From c41cf388cecdd01c73b965852add79245d0aff57 Mon Sep 17 00:00:00 2001 From: Morten Hjorth-Jensen Date: Wed, 20 Aug 2025 16:09:31 +0200 Subject: [PATCH] update --- .../week34-checkpoint.ipynb | 4289 ++++++++++++++--- doc/pub/week34/ipynb/week34.ipynb | 271 +- 2 files changed, 3577 insertions(+), 983 deletions(-) diff --git a/doc/pub/week34/ipynb/.ipynb_checkpoints/week34-checkpoint.ipynb b/doc/pub/week34/ipynb/.ipynb_checkpoints/week34-checkpoint.ipynb index f9e728efd..dbeb11e14 100644 --- a/doc/pub/week34/ipynb/.ipynb_checkpoints/week34-checkpoint.ipynb +++ b/doc/pub/week34/ipynb/.ipynb_checkpoints/week34-checkpoint.ipynb @@ -2,59 +2,114 @@ "cells": [ { "cell_type": "markdown", - "metadata": {}, + "id": "0b9e53e2", + "metadata": { + "editable": true + }, + "source": [ + "\n", + "" + ] + }, + { + "cell_type": "markdown", + "id": "d69d6c55", + "metadata": { + "editable": true + }, "source": [ - "\n", "# Week 34: Introduction to the course, Logistics and Practicalities\n", - "\n", - " \n", - "**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University\n", - "\n", - "Date: **Sep 16, 2020**\n", - "\n", - "Copyright 1999-2020, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n", - "\n", - "\n", - "\n", - "\n", + "**Morten Hjorth-Jensen**, Department of Physics and Center for Computing in Science Education, University of Oslo, Norway\n", "\n", + "Date: **Week 34, August 18-22, 2025**" + ] + }, + { + "cell_type": "markdown", + "id": "ed7a2caa", + "metadata": { + "editable": true + }, + "source": [ "## Overview of first week\n", "\n", - " * Thursday August 20: First lecture: Presentation of the course, aims and content\n", + "1. The sessions on Tuesdays and Wednesdays last four hours for each group (four groups in total) and will include lectures in a flipped mode (promoting active learning) and work on exercices and projects.\n", "\n", - " * Thursday: Second Lecture: Start with simple linear regression and repetition of linear algebra and elements of statistics\n", + "2. The sessions will begin with lectures, discussions, questions and answers about the material to be covered every week. Videos and teaching material will be announced in due time.\n", "\n", - " * Friday August 21: Linear regression \n", + "3. There are four groups:\n", "\n", - " * Computer lab: Wednesdays, 8am-6pm. First time: Wednesday August 26.\n", + " * Tuesdays 815am-12pm and 1215pm-4pm\n", "\n", + " * Wednesdays 815am-12pm and 1215pm-4pm.\n", "\n", + "4. On Mondays we have a regular lecture which will be organized as a mix of active learning sessions and regular lectures. These lectures/active learning sessions start at 215pm and end at 4pm and serve the aims of giving an overview over various topics as well as solving specific problems. These lectures will also be recorded. Lectures can be attended in person or via zoom at \n", + "\n", "\n", + "Videos and learning material with reading suggestions will be made available before each week starts." + ] + }, + { + "cell_type": "markdown", + "id": "1a7fa793", + "metadata": { + "editable": true + }, + "source": [ + "## Schedule first week\n", "\n", - "## Thursday August 20\n", - "\n", - "[Video of Lecture](https://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/h20/forelesningsvideoer/zoom_0.mp4?vrtx=view-as-webpage).\n", + " * August 18: Lecture: Presentation of course, Linear regression, examples and theory \n", "\n", + " * August 19: Introduction to software and repetition of Python Programming, linear algebra and basic elements of statistics. Please select group.\n", "\n", + " * August 20: Introduction to software and repetition of Python Programming, linear algebra and basic elements of statistics. Please select group." + ] + }, + { + "cell_type": "markdown", + "id": "42eed58e", + "metadata": { + "editable": true + }, + "source": [ "## Lectures and ComputerLab\n", "\n", - " * Lectures: Thursday (12.15pm-2pm and Friday (12.15pm-2pm). Due to the present COVID-19 situation all lectures will be online. They will be recorded and posted online at the official UiO [website](https://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/h20/index.html).\n", + " * Mondays: regular lectures/active learning sessions (2.15pm-4pm) \n", + "\n", + " * The sessions on Tuesdays and Wednesdays last four hours and will include partly lectures and discussions in the beginning.\n", "\n", " * Weekly reading assignments and videos needed to solve projects and exercises.\n", "\n", - " * Weekly exercises when not working on projects. You can hand in exercises if you want.\n", + " * Weekly exercises. You can hand in exercises if you want and get an extra score, see below.\n", "\n", " * Detailed lecture notes, exercises, all programs presented, projects etc can be found at the homepage of the course.\n", "\n", - " * Weekly plans and all other information are on the official webpage.\n", - "\n", - " * No final exam, three projects that are graded and have to be approved.\n", - "\n", - "\n", - "\n", - "\n", + " * Weekly plans and all other information are on the official website. This info will also be conveyed via weekly emails.\n", "\n", + " * No final exam, three projects that are graded and have to be approved." + ] + }, + { + "cell_type": "markdown", + "id": "19b78895", + "metadata": { + "editable": true + }, + "source": [ + "## Communication channels\n", "\n", + "* Communications (email and more) via \n", + "" + ] + }, + { + "cell_type": "markdown", + "id": "920c8b78", + "metadata": { + "editable": true + }, + "source": [ "## Course Format\n", "\n", " * Three compulsory projects. Electronic reports only using [Canvas](https://www.uio.no/english/services/it/education/canvas/) to hand in projects and [git](https://git-scm.com/) as version control software and [GitHub](https://github.com/) for repository (or [GitLab](https://about.gitlab.com/)) of all your material.\n", @@ -63,68 +118,166 @@ "\n", "a. For the last project each group/participant submits a proposal or works with suggested (by us) proposals for the project.\n", "\n", - "b. 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\n", - "\n", - "c. Poster session where all participants can study and discuss the other proposals.\n", - "\n", - "d. Based on feedback etc, each group finalizes the report and submits for grading. \n", - "\n", - "\n", - " * 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 [github address](https://github.com/CompPhysics/MachineLearning/tree/master/doc/Programs) of the course.\n", - "\n", - "\n", - "\n", - "\n", + "b. If possible, we would like to organize the last project as a workshop where each group presents this to all other participants of the course\n", "\n", + "c. Based on feedback etc, each group finalizes the report and submits for grading. \n", "\n", + " * Python is the default programming language, but feel free to use C/C++, Julia and/or Fortran or other programming languages. All source codes discussed during the lectures can be found at the webpage and [github address](https://github.com/CompPhysics/MachineLearning/tree/master/doc/Programs) of the course." + ] + }, + { + "cell_type": "markdown", + "id": "66791ac8", + "metadata": { + "editable": true + }, + "source": [ "## Teachers\n", "\n", - "\n", - "**Teachers :**\n", "* Morten Hjorth-Jensen, morten.hjorth-jensen@fys.uio.no\n", "\n", " * **Phone**: +47-48257387\n", "\n", " * **Office**: Department of Physics, University of Oslo, Eastern wing, room FØ470 \n", "\n", - " * **Office hours**: *Anytime*! In Fall Semester 2020 (FS20), as a rule of thumb office hours are planned via computer or telephone. Individual or group office hours will be performed via zoom. Feel free to send an email for planning. In person meetings may also be possible if allowed by the University of Oslo's COVID-19 instructions.\n", + " * **Office hours**: *Anytime*! Individual or group office hours can be arranged either in person or via zoom. Feel free to send an email for planning. \n", "\n", + "* Ida Torkjellsdatter Storehaug, i.t.storehaug@fys.uio.no\n", "\n", - "* Øyvind Sigmundson Schøyen, oyvinssc@student.matnat.uio.no\n", + "* Oskar Leinonen, oskarlei@fys.uio.no\n", "\n", - " * **Office**: Department of Physics, University of Oslo, Eastern wing, room FØ452\n", - "\n", - "\n", - "* Michael Bitney, m.s.bitney@fys.uio.no\n", - "\n", - "* Kristian Wold, kriswold@student.matnat.uio.no\n", - "\n", - "* Nicolai Haug, nicoha@student.matnat.uio.no\n", - "\n", - "* Per-Dimitri Sønsteland, perdimitri.bs@gmail.com\n", + "* Mia-Katrin Ose Kvalsund, m.k.o.kvalsund@fys.uio.no\n", "\n", + "* Karl Henrik Fredly, k.h.fredly@fys.uio.no\n", "\n", + "* Eir Eline Hørlyk, e.e.horlyk@fys.uio.no\n", "\n", + "* Britt S. Haanen, b.s.m.haanen@fys.uio.no" + ] + }, + { + "cell_type": "markdown", + "id": "d006c6f2", + "metadata": { + "editable": true + }, + "source": [ "## Deadlines for projects (tentative)\n", "\n", + "1. Project 1: October 6 (available September 1) graded with feedback)\n", "\n", - "1. Project 1: September 28 (graded with feedback)\n", + "2. Project 2: November 3 (available October 6, graded with feedback)\n", "\n", - "2. Project 2: November 2 (graded with feedback)\n", + "3. Project 3: December 8 (available November 3, graded with feedback)\n", "\n", - "3. Project 3: December 7 (graded with feedback)\n", + "Extra Credit (not mandatory), weekly exercise assignments, 10 in total (due Friday same week), 10% additional score. The extra credit assignments are due each Friday and can be uploaed to **Canvas** in your preferred format (although we prefer jupyter-notebooks). First assignment is for week 35. Each weekly exercise set gives one additional point to the final score, see below on grading." + ] + }, + { + "cell_type": "markdown", + "id": "d2b59d04", + "metadata": { + "editable": true + }, + "source": [ + "## Grading\n", "\n", - "Projects are handed in using **Canvas**. We use Github as repository for codes, benchmark calculations etc. Comments and feedback on projects only via **Canvas**.\n", + "Grades are awarded on a scale from A to F, where A is the best grade and F is a fail. There are three projects which are graded and each project counts 1/3 of the final grade. The total score is thus the average from all three projects.\n", "\n", + "The final number of points is based on the average of all projects and the grade follows the following table:\n", "\n", + " * 92-100 points: A\n", "\n", + " * 77-91 points: B\n", "\n", - "## Recommended textbooks\n", + " * 58-76 points: C\n", "\n", - "* [Trevor Hastie, Robert Tibshirani, Jerome H. Friedman, The Elements of Statistical Learning, Springer](https://www.springer.com/gp/book/9780387848570)\n", + " * 46-57 points: D\n", "\n", - "* [Aurelien Geron, Hands-On Machine Learning with Scikit-Learn, Keras, and TensorFlow, 2nd Edition](https://www.oreilly.com/library/view/hands-on-machine-learning/9781492032632/)\n", + " * 40-45 points: E\n", "\n", + " * 0-39 points: F-failed\n", + "\n", + "In addition you can get an extra score for weekly assignments (10 in total and due each Friday). Each weekly assignment counts 1 point. As an example, this means that if your average after three projects is 88 points and you have handed in and gotten approved four weekly exercises, the total score is 88+4=92, which translates into an A." + ] + }, + { + "cell_type": "markdown", + "id": "2b1dfff8", + "metadata": { + "editable": true + }, + "source": [ + "## Reading material\n", + "\n", + "The lecture notes are collected as a jupyter-book at .\n", + "\n", + "In addition to the lecture notes, we recommend the books of Rasckha et\n", + "al and Goodfellow et al. We will follow these texts closely and the\n", + "weekly reading assignments refer to these texts. The text by Hastie et\n", + "al is also widely used in the Machine Learning community. See next slide for link to textbooks." + ] + }, + { + "cell_type": "markdown", + "id": "39ce46e9", + "metadata": { + "editable": true + }, + "source": [ + "## Main textbooks\n", + "\n", + "* Goodfellow, Bengio, and Courville (GBC), Deep Learning \n", + "\n", + "* Sebastian Raschka, Yuxi Lie, and Vahid Mirjalili (RLM), Machine Learning with PyTorch and Scikit-Learn at , see also \n", + "\n", + "The weekly reading suggestions are all from these two texts. The text by GBC can be accessed chapter by chapter from the abovementioned URL.\n", + "Each chapter of RLM gives access to the pertinent notebooks. These notebooks are highly recommended." + ] + }, + { + "cell_type": "markdown", + "id": "cc1139fe", + "metadata": { + "editable": true + }, + "source": [ + "## Other popular texts\n", + "\n", + "**Other texts.**\n", + "\n", + "* Christopher M. Bishop (CB), Pattern Recognition and Machine Learning\n", + "\n", + "* [Hastie, Tibshirani, and Friedman (HTF), The Elements of Statistical Learning, Springer](https://www.springer.com/gp/book/9780387848570).\n", + "\n", + "* [Aurelien Geron (AG), Hands‑On Machine Learning with Scikit‑Learn and TensorFlow, O'Reilly](https://www.oreilly.com/library/view/hands-on-machine-learning/9781492032632/). This text is very useful since it contains many code examples and hands-on applications of all algorithms discussed in this course.\n", + "\n", + "* [Kevin Murphy (KM), Probabilistic Machine Learning, an Introduction](https://probml.github.io/pml-book/book1.html)\n", + "\n", + "* David Foster (DF), Generative Deep Learning, \n", + "\n", + "* Babcock and Gavras (BG), Generative AI with Python and TensorFlow, " + ] + }, + { + "cell_type": "markdown", + "id": "f56c0bb5", + "metadata": { + "editable": true + }, + "source": [ + "## Reading suggestions week 34\n", + "\n", + "This week: Refresh linear algebra, GBC chapter 2. Install scikit-learn. See lecture notes for week 34 at (these notes)." + ] + }, + { + "cell_type": "markdown", + "id": "4b601e8d", + "metadata": { + "editable": true + }, + "source": [ "## Prerequisites\n", "\n", "Basic knowledge in programming and mathematics, with an emphasis on\n", @@ -135,16 +288,163 @@ "of the corresponding computing and programming courses INF1000/INF1110\n", "or MAT-INF1100/MAT-INF1100L/BIOS1100/KJM-INF1100. Most universities\n", "offer nowadays a basic programming course (often compulsory) where\n", - "Python is the recurring programming language.\n", + "Python is the recurring programming language." + ] + }, + { + "cell_type": "markdown", + "id": "2518a99e", + "metadata": { + "editable": true + }, + "source": [ + "## Topics covered in this course: Statistical analysis and optimization of data\n", "\n", + "The course has two central parts\n", "\n", + "1. Statistical analysis and optimization of data\n", "\n", + "2. Machine learning\n", + "\n", + "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." + ] + }, + { + "cell_type": "markdown", + "id": "a0e1e9ae", + "metadata": { + "editable": true + }, + "source": [ + "## Statistical analysis and optimization of data\n", + "\n", + "We plan to cover the following topics:\n", + "* Basic concepts, expectation values, variance, covariance, correlation functions and errors;\n", + "\n", + "* Simpler models, binomial distribution, the Poisson distribution, simple and multivariate normal distributions;\n", + "\n", + "* Central elements of Bayesian statistics and modeling;\n", + "\n", + "* Gradient methods for data optimization;\n", + "\n", + "* Monte Carlo methods, Markov chains, Gibbs sampling and Metropolis-Hastings sampling (tentative);\n", + "\n", + "* Estimation of errors and resampling techniques such as the cross-validation, blocking, bootstrapping and jackknife methods;\n", + "\n", + "* Principal Component Analysis (PCA) and its mathematical foundation;" + ] + }, + { + "cell_type": "markdown", + "id": "8af343cb", + "metadata": { + "editable": true + }, + "source": [ + "## Machine Learning\n", + "\n", + "* Pre deep-learning revolution (2008 approx)\n", + "\n", + " * Linear Regression and Logistic Regression, classification and regression problems;\n", + "\n", + " * Bayesian linear and logistic regression, kernel regression;\n", + "\n", + " * Decisions trees, Random Forests, Bagging and Boosting methods;\n", + "\n", + " * Support vector machines (only survey);\n", + "\n", + " * Unsupervised learning and dimensionality reduction, from PCA to clustering;" + ] + }, + { + "cell_type": "markdown", + "id": "c0cfedb4", + "metadata": { + "editable": true + }, + "source": [ + "## Deep learning methods\n", + "\n", + "* Deep learning \n", + "\n", + " * Neural networks and deep learning;\n", + "\n", + " * Convolutional neural networks;\n", + "\n", + " * Recurrent neural networks;\n", + "\n", + " * Autoencoders\n", + "\n", + " * Generative methods with an emphasis on Boltzmann Machines, Variational Autoencoders and Generalized Adversarial Networks(covered by FYS5429);\n", + "\n", + "Hands-on demonstrations, exercises and projects aim at deepening your understanding of these topics." + ] + }, + { + "cell_type": "markdown", + "id": "4561f563", + "metadata": { + "editable": true + }, + "source": [ + "## Extremely useful tools, strongly recommended\n", + "\n", + "**and discussed at the lab sessions.**\n", + "\n", + " * GIT for version control, and GitHub or GitLab as repositories, highly recommended. This will be discussed during the first exercise session\n", + "\n", + " * Anaconda and other Python environments, see intro slides and links to programming resources at " + ] + }, + { + "cell_type": "markdown", + "id": "b94446df", + "metadata": { + "editable": true + }, + "source": [ + "## Other courses on Data science and Machine Learning at UiO\n", + "\n", + "* [FYS5419 Quantum Computing and Quantum Machine Learning](https://www.uio.no/studier/emner/matnat/fys/FYS5419/index-eng.html)\n", + "\n", + "* [FYS5429 Advanced Machine Learning for the Physical Sciences](https://www.uio.no/studier/emner/matnat/fys/FYS5429/index-eng.html)\n", + "\n", + "* [STK2100 Machine learning and statistical methods for prediction and classification](http://www.uio.no/studier/emner/matnat/math/STK2100/index-eng.html). \n", + "\n", + "* [IN3050/4050 Introduction to Artificial Intelligence and Machine Learning](https://www.uio.no/studier/emner/matnat/ifi/IN3050/index-eng.html). Introductory course in machine learning and AI with an algorithmic approach. \n", + "\n", + "* [STK-INF3000/4000 Selected Topics in Data Science](http://www.uio.no/studier/emner/matnat/math/STK-INF3000/index-eng.html). The course provides insight into selected contemporary relevant topics within Data Science. \n", + "\n", + "* [IN4080 Natural Language Processing](https://www.uio.no/studier/emner/matnat/ifi/IN4080/index.html). Probabilistic and machine learning techniques applied to natural language processing." + ] + }, + { + "cell_type": "markdown", + "id": "2da0b611", + "metadata": { + "editable": true + }, + "source": [ + "## Other courses on Data science and Machine Learning at UiO, contn\n", + "\n", + "* [STK-IN4300 Statistical learning methods in Data Science](https://www.uio.no/studier/emner/matnat/math/STK-IN4300/index-eng.html). An advanced introduction to statistical and machine learning. For students with a good mathematics and statistics background.\n", + "\n", + "* [IN3310/4310 Deep Learnig for Image Analysis](https://www.uio.no/studier/emner/matnat/ifi/IN4310/index.html)\n", + "\n", + "* [STK4051 Computational Statistics](https://www.uio.no/studier/emner/matnat/math/STK4051/index-eng.html)\n", + "\n", + "* [STK4021 Applied Bayesian Analysis and Numerical Methods](https://www.uio.no/studier/emner/matnat/math/STK4021/index-eng.html)" + ] + }, + { + "cell_type": "markdown", + "id": "cb6368d0", + "metadata": { + "editable": true + }, + "source": [ "## Learning outcomes\n", "\n", - "\n", - "\n", - "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\n", - "\n", "* Learn about basic data analysis, statistical analysis, Bayesian statistics, Monte Carlo sampling, data optimization and machine learning;\n", "\n", "* Be capable of extending the acquired knowledge to other systems and cases;\n", @@ -161,210 +461,20 @@ "\n", "* Reduction of data sets, from PCA to clustering;\n", "\n", - "* Autoencoders and Reinforcement Learning;\n", - "\n", - "* 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).\n", - "\n", - "\n", - "\n", - "## Topics covered in this course: Statistical analysis and optimization of data\n", - "\n", - "The course has two central parts\n", - "\n", - "1. Statistical analysis and optimization of data\n", - "\n", - "2. Machine learning\n", - "\n", - "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\n", - "\n", - "**Statistical analysis and optimization of data.**\n", - "\n", - "\n", - "The following topics will be covered\n", - "* Basic concepts, expectation values, variance, covariance, correlation functions and errors;\n", - "\n", - "* Simpler models, binomial distribution, the Poisson distribution, simple and multivariate normal distributions;\n", - "\n", - "* Central elements of Bayesian statistics and modeling;\n", - "\n", - "* Gradient methods for data optimization, \n", - "\n", - "* Monte Carlo methods, Markov chains, Gibbs sampling and Metropolis-Hastings sampling;\n", - "\n", - "* Estimation of errors and resampling techniques such as the cross-validation, blocking, bootstrapping and jackknife methods;\n", - "\n", - "* Principal Component Analysis (PCA) and its mathematical foundation\n", - "\n", - "\n", - "\n", - "\n", - "## Topics covered in this course: Machine Learning\n", - "\n", - "The following topics will be covered\n", - "* Linear Regression and Logistic Regression;\n", - "\n", - "* Neural networks and deep learning, including convolutional and recurrent neural networks\n", - "\n", - "* Decisions trees, Random Forests, Bagging and Boosting\n", - "\n", - "* Support vector machines\n", - "\n", - "* Bayesian linear and logistic regression\n", - "\n", - "* Boltzmann Machines\n", - "\n", - "* Unsupervised learning Dimensionality reduction, from PCA to cluster models\n", - "\n", - "Hands-on demonstrations, exercises and projects aim at deepening your understanding of these topics.