
|
two main categories. In <em>supervised learning</em> we know the answer to a
|
|
problem, and let the computer deduce the logic behind it. On the other
|
|
hand, <em>unsupervised learning</em> is a method for finding patterns and
|
|
relationship in data sets without any prior knowledge of the system.
|
|
Some authours also operate with a third category, namely
|
|
<em>reinforcement learning</em>. This is a paradigm of learning inspired by
|
|
behavioral psychology, where learning is achieved by trial-and-error,
|
|
solely from rewards and punishment.
|
|
|
|
<p>
|
|
Another way to categorize machine learning tasks is to consider the
|
|
desired output of a system. Some of the most common tasks are:
|
|
|
|
<ul>
|
|
<li> Classification: Outputs are divided into two or more classes. The goal is to produce a model that assigns inputs into one of these classes. An example is to identify digits based on pictures of hand-written ones. Classification is typically supervised learning.</li>
|
|
<li> Regression: Finding a functional relationship between an input data set and a reference data set. The goal is to construct a function that maps input data to continuous output values.</li>
|
|
<li> Clustering: Data are divided into groups with certain common traits, without knowing the different groups beforehand. It is thus a form of unsupervised learning.</li>
|
|
</ul>
|
|
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="essential-elements-of-ml">Essential elements of ML </h2>
|
|
|
|
<p>
|
|
The methods we cover have three main topics in common, irrespective of
|
|
whether we deal with supervised or unsupervised learning.
|
|
<!-- !bpop -->
|
|
|
|
<ul>
|
|
<li> The first ingredient is normally our data set (which can be subdivided into training, validation and test data). Many find the most difficult part of using Machine Learning to be the set up of your data in a meaningful way.</li>
|
|
<li> The second item is a model which is normally a function of some parameters. The model reflects our knowledge of the system (or lack thereof). As an example, if we know that our data show a behavior similar to what would be predicted by a polynomial, fitting our data to a polynomial of some degree would then determin our model.</li>
|
|
<li> The last ingredient is a so-called <b>cost/loss</b> function (or error or risk function) which allows us to present an estimate on how good our model is in reproducing the data it is supposed to train.</li>
|
|
</ul>
|
|
|
|
<!-- !epop -->
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="an-optimization-minimization-problem">An optimization/minimization problem </h2>
|
|
|
|
<p>
|
|
At the heart of basically all Machine Learning algorithms we will encounter so-called minimization or optimization algorithms. A large family of such methods are so-called <b>gradient methods</b>.
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="a-frequentist-approach-to-data-analysis">A Frequentist approach to data analysis </h2>
|
|
|
|
<p>
|
|
When you hear phrases like <b>predictions and estimations</b> and
|
|
<b>correlations and causations</b>, what do you think of? May be you think
|
|
of the difference between classifying new data points and generating
|
|
new data points.
|
|
Or perhaps you consider that correlations represent some kind of symmetric statements like
|
|
if \( A \) is correlated with \( B \), then \( B \) is correlated with
|
|
\( A \). Causation on the other hand is directional, that is if \( A \) causes \( B \), \( B \) does not
|
|
necessarily cause \( A \).
|
|
|
|
<p>
|
|
These concepts are in some sense the difference between machine
|
|
learning and statistics. In machine learning and prediction based
|
|
tasks, we are often interested in developing algorithms that are
|
|
capable of learning patterns from given data in an automated fashion,
|
|
and then using these learned patterns to make predictions or
|
|
assessments of newly given data. In many cases, our primary concern
|
|
is the quality of the predictions or assessments, and we are less
|
|
concerned about the underlying patterns that were learned in order
|
|
to make these predictions.
|
|
|
|
<p>
|
|
In machine learning we normally use <a href="https://en.wikipedia.org/wiki/Frequentist_inference" target="_blank">a so-called frequentist approach</a>,
|
|
where the aim is to make predictions and find correlations. We focus
|
|
less on for example extracting a probability distribution function (PDF). The PDF can be
|
|
used in turn to make estimations and find causations such as given \( A \)
|
|
what is the likelihood of finding \( B \).
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="what-is-a-good-model">What is a good model? </h2>
|
|
|
|
<p>
|
|
In science and engineering we often end up in situations where we want to infer (or learn) a
|
|
quantitative model \( M \) for a given set of sample points \( \boldsymbol{X} \in [x_1, x_2,\dots x_N] \).
|
|
|
|
<p>
|
|
As we will see repeatedely in these lectures, we could try to fit these data points to a model given by a
|
|
straight line, or if we wish to be more sophisticated to a more complex
|
|
function.
|
|
|
|
<p>
|
|
The reason for inferring such a model is that it
|
|
serves many useful purposes. On the one hand, the model can reveal information
|
|
encoded in the data or underlying mechanisms from which the data were generated. For instance, we could discover important
|
|
corelations that relate interesting physics interpretations.
|
|
|
|
<p>
|
|
In addition, it can simplify the representation of the given data set and help
|
|
us in making predictions about future data samples.
|
|
|
|
<p>
|
|
A first important consideration to keep in mind is that inferring the <em>correct</em> model
|
|
for a given data set is an elusive, if not impossible, task. The fundamental difficulty
|
|
is that if we are not specific about what we mean by a <em>correct</em> model, there
|
|
could easily be many different models that fit the given data set <em>equally well</em>.
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="what-is-a-good-model-can-we-define-it">What is a good model? Can we define it? </h2>
|
|
|
|
<p>
|
|
The central question is this: what leads us to say that a model is correct or
|
|
optimal for a given data set? To make the model inference problem well posed, i.e.,
|
|
to guarantee that there is a unique optimal model for the given data, we need to
|
|
impose additional assumptions or restrictions on the class of models considered. To
|
|
this end, we should not be looking for just any model that can describe the data.
|
|
Instead, we should look for a <b>model</b> \( M \) that is the best among a restricted class
|
|
of models. In addition, to make the model inference problem computationally
|
|
tractable, we need to specify how restricted the class of models needs to be. A
|
|
common strategy is to start
|
|
with the simplest possible class of models that is just necessary to describe the data
|
|
or solve the problem at hand. More precisely, the model class should be rich enough
|
|
to contain at least one model that can fit the data to a desired accuracy and yet be
|
|
restricted enough that it is relatively simple to find the best model for the given data.
|
|
|
|
<p>
|
|
Thus, the most popular strategy is to start from the
|
|
simplest class of models and increase the complexity of the models only when the
|
|
simpler models become inadequate. For instance, if we work with a regression problem to fit a set of sample points, one
|
|
may first try the simplest class of models, namely linear models, followed obviously by more complex models.
|
|
|
|
<p>
|
|
How to evaluate which model fits best the data is something we will come back to over and over again in these sets of lectures.
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="software-and-needed-installations">Software and needed installations </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 <b>R</b>
|
|
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>
|
|
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 <b>pip</b> as
|
|
|
|
<ol>
|
|
<li> pip install numpy scipy matplotlib ipython scikit-learn mglearn sympy pandas pillow</li>
|
|
</ol>
|
|
|
|
For Python3, replace <b>pip</b> with <b>pip3</b>.
|
|
|
|
<p>
|
|
For OSX users we recommend, after having installed Xcode, to
|
|
install <b>brew</b>. Brew allows for a seamless installation of additional
|
|
software via for example
|
|
|
|
<ol>
|
|
<li> brew install python3</li>
|
|
</ol>
|
|
|
|
For Linux users, with its variety of distributions like for example the widely popular Ubuntu distribution,
|
|
you can use <b>pip</b> as well and simply install Python as
|
|
|
|
<ol>
|
|
<li> sudo apt-get install python3 (or python for pyhton2.7)</li>
|
|
</ol>
|
|
|
|
etc etc.
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="python-installers">Python installers </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
|
|
|
|
<ul>
|
|
<li> <a href="https://docs.anaconda.com/" target="_blank">Anaconda</a>,</li>
|
|
</ul>
|
|
|
|
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 <b>conda</b>.
|
|
|
|
<ul>
|
|
<li> <a href="https://www.enthought.com/product/canopy/" target="_blank">Enthought canopy</a></li>
|
|
</ul>
|
|
|
|
is a Python
|
|
distribution for scientific and analytic computing distribution and
|
|
analysis environment, available for free and under a commercial
|
|
license.
|
|
|
|
<p>
|
|
Furthermore, <a href="https://colab.research.google.com/notebooks/welcome.ipynb" target="_blank">Google's Colab</a> is a free Jupyter notebook environment that requires
|
|
no setup and runs entirely in the cloud. Try it out!
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="useful-python-libraries">Useful Python libraries </h2>
|
|
Here we list several useful Python libraries we strongly recommend (if you use anaconda many of these are already there)
|
|
|
|
<ul>
|
|
<li> <a href="https://www.numpy.org/" target="_blank">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</li>
|
|
<li> <a href="https://pandas.pydata.org/" target="_blank">The pandas</a> library provides high-performance, easy-to-use data structures and data analysis tools</li>
|
|
<li> <a href="http://xarray.pydata.org/en/stable/" target="_blank">Xarray</a> is a Python package that makes working with labelled multi-dimensional arrays simple, efficient, and fun!</li>
|
|
<li> <a href="https://www.scipy.org/" target="_blank">Scipy</a> (pronounced “Sigh Pie”) is a Python-based ecosystem of open-source software for mathematics, science, and engineering.</li>
|
|
<li> <a href="https://matplotlib.org/" target="_blank">Matplotlib</a> is a Python 2D plotting library which produces publication quality figures in a variety of hardcopy formats and interactive environments across platforms.</li>
|
|
<li> <a href="https://github.com/HIPS/autograd" target="_blank">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</li>
|
|
<li> <a href="https://www.sympy.org/en/index.html" target="_blank">SymPy</a> is a Python library for symbolic mathematics.</li>
|
|
<li> <a href="https://scikit-learn.org/stable/" target="_blank">scikit-learn</a> has simple and efficient tools for machine learning, data mining and data analysis</li>
|
|
<li> <a href="https://www.tensorflow.org/" target="_blank">TensorFlow</a> is a Python library for fast numerical computing created and released by Google</li>
|
|
<li> <a href="https://keras.io/" target="_blank">Keras</a> is a high-level neural networks API, written in Python and capable of running on top of TensorFlow, CNTK, or Theano</li>
|
|
<li> And many more such as <a href="https://pytorch.org/" target="_blank">pytorch</a>, <a href="https://pypi.org/project/Theano/" target="_blank">Theano</a> etc</li>
|
|
</ul>
|
|
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="installing-r-c-cython-or-julia">Installing R, C++, cython or Julia </h2>
|
|
|
|
<p>
|
|
You will also find it convenient to utilize <b>R</b>. We will mainly
|
|
use Python during our lectures and in various projects and exercises.
|
|
Those of you
|
|
already familiar with <b>R</b> should feel free to continue using <b>R</b>, keeping
|
|
however an eye on the parallel Python set ups. Similarly, if you are a
|
|
Python afecionado, feel free to explore <b>R</b> as well. Jupyter/Ipython
|
|
notebook allows you to run <b>R</b> codes interactively in your
|
|
browser. The software library <b>R</b> 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>
|
|
To install <b>R</b> with Jupyter notebook
|
|
<a href="https://mpacer.org/maths/r-kernel-for-ipython-notebook" target="_blank">follow the link here</a>
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="installing-r-c-cython-numba-etc">Installing R, C++, cython, Numba etc </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>
|
|
To add more entropy, <b>cython</b> 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 href="https://numba.pydata.org/" target="_blank">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 <b>PDF</b> file or a Latex file for
|
|
further processing. For example, convert to latex as
|
|
|
|
<p>
|
|
|
|
<!-- code=text typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span>pycod jupyter nbconvert filename.ipynb --to latex
|
|
</pre></div>
|
|
<p>
|
|
And to add more versatility, the Python package <a href="http://www.sympy.org/en/index.html" target="_blank">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>
|
|
Finally, if you wish to use the light mark-up language
|
|
<a href="https://github.com/hplgit/doconce" target="_blank">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 <b>doconce</b>.
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="numpy-examples-and-important-matrix-and-vector-handling-packages">Numpy examples and Important Matrix and vector handling packages </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 <b>Numerical Recipes</b>. 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.
|
|
|
|
<ul>
|
|
<li> LINPACK: package for linear equations and least square problems.</li>
|
|
<li> LAPACK:package for solving symmetric, unsymmetric and generalized eigenvalue problems. From LAPACK's website <a href="http://www.netlib.org" target="_blank"><tt>http://www.netlib.org</tt></a> it is possible to download for free all source codes from this library. Both C/C++ and Fortran versions are available.</li>
|
|
<li> 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 href="http://www.netlib.org" target="_blank"><tt>http://www.netlib.org</tt></a>.</li>
|
|
</ul>
|
|
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="basic-matrix-features">Basic Matrix Features </h2>
|
|
|
|
<p>
|
|
<div class="alert alert-block alert-block alert-text-normal">
|
|
<b>Matrix properties reminder</b>
|
|
<p>
|
|
$$
|
|
\mathbf{A} =
|
|
\begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\
|
|
a_{21} & a_{22} & a_{23} & a_{24} \\
|
|
a_{31} & a_{32} & a_{33} & a_{34} \\
|
|
a_{41} & a_{42} & a_{43} & a_{44}
|
|
\end{bmatrix}\qquad
|
|
\mathbf{I} =
|
|
\begin{bmatrix} 1 & 0 & 0 & 0 \\
|
|
0 & 1 & 0 & 0 \\
|
|
0 & 0 & 1 & 0 \\
|
|
0 & 0 & 0 & 1
|
|
\end{bmatrix}
|
|
$$
|
|
|
|
<p>
|
|
The inverse of a matrix is defined by
|
|
|
|
$$
|
|
\mathbf{A}^{-1} \cdot \mathbf{A} = I
|
|
$$
|
|
|
|
<p>
|
|
<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"> \( a_{ij}=a_{ij}^{ < em>} \) </td> </tr>
|
|
<tr><td align="center"> \( A=A^{\dagger} \) </td> <td align="center"> hermitian </td> <td align="center"> \( a_{ij}=a_{ji}^{ < /em>} \) </td> </tr>
|
|
<tr><td align="center"> \( A=\left(A^{\dagger}\right )^{-1} \) </td> <td align="center"> unitary </td> <td align="center"> \( \sum_k a_{ik}a_{jk}^{ < em>}=\sum_k a_{ki}^{ < /em> } a_{kj}=\delta_{ij} \) </td> </tr>
|
|
</tbody>
|
|
</table>
|
|
|
|
</div>
|
|
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h3 id="some-famous-matrices">Some famous Matrices </h3>
|
|
|
|
<ul>
|
|
<li> Diagonal if \( a_{ij}=0 \) for \( i\ne j \)</li>
|
|
<li> Upper triangular if \( a_{ij}=0 \) for \( i>j \)</li>
|
|
<li> Lower triangular if \( a_{ij}=0 \) for \( i < j \)</li>
|
|
<li> Upper Hessenberg if \( a_{ij}=0 \) for \( i>j+1 \)</li>
|
|
<li> Lower Hessenberg if \( a_{ij}=0 \) for \( i < j+1 \)</li>
|
|
<li> Tridiagonal if \( a_{ij}=0 \) for \( |i -j|>1 \)</li>
|
|
<li> Lower banded with bandwidth \( p \): \( a_{ij}=0 \) for \( i>j+p \)</li>
|
|
<li> Upper banded with bandwidth \( p \): \( a_{ij}=0 \) for \( i < j+p \)</li>
|
|
<li> Banded, block upper triangular, block lower triangular....</li>
|
|
</ul>
|
|
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h3 id="more-basic-matrix-features">More Basic Matrix Features </h3>
|
|
|
|
<p>
|
|
<div class="alert alert-block alert-block alert-text-normal">
|
|
<b>Some Equivalent Statements</b>
|
|
<p>
|
|
For an \( N\times N \) matrix \( \mathbf{A} \) the following properties are all equivalent
|
|
|
|
<ul>
|
|
<li> If the inverse of \( \mathbf{A} \) exists, \( \mathbf{A} \) is nonsingular.</li>
|
|
<li> The equation \( \mathbf{Ax}=0 \) implies \( \mathbf{x}=0 \).</li>
|
|
<li> The rows of \( \mathbf{A} \) form a basis of \( R^N \).</li>
|
|
<li> The columns of \( \mathbf{A} \) form a basis of \( R^N \).</li>
|
|
<li> \( \mathbf{A} \) is a product of elementary matrices.</li>
|
|
<li> \( 0 \) is not eigenvalue of \( \mathbf{A} \).</li>
|
|
</ul>
|
|
</div>
|
|
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="numpy-and-arrays">Numpy and arrays </h2>
|
|
<a href="http://www.numpy.org/" target="_blank">Numpy</a> provides an easy way to handle arrays in Python. The standard way to import this library is as
|
|
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
|
</pre></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>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span>n <span style="color: #666666">=</span> <span style="color: #666666">10</span>
|
|
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>normal(size<span style="color: #666666">=</span>n)
|
|
<span style="color: #008000">print</span>(x)
|
|
</pre></div>
|
|
<p>
|
|
We defined a vector \( x \) with \( n=10 \) elements with its values given by the Normal distribution \( N(0,1) \).
|
|
Another alternative is to declare a vector as follows
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
|
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([<span style="color: #666666">1</span>, <span style="color: #666666">2</span>, <span style="color: #666666">3</span>])
|
|
<span style="color: #008000">print</span>(x)
|
|
</pre></div>
|
|
<p>
|
|
Here we have defined a vector with three elements, with \( x_0=1 \), \( x_1=2 \) and \( x_2=3 \). Note that both Python and C++
|
|
start numbering array elements from \( 0 \) and on. This means that a vector with \( n \) elements has a sequence of entities \( x_0, x_1, x_2, \dots, x_{n-1} \). We could also let (recommended) Numpy to compute the logarithms of a specific array as
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
|
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>log(np<span style="color: #666666">.</span>array([<span style="color: #666666">4</span>, <span style="color: #666666">7</span>, <span style="color: #666666">8</span>]))
|
|
<span style="color: #008000">print</span>(x)
|
|
</pre></div>
|
|
<p>
|
|
In the last example we used Numpy's unary function \( np.log \). 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 <b>log</b> function
|
|
from Python's <b>math</b> module. The looping is done explicitely by the
|
|
<b>np.log</b> function. The alternative, and slower way to compute the
|
|
logarithms of a vector would be to write
|
|
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">math</span> <span style="color: #008000; font-weight: bold">import</span> log
|
|
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([<span style="color: #666666">4</span>, <span style="color: #666666">7</span>, <span style="color: #666666">8</span>])
|
|
<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #666666">0</span>, <span style="color: #008000">len</span>(x)):
|
|
x[i] <span style="color: #666666">=</span> log(x[i])
|
|
<span style="color: #008000">print</span>(x)
|
|
</pre></div>
|
|
<p>
|
|
We note that our code is much longer already and we need to import the <b>log</b> function from the <b>math</b> module.
