From a24b358dfea17213cca988df4eb460820a4eef81 Mon Sep 17 00:00:00 2001 From: Morten Hjorth-Jensen Date: Mon, 4 Aug 2025 12:22:25 +0200 Subject: [PATCH] Update README.md --- README.md | 2479 +---------------------------------------------------- 1 file changed, 13 insertions(+), 2466 deletions(-) diff --git a/README.md b/README.md index a3ce2a894..0b5a5fc57 100644 --- a/README.md +++ b/README.md @@ -96,19 +96,17 @@ possible. ## Instructor information - ## Teaching team Fall 2025 -* Karl Henrik Fredly, k.h.fredly@fys.uio.no * Morten Hjorth-Jensen, mhjensen@uio.no * _Phone_: +47-48257387 * _Office_: Department of Physics, room FØ470 * _Office hours_: *Anytime*! Individual or group office hours can be performed either in person or via zoom. Feel free to send an email for planning. -* Mia-Katrin Ose Kvalsund, m.k.o.kvalsund@fys.uio.no * Ida Torkjellsdatter Storehaug, i.t.storehaug@fys.uio.no - - -## Discord as discussion tool -* Our Discord channel is at https://discord.gg/mKq3zjxP +* Eivind Støland, eivinsto@@fys.uio.no +* Mia-Katrin Ose Kvalsund, m.k.o.kvalsund@fys.uio.no +* Karl Henrik Fredly, k.h.fredly@fys.uio.no +* Eir Eline Hørlyk, e.e.horlyk@fys.uio.no +* Brit S. Haanen, b.s.m.haanen@fys.uio.no ## Practicalities @@ -116,13 +114,11 @@ possible. 1. The sessions on Tuesdays and Wednesdays last four hours for each group (four in total) and will include lectures in a flipped mode (promoting active learning) and work on exercices and projects. The sessions will begin with lectures and questions and answers about the material to be covered every week. 2. There are four groups, Tuesdays 815am-12pm and 1215pm-4pm and Wednesdays 815am-12pm and 1215pm-4pm. Please sign up as soon as possible for one of the groups. Max capacity per group is 30-40 participants. -3. On Mondays we have a regular lecture. These lectures start at 1015am and end at 12pm. These lectures are recorded. +3. On Mondays we have a regular lecture. These lectures start at 215pm and ends at 4pm. These lectures are recorded. 4. Three projects which are graded and count 1/3 each of the final grade; 5. A selected number of weekly assignments. The weekly assignments can be handed in and for all assignments you can get an extra score of 20 points to the final grade. -6. The course is part of the CS Master of Science program, but is open to other bachelor and Master of Science students at the University of Oslo; -7. The course is offered as a so-called _cloned_ course, FYS-STK4155 at the Master of Science level and FYS-STK3155 as a senior undergraduate)course; -8. Videos of teaching material are available via the links at https://compphysics.github.io/MachineLearning/doc/web/course.html; -9. Weekly email with summary of activities will be mailed to all participants; +6. The course is offered as a so-called _cloned_ course, FYS-STK4155 at the Master of Science level and FYS-STK3155 as a senior undergraduate)course; +7. Weekly email with summary of activities will be mailed to all participants; ## Grading @@ -137,15 +133,13 @@ The final number of points is based on the average of all projects (including ev * 40-45 points: E * 0-39 points: F-failed +## Tentaive deadllines for projects -### In summary +* First project, due October 6, counts 1/3 of final grade +* Second project, due November 3, counts 1/3 of final grade +* Third project, due December 8, counts 1/3 of final grade +* Extra Credit (not mandatory), weekly exercise assignments, 10 in total (due Fridays), adds one point per exercises set. -| Activity | Fraction of total grade | -|------|-----| -| First project, _due October 6_ | 1/3 | -| Second project, _due November 3_ | 1/3 | -| Third project, _due December 8_ | 1/3 | -| Extra Credit (not mandatory), weekly exercise assignments, 10 in total (due Fridays)| 10% | @@ -281,2453 +275,6 @@ _General learning book on statistical analysis_: -* Ida Torkjellsdatter Storehaug, i.t.storehaug@fys.uio.no -* Eivind Støland, eivinsto@@fys.uio.no -* Mia-Katrin Ose Kvalsund, m.k.o.kvalsund@fys.uio.no -* Karl Henrik Fredly, k.h.fredly@fys.uio.no -* Eir Eline Hørlyk, e.e.horlyk@fys.uio.no -* Brit S. Haanen, b.s.m.haanen@fys.uio.no - - -!split -===== Deadlines for projects (tentative) ===== - - -o Project 1: October 6 (available September 1) graded with feedback) -o Project 2: November 3 (available October 6, graded with feedback) -o Project 3: December 8 (available November 3, graded with feedback) - - - -!bblock -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. -!eblock - -!split -===== Grading ===== - -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. - -The final number of points is based on the average of all projects and the grade follows the following table: - - * 92-100 points: A - * 77-91 points: B - * 58-76 points: C - * 46-57 points: D - * 40-45 points: E - * 0-39 points: F-failed - -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. - - - -!split -===== Reading material ===== - -!bblock -The lecture notes are collected as a jupyter-book at URL:"https://compphysics.github.io/MachineLearning/doc/LectureNotes/_build/html/intro.html". -The lecture notes can also be retrieved as a standard PDF file at URL:"https://compphysics.github.io/MachineLearning/doc/LectureNotes/MLbook.pdf". - -In addition to the lecture notes, we recommend the books of Rasckha et -al and Goodfellow et al. We will follow these texts closely and the -weekly reading assignments refer to these texts. The text by Hastie et -al is also widely used in the Machine Learning community. See next slide for link to textbooks. - -!eblock - -!split -===== Main textbooks ===== - -!bblock -* Goodfellow, Bengio, and Courville (GBC), Deep Learning URL:"https://www.deeplearningbook.org/" -* Sebastian Raschka, Yuxi Lie, and Vahid Mirjalili (RLM), Machine Learning with PyTorch and Scikit-Learn at URL:"https://www.packtpub.com/product/machine-learning-with-pytorch-and-scikit-learn/9781801819312", see also URL:"https://sebastianraschka.com/blog/2022/ml-pytorch-book.html" -!eblock - -The weekly reading suggestions are all from these two texts. The text by GBC can be accessed chapter by chapter from the abovementioned URL. -Each chapter of RLM gives access to the pertinent notebooks. These notebooks are highly recommended. - - -!split -===== Other popular texts ===== - -!bblock Other texts -* Christopher M. Bishop (CB), Pattern Recognition and Machine Learning -* "Hastie, Tibshirani, and Friedman (HTF), The Elements of Statistical Learning, Springer":"https://www.springer.com/gp/book/9780387848570". -* "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. -* "Kevin Murphy (KM), Probabilistic Machine Learning, an Introduction":"https://probml.github.io/pml-book/book1.html" -* David Foster (DF), Generative Deep Learning, URL:"https://www.oreilly.com/library/view/generative-deep-learning/9781098134174/" -* Babcock and Gavras (BG), Generative AI with Python and TensorFlow, URL:"https://github.com/PacktPublishing/Hands-On-Generative-AI-with-Python-and-TensorFlow-2" - -!eblock - -!split -===== Reading suggestions week 34 ===== - -!bblock -This week: Refresh linear algebra, GBC chapter 2. Install scikit-learn. See lecture notes for week 34 at https://compphysics.github.io/MachineLearning/doc/web/course.html (these notes). -!eblock - - -!split -===== Prerequisites ===== - -Basic knowledge in programming and mathematics, with an emphasis on -linear algebra. Knowledge of Python or/and C++ as programming -languages is strongly recommended and experience with Jupiter notebook -is recommended. Required courses are the equivalents to the University -of Oslo mathematics courses MAT1100, MAT1110, MAT1120 and at least one -of the corresponding computing and programming courses INF1000/INF1110 -or MAT-INF1100/MAT-INF1100L/BIOS1100/KJM-INF1100. Most universities -offer nowadays a basic programming course (often compulsory) where -Python is the recurring programming language. - - - - -!split -===== Topics covered in this course: Statistical analysis and optimization of data ===== - -The course has two central parts - -o Statistical analysis and optimization of data -o Machine learning - -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. - - -!split - -===== Statistical analysis and optimization of data ===== - -We plan to cover the following topics: -* Basic concepts, expectation