\n", - "\n", - "\n", - "\n", - "\n", - "## Extremely useful tools, strongly recommended\n", - "\n", - "**and discussed at the lab sessions.**\n", - "\n", - " * GIT for version control, and GitHub or GitLab as repositories, highly recommended. This will be discussed during the first exercise session\n", - "\n", - " * Anaconda and other Python environments, see intro slides and first exercise session\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Other courses on Data science and Machine Learning at UiO\n", - "\n", - "The link here gives an excellent overview of courses on Machine learning at UiO.\n", - "\n", - "1. [STK2100 Machine learning and statistical methods for prediction and classification](http://www.uio.no/studier/emner/matnat/math/STK2100/index-eng.html). \n", - "\n", - "2. [IN3050 Introduction to Artificial Intelligence and Machine Learning](https://www.uio.no/studier/emner/matnat/ifi/IN3050/index-eng.html). Introductory course in machine learning and AI with an algorithmic approach. \n", - "\n", - "3. [STK-INF3000/4000 Selected Topics in Data Science](http://www.uio.no/studier/emner/matnat/math/STK-INF3000/index-eng.html). The course provides insight into selected contemporary relevant topics within Data Science. \n", - "\n", - "4. [IN4080 Natural Language Processing](https://www.uio.no/studier/emner/matnat/ifi/IN4080/index.html). Probabilistic and machine learning techniques applied to natural language processing. \n", - "\n", - "5. [STK-IN4300 Statistical learning methods in Data Science](https://www.uio.no/studier/emner/matnat/math/STK-IN4300/index-eng.html). An advanced introduction to statistical and machine learning. For students with a good mathematics and statistics background.\n", - "\n", - "6. [INF4490 Biologically Inspired Computing](http://www.uio.no/studier/emner/matnat/ifi/INF4490/). An introduction to self-adapting methods also called artificial intelligence or machine learning. \n", - "\n", - "7. [IN-STK5000 Adaptive Methods for Data-Based Decision Making](https://www.uio.no/studier/emner/matnat/ifi/IN-STK5000/index-eng.html). Methods for adaptive collection and processing of data based on machine learning techniques. \n", - "\n", - "8. [IN5400/INF5860 Machine Learning for Image Analysis](https://www.uio.no/studier/emner/matnat/ifi/IN5400/). An introduction to deep learning with particular emphasis on applications within Image analysis, but useful for other application areas too.\n", - "\n", - "9. [TEK5040 Deep learning for autonomous systems](https://www.uio.no/studier/emner/matnat/its/TEK5040/). 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.\n", - "\n", - "10. [STK4051 Computational Statistics](https://www.uio.no/studier/emner/matnat/math/STK4051/index-eng.html)\n", - "\n", - "11. [STK4021 Applied Bayesian Analysis and Numerical Methods](https://www.uio.no/studier/emner/matnat/math/STK4021/index-eng.html)\n", - "\n", - "## Introduction\n", - "\n", - "Our emphasis throughout this series of lectures \n", - "is on understanding the mathematical aspects of\n", - "different algorithms used in the fields of data analysis and machine learning. \n", - "\n", - "However, where possible we will emphasize the\n", - "importance of using available software. We start thus with a hands-on\n", - "and top-down approach to machine learning. The aim is thus to start with\n", - "relevant data or data we have produced \n", - "and use these to introduce statistical data analysis\n", - "concepts and machine learning algorithms before we delve into the\n", - "algorithms themselves. The examples we will use in the beginning, start with simple\n", - "polynomials with random noise added. We will use the Python\n", - "software package [Scikit-Learn](http://scikit-learn.org/stable/) and\n", - "introduce various machine learning algorithms to make fits of\n", - "the data and predictions. We move thereafter to more interesting\n", - "cases such as data from say experiments (below we will look at experimental nuclear binding energies as an example).\n", - "These are examples where we can easily set up the data and\n", - "then use machine learning algorithms included in for example\n", - "**Scikit-Learn**. \n", - "\n", - "These examples will serve us the purpose of getting\n", - "started. Furthermore, they allow us to catch more than two birds with\n", - "a stone. They will allow us to bring in some programming specific\n", - "topics and tools as well as showing the power of various Python \n", - "libraries for machine learning and statistical data analysis. \n", - "\n", - "Here, we will mainly focus on two\n", - "specific Python packages for Machine Learning, Scikit-Learn and\n", - "Tensorflow (see below for links etc). Moreover, the examples we\n", - "introduce will serve as inputs to many of our discussions later, as\n", - "well as allowing you to set up models and produce your own data and\n", - "get started with programming.\n", - "\n", - "\n", - "## What is Machine Learning?\n", - "\n", - "Statistics, data science and machine learning form important fields of\n", - "research in modern science. They describe how to learn and make\n", - "predictions from data, as well as allowing us to extract important\n", - "correlations about physical process and the underlying laws of motion\n", - "in large data sets. The latter, big data sets, appear frequently in\n", - "essentially all disciplines, from the traditional Science, Technology,\n", - "Mathematics and Engineering fields to Life Science, Law, education\n", - "research, the Humanities and the Social Sciences. \n", - "\n", - "It has become more\n", - "and more common to see research projects on big data in for example\n", - "the Social Sciences where extracting patterns from complicated survey\n", - "data is one of many research directions. Having a solid grasp of data\n", - "analysis and machine learning is thus becoming central to scientific\n", - "computing in many fields, and competences and skills within the fields\n", - "of machine learning and scientific computing are nowadays strongly\n", - "requested by many potential employers. The latter cannot be\n", - "overstated, familiarity with machine learning has almost become a\n", - "prerequisite for many of the most exciting employment opportunities,\n", - "whether they are in bioinformatics, life science, physics or finance,\n", - "in the private or the public sector. This author has had several\n", - "students or met students who have been hired recently based on their\n", - "skills and competences in scientific computing and data science, often\n", - "with marginal knowledge of machine learning.\n", - "\n", - "Machine learning is a subfield of computer science, and is closely\n", - "related to computational statistics. It evolved from the study of\n", - "pattern recognition in artificial intelligence (AI) research, and has\n", - "made contributions to AI tasks like computer vision, natural language\n", - "processing and speech recognition. Many of the methods we will study are also \n", - "strongly rooted in basic mathematics and physics research. \n", - "\n", - "Ideally, machine learning represents the science of giving computers\n", - "the ability to learn without being explicitly programmed. The idea is\n", - "that there exist generic algorithms which can be used to find patterns\n", - "in a broad class of data sets without having to write code\n", - "specifically for each problem. The algorithm will build its own logic\n", - "based on the data. You should however always keep in mind that\n", - "machines and algorithms are to a large extent developed by humans. The\n", - "insights and knowledge we have about a specific system, play a central\n", - "role when we develop a specific machine learning algorithm. \n", - "\n", - "Machine learning is an extremely rich field, in spite of its young\n", - "age. The increases we have seen during the last three decades in\n", - "computational capabilities have been followed by developments of\n", - "methods and techniques for analyzing and handling large date sets,\n", - "relying heavily on statistics, computer science and mathematics. The\n", - "field is rather new and developing rapidly. Popular software packages\n", - "written in Python for machine learning like\n", - "[Scikit-learn](http://scikit-learn.org/stable/),\n", - "[Tensorflow](https://www.tensorflow.org/),\n", - "[PyTorch](http://pytorch.org/) and [Keras](https://keras.io/), all\n", - "freely available at their respective GitHub sites, encompass\n", - "communities of developers in the thousands or more. And the number of\n", - "code developers and contributors keeps increasing. Not all the\n", - "algorithms and methods can be given a rigorous mathematical\n", - "justification, opening up thereby large rooms for experimenting and\n", - "trial and error and thereby exciting new developments. However, a\n", - "solid command of linear algebra, multivariate theory, probability\n", - "theory, statistical data analysis, understanding errors and Monte\n", - "Carlo methods are central elements in a proper understanding of many\n", - "of algorithms and methods we will discuss.\n", - "\n", - "\n", + "* Generative models\n", + "\n", + "* 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." + ] + }, + { + "cell_type": "markdown", + "id": "cd5643ee", + "metadata": { + "editable": true + }, + "source": [ "## Types of Machine Learning\n", "\n", - "\n", "The approaches to machine learning are many, but are often split into\n", "two main categories. In *supervised learning* we know the answer to a\n", "problem, and let the computer deduce the logic behind it. On the other\n", @@ -382,24 +492,255 @@ "\n", " * 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.\n", "\n", - " * Clustering: Data are divided into groups with certain common traits, without knowing the different groups beforehand. It is thus a form of unsupervised learning.\n", + " * Clustering: Data are divided into groups with certain common traits, without knowing the different groups beforehand. It is thus a form of unsupervised learning." + ] + }, + { + "cell_type": "markdown", + "id": "dad2568b", + "metadata": { + "editable": true + }, + "source": [ + "## Essential elements of ML\n", "\n", "The methods we cover have three main topics in common, irrespective of\n", - "whether we deal with supervised or unsupervised learning. The first\n", - "ingredient is normally our data set (which can be subdivided into\n", - "training and test data), the second item is a model which is normally a\n", - "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. \n", + "whether we deal with supervised or unsupervised learning.\n", + "* 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. \n", "\n", - "The last ingredient is a so-called **cost**\n", - "function which allows us to present an estimate on how good our model\n", - "is in reproducing the data it is supposed to train. \n", - "At the heart of basically all ML algorithms there are so-called minimization algorithms, often we end up with various variants of **gradient** methods.\n", + "* 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. \n", "\n", + "* The last ingredient is a so-called **cost/loss** 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." + ] + }, + { + "cell_type": "markdown", + "id": "ccf34c1f", + "metadata": { + "editable": true + }, + "source": [ + "## An optimization/minimization problem\n", "\n", + "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 **gradient methods**." + ] + }, + { + "cell_type": "markdown", + "id": "d9a80c4f", + "metadata": { + "editable": true + }, + "source": [ + "## The plethora of machine learning algorithms/methods\n", "\n", + "1. Deep learning: Neural Networks (NNs), Convolutional NNs, Recurrent NNs, Transformers, Boltzmann machines, autoencoders and variational autoencoders and generative adversarial networks and other generative models \n", "\n", + "2. Bayesian statistics and Bayesian Machine Learning, Bayesian experimental design, Bayesian Regression models, Bayesian neural networks, Gaussian processes and much more\n", "\n", + "3. Dimensionality reduction (Principal component analysis), Clustering Methods and more\n", "\n", + "4. Ensemble Methods, Random forests, bagging and voting methods, gradient boosting approaches \n", + "\n", + "5. Linear and logistic regression, Kernel methods, support vector machines and more\n", + "\n", + "6. Reinforcement Learning; Transfer Learning and more" + ] + }, + { + "cell_type": "markdown", + "id": "bf1454b7", + "metadata": { + "editable": true + }, + "source": [ + "## What Is Generative Modeling?\n", + "\n", + "Generative modeling can be broadly defined as follows:\n", + "\n", + "Generative modeling is a branch of machine learning that involves\n", + "training a model to produce new data that is similar to a given\n", + "dataset.\n", + "\n", + "What does this mean in practice? Suppose we have a dataset containing\n", + "photos of horses. We can train a generative model on this dataset to\n", + "capture the rules that govern the complex relationships between pixels\n", + "in images of horses. Then we can sample from this model to create\n", + "novel, realistic images of horses that did not exist in the original\n", + "dataset." + ] + }, + { + "cell_type": "markdown", + "id": "4a01a6f0", + "metadata": { + "editable": true + }, + "source": [ + "## Example of generative modeling, [taken from Generative Deep Learning by David Foster](https://www.oreilly.com/library/view/generative-deep-learning/9781098134174/ch01.html)\n", + "\n", + "\n", + "\n", + "\n", + "

Figure 1:

\n", + "" + ] + }, + { + "cell_type": "markdown", + "id": "2bf27180", + "metadata": { + "editable": true + }, + "source": [ + "## Generative Versus Discriminative Modeling\n", + "\n", + "In order to truly understand what generative modeling aims to achieve\n", + "and why this is important, it is useful to compare it to its\n", + "counterpart, discriminative modeling. If you have studied machine\n", + "learning, most problems you will have faced will have most likely been\n", + "discriminative in nature." + ] + }, + { + "cell_type": "markdown", + "id": "15ddfa02", + "metadata": { + "editable": true + }, + "source": [ + "## Example of discriminative modeling, [taken from Generative Deep Learning by David Foster](https://www.oreilly.com/library/view/generative-deep-learning/9781098134174/ch01.html)\n", + "\n", + "\n", + "\n", + "\n", + "

Figure 1:

\n", + "" + ] + }, + { + "cell_type": "markdown", + "id": "371a54c3", + "metadata": { + "editable": true + }, + "source": [ + "## Discriminative Modeling\n", + "\n", + "When performing discriminative modeling, each observation in the\n", + "training data has a label. For a binary classification problem such as\n", + "our data could be labeled as ones and zeros. Our model then learns how to\n", + "discriminate between these two groups and outputs the probability that\n", + "a new observation has label 1 or 0\n", + "\n", + "In contrast, generative modeling doesn’t require the dataset to be\n", + "labeled because it concerns itself with generating entirely new\n", + "data (for example an image), rather than trying to predict a label for say a given image." + ] + }, + { + "cell_type": "markdown", + "id": "f9ec1f0a", + "metadata": { + "editable": true + }, + "source": [ + "## A Frequentist approach to data analysis\n", + "\n", + "When you hear phrases like **predictions and estimations** and\n", + "**correlations and causations**, what do you think of? May be you think\n", + "of the difference between classifying new data points and generating\n", + "new data points.\n", + "Or perhaps you consider that correlations represent some kind of symmetric statements like\n", + "if $A$ is correlated with $B$, then $B$ is correlated with\n", + "$A$. Causation on the other hand is directional, that is if $A$ causes $B$, $B$ does not\n", + "necessarily cause $A$.\n", + "\n", + "These concepts are in some sense the difference between machine\n", + "learning and statistics. In machine learning and prediction based\n", + "tasks, we are often interested in developing algorithms that are\n", + "capable of learning patterns from given data in an automated fashion,\n", + "and then using these learned patterns to make predictions or\n", + "assessments of newly given data. In many cases, our primary concern\n", + "is the quality of the predictions or assessments, and we are less\n", + "concerned about the underlying patterns that were learned in order\n", + "to make these predictions.\n", + "\n", + "In machine learning we normally use [a so-called frequentist approach](https://en.wikipedia.org/wiki/Frequentist_inference),\n", + "where the aim is to make predictions and find correlations. We focus\n", + "less on for example extracting a probability distribution function (PDF). The PDF can be\n", + "used in turn to make estimations and find causations such as given $A$\n", + "what is the likelihood of finding $B$." + ] + }, + { + "cell_type": "markdown", + "id": "19f2aef4", + "metadata": { + "editable": true + }, + "source": [ + "## What is a good model?\n", + "\n", + "In science and engineering we often end up in situations where we want to infer (or learn) a\n", + "quantitative model $M$ for a given set of sample points $\\boldsymbol{X} \\in [x_1, x_2,\\dots x_N]$.\n", + "\n", + "As we will see repeatedely in these lectures, we could try to fit these data points to a model given by a\n", + "straight line, or if we wish to be more sophisticated to a more complex\n", + "function.\n", + "\n", + "The reason for inferring such a model is that it\n", + "serves many useful purposes. On the one hand, the model can reveal information\n", + "encoded in the data or underlying mechanisms from which the data were generated. For instance, we could discover important\n", + "corelations that relate interesting physics interpretations.\n", + "\n", + "In addition, it can simplify the representation of the given data set and help\n", + "us in making predictions about future data samples.\n", + "\n", + "A first important consideration to keep in mind is that inferring the *correct* model\n", + "for a given data set is an elusive, if not impossible, task. The fundamental difficulty\n", + "is that if we are not specific about what we mean by a *correct* model, there\n", + "could easily be many different models that fit the given data set *equally well*." + ] + }, + { + "cell_type": "markdown", + "id": "e08bd4e6", + "metadata": { + "editable": true + }, + "source": [ + "## What is a good model? Can we define it?\n", + "\n", + "The central question is this: what leads us to say that a model is correct or\n", + "optimal for a given data set? To make the model inference problem well posed, i.e.,\n", + "to guarantee that there is a unique optimal model for the given data, we need to\n", + "impose additional assumptions or restrictions on the class of models considered. To\n", + "this end, we should not be looking for just any model that can describe the data.\n", + "Instead, we should look for a **model** $M$ that is the best among a restricted class\n", + "of models. In addition, to make the model inference problem computationally\n", + "tractable, we need to specify how restricted the class of models needs to be. A\n", + "common strategy is to start \n", + "with the simplest possible class of models that is just necessary to describe the data\n", + "or solve the problem at hand. More precisely, the model class should be rich enough\n", + "to contain at least one model that can fit the data to a desired accuracy and yet be\n", + "restricted enough that it is relatively simple to find the best model for the given data.\n", + "\n", + "Thus, the most popular strategy is to start from the\n", + "simplest class of models and increase the complexity of the models only when the\n", + "simpler models become inadequate. For instance, if we work with a regression problem to fit a set of sample points, one\n", + "may first try the simplest class of models, namely linear models, followed obviously by more complex models.\n", + "\n", + "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." + ] + }, + { + "cell_type": "markdown", + "id": "729c3f46", + "metadata": { + "editable": true + }, + "source": [ "## Software and needed installations\n", "\n", "We will make extensive use of Python as programming language and its\n", @@ -410,7 +751,6 @@ "Rust, Julia, Fortran etc if you prefer. The focus in these lectures will be\n", "on Python.\n", "\n", - "\n", "If you have Python installed (we strongly recommend Python3) and you feel\n", "pretty familiar with installing different packages, we recommend that\n", "you install the following Python packages via **pip** as \n", @@ -430,9 +770,16 @@ "\n", "1. sudo apt-get install python3 (or python for pyhton2.7)\n", "\n", - "etc etc. \n", - "\n", - "\n", + "etc etc." + ] + }, + { + "cell_type": "markdown", + "id": "6825a222", + "metadata": { + "editable": true + }, + "source": [ "## Python installers\n", "\n", "If you don't want to perform these operations separately and venture\n", @@ -456,8 +803,16 @@ "license.\n", "\n", "Furthermore, [Google's Colab](https://colab.research.google.com/notebooks/welcome.ipynb) is a free Jupyter notebook environment that requires \n", - "no setup and runs entirely in the cloud. Try it out!