|
|
The attentive reader will also notice that the output is \( [1, 1, 2] \). Python interprets automagically our numbers as integers (like the <b>automatic</b> keyword in C++). To change this we could define our array elements to be double precision numbers as
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
|
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>log(np<span style="color: #666666">.</span>array([<span style="color: #666666">4</span>, <span style="color: #666666">7</span>, <span style="color: #666666">8</span>], dtype <span style="color: #666666">=</span> np<span style="color: #666666">.</span>float64))
|
|
<span style="color: #008000">print</span>(x)
|
|
</pre></div>
|
|
<p>
|
|
or simply write them as double precision numbers (Python uses 64 bits as default for floating point type variables), that is
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
|
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>log(np<span style="color: #666666">.</span>array([<span style="color: #666666">4.0</span>, <span style="color: #666666">7.0</span>, <span style="color: #666666">8.0</span>]))
|
|
<span style="color: #008000">print</span>(x)
|
|
</pre></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 <b>itemsize</b> functionality (the array \( x \) is actually an object which inherits the functionalities defined in Numpy) as
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
|
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>log(np<span style="color: #666666">.</span>array([<span style="color: #666666">4.0</span>, <span style="color: #666666">7.0</span>, <span style="color: #666666">8.0</span>]))
|
|
<span style="color: #008000">print</span>(x<span style="color: #666666">.</span>itemsize)
|
|
</pre></div>
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="matrices-in-python">Matrices in Python </h2>
|
|
|
|
<p>
|
|
Having defined vectors, we are now ready to try out matrices. We can
|
|
define a \( 3 \times 3 \) real matrix \( \boldsymbol{A} \) as (recall that we user
|
|
lowercase letters for vectors and uppercase letters for matrices)
|
|
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
|
A <span style="color: #666666">=</span> np<span style="color: #666666">.</span>log(np<span style="color: #666666">.</span>array([ [<span style="color: #666666">4.0</span>, <span style="color: #666666">7.0</span>, <span style="color: #666666">8.0</span>], [<span style="color: #666666">3.0</span>, <span style="color: #666666">10.0</span>, <span style="color: #666666">11.0</span>], [<span style="color: #666666">4.0</span>, <span style="color: #666666">5.0</span>, <span style="color: #666666">7.0</span>] ]))
|
|
<span style="color: #008000">print</span>(A)
|
|
</pre></div>
|
|
<p>
|
|
If we use the <b>shape</b> function we would get \( (3, 3) \) as output, that is verifying that our matrix is a \( 3\times 3 \) 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>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
|
A <span style="color: #666666">=</span> np<span style="color: #666666">.</span>log(np<span style="color: #666666">.</span>array([ [<span style="color: #666666">4.0</span>, <span style="color: #666666">7.0</span>, <span style="color: #666666">8.0</span>], [<span style="color: #666666">3.0</span>, <span style="color: #666666">10.0</span>, <span style="color: #666666">11.0</span>], [<span style="color: #666666">4.0</span>, <span style="color: #666666">5.0</span>, <span style="color: #666666">7.0</span>] ]))
|
|
<span style="color: #408080; font-style: italic"># print the first column, row-major order and elements start with 0</span>
|
|
<span style="color: #008000">print</span>(A[:,<span style="color: #666666">0</span>])
|
|
</pre></div>
|
|
<p>
|
|
We can continue this was by printing out other columns or rows. The example here prints out the second column
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
|
A <span style="color: #666666">=</span> np<span style="color: #666666">.</span>log(np<span style="color: #666666">.</span>array([ [<span style="color: #666666">4.0</span>, <span style="color: #666666">7.0</span>, <span style="color: #666666">8.0</span>], [<span style="color: #666666">3.0</span>, <span style="color: #666666">10.0</span>, <span style="color: #666666">11.0</span>], [<span style="color: #666666">4.0</span>, <span style="color: #666666">5.0</span>, <span style="color: #666666">7.0</span>] ]))
|
|
<span style="color: #408080; font-style: italic"># print the first column, row-major order and elements start with 0</span>
|
|
<span style="color: #008000">print</span>(A[<span style="color: #666666">1</span>,:])
|
|
</pre></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 href="http://www.numpy.org/" target="_blank">Numpy website for more details</a>. Useful functions when defining a matrix are the <b>np.zeros</b> function which declares a matrix of a given dimension and sets all elements to zero
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
|
n <span style="color: #666666">=</span> <span style="color: #666666">10</span>
|
|
<span style="color: #408080; font-style: italic"># define a matrix of dimension 10 x 10 and set all elements to zero</span>
|
|
A <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros( (n, n) )
|
|
<span style="color: #008000">print</span>(A)
|
|
</pre></div>
|
|
<p>
|
|
or initializing all elements to
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
|
n <span style="color: #666666">=</span> <span style="color: #666666">10</span>
|
|
<span style="color: #408080; font-style: italic"># define a matrix of dimension 10 x 10 and set all elements to one</span>
|
|
A <span style="color: #666666">=</span> np<span style="color: #666666">.</span>ones( (n, n) )
|
|
<span style="color: #008000">print</span>(A)
|
|
</pre></div>
|
|
<p>
|
|
or as unitarily distributed random numbers (see the material on random number generators in the statistics part)
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
|
n <span style="color: #666666">=</span> <span style="color: #666666">10</span>
|
|
<span style="color: #408080; font-style: italic"># define a matrix of dimension 10 x 10 and set all elements to random numbers with x \in [0, 1]</span>
|
|
A <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(n, n)
|
|
<span style="color: #008000">print</span>(A)
|
|
</pre></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
|
|
\( \boldsymbol{x}, \boldsymbol{y}, \boldsymbol{z} \) with \( n \) elements each. The covariance matrix is defined as
|
|
$$
|
|
\boldsymbol{\Sigma} = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\
|
|
\sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\
|
|
\sigma_{zx} & \sigma_{zy} & \sigma_{zz}
|
|
\end{bmatrix},
|
|
$$
|
|
|
|
where for example
|
|
$$
|
|
\sigma_{xy} =\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})(y_i- \overline{y}).
|
|
$$
|
|
|
|
The Numpy function <b>np.cov</b> calculates the covariance elements using the factor \( 1/(n-1) \) instead of \( 1/n \) since it assumes we do not have the exact mean values.
|
|
The following simple function uses the <b>np.vstack</b> function which takes each vector of dimension \( 1\times n \) and produces a \( 3\times n \) matrix \( \boldsymbol{W} \)
|
|
$$
|
|
\boldsymbol{W} = \begin{bmatrix} x_0 & y_0 & z_0 \\
|
|
x_1 & y_1 & z_1 \\
|
|
x_2 & y_2 & z_2 \\
|
|
\dots & \dots & \dots \\
|
|
x_{n-2} & y_{n-2} & z_{n-2} \\
|
|
x_{n-1} & y_{n-1} & z_{n-1}
|
|
\end{bmatrix},
|
|
$$
|
|
|
|
<p>
|
|
which in turn is converted into into the \( 3\times 3 \) covariance matrix
|
|
\( \boldsymbol{\Sigma} \) via the Numpy function <b>np.cov()</b>. We note that we can also calculate
|
|
the mean value of each set of samples \( \boldsymbol{x} \) etc using the Numpy
|
|
function <b>np.mean(x)</b>. We can also extract the eigenvalues of the
|
|
covariance matrix through the <b>np.linalg.eig()</b> function.
|
|
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #408080; font-style: italic"># Importing various packages</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
|
|
|
n <span style="color: #666666">=</span> <span style="color: #666666">100</span>
|
|
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>normal(size<span style="color: #666666">=</span>n)
|
|
<span style="color: #008000">print</span>(np<span style="color: #666666">.</span>mean(x))
|
|
y <span style="color: #666666">=</span> <span style="color: #666666">4+3*</span>x<span style="color: #666666">+</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>normal(size<span style="color: #666666">=</span>n)
|
|
<span style="color: #008000">print</span>(np<span style="color: #666666">.</span>mean(y))
|
|
z <span style="color: #666666">=</span> x<span style="color: #666666">**3+</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>normal(size<span style="color: #666666">=</span>n)
|
|
<span style="color: #008000">print</span>(np<span style="color: #666666">.</span>mean(z))
|
|
W <span style="color: #666666">=</span> np<span style="color: #666666">.</span>vstack((x, y, z))
|
|
Sigma <span style="color: #666666">=</span> np<span style="color: #666666">.</span>cov(W)
|
|
<span style="color: #008000">print</span>(Sigma)
|
|
Eigvals, Eigvecs <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>eig(Sigma)
|
|
<span style="color: #008000">print</span>(Eigvals)
|
|
</pre></div>
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">scipy</span> <span style="color: #008000; font-weight: bold">import</span> sparse
|
|
eye <span style="color: #666666">=</span> np<span style="color: #666666">.</span>eye(<span style="color: #666666">4</span>)
|
|
<span style="color: #008000">print</span>(eye)
|
|
sparse_mtx <span style="color: #666666">=</span> sparse<span style="color: #666666">.</span>csr_matrix(eye)
|
|
<span style="color: #008000">print</span>(sparse_mtx)
|
|
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(<span style="color: #666666">-10</span>,<span style="color: #666666">10</span>,<span style="color: #666666">100</span>)
|
|
y <span style="color: #666666">=</span> np<span style="color: #666666">.</span>sin(x)
|
|
plt<span style="color: #666666">.</span>plot(x,y,marker<span style="color: #666666">=</span><span style="color: #BA2121">'x'</span>)
|
|
plt<span style="color: #666666">.</span>show()
|
|
</pre></div>
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="meet-the-pandas">Meet the Pandas </h2>
|
|
|
|
<p>
|
|
<br /><br /><center><p><img src="fig/pandas.jpg" align="bottom" width=600></p></center><br /><br />
|
|
|
|
<p>
|
|
Another useful Python package is
|
|
<a href="https://pandas.pydata.org/" target="_blank">pandas</a>, which is an open source library
|
|
providing high-performance, easy-to-use data structures and data
|
|
analysis tools for Python. <b>pandas</b> stands for panel data, a term borrowed from econometrics and is an efficient library for data analysis with an emphasis on tabular data.
|
|
<b>pandas</b> has two major classes, the <b>DataFrame</b> class with two-dimensional data objects and tabular data organized in columns and the class <b>Series</b> with a focus on one-dimensional data objects. Both classes allow you to index data easily as we will see in the examples below.
|
|
<b>pandas</b> allows you also to perform mathematical operations on the data, spanning from simple reshapings of vectors and matrices to statistical operations.
|
|
|
|
<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 <b>pandas</b>, in particular in connection with classification of data.
|
|
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">pandas</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">pd</span>
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">IPython.display</span> <span style="color: #008000; font-weight: bold">import</span> display
|
|
data <span style="color: #666666">=</span> {<span style="color: #BA2121">'First Name'</span>: [<span style="color: #BA2121">"Frodo"</span>, <span style="color: #BA2121">"Bilbo"</span>, <span style="color: #BA2121">"Aragorn II"</span>, <span style="color: #BA2121">"Samwise"</span>],
|
|
<span style="color: #BA2121">'Last Name'</span>: [<span style="color: #BA2121">"Baggins"</span>, <span style="color: #BA2121">"Baggins"</span>,<span style="color: #BA2121">"Elessar"</span>,<span style="color: #BA2121">"Gamgee"</span>],
|
|
<span style="color: #BA2121">'Place of birth'</span>: [<span style="color: #BA2121">"Shire"</span>, <span style="color: #BA2121">"Shire"</span>, <span style="color: #BA2121">"Eriador"</span>, <span style="color: #BA2121">"Shire"</span>],
|
|
<span style="color: #BA2121">'Date of Birth T.A.'</span>: [<span style="color: #666666">2968</span>, <span style="color: #666666">2890</span>, <span style="color: #666666">2931</span>, <span style="color: #666666">2980</span>]
|
|
}
|
|
data_pandas <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>DataFrame(data)
|
|
display(data_pandas)
|
|
</pre></div>
|
|
<p>
|
|
In the above we have imported <b>pandas</b> with the shorthand <b>pd</b>, the latter has become the standard way we import <b>pandas</b>. We make then a list of various variables
|
|
and reorganize the aboves lists into a <b>DataFrame</b> 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.
|
|
<b>pandas</b> is extremely flexible and we can easily change the above indices by defining a new type of indexing as
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span>data_pandas <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>DataFrame(data,index<span style="color: #666666">=</span>[<span style="color: #BA2121">'Frodo'</span>,<span style="color: #BA2121">'Bilbo'</span>,<span style="color: #BA2121">'Aragorn'</span>,<span style="color: #BA2121">'Sam'</span>])
|
|
display(data_pandas)
|
|
</pre></div>
|
|
<p>
|
|
Thereafter we display the content of the row which begins with the index <b>Aragorn</b>
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span>display(data_pandas<span style="color: #666666">.</span>loc[<span style="color: #BA2121">'Aragorn'</span>])
|
|
</pre></div>
|
|
<p>
|
|
We can easily append data to this, for example
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span>new_hobbit <span style="color: #666666">=</span> {<span style="color: #BA2121">'First Name'</span>: [<span style="color: #BA2121">"Peregrin"</span>],
|
|
<span style="color: #BA2121">'Last Name'</span>: [<span style="color: #BA2121">"Took"</span>],
|
|
<span style="color: #BA2121">'Place of birth'</span>: [<span style="color: #BA2121">"Shire"</span>],
|
|
<span style="color: #BA2121">'Date of Birth T.A.'</span>: [<span style="color: #666666">2990</span>]
|
|
}
|
|
data_pandas<span style="color: #666666">=</span>data_pandas<span style="color: #666666">.</span>append(pd<span style="color: #666666">.</span>DataFrame(new_hobbit, index<span style="color: #666666">=</span>[<span style="color: #BA2121">'Pippin'</span>]))
|
|
display(data_pandas)
|
|
</pre></div>
|
|
<p>
|
|
Here are other examples where we use the <b>DataFrame</b> functionality to handle arrays, now with more interesting features for us, namely numbers. We set up a matrix
|
|
of dimensionality \( 10\times 5 \) 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>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">pandas</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">pd</span>
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">IPython.display</span> <span style="color: #008000; font-weight: bold">import</span> display
|
|
np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">100</span>)
|
|
<span style="color: #408080; font-style: italic"># setting up a 10 x 5 matrix</span>
|
|
rows <span style="color: #666666">=</span> <span style="color: #666666">10</span>
|
|
cols <span style="color: #666666">=</span> <span style="color: #666666">5</span>
|
|
a <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(rows,cols)
|
|
df <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>DataFrame(a)
|
|
display(df)
|
|
<span style="color: #008000">print</span>(df<span style="color: #666666">.</span>mean())
|
|
<span style="color: #008000">print</span>(df<span style="color: #666666">.</span>std())
|
|
display(df<span style="color: #666666">**2</span>)
|
|
</pre></div>
|
|
<p>
|
|
Thereafter we can select specific columns only and plot final results
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span>df<span style="color: #666666">.</span>columns <span style="color: #666666">=</span> [<span style="color: #BA2121">'First'</span>, <span style="color: #BA2121">'Second'</span>, <span style="color: #BA2121">'Third'</span>, <span style="color: #BA2121">'Fourth'</span>, <span style="color: #BA2121">'Fifth'</span>]
|
|
df<span style="color: #666666">.</span>index <span style="color: #666666">=</span> np<span style="color: #666666">.</span>arange(<span style="color: #666666">10</span>)
|
|
|
|
display(df)
|
|
<span style="color: #008000">print</span>(df[<span style="color: #BA2121">'Second'</span>]<span style="color: #666666">.</span>mean() )
|
|
<span style="color: #008000">print</span>(df<span style="color: #666666">.</span>info())
|
|
<span style="color: #008000">print</span>(df<span style="color: #666666">.</span>describe())
|
|
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">pylab</span> <span style="color: #008000; font-weight: bold">import</span> plt, mpl
|
|
plt<span style="color: #666666">.</span>style<span style="color: #666666">.</span>use(<span style="color: #BA2121">'seaborn'</span>)
|
|
mpl<span style="color: #666666">.</span>rcParams[<span style="color: #BA2121">'font.family'</span>] <span style="color: #666666">=</span> <span style="color: #BA2121">'serif'</span>
|
|
|
|
df<span style="color: #666666">.</span>cumsum()<span style="color: #666666">.</span>plot(lw<span style="color: #666666">=2.0</span>, figsize<span style="color: #666666">=</span>(<span style="color: #666666">10</span>,<span style="color: #666666">6</span>))
|
|
plt<span style="color: #666666">.</span>show()
|
|
|
|
|
|
df<span style="color: #666666">.</span>plot<span style="color: #666666">.</span>bar(figsize<span style="color: #666666">=</span>(<span style="color: #666666">10</span>,<span style="color: #666666">6</span>), rot<span style="color: #666666">=15</span>)
|
|
plt<span style="color: #666666">.</span>show()
|
|
</pre></div>
|
|
<p>
|
|
We can produce a \( 4\times 4 \) matrix
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span>b <span style="color: #666666">=</span> np<span style="color: #666666">.</span>arange(<span style="color: #666666">16</span>)<span style="color: #666666">.</span>reshape((<span style="color: #666666">4</span>,<span style="color: #666666">4</span>))
|
|
<span style="color: #008000">print</span>(b)
|
|
df1 <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>DataFrame(b)
|
|
<span style="color: #008000">print</span>(df1)
|
|
</pre></div>
|
|
<p>
|
|
and many other operations.
|
|
|
|
<p>
|
|
The <b>Series</b> class is another important class included in
|
|
<b>pandas</b>. You can view it as a specialization of <b>DataFrame</b> but where
|
|
we have just a single column of data. It shares many of the same features as _DataFrame. As with <b>DataFrame</b>,
|
|
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 href="http://xarray.pydata.org/en/stable/" target="_blank">xarray</a>. <b>xarray</b> has much of the same flexibility as <b>pandas</b>, but allows for the extension to higher dimensions than two. We will see examples later of the usage of both <b>pandas</b> and <b>xarray</b>.