values, variance, covariance, correlation functions and errors; -* Simpler models, binomial distribution, the Poisson distribution, simple and multivariate normal distributions; -* Central elements of Bayesian statistics and modeling; -* Gradient methods for data optimization; -* Monte Carlo methods, Markov chains, Gibbs sampling and Metropolis-Hastings sampling (tentative); -* Estimation of errors and resampling techniques such as the cross-validation, blocking, bootstrapping and jackknife methods; -* Principal Component Analysis (PCA) and its mathematical foundation; - - -!split -===== Machine Learning ===== - -* Pre deep-learning revolution (2008 approx) - * Linear Regression and Logistic Regression, classification and regression problems; - * Bayesian linear and logistic regression, kernel regression; - * Decisions trees, Random Forests, Bagging and Boosting methods; - * Support vector machines (only survey); - * Unsupervised learning and dimensionality reduction, from PCA to clustering; - - -!split -===== Deep learning methods ===== - -* Deep learning - * Neural networks and deep learning; - * Convolutional neural networks; - * Recurrent neural networks; - * Autoencoders - * Generative methods with an emphasis on Boltzmann Machines, Variational Autoencoders and Generalized Adversarial Networks(covered by FYS5429); -Hands-on demonstrations, exercises and projects aim at deepening your understanding of these topics. - -!split -===== Extremely useful tools, strongly recommended ===== - -!bblock and discussed at the lab sessions - * GIT for version control, and GitHub or GitLab as repositories, highly recommended. This will be discussed during the first exercise session - * Anaconda and other Python environments, see intro slides and links to programming resources at URL:"https://computationalscienceuio.github.io/RefreshProgrammingSkills/intro.html" -!eblock - - - - - -!split -===== Other courses on Data science and Machine Learning at UiO ===== - -* "FYS5419 Quantum Computing and Quantum Machine Learning":"https://www.uio.no/studier/emner/matnat/fys/FYS5419/index-eng.html" -* "FYS5429 Advanced Machine Learning for the Physical Sciences":"https://www.uio.no/studier/emner/matnat/fys/FYS5429/index-eng.html" -* "STK2100 Machine learning and statistical methods for prediction and classification":"http://www.uio.no/studier/emner/matnat/math/STK2100/index-eng.html". -* "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. -* "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. -* "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. - - -!split -===== Other courses on Data science and Machine Learning at UiO, contn ===== - -!bblock -* "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. -* "IN3310/4310 Deep Learnig for Image Analysis":"https://www.uio.no/studier/emner/matnat/ifi/IN4310/index.html" -* "STK4051 Computational Statistics":"https://www.uio.no/studier/emner/matnat/math/STK4051/index-eng.html" -* "STK4021 Applied Bayesian Analysis and Numerical Methods":"https://www.uio.no/studier/emner/matnat/math/STK4021/index-eng.html" -!eblock - - - - -!split -===== Learning outcomes ===== - -!bblock -* Learn about basic data analysis, statistical analysis, Bayesian statistics, Monte Carlo sampling, data optimization and machine learning; -* Be capable of extending the acquired knowledge to other systems and cases; -* Have an understanding of central algorithms used in data analysis and machine learning; -* Understand linear methods for regression and classification, from ordinary least squares, via Lasso and Ridge to Logistic regression; -* Learn about neural networks and deep learning methods for supervised and unsupervised learning. Emphasis on feed forward neural networks, convolutional and recurrent neural networks; -* Learn about about decision trees, random forests, bagging and boosting methods; -* Learn about support vector machines and kernel transformations; -* Reduction of data sets, from PCA to clustering; -* Generative models -* 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. - -!eblock - - - -!split -===== Types of Machine Learning ===== - - -The approaches to machine learning are many, but are often split into -two main categories. In *supervised learning* we know the answer to a -problem, and let the computer deduce the logic behind it. On the other -hand, *unsupervised learning* is a method for finding patterns and -relationship in data sets without any prior knowledge of the system. -Some authours also operate with a third category, namely -*reinforcement learning*. This is a paradigm of learning inspired by -behavioral psychology, where learning is achieved by trial-and-error, -solely from rewards and punishment. - -Another way to categorize machine learning tasks is to consider the -desired output of a system. Some of the most common tasks are: - - * Classification: Outputs are divided into two or more classes. The goal is to produce a model that assigns inputs into one of these classes. An example is to identify digits based on pictures of hand-written ones. Classification is typically supervised learning. - - * 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. - - * Clustering: Data are divided into groups with certain common traits, without knowing the different groups beforehand. It is thus a form of unsupervised learning. - - -!split -===== Essential elements of ML ===== - -The methods we cover have three main topics in common, irrespective of -whether we deal with supervised or unsupervised learning. -!bpop -* 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. - -* 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. - -* 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. -!epop - - - - -!split -===== An optimization/minimization problem ===== - -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_. - -!split -===== A Frequentist approach to data analysis ===== - -When you hear phrases like _predictions and estimations_ and -_correlations and causations_, what do you think of? May be you think -of the difference between classifying new data points and generating -new data points. -Or perhaps you consider that correlations represent some kind of symmetric statements like -if $A$ is correlated with $B$, then $B$ is correlated with -$A$. Causation on the other hand is directional, that is if $A$ causes $B$, $B$ does not -necessarily cause $A$. - -These concepts are in some sense the difference between machine -learning and statistics. In machine learning and prediction based -tasks, we are often interested in developing algorithms that are -capable of learning patterns from given data in an automated fashion, -and then using these learned patterns to make predictions or -assessments of newly given data. In many cases, our primary concern -is the quality of the predictions or assessments, and we are less -concerned about the underlying patterns that were learned in order -to make these predictions. - -In machine learning we normally use "a so-called frequentist approach":"https://en.wikipedia.org/wiki/Frequentist_inference", -where the aim is to make predictions and find correlations. We focus -less on for example extracting a probability distribution function (PDF). The PDF can be -used in turn to make estimations and find causations such as given $A$ -what is the likelihood of finding $B$. - - -!split -===== What is a good model? ===== - -In science and engineering we often end up in situations where we want to infer (or learn) a -quantitative model $M$ for a given set of sample points $\bm{X} \in [x_1, x_2,\dots x_N]$. - -As we will see repeatedely in these lectures, we could try to fit these data points to a model given by a -straight line, or if we wish to be more sophisticated to a more complex -function. - -The reason for inferring such a model is that it -serves many useful purposes. On the one hand, the model can reveal information -encoded in the data or underlying mechanisms from which the data were generated. For instance, we could discover important -corelations that relate interesting physics interpretations. - -In addition, it can simplify the representation of the given data set and help -us in making predictions about future data samples. - -A first important consideration to keep in mind is that inferring the *correct* model -for a given data set is an elusive, if not impossible, task. The fundamental difficulty -is that if we are not specific about what we mean by a *correct* model, there -could easily be many different models that fit the given data set *equally well*. - - -!split -===== What is a good model? Can we define it? ===== - - -The central question is this: what leads us to say that a model is correct or -optimal for a given data set? To make the model inference problem well posed, i.e., -to guarantee that there is a unique optimal model for the given data, we need to -impose additional assumptions or restrictions on the class of models considered. To -this end, we should not be looking for just any model that can describe the data. -Instead, we should look for a _model_ $M$ that is the best among a restricted class -of models. In addition, to make the model inference problem computationally -tractable, we need to specify how restricted the class of models needs to be. A -common strategy is to start -with the simplest possible class of models that is just necessary to describe the data -or solve the problem at hand. More precisely, the model class should be rich enough -to contain at least one model that can fit the data to a desired accuracy and yet be -restricted enough that it is relatively simple to find the best model for the given data. - -Thus, the most popular strategy is to start from the -simplest class of models and increase the complexity of the models only when the -simpler models become inadequate. For instance, if we work with a regression problem to fit a set of sample points, one -may first try the simplest class of models, namely linear models, followed obviously by more complex models. - -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. - - - - - - - - -!split -===== Software and needed installations ===== - -We will make extensive use of Python as programming language and its -myriad of available libraries. You will find -Jupyter notebooks invaluable in your work. You can run _R_ -codes in the Jupyter/IPython notebooks, with the immediate benefit of -visualizing your data. You can also use compiled languages like C++, -Rust, Julia, Fortran etc if you prefer. The focus in these lectures will be -on Python. - - -If you have Python installed (we strongly recommend Python3) and you feel -pretty familiar with installing different packages, we recommend that -you install the following Python packages via _pip_ as - -o pip install numpy scipy matplotlib ipython scikit-learn mglearn sympy pandas pillow - -For Python3, replace _pip_ with _pip3_. - -For OSX users we recommend, after having installed Xcode, to -install _brew_. Brew allows for a seamless installation of additional -software via for example - -o brew install python3 - -For Linux users, with its variety of distributions like for example the widely popular Ubuntu distribution, -you can use _pip_ as well and simply install Python as - -o sudo apt-get install python3 (or python for pyhton2.7) - -etc etc. - - -!split -===== Python installers ===== - -If you don't want to perform these operations separately and venture -into the hassle of exploring how to set up dependencies and paths, we -recommend two widely used distrubutions which set up all relevant -dependencies for Python, namely - -* "Anaconda":"https://docs.anaconda.com/", - -which is an open source -distribution of the Python and R programming languages for large-scale -data processing, predictive analytics, and scientific computing, that -aims to simplify package management and deployment. Package versions -are managed by the package management system _conda_. - -* "Enthought canopy":"https://www.enthought.com/product/canopy/" - -is a Python -distribution for scientific and analytic computing distribution and -analysis environment, available for free and under a commercial -license. - -Furthermore, "Google's Colab":"https://colab.research.google.com/notebooks/welcome.ipynb" is a free Jupyter notebook environment that requires -no setup and runs entirely in the cloud. Try it out! - -!split -===== Useful Python libraries ===== -Here we list several useful Python libraries we strongly recommend (if you use anaconda many of these are already there) - -* "NumPy":"https://www.numpy.org/" is a highly popular library for large, multi-dimensional arrays and matrices, along with a large collection of high-level mathematical functions to operate on these arrays -* "The pandas":"https://pandas.pydata.org/" library provides high-performance, easy-to-use data structures and data analysis tools -* "Xarray":"http://xarray.pydata.org/en/stable/" is a Python package that makes working with labelled multi-dimensional arrays simple, efficient, and fun! -* "Scipy":"https://www.scipy.org/" (pronounced “Sigh Pie”) is a Python-based ecosystem of open-source software for mathematics, science, and engineering. -* "Matplotlib":"https://matplotlib.org/" is a Python 2D plotting library which produces publication quality figures in a variety of hardcopy formats and interactive environments across platforms. -* "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 -* "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. -* "SymPy":"https://www.sympy.org/en/index.html" is a Python library for symbolic mathematics. -* "scikit-learn":"https://scikit-learn.org/stable/" has simple and efficient tools for machine learning, data mining and data analysis -* "TensorFlow":"https://www.tensorflow.org/" is a Python library for fast numerical computing created and released by Google -* "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 -* "Pytorch":"https://pytorch.org/", highly recommened -* "Theano":"https://pypi.org/project/Theano/" and many other - -!split -===== Installing R, C++, cython or Julia ===== - -You will also find it convenient to utilize _R_. We will mainly -use Python during our lectures and in various projects and exercises. -Those of you -already familiar with _R_ should feel free to continue using _R_, keeping -however an eye on the parallel Python set ups. Similarly, if you are a -Python afecionado, feel free to explore _R_ as well. Jupyter(Julia, Python and R) /Ipython -notebook allows you to run _R_ codes and _Julia_ codes interactively in your -browser. The software library _R_ is really tailored for statistical data analysis -and allows for an easy usage of the tools and algorithms we will discuss in these -lectures. - -To install _R_ with Jupyter notebook -"follow the link here":"https://mpacer.org/maths/r-kernel-for-ipython-notebook" - - - -!split -===== Installing R, C++, cython, Numba etc ===== - - -For the C++ aficionados, Jupyter/IPython notebook allows you also to -install C++ and run codes written in this language interactively in -the browser. Since we will emphasize writing many of the algorithms -yourself, you can thus opt for either Python or C++ (or Fortran or other compiled languages) as programming -languages. - -To add more entropy, _cython_ can also be used when running your -notebooks. It means that Python with the jupyter notebook -setup allows you to integrate widely popular softwares and tools for -scientific computing. Similarly, the -"Numba Python package":"https://numba.pydata.org/" delivers increased performance -capabilities with minimal rewrites of your codes. With its -versatility, including symbolic operations, Python offers a unique -computational environment. Your jupyter notebook can easily be -converted into a nicely rendered _PDF_ file or a Latex file for -further processing. For example, convert to latex as - -!bc -pycod jupyter nbconvert filename.ipynb --to latex -!ec - -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. - -Finally, we recommend strongly using Autograd or JAX for automatic differentiation. - - -!split -===== Numpy examples and Important Matrix and vector handling packages ===== - -There are several central software libraries for linear algebra and eigenvalue problems. Several of the more -popular ones have been wrapped into ofter software packages like those from the widely used text _Numerical Recipes_. The original source codes in many of the available packages are often taken from the widely used -software package LAPACK, which follows two other popular packages -developed in the 1970s, namely