\n", - "\n", + "no setup and runs entirely in the cloud. Try it out!" + ] + }, + { + "cell_type": "markdown", + "id": "bad98b2e", + "metadata": { + "editable": true + }, + "source": [ "## Useful Python libraries\n", "Here we list several useful Python libraries we strongly recommend (if you use anaconda many of these are already there)\n", "\n", @@ -473,6 +828,8 @@ "\n", "* [Autograd](https://github.com/HIPS/autograd) 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\n", "\n", + "* [JAX](https://jax.readthedocs.io/en/latest/index.html) has now more or less replaced **Autograd**. JAX is Autograd and XLA, brought together for high-performance numerical computing and machine learning research. It provides composable transformations of Python+NumPy programs: differentiate, vectorize, parallelize, Just-In-Time compile to GPU/TPU, and more.\n", + "\n", "* [SymPy](https://www.sympy.org/en/index.html) is a Python library for symbolic mathematics. \n", "\n", "* [scikit-learn](https://scikit-learn.org/stable/) has simple and efficient tools for machine learning, data mining and data analysis\n", @@ -481,8 +838,18 @@ "\n", "* [Keras](https://keras.io/) is a high-level neural networks API, written in Python and capable of running on top of TensorFlow, CNTK, or Theano\n", "\n", - "* And many more such as [pytorch](https://pytorch.org/), [Theano](https://pypi.org/project/Theano/) etc \n", + "* [Pytorch](https://pytorch.org/), highly recommened\n", "\n", + "* [Theano](https://pypi.org/project/Theano/) and many other" + ] + }, + { + "cell_type": "markdown", + "id": "1daa92e1", + "metadata": { + "editable": true + }, + "source": [ "## Installing R, C++, cython or Julia\n", "\n", "You will also find it convenient to utilize **R**. We will mainly\n", @@ -490,20 +857,25 @@ "Those of you\n", "already familiar with **R** should feel free to continue using **R**, keeping\n", "however an eye on the parallel Python set ups. Similarly, if you are a\n", - "Python afecionado, feel free to explore **R** as well. Jupyter/Ipython\n", - "notebook allows you to run **R** codes interactively in your\n", + "Python afecionado, feel free to explore **R** as well. Jupyter(Julia, Python and R) /Ipython\n", + "notebook allows you to run **R** codes and **Julia** codes interactively in your\n", "browser. The software library **R** is really tailored for statistical data analysis\n", "and allows for an easy usage of the tools and algorithms we will discuss in these\n", "lectures.\n", "\n", "To install **R** with Jupyter notebook \n", - "[follow the link here](https://mpacer.org/maths/r-kernel-for-ipython-notebook)\n", - "\n", - "\n", - "\n", + "[follow the link here](https://mpacer.org/maths/r-kernel-for-ipython-notebook)" + ] + }, + { + "cell_type": "markdown", + "id": "1c840067", + "metadata": { + "editable": true + }, + "source": [ "## Installing R, C++, cython, Numba etc\n", "\n", - "\n", "For the C++ aficionados, Jupyter/IPython notebook allows you also to\n", "install C++ and run codes written in this language interactively in\n", "the browser. Since we will emphasize writing many of the algorithms\n", @@ -524,22 +896,33 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "f33d6379", + "metadata": { + "editable": true + }, "source": [ " pycod jupyter nbconvert filename.ipynb --to latex \n" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "e3a03646", + "metadata": { + "editable": true + }, "source": [ "And to add more versatility, the Python package [SymPy](http://www.sympy.org/en/index.html) is a Python library for symbolic mathematics. It aims to become a full-featured computer algebra system (CAS) and is entirely written in Python. \n", "\n", - "Finally, if you wish to use the light mark-up language \n", - "[doconce](https://github.com/hplgit/doconce) you can convert a standard ascii text file into various HTML \n", - "formats, ipython notebooks, latex files, pdf files etc with minimal edits. These lectures were generated using **doconce**.\n", - "\n", - "\n", + "Finally, we recommend strongly using Autograd or JAX for automatic differentiation." + ] + }, + { + "cell_type": "markdown", + "id": "fb6d2495", + "metadata": { + "editable": true + }, + "source": [ "## Numpy examples and Important Matrix and vector handling packages\n", "\n", "There are several central software libraries for linear algebra and eigenvalue problems. Several of the more\n", @@ -551,108 +934,16 @@ "\n", " * LAPACK:package for solving symmetric, unsymmetric and generalized eigenvalue problems. From LAPACK's website it is possible to download for free all source codes from this library. Both C/C++ and Fortran versions are available.\n", "\n", - " * 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 .\n", - "\n", - "## Basic Matrix Features\n", - "\n", - "**Matrix properties reminder.**" + " * 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 ." ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "3101a680", + "metadata": { + "editable": true + }, "source": [ - "$$\n", - "\\mathbf{A} =\n", - " \\begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\\\\n", - " a_{21} & a_{22} & a_{23} & a_{24} \\\\\n", - " a_{31} & a_{32} & a_{33} & a_{34} \\\\\n", - " a_{41} & a_{42} & a_{43} & a_{44}\n", - " \\end{bmatrix}\\qquad\n", - "\\mathbf{I} =\n", - " \\begin{bmatrix} 1 & 0 & 0 & 0 \\\\\n", - " 0 & 1 & 0 & 0 \\\\\n", - " 0 & 0 & 1 & 0 \\\\\n", - " 0 & 0 & 0 & 1\n", - " \\end{bmatrix}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The inverse of a matrix is defined by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbf{A}^{-1} \\cdot \\mathbf{A} = I\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "
Relations Name matrix elements
$A = A^{T}$ symmetric $a_{ij} = a_{ji}$
$A = \\left (A^{T} \\right )^{-1}$ real orthogonal $\\sum_k a_{ik} a_{jk} = \\sum_k a_{ki} a_{kj} = \\delta_{ij}$
$A = A^{ * }$ real matrix $a_{ij} = a_{ij}^{ * }$
$A = A^{\\dagger}$ hermitian $a_{ij} = a_{ji}^{ * }$
$A = \\left (A^{\\dagger} \\right )^{-1}$ unitary $\\sum_k a_{ik} a_{jk}^{ * } = \\sum_k a_{ki}^{ * } a_{kj} = \\delta_{ij}$
\n", - "\n", - "\n", - "\n", - "### Some famous Matrices\n", - "\n", - " * Diagonal if $a_{ij}=0$ for $i\\ne j$\n", - "\n", - " * Upper triangular if $a_{ij}=0$ for $i > j$\n", - "\n", - " * Lower triangular if $a_{ij}=0$ for $i < j$\n", - "\n", - " * Upper Hessenberg if $a_{ij}=0$ for $i > j+1$\n", - "\n", - " * Lower Hessenberg if $a_{ij}=0$ for $i < j+1$\n", - "\n", - " * Tridiagonal if $a_{ij}=0$ for $|i -j| > 1$\n", - "\n", - " * Lower banded with bandwidth $p$: $a_{ij}=0$ for $i > j+p$\n", - "\n", - " * Upper banded with bandwidth $p$: $a_{ij}=0$ for $i < j+p$\n", - "\n", - " * Banded, block upper triangular, block lower triangular....\n", - "\n", - "### More Basic Matrix Features\n", - "\n", - "**Some Equivalent Statements.**\n", - "\n", - "For an $N\\times N$ matrix $\\mathbf{A}$ the following properties are all equivalent\n", - "\n", - " * If the inverse of $\\mathbf{A}$ exists, $\\mathbf{A}$ is nonsingular.\n", - "\n", - " * The equation $\\mathbf{Ax}=0$ implies $\\mathbf{x}=0$.\n", - "\n", - " * The rows of $\\mathbf{A}$ form a basis of $R^N$.\n", - "\n", - " * The columns of $\\mathbf{A}$ form a basis of $R^N$.\n", - "\n", - " * $\\mathbf{A}$ is a product of elementary matrices.\n", - "\n", - " * $0$ is not eigenvalue of $\\mathbf{A}$.\n", - "\n", - "\n", - "\n", "## Numpy and arrays\n", "[Numpy](http://www.numpy.org/) provides an easy way to handle arrays in Python. The standard way to import this library is as" ] @@ -660,7 +951,14 @@ { "cell_type": "code", "execution_count": 1, - "metadata": {}, + "id": "66625f81", + "metadata": { + "collapsed": false, + "editable": true, + "jupyter": { + "outputs_hidden": false + } + }, "outputs": [], "source": [ "import numpy as np" @@ -668,25 +966,62 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "d8edb94a", + "metadata": { + "editable": true + }, "source": [ "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," ] }, { "cell_type": "code", - "execution_count": 2, - "metadata": {}, - "outputs": [], + "execution_count": 3, + "id": "7d44772f", + "metadata": { + "collapsed": false, + "editable": true, + "jupyter": { + "outputs_hidden": false + } + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "[ 0.22889528 0.91634536 0.58026357 -0.20121369 1.63697022 -0.16620476\n", + " -1.88888726 -2.65872949 -0.00563363 1.25894439 -0.07825328 1.43521368\n", + " 0.97253438 -1.64819989 0.98672233 -1.73087706 -0.99593933 -1.34926433\n", + " -1.26715086 -0.95021858 -0.33244987 0.69501728 -0.48914472 1.87699249\n", + " -0.91130987 -0.2889529 -0.74183199 0.74935175 1.34457198 -1.70306077\n", + " -0.21142106 2.10793224 -0.40622194 2.34515157 -0.40761485 1.69110258\n", + " 0.72886914 -0.43700299 -1.14714793 0.15694733 0.66461823 0.89304652\n", + " 1.4543903 -0.37398506 -0.14684297 -0.57722604 -0.44036741 -0.40055163\n", + " 0.21510519 -0.53120826 -0.43941984 0.68555857 0.85074613 -0.8371588\n", + " 1.19968839 0.59476652 1.09647082 0.67324848 0.5365493 -0.11823534\n", + " 0.52454391 1.43031082 0.22692571 -0.87295854 -2.82106202 -0.27436094\n", + " 0.51321315 0.51606513 -1.56397437 -1.23442784 0.43472146 0.74179877\n", + " 0.73562128 1.00883303 -0.05851042 -1.19276526 1.16715641 -0.6401151\n", + " 0.81572064 1.0670422 -0.38749302 -1.67392967 -2.76996862 0.34870472\n", + " 1.3352494 -1.23489158 -1.70233983 0.62477383 0.14297718 0.23248733\n", + " 0.26787549 1.75420536 -1.22660411 -0.27597454 0.28073813 0.42879538\n", + " 1.64568937 -0.19205253 1.9311968 -0.26983672]\n" + ] + } + ], "source": [ - "n = 10\n", + "n = 100\n", "x = np.random.normal(size=n)\n", "print(x)" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "7c3a5d5e", + "metadata": { + "editable": true + }, "source": [ "We defined a vector $x$ with $n=10$ elements with its values given by the Normal distribution $N(0,1)$.\n", "Another alternative is to declare a vector as follows" @@ -694,18 +1029,36 @@ }, { "cell_type": "code", - "execution_count": 3, - "metadata": {}, - "outputs": [], + "execution_count": 5, + "id": "bd150398", + "metadata": { + "collapsed": false, + "editable": true, + "jupyter": { + "outputs_hidden": false + } + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "[1 2 4]\n" + ] + } + ], "source": [ "import numpy as np\n", - "x = np.array([1, 2, 3])\n", + "x = np.array([1, 2, 4])\n", "print(x)" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "62bb511e", + "metadata": { + "editable": true + }, "source": [ "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++\n", "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" @@ -714,7 +1067,14 @@ { "cell_type": "code", "execution_count": 4, - "metadata": {}, + "id": "de9d1b88", + "metadata": { + "collapsed": false, + "editable": true, + "jupyter": { + "outputs_hidden": false + } + }, "outputs": [], "source": [ "import numpy as np\n", @@ -724,7 +1084,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "575f4f67", + "metadata": { + "editable": true + }, "source": [ "In the last example we used Numpy's unary function $np.log$. This function is\n", "highly tuned to compute array elements since the code is vectorized\n", @@ -738,7 +1101,14 @@ { "cell_type": "code", "execution_count": 5, - "metadata": {}, + "id": "eadd4613", + "metadata": { + "collapsed": false, + "editable": true, + "jupyter": { + "outputs_hidden": false + } + }, "outputs": [], "source": [ "import numpy as np\n", @@ -751,7 +1121,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "43c2b238", + "metadata": { + "editable": true + }, "source": [ "We note that our code is much longer already and we need to import the **log** function from the **math** module. \n", "The attentive reader will also notice that the output is $[1, 1, 2]$. Python interprets automagically our numbers as integers (like the **automatic** keyword in C++). To change this we could define our array elements to be double precision numbers as" @@ -760,7 +1133,14 @@ { "cell_type": "code", "execution_count": 6, - "metadata": {}, + "id": "f1153763", + "metadata": { + "collapsed": false, + "editable": true, + "jupyter": { + "outputs_hidden": false + } + }, "outputs": [], "source": [ "import numpy as np\n", @@ -770,7 +1150,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "b897cb85", + "metadata": { + "editable": true + }, "source": [ "or simply write them as double precision numbers (Python uses 64 bits as default for floating point type variables), that is" ] @@ -778,17 +1161,27 @@ { "cell_type": "code", "execution_count": 7, - "metadata": {}, + "id": "3417f7b5", + "metadata": { + "collapsed": false, + "editable": true, + "jupyter": { + "outputs_hidden": false + } + }, "outputs": [], "source": [ "import numpy as np\n", - "x = np.log(np.array([4.0, 7.0, 8.0])\n", + "x = np.log(np.array([4.0, 7.0, 8.0]))\n", "print(x)" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "421de263", + "metadata": { + "editable": true + }, "source": [ "To check the number of bytes (remember that one byte contains eight bits for double precision variables), you can use simple use the **itemsize** functionality (the array $x$ is actually an object which inherits the functionalities defined in Numpy) as" ] @@ -796,29 +1189,46 @@ { "cell_type": "code", "execution_count": 8, - "metadata": {}, + "id": "e9055b4f", + "metadata": { + "collapsed": false, + "editable": true, + "jupyter": { + "outputs_hidden": false + } + }, "outputs": [], "source": [ "import numpy as np\n", - "x = np.log(np.array([4.0, 7.0, 8.0])\n", + "x = np.log(np.array([4.0, 7.0, 8.0]))\n", "print(x.itemsize)" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "52301fe3", + "metadata": { + "editable": true + }, "source": [ "## Matrices in Python\n", "\n", "Having defined vectors, we are now ready to try out matrices. We can\n", - "define a $3 \\times 3 $ real matrix $\\hat{A}$ as (recall that we user\n", + "define a $3 \\times 3 $ real matrix $\\boldsymbol{A}$ as (recall that we user\n", "lowercase letters for vectors and uppercase letters for matrices)" ] }, { "cell_type": "code", "execution_count": 9, - "metadata": {}, + "id": "7f400b7f", + "metadata": { + "collapsed": false, + "editable": true, + "jupyter": { + "outputs_hidden": false + } + }, "outputs": [], "source": [ "import numpy as np\n", @@ -828,7 +1238,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "35677f44", + "metadata": { + "editable": true + }, "source": [ "If we use the **shape** 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" ] @@ -836,7 +1249,14 @@ { "cell_type": "code", "execution_count": 10, - "metadata": {}, + "id": "16b1c27a", + "metadata": { + "collapsed": false, + "editable": true, + "jupyter": { + "outputs_hidden": false + } + }, "outputs": [], "source": [ "import numpy as np\n", @@ -847,7 +1267,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "7566ab17", + "metadata": { + "editable": true + }, "source": [ "We can continue this was by printing out other columns or rows. The example here prints out the second column" ] @@ -855,7 +1278,14 @@ { "cell_type": "code", "execution_count": 11, - "metadata": {}, + "id": "b10affa1", + "metadata": { + "collapsed": false, + "editable": true, + "jupyter": { + "outputs_hidden": false + } + }, "outputs": [], "source": [ "import numpy as np\n", @@ -866,7 +1296,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "c50f969e", + "metadata": { + "editable": true + }, "source": [ "Numpy contains many other functionalities that allow us to slice, subdivide etc etc arrays. We strongly recommend that you look up the [Numpy website for more details](http://www.numpy.org/). Useful functions when defining a matrix are the **np.zeros** function which declares a matrix of a given dimension and sets all elements to zero" ] @@ -874,7 +1307,14 @@ { "cell_type": "code", "execution_count": 12, - "metadata": {}, + "id": "35f1236f", + "metadata": { + "collapsed": false, + "editable": true, + "jupyter": { + "outputs_hidden": false + } + }, "outputs": [], "source": [ "import numpy as np\n", @@ -886,7 +1326,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "02cec13a", + "metadata": { + "editable": true + }, "source": [ "or initializing all elements to" ] @@ -894,7 +1337,14 @@ { "cell_type": "code", "execution_count": 13, - "metadata": {}, + "id": "821bc62d", + "metadata": { + "collapsed": false, + "editable": true, + "jupyter": { + "outputs_hidden": false + } + }, "outputs": [], "source": [ "import numpy as np\n", @@ -906,7 +1356,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "5df31e24", + "metadata": { + "editable": true + }, "source": [ "or as unitarily distributed random numbers (see the material on random number generators in the statistics part)" ] @@ -914,7 +1367,14 @@ { "cell_type": "code", "execution_count": 14, - "metadata": {}, + "id": "d821d3ff", + "metadata": { + "collapsed": false, + "editable": true, + "jupyter": { + "outputs_hidden": false + } + }, "outputs": [], "source": [ "import numpy as np\n", @@ -926,19 +1386,25 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "a99c04f6", + "metadata": { + "editable": true + }, "source": [ "As we will see throughout these lectures, there are several extremely useful functionalities in Numpy.\n", "As an example, consider the discussion of the covariance matrix. Suppose we have defined three vectors\n", - "$\\hat{x}, \\hat{y}, \\hat{z}$ with $n$ elements each. The covariance matrix is defined as" + "$\\boldsymbol{x}, \\boldsymbol{y}, \\boldsymbol{z}$ with $n$ elements each. The covariance matrix is defined as" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "fdb302f5", + "metadata": { + "editable": true + }, "source": [ "$$\n", - "\\hat{\\Sigma} = \\begin{bmatrix} \\sigma_{xx} & \\sigma_{xy} & \\sigma_{xz} \\\\\n", + "\\boldsymbol{\\Sigma} = \\begin{bmatrix} \\sigma_{xx} & \\sigma_{xy} & \\sigma_{xz} \\\\\n", " \\sigma_{yx} & \\sigma_{yy} & \\sigma_{yz} \\\\\n", " \\sigma_{zx} & \\sigma_{zy} & \\sigma_{zz} \n", " \\end{bmatrix},\n", @@ -947,14 +1413,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "4c91d693", + "metadata": { + "editable": true + }, "source": [ "where for example" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "316ccd01", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\sigma_{xy} =\\frac{1}{n} \\sum_{i=0}^{n-1}(x_i- \\overline{x})(y_i- \\overline{y}).\n", @@ -963,34 +1435,40 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "73cf195b", + "metadata": { + "editable": true + }, "source": [ "The Numpy function **np.cov** 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. \n", - "The following simple function uses the **np.vstack** function which takes each vector of dimension $1\\times n$ and produces a $3\\times n$ matrix $\\hat{W}$" + "The following simple function uses the **np.vstack** function which takes each vector of dimension $1\\times n$ and produces a $3\\times n$ matrix $\\boldsymbol{W}$" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "c9d6f1ba", + "metadata": { + "editable": true + }, "source": [ "$$\n", - "\\hat{W} = \\begin{bmatrix} x_0 & y_0 & z_0 \\\\\n", - " x_1 & y_1 & z_1 \\\\\n", - " x_2 & y_2 & z_2 \\\\\n", - " \\dots & \\dots & \\dots \\\\\n", - " x_{n-2} & y_{n-2} & z_{n-2} \\\\\n", - " x_{n-1} & y_{n-1} & z_{n-1}\n", + "\\boldsymbol{W} = \\begin{bmatrix} x_0 & x_1 & x_2 & \\dots & x_{n-2} & x_{n-1} \\\\\n", + " y_0 & y_1 & y_2 & \\dots & y_{n-2} & y_{n-1} \\\\\n", + "\t\t\t z_0 & z_1 & z_2 & \\dots & z_{n-2} & z_{n-1} \\\\\n", " \\end{bmatrix},\n", "$$" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "a5bbc329", + "metadata": { + "editable": true + }, "source": [ "which in turn is converted into into the $3\\times 3$ covariance matrix\n", - "$\\hat{\\Sigma}$ via the Numpy function **np.cov()**. We note that we can also calculate\n", - "the mean value of each set of samples $\\hat{x}$ etc using the Numpy\n", + "$\\boldsymbol{\\Sigma}$ via the Numpy function **np.cov()**. We note that we can also calculate\n", + "the mean value of each set of samples $\\boldsymbol{x}$ etc using the Numpy\n", "function **np.mean(x)**. We can also extract the eigenvalues of the\n", "covariance matrix through the **np.linalg.eig()** function." ] @@ -998,7 +1476,14 @@ { "cell_type": "code", "execution_count": 15, - "metadata": {}, + "id": "74dd353c", + "metadata": { + "collapsed": false, + "editable": true, + "jupyter": { + "outputs_hidden": false + } + }, "outputs": [], "source": [ "# Importing various packages\n", @@ -1021,7 +1506,14 @@ { "cell_type": "code", "execution_count": 16, - "metadata": {}, + "id": "f07645f8", + "metadata": { + "collapsed": false, + "editable": true, + "jupyter": { + "outputs_hidden": false + } + }, "outputs": [], "source": [ "%matplotlib inline\n", @@ -1041,20 +1533,19 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "245ee493", + "metadata": { + "editable": true + }, "source": [ "## Meet the Pandas\n", "\n", - "\n", "\n", "\n", "\n", - "

\n", - "\n", - "\n", + "

Figure 1:

\n", "\n", "\n", - "\n", "Another useful Python package is\n", "[pandas](https://pandas.pydata.org/), which is an open source library\n", "providing high-performance, easy-to-use data structures and data\n", @@ -1067,9 +1558,88 @@ }, { "cell_type": "code", - "execution_count": 17, - "metadata": {}, - "outputs": [], + "execution_count": 1, + "id": "e428bcdc", + "metadata": { + "collapsed": false, + "editable": true, + "jupyter": { + "outputs_hidden": false + } + }, + "outputs": [ + { + "data": { + "text/html": [ + "