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="friday-august-27">Friday August 27 </h2>
|
|
|
|
<p>
|
|
"Video of Lecture August 27, 2021":"https://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/h21/forelesningsvideoer/LectureThursdayAugust27.mp4?vrtx=view-as-webpage
|
|
|
|
<p>
|
|
<a href="https://www.uio.no/studier/emner/matnat/fys/FYS-STK3155/h20/forelesningsvideoer/LectureAug21.mp4?vrtx=view-as-webpage" target="_blank">Video of Lecture from fall 2020</a> and <a href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/NotesAugust21.pdf" target="_blank">Handwritten notes</a>
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h3 id="simple-linear-regression-model-using-_scikit-learn_">Simple linear regression model using <b>scikit-learn</b> </h3>
|
|
|
|
<p>
|
|
We start with perhaps our simplest possible example, using <b>Scikit-Learn</b> to perform linear regression analysis on a data set produced by us.
|
|
|
|
<p>
|
|
What follows is a simple Python code where we have defined a function
|
|
\( y \) in terms of the variable \( x \). Both are defined as vectors with \( 100 \) entries.
|
|
The numbers in the vector \( \boldsymbol{x} \) are given
|
|
by random numbers generated with a uniform distribution with entries
|
|
\( x_i \in [0,1] \) (more about probability distribution functions
|
|
later). These values are then used to define a function \( y(x) \)
|
|
(tabulated again as a vector) with a linear dependence on \( x \) plus a
|
|
random noise added via the normal distribution.
|
|
|
|
<p>
|
|
The Numpy functions are imported used the <b>import numpy as np</b>
|
|
statement and the random number generator for the uniform distribution
|
|
is called using the function <b>np.random.rand()</b>, where we specificy
|
|
that we want \( 100 \) random variables. Using Numpy we define
|
|
automatically an array with the specified number of elements, \( 100 \) in
|
|
our case. With the Numpy function <b>randn()</b> we can compute random
|
|
numbers with the normal distribution (mean value \( \mu \) equal to zero and
|
|
variance \( \sigma^2 \) set to one) and produce the values of \( y \) assuming a linear
|
|
dependence as function of \( x \)
|
|
|
|
$$
|
|
y = 2x+N(0,1),
|
|
$$
|
|
|
|
<p>
|
|
where \( N(0,1) \) represents random numbers generated by the normal
|
|
distribution. From <b>Scikit-Learn</b> we import then the
|
|
<b>LinearRegression</b> functionality and make a prediction \( \tilde{y} =
|
|
\alpha + \beta x \) using the function <b>fit(x,y)</b>. We call the set of
|
|
data \( (\boldsymbol{x},\boldsymbol{y}) \) for our training data. The Python package
|
|
<b>scikit-learn</b> has also a functionality which extracts the above
|
|
fitting parameters \( \alpha \) and \( \beta \) (see below). Later we will
|
|
distinguish between training data and test data.
|
|
|
|
<p>
|
|
For plotting we use the Python package
|
|
<a href="https://matplotlib.org/" target="_blank">matplotlib</a> which produces publication
|
|
quality figures. Feel free to explore the extensive
|
|
<a href="https://matplotlib.org/gallery/index.html" target="_blank">gallery</a> of examples. In
|
|
this example we plot our original values of \( x \) and \( y \) as well as the
|
|
prediction <b>ypredict</b> (\( \tilde{y} \)), which attempts at fitting our
|
|
data with a straight line.
|
|
|
|
<p>
|
|
The Python code follows here.
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #408080; font-style: italic"># Importing various packages</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> LinearRegression
|
|
|
|
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(<span style="color: #666666">100</span>,<span style="color: #666666">1</span>)
|
|
y <span style="color: #666666">=</span> <span style="color: #666666">2*</span>x<span style="color: #666666">+</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(<span style="color: #666666">100</span>,<span style="color: #666666">1</span>)
|
|
linreg <span style="color: #666666">=</span> LinearRegression()
|
|
linreg<span style="color: #666666">.</span>fit(x,y)
|
|
xnew <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([[<span style="color: #666666">0</span>],[<span style="color: #666666">1</span>]])
|
|
ypredict <span style="color: #666666">=</span> linreg<span style="color: #666666">.</span>predict(xnew)
|
|
|
|
plt<span style="color: #666666">.</span>plot(xnew, ypredict, <span style="color: #BA2121">"r-"</span>)
|
|
plt<span style="color: #666666">.</span>plot(x, y ,<span style="color: #BA2121">'ro'</span>)
|
|
plt<span style="color: #666666">.</span>axis([<span style="color: #666666">0</span>,<span style="color: #666666">1.0</span>,<span style="color: #666666">0</span>, <span style="color: #666666">5.0</span>])
|
|
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">r'$x$'</span>)
|
|
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">r'$y$'</span>)
|
|
plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">r'Simple Linear Regression'</span>)
|
|
plt<span style="color: #666666">.</span>show()
|
|
</pre></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 \( x \) and the normal distribution. Try to change the
|
|
function \( y \) to
|
|
|
|
$$
|
|
y = 10x+0.01 \times N(0,1),
|
|
$$
|
|
|
|
<p>
|
|
where \( x \) 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 <b>cost</b> function.
|
|
|
|
<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 \( \chi^2 \)
|
|
function (a variant of the mean-squared error (MSE))
|
|
|
|
$$ \chi^2 = \frac{1}{n}
|
|
\sum_{i=0}^{n-1}\frac{(y_i-\tilde{y}_i)^2}{\sigma_i^2},
|
|
$$
|
|
|
|
<p>
|
|
where \( \sigma_i^2 \) is the variance (to be defined later) of the entry
|
|
\( y_i \). We may not know the explicit value of \( \sigma_i^2 \), it serves
|
|
however the aim of scaling the equations and make the cost function
|
|
dimensionless.
|
|
|
|
<p>
|
|
Minimizing the cost function is a central aspect of
|
|
our discussions to come. Finding its minima as function of the model
|
|
parameters (\( \alpha \) and \( \beta \) 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 <b>gradient</b> 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 \( \chi^2 \) function becomes smaller.
|
|
|
|
<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
|
|
|
|
$$
|
|
\epsilon_{\mathrm{relative}}= \frac{\vert \boldsymbol{y} -\boldsymbol{\tilde{y}}\vert}{\vert \boldsymbol{y}\vert}.
|
|
$$
|
|
|
|
<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>
|
|
We can modify easily the above Python code and plot the relative error instead
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> LinearRegression
|
|
|
|
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(<span style="color: #666666">100</span>,<span style="color: #666666">1</span>)
|
|
y <span style="color: #666666">=</span> <span style="color: #666666">5*</span>x<span style="color: #666666">+0.01*</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(<span style="color: #666666">100</span>,<span style="color: #666666">1</span>)
|
|
linreg <span style="color: #666666">=</span> LinearRegression()
|
|
linreg<span style="color: #666666">.</span>fit(x,y)
|
|
ypredict <span style="color: #666666">=</span> linreg<span style="color: #666666">.</span>predict(x)
|
|
|
|
plt<span style="color: #666666">.</span>plot(x, np<span style="color: #666666">.</span>abs(ypredict<span style="color: #666666">-</span>y)<span style="color: #666666">/</span><span style="color: #008000">abs</span>(y), <span style="color: #BA2121">"ro"</span>)
|
|
plt<span style="color: #666666">.</span>axis([<span style="color: #666666">0</span>,<span style="color: #666666">1.0</span>,<span style="color: #666666">0.0</span>, <span style="color: #666666">0.5</span>])
|
|
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">r'$x$'</span>)
|
|
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">r'$\epsilon_{\mathrm</span><span style="color: #BB6688; font-weight: bold">{relative}</span><span style="color: #BA2121">}$'</span>)
|
|
plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">r'Relative error'</span>)
|
|
plt<span style="color: #666666">.</span>show()
|
|
</pre></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>
|
|
As mentioned above, <b>Scikit-Learn</b> has an impressive functionality.
|
|
We can for example extract the values of \( \alpha \) and \( \beta \) and
|
|
their error estimates, or the variance and standard deviation and many
|
|
other properties from the statistical data analysis.
|
|
|
|
<p>
|
|
Here we show an
|
|
example of the functionality of <b>Scikit-Learn</b>.
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> LinearRegression
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.metrics</span> <span style="color: #008000; font-weight: bold">import</span> mean_squared_error, r2_score, mean_squared_log_error, mean_absolute_error
|
|
|
|
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(<span style="color: #666666">100</span>,<span style="color: #666666">1</span>)
|
|
y <span style="color: #666666">=</span> <span style="color: #666666">2.0+</span> <span style="color: #666666">5*</span>x<span style="color: #666666">+0.5*</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(<span style="color: #666666">100</span>,<span style="color: #666666">1</span>)
|
|
linreg <span style="color: #666666">=</span> LinearRegression()
|
|
linreg<span style="color: #666666">.</span>fit(x,y)
|
|
ypredict <span style="color: #666666">=</span> linreg<span style="color: #666666">.</span>predict(x)
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">'The intercept alpha: </span><span style="color: #BB6622; font-weight: bold">\n</span><span style="color: #BA2121">'</span>, linreg<span style="color: #666666">.</span>intercept_)
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">'Coefficient beta : </span><span style="color: #BB6622; font-weight: bold">\n</span><span style="color: #BA2121">'</span>, linreg<span style="color: #666666">.</span>coef_)
|
|
<span style="color: #408080; font-style: italic"># The mean squared error </span>
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">"Mean squared error: </span><span style="color: #BB6688; font-weight: bold">%.2f</span><span style="color: #BA2121">"</span> <span style="color: #666666">%</span> mean_squared_error(y, ypredict))
|
|
<span style="color: #408080; font-style: italic"># Explained variance score: 1 is perfect prediction </span>
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">'Variance score: </span><span style="color: #BB6688; font-weight: bold">%.2f</span><span style="color: #BA2121">'</span> <span style="color: #666666">%</span> r2_score(y, ypredict))
|
|
<span style="color: #408080; font-style: italic"># Mean squared log error </span>
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">'Mean squared log error: </span><span style="color: #BB6688; font-weight: bold">%.2f</span><span style="color: #BA2121">'</span> <span style="color: #666666">%</span> mean_squared_log_error(y, ypredict) )
|
|
<span style="color: #408080; font-style: italic"># Mean absolute error </span>
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">'Mean absolute error: </span><span style="color: #BB6688; font-weight: bold">%.2f</span><span style="color: #BA2121">'</span> <span style="color: #666666">%</span> mean_absolute_error(y, ypredict))
|
|
plt<span style="color: #666666">.</span>plot(x, ypredict, <span style="color: #BA2121">"r-"</span>)
|
|
plt<span style="color: #666666">.</span>plot(x, y ,<span style="color: #BA2121">'ro'</span>)
|
|
plt<span style="color: #666666">.</span>axis([<span style="color: #666666">0.0</span>,<span style="color: #666666">1.0</span>,<span style="color: #666666">1.5</span>, <span style="color: #666666">7.0</span>])
|
|
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">r'$x$'</span>)
|
|
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">r'$y$'</span>)
|
|
plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">r'Linear Regression fit '</span>)
|
|
plt<span style="color: #666666">.</span>show()
|
|
</pre></div>
|
|
<p>
|
|
The function <b>coef</b> gives us the parameter \( \beta \) of our fit while <b>intercept</b> yields
|
|
\( \alpha \). Depending on the constant in front of the normal distribution, we get values near or far from \( \alpha =2 \) and \( \beta =5 \). Try to play around with different parameters in front of the normal distribution. The function <b>meansquarederror</b> gives us the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error or loss defined as
|
|
$$ MSE(\boldsymbol{y},\boldsymbol{\tilde{y}}) = \frac{1}{n}
|
|
\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2,
|
|
$$
|
|
|
|
<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 \( \chi^2 \) function defined above.
|
|
|
|
<p>
|
|
The <b>r2score</b> function computes \( R^2 \), 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 \( \boldsymbol{y} \),
|
|
disregarding the input features, would get a \( R^2 \) score of \( 0.0 \).
|
|
|
|
<p>
|
|
If \( \tilde{\boldsymbol{y}}_i \) is the predicted value of the \( i-th \) sample and \( y_i \) is the corresponding true value, then the score \( R^2 \) is defined as
|
|
$$
|
|
R^2(\boldsymbol{y}, \tilde{\boldsymbol{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2},
|
|
$$
|
|
|
|
where we have defined the mean value of \( \boldsymbol{y} \) as
|
|
$$
|
|
\bar{y} = \frac{1}{n} \sum_{i=0}^{n - 1} y_i.
|
|
$$
|
|
|
|
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 \( l1 \)-norm loss. In our discussion above we presented the relative error.
|
|
The MAE is defined as follows
|
|
$$
|
|
\text{MAE}(\boldsymbol{y}, \boldsymbol{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n-1} \left| y_i - \tilde{y}_i \right|.
|
|
$$
|
|
|
|
We present the
|
|
squared logarithmic (quadratic) error
|
|
$$
|
|
\text{MSLE}(\boldsymbol{y}, \boldsymbol{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n - 1} (\log_e (1 + y_i) - \log_e (1 + \tilde{y}_i) )^2,
|
|
$$
|
|
|
|
<p>
|
|
where \( \log_e (x) \) stands for the natural logarithm of \( x \). 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>
|
|
Finally, another cost function is the Huber cost function used in robust regression.
|
|
|
|
<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>
|
|
The Huber cost function is defined as
|
|
$$
|
|
H_{\delta}(\boldsymbol{a})=\left\{\begin{array}{cc}\frac{1}{2} \boldsymbol{a}^{2}& \text{for }|\boldsymbol{a}|\leq \delta\\ \delta (|\boldsymbol{a}|-\frac{1}{2}\delta ),&\text{otherwise}.\end{array}\right.
|
|
$$
|
|
|
|
Here \( \boldsymbol{a}=\boldsymbol{y} - \boldsymbol{\tilde{y}} \).
|
|
|
|
<p>
|
|
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 \( x \)-dependence we study now a cubic polynomial and use the
|
|
polynomial regression analysis tools of scikit-learn.
|
|
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">random</span>
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> Ridge
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> PolynomialFeatures
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.pipeline</span> <span style="color: #008000; font-weight: bold">import</span> make_pipeline
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> LinearRegression
|
|
|
|
x<span style="color: #666666">=</span>np<span style="color: #666666">.</span>linspace(<span style="color: #666666">0.02</span>,<span style="color: #666666">0.98</span>,<span style="color: #666666">200</span>)
|
|
noise <span style="color: #666666">=</span> np<span style="color: #666666">.</span>asarray(random<span style="color: #666666">.</span>sample((<span style="color: #008000">range</span>(<span style="color: #666666">200</span>)),<span style="color: #666666">200</span>))
|
|
y<span style="color: #666666">=</span>x<span style="color: #666666">**3*</span>noise
|
|
yn<span style="color: #666666">=</span>x<span style="color: #666666">**3*100</span>
|
|
poly3 <span style="color: #666666">=</span> PolynomialFeatures(degree<span style="color: #666666">=3</span>)
|
|
X <span style="color: #666666">=</span> poly3<span style="color: #666666">.</span>fit_transform(x[:,np<span style="color: #666666">.</span>newaxis])
|
|
clf3 <span style="color: #666666">=</span> LinearRegression()
|
|
clf3<span style="color: #666666">.</span>fit(X,y)
|
|
|
|
Xplot<span style="color: #666666">=</span>poly3<span style="color: #666666">.</span>fit_transform(x[:,np<span style="color: #666666">.</span>newaxis])
|
|
poly3_plot<span style="color: #666666">=</span>plt<span style="color: #666666">.</span>plot(x, clf3<span style="color: #666666">.</span>predict(Xplot), label<span style="color: #666666">=</span><span style="color: #BA2121">'Cubic Fit'</span>)
|
|
plt<span style="color: #666666">.</span>plot(x,yn, color<span style="color: #666666">=</span><span style="color: #BA2121">'red'</span>, label<span style="color: #666666">=</span><span style="color: #BA2121">"True Cubic"</span>)
|
|
plt<span style="color: #666666">.</span>scatter(x, y, label<span style="color: #666666">=</span><span style="color: #BA2121">'Data'</span>, color<span style="color: #666666">=</span><span style="color: #BA2121">'orange'</span>, s<span style="color: #666666">=15</span>)
|
|
plt<span style="color: #666666">.</span>legend()
|
|
plt<span style="color: #666666">.</span>show()
|
|
|
|
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">error</span>(a):
|
|
<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> y:
|
|
err<span style="color: #666666">=</span>(y<span style="color: #666666">-</span>yn)<span style="color: #666666">/</span>yn
|
|
<span style="color: #008000; font-weight: bold">return</span> <span style="color: #008000">abs</span>(np<span style="color: #666666">.</span>sum(err))<span style="color: #666666">/</span><span style="color: #008000">len</span>(err)
|
|
|
|
<span style="color: #008000">print</span> (error(y))
|
|
</pre></div>
|
|
|
|
<h3 id="to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies">To our real data: nuclear binding energies. Brief reminder on masses and binding energies </h3>
|
|
|
|
<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 \( M(N, Z) \) of the neutral atom with
|
|
atomic mass number \( A \) and charge \( Z \). The number of neutrons is \( N \). There are indeed several sophisticated experiments worldwide which allow us to measure this quantity to high precision (parts per million even).
|
|
|
|
<p>
|
|
Atomic masses are usually tabulated in terms of the mass excess defined by
|
|
$$
|
|
\Delta M(N, Z) = M(N, Z) - uA,
|
|
$$
|
|
|
|
where \( u \) is the Atomic Mass Unit
|
|
$$
|
|
u = M(^{12}\mathrm{C})/12 = 931.4940954(57) \hspace{0.1cm} \mathrm{MeV}/c^2.
|
|
$$
|
|
|
|
The nucleon masses are
|
|
$$
|
|
m_p = 1.00727646693(9)u,
|
|
$$
|
|
|
|
and
|
|
$$
|
|
m_n = 939.56536(8)\hspace{0.1cm} \mathrm{MeV}/c^2 = 1.0086649156(6)u.
|
|
$$
|
|
|
|
<p>
|
|
In the <a href="http://nuclearmasses.org/resources_folder/Wang_2017_Chinese_Phys_C_41_030003.pdf" target="_blank">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>
|
|
The nuclear binding energy is defined as the energy required to break
|
|
up a given nucleus into its constituent parts of \( N \) neutrons and \( Z \)
|
|
protons. In terms of the atomic masses \( M(N, Z) \) the binding energy is
|
|
defined by
|
|
|
|
$$
|
|
BE(N, Z) = ZM_H c^2 + Nm_n c^2 - M(N, Z)c^2 ,
|
|
$$
|
|
|
|
where \( M_H \) is the mass of the hydrogen atom and \( m_n \) is the mass of the neutron.
|
|
In terms of the mass excess the binding energy is given by
|
|
$$
|
|
BE(N, Z) = Z\Delta_H c^2 + N\Delta_n c^2 -\Delta(N, Z)c^2 ,
|
|
$$
|
|
|
|
where \( \Delta_H c^2 = 7.2890 \) MeV and \( \Delta_n c^2 = 8.0713 \) MeV.
|
|
|
|
<p>
|
|
A popular and physically intuitive model which can be used to parametrize
|
|
the experimental binding energies as function of \( A \), is the so-called
|
|
<b>liquid drop model</b>. The ansatz is based on the following expression
|
|
|
|
$$
|
|
BE(N,Z) = a_1A-a_2A^{2/3}-a_3\frac{Z^2}{A^{1/3}}-a_4\frac{(N-Z)^2}{A},
|
|
$$
|
|
|
|
<p>
|
|
where \( A \) stands for the number of nucleons and the $a_i$s are parameters which are determined by a fit
|
|
to the experimental data.
|
|
|
|
<p>
|
|
To arrive at the above expression we have assumed that we can make the following assumptions:
|
|
|
|
<ul>
|
|
<li> There is a volume term \( a_1A \) 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.</li>
|
|
<li> There is a surface energy term \( a_2A^{2/3} \). 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.</li>
|
|
<li> There is a Coulomb energy term \( a_3\frac{Z^2}{A^{1/3}} \). The electric repulsion between each pair of protons in a nucleus yields less binding.</li>
|
|
<li> There is an asymmetry term \( a_4\frac{(N-Z)^2}{A} \). 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.</li>
|
|
</ul>
|
|
|
|
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.
|
|
|
|
<h3 id="organizing-our-data">Organizing our data </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>
|
|
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 <b>scikit-learn</b>.