EISPACK and LINPACK. We describe them shortly here. - - * LINPACK: package for linear equations and least square problems. - * LAPACK:package for solving symmetric, unsymmetric and generalized eigenvalue problems. From LAPACK's website URL: "http://www.netlib.org" it is possible to download for free all source codes from this library. Both C/C++ and Fortran versions are available. - * 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 URL: "http://www.netlib.org". - - -!split -===== Numpy and arrays ===== -"Numpy":"http://www.numpy.org/" provides an easy way to handle arrays in Python. The standard way to import this library is as - -!bc pycod -import numpy as np -!ec -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, -!bc pycod -n = 10 -x = np.random.normal(size=n) -print(x) -!ec -We defined a vector $x$ with $n=10$ elements with its values given by the Normal distribution $N(0,1)$. -Another alternative is to declare a vector as follows -!bc pycod -import numpy as np -x = np.array([1, 2, 3]) -print(x) -!ec -Here we have defined a vector with three elements, with $x_0=1$, $x_1=2$ and $x_2=3$. Note that both Python and C++ -start numbering array elements from $0$ and on. This means that a vector with $n$ elements has a sequence of entities $x_0, x_1, x_2, \dots, x_{n-1}$. We could also let (recommended) Numpy to compute the logarithms of a specific array as -!bc pycod -import numpy as np -x = np.log(np.array([4, 7, 8])) -print(x) -!ec - -In the last example we used Numpy's unary function $np.log$. This function is -highly tuned to compute array elements since the code is vectorized -and does not require looping. We normaly recommend that you use the -Numpy intrinsic functions instead of the corresponding _log_ function -from Python's _math_ module. The looping is done explicitely by the -_np.log_ function. The alternative, and slower way to compute the -logarithms of a vector would be to write - -!bc pycod -import numpy as np -from math import log -x = np.array([4, 7, 8]) -for i in range(0, len(x)): - x[i] = log(x[i]) -print(x) -!ec -We note that our code is much longer already and we need to import the _log_ function from the _math_ module. -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 -!bc pycod -import numpy as np -x = np.log(np.array([4, 7, 8], dtype = np.float64)) -print(x) -!ec -or simply write them as double precision numbers (Python uses 64 bits as default for floating point type variables), that is -!bc pycod -import numpy as np -x = np.log(np.array([4.0, 7.0, 8.0])) -print(x) -!ec -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 -!bc pycod -import numpy as np -x = np.log(np.array([4.0, 7.0, 8.0])) -print(x.itemsize) -!ec - -!split -===== Matrices in Python ===== - -Having defined vectors, we are now ready to try out matrices. We can -define a $3 \times 3 $ real matrix $\bm{A}$ as (recall that we user -lowercase letters for vectors and uppercase letters for matrices) - -!bc pycod -import numpy as np -A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ])) -print(A) -!ec -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 -!bc pycod -import numpy as np -A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ])) -# print the first column, row-major order and elements start with 0 -print(A[:,0]) -!ec -We can continue this was by printing out other columns or rows. The example here prints out the second column -!bc pycod -import numpy as np -A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ])) -# print the first column, row-major order and elements start with 0 -print(A[1,:]) -!ec -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 -!bc pycod -import numpy as np -n = 10 -# define a matrix of dimension 10 x 10 and set all elements to zero -A = np.zeros( (n, n) ) -print(A) -!ec -or initializing all elements to -!bc pycod -import numpy as np -n = 10 -# define a matrix of dimension 10 x 10 and set all elements to one -A = np.ones( (n, n) ) -print(A) -!ec -or as unitarily distributed random numbers (see the material on random number generators in the statistics part) -!bc pycod -import numpy as np -n = 10 -# define a matrix of dimension 10 x 10 and set all elements to random numbers with x \in [0, 1] -A = np.random.rand(n, n) -print(A) -!ec - -As we will see throughout these lectures, there are several extremely useful functionalities in Numpy. -As an example, consider the discussion of the covariance matrix. Suppose we have defined three vectors -$\bm{x}, \bm{y}, \bm{z}$ with $n$ elements each. The covariance matrix is defined as -!bt -\[ -\bm{\Sigma} = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\ - \sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\ - \sigma_{zx} & \sigma_{zy} & \sigma_{zz} - \end{bmatrix}, -\] -!et -where for example -!bt -\[ -\sigma_{xy} =\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})(y_i- \overline{y}). -\] -!et -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. -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 $\bm{W}$ -!bt -\[ -\bm{W} = \begin{bmatrix} x_0 & x_1 & x_2 & \dots & x_{n-2} & x_{n-1} \\ - y_0 & y_1 & y_2 & \dots & y_{n-2} & y_{n-1} \\ - z_0 & z_1 & z_2 & \dots & z_{n-2} & z_{n-1} \\ - \end{bmatrix}, -\] -!et - -which in turn is converted into into the $3\times 3$ covariance matrix -$\bm{\Sigma}$ via the Numpy function _np.cov()_. We note that we can also calculate -the mean value of each set of samples $\bm{x}$ etc using the Numpy -function _np.mean(x)_. We can also extract the eigenvalues of the -covariance matrix through the _np.linalg.eig()_ function. - -!bc pycod -# Importing various packages -import numpy as np - -n = 100 -x = np.random.normal(size=n) -print(np.mean(x)) -y = 4+3*x+np.random.normal(size=n) -print(np.mean(y)) -z = x**3+np.random.normal(size=n) -print(np.mean(z)) -W = np.vstack((x, y, z)) -Sigma = np.cov(W) -print(Sigma) -Eigvals, Eigvecs = np.linalg.eig(Sigma) -print(Eigvals) -!ec - - -!bc pycod -import numpy as np -import matplotlib.pyplot as plt -from scipy import sparse -eye = np.eye(4) -print(eye) -sparse_mtx = sparse.csr_matrix(eye) -print(sparse_mtx) -x = np.linspace(-10,10,100) -y = np.sin(x) -plt.plot(x,y,marker='x') -plt.show() -!ec - -!split -===== Meet the Pandas ===== - - -FIGURE: [fig/pandas.jpg, width=600 frac=0.8] - -Another useful Python package is -"pandas":"https://pandas.pydata.org/", which is an open source library -providing high-performance, easy-to-use data structures and data -analysis tools for Python. _pandas_ stands for panel data, a term borrowed from econometrics and is an efficient library for data analysis with an emphasis on tabular data. -_pandas_ has two major classes, the _DataFrame_ class with two-dimensional data objects and tabular data organized in columns and the class _Series_ with a focus on one-dimensional data objects. Both classes allow you to index data easily as we will see in the examples below. -_pandas_ allows you also to perform mathematical operations on the data, spanning from simple reshapings of vectors and matrices to statistical operations. - -The following simple example shows how we can, in an easy way make tables of our data. Here we define a data set which includes names, place of birth and date of birth, and displays the data in an easy to read way. We will see repeated use of _pandas_, in particular in connection with classification of data. - -!bc pycod -import pandas as pd -from IPython.display import display -data = {'First Name': ["Frodo", "Bilbo", "Aragorn II", "Samwise"], - 'Last Name': ["Baggins", "Baggins","Elessar","Gamgee"], - 'Place of birth': ["Shire", "Shire", "Eriador", "Shire"], - 'Date of Birth T.A.': [2968, 2890, 2931, 2980] - } -data_pandas = pd.DataFrame(data) -display(data_pandas) -!ec - -In the above we have imported _pandas_ with the shorthand _pd_, the latter has become the standard way we import _pandas_. We make then a list of various variables -and reorganize the aboves lists into a _DataFrame_ and then print out a neat table with specific column labels as *Name*, *place of birth* and *date of birth*. -Displaying these results, we see that the indices are given by the default numbers from zero to three. -_pandas_ is extremely flexible and we can easily change the above indices by defining a new type of indexing as -!bc pycod -data_pandas = pd.DataFrame(data,index=['Frodo','Bilbo','Aragorn','Sam']) -display(data_pandas) -!ec -Thereafter we display the content of the row which begins with the index _Aragorn_ -!bc pycod -display(data_pandas.loc['Aragorn']) -!ec - -We can easily append data to this, for example -!bc pycod -new_hobbit = {'First Name': ["Peregrin"], - 'Last Name': ["Took"], - 'Place of birth': ["Shire"], - 'Date of Birth T.A.': [2990] - } -data_pandas=data_pandas.append(pd.DataFrame(new_hobbit, index=['Pippin'])) -display(data_pandas) -!ec - - -Here are other examples where we use the _DataFrame_ functionality to handle arrays, now with more interesting features for us, namely numbers. We set up a matrix -of dimensionality $10\times 5$ and compute the mean value and standard deviation of each column. Similarly, we can perform mathematial operations like squaring the matrix elements and many other operations. -!bc pycod -import numpy as np -import pandas as pd -from IPython.display import display -np.random.seed(100) -# setting up a 10 x 5 matrix -rows = 10 -cols = 5 -a = np.random.randn(rows,cols) -df = pd.DataFrame(a) -display(df) -print(df.mean()) -print(df.std()) -display(df**2) -!ec - -Thereafter we can select specific columns only and plot final results -!bc pycod -df.columns = ['First', 'Second', 'Third', 'Fourth', 'Fifth'] -df.index = np.arange(10) - -display(df) -print(df['Second'].mean() ) -print(df.info()) -print(df.describe()) - -from pylab import plt, mpl -plt.style.use('seaborn') -mpl.rcParams['font.family'] = 'serif' - -df.cumsum().plot(lw=2.0, figsize=(10,6)) -plt.show() - - -df.plot.bar(figsize=(10,6), rot=15) -plt.show() -!ec -We can produce a $4\times 4$ matrix -!bc pycod -b = np.arange(16).reshape((4,4)) -print(b) -df1 = pd.DataFrame(b) -print(df1) -!ec -and many other operations. - -The _Series_ class is another important class included in -_pandas_. You can view it as a specialization of _DataFrame_ but where -we have just a single column of data. It shares many of the same features as _DataFrame_. As with _DataFrame_, -most operations are vectorized, achieving thereby a high performance when dealing with computations of arrays, in particular labeled arrays. -As we will see below it leads also to a very concice code close to the mathematical operations we may be interested in. -For multidimensional arrays, we recommend strongly "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_. - -!split -===== Pandas AI ===== - -Try out "Pandas AI":"https://pandas-ai.com/" - - -!split -=== Simple linear regression model using _scikit-learn_ === - -We start with perhaps our simplest possible example, using _Scikit-Learn_ to perform linear regression analysis on a data set produced by us. - -What follows is a simple Python code where we have defined a function -$y$ in terms of the variable $x$. Both are defined as vectors with $100$ entries. -The numbers in the vector $\bm{x}$ are given -by random numbers generated with a uniform distribution with entries -$x_i \in [0,1]$ (more about probability distribution functions -later). These values are then used to define a function $y(x)$ -(tabulated again as a vector) with a linear dependence on $x$ plus a -random noise added via the normal distribution. - - -The Numpy functions are imported used the _import numpy as np_ -statement and the random number generator for the uniform distribution -is called using the function _np.random.rand()_, where we specificy -that we want $100$ random variables. Using Numpy we define -automatically an array with the specified number of elements, $100$ in -our case. With the Numpy function _randn()_ we can compute random -numbers with the normal distribution (mean value $\mu$ equal to zero and -variance $\sigma^2$ set to one) and produce the values of $y$ assuming a linear -dependence as function of $x$ - -!bt -\[ -y = 2x+N(0,1), -\] -!et - -where $N(0,1)$ represents random numbers generated by the normal -distribution. From _Scikit-Learn_ we import then the -_LinearRegression_ functionality and make a prediction $\tilde{y} = -\alpha + \beta x$ using the function _fit(x,y)_. We call the set of -data $(\bm{x},\bm{y})$ for our training data. The Python package -_scikit-learn_ has also a functionality which extracts the above -fitting parameters $\alpha$ and $\beta$ (see below). Later we will -distinguish between training data and test data. - -For plotting we use the Python package -"matplotlib":"https://matplotlib.org/" which produces publication -quality figures. Feel free to explore the extensive -"gallery":"https://matplotlib.org/gallery/index.html" of examples. In -this example we plot our original values of $x$ and $y$ as well as the -prediction _ypredict_ ($\tilde{y}$), which attempts at fitting our -data with a straight line. - -The Python code follows here. -!bc pycod -# Importing various packages -import numpy as np -import matplotlib.pyplot as plt -from sklearn.linear_model import LinearRegression - -x = np.random.rand(100,1) -y = 2*x+np.random.randn(100,1) -linreg = LinearRegression() -linreg.fit(x,y) -xnew = np.array([[0],[1]]) -ypredict = linreg.predict(xnew) - -plt.plot(xnew, ypredict, "r-") -plt.plot(x, y ,'ro') -plt.axis([0,1.0,0, 5.0]) -plt.xlabel(r'$x$') -plt.ylabel(r'$y$') -plt.title(r'Simple Linear Regression') -plt.show() -!ec - -This example serves several aims. It allows us to demonstrate several -aspects of data analysis and later machine learning algorithms. The -immediate visualization shows that our linear fit is not -impressive. It goes through the data points, but there are many -outliers which are not reproduced by our linear regression. We could -now play around with this small program and change for example the -factor in front of $x$ and the normal distribution. Try to change the -function $y$ to - -!bt -\[ -y = 10x+0.01 \times N(0,1), -\] -!et - -where $x$ is defined as before. Does the fit look better? Indeed, by -reducing the role of the noise given by the normal distribution we see immediately that -our linear prediction seemingly reproduces better the training -set. However, this testing 'by the eye' is obviouly not satisfactory in the -long run. Here we have only defined the training data and our model, and -have not discussed a more rigorous approach to the _cost_ function. - -We need more rigorous criteria in defining whether we have succeeded or -not in modeling our training data. You will be surprised to see that -many scientists seldomly venture beyond this 'by the eye' approach. A -standard approach for the *cost* function is the so-called $\chi^2$ -function (a variant of the mean-squared error (MSE)) - -!bt -\[ \chi^2 = \frac{1}{n} -\sum_{i=0}^{n-1}\frac{(y_i-\tilde{y}_i)^2}{\sigma_i^2}, -\] -!et - -where $\sigma_i^2$ is the variance (to be defined later) of the entry -$y_i$. We may not know the explicit value of $\sigma_i^2$, it serves -however the aim of scaling the equations and make the cost function -dimensionless. - -Minimizing the cost function is a central aspect of -our discussions to come. Finding its minima as function of the model -parameters ($\alpha$ and $\beta$ in our case) will be a recurring -theme in these series of lectures. Essentially all machine learning -algorithms we will discuss center around the minimization of the -chosen cost function. This depends in turn on our specific -model for describing the data, a typical situation in supervised -learning. Automatizing the search for the minima of the cost function is a -central ingredient in all algorithms. Typical methods which are -employed are various variants of _gradient_ methods. These will be -discussed in more detail later. Again, you'll be surprised to hear that -many practitioners minimize the above function ''by the eye', popularly dubbed as -'chi by the eye'. That is, change a parameter and see (visually and numerically) that -the $\chi^2$ function becomes smaller. - -There are many ways to define the cost function. A simpler approach is to look at the relative difference between the training data and the predicted data, that is we define -the relative error (why would we prefer the MSE instead of the relative error?) as - -!bt -\[ -\epsilon_{\mathrm{relative}}= \frac{\vert \bm{y} -\bm{\tilde{y}}\vert}{\vert \bm{y}\vert}. -\] -!et - -The squared cost function results in an arithmetic mean-unbiased -estimator, and