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" + ], + "text/plain": [ + " 0 1 2 3 4\n", + "0 3.061679 0.117430 1.329492 0.063724 0.962990\n", + "1 0.264421 0.048920 1.144993 0.035909 0.065026\n", + "2 0.209789 0.189367 0.340583 0.667239 0.452553\n", + "3 0.010902 0.282259 1.060349 0.191963 1.250636\n", + "4 2.621102 2.376547 0.063443 0.709698 0.034047\n", + "5 0.878123 0.534362 1.853835 0.106431 0.003100\n", + "6 0.049462 2.082875 0.572069 0.666597 0.563167\n", + "7 0.207888 1.415201 2.858185 1.839818 1.518895\n", + "8 0.296414 0.446453 0.000054 0.375694 1.689345\n", + "9 3.003620 0.966899 0.127812 2.603636 2.162999" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], "source": [ "import numpy as np\n", "import pandas as pd\n", @@ -1171,16 +2060,38 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "a83eb591", + "metadata": { + "editable": true + }, "source": [ "Thereafter we can select specific columns only and plot final results" ] }, { "cell_type": "code", - "execution_count": 22, - "metadata": {}, - "outputs": [], + "execution_count": 6, + "id": "2f42295f", + "metadata": { + "collapsed": false, + "editable": true, + "jupyter": { + "outputs_hidden": false + } + }, + "outputs": [ + { + "ename": "NameError", + "evalue": "name 'df' is not defined", + "output_type": "error", + "traceback": [ + "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", + "\u001b[0;31mNameError\u001b[0m Traceback (most recent call last)", + "Cell \u001b[0;32mIn[6], line 1\u001b[0m\n\u001b[0;32m----> 1\u001b[0m \u001b[43mdf\u001b[49m\u001b[38;5;241m.\u001b[39mcolumns \u001b[38;5;241m=\u001b[39m [\u001b[38;5;124m'\u001b[39m\u001b[38;5;124mFirst\u001b[39m\u001b[38;5;124m'\u001b[39m, \u001b[38;5;124m'\u001b[39m\u001b[38;5;124mSecond\u001b[39m\u001b[38;5;124m'\u001b[39m, \u001b[38;5;124m'\u001b[39m\u001b[38;5;124mThird\u001b[39m\u001b[38;5;124m'\u001b[39m, \u001b[38;5;124m'\u001b[39m\u001b[38;5;124mFourth\u001b[39m\u001b[38;5;124m'\u001b[39m, \u001b[38;5;124m'\u001b[39m\u001b[38;5;124mFifth\u001b[39m\u001b[38;5;124m'\u001b[39m]\n\u001b[1;32m 2\u001b[0m df\u001b[38;5;241m.\u001b[39mindex \u001b[38;5;241m=\u001b[39m np\u001b[38;5;241m.\u001b[39marange(\u001b[38;5;241m10\u001b[39m)\n\u001b[1;32m 4\u001b[0m display(df)\n", + "\u001b[0;31mNameError\u001b[0m: name 'df' is not defined" + ] + } + ], "source": [ "df.columns = ['First', 'Second', 'Third', 'Fourth', 'Fifth']\n", "df.index = np.arange(10)\n", @@ -1204,7 +2115,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "fb1b0336", + "metadata": { + "editable": true + }, "source": [ "We can produce a $4\\times 4$ matrix" ] @@ -1212,7 +2126,14 @@ { "cell_type": "code", "execution_count": 23, - "metadata": {}, + "id": "0438c751", + "metadata": { + "collapsed": false, + "editable": true, + "jupyter": { + "outputs_hidden": false + } + }, "outputs": [], "source": [ "b = np.arange(16).reshape((4,4))\n", @@ -1223,72 +2144,53 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "f6d3d4c8", + "metadata": { + "editable": true + }, "source": [ "and many other operations. \n", "\n", "The **Series** class is another important class included in\n", "**pandas**. You can view it as a specialization of **DataFrame** but where\n", - "we have just a single column of data. It shares many of the same features as _DataFrame. As with **DataFrame**,\n", + "we have just a single column of data. It shares many of the same features as **DataFrame**. As with **DataFrame**,\n", "most operations are vectorized, achieving thereby a high performance when dealing with computations of arrays, in particular labeled arrays.\n", "As we will see below it leads also to a very concice code close to the mathematical operations we may be interested in.\n", - "For multidimensional arrays, we recommend strongly [xarray](http://xarray.pydata.org/en/stable/). **xarray** has much of the same flexibility as **pandas**, but allows for the extension to higher dimensions than two. We will see examples later of the usage of both **pandas** and **xarray**. \n", - "\n", - "\n", - "## Friday August 21\n", - "\n", - "[Video of Lecture](https://www.uio.no/studier/emner/matnat/fys/FYS-STK3155/h20/forelesningsvideoer/LectureAug21.mp4?vrtx=view-as-webpage) and [Handwritten notes](https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/NotesAugust21.pdf)\n", - "\n", - "\n", - "\n", - "\n", - "## Reading Data and fitting\n", - "\n", - "In order to study various Machine Learning algorithms, we need to\n", - "access data. Acccessing data is an essential step in all machine\n", - "learning algorithms. In particular, setting up the so-called **design\n", - "matrix** (to be defined below) is often the first element we need in\n", - "order to perform our calculations. To set up the design matrix means\n", - "reading (and later, when the calculations are done, writing) data\n", - "in various formats, The formats span from reading files from disk,\n", - "loading data from databases and interacting with online sources\n", - "like web application programming interfaces (APIs).\n", - "\n", - "In handling various input formats, as discussed above, we will mainly stay with **pandas**,\n", - "a Python package which allows us, in a seamless and painless way, to\n", - "deal with a multitude of formats, from standard **csv** (comma separated\n", - "values) files, via **excel**, **html** to **hdf5** formats. With **pandas**\n", - "and the **DataFrame** and **Series** functionalities we are able to convert text data\n", - "into the calculational formats we need for a specific algorithm. And our code is going to be \n", - "pretty close the basic mathematical expressions.\n", - "\n", - "Our first data set is going to be a classic from nuclear physics, namely all\n", - "available data on binding energies. Don't be intimidated if you are not familiar with nuclear physics. It serves simply as an example here of a data set. \n", - "\n", - "We will show some of the\n", - "strengths of packages like **Scikit-Learn** in fitting nuclear binding energies to\n", - "specific functions using linear regression first. Then, as a teaser, we will show you how \n", - "you can easily implement other algorithms like decision trees and random forests and neural networks.\n", - "\n", - "But before we really start with nuclear physics data, let's just look at some simpler polynomial fitting cases, such as,\n", - "(don't be offended) fitting straight lines!\n", - "\n", - "## Friday August 21\n", + "For multidimensional arrays, we recommend strongly [xarray](http://xarray.pydata.org/en/stable/). **xarray** has much of the same flexibility as **pandas**, but allows for the extension to higher dimensions than two. We will see examples later of the usage of both **pandas** and **xarray**." + ] + }, + { + "cell_type": "markdown", + "id": "d552a0d0", + "metadata": { + "editable": true + }, + "source": [ + "## Pandas AI\n", "\n", + "Try out [Pandas AI](https://pandas-ai.com/)" + ] + }, + { + "cell_type": "markdown", + "id": "b96bd9e5", + "metadata": { + "editable": true + }, + "source": [ "### Simple linear regression model using **scikit-learn**\n", "\n", "We start with perhaps our simplest possible example, using **Scikit-Learn** to perform linear regression analysis on a data set produced by us. \n", "\n", "What follows is a simple Python code where we have defined a function\n", "$y$ in terms of the variable $x$. Both are defined as vectors with $100$ entries. \n", - "The numbers in the vector $\\hat{x}$ are given\n", + "The numbers in the vector $\\boldsymbol{x}$ are given\n", "by random numbers generated with a uniform distribution with entries\n", "$x_i \\in [0,1]$ (more about probability distribution functions\n", "later). These values are then used to define a function $y(x)$\n", "(tabulated again as a vector) with a linear dependence on $x$ plus a\n", "random noise added via the normal distribution.\n", "\n", - "\n", "The Numpy functions are imported used the **import numpy as np**\n", "statement and the random number generator for the uniform distribution\n", "is called using the function **np.random.rand()**, where we specificy\n", @@ -1302,7 +2204,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "41318c5c", + "metadata": { + "editable": true + }, "source": [ "$$\n", "y = 2x+N(0,1),\n", @@ -1311,13 +2216,16 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "789c6fe4", + "metadata": { + "editable": true + }, "source": [ "where $N(0,1)$ represents random numbers generated by the normal\n", "distribution. From **Scikit-Learn** we import then the\n", "**LinearRegression** functionality and make a prediction $\\tilde{y} =\n", "\\alpha + \\beta x$ using the function **fit(x,y)**. We call the set of\n", - "data $(\\hat{x},\\hat{y})$ for our training data. The Python package\n", + "data $(\\boldsymbol{x},\\boldsymbol{y})$ for our training data. The Python package\n", "**scikit-learn** has also a functionality which extracts the above\n", "fitting parameters $\\alpha$ and $\\beta$ (see below). Later we will\n", "distinguish between training data and test data.\n", @@ -1336,7 +2244,14 @@ { "cell_type": "code", "execution_count": 24, - "metadata": {}, + "id": "8c7472f2", + "metadata": { + "collapsed": false, + "editable": true, + "jupyter": { + "outputs_hidden": false + } + }, "outputs": [], "source": [ "# Importing various packages\n", @@ -1362,7 +2277,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "25cef2d3", + "metadata": { + "editable": true + }, "source": [ "This example serves several aims. It allows us to demonstrate several\n", "aspects of data analysis and later machine learning algorithms. The\n", @@ -1376,7 +2294,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "58a04438", + "metadata": { + "editable": true + }, "source": [ "$$\n", "y = 10x+0.01 \\times N(0,1),\n", @@ -1385,7 +2306,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "55191297", + "metadata": { + "editable": true + }, "source": [ "where $x$ is defined as before. Does the fit look better? Indeed, by\n", "reducing the role of the noise given by the normal distribution we see immediately that\n", @@ -1403,7 +2327,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "2951bf64", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\chi^2 = \\frac{1}{n}\n", @@ -1413,7 +2340,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "1109ceeb", + "metadata": { + "editable": true + }, "source": [ "where $\\sigma_i^2$ is the variance (to be defined later) of the entry\n", "$y_i$. We may not know the explicit value of $\\sigma_i^2$, it serves\n", @@ -1441,16 +2371,22 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "7eeca264", + "metadata": { + "editable": true + }, "source": [ "$$\n", - "\\epsilon_{\\mathrm{relative}}= \\frac{\\vert \\hat{y} -\\hat{\\tilde{y}}\\vert}{\\vert \\hat{y}\\vert}.\n", + "\\epsilon_{\\mathrm{relative}}= \\frac{\\vert \\boldsymbol{y} -\\boldsymbol{\\tilde{y}}\\vert}{\\vert \\boldsymbol{y}\\vert}.\n", "$$" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "fd8b7402", + "metadata": { + "editable": true + }, "source": [ "The squared cost function results in an arithmetic mean-unbiased\n", "estimator, and the absolute-value cost function results in a\n", @@ -1464,9 +2400,27 @@ }, { "cell_type": "code", - "execution_count": 25, - "metadata": {}, - "outputs": [], + "execution_count": 7, + "id": "0de5719f", + "metadata": { + "collapsed": false, + "editable": true, + "jupyter": { + "outputs_hidden": false + } + }, + "outputs": [ + { + "data": { + "image/png": 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", + "text/plain": [ + "
" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", @@ -1488,12 +2442,15 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "ce194339", + "metadata": { + "editable": true + }, "source": [ "Depending on the parameter in front of the normal distribution, we may\n", "have a small or larger relative error. Try to play around with\n", "different training data sets and study (graphically) the value of the\n", - "relative error.\n", + "relative error $x$.\n", "\n", "As mentioned above, **Scikit-Learn** has an impressive functionality.\n", "We can for example extract the values of $\\alpha$ and $\\beta$ and\n", @@ -1506,9 +2463,41 @@ }, { "cell_type": "code", - "execution_count": 26, - "metadata": {}, - "outputs": [], + "execution_count": 10, + "id": "cde68f17", + "metadata": { + "collapsed": false, + "editable": true, + "jupyter": { + "outputs_hidden": false + } + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "The intercept alpha: \n", + " [2.]\n", + "Coefficient beta : \n", + " [[5.]]\n", + "Mean squared error: 0.00\n", + "Variance score: 1.00\n", + "Mean squared log error: 0.00\n", + "Mean absolute error: 0.00\n" + ] + }, + { + "data": { + "image/png": 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", + "text/plain": [ + "
" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], "source": [ "import numpy as np \n", "import matplotlib.pyplot as plt \n", @@ -1516,7 +2505,7 @@ "from sklearn.metrics import mean_squared_error, r2_score, mean_squared_log_error, mean_absolute_error\n", "\n", "x = np.random.rand(100,1)\n", - "y = 2.0+ 5*x+0.5*np.random.randn(100,1)\n", + "y = 2.0+ 5*x#+0.5*np.random.randn(100,1)\n", "linreg = LinearRegression()\n", "linreg.fit(x,y)\n", "ypredict = linreg.predict(x)\n", @@ -1541,25 +2530,34 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "f48a129c", + "metadata": { + "editable": true + }, "source": [ "The function **coef** gives us the parameter $\\beta$ of our fit while **intercept** yields \n", - "$\\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 **meansquarederror** gives us the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error or loss defined as" + "$\\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 **meansquarederror** gives us the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error or loss defined as" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "c337129d", + "metadata": { + "editable": true + }, "source": [ "$$\n", - "MSE(\\hat{y},\\hat{\\tilde{y}}) = \\frac{1}{n}\n", + "MSE(\\boldsymbol{y},\\boldsymbol{\\tilde{y}}) = \\frac{1}{n}\n", "\\sum_{i=0}^{n-1}(y_i-\\tilde{y}_i)^2,\n", "$$" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "d6462e3c", + "metadata": { + "editable": true + }, "source": [ "The smaller the value, the better the fit. Ideally we would like to\n", "have an MSE equal zero. The attentive reader has probably recognized\n", @@ -1569,31 +2567,40 @@ "determination. It provides a measure of how well future samples are\n", "likely to be predicted by the model. Best possible score is 1.0 and it\n", "can be negative (because the model can be arbitrarily worse). A\n", - "constant model that always predicts the expected value of $\\hat{y}$,\n", + "constant model that always predicts the expected value of $\\boldsymbol{y}$,\n", "disregarding the input features, would get a $R^2$ score of $0.0$.\n", "\n", - "If $\\tilde{\\hat{y}}_i$ is the predicted value of the $i-th$ sample and $y_i$ is the corresponding true value, then the score $R^2$ is defined as" + "If $\\tilde{\\boldsymbol{y}}_i$ is the predicted value of the $i-th$ sample and $y_i$ is the corresponding true value, then the score $R^2$ is defined as" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "c2a5a5fd", + "metadata": { + "editable": true + }, "source": [ "$$\n", - "R^2(\\hat{y}, \\tilde{\\hat{y}}) = 1 - \\frac{\\sum_{i=0}^{n - 1} (y_i - \\tilde{y}_i)^2}{\\sum_{i=0}^{n - 1} (y_i - \\bar{y})^2},\n", + "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},\n", "$$" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "d19d16f8", + "metadata": { + "editable": true + }, "source": [ - "where we have defined the mean value of $\\hat{y}$ as" + "where we have defined the mean value of $\\boldsymbol{y}$ as" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "7739ec09", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\bar{y} = \\frac{1}{n} \\sum_{i=0}^{n - 1} y_i.\n", @@ -1602,7 +2609,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "7f433018", + "metadata": { + "editable": true + }, "source": [ "Another quantity taht we will meet again in our discussions of regression analysis is \n", " 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.\n", @@ -1611,16 +2621,22 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "20361838", + "metadata": { + "editable": true + }, "source": [ "$$\n", - "\\text{MAE}(\\hat{y}, \\hat{\\tilde{y}}) = \\frac{1}{n} \\sum_{i=0}^{n-1} \\left| y_i - \\tilde{y}_i \\right|.\n", + "\\text{MAE}(\\boldsymbol{y}, \\boldsymbol{\\tilde{y}}) = \\frac{1}{n} \\sum_{i=0}^{n-1} \\left| y_i - \\tilde{y}_i \\right|.\n", "$$" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "02d95914", + "metadata": { + "editable": true + }, "source": [ "We present the \n", "squared logarithmic (quadratic) error" @@ -1628,23 +2644,28 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "524bb980", + "metadata": { + "editable": true + }, "source": [ "$$\n", - "\\text{MSLE}(\\hat{y}, \\hat{\\tilde{y}}) = \\frac{1}{n} \\sum_{i=0}^{n - 1} (\\log_e (1 + y_i) - \\log_e (1 + \\tilde{y}_i) )^2,\n", + "\\text{MSLE}(\\boldsymbol{y}, \\boldsymbol{\\tilde{y}}) = \\frac{1}{n} \\sum_{i=0}^{n - 1} (\\log_e (1 + y_i) - \\log_e (1 + \\tilde{y}_i) )^2,\n", "$$" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "6cb0b956", + "metadata": { + "editable": true + }, "source": [ "where $\\log_e (x)$ stands for the natural logarithm of $x$. This error\n", "estimate is best to use when targets having exponential growth, such\n", "as population counts, average sales of a commodity over a span of\n", "years etc. \n", "\n", - "\n", "Finally, another cost function is the Huber cost function used in robust regression.\n", "\n", "The rationale behind this possible cost function is its reduced\n", @@ -1657,64 +2678,35 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "142be577", + "metadata": { + "editable": true + }, "source": [ "$$\n", - "H_{\\delta}(a)={\\begin{cases}{\\frac {1}{2}}{a^{2}}&{\\text{for }}|a|\\leq \\delta ,\\\\\\delta (|a|-{\\frac {1}{2}}\\delta ),&{\\text{otherwise.}}\\end{cases}}}.\n", + "H_{\\delta}(\\boldsymbol{a})=\\left\\{\\begin{array}{cc}\\frac{1}{2} \\boldsymbol{a}^{2}& \\text{for }|\\boldsymbol{a}|\\leq \\delta\\\\ \\delta (|\\boldsymbol{a}|-\\frac{1}{2}\\delta ),&\\text{otherwise}.\\end{array}\\right.\n", "$$" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "4e765569", + "metadata": { + "editable": true + }, "source": [ - "Here $a=\\boldsymbol{y} - \\boldsymbol{\\tilde{y}}$.\n", - "We will discuss in more\n", - "detail these and other functions in the various lectures. We conclude this part with another example. Instead of \n", - "a linear $x$-dependence we study now a cubic polynomial and use the polynomial regression analysis tools of scikit-learn." - ] - }, - { - "cell_type": "code", - "execution_count": 27, - "metadata": {}, - "outputs": [], - "source": [ - "import matplotlib.pyplot as plt\n", - "import numpy as np\n", - "import random\n", - "from sklearn.linear_model import Ridge\n", - "from sklearn.preprocessing import PolynomialFeatures\n", - "from sklearn.pipeline import make_pipeline\n", - "from sklearn.linear_model import LinearRegression\n", + "Here $\\boldsymbol{a}=\\boldsymbol{y} - \\boldsymbol{\\tilde{y}}$.\n", "\n", - "x=np.linspace(0.02,0.98,200)\n", - "noise = np.asarray(random.sample((range(200)),200))\n", - "y=x**3*noise\n", - "yn=x**3*100\n", - "poly3 = PolynomialFeatures(degree=3)\n", - "X = poly3.fit_transform(x[:,np.newaxis])\n", - "clf3 = LinearRegression()\n", - "clf3.fit(X,y)\n", - "\n", - "Xplot=poly3.fit_transform(x[:,np.newaxis])\n", - "poly3_plot=plt.plot(x, clf3.predict(Xplot), label='Cubic Fit')\n", - "plt.plot(x,yn, color='red', label=\"True Cubic\")\n", - "plt.scatter(x, y, label='Data', color='orange', s=15)\n", - "plt.legend()\n", - "plt.show()\n", - "\n", - "def error(a):\n", - " for i in y:\n", - " err=(y-yn)/yn\n", - " return abs(np.sum(err))/len(err)\n", - "\n", - "print (error(y))" + "We will discuss in more detail these and other functions in the\n", + "various lectures and lab sessions." ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "f79ebfdf", + "metadata": { + "editable": true + }, "source": [ "### To our real data: nuclear binding energies. Brief reminder on masses and binding energies\n", "\n", @@ -1728,7 +2720,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "eb09c88a", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\Delta M(N, Z) = M(N, Z) - uA,\n", @@ -1737,14 +2732,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "bf69319c", + "metadata": { + "editable": true + }, "source": [ "where $u$ is the Atomic Mass Unit" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "eec8df19", + "metadata": { + "editable": true + }, "source": [ "$$\n", "u = M(^{12}\\mathrm{C})/12 = 931.4940954(57) \\hspace{0.1cm} \\mathrm{MeV}/c^2.\n", @@ -1753,14 +2754,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "313f9634", + "metadata": { + "editable": true + }, "source": [ "The nucleon masses are" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "83b87f63", + "metadata": { + "editable": true + }, "source": [ "$$\n", "m_p = 1.00727646693(9)u,\n", @@ -1769,14 +2776,20 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "be859efd", + "metadata": { + "editable": true + }, "source": [ "and" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "a976e227", + "metadata": { + "editable": true + }, "source": [ "$$\n", "m_n = 939.56536(8)\\hspace{0.1cm} \\mathrm{MeV}/c^2 = 1.0086649156(6)u.\n", @@ -1785,7 +2798,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "9f1dc803", + "metadata": { + "editable": true + }, "source": [ "In the [2016 mass evaluation of by W.J.Huang, G.Audi, M.Wang, F.G.Kondev, S.Naimi and X.Xu](http://nuclearmasses.org/resources_folder/Wang_2017_Chinese_Phys_C_41_030003.pdf)\n", "there are data on masses and decays of 3437 nuclei.\n", @@ -1798,7 +2814,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "4fe2c4e9", + "metadata": { + "editable": true + }, "source": [ "$$\n", "BE(N, Z) = ZM_H c^2 + Nm_n c^2 - M(N, Z)c^2 ,\n", @@ -1807,7 +2826,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "bd8e264d", + "metadata": { + "editable": true + }, "source": [ "where $M_H$ is the mass of the hydrogen atom and $m_n$ is the mass of the neutron.