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #408080; font-style: italic"># Common imports</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">pandas</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">pd</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">skl</span>
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.metrics</span> <span style="color: #008000; font-weight: bold">import</span> mean_squared_error, r2_score, mean_absolute_error
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">os</span>
|
|
|
|
<span style="color: #408080; font-style: italic"># Where to save the figures and data files</span>
|
|
PROJECT_ROOT_DIR <span style="color: #666666">=</span> <span style="color: #BA2121">"Results"</span>
|
|
FIGURE_ID <span style="color: #666666">=</span> <span style="color: #BA2121">"Results/FigureFiles"</span>
|
|
DATA_ID <span style="color: #666666">=</span> <span style="color: #BA2121">"DataFiles/"</span>
|
|
|
|
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(PROJECT_ROOT_DIR):
|
|
os<span style="color: #666666">.</span>mkdir(PROJECT_ROOT_DIR)
|
|
|
|
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(FIGURE_ID):
|
|
os<span style="color: #666666">.</span>makedirs(FIGURE_ID)
|
|
|
|
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(DATA_ID):
|
|
os<span style="color: #666666">.</span>makedirs(DATA_ID)
|
|
|
|
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">image_path</span>(fig_id):
|
|
<span style="color: #008000; font-weight: bold">return</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>join(FIGURE_ID, fig_id)
|
|
|
|
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">data_path</span>(dat_id):
|
|
<span style="color: #008000; font-weight: bold">return</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>join(DATA_ID, dat_id)
|
|
|
|
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">save_fig</span>(fig_id):
|
|
plt<span style="color: #666666">.</span>savefig(image_path(fig_id) <span style="color: #666666">+</span> <span style="color: #BA2121">".png"</span>, <span style="color: #008000">format</span><span style="color: #666666">=</span><span style="color: #BA2121">'png'</span>)
|
|
|
|
infile <span style="color: #666666">=</span> <span style="color: #008000">open</span>(data_path(<span style="color: #BA2121">"MassEval2016.dat"</span>),<span style="color: #BA2121">'r'</span>)
|
|
</pre></div>
|
|
<p>
|
|
Before we proceed, we define also a function for making our plots. You can obviously avoid this and simply set up various <b>matplotlib</b> 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>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">pylab</span> <span style="color: #008000; font-weight: bold">import</span> plt, mpl
|
|
plt<span style="color: #666666">.</span>style<span style="color: #666666">.</span>use(<span style="color: #BA2121">'seaborn'</span>)
|
|
mpl<span style="color: #666666">.</span>rcParams[<span style="color: #BA2121">'font.family'</span>] <span style="color: #666666">=</span> <span style="color: #BA2121">'serif'</span>
|
|
|
|
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">MakePlot</span>(x,y, styles, labels, axlabels):
|
|
plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">10</span>,<span style="color: #666666">6</span>))
|
|
<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #008000">len</span>(x)):
|
|
plt<span style="color: #666666">.</span>plot(x[i], y[i], styles[i], label <span style="color: #666666">=</span> labels[i])
|
|
plt<span style="color: #666666">.</span>xlabel(axlabels[<span style="color: #666666">0</span>])
|
|
plt<span style="color: #666666">.</span>ylabel(axlabels[<span style="color: #666666">1</span>])
|
|
plt<span style="color: #666666">.</span>legend(loc<span style="color: #666666">=0</span>)
|
|
</pre></div>
|
|
<p>
|
|
Our next step is to read the data on experimental binding energies and
|
|
reorganize them as functions of the mass number \( A \), the number of
|
|
protons \( Z \) and neutrons \( N \) using <b>pandas</b>. 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>
|
|
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 <b>pandas</b>. The file begins with some basic format information.
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #BA2121; font-style: italic">""" </span>
|
|
<span style="color: #BA2121; font-style: italic">This is taken from the data file of the mass 2016 evaluation. </span>
|
|
<span style="color: #BA2121; font-style: italic">All files are 3436 lines long with 124 character per line. </span>
|
|
<span style="color: #BA2121; font-style: italic"> Headers are 39 lines long. </span>
|
|
<span style="color: #BA2121; font-style: italic"> col 1 : Fortran character control: 1 = page feed 0 = line feed </span>
|
|
<span style="color: #BA2121; font-style: italic"> 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 style="color: #BA2121; font-style: italic"> These formats are reflected in the pandas widths variable below, see the statement </span>
|
|
<span style="color: #BA2121; font-style: italic"> 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 style="color: #BA2121; font-style: italic"> Pandas has also a variable header, with length 39 in this case. </span>
|
|
<span style="color: #BA2121; font-style: italic">"""</span>
|
|
</pre></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 <b>pandas</b> DataFrame structure.
|
|
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #408080; font-style: italic"># Read the experimental data with Pandas</span>
|
|
Masses <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>read_fwf(infile, usecols<span style="color: #666666">=</span>(<span style="color: #666666">2</span>,<span style="color: #666666">3</span>,<span style="color: #666666">4</span>,<span style="color: #666666">6</span>,<span style="color: #666666">11</span>),
|
|
names<span style="color: #666666">=</span>(<span style="color: #BA2121">'N'</span>, <span style="color: #BA2121">'Z'</span>, <span style="color: #BA2121">'A'</span>, <span style="color: #BA2121">'Element'</span>, <span style="color: #BA2121">'Ebinding'</span>),
|
|
widths<span style="color: #666666">=</span>(<span style="color: #666666">1</span>,<span style="color: #666666">3</span>,<span style="color: #666666">5</span>,<span style="color: #666666">5</span>,<span style="color: #666666">5</span>,<span style="color: #666666">1</span>,<span style="color: #666666">3</span>,<span style="color: #666666">4</span>,<span style="color: #666666">1</span>,<span style="color: #666666">13</span>,<span style="color: #666666">11</span>,<span style="color: #666666">11</span>,<span style="color: #666666">9</span>,<span style="color: #666666">1</span>,<span style="color: #666666">2</span>,<span style="color: #666666">11</span>,<span style="color: #666666">9</span>,<span style="color: #666666">1</span>,<span style="color: #666666">3</span>,<span style="color: #666666">1</span>,<span style="color: #666666">12</span>,<span style="color: #666666">11</span>,<span style="color: #666666">1</span>),
|
|
header<span style="color: #666666">=39</span>,
|
|
index_col<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">False</span>)
|
|
|
|
<span style="color: #408080; font-style: italic"># Extrapolated values are indicated by '#' in place of the decimal place, so</span>
|
|
<span style="color: #408080; font-style: italic"># the Ebinding column won't be numeric. Coerce to float and drop these entries.</span>
|
|
Masses[<span style="color: #BA2121">'Ebinding'</span>] <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>to_numeric(Masses[<span style="color: #BA2121">'Ebinding'</span>], errors<span style="color: #666666">=</span><span style="color: #BA2121">'coerce'</span>)
|
|
Masses <span style="color: #666666">=</span> Masses<span style="color: #666666">.</span>dropna()
|
|
<span style="color: #408080; font-style: italic"># Convert from keV to MeV.</span>
|
|
Masses[<span style="color: #BA2121">'Ebinding'</span>] <span style="color: #666666">/=</span> <span style="color: #666666">1000</span>
|
|
|
|
<span style="color: #408080; font-style: italic"># Group the DataFrame by nucleon number, A.</span>
|
|
Masses <span style="color: #666666">=</span> Masses<span style="color: #666666">.</span>groupby(<span style="color: #BA2121">'A'</span>)
|
|
<span style="color: #408080; font-style: italic"># Find the rows of the grouped DataFrame with the maximum binding energy.</span>
|
|
Masses <span style="color: #666666">=</span> Masses<span style="color: #666666">.</span>apply(<span style="color: #008000; font-weight: bold">lambda</span> t: t[t<span style="color: #666666">.</span>Ebinding<span style="color: #666666">==</span>t<span style="color: #666666">.</span>Ebinding<span style="color: #666666">.</span>max()])
|
|
</pre></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 <b>pandas</b>. 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 <b>numpy</b> and
|
|
<b>Scikit-Learn</b> afterwards.
|
|
|
|
<p>
|
|
Now we define five variables which contain
|
|
the number of nucleons \( A \), the number of protons \( Z \) and the number of neutrons \( N \), the element name and finally the energies themselves.
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span>A <span style="color: #666666">=</span> Masses[<span style="color: #BA2121">'A'</span>]
|
|
Z <span style="color: #666666">=</span> Masses[<span style="color: #BA2121">'Z'</span>]
|
|
N <span style="color: #666666">=</span> Masses[<span style="color: #BA2121">'N'</span>]
|
|
Element <span style="color: #666666">=</span> Masses[<span style="color: #BA2121">'Element'</span>]
|
|
Energies <span style="color: #666666">=</span> Masses[<span style="color: #BA2121">'Ebinding'</span>]
|
|
<span style="color: #008000">print</span>(Masses)
|
|
</pre></div>
|
|
<p>
|
|
The next step, and we will define this mathematically later, is to set up the so-called <b>design matrix</b>. We will throughout call this matrix \( \boldsymbol{X} \).
|
|
It has dimensionality \( p\times n \), where \( n \) is the number of data points and \( p \) are the so-called predictors. In our case here they are given by the number of polynomials in \( A \) we wish to include in the fit.
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #408080; font-style: italic"># Now we set up the design matrix X</span>
|
|
X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((<span style="color: #008000">len</span>(A),<span style="color: #666666">5</span>))
|
|
X[:,<span style="color: #666666">0</span>] <span style="color: #666666">=</span> <span style="color: #666666">1</span>
|
|
X[:,<span style="color: #666666">1</span>] <span style="color: #666666">=</span> A
|
|
X[:,<span style="color: #666666">2</span>] <span style="color: #666666">=</span> A<span style="color: #666666">**</span>(<span style="color: #666666">2.0/3.0</span>)
|
|
X[:,<span style="color: #666666">3</span>] <span style="color: #666666">=</span> A<span style="color: #666666">**</span>(<span style="color: #666666">-1.0/3.0</span>)
|
|
X[:,<span style="color: #666666">4</span>] <span style="color: #666666">=</span> A<span style="color: #666666">**</span>(<span style="color: #666666">-1.0</span>)
|
|
</pre></div>
|
|
<p>
|
|
With <b>scikitlearn</b> we are now ready to use linear regression and fit our data.
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span>clf <span style="color: #666666">=</span> skl<span style="color: #666666">.</span>LinearRegression()<span style="color: #666666">.</span>fit(X, Energies)
|
|
fity <span style="color: #666666">=</span> clf<span style="color: #666666">.</span>predict(X)
|
|
</pre></div>
|
|
<p>
|
|
Pretty simple!
|
|
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>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #408080; font-style: italic"># The mean squared error </span>
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">"Mean squared error: </span><span style="color: #BB6688; font-weight: bold">%.2f</span><span style="color: #BA2121">"</span> <span style="color: #666666">%</span> mean_squared_error(Energies, fity))
|
|
<span style="color: #408080; font-style: italic"># Explained variance score: 1 is perfect prediction </span>
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">'Variance score: </span><span style="color: #BB6688; font-weight: bold">%.2f</span><span style="color: #BA2121">'</span> <span style="color: #666666">%</span> r2_score(Energies, fity))
|
|
<span style="color: #408080; font-style: italic"># Mean absolute error </span>
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">'Mean absolute error: </span><span style="color: #BB6688; font-weight: bold">%.2f</span><span style="color: #BA2121">'</span> <span style="color: #666666">%</span> mean_absolute_error(Energies, fity))
|
|
<span style="color: #008000">print</span>(clf<span style="color: #666666">.</span>coef_, clf<span style="color: #666666">.</span>intercept_)
|
|
|
|
Masses[<span style="color: #BA2121">'Eapprox'</span>] <span style="color: #666666">=</span> fity
|
|
<span style="color: #408080; font-style: italic"># Generate a plot comparing the experimental with the fitted values values.</span>
|
|
fig, ax <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>subplots()
|
|
ax<span style="color: #666666">.</span>set_xlabel(<span style="color: #BA2121">r'$A = N + Z$'</span>)
|
|
ax<span style="color: #666666">.</span>set_ylabel(<span style="color: #BA2121">r'$E_\mathrm</span><span style="color: #BB6688; font-weight: bold">{bind}</span><span style="color: #BA2121">\,/\mathrm</span><span style="color: #BB6688; font-weight: bold">{MeV}</span><span style="color: #BA2121">$'</span>)
|
|
ax<span style="color: #666666">.</span>plot(Masses[<span style="color: #BA2121">'A'</span>], Masses[<span style="color: #BA2121">'Ebinding'</span>], alpha<span style="color: #666666">=0.7</span>, lw<span style="color: #666666">=2</span>,
|
|
label<span style="color: #666666">=</span><span style="color: #BA2121">'Ame2016'</span>)
|
|
ax<span style="color: #666666">.</span>plot(Masses[<span style="color: #BA2121">'A'</span>], Masses[<span style="color: #BA2121">'Eapprox'</span>], alpha<span style="color: #666666">=0.7</span>, lw<span style="color: #666666">=2</span>, c<span style="color: #666666">=</span><span style="color: #BA2121">'m'</span>,
|
|
label<span style="color: #666666">=</span><span style="color: #BA2121">'Fit'</span>)
|
|
ax<span style="color: #666666">.</span>legend()
|
|
save_fig(<span style="color: #BA2121">"Masses2016"</span>)
|
|
plt<span style="color: #666666">.</span>show()
|
|
</pre></div>
|
|
|
|
<h3 id="seeing-the-wood-for-the-trees">Seeing the wood for the trees </h3>
|
|
|
|
<p>
|
|
As a teaser, let us now see how we can do this with decision trees using <b>scikit-learn</b>. Later we will switch to so-called <b>random forests</b>!
|
|
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #408080; font-style: italic">#Decision Tree Regression</span>
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.tree</span> <span style="color: #008000; font-weight: bold">import</span> DecisionTreeRegressor
|
|
regr_1<span style="color: #666666">=</span>DecisionTreeRegressor(max_depth<span style="color: #666666">=5</span>)
|
|
regr_2<span style="color: #666666">=</span>DecisionTreeRegressor(max_depth<span style="color: #666666">=7</span>)
|
|
regr_3<span style="color: #666666">=</span>DecisionTreeRegressor(max_depth<span style="color: #666666">=9</span>)
|
|
regr_1<span style="color: #666666">.</span>fit(X, Energies)
|
|
regr_2<span style="color: #666666">.</span>fit(X, Energies)
|
|
regr_3<span style="color: #666666">.</span>fit(X, Energies)
|
|
|
|
|
|
y_1 <span style="color: #666666">=</span> regr_1<span style="color: #666666">.</span>predict(X)
|
|
y_2 <span style="color: #666666">=</span> regr_2<span style="color: #666666">.</span>predict(X)
|
|
y_3<span style="color: #666666">=</span>regr_3<span style="color: #666666">.</span>predict(X)
|
|
Masses[<span style="color: #BA2121">'Eapprox'</span>] <span style="color: #666666">=</span> y_3
|
|
<span style="color: #408080; font-style: italic"># Plot the results</span>
|
|
plt<span style="color: #666666">.</span>figure()
|
|
plt<span style="color: #666666">.</span>plot(A, Energies, color<span style="color: #666666">=</span><span style="color: #BA2121">"blue"</span>, label<span style="color: #666666">=</span><span style="color: #BA2121">"Data"</span>, linewidth<span style="color: #666666">=2</span>)
|
|
plt<span style="color: #666666">.</span>plot(A, y_1, color<span style="color: #666666">=</span><span style="color: #BA2121">"red"</span>, label<span style="color: #666666">=</span><span style="color: #BA2121">"max_depth=5"</span>, linewidth<span style="color: #666666">=2</span>)
|
|
plt<span style="color: #666666">.</span>plot(A, y_2, color<span style="color: #666666">=</span><span style="color: #BA2121">"green"</span>, label<span style="color: #666666">=</span><span style="color: #BA2121">"max_depth=7"</span>, linewidth<span style="color: #666666">=2</span>)
|
|
plt<span style="color: #666666">.</span>plot(A, y_3, color<span style="color: #666666">=</span><span style="color: #BA2121">"m"</span>, label<span style="color: #666666">=</span><span style="color: #BA2121">"max_depth=9"</span>, linewidth<span style="color: #666666">=2</span>)
|
|
|
|
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">"$A$"</span>)
|
|
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">"$E$[MeV]"</span>)
|
|
plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">"Decision Tree Regression"</span>)
|
|
plt<span style="color: #666666">.</span>legend()
|
|
save_fig(<span style="color: #BA2121">"Masses2016Trees"</span>)
|
|
plt<span style="color: #666666">.</span>show()
|
|
<span style="color: #008000">print</span>(Masses)
|
|
<span style="color: #008000">print</span>(np<span style="color: #666666">.</span>mean( (Energies<span style="color: #666666">-</span>y_1)<span style="color: #666666">**2</span>))
|
|
</pre></div>
|
|
|
|
<h3 id="and-what-about-using-neural-networks">And what about using neural networks? </h3>
|
|
|
|
The <b>seaborn</b> package allows us to visualize data in an efficient way. Note that we use <b>scikit-learn</b>'s multi-layer perceptron (or feed forward neural network)
|
|
functionality.
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.neural_network</span> <span style="color: #008000; font-weight: bold">import</span> MLPRegressor
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.metrics</span> <span style="color: #008000; font-weight: bold">import</span> accuracy_score
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">seaborn</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">sns</span>
|
|
|
|
X_train <span style="color: #666666">=</span> X
|
|
Y_train <span style="color: #666666">=</span> Energies
|
|
n_hidden_neurons <span style="color: #666666">=</span> <span style="color: #666666">100</span>
|
|
epochs <span style="color: #666666">=</span> <span style="color: #666666">100</span>
|
|
<span style="color: #408080; font-style: italic"># store models for later use</span>
|
|
eta_vals <span style="color: #666666">=</span> np<span style="color: #666666">.</span>logspace(<span style="color: #666666">-5</span>, <span style="color: #666666">1</span>, <span style="color: #666666">7</span>)
|
|
lmbd_vals <span style="color: #666666">=</span> np<span style="color: #666666">.</span>logspace(<span style="color: #666666">-5</span>, <span style="color: #666666">1</span>, <span style="color: #666666">7</span>)
|
|
<span style="color: #408080; font-style: italic"># store the models for later use</span>
|
|
DNN_scikit <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((<span style="color: #008000">len</span>(eta_vals), <span style="color: #008000">len</span>(lmbd_vals)), dtype<span style="color: #666666">=</span><span style="color: #008000">object</span>)
|
|
train_accuracy <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((<span style="color: #008000">len</span>(eta_vals), <span style="color: #008000">len</span>(lmbd_vals)))
|
|
sns<span style="color: #666666">.</span>set()
|
|
<span style="color: #008000; font-weight: bold">for</span> i, eta <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">enumerate</span>(eta_vals):
|
|
<span style="color: #008000; font-weight: bold">for</span> j, lmbd <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">enumerate</span>(lmbd_vals):
|
|
dnn <span style="color: #666666">=</span> MLPRegressor(hidden_layer_sizes<span style="color: #666666">=</span>(n_hidden_neurons), activation<span style="color: #666666">=</span><span style="color: #BA2121">'logistic'</span>,
|
|
alpha<span style="color: #666666">=</span>lmbd, learning_rate_init<span style="color: #666666">=</span>eta, max_iter<span style="color: #666666">=</span>epochs)
|
|
dnn<span style="color: #666666">.</span>fit(X_train, Y_train)
|
|
DNN_scikit[i][j] <span style="color: #666666">=</span> dnn
|
|
train_accuracy[i][j] <span style="color: #666666">=</span> dnn<span style="color: #666666">.</span>score(X_train, Y_train)
|
|
|
|
fig, ax <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>subplots(figsize <span style="color: #666666">=</span> (<span style="color: #666666">10</span>, <span style="color: #666666">10</span>))
|
|
sns<span style="color: #666666">.</span>heatmap(train_accuracy, annot<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>, ax<span style="color: #666666">=</span>ax, cmap<span style="color: #666666">=</span><span style="color: #BA2121">"viridis"</span>)
|
|
ax<span style="color: #666666">.</span>set_title(<span style="color: #BA2121">"Training Accuracy"</span>)
|
|
ax<span style="color: #666666">.</span>set_ylabel(<span style="color: #BA2121">"$\eta$"</span>)
|
|
ax<span style="color: #666666">.</span>set_xlabel(<span style="color: #BA2121">"$\lambda$"</span>)
|
|
plt<span style="color: #666666">.</span>show()
|
|
</pre></div>
|
|
|
|
<h2 id="a-first-summary">A first summary </h2>
|
|
|
|
<p>
|
|
The aim behind these introductory words was to present to you various
|
|
Python libraries and their functionalities, in particular libraries like
|
|
<b>numpy</b>, <b>pandas</b>, <b>xarray</b> and <b>matplotlib</b> and other that make our life much easier
|
|
in handling various data sets and visualizing data.
|
|
|
|
<p>
|
|
Furthermore,
|
|
<b>Scikit-Learn</b> allows us with few lines of code to implement popular
|
|
Machine Learning algorithms for supervised learning. Later we will meet <b>Tensorflow</b>, a powerful library for deep learning.
|
|
Now it is time to dive more into the details of various methods. We will start with linear regression and try to take a deeper look at what it entails.