the absolute-value cost function results in a -median-unbiased estimator (in the one-dimensional case, and a -geometric median-unbiased estimator for the multi-dimensional -case). The squared cost function has the disadvantage that it has the tendency -to be dominated by outliers. - -We can modify easily the above Python code and plot the relative error instead -!bc pycod -import numpy as np -import matplotlib.pyplot as plt -from sklearn.linear_model import LinearRegression - -x = np.random.rand(100,1) -y = 5*x+0.01*np.random.randn(100,1) -linreg = LinearRegression() -linreg.fit(x,y) -ypredict = linreg.predict(x) - -plt.plot(x, np.abs(ypredict-y)/abs(y), "ro") -plt.axis([0,1.0,0.0, 0.5]) -plt.xlabel(r'$x$') -plt.ylabel(r'$\epsilon_{\mathrm{relative}}$') -plt.title(r'Relative error') -plt.show() -!ec - -Depending on the parameter in front of the normal distribution, we may -have a small or larger relative error. Try to play around with -different training data sets and study (graphically) the value of the -relative error. - -As mentioned above, _Scikit-Learn_ has an impressive functionality. -We can for example extract the values of $\alpha$ and $\beta$ and -their error estimates, or the variance and standard deviation and many -other properties from the statistical data analysis. - -Here we show an -example of the functionality of _Scikit-Learn_. -!bc pycod -import numpy as np -import matplotlib.pyplot as plt -from sklearn.linear_model import LinearRegression -from sklearn.metrics import mean_squared_error, r2_score, mean_squared_log_error, mean_absolute_error - -x = np.random.rand(100,1) -y = 2.0+ 5*x+0.5*np.random.randn(100,1) -linreg = LinearRegression() -linreg.fit(x,y) -ypredict = linreg.predict(x) -print('The intercept alpha: \n', linreg.intercept_) -print('Coefficient beta : \n', linreg.coef_) -# The mean squared error -print("Mean squared error: %.2f" % mean_squared_error(y, ypredict)) -# Explained variance score: 1 is perfect prediction -print('Variance score: %.2f' % r2_score(y, ypredict)) -# Mean squared log error -print('Mean squared log error: %.2f' % mean_squared_log_error(y, ypredict) ) -# Mean absolute error -print('Mean absolute error: %.2f' % mean_absolute_error(y, ypredict)) -plt.plot(x, ypredict, "r-") -plt.plot(x, y ,'ro') -plt.axis([0.0,1.0,1.5, 7.0]) -plt.xlabel(r'$x$') -plt.ylabel(r'$y$') -plt.title(r'Linear Regression fit ') -plt.show() - -!ec -The function _coef_ gives us the parameter $\beta$ of our fit while _intercept_ yields -$\alpha$. Depending on the constant in front of the normal distribution, we get values near or far from $\alpha =2$ and $\beta =5$. Try to play around with different parameters in front of the normal distribution. The function _meansquarederror_ gives us the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error or loss defined as -!bt -\[ MSE(\bm{y},\bm{\tilde{y}}) = \frac{1}{n} -\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2, -\] -!et - -The smaller the value, the better the fit. Ideally we would like to -have an MSE equal zero. The attentive reader has probably recognized -this function as being similar to the $\chi^2$ function defined above. - -The _r2score_ function computes $R^2$, the coefficient of -determination. It provides a measure of how well future samples are -likely to be predicted by the model. Best possible score is 1.0 and it -can be negative (because the model can be arbitrarily worse). A -constant model that always predicts the expected value of $\bm{y}$, -disregarding the input features, would get a $R^2$ score of $0.0$. - -If $\tilde{\bm{y}}_i$ 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 -!bt -\[ -R^2(\bm{y}, \tilde{\bm{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2}, -\] -!et -where we have defined the mean value of $\bm{y}$ as -!bt -\[ -\bar{y} = \frac{1}{n} \sum_{i=0}^{n - 1} y_i. -\] -!et -Another quantity taht we will meet again in our discussions of regression analysis is - the mean absolute error (MAE), a risk metric corresponding to the expected value of the absolute error loss or what we call the $l1$-norm loss. In our discussion above we presented the relative error. -The MAE is defined as follows -!bt -\[ -\text{MAE}(\bm{y}, \bm{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n-1} \left| y_i - \tilde{y}_i \right|. -\] -!et -We present the -squared logarithmic (quadratic) error -!bt -\[ -\text{MSLE}(\bm{y}, \bm{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n - 1} (\log_e (1 + y_i) - \log_e (1 + \tilde{y}_i) )^2, -\] -!et - -where $\log_e (x)$ stands for the natural logarithm of $x$. This error -estimate is best to use when targets having exponential growth, such -as population counts, average sales of a commodity over a span of -years etc. - - -Finally, another cost function is the Huber cost function used in robust regression. - -The rationale behind this possible cost function is its reduced -sensitivity to outliers in the data set. In our discussions on -dimensionality reduction and normalization of data we will meet other -ways of dealing with outliers. - -The Huber cost function is defined as -!bt -\[ -H_{\delta}(\bm{a})=\left\{\begin{array}{cc}\frac{1}{2} \bm{a}^{2}& \text{for }|\bm{a}|\leq \delta\\ \delta (|\bm{a}|-\frac{1}{2}\delta ),&\text{otherwise}.\end{array}\right. -\] -!et -Here $\bm{a}=\bm{y} - \bm{\tilde{y}}$. - - -We will discuss in more detail these and other functions in the -various lectures and lab sessions. - - - - -=== To our real data: nuclear binding energies. Brief reminder on masses and binding energies === - -Let us now dive into nuclear physics and remind ourselves briefly about some basic features about binding -energies. A basic quantity which can be measured for the ground -states of nuclei is the atomic mass $M(N, Z)$ of the neutral atom with -atomic mass number $A$ and charge $Z$. The number of neutrons is $N$. There are indeed several sophisticated experiments worldwide which allow us to measure this quantity to high precision (parts per million even). - -Atomic masses are usually tabulated in terms of the mass excess defined by -!bt -\[ -\Delta M(N, Z) = M(N, Z) - uA, -\] -!et -where $u$ is the Atomic Mass Unit -!bt -\[ -u = M(^{12}\mathrm{C})/12 = 931.4940954(57) \hspace{0.1cm} \mathrm{MeV}/c^2. -\] -!et -The nucleon masses are -!bt -\[ -m_p = 1.00727646693(9)u, -\] -!et -and -!bt -\[ -m_n = 939.56536(8)\hspace{0.1cm} \mathrm{MeV}/c^2 = 1.0086649156(6)u. -\] -!et - -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" -there are data on masses and decays of 3437 nuclei. - -The nuclear binding energy is defined as the energy required to break -up a given nucleus into its constituent parts of $N$ neutrons and $Z$ -protons. In terms of the atomic masses $M(N, Z)$ the binding energy is -defined by - - -!bt -\[ -BE(N, Z) = ZM_H c^2 + Nm_n c^2 - M(N, Z)c^2 , -\] -!et -where $M_H$ is the mass of the hydrogen atom and $m_n$ is the mass of the neutron. -In terms of the mass excess the binding energy is given by -!bt -\[ -BE(N, Z) = Z\Delta_H c^2 + N\Delta_n c^2 -\Delta(N, Z)c^2 , -\] -!et -where $\Delta_H c^2 = 7.2890$ MeV and $\Delta_n c^2 = 8.0713$ MeV. - - -A popular and physically intuitive model which can be used to parametrize -the experimental binding energies as function of $A$, is the so-called -_liquid drop model_. The ansatz is based on the following expression - -!bt -\[ -BE(N,Z) = a_1A-a_2A^{2/3}-a_3\frac{Z^2}{A^{1/3}}-a_4\frac{(N-Z)^2}{A}, -\] -!et - -where $A$ stands for the number of nucleons and the $a_i$s are parameters which are determined by a fit -to the experimental data. - - - - -To arrive at the above expression we have assumed that we can make the following assumptions: - - * 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. - - * There is a surface energy term $a_2A^{2/3}$. The assumption here is that a nucleon at the surface of a nucleus interacts with fewer other nucleons than one in the interior of the nucleus and hence its binding energy is less. This surface energy term takes that into account and is therefore negative and is proportional to the surface area. - - - * There is a Coulomb energy term $a_3\frac{Z^2}{A^{1/3}}$. The electric repulsion between each pair of protons in a nucleus yields less binding. - - * There is an asymmetry term $a_4\frac{(N-Z)^2}{A}$. This term is associated