\n", "In terms of the mass excess the binding energy is given by" @@ -1815,7 +2837,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "9a0e9ac7", + "metadata": { + "editable": true + }, "source": [ "$$\n", "BE(N, Z) = Z\\Delta_H c^2 + N\\Delta_n c^2 -\\Delta(N, Z)c^2 ,\n", @@ -1824,11 +2849,13 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "9b20bc14", + "metadata": { + "editable": true + }, "source": [ "where $\\Delta_H c^2 = 7.2890$ MeV and $\\Delta_n c^2 = 8.0713$ MeV.\n", "\n", - "\n", "A popular and physically intuitive model which can be used to parametrize \n", "the experimental binding energies as function of $A$, is the so-called \n", "**liquid drop model**. The ansatz is based on the following expression" @@ -1836,7 +2863,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "5fae8c2f", + "metadata": { + "editable": true + }, "source": [ "$$\n", "BE(N,Z) = a_1A-a_2A^{2/3}-a_3\\frac{Z^2}{A^{1/3}}-a_4\\frac{(N-Z)^2}{A},\n", @@ -1845,14 +2875,14 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "c955d871", + "metadata": { + "editable": true + }, "source": [ "where $A$ stands for the number of nucleons and the $a_i$s are parameters which are determined by a fit \n", "to the experimental data. \n", "\n", - "\n", - "\n", - "\n", "To arrive at the above expression we have assumed that we can make the following assumptions:\n", "\n", " * 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.\n", @@ -1865,23 +2895,36 @@ "\n", "We could also add a so-called pairing term, which is a correction term that\n", "arises from the tendency of proton pairs and neutron pairs to\n", - "occur. An even number of particles is more stable than an odd number. \n", - "\n", - "\n", + "occur. An even number of particles is more stable than an odd number." + ] + }, + { + "cell_type": "markdown", + "id": "10feca33", + "metadata": { + "editable": true + }, + "source": [ "### Organizing our data\n", "\n", "Let us start with reading and organizing our data. \n", "We start with the compilation of masses and binding energies from 2016.\n", "After having downloaded this file to our own computer, we are now ready to read the file and start structuring our data.\n", "\n", - "\n", "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 **scikit-learn**." ] }, { "cell_type": "code", - "execution_count": 1, - "metadata": {}, + "execution_count": 4, + "id": "cd57c0cd", + "metadata": { + "collapsed": false, + "editable": true, + "jupyter": { + "outputs_hidden": false + } + }, "outputs": [], "source": [ "# Common imports\n", @@ -1921,33 +2964,10 @@ }, { "cell_type": "markdown", - "metadata": {}, - "source": [ - "Before we proceed, we define also a function for making our plots. You can obviously avoid this and simply set up various **matplotlib** 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." - ] - }, - { - "cell_type": "code", - "execution_count": 2, - "metadata": {}, - "outputs": [], - "source": [ - "from pylab import plt, mpl\n", - "plt.style.use('seaborn')\n", - "mpl.rcParams['font.family'] = 'serif'\n", - "\n", - "def MakePlot(x,y, styles, labels, axlabels):\n", - " plt.figure(figsize=(10,6))\n", - " for i in range(len(x)):\n", - " plt.plot(x[i], y[i], styles[i], label = labels[i])\n", - " plt.xlabel(axlabels[0])\n", - " plt.ylabel(axlabels[1])\n", - " plt.legend(loc=0)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, + "id": "ed4e2094", + "metadata": { + "editable": true + }, "source": [ "Our next step is to read the data on experimental binding energies and\n", "reorganize them as functions of the mass number $A$, the number of\n", @@ -1955,26 +2975,21 @@ "always useful (unless you have a binary file or other types of compressed\n", "data) to actually open the file and simply take a look at it!\n", "\n", - "\n", "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 **pandas**. The file begins with some basic format information." ] }, { "cell_type": "code", - "execution_count": 3, - "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "' \\nThis is taken from the data file of the mass 2016 evaluation. \\nAll files are 3436 lines long with 124 character per line. \\n Headers are 39 lines long. \\n col 1 : Fortran character control: 1 = page feed 0 = line feed \\n 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 \\n These formats are reflected in the pandas widths variable below, see the statement \\n widths=(1,3,5,5,5,1,3,4,1,13,11,11,9,1,2,11,9,1,3,1,12,11,1), \\n Pandas has also a variable header, with length 39 in this case. \\n'" - ] - }, - "execution_count": 3, - "metadata": {}, - "output_type": "execute_result" + "execution_count": 28, + "id": "309536df", + "metadata": { + "collapsed": false, + "editable": true, + "jupyter": { + "outputs_hidden": false } - ], + }, + "outputs": [], "source": [ "\"\"\" \n", "This is taken from the data file of the mass 2016 evaluation. \n", @@ -1990,7 +3005,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "27597f0a", + "metadata": { + "editable": true + }, "source": [ "The data we are interested in are in columns 2, 3, 4 and 11, giving us\n", "the number of neutrons, protons, mass numbers and binding energies,\n", @@ -2000,8 +3018,15 @@ }, { "cell_type": "code", - "execution_count": 4, - "metadata": {}, + "execution_count": 5, + "id": "1c60fe5f", + "metadata": { + "collapsed": false, + "editable": true, + "jupyter": { + "outputs_hidden": false + } + }, "outputs": [], "source": [ "# Read the experimental data with Pandas\n", @@ -2026,7 +3051,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "36972bcb", + "metadata": { + "editable": true + }, "source": [ "We have now read in the data, grouped them according to the variables we are interested in. \n", "We see how easy it is to reorganize the data using **pandas**. If we\n", @@ -2042,8 +3070,15 @@ }, { "cell_type": "code", - "execution_count": 5, - "metadata": {}, + "execution_count": 6, + "id": "85ffeaf7", + "metadata": { + "collapsed": false, + "editable": true, + "jupyter": { + "outputs_hidden": false + } + }, "outputs": [ { "name": "stdout", @@ -2051,19 +3086,19 @@ "text": [ " N Z A Element Ebinding\n", "A \n", - "1 0 0 1 1 H 0.000000\n", - "2 1 1 1 2 H 1.112283\n", - "3 2 2 1 3 H 2.827265\n", - "4 6 2 2 4 He 7.073915\n", - "5 9 3 2 5 He 5.512132\n", + "4 0 1 3 4 Li 1.153760\n", + "5 2 3 2 5 He 5.512132\n", + "6 7 3 3 6 Li 5.332331\n", + "7 12 4 3 7 Li 5.606439\n", + "8 17 4 4 8 Be 7.062435\n", "... ... ... ... ... ...\n", - "264 3304 156 108 264 Hs 7.298375\n", - "265 3310 157 108 265 Hs 7.296247\n", - "266 3317 158 108 266 Hs 7.298273\n", - "269 3338 159 110 269 Ds 7.250154\n", - "270 3344 160 110 270 Ds 7.253775\n", + "264 3297 156 108 264 Hs 7.298375\n", + "265 3303 157 108 265 Hs 7.296247\n", + "266 3310 158 108 266 Hs 7.298273\n", + "269 3331 159 110 269 Ds 7.250154\n", + "270 3337 160 110 270 Ds 7.253775\n", "\n", - "[267 rows x 5 columns]\n" + "[264 rows x 5 columns]\n" ] } ], @@ -2078,7 +3113,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "799627e9", + "metadata": { + "editable": true + }, "source": [ "The next step, and we will define this mathematically later, is to set up the so-called **design matrix**. We will throughout call this matrix $\\boldsymbol{X}$.\n", "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." @@ -2086,8 +3124,15 @@ }, { "cell_type": "code", - "execution_count": 6, - "metadata": {}, + "execution_count": 9, + "id": "365fbba9", + "metadata": { + "collapsed": false, + "editable": true, + "jupyter": { + "outputs_hidden": false + } + }, "outputs": [], "source": [ "# Now we set up the design matrix X\n", @@ -2101,15 +3146,25 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "0d3c5663", + "metadata": { + "editable": true + }, "source": [ "With **scikitlearn** we are now ready to use linear regression and fit our data." ] }, { "cell_type": "code", - "execution_count": 7, - "metadata": {}, + "execution_count": 10, + "id": "60ba302b", + "metadata": { + "collapsed": false, + "editable": true, + "jupyter": { + "outputs_hidden": false + } + }, "outputs": [], "source": [ "clf = skl.LinearRegression().fit(X, Energies)\n", @@ -2118,7 +3173,10 @@ }, { "cell_type": "markdown", - "metadata": {}, + "id": "af13881a", + "metadata": { + "editable": true + }, "source": [ "Pretty simple! \n", "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." @@ -2126,25 +3184,32 @@ }, { "cell_type": "code", - "execution_count": 8, - "metadata": {}, + "execution_count": 11, + "id": "a5cba204", + "metadata": { + "collapsed": false, + "editable": true, + "jupyter": { + "outputs_hidden": false + } + }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "Mean squared error: 0.04\n", + "Mean squared error: 0.02\n", "Variance score: 0.95\n", "Mean absolute error: 0.05\n", - "[ 0.00000000e+00 7.06492086e-03 -1.73091052e-01 -1.66020213e+01\n", - " 1.17385778e+00] 15.212327334149494\n" + "[ 0.00000000e+00 -2.96611194e-02 2.01719003e-01 1.08078025e+01\n", + " -4.03097597e+01] 5.294399745619595\n" ] }, { "data": { - "image/png": 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\n", + "image/png": 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", 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" ] }, "metadata": {}, @@ -2176,87 +3241,10 @@ }, { "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Seeing the wood for the trees\n", - "\n", - "As a teaser, let us now see how we can do this with decision trees using **scikit-learn**. Later we will switch to so-called **random forests**!" - ] - }, - { - "cell_type": "code", - "execution_count": 10, - "metadata": {}, - "outputs": [ - { - "data": { - "image/png": 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9exZXHriMA8kHeO6c51qosvbhyi+upG9eXyqeK+Xl8S83a5vPcv/Hssu/auXKRCTSKayDzLxzB/Yvlh33dsVGJd9VbiYKK3PsUzFhanIbqyOGpO+6g8fM1m1mnE44/eAQAL5K3kb1vnTsNoOzzvISF+ffJjraSk2N57jra2ler39im9XqH/KuqYH4eCgqNoEPMrMMCgtNuFz+ofiychMet3/4vazchM8LWVkGDgfU1JhITTWw2Y48jscDeXkmystNbOm/mb55fblh6Q1c/sVVR6xbN+RuNhtYzFBhquH9Ee+zr3I+vS0Zrdwix88z4hQ8Q4aFugwRaQEmwzCMUBfRmIKCihbbV0ZGQovu72SkjB+L9bvj7x2/1wemXgNn7IbP/9b0+tV05Gsew8WRIWKjlNO4EgvO466jvdvMfRQw8bi2OZj9CpMOfNno70xAHNuxUtMC1R0fX0IiRVtyaPRbSgi0pf8Pw4Haq/naU1tlZDQ+Z0Y96yCz7M4BoPrqa/0zspppacparvxiEJcvv4TPY6Lxef29QsMAX+3Xrbq+tgFYiMaCjRqKcVKMxQKdO/uoroH0bjm4+h3ZcwSIibZRXXP4md7I0cPYQ5ea1zj8G2xlJZSU+Hvvu3eZKS4xkdvNx5hdl9DhwFWsp/H2BPCZq4jttRyfpYT4BIPOnf17t1jqJ7dZo5z1E91aQNSSRZjLyzDv3YOvZ6+W27GIhITCOpgqKzFVVWFEReF4dAFH+9fZ6zJwVRnEJNdPdV495yZ+98nNWAxLYB5XU1G/LTaRPm+eQl6+jcGDvXTqVh9BR5ucFZORgKOdfENtSTagQ+3rjrU/i6qLmPnrG5m2ahqZlh5UVZmocfq/NHk8ACaS3Va61MRSs+1cwD83r3j1kfvfl1LFJ9duIz+jmoR46N7dR1U1xMQQOEVxPKK9aZz+RRlDduUorEXaAYV1EJmLCgHwpaUfNag9LoP/DN5Bl9JyHOMzyJwQx6o1cMW6H2AxLKzolMyC/f1ISfFx110uYmIMhg71ER0Fr75mpV8/H2NG+yguNdF3sA2bzcQgQn8Ouj1Ki0nDc76HXwz8xVHXMXvN/OjTH3H++vOxehv/383usdO5JJbrnxiGx9y8P6uNXTdy/6X3U5JQ0vgKfcHeEzbvXE/S2ec0a58i0nbpnHUQWdetIeX8s3EPHU7pfxtOMrt72S/526bnuWL5Fdz68a2Nbl8UX8rp6yew9js7vXv7SEtr+T+6ttJW4eKgsYenVzyH12j6Uq86huH/rlZaCsVFZpJsZka9NJRT1nbDYjT/poL50dV8l1SCyWQQEwtmE1isEBtj4Kjex/qOm+k1ZBc/++37J/LRWpz+bh0ftVfztae20jnrNqCuZ22kpzdYXuOpwXjSxyvrXyGj3D8h7PEpj9OlsAvxNbVXQZsg7dpMzkq0MGZM84NBWtegDoOYc/ofTn5HV4HhMTj8ZHl1tf8Wr5mZBlu2mFm1ykKq1U23hVvJKoXMmpij7LAzI78/jYNr83hp4U5sNhMpyQZmM/jS7GRekoSRbCW+u43eo6wter5cRFqewjqITIWHDIMf4tNla7nm86uw+vx/HFv6Z7HwxX/g8fgvFzrqXbqkXTFZj0zMWBsMHg5gIiPb4MyJHsCE77Y+VC4rx1fto6TYxLbtZnw+2LvXzPbtZnqxk65l39G5qBcdKPHffKau47EbWJcPQA0+/tl5O0UJpf6eee2kt6LkEj4/83PKk8uwWMBaO6G8d3IfnjpnIVGWqNZvEBEJUFgHkbmgAABfesPLqbY/eYCuvh6s7b6fc/82hekD/FPHrPrTkaMwR5tJmJQMQBL1d2WrY3J0561Lx/OrC4YS5a4PVpNhol9uP07dcSp2j52uhV0Zsa9vo8e46PNzqIyqbLAsLzmP5697lZ/ddX0LfhoRaYriIIgCE8xqw3rPahfb3qlm2Dp/T9tzRRTZg/RHIifPiE/g5r3JnL17Hblvv42vQ2aj6/lKDPKWAE4oLYPcXBMYJrK2eum/GxJrEhusn5iXSN/58Nmja8nPMOM6z0TyEEju5SOhl39koENsJtnxHRs5moicKCVDEJkLa3vWGRm4awy2X7yNLE8NkEBORg5Tbx0f2gKlXfH16EWf1YVklEThHjik8ZXSgbuOsn2lF5/LwOmEDestlJXCirce44KVp5DujCc91we1N+jx4WNFvxXsyNxBaXw5M38zi+F9dM90kZaisA4ic1Ehd50LLxfOZtiv3+Nez68oji3luy6b+Wz8V0yN+2GoS5R2xNu9B7bVq4j5yxPYP/n4pPZ1bu3PUcPKmDn2eSzFqfTfOoJh348hyhVNx5JOjPt+HOO+HweA45N81hl7MUfXkDH8ezp0raAm1U6c7/BHpwSXLymJ6ptvPbGL10VCSGEdRBWl+XQof4hnvxhAeYz/OVIvTfg7i0cv5jdj5oS4OmlvPP0HAhD18Yfw8Yctss8BwFIA9gIbgBcBcJJCAWdTaovj8w6jGLB/sH8DF5QsyyI/Kq9hbbYS8jo/R1XixqMeq0MlXL6ZZtwF//gYScnU3HBTC+9VpHUprIPoM5+L0dtPAyCpOgmHyczsuX9ifubjJEenhLg6aW9qbrwJIzYWU2Vl0yu3gOTa/17hLf5c8BgJNQlM2DyBC9dcSJSzU4N1o5yd6LXlcQoSChosd0Q7+GTIJ+zI3IEj2kHJxX2YYR7eIvXZvl5L1Hv/xrrpmxbZn0gwKayDxTDY7RxOKpAXW0VCVQIHJ3dmTLf0JjcVORFGfAI1N90S9OPe6pmJa90jlDiLqbioirfKF2GrsmPCRs4uDx6XiYGru3PuygFkVDS8MiKjIoOb/3dz4P2u1P08bowgLs5ESopBTAyYEy10uiWNnmdFHddljZ5lS/1hfQIP0hEJNYV1kJgcFSSWnAlA/lWxTHxgGGaz7kQh7U+0NZpZo2cfsfzwu0x5HV585V4qK2HbNjMVFSbKN9Tg+7gIc6Wb5JqddC/uRHcOQgmwr35fVf/N5eOYakwm/yWOeZ3L2XzaAWI7O4mPB3OSB2fn6gbHNxlVdBgLt2z5tv42ciJhQmEdJKveX0evvEE4rTVMvmOUgloiniXegiXeQjIwqk/twivi4EH/5K/F363n8b88Qkplw1NEA/YPYNKGSXSujg0sy/w2lmHfZjVYb0WfFXwy5BN8Jl9gmbPb6eQkruH3+/fh69ylVT6XSGtQWAfBPx5aRveFbuKAXdlfc0r2uFCXJNLmXdz/EhwzHexz7G2w3Av8t3wp1jIblQ4TRfst9FrVkR7b0/C6TPh8kF0Vw9htYxm7bewR+y2LKWPJ5E244spgpJWM86xkjISoeP8X6HhbAh1iOxyxnUgoBS2sn3/+efbv309KSgq7d+/mgQceIDo6OliHDzrb559h/fhDCqoM+r86mThXKrs7bGJst5XAbaEuT6TNM5lMzBh4YndKK8tx8/39BVTvdlFZCWWlJqqqTHS3fk1WRWeSqgGcsNMJb4Db5OWbTlvYl7YPt8WNr29nLvjBNaR2MNFxtB2zRSNhElpBCeuCggIWLlzIypUrMZvNzJw5k48++oiLLrooGIcPidd/sZYEx6lUJa6klyuOXZkbuDb/TlwX/vioz5IWkZaR1MPG6L8eeRc19wsf8fv//YHC2G7YPZ0Y+v0Quh/sTGZ5OoP2DWLQvkH+FdcBr26mDNgdVcXOrDxKkyrI7ZFHRadiomLszLjiCkZ0HxbUzyWRKyhhHRMTg81mw+FwkJiYSFVVFX369Gl6wzBwYLMHT41Bl5G2+mXf7qT33jMxY4aSbgD4hh2kasLDOKfpxicioRIzdCgL79mKEb0HX1Y28BmkgCsxmmJHX5zuRDYm29hpG0CqI524mjgyKjI4ZXdP/w421oez8/FKVkW9gslShjt+K0b8PkxmL0bSDkx2B9EeM+Pzo7H7mtcrN2w2Kn87F9fkC1rhk0u4C9rzrBcvXsw777xDRkYGhmFw3333EXeMuwh5PF6sVkswSjthPp/B69FfEetxMz73NJKy/N99/nbXQnr8uf7hCHtT8rgy/zJstrb9eUTavaoq6NUL8vKOuVpOMnyTCRhgdnfC7OqK2dURS81gTJ4USu3xpJf1anRbHz4KEgvwWDwUJ23j3H07iPV4iGc7cezEhBcbZY3f7GXsWPjyy5P+mNL+BKVn/d133/HXv/6VRYsWYbVamTdvHk899RSzZs066jYlJVUtdvzWejB57kY3WW7/5SHLFhZy2kz/l489a3bTg77szthKud2G67IMSktb7vO0pvb0EPdgUHs1X5tpqxVfY84/dlgnAseaBuo1vDy+4jV2fH+AhIMpdN3encSyBOzOKHoXdCSz3P/glE4lnchjwhHbl8aUsT8tF6/Jy+4Oe9jbYQ82TxVj8g5w6sKt2DJTGXpOKuVVwbmhTbhrM3+3WkBGRkKjy4MS1vn5+SQnJ2OtfeZjRkYGBw4cCMahW9XOT2qou6gkf2kV1Ia1pcD/OY2sA1z5yS9DVJ2INCouDl/PxnvFzWUCftHrd43+rrLYR9F2D7v3HWDJu09jL3UR5Y5i8N7BdCjrgN1jJ7k6ieR9SQAM3Tu4wfaun+TiIpdlZi+FcRX4TAZ5qSXkZxZhWHyUdiilvGMJ1mg4bfAELpowhdh4TYBr74IS1meeeSafffYZ8+bNIyEhgW3btjF79pE3TQg3xWvqw9q32QFk4PF5SCv1X+/ZtZcu/xCJNHGpZuJG2+k6uhtnTH+IMmdpg997fQZ7V3kp2+bDU+WjbKULX66H2Io1VJqqSKpMw+Kz0KGsA9kV/meWdypPgV09Gz1eDuuosrjZ2rGQHeN34Z5QwOWnX8qQ9KGt/lkleIIS1haLhTlz2t+DKjxb6++QlF6az183fkxe+UHOPzgRgL5Te4SqNBFpA0wmU6P3/U+bCExsuMy8E9znjGBFTzuevv0p8UbjLkvA8FkwFXbCVJkKXgt2RyZWZwpOiwFGIukV6cR6bQzfm83wl7PxvexjR9Ymanwf0zc+l+xOu7FZvc0tmOqrr8V10fST//DSonRTlJMQm19/EVaqy8Zj7/6RGGcM0zyTccTkETP6FHzH2F5EpI6vZy/Sug9l+saNsPHoTyM71JZ0+GiAGYs7meSiiSSVnEF05UD65PUD+rHvIGzdV8n6HqvJyfyevlUr+PWa3cfcp+X7LRQrrNucoM0GP14tOVmgNSYfuGsMNnVbg90ws7njdgbl9qbC7iTBFQVAlG0Fvff9LOzuP9yeJmoEg9qr+dRWTTMVFWHdsA6A5KRYSsuOf2JqWambucvew7Y7hlHbR9E7v3eD3+9JKCLWEo+1q5us8x106OfGUnt/qqTbf0pysYOib7biy8xqZO9tU3v6uxXSCWbt0Y7PnNgNM3lJeZT2SodcSHBF4QNsVJGd/F3YBbWIhJaRlob77HP9bzIScJ9AAMUCD18yhZKaYnyGwZ4Vbna96aJs0zaGbI+ma0Waf8XSKGo2xpNj8vJd5+/4bOBnVI8bT+ea73lg3WqYfGHLfTA5aQrrZor+6U8w/e9jYg3/uR9v1GnALPal53D7N0+xz3IliZbvybIsw1pdjnP0RZSHtmQRiWAp0akApJ0NI84G6MoTnzzK0g8/xVpjZey3pzEiZxhpFakM3juYwYfMSv941QGqrGup7gi+sRDT3yB9tI8o/wR2BqcPJdrafm8X3RYprJvD42HxZ2cR476ICx0ziDK85HTtQQ8g0dhGYul+BvJn/xMGAMNkwnXOpJCWLCJyuDt+cCd3/ODOBsscBV7WP15GyfJKTIWbyCiNJas82//LYmATgIlqm4uN3TZSY6vh3fT3GTf5J/Ttn0rHYVZsCbrhU2tTWDdD+d6DdCnoD8DLc9/i4iuGYRm7EoDqaedTeMt9DTew2zHiGz/vICLSlsRnWDjj/lQgFfMBC8t/OJzn+vWlNHMgXXd0o/euHmSUpdC1pANjto+p3/DzXBzkssnkY1uHCjzRXio61FAyqpSodA/23jWQ6QbgrM7jGZKh+6ifDIV1M+Rtqb+By/ZV+WwbbaVrSRbVtmom3XI6RmpSCKsTEWkZvuyOXFCRwkXvbqT6muEYvUuh99cAOIpjyc9JI0jEbI4AACAASURBVN/lZqmlht65/bF5bWSXZDMov/bfwN3A6voHqGzL2kaVvYr/Jf2HfNN7DCCexIxKMrsVYzKfXK2eESNxXnblye0kjCism6EoJ59E/LcPTNjuYvmL33Iqdr7pspNTU88IcXUiIi3HfdpYLIvfJuaf/2iwPBboAAwBeifD86dAjRW+daaQVNYfs2EjpXQAKaWDMRtWEhw96JPX8IFNpbX/bY4pxGmrpCzuIDs7rsZpd+CIKaY0aRP3rKhk+LHvBhtQdOb42geytH8K62YoP1AcCOs+B7LYaRwEOpPf3xXawkREWpjjDw/jPm0cJq/nqOukA/cc9bf+y81qKr/hwPdxVDrg7cotxJQkY/VaGbBvAOmOdBKq00kv70avA6MabL07poxt8W586eXYE9xEpzmJG1BIUgcXFqv/IpuoxYvo8vVWrF+tjJgbuCism6GqoP7mJxkV6SRUOwHoNOXI5+WKiIQzIzOTmh/9uEX2VXel9q8MH2vyVlPlqcTwwZ4tDoryTHjXW4jZbMHlMBFfYKZ7hUFSde2QuiM9sB/fUigBiuKLWNNrDeUx5+AeN4LLF39B79MuIi2l/d9+SmHdDK7y6gbvoz1RLO/7JRdNuTxEFYmIhA+zyczo7EMmp3Wr/Xl9w/UcNVXMfnM25WuL6ZXTG7PHSrf8Lgza1we710aaI43zNpzXYJtd765ni8XDtsxCitIrKOtYQfGIPKwda+jXowfXj/oR5pM9Qd4GKKybweNwN3i/OfsgiY9nkRWfGaKKRETan/joWJ6Y8RjMOPJ3LpfB9o+cHPiokvL8Slw5r9Dp4EiiPFHEumIZkpsFuVmwEfigdhuLi793XUx84kBis0zEjrXReYqFmDQzFpOFrLhsTGFy8yqFdTMYtR3rrwd8Qc/LbmLqjBHEJof/NzURkXBht5sYODWagVOjgTSiz3+XdxIeYPuPrsMbOxj3ihjYH01iTgKd9idh9ZpIdNkYndMNqIQNwIdQPdfLxuxt/mvGp7r540WzTrgmX0IixMa21Ec8JoV1M5hc/m9e3jgvp98WH+JqRETEMmocVz+7Ae78x1HXWdYxnVd6j8FtiiatogN99w+h+8G+9M/tT//c/hz47gCfPzeVobvOIpl1pLLuuGrwxSdQsmJtUO6jrrBuBrPL5v8Zo+YSEWkLai6/Gvt/P8JUUYHFbMLrO/KZVOM8MG7LmkOWvI0nLYqS6j6s4XKyS/tC6bPsAfZwDSv7fsow/s2k0gNH7Ksxvs6dMeLiWuYDNUHp0wxWtx0AW5I9xJWIiAiAd8hQSlb4e8IZGQkUH8dDTyxAl01uvr9wOR0qE1jdczVD9gzhtK3+B40/m1lM/MQRxKVZ8fhM2NKtnHpDLDEJoTu/rbBuBrs7xv8zJTjfoEREpHV1HGwjeukZFG7zcsM5I8n5qoYP7/uQUzalMDA/FV5t+NzvDQ9Xs673FjaN3Mz+kVvxxXjoFN+Zx8/+C1GWqFavV2HdDHa3fwJBXFZyiCsREZGWktrNSmo3fwz2GB3NrR9czP82LOdff/4HPfd1wuKzYDbMdCnsQo+CHoz/dgTjvx1B9avVuKwuymLL2fbmAQYP6t7qtSqsmyHa5e9RJ3VOC3ElIiLSms4eNo7Rfx3GpsJvwKg/D56z2kXxKxbS1ljoXh5DjDsGnyuFDrbOQalLYd0MsU7/DPAOfSPjHrQiIpEs3hbPadljGy68uPY/4OBWD4YP+na3YIsOznlshXUT3NVOYp3+nnV2P91eVEQk0nXoG/zo1J09mpD33T7MmKmyVxIT3/qTCERERA4XsWH9wR/e5LMn3z9iudvpxuvxBt4XbPM/q60qynHEuiIiIsEQkWFdll9KxjNZmB5t+PG9Hi8fjXyd90f8M7CsdE8hANX2yqDWKCIiUiciz1mX55UQ7Y7B6rU1XH6ghO4H+wNQsqeAlK4ZVOSXkklnnLaqUJQqIiISmT1rd7X/KVpWn5XqiprA8oqC8sDr3Wt2AuAs9oe0y9bwMZkiIiLBEpFh7XHWn5N2FJQFXlccrA/rwm25ALjLnQB4ba4gVSciItJQZIa1q/751If2piuL6u8tW763CABflQcAr73hM61FRESCJSLD2ueu71lXFddPHKspqz8vXVPoX26q9F/w7os58okuIiIiwRCRYe1x1veSK0vqw9p5SFj7yv3D3rbqaAAsKQ0no4mIiARLZIa1yxN4XVNeH9DOSmf9SrXzyWKrEwFI6JIalNpEREQOF5lhfUjP2lVeP8vbXVn/2lLjv6otscr/8I6sYV2CVJ2IiEhDERnWPk99z/rQ3rSnqj7Eba4YnNVuUhzpAPQ7e0jwChQRETlERIa111U/wcxTWX+dtbe6PqyjnXFs/XwLFsNCaWwxiR31eEwREQmNiAxrn7u+Z+2prr9+2uusXx7jSmDPqu0AlMcWBa84ERGRw0RkWHsPuXTLU1Mf0MYhPe64miTKdx4EoDKmJHjFiYiIHCYiw/rQ66x9h/SmDbcv8DqhJhH3Af+Ttlwxui+4iIiETkSG9aGPwPQdcutRDrtJWVSx/xprX5wHERGRUInIsD60Z20c8prDMjm9tBMA9nR7MMoSERFpVESGtddTP9xtuOtvI2ryNlwvq9R/bXViF80EFxGR0InIsDY8jfemTd7Gm6PziO6tW5CIiMgxRGRY+xqE9SE9a4+/ObyHdLErosvpPX5g0GoTERE5XGSGtbc+jE0eU+C12WcBIC85N7DMPfRNojM1DC4iIqFjDXUBoWAccs7a5Kv/vmLx+psjb9h+tqVaOXvmKM4c9lTQ6xMRETlURIa1z1s/9G32Htmzjkmxcd0zU4Nel4iISGMichjc8Nb3rM1eS+C1xet/ZrUtVpdqiYhI26Gw9tWHtdXnH2iwxkUHvSYREZGjidCwPnQYvP5MgMXr71FHJyisRUSk7YjQsK7vWVt89WFtqw1re0JM0GsSERE5msgMa6O+Z205pGdtrT1nHZsSH/SaREREjiYyw/qQS7esPlvgtc0TBUBsekLQaxIRETmayAxr36E96/qwttYOg8enqWctIiJtR0SGNYeEtfWQsLZ7/GGdkJUc9JJERESOJuLD2lYb1u4aF/bannVih6SQlCUiItKYoN3BbOfOnbz33ntERUWxevVqbr/9doYOHRqswzdgNOhZ+wO6PK8UAKfVicVqaXQ7ERGRUAhKWHu9XubNm8czzzyD2Wxm2rRpWK0hvNPpIQ/dqrtcq+JgGQBuiysUFYmIiBxVUBLzm2++wTAMXnrpJWpqakhOTubyyy8PxqEbd8ilW3XnqR2FDkyAy+oMUVEiIiKNC0pY5+bmsn79eh555BESEhK46667sNlsXHLJJUfdJiUlFmsLDkdnZNRfjmWi/uEdVp+V+LhoTE43YMFjdTVYNxJF+uc/Xmqv5lNbHR+1V/O197YKSljHxcXRs2dPEhL8jTly5Ei++uqrY4Z1SUlVix0/IyOBgoKKwHvfIddZA+zcVkDh/mLSycBjdjdYN9Ic3lZybGqv5lNbHR+1V/O1p7Y62peOoMwGHzZsGKWlpXi9/pPFubm5dO/ePRiHbpzR8G15kQNnuf/Lgc5Zi4hIWxOUnnVycjJ33XUXDz74ICkpKRQXF/Ozn/0sGIdu3GFhXVnkwO2oAcBrcYegIBERkaML2pTsc889l3PPPTdYhzu2hqPgVJdU4Kr0TyxTWIuISFsTmTdFOaxnXVNWjafaP/ztNSusRUSkbVFYAzXllXhr/CHttXgb2UBERCR0IjKsTYeFtau8Bl+NP6R9Fk8IKhIRETm6iAzrw89Zuyud+Ny1YW1Wz1pERNqWyAxrw9TgrafaieHyJ7hhMRrbQkREJGQiMqwPHwb3VrnB41/os/ga2UJERCR0IjKsD+9Z+5xuqDtVHcLni4iIiDQmQsO64Vuf00tUhf+BHiTbg1+PiIjIMURkWJsO61kbTjfJFdkAJJ0+MBQliYiIHFVEhnXdMHi1rRoAe1kZHUq6AjD68tEhK0tERKQxERnWdT3rwqTdAGTsH4TNZyMv6QBZnRNDWZqIiMgRIjqsXQlrAcgu6QZAXkpRyGoSERE5mogM67ph8FhzEQeT9gcWl3TU5DIREWl7IjKs63rWpdn9ONBhR2B5VP/0UJUkIiJyVBEd1u64RKomDg0s73dW11CVJCIiclQRGtb+j20ymTj/phFU26opjSlnzNm9QlyZiIjIkSLyfl11PWuTxUznHqlsnJ+IPdpKVHRENoeIiLRxEZpOdWHt/znlmqHHWllERCSkInQYvL5nLSIi0tZFZFrVnbM2WyPy44uISJiJyLRSz1pERMJJRKZVoGetsBYRkTAQkWllrrt0y2oJcSUiIiJNi8iwrhsGNyusRUQkDERoWGuCmYiIhI+ITCuFtYiIhJMmb4qyePHiJneSkpLC+PHjW6SgYKgP6wi9J4yIiISVJtPqmWeeYcSIEcdcp7S0NKzC2lw7oGCxqWctIiJtX5NhPWbMGObOnXvMdR588MEWKygYTD71rEVEJHw02bXctWsXLpfrmOv8/Oc/b7GCgqHu0i31rEVEJBw02bV0u93cd999xMfHM336dAYNGnTEOnFxca1SXGsJnLO2qWctIiJtX5Np9eCDD9K9e3cKCwtZtGgRzz33HKeeeioXXnghSUlJwaixxQV61nZdZy0iIm1fk2HdvXt3ANLT0/nxj38MwNq1a5k5cybZ2dn8+c9/btUCW4Nmg4uISDg5rrQqKyvj3//+N2+88Qa7d++mU6dOrVVXq6rrWVujbCGuREREpGlNhvXTTz/N8OHDeeONN/jkk0/o1asXV1xxBRdddBEJCQnBqLHF1d1u1GpXz1pERNq+JtPqySefJCEhgalTp/Lqq68yYMCAYNTVqkzqWYuISBhpMqwnTpzIo48+it1uD0Y9QRGYYKawFhGRMNDkhcbz5s1rV0ENYDb8s8BtURoGFxGRtq/JsK47L11UVMTdd9/Nz3/+c6qqqrjvvvsoKytr9QJbg9lXNwyusBYRkbav2bfwmj9/Pqeeeip2u53Y2FiuvPJK/vjHP7Zmba2mbhjcFqNhcBERafuaHdaZmZlcdtllxMbGAjBw4MCwnQ1ef+lW+xreFxGR9qnZYV1aWgqAyeS/7Kmqqoq9e/e2TlWtzOzzn7O2R6tnLSIibV+zT9qOHTuWqVOn4nK5uOWWW9i0aRO//e1vW7O2VhPoWUdHhbgSERGRpjU7rCdPnky/fv1YsWIFhmFwzz330LNnz9asrdXUTTCzx2oYXERE2r4mw/qf//wn11xzDQA9e/ZsENBvvPEGl112WetV1wq8Hi9mNBtcRETCR5Np9fe//51169Y1+rvNmzeHXVi7qvzP5vbhw2LVU7dERKTtazKsU1JS2LhxI1OnTsViaRhue/bsabXCWour2gmAz+wLcSUiIiLN02RYv/baayxfvpwlS5YwZswYpk2bFgjtwYMHt3qBLc1TXduzNimsRUQkPDTr0q1x48Yxf/58OnbsyG9+8xveeOMNPB4PEyZMaOXyWp6rxg2Az+wNcSUiIiLN0+zrrMF/+davfvUrFi9ezE033dRaNbUqj1M9axERCS/NDuuDBw9y//33M2nSJLKysrjvvvtas65W466u7VkrrEVEJEw0ec46Pz+fhQsX8vbbb3Puuefy1ltvBS7f2rVrF927d2/tGluUz+MPaU0wExGRcNFkWJ9zzjlkZ2dz77330r17d4qKiigqKsIwDF566SWefPLJYNTZYtxODwA+k85Zi4hIeGgyrEeMGMH06dMB2L9/f4PfORyO4zpYTU0Nl112GWeccQZ33333cW3bUrxONyY0DC4iIuGjybCeOXMmY8eObfR3PXr0OK6DPfbYYwwcOPC4tmlpHqcbGxYMhbWIiISJJieYHS2oAYYPH97sAy1evJhTTjmFzp07N3ub