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="why-linear-regression-aka-ordinary-least-squares-and-family">Why Linear Regression (aka Ordinary Least Squares and family) </h2>
|
|
|
|
<p>
|
|
Fitting a continuous function with linear parameterization in terms of the parameters \( \boldsymbol{\beta} \).
|
|
|
|
<ul>
|
|
<li> Method of choice for fitting a continuous function!</li>
|
|
<li> Gives an excellent introduction to central Machine Learning features with <b>understandable pedagogical</b> links to other methods like <b>Neural Networks</b>, <b>Support Vector Machines</b> etc</li>
|
|
<li> Analytical expression for the fitting parameters \( \boldsymbol{\beta} \)</li>
|
|
<li> Analytical expressions for statistical propertiers like mean values, variances, confidence intervals and more</li>
|
|
<li> Analytical relation with probabilistic interpretations</li>
|
|
<li> Easy to introduce basic concepts like bias-variance tradeoff, cross-validation, resampling and regularization techniques and many other ML topics</li>
|
|
<li> Easy to code! And links well with classification problems and logistic regression and neural networks</li>
|
|
<li> Allows for <b>easy</b> hands-on understanding of gradient descent methods</li>
|
|
<li> and many more features</li>
|
|
</ul>
|
|
|
|
For more discussions of Ridge and Lasso regression, <a href="https://arxiv.org/abs/1509.09169" target="_blank">Wessel van Wieringen's</a> article is highly recommended.
|
|
Similarly, <a href="https://arxiv.org/abs/1803.08823" target="_blank">Mehta et al's article</a> is also recommended.
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="regression-analysis-overarching-aims">Regression analysis, overarching aims </h2>
|
|
<div class="alert alert-block alert-block alert-text-normal">
|
|
<b></b>
|
|
<p>
|
|
|
|
<p>
|
|
Regression modeling deals with the description of the sampling distribution of a given random variable \( y \) and how it varies as function of another variable or a set of such variables \( \boldsymbol{x} =[x_0, x_1,\dots, x_{n-1}]^T \).
|
|
The first variable is called the <b>dependent</b>, the <b>outcome</b> or the <b>response</b> variable while the set of variables \( \boldsymbol{x} \) is called the independent variable, or the predictor variable or the explanatory variable.
|
|
|
|
<p>
|
|
A regression model aims at finding a likelihood function \( p(\boldsymbol{y}\vert \boldsymbol{x}) \), that is the conditional distribution for \( \boldsymbol{y} \) with a given \( \boldsymbol{x} \). The estimation of \( p(\boldsymbol{y}\vert \boldsymbol{x}) \) is made using a data set with
|
|
|
|
<ul>
|
|
<li> \( n \) cases \( i = 0, 1, 2, \dots, n-1 \)</li>
|
|
<li> Response (target, dependent or outcome) variable \( y_i \) with \( i = 0, 1, 2, \dots, n-1 \)</li>
|
|
<li> \( p \) so-called explanatory (independent or predictor) variables \( \boldsymbol{x}_i=[x_{i0}, x_{i1}, \dots, x_{ip-1}] \) with \( i = 0, 1, 2, \dots, n-1 \) and explanatory variables running from \( 0 \) to \( p-1 \). See below for more explicit examples.</li>
|
|
</ul>
|
|
|
|
The goal of the regression analysis is to extract/exploit relationship between \( \boldsymbol{y} \) and \( \boldsymbol{x} \) in or to infer causal dependencies, approximations to the likelihood functions, functional relationships and to make predictions, making fits and many other things.
|
|
</div>
|
|
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="regression-analysis-overarching-aims-ii">Regression analysis, overarching aims II </h2>
|
|
<div class="alert alert-block alert-block alert-text-normal">
|
|
<b></b>
|
|
<p>
|
|
|
|
<p>
|
|
Consider an experiment in which \( p \) characteristics of \( n \) samples are
|
|
measured. The data from this experiment, for various explanatory variables \( p \) are normally represented by a matrix
|
|
\( \mathbf{X} \).
|
|
|
|
<p>
|
|
The matrix \( \mathbf{X} \) is called the <em>design
|
|
matrix</em>. Additional information of the samples is available in the
|
|
form of \( \boldsymbol{y} \) (also as above). The variable \( \boldsymbol{y} \) is
|
|
generally referred to as the <em>response variable</em>. The aim of
|
|
regression analysis is to explain \( \boldsymbol{y} \) in terms of
|
|
\( \boldsymbol{X} \) through a functional relationship like \( y_i =
|
|
f(\mathbf{X}_{i,\ast}) \). When no prior knowledge on the form of
|
|
\( f(\cdot) \) is available, it is common to assume a linear relationship
|
|
between \( \boldsymbol{X} \) and \( \boldsymbol{y} \). This assumption gives rise to
|
|
the <em>linear regression model</em> where \( \boldsymbol{\beta} = [\beta_0, \ldots,
|
|
\beta_{p-1}]^{T} \) are the <em>regression parameters</em>.
|
|
|
|
<p>
|
|
Linear regression gives us a set of analytical equations for the parameters \( \beta_j \).
|
|
|
|
|
|
</div>
|
|
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="examples">Examples </h2>
|
|
<div class="alert alert-block alert-block alert-text-normal">
|
|
<b></b>
|
|
<p>
|
|
In order to understand the relation among the predictors \( p \), the set of data \( n \) and the target (outcome, output etc) \( \boldsymbol{y} \),
|
|
consider the model we discussed for describing nuclear binding energies.
|
|
|
|
<p>
|
|
There we assumed that we could parametrize the data using a polynomial approximation based on the liquid drop model.
|
|
Assuming
|
|
$$
|
|
BE(A) = a_0+a_1A+a_2A^{2/3}+a_3A^{-1/3}+a_4A^{-1},
|
|
$$
|
|
|
|
we have five predictors, that is the intercept, the \( A \) dependent term, the \( A^{2/3} \) term and the \( A^{-1/3} \) and \( A^{-1} \) terms.
|
|
This gives \( p=0,1,2,3,4 \). Furthermore we have \( n \) entries for each predictor. It means that our design matrix is a
|
|
\( p\times n \) matrix \( \boldsymbol{X} \).
|
|
|
|
<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 href="https://www.sciencedirect.com/science/article/pii/S0957417407006719?via%3Dihub" target="_blank">credit card default data from Taiwan</a>. The data set contains data on \( n=30000 \) credit card holders with predictors like gender, marital status, age, profession, education, etc. In total there are \( 24 \) such predictors or attributes leading to a design matrix of dimensionality \( 24 \times 30000 \). This is however a classification problem and we will come back to it when we discuss Logistic Regression.
|
|
|
|
|
|
</div>
|
|
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="general-linear-models">General linear models </h2>
|
|
<div class="alert alert-block alert-block alert-text-normal">
|
|
<b></b>
|
|
<p>
|
|
Before we proceed let us study a case from linear algebra where we aim at fitting a set of data \( \boldsymbol{y}=[y_0,y_1,\dots,y_{n-1}] \). 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 \( \boldsymbol{x}=[x_0,x_1,\dots,x_{n-1}] \), that is \( y_i = y(x_i) \) with \( i=0,1,2,\dots,n-1 \). The variables \( x_i \) could represent physical quantities like time, temperature, position etc. We assume that \( y(x) \) is a smooth function.
|
|
|
|
<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 \( y \) 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 \( n-1 \) with \( n \) points, that is
|
|
$$
|
|
y=y(x) \rightarrow y(x_i)=\tilde{y}_i+\epsilon_i=\sum_{j=0}^{n-1} \beta_j x_i^j+\epsilon_i,
|
|
$$
|
|
|
|
where \( \epsilon_i \) is the error in our approximation.
|
|
|
|
|
|
</div>
|
|
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="rewriting-the-fitting-procedure-as-a-linear-algebra-problem">Rewriting the fitting procedure as a linear algebra problem </h2>
|
|
<div class="alert alert-block alert-block alert-text-normal">
|
|
<b></b>
|
|
<p>
|
|
For every set of values \( y_i,x_i \) we have thus the corresponding set of equations
|
|
$$
|
|
\begin{align*}
|
|
y_0&=\beta_0+\beta_1x_0^1+\beta_2x_0^2+\dots+\beta_{n-1}x_0^{n-1}+\epsilon_0\\
|
|
y_1&=\beta_0+\beta_1x_1^1+\beta_2x_1^2+\dots+\beta_{n-1}x_1^{n-1}+\epsilon_1\\
|
|
y_2&=\beta_0+\beta_1x_2^1+\beta_2x_2^2+\dots+\beta_{n-1}x_2^{n-1}+\epsilon_2\\
|
|
\dots & \dots \\
|
|
y_{n-1}&=\beta_0+\beta_1x_{n-1}^1+\beta_2x_{n-1}^2+\dots+\beta_{n-1}x_{n-1}^{n-1}+\epsilon_{n-1}.\\
|
|
\end{align*}
|
|
$$
|
|
</div>
|
|
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="rewriting-the-fitting-procedure-as-a-linear-algebra-problem-more-details">Rewriting the fitting procedure as a linear algebra problem, more details </h2>
|
|
<div class="alert alert-block alert-block alert-text-normal">
|
|
<b></b>
|
|
<p>
|
|
Defining the vectors
|
|
$$
|
|
\boldsymbol{y} = [y_0,y_1, y_2,\dots, y_{n-1}]^T,
|
|
$$
|
|
|
|
and
|
|
$$
|
|
\boldsymbol{\beta} = [\beta_0,\beta_1, \beta_2,\dots, \beta_{n-1}]^T,
|
|
$$
|
|
|
|
and
|
|
$$
|
|
\boldsymbol{\epsilon} = [\epsilon_0,\epsilon_1, \epsilon_2,\dots, \epsilon_{n-1}]^T,
|
|
$$
|
|
|
|
and the design matrix
|
|
$$
|
|
\boldsymbol{X}=
|
|
\begin{bmatrix}
|
|
1& x_{0}^1 &x_{0}^2& \dots & \dots &x_{0}^{n-1}\\
|
|
1& x_{1}^1 &x_{1}^2& \dots & \dots &x_{1}^{n-1}\\
|
|
1& x_{2}^1 &x_{2}^2& \dots & \dots &x_{2}^{n-1}\\
|
|
\dots& \dots &\dots& \dots & \dots &\dots\\
|
|
1& x_{n-1}^1 &x_{n-1}^2& \dots & \dots &x_{n-1}^{n-1}\\
|
|
\end{bmatrix}
|
|
$$
|
|
|
|
we can rewrite our equations as
|
|
$$
|
|
\boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta}+\boldsymbol{\epsilon}.
|
|
$$
|
|
|
|
The above design matrix is called a <a href="https://en.wikipedia.org/wiki/Vandermonde_matrix" target="_blank">Vandermonde matrix</a>.
|
|
</div>
|
|
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="generalizing-the-fitting-procedure-as-a-linear-algebra-problem">Generalizing the fitting procedure as a linear algebra problem </h2>
|
|
<div class="alert alert-block alert-block alert-text-normal">
|
|
<b></b>
|
|
<p>
|
|
|
|
<p>
|
|
We are obviously not limited to the above polynomial expansions. We
|
|
could replace the various powers of \( x \) with elements of Fourier
|
|
series or instead of \( x_i^j \) we could have \( \cos{(j x_i)} \) or \( \sin{(j
|
|
x_i)} \), or time series or other orthogonal functions. For every set
|
|
of values \( y_i,x_i \) we can then generalize the equations to
|
|
|
|
$$
|
|
\begin{align*}
|
|
y_0&=\beta_0x_{00}+\beta_1x_{01}+\beta_2x_{02}+\dots+\beta_{n-1}x_{0n-1}+\epsilon_0\\
|
|
y_1&=\beta_0x_{10}+\beta_1x_{11}+\beta_2x_{12}+\dots+\beta_{n-1}x_{1n-1}+\epsilon_1\\
|
|
y_2&=\beta_0x_{20}+\beta_1x_{21}+\beta_2x_{22}+\dots+\beta_{n-1}x_{2n-1}+\epsilon_2\\
|
|
\dots & \dots \\
|
|
y_{i}&=\beta_0x_{i0}+\beta_1x_{i1}+\beta_2x_{i2}+\dots+\beta_{n-1}x_{in-1}+\epsilon_i\\
|
|
\dots & \dots \\
|
|
y_{n-1}&=\beta_0x_{n-1,0}+\beta_1x_{n-1,2}+\beta_2x_{n-1,2}+\dots+\beta_{n-1}x_{n-1,n-1}+\epsilon_{n-1}.\\
|
|
\end{align*}
|
|
$$
|
|
|
|
<p>
|
|
<b>Note that we have \( p=n \) here. The matrix is symmetric. This is generally not the case!</b>
|
|
</div>
|
|
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="generalizing-the-fitting-procedure-as-a-linear-algebra-problem">Generalizing the fitting procedure as a linear algebra problem </h2>
|
|
<div class="alert alert-block alert-block alert-text-normal">
|
|
<b></b>
|
|
<p>
|
|
We redefine in turn the matrix \( \boldsymbol{X} \) as
|
|
$$
|
|
\boldsymbol{X}=
|
|
\begin{bmatrix}
|
|
x_{00}& x_{01} &x_{02}& \dots & \dots &x_{0,n-1}\\
|
|
x_{10}& x_{11} &x_{12}& \dots & \dots &x_{1,n-1}\\
|
|
x_{20}& x_{21} &x_{22}& \dots & \dots &x_{2,n-1}\\
|
|
\dots& \dots &\dots& \dots & \dots &\dots\\
|
|
x_{n-1,0}& x_{n-1,1} &x_{n-1,2}& \dots & \dots &x_{n-1,n-1}\\
|
|
\end{bmatrix}
|
|
$$
|
|
|
|
and without loss of generality we rewrite again our equations as
|
|
$$
|
|
\boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta}+\boldsymbol{\epsilon}.
|
|
$$
|
|
|
|
The left-hand side of this equation is kwown. Our error vector \( \boldsymbol{\epsilon} \) and the parameter vector \( \boldsymbol{\beta} \) are our unknow quantities. How can we obtain the optimal set of \( \beta_i \) values?
|
|
</div>
|
|
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="optimizing-our-parameters">Optimizing our parameters </h2>
|
|
<div class="alert alert-block alert-block alert-text-normal">
|
|
<b></b>
|
|
<p>
|
|
We have defined the matrix \( \boldsymbol{X} \) via the equations
|
|
$$
|
|
\begin{align*}
|
|
y_0&=\beta_0x_{00}+\beta_1x_{01}+\beta_2x_{02}+\dots+\beta_{n-1}x_{0n-1}+\epsilon_0\\
|
|
y_1&=\beta_0x_{10}+\beta_1x_{11}+\beta_2x_{12}+\dots+\beta_{n-1}x_{1n-1}+\epsilon_1\\
|
|
y_2&=\beta_0x_{20}+\beta_1x_{21}+\beta_2x_{22}+\dots+\beta_{n-1}x_{2n-1}+\epsilon_1\\
|
|
\dots & \dots \\
|
|
y_{i}&=\beta_0x_{i0}+\beta_1x_{i1}+\beta_2x_{i2}+\dots+\beta_{n-1}x_{in-1}+\epsilon_1\\
|
|
\dots & \dots \\
|
|
y_{n-1}&=\beta_0x_{n-1,0}+\beta_1x_{n-1,2}+\beta_2x_{n-1,2}+\dots+\beta_{n-1}x_{n-1,n-1}+\epsilon_{n-1}.\\
|
|
\end{align*}
|
|
$$
|
|
|
|
<p>
|
|
As we noted above, we stayed with a system with the design matrix
|
|
\( \boldsymbol{X}\in {\mathbb{R}}^{n\times n} \), that is we have \( p=n \). For reasons to come later (algorithmic arguments) we will hereafter define
|
|
our matrix as \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \), with the predictors refering to the column numbers and the entries \( n \) being the row elements.
|
|
|
|
|
|
</div>
|
|
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="our-model-for-the-nuclear-binding-energies">Our model for the nuclear binding energies </h2>
|
|
|
|
<p>
|
|
In our <a href="https://compphysics.github.io/MachineLearning/doc/pub/How2ReadData/html/How2ReadData.html" target="_blank">introductory notes</a> we looked at the so-called <a href="https://en.wikipedia.org/wiki/Semi-empirical_mass_formula" target="_blank">liquid drop model</a>. Let us remind ourselves about what we did by looking at the code.
|
|
|
|
<p>
|
|
We restate the parts of the code we are most interested in.