1uBx1Q2DK6xFRCQ8NBnWy5cvb3InK1euPObvt2/fzs6dO5k0aVLzK2slHqd/NrihCWYiIhImmhwG//jjjxk3btwx1/n000857bTTjrkPu93OwoULWbt2LW63mxdffJEbbrjhqNukpMRibcF7d2dkJAAQY/d/ZJ/JF1gmDaldjo/aq/nUVsdH7dV87b2tmgzrV199lddeew0AwzAwmUyB39W9HzZs2DH3MXPmzMBrp9NJVVXVMYMaoKSkqqnSmi0jI4GCggoAyksrSSUaw+QLLJN6h7aVNE3t1Xxqq+Oj9mq+9tRWR/vS0axLtxYsWAD4gzYqKuqEi/jwww9ZvXo1brebd999l6lTp57wvk6Uz61z1iIiEl6aDOtDL9e66qqrePvttwPvmxrKPtx5553Heeedd3wVtjCv2399tWaDi4hIuGhygplhGIHXPl/DgFuyZEnLV9TKfLVhrZ61iIiEiybD+tBz1Ie+hoZBHi68HvWsRUQkvDQ5DL57925mz57NkCFDqK6uxu12Y7PZgCPDOxz4NAwuIiJhpsmwXrhwIZs3b2b9+vXYbDZGjhxJp06d6NOnD/n5+cGosUV5ax/kYRB+owIiIhKZmgzrU089lVNPPTXw3ul0smXLFjZv3kxSUlKrFtcaDA2Di4hImGkyrA8XFRXFsGHDmry2uq3yBcJaPWsREQkPTU4wa298Xs0GFxGR8BJxYW14/T1qDYOLiEi4iMCwrp1gpmFwEREJExEX1r662eDqWYuISJiIuLBWz1pERMJNBIa1zlmLiEh4icCw1k1RREQkvEReWNfdz1zD4CIiEiYiL6w9OmctIiLhJfLC2ld3zlphLSIi4SHiwhqfhsFFRCS8RGxYaza4iIiEi4gLaw2Di4hIuIm4sMZb+9MU0ipERESaLfLC2lDPWkREwkvEhbURmGAW2jpERESaK+LCmrp5ZepZi4hImIi8sK7NaEM9axERCRMRG9bqWYuISLiIvLCuGwaPvE8uIiJhKvIiS8PgIiISZiI2rDUbXEREwkXEhbVJYS0iImEm4sJa56xFRCTcRF5k1Z2sjrxPLiIiYSriIkvD4CIiEm4iLqzx+VPaZFVai4hIeIi4sDZ5az+yNeI+uoiIhKmISyyzzwKAyRZxH11ERMJUxCVWXVhboiwhrkRERKR5IjaszXZriCsRERFpnogLa0ttWFujbSGuREREpHkiL6y9/pC2xNhDXImIiEjzRFxYm33+4W9brMJaRETCQ8SFtaU2rO2xUSGuREREpHkiLqyttcPgtrjoEFciIiLSPJEX1j5/WEclxoS4EhERkeaJuLCum2AWkxQX4kpERESaJ+LCum4YPCZZYS0iIuEh4sLa5vXPAo9Pjw9xJSIiIs0TeWHt8fes49ITQlyJiIhI80RUWLudbmy1E8zi0xTWIiISHiIqrB1FFQC4LC4sVj3IQ0REwkNkhXVBGQBuqyvElYiIiDRfRIV1ZXE1AB6LO8SViIiINF9EhXVNqQMAj1lhLSIi4SPCwroSUM9aRETCS0SFtdOhYXAREQk/1mAcZM+ePTz22GMMHDiQvLw8kpOTue2224Jx6AZcDicAXg2Di4hIGAlKWJeWljJlyhTOOeccAKZMmcKECRMYPHhwMA4f4KqqC2tPUI8rIiJyMoIS1kOHDm3w3ufzERMT/KdeuStrw9qisBYRkfARlLA+1Mcff8wZZ5xBr169jrleSkos1ha8cUlGRgJmnwH4e9YZGbqD2dGobY6P2qv51FbHR+3VfO29rYIa1itXrmTVqlXMnj27yXVLSqpa7LgZGQkUFFRQU1EDgM/soaCgosX2357UtZU0j9qr+dRWx0ft1Xztqa2O9qUjaLPBly5dyhdffMFvfvMbCgoK+Prrr4N16ACvyz/87TN7g35sERGRExWUsN60aRN33nknGzZs4LrrruOnP/0pOTk5wTh0A4bTH9I+i8JaRETCR1CGwQcPHhySnvThfO7asDb7QlyJiIhI80XUTVHw+CeYGQprEREJIxEV1kZdWFuMEFciIiLSfBEV1tReXq2wFhGRcBJZYV03r6zlLt8WERFpdREV1iavyf8i6LeCEREROXERGdYmqynElYiIiDRfZIW1r/bjKqxFRCSMRFRYm2vD2mzTSWsREQkfkRXWXn9Im6MU1iIiEj4iaqqV2Vcb1vaI+tgiIkH17beb+MtfnsDj8TBq1BicTv/jia+99kYSEo7+dKzXX/8Xl19+dbDKDCuR1bP2+UPaEq2wFhFpLQMHDmbEiJEMHjyUm276CT/96R2MGDGSn//8Vjwez1G3e/31V4JYZXiJqNSy1IV1lC3ElYiItL6rr47hv/9t2X/mzznHw7/+VX3c240dO46//e05Vq1aweLFbzJs2Aj27NnNueeez6hRY/jkk49xOCr461+fpVu37px++pnMmXPvEetFqsgKa6//49riokJciYhI5MnKyiY3dz+XX341o0aNoby8jF/+8nZGjRrDD35wLk8//QQ33fQTAGpqahpdL1JFVljX9qztsQprEWn/TqQH3Jry8g5w/vkX8PXXa9m8+RssFiulpSWNrmsYRrPWixQRdc7a4vMPf9vjFdYiIsG0evVKXC4X+/btpbCwgBtuuJmrrprRYB2z2YxhGGzduoV33ll81PUiUUT1rK1ef1hHxceEuBIRkfZry5Zv2bDha9xuNy+++DxOpxOPx8MTTzxNSUkJS5d+wlNPPU5iYiIOh4OlSz9hwoQfcPrpZ7BgwWP4fD6mTbv0qOtFIpNhGG3yEVQFBRUttq+MjAQKCipY1vM/pDk64H7Wy/Dpo1ts/+1JXVtJ86i9mk9tdXzUXs3XntoqI6PxS9siahjcWjsMHp0cH+JKREREmi+iwtrmsQMQl6phcBERCR8RFdZ156zjM5JCXImIiEjzRUxYez1e7F5/zzo+7ei3uxMREWlrIiasHUX+yQdusxub7mAmIiJhJHLCuqAMALfVHeJKREREjk/EhHXZAX9Yu6w1Ia5ERETk+ERMWJcf8N+qzmV1hrgSERFpSVu3buWOO27l/fffOeF9vP76vwKvP/roP5x//sQT2s+BA7lcffWl3HbbLdx22y08+eSjJ1zToSLmDmZVxQ4SiFfPWkQiRuLVPyTqvx+16D6d50yi/F9vtug+T1bfvn0ZNmzESe3j9ddfCTxLe9KkySxc+JcT3teMGTcwZcqFJ1XP4SImrGtKHAC41bMWEWlV7733b5599imuvPIaduzYRmlpGVOmXMhXX61g3769zJ//KB999AE5OTtITU0jL+8Ad911Lzk5O/jznx8mJiaG3/zm98yadScXXTSNadN+2OhxXnrpRXbt2snQoYPYtSuH7OyOAOzcuYN//vPv9OrVm927d3HddT/CYrHy+ON/xO12M3jwUPbu3UP37j259tobWLLk7cDjOQcNGsJpp50OwOLFb/Htt5soKSlm3rxHsFgsLF78FsuWLT2ilkGDBgeeGLZ8+TJKS0uorKzknHPOo0ePnifdphET1s7SKgA8FoW1iESGUPWAL7jgIv7zn3fp27c/V199Hffe+yuqqyu59977eOyxP7F69SoyMjpw8cWXYDabeeyxP/LVVys5/fQzeOihP/Gzn/2YpUs/4corZzBp0vmNHmPnzu18+OF7vPzyG2RkJLBhwy8Cv3v44fu57bZfMGTIMNatW8OCBY/x0EN/4swzJ7BmzVfccMPNAMyYcRnjxp3BxRdfwksv/S0QtnVGjhzFtGmXMmvWL9i2bSv9+w9g2rRLmTbt0qN+9uTkFG666VZ69uxFcXERP/nJjbzwwj9JSDi5S4YjJqzdDn9IezQbXEQkKDp16gxAfHwCHTv6XyckJFBVVUWHDpn85S9PkJSUTE5ODn379gcgJSWVG274MU8//QSvv77kqPvOyckJ7B+gY8dOgdc7dmzjq69Wsn7917hcTmJjYxpdr2PHzuTk5NCzZ+9Gj9G5cxcAkpKSqaqqBGDRojdZuvR/R6w7aNBgbrnlp8TExNCzZy8AUlPTSElJZfv2rYwYMfIYLdW0iAlrT5ULAK9FYS0iEmq//e3dvPjiK2RlZQWCEMDlcvHtt5uYOvViFix4lDvvnNXo9t2792Dfvr2B97m5+wNB3Lt3X8aPP5vevfvgcrlYtuzTBuvV2b9/L9279wDAZPLPt966dUvgi4PJZDriuNOn/5Dp0xsflgcCIwq9evXG4/FQUHAwMDx/MiImrH01HgC8Vk+IKxERad9Wr15Jfn4e77//DmecMZ4dO7bx4Yfvk56ewYYNX7Nz53bOOWcSjzzyMEOHDmPTpo3s2bOLbt168PLLf2P48FM45ZRT+dnPbsFisXD77b88Ijh79erNpEmTmTNnNgMH9qO8vIzly5cxcuQo7rnnd7z66stkZ3fk4MF8Jk2aEtjO5XLx0kt/Y/v2bUyaNJlevfy96n79+vPMMwuIiopi3759OBwO3n13CX369AvUP3TocKzWY8dmhw6Z/P3vf6VPn77s27eXm2++lays7JNu04h5ROYjo+ZwypqJfN3/M2Ys+2WL7bu9aU+PmgsGtVfzqa2Oj9qr+ZrbVu+//w4HDuQecW66LdEjMmtHvw2rL7R1iIhI0OXn57F8+bJAzz7cRMwwuKnuVLVuCy4iElY+/fS/VFdXN1g2YcIPiI2NbfY+MjOzeOCBP7Z0aUETOWHt8Q8imKKOnDAgIiJt18SJ54S6hJCLmGFwi8f/vcQUZQlxJSIiIscnYsLaXBvWlhiNg4uISHiJmLC2ev0hbY2PCnElIiIixyeCwtoOQFRCdIgrERGRltSWnrr1/vvv8KMfzQg8deuyyy4+qbrqRE5Ye/w96qik5s8eFBGRtq+lnrpVZ9KkycTHx5/Qfrp27caDD/6JBQsWsmDBQnr16sWECT84qdoggmaD2z3+HnVcemKIKxERCY6r3/0h/93Tso/IPKfrJP419dgPCInkp24NHjw0sGzbtu/p0qXbcV1idjQRFNb+y2NmbAAACmJJREFUnnVchsJaRKQ1RfJTtw715puvceONPz7BVmwogsLa37NOzE4OcSUiIsHRVA+4tUXiU7fqFBcX4Xa7W+S+4BBBYR3l9od1Stf0EFciIiLt9albdRYtepOLL25eD7w5IiKsHcUO7F47XpOXxA5JoS5HRKRdi+SnbtUdY8uWb1v0gSER8dStit0F7Bm1h8ooB6P3jm+x/bZHetLP8VF7NZ/a6viovZpPT91qJ4r3FgLgtNaEuBIREQkFPXUrDJTllmLBjtvqDHUpIiJynPTUrQgJ64r8UpLpgMumnrWISLjRU7ciZBi8qsQBgNuinrWIiISfiAjrmrIqADwWV4grEREROX4REdauCn+P2mNVWIuISPiJiLD2VvpD2mvxhLgSERGR4xcRYe2p9oe0z6qwFhFpb9rSIzJdLhd//OOD/P3vf+WRRx7mv//98IRrOlTQZoN/+eWXfPTRR6SlpWEymbjtttuCdWgMpxcAn9UXtGOKiEhwtNQjMi+//GrA/4jMhQv/ckL7Wbz4TWw2O9dffxMej4crrpjG8OEjSU8/uVtdByWsq6urmTNnDu+99x52u53bb7+dFStWMHbs2GAcHmpPVRvWNnmzNhGRVrH76m04/lveovuMPyeRbv/qc8x1IvkRmXv37g3co9xqtZKSkso336w/6cvPghLW69evp2PHjtjtdgBOOeUUlv5/e/ca01SahwH8gdI60nakLc0gKArCsGG0FkczJDXiBRI0M7sSNyZC/OIlMYYx8fJhlTWZaDQx6oZAjCvBZMyqiTFGY8YYQCKsQrhMAKOzIuEimsHLmTLSllJr4ewHmK4oVVhNz2v7/L5Imzbnn4c3PD3N8bw1NcEr61dj/2qCczgionAWzltkWiwLUVs7unGI0+nE48e9GBwcDPj6yQpKWdvtdmi1Wv9jnU4Hu93+zvcYDNGIilJ9lONHzBi9AfznKcaA912l/2FGU8O8Jo9ZTc2H5mWuWvSRJpk6jSYKCxakwWzWIzbWiPT0L2E26xEXFwuVagRxcUb8+OM/YTAY8OuvjzA8bIXZrIfZrMeOHd/j6NGjuHHjhv8k703NzU+QlDTXn1FKShL0+s9gNuvR3d2JX35pRWfnf+DxeGAwfA6zWQ+9/jOkpCT535OUNBd2+xN8800GVKrIcXmrVJHIyEhHREQEvvjCDI1Ghtmsx/nz51FZWfnWPAsXLsTOnTuxYcNfERk5jEuXzkGr1cJisSAtLfmDf5dBKWuTyTTuk4XL5YLJZHrne37/3f3Rjl/477+j9l+38JdvN/HG+O/BzQOmhnlNHrOamk89L6/Xh/7+QUyb5oTH8wovXrghSU4MDr6E0+nBoUPf+7fIfP68H06nB5LkhNfrRUPDz1iz5s/44YeDAbfINBpnoru7B5LkhNmsR2dnD2JizJAkJ+bNS8XixbZxW2RKkhNOpwednT3+XLu7e2A0zoQkOTEyMrqB1B9bZA4Pj+C330ZvqPX6/Dk53yEn57sJZ5IkJxyOAXz11deIi4uDz+fD1as/Ye7cP036dxmo1INS1larFX19ffB6vdBoNGhpaUF+fn4wDg0AUEWpsODbr4N2PCKicBbOW2RKkoTS0n/Aal0Et9uN3bv/BrVa/cGZBm2LzLq6OlRUVMBgMECtVr/3avCP+YnyU/+EGkzMamqY1+Qxq6lhXpMXDltkBu2/btlsNthstmAdjoiIyO+PLTKdTie6uzuRnJyi9EhTEha7bhER0aeLW2SyrImISHDcIjNMbjdKRET0KWNZExERCY5lTUREJDiWNRERkeBY1kRERIJjWRMREQmOZU1ERCS4oN1ulIiIiP4/PLMmIiISHMuaiIhIcCxrIiIiwbGsiYiIBMeyJiIiEhzLmoiISHAhvUVmfX09KisrYTKZEBERgcLCQqVHEs769esxbdo0AEBkZCTOnDmDFy9e4Pjx45g9ezYePnyIXbt2ITY2VuFJlSFJEoqLi9He3o5Lly4BwDvzKS8vh8vlgsPhgM1mw6pVq5QcP6gmyqq0tBRNTU3+12zbtg02mw1AeGf16NEjFBcXIz09HU+fPkVMTAwKCwu5tiYQKKuwW1tyiHK73XJ2drb88uVLWZZlubCwUK6vr1d4KvGUlJS89dz+/fvla9euybIsy9XV1fKePXuCPZYwrl+/LldXV8t5eXn+5wLl09bWJm/ZskWWZVn2er1yTk6OPDAwEPyhFTJRVhOtL1lmVnfu3JGrqqr8j1evXi3fvXuXa2sCgbIKt7UVsl+Dt7W1IT4+HhqNBgCwaNEi1NTUKDuUgDo6OlBWVobS0lJ/PrW1tcjIyAAwmlttba2CEyorNzcXWq123HOB8rl58yasVisAQK1WIzk5Gc3NzcEdWEETZQUAJ0+exOnTp1FWVoahoSEAzMpisSA7O9v/eGRkBNOnT+famkCgrIDwWlsh+zW43W4f94dDp9PBbrcrOJGYtm7dCovFguHhYRQUFECr1Y7LTqfTYWBgAD6fD1FRIbtcpiRQPv39/UhOTva/TqfTob+/X6kxhZCbm4uEhARER0fj3LlzOHjwIA4fPsysXlNVVYWlS5di3rx5XFvv8XpW4ba2QvbM2mQyYXBw0P/Y5XLBZDIpOJGYLBYLAEClUmHx4sVobGwcl53L5cKMGTNY1K8JlI/RaHxrzRmNRqXGFEJqaiqio6MBAJmZmWhoaAAAZjWmoaEBjY2N2LdvHwCurXd5M6twW1shW9ZWqxV9fX3wer0AgJaWFixfvlzZoQTT1dWFixcv+h/39vYiMTERWVlZaG1tBTCaW1ZWllIjCilQPitWrEBbWxsAwOfzoaurC0uWLFFsThEcOXLE/3Nvby/mzJkDgFkBQE1NDW7fvo2ioiJIkoTW1laurQAmyirc1lZIb+RRV1eHiooKGAwGqNVqXg3+hmfPnuHAgQNIT0+Hy+WCz+fD3r174XA4cOzYMcTHx+Px48fYvXt32F4N3tTUhCtXruDWrVvYsGEDNm3aBI/HEzCf8vJyOBwODAwMYNmyZaFxFeokTZTViRMnMDQ0BJPJhI6ODuzYsQNJSUkAwjure/fuYePGjZg/fz4AwO12o6CgACtXruTaekOgrHp6esJqbYV0WRMREYWCkP0anIiIKFSwrImIiATHsiYiIhIcy5qIiEhwLGsiIiLBsayJiIgEx7ImIiISHMuaiMZpbm5GWloaLl++rPQoRDSGZU1EfiMjIzh06BAyMzPR3t6u9DhENIZlTUR+Fy5cQGJiIvLz83H//n2lxyGiMdxKiYgAAA6HA6dOncLZs2fh8/nw4MEDpUciojEsayICAJSUlGDt2rWYNWsWZFmG1+vFkydPMHPmTKVHIwp7LGsi8m+XGhMTg6tXrwIAXr16hfb2dpY1kQC46xYRYfPmzVizZg3WrVvnf66oqAgJCQnYvn27gpMREcALzIjCXnV1NZ4/f468vLxxz6empvKKcCJB8MyaiIhIcDyzJiIiEhzLmoiISHAsayIiIsGxrImIiATHsiYiIhIcy5qIiEhwLGsiIiLBsayJiIgE91+oOF6z2zhyCwAAAABJRU5ErkJggg==\n", 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" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - " N Z A Element Ebinding Eapprox\n", - "A \n", - "1 0 0 1 1 H 0.000000 0.000000\n", - "2 1 1 1 2 H 1.112283 1.112283\n", - "3 2 2 1 3 H 2.827265 2.827265\n", - "4 6 2 2 4 He 7.073915 7.073915\n", - "5 9 3 2 5 He 5.512132 5.422231\n", - "... ... ... ... ... ... ...\n", - "264 3304 156 108 264 Hs 7.298375 7.387331\n", - "265 3310 157 108 265 Hs 7.296247 7.387331\n", - "266 3317 158 108 266 Hs 7.298273 7.387331\n", - "269 3338 159 110 269 Ds 7.250154 7.387331\n", - "270 3344 160 110 270 Ds 7.253775 7.387331\n", - "\n", - "[267 rows x 6 columns]\n", - "0.009883615646716182\n" - ] - } - ], - "source": [ - "\n", - "#Decision Tree Regression\n", - "from sklearn.tree import DecisionTreeRegressor\n", - "regr_1=DecisionTreeRegressor(max_depth=5)\n", - "regr_2=DecisionTreeRegressor(max_depth=7)\n", - "regr_3=DecisionTreeRegressor(max_depth=9)\n", - "regr_1.fit(X, Energies)\n", - "regr_2.fit(X, Energies)\n", - "regr_3.fit(X, Energies)\n", - "\n", - "\n", - "y_1 = regr_1.predict(X)\n", - "y_2 = regr_2.predict(X)\n", - "y_3=regr_3.predict(X)\n", - "Masses['Eapprox'] = y_1\n", - "# Plot the results\n", - "plt.figure()\n", - "plt.plot(A, Energies, color=\"blue\", label=\"Data\", linewidth=2)\n", - "plt.plot(A, y_1, color=\"red\", label=\"max_depth=5\", linewidth=2)\n", - "plt.plot(A, y_2, color=\"green\", label=\"max_depth=7\", linewidth=2)\n", - "plt.plot(A, y_3, color=\"m\", label=\"max_depth=9\", linewidth=2)\n", - "\n", - "plt.xlabel(\"$A$\")\n", - "plt.ylabel(\"$E$[MeV]\")\n", - "plt.title(\"Decision Tree Regression\")\n", - "plt.legend()\n", - "save_fig(\"Masses2016Trees\")\n", - "plt.show()\n", - "print(Masses)\n", - "print(np.mean( (Energies-y_1)**2))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, + "id": "db2d6c19", + "metadata": { + "editable": true + }, "source": [ "### And what about using neural networks?\n", "\n", @@ -2266,44 +3254,59 @@ }, { "cell_type": "code", - "execution_count": 37, - "metadata": {}, + "execution_count": 34, + "id": "66c27da6", + "metadata": { + "collapsed": false, + "editable": true, + "jupyter": { + "outputs_hidden": false + } + }, "outputs": [], "source": [ "from sklearn.neural_network import MLPRegressor\n", "from sklearn.metrics import accuracy_score\n", "import seaborn as sns\n", "\n", + "\n", "X_train = X\n", "Y_train = Energies\n", - "n_hidden_neurons = 100\n", + "n_hidden_neurons = 50\n", "epochs = 100\n", "# store models for later use\n", - "eta_vals = np.logspace(-5, 1, 7)\n", - "lmbd_vals = np.logspace(-5, 1, 7)\n", + "eta_vals = np.logspace(-3, 0, 4)\n", + "lmbd_vals = np.logspace(-3, 0, 4)\n", "# store the models for later use\n", "DNN_scikit = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)\n", "train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))\n", "sns.set()\n", "for i, eta in enumerate(eta_vals):\n", " for j, lmbd in enumerate(lmbd_vals):\n", - " dnn = MLPRegressor(hidden_layer_sizes=(n_hidden_neurons), activation='logistic',\n", + " dnn = MLPRegressor(hidden_layer_sizes=(n_hidden_neurons), activation='relu', solver='adam',\n", " alpha=lmbd, learning_rate_init=eta, max_iter=epochs)\n", " dnn.fit(X_train, Y_train)\n", " DNN_scikit[i][j] = dnn\n", " train_accuracy[i][j] = dnn.score(X_train, Y_train)\n", - "\n", + " fity = dnn.predict(X_train)\n", + " MSE = mean_squared_error(Y_train, fity)\n", + " print(\"Mean squared error: %.2f\" % mean_squared_error(Y_train, fity))\n", + " train_accuracy[i][j] = MSE\n", "fig, ax = plt.subplots(figsize = (10, 10))\n", "sns.heatmap(train_accuracy, annot=True, ax=ax, cmap=\"viridis\")\n", "ax.set_title(\"Training Accuracy\")\n", "ax.set_ylabel(\"$\\eta$\")\n", "ax.set_xlabel(\"$\\lambda$\")\n", - "plt.show()" + "plt.show()\n", + "print(train_accuracy)" ] }, { "cell_type": "markdown", - "metadata": {}, + "id": "50a1e770", + "metadata": { + "editable": true + }, "source": [ "## A first summary\n", "\n", @@ -2317,11 +3320,1701 @@ "Machine Learning algorithms for supervised learning. Later we will meet **Tensorflow**, a powerful library for deep learning. \n", "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." ] + }, + { + "cell_type": "markdown", + "id": "5af218e3", + "metadata": { + "editable": true + }, + "source": [ + "## Why Linear Regression (aka Ordinary Least Squares and family)\n", + "\n", + "Fitting a continuous function with linear parameterization in terms of the parameters $\\boldsymbol{\\theta}$.\n", + "* Method of choice for fitting a continuous function!\n", + "\n", + "* Gives an excellent introduction to central Machine Learning features with **understandable pedagogical** links to other methods like **Neural Networks**, **Support Vector Machines** etc\n", + "\n", + "* Analytical expression for the fitting parameters $\\boldsymbol{\\theta}$\n", + "\n", + "* Analytical expressions for statistical propertiers like mean values, variances, confidence intervals and more\n", + "\n", + "* Analytical relation with probabilistic interpretations \n", + "\n", + "* Easy to introduce basic concepts like bias-variance tradeoff, cross-validation, resampling and regularization techniques and many other ML topics\n", + "\n", + "* Easy to code! And links well with classification problems and logistic regression and neural networks\n", + "\n", + "* Allows for **easy** hands-on understanding of gradient descent methods\n", + "\n", + "* and many more features\n", + "\n", + "For more discussions of Ridge and Lasso regression, [Wessel van Wieringen's](https://arxiv.org/abs/1509.09169) article is highly recommended.\n", + "Similarly, [Mehta et al's article](https://arxiv.org/abs/1803.08823) is also recommended." + ] + }, + { + "cell_type": "markdown", + "id": "55450c7b", + "metadata": { + "editable": true + }, + "source": [ + "## Regression analysis, overarching aims\n", + "\n", + "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$. \n", + "The first variable $y$ is called the the **outcome** or the **response** variable, or simply just the **outputs**.\n", + "\n", + "The set of variables $\\boldsymbol{x}$ is called the independent variable, or the predictor variable or the explanatory variable, or simply just the **inputs**. **We will throughout the course just use inputs and outputs as names**.\n", + "\n", + "A regression model aims at finding a likelihood function $p(\\boldsymbol{y}\\vert \\boldsymbol{x})$ or in the more traditional sense a function $\\boldsymbol{y}(\\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 \n", + "* $n$ cases $i = 0, 1, 2, \\dots, n-1$ \n", + "\n", + "* Response (our output) variable $y_i$ with $i = 0, 1, 2, \\dots, n-1$ \n", + "\n", + "* $p$ so-called explanatory (independent or predictor or feature) 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$. These are the inputs. See below for more explicit examples. \n", + "\n", + " The goal of the regression analysis is to extract/exploit relationship between $\\boldsymbol{y}$ and $\\boldsymbol{x}$ in order to infer specific dependencies, approximations to the likelihood functions, functional relationships and to make predictions, making fits and many other things." + ] + }, + { + "cell_type": "markdown", + "id": "be42ca6d", + "metadata": { + "editable": true + }, + "source": [ + "## Regression analysis, overarching aims II\n", + "\n", + "Consider an experiment in which $p$ characteristics/features of $n$ samples are\n", + "measured. The data from this experiment, for various explanatory variables $p$ are normally represented by a matrix \n", + "$\\mathbf{X}$.\n", + "\n", + "The matrix $\\mathbf{X}$ is called the *design\n", + "matrix*. Additional information of the samples is available in the\n", + "form of $\\boldsymbol{y}$ (also as above). The variable $\\boldsymbol{y}$ is\n", + "generally referred to as the *response variable*. The aim of\n", + "regression analysis is to explain $\\boldsymbol{y}$ in terms of\n", + "$\\boldsymbol{X}$ through a functional relationship like $y_i =\n", + "f(\\mathbf{X}_{i,\\ast})$. When no prior knowledge on the form of\n", + "$f(\\cdot)$ is available, it is common to assume a linear relationship\n", + "between $\\boldsymbol{X}$ and $\\boldsymbol{y}$. This assumption gives rise to\n", + "the *linear regression model* where $\\boldsymbol{\\theta} = [\\theta_0, \\ldots,\n", + "\\theta_{p-1}]^{T}$ are the *regression parameters*. \n", + "\n", + "Linear regression gives us a set of analytical equations for the parameters $\\theta_j$." + ] + }, + { + "cell_type": "markdown", + "id": "4b7a4ceb", + "metadata": { + "editable": true + }, + "source": [ + "## Examples\n", + "In order to understand the relation among the predictors (or features or properties) $p$, the set of data $n$ and the target (outcome, output etc) $\\boldsymbol{y}$,\n", + "consider the model we discussed for describing nuclear binding energies. \n", + "\n", + "There we assumed that we could parametrize the data using a polynomial approximation based on the liquid drop model.\n", + "Assuming" + ] + }, + { + "cell_type": "markdown", + "id": "e00d9920", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "BE(A) = a_0+a_1A+a_2A^{2/3}+a_3A^{-1/3}+a_4A^{-1},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "6243c073", + "metadata": { + "editable": true + }, + "source": [ + "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.\n", + "This gives $p=0,1,2,3,4$. Furthermore we have $n$ entries for each predictor. It means that our design matrix is a \n", + "$p\\times n$ matrix $\\boldsymbol{X}$.\n", + "\n", + "Here the predictors are based on a model we have made. A popular data set which is widely encountered in ML applications is the\n", + "so-called [credit card default data from Taiwan](https://www.sciencedirect.com/science/article/pii/S0957417407006719?via%3Dihub). 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." + ] + }, + { + "cell_type": "markdown", + "id": "009ea7de", + "metadata": { + "editable": true + }, + "source": [ + "## General linear models and linear algebra\n", + "Before we proceed let us study a case 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. \n", + "\n", + "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" + ] + }, + { + "cell_type": "markdown", + "id": "712c684d", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "y=y(x) \\rightarrow y(x_i)=\\tilde{y}_i+\\epsilon_i=\\sum_{j=0}^{n-1} \\theta_j x_i^j+\\epsilon_i,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "b4bcd2a5", + "metadata": { + "editable": true + }, + "source": [ + "where $\\epsilon_i$ is the error in our approximation." + ] + }, + { + "cell_type": "markdown", + "id": "143b6359", + "metadata": { + "editable": true + }, + "source": [ + "## Rewriting the fitting procedure as a linear algebra problem\n", + "For every set of values $y_i,x_i$ we have thus the corresponding set of equations" + ] + }, + { + "cell_type": "markdown", + "id": "d2e56068", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\begin{align*}\n", + "y_0&=\\theta_0+\\theta_1x_0^1+\\theta_2x_0^2+\\dots+\\theta_{n-1}x_0^{n-1}+\\epsilon_0\\\\\n", + "y_1&=\\theta_0+\\theta_1x_1^1+\\theta_2x_1^2+\\dots+\\theta_{n-1}x_1^{n-1}+\\epsilon_1\\\\\n", + "y_2&=\\theta_0+\\theta_1x_2^1+\\theta_2x_2^2+\\dots+\\theta_{n-1}x_2^{n-1}+\\epsilon_2\\\\\n", + "\\dots & \\dots \\\\\n", + "y_{n-1}&=\\theta_0+\\theta_1x_{n-1}^1+\\theta_2x_{n-1}^2+\\dots+\\theta_{n-1}x_{n-1}^{n-1}+\\epsilon_{n-1}.