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #408080; font-style: italic"># Common imports</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">pandas</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">pd</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">IPython.display</span> <span style="color: #008000; font-weight: bold">import</span> display
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">os</span>
|
|
|
|
<span style="color: #408080; font-style: italic"># Where to save the figures and data files</span>
|
|
PROJECT_ROOT_DIR <span style="color: #666666">=</span> <span style="color: #BA2121">"Results"</span>
|
|
FIGURE_ID <span style="color: #666666">=</span> <span style="color: #BA2121">"Results/FigureFiles"</span>
|
|
DATA_ID <span style="color: #666666">=</span> <span style="color: #BA2121">"DataFiles/"</span>
|
|
|
|
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(PROJECT_ROOT_DIR):
|
|
os<span style="color: #666666">.</span>mkdir(PROJECT_ROOT_DIR)
|
|
|
|
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(FIGURE_ID):
|
|
os<span style="color: #666666">.</span>makedirs(FIGURE_ID)
|
|
|
|
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(DATA_ID):
|
|
os<span style="color: #666666">.</span>makedirs(DATA_ID)
|
|
|
|
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">image_path</span>(fig_id):
|
|
<span style="color: #008000; font-weight: bold">return</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>join(FIGURE_ID, fig_id)
|
|
|
|
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">data_path</span>(dat_id):
|
|
<span style="color: #008000; font-weight: bold">return</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>join(DATA_ID, dat_id)
|
|
|
|
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">save_fig</span>(fig_id):
|
|
plt<span style="color: #666666">.</span>savefig(image_path(fig_id) <span style="color: #666666">+</span> <span style="color: #BA2121">".png"</span>, <span style="color: #008000">format</span><span style="color: #666666">=</span><span style="color: #BA2121">'png'</span>)
|
|
|
|
infile <span style="color: #666666">=</span> <span style="color: #008000">open</span>(data_path(<span style="color: #BA2121">"MassEval2016.dat"</span>),<span style="color: #BA2121">'r'</span>)
|
|
|
|
|
|
<span style="color: #408080; font-style: italic"># Read the experimental data with Pandas</span>
|
|
Masses <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>read_fwf(infile, usecols<span style="color: #666666">=</span>(<span style="color: #666666">2</span>,<span style="color: #666666">3</span>,<span style="color: #666666">4</span>,<span style="color: #666666">6</span>,<span style="color: #666666">11</span>),
|
|
names<span style="color: #666666">=</span>(<span style="color: #BA2121">'N'</span>, <span style="color: #BA2121">'Z'</span>, <span style="color: #BA2121">'A'</span>, <span style="color: #BA2121">'Element'</span>, <span style="color: #BA2121">'Ebinding'</span>),
|
|
widths<span style="color: #666666">=</span>(<span style="color: #666666">1</span>,<span style="color: #666666">3</span>,<span style="color: #666666">5</span>,<span style="color: #666666">5</span>,<span style="color: #666666">5</span>,<span style="color: #666666">1</span>,<span style="color: #666666">3</span>,<span style="color: #666666">4</span>,<span style="color: #666666">1</span>,<span style="color: #666666">13</span>,<span style="color: #666666">11</span>,<span style="color: #666666">11</span>,<span style="color: #666666">9</span>,<span style="color: #666666">1</span>,<span style="color: #666666">2</span>,<span style="color: #666666">11</span>,<span style="color: #666666">9</span>,<span style="color: #666666">1</span>,<span style="color: #666666">3</span>,<span style="color: #666666">1</span>,<span style="color: #666666">12</span>,<span style="color: #666666">11</span>,<span style="color: #666666">1</span>),
|
|
header<span style="color: #666666">=39</span>,
|
|
index_col<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">False</span>)
|
|
|
|
<span style="color: #408080; font-style: italic"># Extrapolated values are indicated by '#' in place of the decimal place, so</span>
|
|
<span style="color: #408080; font-style: italic"># the Ebinding column won't be numeric. Coerce to float and drop these entries.</span>
|
|
Masses[<span style="color: #BA2121">'Ebinding'</span>] <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>to_numeric(Masses[<span style="color: #BA2121">'Ebinding'</span>], errors<span style="color: #666666">=</span><span style="color: #BA2121">'coerce'</span>)
|
|
Masses <span style="color: #666666">=</span> Masses<span style="color: #666666">.</span>dropna()
|
|
<span style="color: #408080; font-style: italic"># Convert from keV to MeV.</span>
|
|
Masses[<span style="color: #BA2121">'Ebinding'</span>] <span style="color: #666666">/=</span> <span style="color: #666666">1000</span>
|
|
|
|
<span style="color: #408080; font-style: italic"># Group the DataFrame by nucleon number, A.</span>
|
|
Masses <span style="color: #666666">=</span> Masses<span style="color: #666666">.</span>groupby(<span style="color: #BA2121">'A'</span>)
|
|
<span style="color: #408080; font-style: italic"># Find the rows of the grouped DataFrame with the maximum binding energy.</span>
|
|
Masses <span style="color: #666666">=</span> Masses<span style="color: #666666">.</span>apply(<span style="color: #008000; font-weight: bold">lambda</span> t: t[t<span style="color: #666666">.</span>Ebinding<span style="color: #666666">==</span>t<span style="color: #666666">.</span>Ebinding<span style="color: #666666">.</span>max()])
|
|
A <span style="color: #666666">=</span> Masses[<span style="color: #BA2121">'A'</span>]
|
|
Z <span style="color: #666666">=</span> Masses[<span style="color: #BA2121">'Z'</span>]
|
|
N <span style="color: #666666">=</span> Masses[<span style="color: #BA2121">'N'</span>]
|
|
Element <span style="color: #666666">=</span> Masses[<span style="color: #BA2121">'Element'</span>]
|
|
Energies <span style="color: #666666">=</span> Masses[<span style="color: #BA2121">'Ebinding'</span>]
|
|
|
|
<span style="color: #408080; font-style: italic"># Now we set up the design matrix X</span>
|
|
X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((<span style="color: #008000">len</span>(A),<span style="color: #666666">5</span>))
|
|
X[:,<span style="color: #666666">0</span>] <span style="color: #666666">=</span> <span style="color: #666666">1</span>
|
|
X[:,<span style="color: #666666">1</span>] <span style="color: #666666">=</span> A
|
|
X[:,<span style="color: #666666">2</span>] <span style="color: #666666">=</span> A<span style="color: #666666">**</span>(<span style="color: #666666">2.0/3.0</span>)
|
|
X[:,<span style="color: #666666">3</span>] <span style="color: #666666">=</span> A<span style="color: #666666">**</span>(<span style="color: #666666">-1.0/3.0</span>)
|
|
X[:,<span style="color: #666666">4</span>] <span style="color: #666666">=</span> A<span style="color: #666666">**</span>(<span style="color: #666666">-1.0</span>)
|
|
<span style="color: #408080; font-style: italic"># Then nice printout using pandas</span>
|
|
DesignMatrix <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>DataFrame(X)
|
|
DesignMatrix<span style="color: #666666">.</span>index <span style="color: #666666">=</span> A
|
|
DesignMatrix<span style="color: #666666">.</span>columns <span style="color: #666666">=</span> [<span style="color: #BA2121">'1'</span>, <span style="color: #BA2121">'A'</span>, <span style="color: #BA2121">'A^(2/3)'</span>, <span style="color: #BA2121">'A^(-1/3)'</span>, <span style="color: #BA2121">'1/A'</span>]
|
|
display(DesignMatrix)
|
|
</pre></div>
|
|
<p>
|
|
With \( \boldsymbol{\beta}\in {\mathbb{R}}^{p\times 1} \), it means that we will hereafter write our equations for the approximation as
|
|
$$
|
|
\boldsymbol{\tilde{y}}= \boldsymbol{X}\boldsymbol{\beta},
|
|
$$
|
|
|
|
throughout these lectures.
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="optimizing-our-parameters-more-details">Optimizing our parameters, more details </h2>
|
|
<div class="alert alert-block alert-block alert-text-normal">
|
|
<b></b>
|
|
<p>
|
|
With the above we use the design matrix to define the approximation \( \boldsymbol{\tilde{y}} \) via the unknown quantity \( \boldsymbol{\beta} \) as
|
|
$$
|
|
\boldsymbol{\tilde{y}}= \boldsymbol{X}\boldsymbol{\beta},
|
|
$$
|
|
|
|
and in order to find the optimal parameters \( \beta_i \) instead of solving the above linear algebra problem, we define a function which gives a measure of the spread between the values \( y_i \) (which represent hopefully the exact values) and the parameterized values \( \tilde{y}_i \), namely
|
|
$$
|
|
C(\boldsymbol{\beta})=\frac{1}{n}\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2=\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{\tilde{y}}\right)^T\left(\boldsymbol{y}-\boldsymbol{\tilde{y}}\right)\right\},
|
|
$$
|
|
|
|
or using the matrix \( \boldsymbol{X} \) and in a more compact matrix-vector notation as
|
|
$$
|
|
C(\boldsymbol{\beta})=\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right\}.
|
|
$$
|
|
|
|
This function is one possible way to define the so-called cost function.
|
|
|
|
<p>
|
|
It is also common to define
|
|
the function \( C \) as
|
|
|
|
$$
|
|
C(\boldsymbol{\beta})=\frac{1}{2n}\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2,
|
|
$$
|
|
|
|
since when taking the first derivative with respect to the unknown parameters \( \beta \), the factor of \( 2 \) cancels out.
|
|
</div>
|
|
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="interpretations-and-optimizing-our-parameters">Interpretations and optimizing our parameters </h2>
|
|
<div class="alert alert-block alert-block alert-text-normal">
|
|
<b></b>
|
|
<p>
|
|
|
|
<p>
|
|
The function
|
|
$$
|
|
C(\boldsymbol{\beta})=\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right\},
|
|
$$
|
|
|
|
can be linked to the variance of the quantity \( y_i \) 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 \( y_i \) as a mean value
|
|
$$
|
|
y_{i}=\langle y_i \rangle = \beta_0x_{i,0}+\beta_1x_{i,1}+\beta_2x_{i,2}+\dots+\beta_{n-1}x_{i,n-1}+\epsilon_i,
|
|
$$
|
|
|
|
<p>
|
|
where \( \langle y_i \rangle \) is the mean value. Keep in mind also that
|
|
till now we have treated \( y_i \) as the exact value. Normally, the
|
|
response (dependent or outcome) variable \( y_i \) 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 \( y_i \) as our exact value for the response variable.
|
|
|
|
<p>
|
|
In order to find the parameters \( \beta_i \) we will then minimize the spread of \( C(\boldsymbol{\beta}) \), that is we are going to solve the problem
|
|
$$
|
|
{\displaystyle \min_{\boldsymbol{\beta}\in
|
|
{\mathbb{R}}^{p}}}\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right\}.
|
|
$$
|
|
|
|
In practical terms it means we will require
|
|
$$
|
|
\frac{\partial C(\boldsymbol{\beta})}{\partial \beta_j} = \frac{\partial }{\partial \beta_j}\left[ \frac{1}{n}\sum_{i=0}^{n-1}\left(y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}\right)^2\right]=0,
|
|
$$
|
|
|
|
which results in
|
|
$$
|
|
\frac{\partial C(\boldsymbol{\beta})}{\partial \beta_j} = -\frac{2}{n}\left[ \sum_{i=0}^{n-1}x_{ij}\left(y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}\right)\right]=0,
|
|
$$
|
|
|
|
or in a matrix-vector form as
|
|
$$
|
|
\frac{\partial C(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = 0 = \boldsymbol{X}^T\left( \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right).
|
|
$$
|
|
|
|
|
|
</div>
|
|
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="interpretations-and-optimizing-our-parameters">Interpretations and optimizing our parameters </h2>
|
|
<div class="alert alert-block alert-block alert-text-normal">
|
|
<b></b>
|
|
<p>
|
|
We can rewrite
|
|
$$
|
|
\frac{\partial C(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = 0 = \boldsymbol{X}^T\left( \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right),
|
|
$$
|
|
|
|
as
|
|
$$
|
|
\boldsymbol{X}^T\boldsymbol{y} = \boldsymbol{X}^T\boldsymbol{X}\boldsymbol{\beta},
|
|
$$
|
|
|
|
and if the matrix \( \boldsymbol{X}^T\boldsymbol{X} \) is invertible we have the solution
|
|
$$
|
|
\boldsymbol{\beta} =\left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}.
|
|
$$
|
|
|
|
<p>
|
|
We note also that since our design matrix is defined as \( \boldsymbol{X}\in
|
|
{\mathbb{R}}^{n\times p} \), the product \( \boldsymbol{X}^T\boldsymbol{X} \in
|
|
{\mathbb{R}}^{p\times p} \). In the above case we have that \( p \ll n \),
|
|
in our case \( p=5 \) meaning that we end up with inverting a small
|
|
\( 5\times 5 \) 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 <b>LU</b> decomposition or <b>Singular Value Decomposition</b> (SVD) for finding the inverse of the matrix
|
|
\( \boldsymbol{X}^T\boldsymbol{X} \).
|
|
</div>
|
|
|
|
|
|
<p>
|
|
<div class="alert alert-block alert-block alert-text-normal">
|
|
<b></b>
|
|
<p>
|
|
<b>Small question</b>: Do you think the example we have at hand here (the nuclear binding energies) can lead to problems in inverting the matrix \( \boldsymbol{X}^T\boldsymbol{X} \)? What kind of problems can we expect?
|
|
</div>
|
|
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="some-useful-matrix-and-vector-expressions">Some useful matrix and vector expressions </h2>
|
|
|
|
<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.
|
|
|
|
$$
|
|
\frac{\partial (\boldsymbol{b}^T\boldsymbol{a})}{\partial \boldsymbol{a}} = \boldsymbol{b},
|
|
$$
|
|
|
|
$$
|
|
\frac{\partial (\boldsymbol{a}^T\boldsymbol{A}\boldsymbol{a})}{\partial \boldsymbol{a}} = (\boldsymbol{A}+\boldsymbol{A}^T)\boldsymbol{a},
|
|
$$
|
|
|
|
$$
|
|
\frac{\partial tr(\boldsymbol{B}\boldsymbol{A})}{\partial \boldsymbol{A}} = \boldsymbol{B}^T,
|
|
$$
|
|
|
|
$$
|
|
\frac{\partial \log{\vert\boldsymbol{A}\vert}}{\partial \boldsymbol{A}} = (\boldsymbol{A}^{-1})^T.
|
|
$$
|
|
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="interpretations-and-optimizing-our-parameters">Interpretations and optimizing our parameters </h2>
|
|
<div class="alert alert-block alert-block alert-text-normal">
|
|
<b></b>
|
|
<p>
|
|
The residuals \( \boldsymbol{\epsilon} \) are in turn given by
|
|
$$
|
|
\boldsymbol{\epsilon} = \boldsymbol{y}-\boldsymbol{\tilde{y}} = \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta},
|
|
$$
|
|
|
|
and with
|
|
$$
|
|
\boldsymbol{X}^T\left( \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)= 0,
|
|
$$
|
|
|
|
we have
|
|
$$
|
|
\boldsymbol{X}^T\boldsymbol{\epsilon}=\boldsymbol{X}^T\left( \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)= 0,
|
|
$$
|
|
|
|
meaning that the solution for \( \boldsymbol{\beta} \) is the one which minimizes the residuals. Later we will link this with the maximum likelihood approach.
|
|
|
|
|
|
</div>
|
|
|
|
|
|
<p>
|
|
Let us now return to our nuclear binding energies and simply code the above equations.
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="own-code-for-ordinary-least-squares">Own code for Ordinary Least Squares </h2>
|
|
|
|
<p>
|
|
It is rather straightforward to implement the matrix inversion and obtain the parameters \( \boldsymbol{\beta} \). After having defined the matrix \( \boldsymbol{X} \) we simply need to
|
|
write
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #408080; font-style: italic"># matrix inversion to find beta</span>
|
|
beta <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>inv(X<span style="color: #666666">.</span>T<span style="color: #666666">.</span>dot(X))<span style="color: #666666">.</span>dot(X<span style="color: #666666">.</span>T)<span style="color: #666666">.</span>dot(Energies)
|
|
<span style="color: #408080; font-style: italic"># and then make the prediction</span>
|
|
ytilde <span style="color: #666666">=</span> X <span style="color: #666666">@</span> beta
|
|
</pre></div>
|
|
<p>
|
|
Alternatively, you can use the least squares functionality in <b>Numpy</b> as
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span>fit <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>lstsq(X, Energies, rcond <span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">None</span>)[<span style="color: #666666">0</span>]
|
|
ytildenp <span style="color: #666666">=</span> np<span style="color: #666666">.</span>dot(fit,X<span style="color: #666666">.</span>T)
|
|
</pre></div>
|
|
<p>
|
|
And finally we plot our fit with and compare with data
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span>Masses[<span style="color: #BA2121">'Eapprox'</span>] <span style="color: #666666">=</span> ytilde
|
|
<span style="color: #408080; font-style: italic"># Generate a plot comparing the experimental with the fitted values values.</span>
|
|
fig, ax <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>subplots()
|
|
ax<span style="color: #666666">.</span>set_xlabel(<span style="color: #BA2121">r'$A = N + Z$'</span>)
|
|
ax<span style="color: #666666">.</span>set_ylabel(<span style="color: #BA2121">r'$E_\mathrm</span><span style="color: #BB6688; font-weight: bold">{bind}</span><span style="color: #BA2121">\,/\mathrm</span><span style="color: #BB6688; font-weight: bold">{MeV}</span><span style="color: #BA2121">$'</span>)
|
|
ax<span style="color: #666666">.</span>plot(Masses[<span style="color: #BA2121">'A'</span>], Masses[<span style="color: #BA2121">'Ebinding'</span>], alpha<span style="color: #666666">=0.7</span>, lw<span style="color: #666666">=2</span>,
|
|
label<span style="color: #666666">=</span><span style="color: #BA2121">'Ame2016'</span>)
|
|
ax<span style="color: #666666">.</span>plot(Masses[<span style="color: #BA2121">'A'</span>], Masses[<span style="color: #BA2121">'Eapprox'</span>], alpha<span style="color: #666666">=0.7</span>, lw<span style="color: #666666">=2</span>, c<span style="color: #666666">=</span><span style="color: #BA2121">'m'</span>,
|
|
label<span style="color: #666666">=</span><span style="color: #BA2121">'Fit'</span>)
|
|
ax<span style="color: #666666">.</span>legend()
|
|
save_fig(<span style="color: #BA2121">"Masses2016OLS"</span>)
|
|
plt<span style="color: #666666">.</span>show()
|
|
</pre></div>
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="adding-error-analysis-and-training-set-up">Adding error analysis and training set up </h2>
|
|
|
|
<p>
|
|
We can easily test our fit by computing the \( R2 \) score that we discussed in connection with the functionality of <b>Scikit-Learn</b> in the introductory slides.
|
|
Since we are not using <b>Scikit-Learn</b> here we can define our own \( R2 \) function as
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">R2</span>(y_data, y_model):
|
|
<span style="color: #008000; font-weight: bold">return</span> <span style="color: #666666">1</span> <span style="color: #666666">-</span> np<span style="color: #666666">.</span>sum((y_data <span style="color: #666666">-</span> y_model) <span style="color: #666666">**</span> <span style="color: #666666">2</span>) <span style="color: #666666">/</span> np<span style="color: #666666">.</span>sum((y_data <span style="color: #666666">-</span> np<span style="color: #666666">.</span>mean(y_data)) <span style="color: #666666">**</span> <span style="color: #666666">2</span>)
|
|
</pre></div>
|
|
<p>
|
|
and we would be using it as
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000">print</span>(R2(Energies,ytilde))
|
|
</pre></div>
|
|
<p>
|
|
We can easily add our <b>MSE</b> score as
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">MSE</span>(y_data,y_model):
|
|
n <span style="color: #666666">=</span> np<span style="color: #666666">.</span>size(y_model)
|
|
<span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>sum((y_data<span style="color: #666666">-</span>y_model)<span style="color: #666666">**2</span>)<span style="color: #666666">/</span>n
|
|
|
|
<span style="color: #008000">print</span>(MSE(Energies,ytilde))
|
|
</pre></div>
|
|
<p>
|
|
and finally the relative error as
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">RelativeError</span>(y_data,y_model):
|
|
<span style="color: #008000; font-weight: bold">return</span> <span style="color: #008000">abs</span>((y_data<span style="color: #666666">-</span>y_model)<span style="color: #666666">/</span>y_data)
|
|
<span style="color: #008000">print</span>(RelativeError(Energies, ytilde))
|
|
</pre></div>
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="the-chi-2-function">The \( \chi^2 \) function </h2>
|
|
<div class="alert alert-block alert-block alert-text-normal">
|
|
<b></b>
|
|
<p>
|
|
|
|
<p>
|
|
Normally, the response (dependent or outcome) variable \( y_i \) 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 \( y_i \) as our exact value for the
|
|
response variable.
|
|
|
|
<p>
|
|
Introducing the standard deviation \( \sigma_i \) for each measurement
|
|
\( y_i \), we define now the \( \chi^2 \) function (omitting the \( 1/n \) term)
|
|
as
|
|
|
|
$$
|
|
\chi^2(\boldsymbol{\beta})=\frac{1}{n}\sum_{i=0}^{n-1}\frac{\left(y_i-\tilde{y}_i\right)^2}{\sigma_i^2}=\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{\tilde{y}}\right)^T\frac{1}{\boldsymbol{\Sigma^2}}\left(\boldsymbol{y}-\boldsymbol{\tilde{y}}\right)\right\},
|
|
$$
|
|
|
|
where the matrix \( \boldsymbol{\Sigma} \) is a diagonal matrix with \( \sigma_i \) as matrix elements.
|
|
|
|
|
|
</div>
|
|
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="the-chi-2-function">The \( \chi^2 \) function </h2>
|
|
<div class="alert alert-block alert-block alert-text-normal">
|
|
<b></b>
|
|
<p>
|
|
|
|
<p>
|
|
In order to find the parameters \( \beta_i \) we will then minimize the spread of \( \chi^2(\boldsymbol{\beta}) \) by requiring
|
|
$$
|
|
\frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \beta_j} = \frac{\partial }{\partial \beta_j}\left[ \frac{1}{n}\sum_{i=0}^{n-1}\left(\frac{y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}}{\sigma_i}\right)^2\right]=0,
|
|
$$
|
|
|
|
which results in
|
|
$$
|
|
\frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \beta_j} = -\frac{2}{n}\left[ \sum_{i=0}^{n-1}\frac{x_{ij}}{\sigma_i}\left(\frac{y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}}{\sigma_i}\right)\right]=0,
|
|
$$
|
|
|
|
or in a matrix-vector form as
|
|
$$
|
|
\frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = 0 = \boldsymbol{A}^T\left( \boldsymbol{b}-\boldsymbol{A}\boldsymbol{\beta}\right).
|
|
$$
|
|
|
|
where we have defined the matrix \( \boldsymbol{A} =\boldsymbol{X}/\boldsymbol{\Sigma} \) with matrix elements \( a_{ij} = x_{ij}/\sigma_i \) and the vector \( \boldsymbol{b} \) with elements \( b_i = y_i/\sigma_i \).
|
|
</div>
|
|
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="the-chi-2-function">The \( \chi^2 \) function </h2>
|
|
<div class="alert alert-block alert-block alert-text-normal">
|
|
<b></b>
|
|
<p>
|
|
|
|
<p>
|
|
We can rewrite
|
|
$$
|
|
\frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = 0 = \boldsymbol{A}^T\left( \boldsymbol{b}-\boldsymbol{A}\boldsymbol{\beta}\right),
|
|
$$
|
|
|
|
as
|
|
$$
|
|
\boldsymbol{A}^T\boldsymbol{b} = \boldsymbol{A}^T\boldsymbol{A}\boldsymbol{\beta},
|
|
$$
|
|
|
|
and if the matrix \( \boldsymbol{A}^T\boldsymbol{A} \) is invertible we have the solution
|
|
$$
|
|
\boldsymbol{\beta} =\left(\boldsymbol{A}^T\boldsymbol{A}\right)^{-1}\boldsymbol{A}^T\boldsymbol{b}.
|
|
$$
|
|
</div>
|
|
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="the-chi-2-function">The \( \chi^2 \) function </h2>
|
|
<div class="alert alert-block alert-block alert-text-normal">
|
|
<b></b>
|
|
<p>
|
|
|
|
<p>
|
|
If we then introduce the matrix
|
|
$$
|
|
\boldsymbol{H} = \left(\boldsymbol{A}^T\boldsymbol{A}\right)^{-1},
|
|
$$
|
|
|
|
we have then the following expression for the parameters \( \beta_j \) (the matrix elements of \( \boldsymbol{H} \) are \( h_{ij} \))
|
|
$$
|
|
\beta_j = \sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}\frac{y_i}{\sigma_i}\frac{x_{ik}}{\sigma_i} = \sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}b_ia_{ik}
|
|
$$
|
|
|
|
We state without proof the expression for the uncertainty in the parameters \( \beta_j \) as (we leave this as an exercise)
|
|
$$
|
|
\sigma^2(\beta_j) = \sum_{i=0}^{n-1}\sigma_i^2\left( \frac{\partial \beta_j}{\partial y_i}\right)^2,
|
|
$$
|
|
|
|
resulting in
|
|
$$
|
|
\sigma^2(\beta_j) = \left(\sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}a_{ik}\right)\left(\sum_{l=0}^{p-1}h_{jl}\sum_{m=0}^{n-1}a_{ml}\right) = h_{jj}!
|
|
$$
|
|
</div>
|
|
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="the-chi-2-function">The \( \chi^2 \) function </h2>
|
|
<div class="alert alert-block alert-block alert-text-normal">
|
|
<b></b>
|
|
<p>
|
|
The first step here is to approximate the function \( y \) with a first-order polynomial, that is we write
|
|
$$
|
|
y=y(x) \rightarrow y(x_i) \approx \beta_0+\beta_1 x_i.
|
|
$$
|
|
|
|
By computing the derivatives of \( \chi^2 \) with respect to \( \beta_0 \) and \( \beta_1 \) show that these are given by
|
|
$$
|
|
\frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \beta_0} = -2\left[ \frac{1}{n}\sum_{i=0}^{n-1}\left(\frac{y_i-\beta_0-\beta_1x_{i}}{\sigma_i^2}\right)\right]=0,
|
|
$$
|
|
|
|
and
|
|
$$
|
|
\frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \beta_1} = -\frac{2}{n}\left[ \sum_{i=0}^{n-1}x_i\left(\frac{y_i-\beta_0-\beta_1x_{i}}{\sigma_i^2}\right)\right]=0.
|
|
$$
|
|
</div>
|
|
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="the-chi-2-function">The \( \chi^2 \) function </h2>
|
|
<div class="alert alert-block alert-block alert-text-normal">
|
|
<b></b>
|
|
<p>
|
|
|
|
<p>
|
|
For a linear fit (a first-order polynomial) we don't need to invert a matrix!!
|
|
Defining
|
|
$$
|
|
\gamma = \sum_{i=0}^{n-1}\frac{1}{\sigma_i^2},
|
|
$$
|
|
|
|
|
|
$$
|
|
\gamma_x = \sum_{i=0}^{n-1}\frac{x_{i}}{\sigma_i^2},
|
|
$$
|
|
|
|
|
|
$$
|
|
\gamma_y = \sum_{i=0}^{n-1}\left(\frac{y_i}{\sigma_i^2}\right),
|
|
$$
|
|
|
|
|
|
$$
|
|
\gamma_{xx} = \sum_{i=0}^{n-1}\frac{x_ix_{i}}{\sigma_i^2},
|
|
$$
|
|
|
|
|
|
$$
|
|
\gamma_{xy} = \sum_{i=0}^{n-1}\frac{y_ix_{i}}{\sigma_i^2},
|
|
$$
|
|
|
|
<p>
|
|
we obtain
|
|
|
|
$$
|
|
\beta_0 = \frac{\gamma_{xx}\gamma_y-\gamma_x\gamma_y}{\gamma\gamma_{xx}-\gamma_x^2},
|
|
$$
|
|
|
|
|
|
$$
|
|
\beta_1 = \frac{\gamma_{xy}\gamma-\gamma_x\gamma_y}{\gamma\gamma_{xx}-\gamma_x^2}.
|
|
$$
|
|
|
|
<p>
|
|
This approach (different linear and non-linear regression) suffers
|
|
often from both being underdetermined and overdetermined in the
|
|
unknown coefficients \( \beta_i \). A better approach is to use the
|
|
Singular Value Decomposition (SVD) method discussed next week.
|
|
|
|
|
|
</div>
|
|
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="fitting-an-equation-of-state-for-dense-nuclear-matter">Fitting an Equation of State for Dense Nuclear Matter </h2>
|
|
|
|
<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 href="https://www.sciencedirect.com/science/article/pii/S0370157399001106" target="_blank">the addition of three-body
|
|
forces</a>. This
|
|
time the file is presented as a standard <b>csv</b> file.
|
|
|
|
<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
|
|
<b>pandas</b> again, rather extensively in order to organize our data.
|
|
|
|
<p>
|
|
The difference now is that we use <b>Scikit-Learn's</b> regression tools
|
|
instead of our own matrix inversion implementation. Furthermore, we
|
|
sneak in <b>Ridge</b> regression (to be discussed below) which includes a
|
|
hyperparameter \( \lambda \), also to be explained below.
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="the-code">The code </h2>
|
|
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #408080; font-style: italic"># Common imports</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">os</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">pandas</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">pd</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">skl</span>
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.metrics</span> <span style="color: #008000; font-weight: bold">import</span> mean_squared_error, r2_score, mean_absolute_error
|
|
|
|
<span style="color: #408080; font-style: italic"># Where to save the figures and data files</span>
|
|
PROJECT_ROOT_DIR <span style="color: #666666">=</span> <span style="color: #BA2121">"Results"</span>
|
|
FIGURE_ID <span style="color: #666666">=</span> <span style="color: #BA2121">"Results/FigureFiles"</span>
|
|
DATA_ID <span style="color: #666666">=</span> <span style="color: #BA2121">"DataFiles/"</span>
|
|
|
|
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(PROJECT_ROOT_DIR):
|
|
os<span style="color: #666666">.</span>mkdir(PROJECT_ROOT_DIR)
|
|
|
|
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(FIGURE_ID):
|
|
os<span style="color: #666666">.</span>makedirs(FIGURE_ID)
|
|
|
|
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(DATA_ID):
|
|
os<span style="color: #666666">.</span>makedirs(DATA_ID)
|
|
|
|
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">image_path</span>(fig_id):
|
|
<span style="color: #008000; font-weight: bold">return</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>join(FIGURE_ID, fig_id)
|
|
|
|
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">data_path</span>(dat_id):
|
|
<span style="color: #008000; font-weight: bold">return</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>join(DATA_ID, dat_id)
|
|
|
|
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">save_fig</span>(fig_id):
|
|
plt<span style="color: #666666">.</span>savefig(image_path(fig_id) <span style="color: #666666">+</span> <span style="color: #BA2121">".png"</span>, <span style="color: #008000">format</span><span style="color: #666666">=</span><span style="color: #BA2121">'png'</span>)
|
|
|
|
infile <span style="color: #666666">=</span> <span style="color: #008000">open</span>(data_path(<span style="color: #BA2121">"EoS.csv"</span>),<span style="color: #BA2121">'r'</span>)
|
|
|
|
<span style="color: #408080; font-style: italic"># Read the EoS data as csv file and organize the data into two arrays with density and energies</span>
|
|
EoS <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>read_csv(infile, names<span style="color: #666666">=</span>(<span style="color: #BA2121">'Density'</span>, <span style="color: #BA2121">'Energy'</span>))
|
|
EoS[<span style="color: #BA2121">'Energy'</span>] <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>to_numeric(EoS[<span style="color: #BA2121">'Energy'</span>], errors<span style="color: #666666">=</span><span style="color: #BA2121">'coerce'</span>)
|
|
EoS <span style="color: #666666">=</span> EoS<span style="color: #666666">.</span>dropna()
|
|
Energies <span style="color: #666666">=</span> EoS[<span style="color: #BA2121">'Energy'</span>]
|
|
Density <span style="color: #666666">=</span> EoS[<span style="color: #BA2121">'Density'</span>]
|
|
<span style="color: #408080; font-style: italic"># The design matrix now as function of various polytrops</span>
|
|
X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((<span style="color: #008000">len</span>(Density),<span style="color: #666666">4</span>))
|
|
X[:,<span style="color: #666666">3</span>] <span style="color: #666666">=</span> Density<span style="color: #666666">**</span>(<span style="color: #666666">4.0/3.0</span>)
|
|
X[:,<span style="color: #666666">2</span>] <span style="color: #666666">=</span> Density
|
|
X[:,<span style="color: #666666">1</span>] <span style="color: #666666">=</span> Density<span style="color: #666666">**</span>(<span style="color: #666666">2.0/3.0</span>)
|
|
X[:,<span style="color: #666666">0</span>] <span style="color: #666666">=</span> <span style="color: #666666">1</span>
|
|
|
|
<span style="color: #408080; font-style: italic"># We use now Scikit-Learn's linear regressor and ridge regressor</span>
|
|
<span style="color: #408080; font-style: italic"># OLS part</span>
|
|
clf <span style="color: #666666">=</span> skl<span style="color: #666666">.</span>LinearRegression()<span style="color: #666666">.</span>fit(X, Energies)
|
|
ytilde <span style="color: #666666">=</span> clf<span style="color: #666666">.</span>predict(X)
|
|
EoS[<span style="color: #BA2121">'Eols'</span>] <span style="color: #666666">=</span> ytilde
|
|
<span style="color: #408080; font-style: italic"># The mean squared error </span>
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">"Mean squared error: </span><span style="color: #BB6688; font-weight: bold">%.2f</span><span style="color: #BA2121">"</span> <span style="color: #666666">%</span> mean_squared_error(Energies, ytilde))
|
|
<span style="color: #408080; font-style: italic"># Explained variance score: 1 is perfect prediction </span>
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">'Variance score: </span><span style="color: #BB6688; font-weight: bold">%.2f</span><span style="color: #BA2121">'</span> <span style="color: #666666">%</span> r2_score(Energies, ytilde))
|
|
<span style="color: #408080; font-style: italic"># Mean absolute error </span>
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">'Mean absolute error: </span><span style="color: #BB6688; font-weight: bold">%.2f</span><span style="color: #BA2121">'</span> <span style="color: #666666">%</span> mean_absolute_error(Energies, ytilde))
|
|
<span style="color: #008000">print</span>(clf<span style="color: #666666">.</span>coef_, clf<span style="color: #666666">.</span>intercept_)
|
|
|
|
<span style="color: #408080; font-style: italic"># The Ridge regression with a hyperparameter lambda = 0.1</span>
|
|
_lambda <span style="color: #666666">=</span> <span style="color: #666666">0.1</span>
|
|
clf_ridge <span style="color: #666666">=</span> skl<span style="color: #666666">.</span>Ridge(alpha<span style="color: #666666">=</span>_lambda)<span style="color: #666666">.</span>fit(X, Energies)
|
|
yridge <span style="color: #666666">=</span> clf_ridge<span style="color: #666666">.</span>predict(X)
|
|
EoS[<span style="color: #BA2121">'Eridge'</span>] <span style="color: #666666">=</span> yridge
|
|
<span style="color: #408080; font-style: italic"># The mean squared error </span>
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">"Mean squared error: </span><span style="color: #BB6688; font-weight: bold">%.2f</span><span style="color: #BA2121">"</span> <span style="color: #666666">%</span> mean_squared_error(Energies, yridge))
|
|
<span style="color: #408080; font-style: italic"># Explained variance score: 1 is perfect prediction </span>
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">'Variance score: </span><span style="color: #BB6688; font-weight: bold">%.2f</span><span style="color: #BA2121">'</span> <span style="color: #666666">%</span> r2_score(Energies, yridge))
|
|
<span style="color: #408080; font-style: italic"># Mean absolute error </span>
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">'Mean absolute error: </span><span style="color: #BB6688; font-weight: bold">%.2f</span><span style="color: #BA2121">'</span> <span style="color: #666666">%</span> mean_absolute_error(Energies, yridge))
|
|
<span style="color: #008000">print</span>(clf_ridge<span style="color: #666666">.</span>coef_, clf_ridge<span style="color: #666666">.</span>intercept_)
|
|
|
|
fig, ax <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>subplots()
|
|
ax<span style="color: #666666">.</span>set_xlabel(<span style="color: #BA2121">r'$\rho[\mathrm</span><span style="color: #BB6688; font-weight: bold">{fm}</span><span style="color: #BA2121">^{-3}]$'</span>)
|
|
ax<span style="color: #666666">.</span>set_ylabel(<span style="color: #BA2121">r'Energy per particle'</span>)
|
|
ax<span style="color: #666666">.</span>plot(EoS[<span style="color: #BA2121">'Density'</span>], EoS[<span style="color: #BA2121">'Energy'</span>], alpha<span style="color: #666666">=0.7</span>, lw<span style="color: #666666">=2</span>,
|
|
label<span style="color: #666666">=</span><span style="color: #BA2121">'Theoretical data'</span>)
|
|
ax<span style="color: #666666">.</span>plot(EoS[<span style="color: #BA2121">'Density'</span>], EoS[<span style="color: #BA2121">'Eols'</span>], alpha<span style="color: #666666">=0.7</span>, lw<span style="color: #666666">=2</span>, c<span style="color: #666666">=</span><span style="color: #BA2121">'m'</span>,
|
|
label<span style="color: #666666">=</span><span style="color: #BA2121">'OLS'</span>)
|
|
ax<span style="color: #666666">.</span>plot(EoS[<span style="color: #BA2121">'Density'</span>], EoS[<span style="color: #BA2121">'Eridge'</span>], alpha<span style="color: #666666">=0.7</span>, lw<span style="color: #666666">=2</span>, c<span style="color: #666666">=</span><span style="color: #BA2121">'g'</span>,
|
|
label<span style="color: #666666">=</span><span style="color: #BA2121">'Ridge $\lambda = 0.1$'</span>)
|
|
ax<span style="color: #666666">.</span>legend()
|
|
save_fig(<span style="color: #BA2121">"EoSfitting"</span>)
|
|
plt<span style="color: #666666">.</span>show()
|
|
</pre></div>
|
|
<p>
|
|
The above simple polynomial in density \( \rho \) gives an excellent fit
|
|
to the data.