\\\\\n", + "\\end{align*}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "2541505d", + "metadata": { + "editable": true + }, + "source": [ + "## Rewriting the fitting procedure as a linear algebra problem, more details\n", + "Defining the vectors" + ] + }, + { + "cell_type": "markdown", + "id": "725efee7", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\boldsymbol{y} = [y_0,y_1, y_2,\\dots, y_{n-1}]^T,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "cb15e9f1", + "metadata": { + "editable": true + }, + "source": [ + "and" + ] + }, + { + "cell_type": "markdown", + "id": "87026e15", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\boldsymbol{\\theta} = [\\theta_0,\\theta_1, \\theta_2,\\dots, \\theta_{n-1}]^T,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "cd09c85a", + "metadata": { + "editable": true + }, + "source": [ + "and" + ] + }, + { + "cell_type": "markdown", + "id": "c7845cfd", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\boldsymbol{\\epsilon} = [\\epsilon_0,\\epsilon_1, \\epsilon_2,\\dots, \\epsilon_{n-1}]^T,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "20e94324", + "metadata": { + "editable": true + }, + "source": [ + "and the design matrix" + ] + }, + { + "cell_type": "markdown", + "id": "46cc127e", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\boldsymbol{X}=\n", + "\\begin{bmatrix} \n", + "1& x_{0}^1 &x_{0}^2& \\dots & \\dots &x_{0}^{n-1}\\\\\n", + "1& x_{1}^1 &x_{1}^2& \\dots & \\dots &x_{1}^{n-1}\\\\\n", + "1& x_{2}^1 &x_{2}^2& \\dots & \\dots &x_{2}^{n-1}\\\\ \n", + "\\dots& \\dots &\\dots& \\dots & \\dots &\\dots\\\\\n", + "1& x_{n-1}^1 &x_{n-1}^2& \\dots & \\dots &x_{n-1}^{n-1}\\\\\n", + "\\end{bmatrix}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "b83e3f3f", + "metadata": { + "editable": true + }, + "source": [ + "we can rewrite our equations as" + ] + }, + { + "cell_type": "markdown", + "id": "d3c3d5b8", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\boldsymbol{y} = \\boldsymbol{X}\\boldsymbol{\\theta}+\\boldsymbol{\\epsilon}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "c6e741c3", + "metadata": { + "editable": true + }, + "source": [ + "The above design matrix is called a [Vandermonde matrix](https://en.wikipedia.org/wiki/Vandermonde_matrix)." + ] + }, + { + "cell_type": "markdown", + "id": "510abc9b", + "metadata": { + "editable": true + }, + "source": [ + "## Generalizing the fitting procedure as a linear algebra problem\n", + "\n", + "We are obviously not limited to the above polynomial expansions. We\n", + "could replace the various powers of $x$ with elements of Fourier\n", + "series or instead of $x_i^j$ we could have $\\cos{(j x_i)}$ or $\\sin{(j\n", + "x_i)}$, or time series or other orthogonal functions. For every set\n", + "of values $y_i,x_i$ we can then generalize the equations to" + ] + }, + { + "cell_type": "markdown", + "id": "1a6464a4", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\begin{align*}\n", + "y_0&=\\theta_0x_{00}+\\theta_1x_{01}+\\theta_2x_{02}+\\dots+\\theta_{n-1}x_{0n-1}+\\epsilon_0\\\\\n", + "y_1&=\\theta_0x_{10}+\\theta_1x_{11}+\\theta_2x_{12}+\\dots+\\theta_{n-1}x_{1n-1}+\\epsilon_1\\\\\n", + "y_2&=\\theta_0x_{20}+\\theta_1x_{21}+\\theta_2x_{22}+\\dots+\\theta_{n-1}x_{2n-1}+\\epsilon_2\\\\\n", + "\\dots & \\dots \\\\\n", + "y_{i}&=\\theta_0x_{i0}+\\theta_1x_{i1}+\\theta_2x_{i2}+\\dots+\\theta_{n-1}x_{in-1}+\\epsilon_i\\\\\n", + "\\dots & \\dots \\\\\n", + "y_{n-1}&=\\theta_0x_{n-1,0}+\\theta_1x_{n-1,2}+\\theta_2x_{n-1,2}+\\dots+\\theta_{n-1}x_{n-1,n-1}+\\epsilon_{n-1}.\\\\\n", + "\\end{align*}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "0f403e2d", + "metadata": { + "editable": true + }, + "source": [ + "**Note that we have $p=n$ here. The matrix is symmetric. This is generally not the case!**" + ] + }, + { + "cell_type": "markdown", + "id": "4a8138dc", + "metadata": { + "editable": true + }, + "source": [ + "## Generalizing the fitting procedure as a linear algebra problem\n", + "We redefine in turn the matrix $\\boldsymbol{X}$ as" + ] + }, + { + "cell_type": "markdown", + "id": "726cbaa4", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\boldsymbol{X}=\n", + "\\begin{bmatrix} \n", + "x_{00}& x_{01} &x_{02}& \\dots & \\dots &x_{0,n-1}\\\\\n", + "x_{10}& x_{11} &x_{12}& \\dots & \\dots &x_{1,n-1}\\\\\n", + "x_{20}& x_{21} &x_{22}& \\dots & \\dots &x_{2,n-1}\\\\ \n", + "\\dots& \\dots &\\dots& \\dots & \\dots &\\dots\\\\\n", + "x_{n-1,0}& x_{n-1,1} &x_{n-1,2}& \\dots & \\dots &x_{n-1,n-1}\\\\\n", + "\\end{bmatrix}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "6b7adfb6", + "metadata": { + "editable": true + }, + "source": [ + "and without loss of generality we rewrite again our equations as" + ] + }, + { + "cell_type": "markdown", + "id": "84165f3e", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\boldsymbol{y} = \\boldsymbol{X}\\boldsymbol{\\theta}+\\boldsymbol{\\epsilon}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "ebf366ef", + "metadata": { + "editable": true + }, + "source": [ + "The left-hand side of this equation is kwown. Our error vector $\\boldsymbol{\\epsilon}$ and the parameter vector $\\boldsymbol{\\theta}$ are our unknow quantities. How can we obtain the optimal set of $\\theta_i$ values?" + ] + }, + { + "cell_type": "markdown", + "id": "02ffad05", + "metadata": { + "editable": true + }, + "source": [ + "## Optimizing our parameters\n", + "We have defined the matrix $\\boldsymbol{X}$ via the equations" + ] + }, + { + "cell_type": "markdown", + "id": "d54237d5", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\begin{align*}\n", + "y_0&=\\theta_0x_{00}+\\theta_1x_{01}+\\theta_2x_{02}+\\dots+\\theta_{n-1}x_{0n-1}+\\epsilon_0\\\\\n", + "y_1&=\\theta_0x_{10}+\\theta_1x_{11}+\\theta_2x_{12}+\\dots+\\theta_{n-1}x_{1n-1}+\\epsilon_1\\\\\n", + "y_2&=\\theta_0x_{20}+\\theta_1x_{21}+\\theta_2x_{22}+\\dots+\\theta_{n-1}x_{2n-1}+\\epsilon_1\\\\\n", + "\\dots & \\dots \\\\\n", + "y_{i}&=\\theta_0x_{i0}+\\theta_1x_{i1}+\\theta_2x_{i2}+\\dots+\\theta_{n-1}x_{in-1}+\\epsilon_1\\\\\n", + "\\dots & \\dots \\\\\n", + "y_{n-1}&=\\theta_0x_{n-1,0}+\\theta_1x_{n-1,2}+\\theta_2x_{n-1,2}+\\dots+\\theta_{n-1}x_{n-1,n-1}+\\epsilon_{n-1}.\\\\\n", + "\\end{align*}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "8a7e9dbb", + "metadata": { + "editable": true + }, + "source": [ + "As we noted above, we stayed with a system with the design matrix \n", + " $\\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 \n", + "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." + ] + }, + { + "cell_type": "markdown", + "id": "d4d5ff52", + "metadata": { + "editable": true + }, + "source": [ + "## Our model for the nuclear binding energies\n", + "\n", + "In our [introductory notes](https://compphysics.github.io/MachineLearning/doc/pub/How2ReadData/html/How2ReadData.html) we looked at the so-called [liquid drop model](https://en.wikipedia.org/wiki/Semi-empirical_mass_formula). Let us remind ourselves about what we did by looking at the code.\n", + "\n", + "We restate the parts of the code we are most interested in." + ] + }, + { + "cell_type": "code", + "execution_count": 35, + "id": "297415bf", + "metadata": { + "collapsed": false, + "editable": true, + "jupyter": { + "outputs_hidden": false + } + }, + "outputs": [], + "source": [ + "# Common imports\n", + "import numpy as np\n", + "import pandas as pd\n", + "import matplotlib.pyplot as plt\n", + "from IPython.display import display\n", + "import os\n", + "\n", + "# Where to save the figures and data files\n", + "PROJECT_ROOT_DIR = \"Results\"\n", + "FIGURE_ID = \"Results/FigureFiles\"\n", + "DATA_ID = \"DataFiles/\"\n", + "\n", + "if not os.path.exists(PROJECT_ROOT_DIR):\n", + " os.mkdir(PROJECT_ROOT_DIR)\n", + "\n", + "if not os.path.exists(FIGURE_ID):\n", + " os.makedirs(FIGURE_ID)\n", + "\n", + "if not os.path.exists(DATA_ID):\n", + " os.makedirs(DATA_ID)\n", + "\n", + "def image_path(fig_id):\n", + " return os.path.join(FIGURE_ID, fig_id)\n", + "\n", + "def data_path(dat_id):\n", + " return os.path.join(DATA_ID, dat_id)\n", + "\n", + "def save_fig(fig_id):\n", + " plt.savefig(image_path(fig_id) + \".png\", format='png')\n", + "\n", + "infile = open(data_path(\"MassEval2016.dat\"),'r')\n", + "\n", + "\n", + "# Read the experimental data with Pandas\n", + "Masses = pd.read_fwf(infile, usecols=(2,3,4,6,11),\n", + " names=('N', 'Z', 'A', 'Element', 'Ebinding'),\n", + " widths=(1,3,5,5,5,1,3,4,1,13,11,11,9,1,2,11,9,1,3,1,12,11,1),\n", + " header=39,\n", + " index_col=False)\n", + "\n", + "# Extrapolated values are indicated by '#' in place of the decimal place, so\n", + "# the Ebinding column won't be numeric. Coerce to float and drop these entries.\n", + "Masses['Ebinding'] = pd.to_numeric(Masses['Ebinding'], errors='coerce')\n", + "Masses = Masses.dropna()\n", + "# Convert from keV to MeV.\n", + "Masses['Ebinding'] /= 1000\n", + "\n", + "# Group the DataFrame by nucleon number, A.\n", + "Masses = Masses.groupby('A')\n", + "# Find the rows of the grouped DataFrame with the maximum binding energy.\n", + "Masses = Masses.apply(lambda t: t[t.Ebinding==t.Ebinding.max()])\n", + "A = Masses['A']\n", + "Z = Masses['Z']\n", + "N = Masses['N']\n", + "Element = Masses['Element']\n", + "Energies = Masses['Ebinding']\n", + "\n", + "# Now we set up the design matrix X\n", + "X = np.zeros((len(A),5))\n", + "X[:,0] = 1\n", + "X[:,1] = A\n", + "X[:,2] = A**(2.0/3.0)\n", + "X[:,3] = A**(-1.0/3.0)\n", + "X[:,4] = A**(-1.0)\n", + "# Then nice printout using pandas\n", + "DesignMatrix = pd.DataFrame(X)\n", + "DesignMatrix.index = A\n", + "DesignMatrix.columns = ['1', 'A', 'A^(2/3)', 'A^(-1/3)', '1/A']\n", + "display(DesignMatrix)" + ] + }, + { + "cell_type": "markdown", + "id": "2ce99829", + "metadata": { + "editable": true + }, + "source": [ + "With $\\boldsymbol{\\theta}\\in {\\mathbb{R}}^{p\\times 1}$, it means that we will hereafter write our equations for the approximation as" + ] + }, + { + "cell_type": "markdown", + "id": "f6a44999", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\boldsymbol{\\tilde{y}}= \\boldsymbol{X}\\boldsymbol{\\theta},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "990c34c3", + "metadata": { + "editable": true + }, + "source": [ + "throughout these lectures." + ] + }, + { + "cell_type": "markdown", + "id": "a06f9457", + "metadata": { + "editable": true + }, + "source": [ + "## Optimizing our parameters, more details\n", + "With the above we use the design matrix to define the approximation $\\boldsymbol{\\tilde{y}}$ via the unknown quantity $\\boldsymbol{\\theta}$ as" + ] + }, + { + "cell_type": "markdown", + "id": "19cd9843", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\boldsymbol{\\tilde{y}}= \\boldsymbol{X}\\boldsymbol{\\theta},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "cc8c11ab", + "metadata": { + "editable": true + }, + "source": [ + "and in order to find the optimal parameters $\\theta_i$ instead of solving the above linear algebra problem, we define a function which gives a measure of the spread between the values $y_i$ (which represent hopefully the exact values) and the parameterized values $\\tilde{y}_i$, namely" + ] + }, + { + "cell_type": "markdown", + "id": "d3d4e384", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "C(\\boldsymbol{\\theta})=\\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\\},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "6187929e", + "metadata": { + "editable": true + }, + "source": [ + "or using the matrix $\\boldsymbol{X}$ and in a more compact matrix-vector notation as" + ] + }, + { + "cell_type": "markdown", + "id": "e0ba2cb1", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "C(\\boldsymbol{\\theta})=\\frac{1}{n}\\left\\{\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\theta}\\right)^T\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\theta}\\right)\\right\\}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "468fec32", + "metadata": { + "editable": true + }, + "source": [ + "This function is one possible way to define the so-called cost function.\n", + "\n", + "It is also common to define\n", + "the function $C$ as" + ] + }, + { + "cell_type": "markdown", + "id": "ba618e10", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "C(\\boldsymbol{\\theta})=\\frac{1}{2n}\\sum_{i=0}^{n-1}\\left(y_i-\\tilde{y}_i\\right)^2,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "a74d3f1c", + "metadata": { + "editable": true + }, + "source": [ + "since when taking the first derivative with respect to the unknown parameters $\\theta$, the factor of $2$ cancels out." + ] + }, + { + "cell_type": "markdown", + "id": "fb434398", + "metadata": { + "editable": true + }, + "source": [ + "## Interpretations and optimizing our parameters\n", + "\n", + "The function" + ] + }, + { + "cell_type": "markdown", + "id": "42558cc8", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "C(\\boldsymbol{\\theta})=\\frac{1}{n}\\left\\{\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\theta}\\right)^T\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\theta}\\right)\\right\\},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "d851da5b", + "metadata": { + "editable": true + }, + "source": [ + "can be linked to the variance of the quantity $y_i$ if we interpret the latter as the mean value. \n", + "When linking (see the discussion below) with the maximum likelihood approach below, we will indeed interpret $y_i$ as a mean value" + ] + }, + { + "cell_type": "markdown", + "id": "49533cb3", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "y_{i}=\\langle y_i \\rangle = \\theta_0x_{i,0}+\\theta_1x_{i,1}+\\theta_2x_{i,2}+\\dots+\\theta_{n-1}x_{i,n-1}+\\epsilon_i,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "a62f6a4c", + "metadata": { + "editable": true + }, + "source": [ + "where $\\langle y_i \\rangle$ is the mean value. Keep in mind also that\n", + "till now we have treated $y_i$ as the exact value. Normally, the\n", + "response (dependent or outcome) variable $y_i$ the outcome of a\n", + "numerical experiment or another type of experiment and is thus only an\n", + "approximation to the true value. It is then always accompanied by an\n", + "error estimate, often limited to a statistical error estimate given by\n", + "the standard deviation discussed earlier. In the discussion here we\n", + "will treat $y_i$ as our exact value for the response variable.\n", + "\n", + "In order to find the parameters $\\theta_i$ we will then minimize the spread of $C(\\boldsymbol{\\theta})$, that is we are going to solve the problem" + ] + }, + { + "cell_type": "markdown", + "id": "381182cd", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "{\\displaystyle \\min_{\\boldsymbol{\\theta}\\in\n", + "{\\mathbb{R}}^{p}}}\\frac{1}{n}\\left\\{\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\theta}\\right)^T\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\theta}\\right)\\right\\}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "04b172c8", + "metadata": { + "editable": true + }, + "source": [ + "In practical terms it means we will require" + ] + }, + { + "cell_type": "markdown", + "id": "265d8614", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\frac{\\partial C(\\boldsymbol{\\theta})}{\\partial \\theta_j} = \\frac{\\partial }{\\partial \\theta_j}\\left[ \\frac{1}{n}\\sum_{i=0}^{n-1}\\left(y_i-\\theta_0x_{i,0}-\\theta_1x_{i,1}-\\theta_2x_{i,2}-\\dots-\\theta_{n-1}x_{i,n-1}\\right)^2\\right]=0,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "68c1005e", + "metadata": { + "editable": true + }, + "source": [ + "which results in" + ] + }, + { + "cell_type": "markdown", + "id": "61fb0fbc", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\frac{\\partial C(\\boldsymbol{\\theta})}{\\partial \\theta_j} = -\\frac{2}{n}\\left[ \\sum_{i=0}^{n-1}x_{ij}\\left(y_i-\\theta_0x_{i,0}-\\theta_1x_{i,1}-\\theta_2x_{i,2}-\\dots-\\theta_{n-1}x_{i,n-1}\\right)\\right]=0,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "acb3fa59", + "metadata": { + "editable": true + }, + "source": [ + "or in a matrix-vector form as" + ] + }, + { + "cell_type": "markdown", + "id": "82f920b5", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\frac{\\partial C(\\boldsymbol{\\theta})}{\\partial \\boldsymbol{\\theta}} = 0 = \\boldsymbol{X}^T\\left( \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\theta}\\right).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "26de4671", + "metadata": { + "editable": true + }, + "source": [ + "## Interpretations and optimizing our parameters\n", + "We can rewrite" + ] + }, + { + "cell_type": "markdown", + "id": "071a86ff", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\frac{\\partial C(\\boldsymbol{\\theta})}{\\partial \\boldsymbol{\\theta}} = 0 = \\boldsymbol{X}^T\\left( \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\theta}\\right),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "fdb51c1d", + "metadata": { + "editable": true + }, + "source": [ + "as" + ] + }, + { + "cell_type": "markdown", + "id": "5532b00a", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\boldsymbol{X}^T\\boldsymbol{y} = \\boldsymbol{X}^T\\boldsymbol{X}\\boldsymbol{\\theta},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "a7c57982", + "metadata": { + "editable": true + }, + "source": [ + "and if the matrix $\\boldsymbol{X}^T\\boldsymbol{X}$ is invertible we have the solution" + ] + }, + { + "cell_type": "markdown", + "id": "022ce912", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\boldsymbol{\\theta} =\\left(\\boldsymbol{X}^T\\boldsymbol{X}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "12518c37", + "metadata": { + "editable": true + }, + "source": [ + "We note also that since our design matrix is defined as $\\boldsymbol{X}\\in\n", + "{\\mathbb{R}}^{n\\times p}$, the product $\\boldsymbol{X}^T\\boldsymbol{X} \\in\n", + "{\\mathbb{R}}^{p\\times p}$. In the above case we have that $p \\ll n$,\n", + "in our case $p=5$ meaning that we end up with inverting a small\n", + "$5\\times 5$ matrix. This is a rather common situation, in many cases we end up with low-dimensional\n", + "matrices to invert. The methods discussed here and for many other\n", + "supervised learning algorithms like classification with logistic\n", + "regression or support vector machines, exhibit dimensionalities which\n", + "allow for the usage of direct linear algebra methods such as **LU** decomposition or **Singular Value Decomposition** (SVD) for finding the inverse of the matrix\n", + "$\\boldsymbol{X}^T\\boldsymbol{X}$.\n", + "\n", + "**Small question**: Do you think the example we have at hand here (the nuclear binding energies) can lead to problems in inverting the matrix $\\boldsymbol{X}^T\\boldsymbol{X}$? What kind of problems can we expect?" + ] + }, + { + "cell_type": "markdown", + "id": "89f9bbda", + "metadata": { + "editable": true + }, + "source": [ + "## Interpretations and optimizing our parameters\n", + "The residuals $\\boldsymbol{\\epsilon}$ are in turn given by" + ] + }, + { + "cell_type": "markdown", + "id": "51458daa", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\boldsymbol{\\epsilon} = \\boldsymbol{y}-\\boldsymbol{\\tilde{y}} = \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\theta},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "eaf4a93e", + "metadata": { + "editable": true + }, + "source": [ + "and with" + ] + }, + { + "cell_type": "markdown", + "id": "2fea3537", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\boldsymbol{X}^T\\left( \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\theta}\\right)= 0,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "ca4f1123", + "metadata": { + "editable": true + }, + "source": [ + "we have" + ] + }, + { + "cell_type": "markdown", + "id": "09de659a", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\boldsymbol{X}^T\\boldsymbol{\\epsilon}=\\boldsymbol{X}^T\\left( \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\theta}\\right)= 0,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "77f9fd08", + "metadata": { + "editable": true + }, + "source": [ + "meaning that the solution for $\\boldsymbol{\\theta}$ is the one which minimizes the residuals. Later we will link this with the maximum likelihood approach.\n", + "\n", + "Let us now return to our nuclear binding energies and simply code the above equations." + ] + }, + { + "cell_type": "markdown", + "id": "beab9d54", + "metadata": { + "editable": true + }, + "source": [ + "## Own code for Ordinary Least Squares\n", + "\n", + "It is rather straightforward to implement the matrix inversion and obtain the parameters $\\boldsymbol{\\theta}$. After having defined the matrix $\\boldsymbol{X}$ we simply need to \n", + "write" + ] + }, + { + "cell_type": "code", + "execution_count": 36, + "id": "ce0d14e1", + "metadata": { + "collapsed": false, + "editable": true, + "jupyter": { + "outputs_hidden": false + } + }, + "outputs": [], + "source": [ + "# matrix inversion to find beta\n", + "beta = np.linalg.inv(X.T.dot(X)).dot(X.T).dot(Energies)\n", + "# and then make the prediction\n", + "ytilde = X @ beta" + ] + }, + { + "cell_type": "markdown", + "id": "20bbbdf2", + "metadata": { + "editable": true + }, + "source": [ + "Alternatively, you can use the least squares functionality in **Numpy** as" + ] + }, + { + "cell_type": "code", + "execution_count": 37, + "id": "d1b21aae", + "metadata": { + "collapsed": false, + "editable": true, + "jupyter": { + "outputs_hidden": false + } + }, + "outputs": [], + "source": [ + "fit = np.linalg.lstsq(X, Energies, rcond =None)[0]\n", + "ytildenp = np.dot(fit,X.T)" + ] + }, + { + "cell_type": "markdown", + "id": "02bb27d6", + "metadata": { + "editable": true + }, + "source": [ + "And finally we plot our fit with and compare with data" + ] + }, + { + "cell_type": "code", + "execution_count": 38, + "id": "98630c99", + "metadata": { + "collapsed": false, + "editable": true, + "jupyter": { + "outputs_hidden": false + } + }, + "outputs": [], + "source": [ + "Masses['Eapprox'] = ytilde\n", + "# Generate a plot comparing the experimental with the fitted values values.\n", + "fig, ax = plt.subplots()\n", + "ax.set_xlabel(r'$A = N + Z$')\n", + "ax.set_ylabel(r'$E_\\mathrm{bind}\\,/\\mathrm{MeV}$')\n", + "ax.plot(Masses['A'], Masses['Ebinding'], alpha=0.7, lw=2,\n", + " label='Ame2016')\n", + "ax.plot(Masses['A'], Masses['Eapprox'], alpha=0.7, lw=2, c='m',\n", + " label='Fit')\n", + "ax.legend()\n", + "save_fig(\"Masses2016OLS\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "9af04036", + "metadata": { + "editable": true + }, + "source": [ + "## Adding error analysis and training set up\n", + "\n", + "We can easily test our fit by computing the $R2$ score that we discussed in connection with the functionality of **Scikit-Learn** in the introductory slides.