|
|
|
|
<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>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="splitting-our-data-in-training-and-test-data">Splitting our Data in Training and Test data </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). <b>Scikit-Learn</b> 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 \( 2/3 \) to \( 4/5 \) 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 <b>bias-variance</b> 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>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">os</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">pandas</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">pd</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
|
|
<span style="color: #408080; font-style: italic"># Where to save the figures and data files</span>
|
|
PROJECT_ROOT_DIR <span style="color: #666666">=</span> <span style="color: #BA2121">"Results"</span>
|
|
FIGURE_ID <span style="color: #666666">=</span> <span style="color: #BA2121">"Results/FigureFiles"</span>
|
|
DATA_ID <span style="color: #666666">=</span> <span style="color: #BA2121">"DataFiles/"</span>
|
|
|
|
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(PROJECT_ROOT_DIR):
|
|
os<span style="color: #666666">.</span>mkdir(PROJECT_ROOT_DIR)
|
|
|
|
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(FIGURE_ID):
|
|
os<span style="color: #666666">.</span>makedirs(FIGURE_ID)
|
|
|
|
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(DATA_ID):
|
|
os<span style="color: #666666">.</span>makedirs(DATA_ID)
|
|
|
|
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">image_path</span>(fig_id):
|
|
<span style="color: #008000; font-weight: bold">return</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>join(FIGURE_ID, fig_id)
|
|
|
|
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">data_path</span>(dat_id):
|
|
<span style="color: #008000; font-weight: bold">return</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>join(DATA_ID, dat_id)
|
|
|
|
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">save_fig</span>(fig_id):
|
|
plt<span style="color: #666666">.</span>savefig(image_path(fig_id) <span style="color: #666666">+</span> <span style="color: #BA2121">".png"</span>, <span style="color: #008000">format</span><span style="color: #666666">=</span><span style="color: #BA2121">'png'</span>)
|
|
|
|
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">R2</span>(y_data, y_model):
|
|
<span style="color: #008000; font-weight: bold">return</span> <span style="color: #666666">1</span> <span style="color: #666666">-</span> np<span style="color: #666666">.</span>sum((y_data <span style="color: #666666">-</span> y_model) <span style="color: #666666">**</span> <span style="color: #666666">2</span>) <span style="color: #666666">/</span> np<span style="color: #666666">.</span>sum((y_data <span style="color: #666666">-</span> np<span style="color: #666666">.</span>mean(y_data)) <span style="color: #666666">**</span> <span style="color: #666666">2</span>)
|
|
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">MSE</span>(y_data,y_model):
|
|
n <span style="color: #666666">=</span> np<span style="color: #666666">.</span>size(y_model)
|
|
<span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>sum((y_data<span style="color: #666666">-</span>y_model)<span style="color: #666666">**2</span>)<span style="color: #666666">/</span>n
|
|
|
|
infile <span style="color: #666666">=</span> <span style="color: #008000">open</span>(data_path(<span style="color: #BA2121">"EoS.csv"</span>),<span style="color: #BA2121">'r'</span>)
|
|
|
|
<span style="color: #408080; font-style: italic"># Read the EoS data as csv file and organized into two arrays with density and energies</span>
|
|
EoS <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>read_csv(infile, names<span style="color: #666666">=</span>(<span style="color: #BA2121">'Density'</span>, <span style="color: #BA2121">'Energy'</span>))
|
|
EoS[<span style="color: #BA2121">'Energy'</span>] <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>to_numeric(EoS[<span style="color: #BA2121">'Energy'</span>], errors<span style="color: #666666">=</span><span style="color: #BA2121">'coerce'</span>)
|
|
EoS <span style="color: #666666">=</span> EoS<span style="color: #666666">.</span>dropna()
|
|
Energies <span style="color: #666666">=</span> EoS[<span style="color: #BA2121">'Energy'</span>]
|
|
Density <span style="color: #666666">=</span> EoS[<span style="color: #BA2121">'Density'</span>]
|
|
<span style="color: #408080; font-style: italic"># The design matrix now as function of various polytrops</span>
|
|
X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((<span style="color: #008000">len</span>(Density),<span style="color: #666666">5</span>))
|
|
X[:,<span style="color: #666666">0</span>] <span style="color: #666666">=</span> <span style="color: #666666">1</span>
|
|
X[:,<span style="color: #666666">1</span>] <span style="color: #666666">=</span> Density<span style="color: #666666">**</span>(<span style="color: #666666">2.0/3.0</span>)
|
|
X[:,<span style="color: #666666">2</span>] <span style="color: #666666">=</span> Density
|
|
X[:,<span style="color: #666666">3</span>] <span style="color: #666666">=</span> Density<span style="color: #666666">**</span>(<span style="color: #666666">4.0/3.0</span>)
|
|
X[:,<span style="color: #666666">4</span>] <span style="color: #666666">=</span> Density<span style="color: #666666">**</span>(<span style="color: #666666">5.0/3.0</span>)
|
|
<span style="color: #408080; font-style: italic"># We split the data in test and training data</span>
|
|
X_train, X_test, y_train, y_test <span style="color: #666666">=</span> train_test_split(X, Energies, test_size<span style="color: #666666">=0.2</span>)
|
|
<span style="color: #408080; font-style: italic"># matrix inversion to find beta</span>
|
|
beta <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>inv(X_train<span style="color: #666666">.</span>T<span style="color: #666666">.</span>dot(X_train))<span style="color: #666666">.</span>dot(X_train<span style="color: #666666">.</span>T)<span style="color: #666666">.</span>dot(y_train)
|
|
<span style="color: #408080; font-style: italic"># and then make the prediction</span>
|
|
ytilde <span style="color: #666666">=</span> X_train <span style="color: #666666">@</span> beta
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">"Training R2"</span>)
|
|
<span style="color: #008000">print</span>(R2(y_train,ytilde))
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">"Training MSE"</span>)
|
|
<span style="color: #008000">print</span>(MSE(y_train,ytilde))
|
|
ypredict <span style="color: #666666">=</span> X_test <span style="color: #666666">@</span> beta
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">"Test R2"</span>)
|
|
<span style="color: #008000">print</span>(R2(y_test,ypredict))
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">"Test MSE"</span>)
|
|
<span style="color: #008000">print</span>(MSE(y_test,ypredict))
|
|
</pre></div>
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="exercises-for-week-36">Exercises for week 36 </h2>
|
|
Here are three possible exercises for week 36 and the lab sessions of Wednesday September 1..
|
|
|
|
<p>
|
|
<!-- --- begin exercise --- -->
|
|
|
|
<h2 id="exercise-1-setting-up-various-python-environments">Exercise 1: Setting up various Python environments </h2>
|
|
|
|
<p>
|
|
The first exercise here is of a mere technical art. We want you to have
|
|
|
|
<ul>
|
|
<li> git as a version control software and to establish a user account on a provider like GitHub. Other providers like GitLab etc are equally fine. You can also use the University of Oslo <a href="https://www.uio.no/tjenester/it/maskin/filer/versjonskontroll/github.html" target="_blank">GitHub facilities</a>.</li>
|
|
<li> Install various Python packages</li>
|
|
</ul>
|
|
|
|
We will make extensive use of Python as programming language and its
|
|
myriad of available libraries. You will find
|
|
IPython/Jupyter notebooks invaluable in your work. You can run <b>R</b>
|
|
codes in the Jupyter/IPython notebooks, with the immediate benefit of
|
|
visualizing your data. You can also use compiled languages like C++,
|
|
Rust, Fortran etc if you prefer. The focus in these lectures will be
|
|
on Python.
|
|
|
|
<p>
|
|
If you have Python installed (we recommend Python3) and you feel
|
|
pretty familiar with installing different packages, we recommend that
|
|
you install the following Python packages via <b>pip</b> as
|
|
|
|
<ol>
|
|
<li> pip install numpy scipy matplotlib ipython scikit-learn sympy pandas pillow</li>
|
|
</ol>
|
|
|
|
For <b>Tensorflow</b>, we recommend following the instructions in the text of
|
|
<a href="http://shop.oreilly.com/product/0636920052289.do" target="_blank">Aurelien Geron, Hands‑On Machine Learning with Scikit‑Learn and TensorFlow, O'Reilly</a>
|
|
|
|
<p>
|
|
We will come back to <b>tensorflow</b> later.
|
|
|
|
<p>
|
|
For Python3, replace <b>pip</b> with <b>pip3</b>.
|
|
|
|
<p>
|
|
For OSX users we recommend, after having installed Xcode, to
|
|
install <b>brew</b>. Brew allows for a seamless installation of additional
|
|
software via for example
|
|
|
|
<ol>
|
|
<li> brew install python3</li>
|
|
</ol>
|
|
|
|
For Linux users, with its variety of distributions like for example the widely popular Ubuntu distribution,
|
|
you can use <b>pip</b> as well and simply install Python as
|
|
|
|
<ol>
|
|
<li> sudo apt-get install python3 (or python for Python2.7)</li>
|
|
</ol>
|
|
|
|
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
|
|
|
|
<ul>
|
|
<li> <a href="https://docs.anaconda.com/" target="_blank">Anaconda</a>,</li>
|
|
</ul>
|
|
|
|
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 <b>conda</b>.
|
|
|
|
<ul>
|
|
<li> <a href="https://www.enthought.com/product/canopy/" target="_blank">Enthought canopy</a></li>
|
|
</ul>
|
|
|
|
is a Python
|
|
distribution for scientific and analytic computing distribution and
|
|
analysis environment, available for free and under a commercial
|
|
license.
|
|
|
|
<p>
|
|
We recommend using <b>Anaconda</b> if you are not too familiar with setting paths in a terminal environment.
|
|
|
|
<p>
|
|
<!-- --- end exercise --- -->
|
|
|
|
<p>
|
|
<!-- --- begin exercise --- -->
|
|
|
|
<h2 id="exercise-2-making-your-own-data-and-exploring-scikit-learn">Exercise 2: making your own data and exploring scikit-learn </h2>
|
|
|
|
<p>
|
|
We will generate our own dataset for a function \( y(x) \) where \( x \in [0,1] \) and defined by random numbers computed with the uniform distribution. The function \( y \) is a quadratic polynomial in \( x \) with added stochastic noise according to the normal distribution \( \cal {N}(0,1) \).
|
|
The following simple Python instructions define our \( x \) and \( y \) values (with 100 data points).
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span>x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(<span style="color: #666666">100</span>,<span style="color: #666666">1</span>)
|
|
y <span style="color: #666666">=</span> <span style="color: #666666">2.0+5*</span>x<span style="color: #666666">*</span>x<span style="color: #666666">+0.1*</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(<span style="color: #666666">100</span>,<span style="color: #666666">1</span>)
|
|
</pre></div>
|
|
<ol>
|
|
<li> Write your own code (following the examples under the <a href="https://compphysics.github.io/MachineLearning/doc/LectureNotes/_build/html/chapter1.html" target="_blank">regression notes</a>) for computing the parametrization of the data set fitting a second-order polynomial.</li>
|
|
<li> Use thereafter <b>scikit-learn</b> (see again the examples in the regression slides) and compare with your own code.</li>
|
|
<li> Using scikit-learn, compute also the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error defined as</li>
|
|
</ol>
|
|
|
|
$$ MSE(\boldsymbol{y},\boldsymbol{\tilde{y}}) = \frac{1}{n}
|
|
\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2,
|
|
$$
|
|
|
|
and the \( R^2 \) score function.
|
|
If \( \tilde{\boldsymbol{y}}_i \) is the predicted value of the \( i-th \) sample and \( y_i \) is the corresponding true value, then the score \( R^2 \) is defined as
|
|
$$
|
|
R^2(\boldsymbol{y}, \tilde{\boldsymbol{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2},
|
|
$$
|
|
|
|
where we have defined the mean value of \( \boldsymbol{y} \) as
|
|
$$
|
|
\bar{y} = \frac{1}{n} \sum_{i=0}^{n - 1} y_i.
|
|
$$
|
|
|
|
You can use the functionality included in scikit-learn. If you feel for it, you can use your own program and define functions which compute the above two functions.
|
|
Discuss the meaning of these results. Try also to vary the coefficient in front of the added stochastic noise term and discuss the quality of the fits.
|
|
|
|
<p>
|
|
<!-- --- begin solution of exercise --- -->
|
|
<b>Solution.</b>
|
|
The code here is an example of where we define our own design matrix and fit parameters \( \beta \).
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">os</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">pandas</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">pd</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
|
|
|
|
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">save_fig</span>(fig_id):
|
|
plt<span style="color: #666666">.</span>savefig(image_path(fig_id) <span style="color: #666666">+</span> <span style="color: #BA2121">".png"</span>, <span style="color: #008000">format</span><span style="color: #666666">=</span><span style="color: #BA2121">'png'</span>)
|
|
|
|
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">R2</span>(y_data, y_model):
|
|
<span style="color: #008000; font-weight: bold">return</span> <span style="color: #666666">1</span> <span style="color: #666666">-</span> np<span style="color: #666666">.</span>sum((y_data <span style="color: #666666">-</span> y_model) <span style="color: #666666">**</span> <span style="color: #666666">2</span>) <span style="color: #666666">/</span> np<span style="color: #666666">.</span>sum((y_data <span style="color: #666666">-</span> np<span style="color: #666666">.</span>mean(y_data)) <span style="color: #666666">**</span> <span style="color: #666666">2</span>)
|
|
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">MSE</span>(y_data,y_model):
|
|
n <span style="color: #666666">=</span> np<span style="color: #666666">.</span>size(y_model)
|
|
<span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>sum((y_data<span style="color: #666666">-</span>y_model)<span style="color: #666666">**2</span>)<span style="color: #666666">/</span>n
|
|
|
|
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(<span style="color: #666666">100</span>)
|
|
y <span style="color: #666666">=</span> <span style="color: #666666">2.0+5*</span>x<span style="color: #666666">*</span>x<span style="color: #666666">+0.1*</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(<span style="color: #666666">100</span>)
|
|
|
|
|
|
<span style="color: #408080; font-style: italic"># The design matrix now as function of a given polynomial</span>
|
|
X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((<span style="color: #008000">len</span>(x),<span style="color: #666666">3</span>))
|
|
X[:,<span style="color: #666666">0</span>] <span style="color: #666666">=</span> <span style="color: #666666">1.0</span>
|
|
X[:,<span style="color: #666666">1</span>] <span style="color: #666666">=</span> x
|
|
X[:,<span style="color: #666666">2</span>] <span style="color: #666666">=</span> x<span style="color: #666666">**2</span>
|
|
<span style="color: #408080; font-style: italic"># We split the data in test and training data</span>
|
|
X_train, X_test, y_train, y_test <span style="color: #666666">=</span> train_test_split(X, y, test_size<span style="color: #666666">=0.2</span>)
|
|
<span style="color: #408080; font-style: italic"># matrix inversion to find beta</span>
|
|
beta <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>inv(X_train<span style="color: #666666">.</span>T <span style="color: #666666">@</span> X_train) <span style="color: #666666">@</span> X_train<span style="color: #666666">.</span>T <span style="color: #666666">@</span> y_train
|
|
<span style="color: #008000">print</span>(beta)
|
|
<span style="color: #408080; font-style: italic"># and then make the prediction</span>
|
|
ytilde <span style="color: #666666">=</span> X_train <span style="color: #666666">@</span> beta
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">"Training R2"</span>)
|
|
<span style="color: #008000">print</span>(R2(y_train,ytilde))
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">"Training MSE"</span>)
|
|
<span style="color: #008000">print</span>(MSE(y_train,ytilde))
|
|
ypredict <span style="color: #666666">=</span> X_test <span style="color: #666666">@</span> beta
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">"Test R2"</span>)
|
|
<span style="color: #008000">print</span>(R2(y_test,ypredict))
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">"Test MSE"</span>)
|
|
<span style="color: #008000">print</span>(MSE(y_test,ypredict))
|
|
</pre></div>
|
|
<p>
|
|
<!-- --- end solution of exercise --- -->
|
|
|
|
<p>
|
|
<!-- --- end exercise --- -->
|
|
|
|
<p>
|
|
<!-- --- begin exercise --- -->
|
|
|
|
<h2 id="exercise-3-normalizing-our-data">Exercise 3: Normalizing our data </h2>
|
|
|
|
<p>
|
|
A much used approach before starting to train the data is to preprocess 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>
|
|
<b>Scikit-Learn</b> has several functions which allow us to rescale the
|
|
data, normally resulting in much better results in terms of various
|
|
accuracy scores. The <b>StandardScaler</b> function in <b>Scikit-Learn</b>
|
|
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 <b>Scikit-Learn</b> is the <b>MinMaxScaler</b> which
|
|
ensures that all features are exactly between \( 0 \) and \( 1 \). The
|
|
|
|
<p>
|
|
The <b>Normalizer</b> 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>
|
|
The <b>RobustScaler</b> 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>
|
|
It also common to split the data in a <b>training</b> set and a <b>testing</b> set. A typical split is to use \( 80\% \) of the data for training and the rest
|
|
for testing. This can be done as follows with our design matrix \( \boldsymbol{X} \) and data \( \boldsymbol{y} \) (remember to import <b>scikit-learn</b>)
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #408080; font-style: italic"># split in training and test data</span>
|
|
X_train, X_test, y_train, y_test <span style="color: #666666">=</span> train_test_split(X,y,test_size<span style="color: #666666">=0.2</span>)
|
|
</pre></div>
|
|
<p>
|
|
Then we can use the standard scaler to scale our data as
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span>scaler <span style="color: #666666">=</span> StandardScaler()
|
|
scaler<span style="color: #666666">.</span>fit(X_train)
|
|
X_train_scaled <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>transform(X_train)
|
|
X_test_scaled <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>transform(X_test)
|
|
</pre></div>
|
|
<p>
|
|
In this exercise we want you to to compute the MSE for the training
|
|
data and the test data as function of the complexity of a polynomial,
|
|
that is the degree of a given polynomial. We want you also to compute the \( R2 \) score as function of the complexity of the model for both training data and test data. You should also run the calculation with and without scaling.
|
|
|
|
<p>
|
|
One of
|
|
the aims is to reproduce Figure 2.11 of <a href="https://github.com/CompPhysics/MLErasmus/blob/master/doc/Textbooks/elementsstat.pdf" target="_blank">Hastie et al</a>.
|
|
|
|
<p>
|
|
Our data is defined by \( x\in [-3,3] \) with a total of for example \( 100 \) data points.
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed()
|
|
n <span style="color: #666666">=</span> <span style="color: #666666">100</span>
|
|
maxdegree <span style="color: #666666">=</span> <span style="color: #666666">14</span>
|
|
<span style="color: #408080; font-style: italic"># Make data set.</span>
|
|
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(<span style="color: #666666">-3</span>, <span style="color: #666666">3</span>, n)<span style="color: #666666">.</span>reshape(<span style="color: #666666">-1</span>, <span style="color: #666666">1</span>)
|
|
y <span style="color: #666666">=</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>x<span style="color: #666666">**2</span>) <span style="color: #666666">+</span> <span style="color: #666666">1.5</span> <span style="color: #666666">*</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>(x<span style="color: #666666">-2</span>)<span style="color: #666666">**2</span>)<span style="color: #666666">+</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>normal(<span style="color: #666666">0</span>, <span style="color: #666666">0.1</span>, x<span style="color: #666666">.</span>shape)
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</pre></div>
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<p>
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where \( y \) is the function we want to fit with a given polynomial.
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<p>
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<b>a)</b>
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Write a first code which sets up a design matrix \( X \) defined by a fifth-order polynomial. Scale your data and split it in training and test data.
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<p>
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<b>b)</b>
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Perform an ordinary least squares and compute the means squared error and the \( R2 \) factor for the training data and the test data, with and without scaling.
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<p>
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<b>c)</b>
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Add now a model which allows you to make polynomials up to degree \( 15 \). Perform a standard OLS fitting of the training data and compute the MSE and \( R2 \) for the training and test data and plot both test and training data MSE and \( R2 \) as functions of the polynomial degree. Compare what you see with Figure 2.11 of Hastie et al. Comment your results. For which polynomial degree do you find an optimal MSE (smallest value)?
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<p>
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