\n", + "Since we are not using **Scikit-Learn** here we can define our own $R2$ function as" + ] + }, + { + "cell_type": "code", + "execution_count": 39, + "id": "2d8cd3da", + "metadata": { + "collapsed": false, + "editable": true, + "jupyter": { + "outputs_hidden": false + } + }, + "outputs": [], + "source": [ + "def R2(y_data, y_model):\n", + " return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)" + ] + }, + { + "cell_type": "markdown", + "id": "71899eed", + "metadata": { + "editable": true + }, + "source": [ + "and we would be using it as" + ] + }, + { + "cell_type": "code", + "execution_count": 40, + "id": "f8cc2dd0", + "metadata": { + "collapsed": false, + "editable": true, + "jupyter": { + "outputs_hidden": false + } + }, + "outputs": [], + "source": [ + "print(R2(Energies,ytilde))" + ] + }, + { + "cell_type": "markdown", + "id": "44f3dd69", + "metadata": { + "editable": true + }, + "source": [ + "We can easily add our **MSE** score as" + ] + }, + { + "cell_type": "code", + "execution_count": 41, + "id": "acdf8d3d", + "metadata": { + "collapsed": false, + "editable": true, + "jupyter": { + "outputs_hidden": false + } + }, + "outputs": [], + "source": [ + "def MSE(y_data,y_model):\n", + " n = np.size(y_model)\n", + " return np.sum((y_data-y_model)**2)/n\n", + "\n", + "print(MSE(Energies,ytilde))" + ] + }, + { + "cell_type": "markdown", + "id": "26d00774", + "metadata": { + "editable": true + }, + "source": [ + "and finally the relative error as" + ] + }, + { + "cell_type": "code", + "execution_count": 42, + "id": "fcf52c3c", + "metadata": { + "collapsed": false, + "editable": true, + "jupyter": { + "outputs_hidden": false + } + }, + "outputs": [], + "source": [ + "def RelativeError(y_data,y_model):\n", + " return abs((y_data-y_model)/y_data)\n", + "print(RelativeError(Energies, ytilde))" + ] + }, + { + "cell_type": "markdown", + "id": "82083461", + "metadata": { + "editable": true + }, + "source": [ + "## The $\\chi^2$ function\n", + "\n", + "Normally, the response (dependent or outcome) variable $y_i$ is the\n", + "outcome of a numerical experiment or another type of experiment and is\n", + "thus only an approximation to the true value. It is then always\n", + "accompanied by an error estimate, often limited to a statistical error\n", + "estimate given by the standard deviation discussed earlier. In the\n", + "discussion here we will treat $y_i$ as our exact value for the\n", + "response variable.\n", + "\n", + "Introducing the standard deviation $\\sigma_i$ for each measurement\n", + "$y_i$, we define now the $\\chi^2$ function (omitting the $1/n$ term)\n", + "as" + ] + }, + { + "cell_type": "markdown", + "id": "c247a29f", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\chi^2(\\boldsymbol{\\theta})=\\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\\},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "5c6266aa", + "metadata": { + "editable": true + }, + "source": [ + "where the matrix $\\boldsymbol{\\Sigma}$ is a diagonal matrix with $\\sigma_i$ as matrix elements." + ] + }, + { + "cell_type": "markdown", + "id": "c36f8d5c", + "metadata": { + "editable": true + }, + "source": [ + "## The $\\chi^2$ function\n", + "\n", + "In order to find the parameters $\\theta_i$ we will then minimize the spread of $\\chi^2(\\boldsymbol{\\theta})$ by requiring" + ] + }, + { + "cell_type": "markdown", + "id": "5b1a5174", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\frac{\\partial \\chi^2(\\boldsymbol{\\theta})}{\\partial \\theta_j} = \\frac{\\partial }{\\partial \\theta_j}\\left[ \\frac{1}{n}\\sum_{i=0}^{n-1}\\left(\\frac{y_i-\\theta_0x_{i,0}-\\theta_1x_{i,1}-\\theta_2x_{i,2}-\\dots-\\theta_{n-1}x_{i,n-1}}{\\sigma_i}\\right)^2\\right]=0,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "81c82695", + "metadata": { + "editable": true + }, + "source": [ + "which results in" + ] + }, + { + "cell_type": "markdown", + "id": "7dc8a3bc", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\frac{\\partial \\chi^2(\\boldsymbol{\\theta})}{\\partial \\theta_j} = -\\frac{2}{n}\\left[ \\sum_{i=0}^{n-1}\\frac{x_{ij}}{\\sigma_i}\\left(\\frac{y_i-\\theta_0x_{i,0}-\\theta_1x_{i,1}-\\theta_2x_{i,2}-\\dots-\\theta_{n-1}x_{i,n-1}}{\\sigma_i}\\right)\\right]=0,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "41789a38", + "metadata": { + "editable": true + }, + "source": [ + "or in a matrix-vector form as" + ] + }, + { + "cell_type": "markdown", + "id": "4a2e42bf", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\frac{\\partial \\chi^2(\\boldsymbol{\\theta})}{\\partial \\boldsymbol{\\theta}} = 0 = \\boldsymbol{A}^T\\left( \\boldsymbol{b}-\\boldsymbol{A}\\boldsymbol{\\theta}\\right).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "acd67ee6", + "metadata": { + "editable": true + }, + "source": [ + "where we have defined the matrix $\\boldsymbol{A} =\\boldsymbol{X}/\\boldsymbol{\\Sigma}$ with matrix elements $a_{ij} = x_{ij}/\\sigma_i$ and the vector $\\boldsymbol{b}$ with elements $b_i = y_i/\\sigma_i$." + ] + }, + { + "cell_type": "markdown", + "id": "ee90b0eb", + "metadata": { + "editable": true + }, + "source": [ + "## The $\\chi^2$ function\n", + "\n", + "We can rewrite" + ] + }, + { + "cell_type": "markdown", + "id": "54eaab7f", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\frac{\\partial \\chi^2(\\boldsymbol{\\theta})}{\\partial \\boldsymbol{\\theta}} = 0 = \\boldsymbol{A}^T\\left( \\boldsymbol{b}-\\boldsymbol{A}\\boldsymbol{\\theta}\\right),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "f941f1ac", + "metadata": { + "editable": true + }, + "source": [ + "as" + ] + }, + { + "cell_type": "markdown", + "id": "89e02a88", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\boldsymbol{A}^T\\boldsymbol{b} = \\boldsymbol{A}^T\\boldsymbol{A}\\boldsymbol{\\theta},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "3d1f3b65", + "metadata": { + "editable": true + }, + "source": [ + "and if the matrix $\\boldsymbol{A}^T\\boldsymbol{A}$ is invertible we have the solution" + ] + }, + { + "cell_type": "markdown", + "id": "1dda0ac2", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\boldsymbol{\\theta} =\\left(\\boldsymbol{A}^T\\boldsymbol{A}\\right)^{-1}\\boldsymbol{A}^T\\boldsymbol{b}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "aafe7aae", + "metadata": { + "editable": true + }, + "source": [ + "## The $\\chi^2$ function\n", + "\n", + "If we then introduce the matrix" + ] + }, + { + "cell_type": "markdown", + "id": "37f9c1fa", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\boldsymbol{H} = \\left(\\boldsymbol{A}^T\\boldsymbol{A}\\right)^{-1},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "373adac2", + "metadata": { + "editable": true + }, + "source": [ + "we have then the following expression for the parameters $\\theta_j$ (the matrix elements of $\\boldsymbol{H}$ are $h_{ij}$)" + ] + }, + { + "cell_type": "markdown", + "id": "bdd6d800", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\theta_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}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "913df005", + "metadata": { + "editable": true + }, + "source": [ + "We state without proof the expression for the uncertainty in the parameters $\\theta_j$ as (we leave this as an exercise)" + ] + }, + { + "cell_type": "markdown", + "id": "76f6737e", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\sigma^2(\\theta_j) = \\sum_{i=0}^{n-1}\\sigma_i^2\\left( \\frac{\\partial \\theta_j}{\\partial y_i}\\right)^2,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "48f6aa66", + "metadata": { + "editable": true + }, + "source": [ + "resulting in" + ] + }, + { + "cell_type": "markdown", + "id": "513a1986", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\sigma^2(\\theta_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}!\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "20fbbe07", + "metadata": { + "editable": true + }, + "source": [ + "## The $\\chi^2$ function\n", + "The first step here is to approximate the function $y$ with a first-order polynomial, that is we write" + ] + }, + { + "cell_type": "markdown", + "id": "4dd97daa", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "y=y(x) \\rightarrow y(x_i) \\approx \\theta_0+\\theta_1 x_i.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "032afd0e", + "metadata": { + "editable": true + }, + "source": [ + "By computing the derivatives of $\\chi^2$ with respect to $\\theta_0$ and $\\theta_1$ show that these are given by" + ] + }, + { + "cell_type": "markdown", + "id": "25eac667", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\frac{\\partial \\chi^2(\\boldsymbol{\\theta})}{\\partial \\theta_0} = -2\\left[ \\frac{1}{n}\\sum_{i=0}^{n-1}\\left(\\frac{y_i-\\theta_0-\\theta_1x_{i}}{\\sigma_i^2}\\right)\\right]=0,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "18c343cd", + "metadata": { + "editable": true + }, + "source": [ + "and" + ] + }, + { + "cell_type": "markdown", + "id": "9864986f", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\frac{\\partial \\chi^2(\\boldsymbol{\\theta})}{\\partial \\theta_1} = -\\frac{2}{n}\\left[ \\sum_{i=0}^{n-1}x_i\\left(\\frac{y_i-\\theta_0-\\theta_1x_{i}}{\\sigma_i^2}\\right)\\right]=0.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "1ecdde02", + "metadata": { + "editable": true + }, + "source": [ + "## The $\\chi^2$ function\n", + "\n", + "For a linear fit (a first-order polynomial) we don't need to invert a matrix!! \n", + "Defining" + ] + }, + { + "cell_type": "markdown", + "id": "9d62dddc", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\gamma = \\sum_{i=0}^{n-1}\\frac{1}{\\sigma_i^2},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "db398f5b", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\gamma_x = \\sum_{i=0}^{n-1}\\frac{x_{i}}{\\sigma_i^2},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "2bbf293c", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\gamma_y = \\sum_{i=0}^{n-1}\\left(\\frac{y_i}{\\sigma_i^2}\\right),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "d82e1506", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\gamma_{xx} = \\sum_{i=0}^{n-1}\\frac{x_ix_{i}}{\\sigma_i^2},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "1ddd417c", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\gamma_{xy} = \\sum_{i=0}^{n-1}\\frac{y_ix_{i}}{\\sigma_i^2},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "e7904873", + "metadata": { + "editable": true + }, + "source": [ + "we obtain" + ] + }, + { + "cell_type": "markdown", + "id": "19604164", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\theta_0 = \\frac{\\gamma_{xx}\\gamma_y-\\gamma_x\\gamma_y}{\\gamma\\gamma_{xx}-\\gamma_x^2},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "0f262e03", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\theta_1 = \\frac{\\gamma_{xy}\\gamma-\\gamma_x\\gamma_y}{\\gamma\\gamma_{xx}-\\gamma_x^2}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "cdb18e0d", + "metadata": { + "editable": true + }, + "source": [ + "This approach (different linear and non-linear regression) suffers\n", + "often from both being underdetermined and overdetermined in the\n", + "unknown coefficients $\\theta_i$. A better approach is to use the\n", + "Singular Value Decomposition (SVD) method discussed next week." + ] } ], "metadata": { "kernelspec": { - "display_name": "Python 3", + "display_name": "Python 3 (ipykernel)", "language": "python", "name": "python3" }, @@ -2335,9 +5028,9 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.6.8" + "version": "3.9.15" } }, "nbformat": 4, - "nbformat_minor": 2 + "nbformat_minor": 5 } diff --git a/doc/pub/week34/ipynb/week34.ipynb b/doc/pub/week34/ipynb/week34.ipynb index 1bad18d69..dbeb11e14 100644 --- a/doc/pub/week34/ipynb/week34.ipynb +++ b/doc/pub/week34/ipynb/week34.ipynb @@ -976,7 +976,7 @@ }, { "cell_type": "code", - "execution_count": 2, + "execution_count": 3, "id": "7d44772f", "metadata": { "collapsed": false, @@ -985,9 +985,33 @@ "outputs_hidden": false } }, - "outputs": [], + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "[ 0.22889528 0.91634536 0.58026357 -0.20121369 1.63697022 -0.16620476\n", + " -1.88888726 -2.65872949 -0.00563363 1.25894439 -0.07825328 1.43521368\n", + " 0.97253438 -1.64819989 0.98672233 -1.73087706 -0.99593933 -1.34926433\n", + " -1.26715086 -0.95021858 -0.33244987 0.69501728 -0.48914472 1.87699249\n", + " -0.91130987 -0.2889529 -0.74183199 0.74935175 1.34457198 -1.70306077\n", + " -0.21142106 2.10793224 -0.40622194 2.34515157 -0.40761485 1.69110258\n", + " 0.72886914 -0.43700299 -1.14714793 0.15694733 0.66461823 0.89304652\n", + " 1.4543903 -0.37398506 -0.14684297 -0.57722604 -0.44036741 -0.40055163\n", + " 0.21510519 -0.53120826 -0.43941984 0.68555857 0.85074613 -0.8371588\n", + " 1.19968839 0.59476652 1.09647082 0.67324848 0.5365493 -0.11823534\n", + " 0.52454391 1.43031082 0.22692571 -0.87295854 -2.82106202 -0.27436094\n", + " 0.51321315 0.51606513 -1.56397437 -1.23442784 0.43472146 0.74179877\n", + " 0.73562128 1.00883303 -0.05851042 -1.19276526 1.16715641 -0.6401151\n", + " 0.81572064 1.0670422 -0.38749302 -1.67392967 -2.76996862 0.34870472\n", + " 1.3352494 -1.23489158 -1.70233983 0.62477383 0.14297718 0.23248733\n", + " 0.26787549 1.75420536 -1.22660411 -0.27597454 0.28073813 0.42879538\n", + " 1.64568937 -0.19205253 1.9311968 -0.26983672]\n" + ] + } + ], "source": [ - "n = 10\n", + "n = 100\n", "x = np.random.normal(size=n)\n", "print(x)" ] @@ -1005,7 +1029,7 @@ }, { "cell_type": "code", - "execution_count": 3, + "execution_count": 5, "id": "bd150398", "metadata": { "collapsed": false, @@ -1014,10 +1038,18 @@ "outputs_hidden": false } }, - "outputs": [], + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "[1 2 4]\n" + ] + } + ], "source": [ "import numpy as np\n", - "x = np.array([1, 2, 3])\n", + "x = np.array([1, 2, 4])\n", "print(x)" ] }, @@ -2038,7 +2070,7 @@ }, { "cell_type": "code", - "execution_count": 3, + "execution_count": 6, "id": "2f42295f", "metadata": { "collapsed": false, @@ -2049,181 +2081,14 @@ }, "outputs": [ { - "data": { - "text/html": [ - "
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"traceback": [ "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", - "\u001b[0;31mFileNotFoundError\u001b[0m Traceback (most recent call last)", - "File \u001b[0;32m~/miniforge3/envs/myenv/lib/python3.9/site-packages/matplotlib/style/core.py:137\u001b[0m, in \u001b[0;36muse\u001b[0;34m(style)\u001b[0m\n\u001b[1;32m 136\u001b[0m \u001b[38;5;28;01mtry\u001b[39;00m:\n\u001b[0;32m--> 137\u001b[0m style \u001b[38;5;241m=\u001b[39m \u001b[43m_rc_params_in_file\u001b[49m\u001b[43m(\u001b[49m\u001b[43mstyle\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 138\u001b[0m \u001b[38;5;28;01mexcept\u001b[39;00m \u001b[38;5;167;01mOSError\u001b[39;00m \u001b[38;5;28;01mas\u001b[39;00m err:\n", - "File \u001b[0;32m~/miniforge3/envs/myenv/lib/python3.9/site-packages/matplotlib/__init__.py:866\u001b[0m, in \u001b[0;36m_rc_params_in_file\u001b[0;34m(fname, transform, fail_on_error)\u001b[0m\n\u001b[1;32m 865\u001b[0m rc_temp \u001b[38;5;241m=\u001b[39m {}\n\u001b[0;32m--> 866\u001b[0m \u001b[38;5;28;01mwith\u001b[39;00m _open_file_or_url(fname) \u001b[38;5;28;01mas\u001b[39;00m fd:\n\u001b[1;32m 867\u001b[0m \u001b[38;5;28;01mtry\u001b[39;00m:\n", - "File \u001b[0;32m~/miniforge3/envs/myenv/lib/python3.9/contextlib.py:119\u001b[0m, in \u001b[0;36m_GeneratorContextManager.__enter__\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m 118\u001b[0m \u001b[38;5;28;01mtry\u001b[39;00m:\n\u001b[0;32m--> 119\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[38;5;28;43mnext\u001b[39;49m\u001b[43m(\u001b[49m\u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mgen\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 120\u001b[0m \u001b[38;5;28;01mexcept\u001b[39;00m \u001b[38;5;167;01mStopIteration\u001b[39;00m:\n", - "File \u001b[0;32m~/miniforge3/envs/myenv/lib/python3.9/site-packages/matplotlib/__init__.py:843\u001b[0m, in \u001b[0;36m_open_file_or_url\u001b[0;34m(fname)\u001b[0m\n\u001b[1;32m 842\u001b[0m fname \u001b[38;5;241m=\u001b[39m os\u001b[38;5;241m.\u001b[39mpath\u001b[38;5;241m.\u001b[39mexpanduser(fname)\n\u001b[0;32m--> 843\u001b[0m \u001b[38;5;28;01mwith\u001b[39;00m \u001b[38;5;28;43mopen\u001b[39;49m\u001b[43m(\u001b[49m\u001b[43mfname\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mencoding\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[38;5;124;43m'\u001b[39;49m\u001b[38;5;124;43mutf-8\u001b[39;49m\u001b[38;5;124;43m'\u001b[39;49m\u001b[43m)\u001b[49m \u001b[38;5;28;01mas\u001b[39;00m f:\n\u001b[1;32m 844\u001b[0m \u001b[38;5;28;01myield\u001b[39;00m f\n", - "\u001b[0;31mFileNotFoundError\u001b[0m: [Errno 2] No such file or directory: 'seaborn'", - "\nThe above exception was the direct cause of the following exception:\n", - "\u001b[0;31mOSError\u001b[0m Traceback (most recent call last)", - "Cell \u001b[0;32mIn[3], line 10\u001b[0m\n\u001b[1;32m 7\u001b[0m \u001b[38;5;28mprint\u001b[39m(df\u001b[38;5;241m.\u001b[39mdescribe())\n\u001b[1;32m 9\u001b[0m \u001b[38;5;28;01mfrom\u001b[39;00m \u001b[38;5;21;01mpylab\u001b[39;00m \u001b[38;5;28;01mimport\u001b[39;00m plt, mpl\n\u001b[0;32m---> 10\u001b[0m \u001b[43mplt\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mstyle\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43muse\u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;124;43m'\u001b[39;49m\u001b[38;5;124;43mseaborn\u001b[39;49m\u001b[38;5;124;43m'\u001b[39;49m\u001b[43m)\u001b[49m\n\u001b[1;32m 11\u001b[0m mpl\u001b[38;5;241m.\u001b[39mrcParams[\u001b[38;5;124m'\u001b[39m\u001b[38;5;124mfont.family\u001b[39m\u001b[38;5;124m'\u001b[39m] \u001b[38;5;241m=\u001b[39m \u001b[38;5;124m'\u001b[39m\u001b[38;5;124mserif\u001b[39m\u001b[38;5;124m'\u001b[39m\n\u001b[1;32m 13\u001b[0m df\u001b[38;5;241m.\u001b[39mcumsum()\u001b[38;5;241m.\u001b[39mplot(lw\u001b[38;5;241m=\u001b[39m\u001b[38;5;241m2.0\u001b[39m, figsize\u001b[38;5;241m=\u001b[39m(\u001b[38;5;241m10\u001b[39m,\u001b[38;5;241m6\u001b[39m))\n", - "File \u001b[0;32m~/miniforge3/envs/myenv/lib/python3.9/site-packages/matplotlib/style/core.py:139\u001b[0m, in \u001b[0;36muse\u001b[0;34m(style)\u001b[0m\n\u001b[1;32m 137\u001b[0m style \u001b[38;5;241m=\u001b[39m _rc_params_in_file(style)\n\u001b[1;32m 138\u001b[0m \u001b[38;5;28;01mexcept\u001b[39;00m \u001b[38;5;167;01mOSError\u001b[39;00m \u001b[38;5;28;01mas\u001b[39;00m err:\n\u001b[0;32m--> 139\u001b[0m \u001b[38;5;28;01mraise\u001b[39;00m \u001b[38;5;167;01mOSError\u001b[39;00m(\n\u001b[1;32m 140\u001b[0m \u001b[38;5;124mf\u001b[39m\u001b[38;5;124m\"\u001b[39m\u001b[38;5;132;01m{\u001b[39;00mstyle\u001b[38;5;132;01m!r}\u001b[39;00m\u001b[38;5;124m is not a valid package style, path of style \u001b[39m\u001b[38;5;124m\"\u001b[39m\n\u001b[1;32m 141\u001b[0m \u001b[38;5;124mf\u001b[39m\u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mfile, URL of style file, or library style name (library \u001b[39m\u001b[38;5;124m\"\u001b[39m\n\u001b[1;32m 142\u001b[0m \u001b[38;5;124mf\u001b[39m\u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mstyles are listed in `style.available`)\u001b[39m\u001b[38;5;124m\"\u001b[39m) \u001b[38;5;28;01mfrom\u001b[39;00m \u001b[38;5;21;01merr\u001b[39;00m\n\u001b[1;32m 143\u001b[0m filtered \u001b[38;5;241m=\u001b[39m {}\n\u001b[1;32m 144\u001b[0m \u001b[38;5;28;01mfor\u001b[39;00m k \u001b[38;5;129;01min\u001b[39;00m style: \u001b[38;5;66;03m# don't trigger RcParams.__getitem__('backend')\u001b[39;00m\n", - "\u001b[0;31mOSError\u001b[0m: 'seaborn' is not a valid package style, path of style file, URL of style file, or library style name (library styles are listed in `style.available`)" + "\u001b[0;31mNameError\u001b[0m Traceback (most recent call last)", + "Cell \u001b[0;32mIn[6], line 1\u001b[0m\n\u001b[0;32m----> 1\u001b[0m \u001b[43mdf\u001b[49m\u001b[38;5;241m.\u001b[39mcolumns \u001b[38;5;241m=\u001b[39m [\u001b[38;5;124m'\u001b[39m\u001b[38;5;124mFirst\u001b[39m\u001b[38;5;124m'\u001b[39m, \u001b[38;5;124m'\u001b[39m\u001b[38;5;124mSecond\u001b[39m\u001b[38;5;124m'\u001b[39m, \u001b[38;5;124m'\u001b[39m\u001b[38;5;124mThird\u001b[39m\u001b[38;5;124m'\u001b[39m, \u001b[38;5;124m'\u001b[39m\u001b[38;5;124mFourth\u001b[39m\u001b[38;5;124m'\u001b[39m, \u001b[38;5;124m'\u001b[39m\u001b[38;5;124mFifth\u001b[39m\u001b[38;5;124m'\u001b[39m]\n\u001b[1;32m 2\u001b[0m df\u001b[38;5;241m.\u001b[39mindex \u001b[38;5;241m=\u001b[39m np\u001b[38;5;241m.\u001b[39marange(\u001b[38;5;241m10\u001b[39m)\n\u001b[1;32m 4\u001b[0m display(df)\n", + "\u001b[0;31mNameError\u001b[0m: name 'df' is not defined" ] } ], @@ -2535,7 +2400,7 @@ }, { "cell_type": "code", - "execution_count": 25, + "execution_count": 7, "id": "0de5719f", "metadata": { "collapsed": false, @@ -2544,7 +2409,18 @@ "outputs_hidden": false } }, - "outputs": [], + "outputs": [ + { + "data": { + "image/png": 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", + "text/plain": [ + "
" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", @@ -2574,7 +2450,7 @@ "Depending on the parameter in front of the normal distribution, we may\n", "have a small or larger relative error. Try to play around with\n", "different training data sets and study (graphically) the value of the\n", - "relative error.\n", + "relative error $x$.\n", "\n", "As mentioned above, **Scikit-Learn** has an impressive functionality.\n", "We can for example extract the values of $\\alpha$ and $\\beta$ and\n", @@ -2587,7 +2463,7 @@ }, { "cell_type": "code", - "execution_count": 26, + "execution_count": 10, "id": "cde68f17", "metadata": { "collapsed": false, @@ -2596,7 +2472,32 @@ "outputs_hidden": false } }, - "outputs": [], + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "The intercept alpha: \n", + " [2.]\n", + "Coefficient beta : \n", + " [[5.]]\n", + "Mean squared error: 0.00\n", + "Variance score: 1.00\n", + "Mean squared log error: 0.00\n", + "Mean absolute error: 0.00\n" + ] + }, + { + "data": { + "image/png": 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", + "text/plain": [ + "
" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], "source": [ "import numpy as np \n", "import matplotlib.pyplot as plt \n", @@ -2604,7 +2505,7 @@ "from sklearn.metrics import mean_squared_error, r2_score, mean_squared_log_error, mean_absolute_error\n", "\n", "x = np.random.rand(100,1)\n", - "y = 2.0+ 5*x+0.5*np.random.randn(100,1)\n", + "y = 2.0+ 5*x#+0.5*np.random.randn(100,1)\n", "linreg = LinearRegression()\n", "linreg.fit(x,y)\n", "ypredict = linreg.predict(x)\n",