with the Pauli exclusion principle and reflects the fact that the proton-neutron interaction is more attractive on the average than the neutron-neutron and proton-proton interactions. - -We could also add a so-called pairing term, which is a correction term that -arises from the tendency of proton pairs and neutron pairs to -occur. An even number of particles is more stable than an odd number. - - -=== Organizing our data === - -Let us start with reading and organizing our data. -We start with the compilation of masses and binding energies from 2016. -After having downloaded this file to our own computer, we are now ready to read the file and start structuring our data. - - -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_. -!bc pycod -# Common imports -import numpy as np -import pandas as pd -import matplotlib.pyplot as plt -import sklearn.linear_model as skl -from sklearn.model_selection import train_test_split -from sklearn.metrics import mean_squared_error, r2_score, mean_absolute_error -import os - -# Where to save the figures and data files -PROJECT_ROOT_DIR = "Results" -FIGURE_ID = "Results/FigureFiles" -DATA_ID = "DataFiles/" - -if not os.path.exists(PROJECT_ROOT_DIR): - os.mkdir(PROJECT_ROOT_DIR) - -if not os.path.exists(FIGURE_ID): - os.makedirs(FIGURE_ID) - -if not os.path.exists(DATA_ID): - os.makedirs(DATA_ID) - -def image_path(fig_id): - return os.path.join(FIGURE_ID, fig_id) - -def data_path(dat_id): - return os.path.join(DATA_ID, dat_id) - -def save_fig(fig_id): - plt.savefig(image_path(fig_id) + ".png", format='png') - -infile = open(data_path("MassEval2016.dat"),'r') -!ec - - -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. -!bc pycod -from pylab import plt, mpl -plt.style.use('seaborn') -mpl.rcParams['font.family'] = 'serif' - -def MakePlot(x,y, styles, labels, axlabels): - plt.figure(figsize=(10,6)) - for i in range(len(x)): - plt.plot(x[i], y[i], styles[i], label = labels[i]) - plt.xlabel(axlabels[0]) - plt.ylabel(axlabels[1]) - plt.legend(loc=0) -!ec - -Our next step is to read the data on experimental binding energies and -reorganize them as functions of the mass number $A$, the number of -protons $Z$ and neutrons $N$ using _pandas_. Before we do this it is -always useful (unless you have a binary file or other types of compressed -data) to actually open the file and simply take a look at it! - - -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. -!bc pycod -""" -This is taken from the data file of the mass 2016 evaluation. -All files are 3436 lines long with 124 character per line. - Headers are 39 lines long. - col 1 : Fortran character control: 1 = page feed 0 = line feed - 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 - These formats are reflected in the pandas widths variable below, see the statement - widths=(1,3,5,5,5,1,3,4,1,13,11,11,9,1,2,11,9,1,3,1,12,11,1), - Pandas has also a variable header, with length 39 in this case. -""" -!ec - -The data we are interested in are in columns 2, 3, 4 and 11, giving us -the number of neutrons, protons, mass numbers and binding energies, -respectively. We add also for the sake of completeness the element name. The data are in fixed-width formatted lines and we will -covert them into the _pandas_ DataFrame structure. - -!bc pycod -# Read the experimental data with Pandas -Masses = pd.read_fwf(infile, usecols=(2,3,4,6,11), - names=('N', 'Z', 'A', 'Element', 'Ebinding'), - widths=(1,3,5,5,5,1,3,4,1,13,11,11,9,1,2,11,9,1,3,1,12,11,1), - header=39, - index_col=False) - -# Extrapolated values are indicated by '#' in place of the decimal place, so -# the Ebinding column won't be numeric. Coerce to float and drop these entries. -Masses['Ebinding'] = pd.to_numeric(Masses['Ebinding'], errors='coerce') -Masses = Masses.dropna() -# Convert from keV to MeV. -Masses['Ebinding'] /= 1000 - -# Group the DataFrame by nucleon number, A. -Masses = Masses.groupby('A') -# Find the rows of the grouped DataFrame with the maximum binding energy. -Masses = Masses.apply(lambda t: t[t.Ebinding==t.Ebinding.max()]) -!ec - -We have now read in the data, grouped them according to the variables we are interested in. -We see how easy it is to reorganize the data using _pandas_. If we -were to do these operations in C/C++ or Fortran, we would have had to -write various functions/subroutines which perform the above -reorganizations for us. Having reorganized the data, we can now start -to make some simple fits using both the functionalities in _numpy_ and -_Scikit-Learn_ afterwards. - -Now we define five variables which contain -the number of nucleons $A$, the number of protons $Z$ and the number of neutrons $N$, the element name and finally the energies themselves. -!bc pycod -A = Masses['A'] -Z = Masses['Z'] -N = Masses['N'] -Element = Masses['Element'] -Energies = Masses['Ebinding'] -print(Masses) -!ec -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 $\bm{X}$. -It has dimensionality $p\times n$, where $n$ is the number of data points and $p$ are the so-called predictors. In our case here they are given by the number of polynomials in $A$ we wish to include in the fit. -!bc pycod -# Now we set up the design matrix X -X = np.zeros((len(A),5)) -X[:,0] = 1 -X[:,1] = A -X[:,2] = A**(2.0/3.0) -X[:,3] = A**(-1.0/3.0) -X[:,4] = A**(-1.0) -!ec -With _scikitlearn_ we are now ready to use linear regression and fit our data. -!bc pycod -clf = skl.LinearRegression().fit(X, Energies) -fity = clf.predict(X) -!ec -Pretty simple! -Now we can print measures of how our fit is doing, the coefficients from the fits and plot the final fit together with our data. -!bc pycod -# The mean squared error -print("Mean squared error: %.2f" % mean_squared_error(Energies, fity)) -# Explained variance score: 1 is perfect prediction -print('Variance score: %.2f' % r2_score(Energies, fity)) -# Mean absolute error -print('Mean absolute error: %.2f' % mean_absolute_error(Energies, fity)) -print(clf.coef_, clf.intercept_) - -Masses['Eapprox'] = fity -# Generate a plot comparing the experimental with the fitted values values. -fig, ax = plt.subplots() -ax.set_xlabel(r'$A = N + Z$') -ax.set_ylabel(r'$E_\mathrm{bind}\,/\mathrm{MeV}$') -ax.plot(Masses['A'], Masses['Ebinding'], alpha=0.7, lw=2, - label='Ame2016') -ax.plot(Masses['A'], Masses['Eapprox'], alpha=0.7, lw=2, c='m', - label='Fit') -ax.legend() -save_fig("Masses2016") -plt.show() -!ec - - - -=== And what about using neural networks? === -The _seaborn_ package allows us to visualize data in an efficient way. Note that we use _scikit-learn_'s multi-layer perceptron (or feed forward neural network) -functionality. -!bc pycod -from sklearn.neural_network import MLPRegressor -from sklearn.metrics import accuracy_score -import seaborn as sns - - -X_train = X -Y_train = Energies -n_hidden_neurons = 50 -epochs = 100 -# store models for later use -eta_vals = np.logspace(-3, 0, 4) -lmbd_vals = np.logspace(-3, 0, 4) -# store the models for later use -DNN_scikit = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object) -train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals))) -sns.set() -for i, eta in enumerate(eta_vals): - for j, lmbd in enumerate(lmbd_vals): - dnn = MLPRegressor(hidden_layer_sizes=(n_hidden_neurons), activation='relu', solver='adam', - alpha=lmbd, learning_rate_init=eta, max_iter=epochs) - dnn.fit(X_train, Y_train) - DNN_scikit[i][j] = dnn - train_accuracy[i][j] = dnn.score(X_train, Y_train) - fity = dnn.predict(X_train) - MSE = mean_squared_error(Y_train, fity) - print("Mean squared error: %.2f" % mean_squared_error(Y_train, fity)) - train_accuracy[i][j] = MSE -fig, ax = plt.subplots(figsize = (10, 10)) -sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis") -ax.set_title("Training Accuracy") -ax.set_ylabel("$\eta$") -ax.set_xlabel("$\lambda$") -plt.show() -print(train_accuracy) - - -!ec - - - - - - -===== A first summary ===== - -The aim behind these introductory words was to present to you various -Python libraries and their functionalities, in particular libraries like -_numpy_, _pandas_, _xarray_ and _matplotlib_ and other that make our life much easier -in handling various data sets and visualizing data. - -Furthermore, -_Scikit-Learn_ allows us with few lines of code to implement popular -Machine Learning algorithms for supervised learning. Later we will meet _Tensorflow_, a powerful library for deep learning. -Now it is time to dive more into the details of various methods. We will start with linear regression and try to take a deeper look at what it entails. - - -!split -===== Why Linear Regression (aka Ordinary Least Squares and family) ===== - -Fitting a continuous function with linear parameterization in terms of the parameters $\bm{\beta}$. -* Method of choice for fitting a continuous function! -* Gives an excellent introduction to central Machine Learning features with _understandable pedagogical_ links to other methods like _Neural Networks_, _Support Vector Machines_ etc -* Analytical expression for the fitting parameters $\bm{\beta}$ -* Analytical expressions for statistical propertiers like mean values, variances, confidence intervals and more -* Analytical relation with probabilistic interpretations -* Easy to introduce basic concepts like bias-variance tradeoff, cross-validation, resampling and regularization techniques and many other ML topics -* Easy to code! And links well with classification problems and logistic regression and neural networks -* Allows for _easy_ hands-on understanding of gradient descent methods -* and many more features - -For more discussions of Ridge and Lasso regression, "Wessel van Wieringen's":"https://arxiv.org/abs/1509.09169" article is highly recommended. -Similarly, "Mehta et al's article":"https://arxiv.org/abs/1803.08823" is also recommended. - - -!split -===== Regression analysis, overarching aims ===== -!bblock - -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 $\bm{x} =[x_0, x_1,\dots, x_{n-1}]^T$. -The first variable is called the _dependent_, the _outcome_ or the _response_ variable while the set of variables $\bm{x}$ is called the independent variable, or the predictor variable or the explanatory variable, or simply just the _inputs_. - -A regression model aims at finding a likelihood function $p(\bm{y}\vert \bm{x})$ or in the more traditional sense a function $\bm{y}(\bm{x})$, that is the conditional distribution for $\bm{y}$ with a given $\bm{x}$. The estimation of $p(\bm{y}\vert \bm{x})$ is made using a data set with -* $n$ cases $i = 0, 1, 2, \dots, n-1$ -* Response (target, dependent or outcome) variable $y_i$ with $i = 0, 1, 2, \dots, n-1$ -* $p$ so-called explanatory (independent or predictor or feature) variables $\bm{x}_i=[x_{i0}, x_{i1}, \dots, x_{ip-1}]$ with $i = 0, 1, 2, \dots, n-1$ and explanatory variables running from $0$ to $p-1$. See below for more explicit examples. - The goal of the regression analysis is to extract/exploit relationship between $\bm{y}$ and $\bm{x}$ in order to infer specific dependencies, approximations to the likelihood functions, functional relationships and to make predictions, making fits and many other things. -!eblock - -!split -===== Regression analysis, overarching aims II ===== -!bblock - - -Consider an experiment in which $p$ characteristics/features of $n$ samples are -measured. The data from this experiment, for various explanatory variables $p$ are normally represented by a matrix -$\mathbf{X}$. - -The matrix $\mathbf{X}$ is called the *design -matrix*. Additional information of the samples is available in the -form of $\bm{y}$ (also as above). The variable $\bm{y}$ is -generally referred to as the *response variable*. The aim of -regression analysis is to explain $\bm{y}$ in terms of -$\bm{X}$ through a functional relationship like $y_i = -f(\mathbf{X}_{i,\ast})$. When no prior knowledge on the form of -$f(\cdot)$ is available, it is common to assume a linear relationship -between $\bm{X}$ and $\bm{y}$. This assumption gives rise to -the *linear regression model* where $\bm{\beta} = [\beta_0, \ldots, -\beta_{p-1}]^{T}$ are the *regression parameters*. - -Linear regression gives us a set of analytical equations for the parameters $\beta_j$. - -!eblock - - - -!split -===== Examples ===== -!bblock -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) $\bm{y}$, -consider the model we discussed for describing nuclear binding energies. - -There we assumed that we could parametrize the data using a polynomial approximation based on the liquid drop model. -Assuming -!bt -\[ -BE(A) = a_0+a_1A+a_2A^{2/3}+a_3A^{-1/3}+a_4A^{-1}, -\] -!et -we have five predictors, that is the intercept, the $A$ dependent term, the $A^{2/3}$ term and the $A^{-1/3}$ and $A^{-1}$ terms. -This gives $p=0,1,2,3,4$. Furthermore we have $n$ entries for each predictor. It means that our design matrix is a -$p\times n$ matrix $\bm{X}$. - -Here the predictors are based on a model we have made. A popular data set which is widely encountered in ML applications is the -so-called "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. - -!eblock - - - - - -!split -===== General linear models and linear algebra ===== -!bblock -Before we proceed let us study a case where we aim at fitting a set of data $\bm{y}=[y_0,y_1,\dots,y_{n-1}]$. 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 $\bm{x}=[x_0,x_1,\dots,x_{n-1}]$, 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. - -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 -!bt -\[ -y=y(x) \rightarrow y(x_i)=\tilde{y}_i+\epsilon_i=\sum_{j=0}^{n-1} \beta_j x_i^j+\epsilon_i, -\] -!et -where $\epsilon_i$ is the error in our approximation. - -!eblock - - -!split -===== Rewriting the fitting procedure as a linear algebra problem ===== -!bblock -For every set of values $y_i,x_i$ we have thus the corresponding set of equations -!bt -\begin{align*} -y_0&=\beta_0+\beta_1x_0^1+\beta_2x_0^2+\dots+\beta_{n-1}x_0^{n-1}+\epsilon_0\\ -y_1&=\beta_0+\beta_1x_1^1+\beta_2x_1^2+\dots+\beta_{n-1}x_1^{n-1}+\epsilon_1\\ -y_2&=\beta_0+\beta_1x_2^1+\beta_2x_2^2+\dots+\beta_{n-1}x_2^{n-1}+\epsilon_2\\ -\dots & \dots \\ -y_{n-1}&=\beta_0+\beta_1x_{n-1}^1+\beta_2x_{n-1}^2+\dots+\beta_{n-1}x_{n-1}^{n-1}+\epsilon_{n-1}.\\ -\end{align*} -!et -!eblock - - -!split -===== Rewriting the fitting procedure as a linear algebra problem, more details ===== -!bblock -Defining the vectors -!bt -\[ -\bm{y} = [y_0,y_1, y_2,\dots, y_{n-1}]^T, -\] -!et -and -!bt -\[ -\bm{\beta} = [\beta_0,\beta_1, \beta_2,\dots, \beta_{n-1}]^T, -\] -!et -and -!bt -\[ -\bm{\epsilon} = [\epsilon_0,\epsilon_1, \epsilon_2,\dots, \epsilon_{n-1}]^T, -\] -!et -and the design matrix -!bt -\[ -\bm{X}= -\begin{bmatrix} -1& x_{0}^1 &x_{0}^2& \dots & \dots &x_{0}^{n-1}\\ -1& x_{1}^1 &x_{1}^2& \dots & \dots &x_{1}^{n-1}\\ -1& x_{2}^1 &x_{2}^2& \dots & \dots &x_{2}^{n-1}\\ -\dots& \dots &\dots& \dots & \dots &\dots\\ -1& x_{n-1}^1 &x_{n-1}^2& \dots & \dots &x_{n-1}^{n-1}\\ -\end{bmatrix} -\] -!et -we can rewrite our equations as -!bt -\[ -\bm{y} = \bm{X}\bm{\beta}+\bm{\epsilon}. -\] -!et -The above design matrix is called a "Vandermonde matrix":"https://en.wikipedia.org/wiki/Vandermonde_matrix". -!eblock - - -!split -===== Generalizing the fitting procedure as a linear algebra problem ===== -!bblock - -We are obviously not limited to the above polynomial expansions. We -could replace the various powers of $x$ with elements of Fourier -series or instead of $x_i^j$ we could have $\cos{(j x_i)}$ or $\sin{(j -x_i)}$, or time series or other orthogonal functions. For every set -of values $y_i,x_i$ we can then generalize the equations to - -!bt -\begin{align*} -y_0&=\beta_0x_{00}+\beta_1x_{01}+\beta_2x_{02}+\dots+\beta_{n-1}x_{0n-1}+\epsilon_0\\ -y_1&=\beta_0x_{10}+\beta_1x_{11}+\beta_2x_{12}+\dots+\beta_{n-1}x_{1n-1}+\epsilon_1\\ -y_2&=\beta_0x_{20}+\beta_1x_{21}+\beta_2x_{22}+\dots+\beta_{n-1}x_{2n-1}+\epsilon_2\\ -\dots & \dots \\ -y_{i}&=\beta_0x_{i0}+\beta_1x_{i1}+\beta_2x_{i2}+\dots+\beta_{n-1}x_{in-1}+\epsilon_i\\ -\dots & \dots \\ -y_{n-1}&=\beta_0x_{n-1,0}+\beta_1x_{n-1,2}+\beta_2x_{n-1,2}+\dots+\beta_{n-1}x_{n-1,n-1}+\epsilon_{n-1}.\\ -\end{align*} -!et - -_Note that we have $p=n$ here. The matrix is symmetric. This is generally not the case!_ -!eblock - - -!split -===== Generalizing the fitting procedure as a linear algebra problem ===== -!bblock -We redefine in turn the matrix $\bm{X}$ as -!bt -\[ -\bm{X}= -\begin{bmatrix} -x_{00}& x_{01} &x_{02}& \dots & \dots &x_{0,n-1}\\ -x_{10}& x_{11} &x_{12}& \dots & \dots &x_{1,n-1}\\ -x_{20}& x_{21} &x_{22}& \dots & \dots &x_{2,n-1}\\ -\dots& \dots &\dots& \dots & \dots &\dots\\ -x_{n-1,0}& x_{n-1,1} &x_{n-1,2}& \dots & \dots &x_{n-1,n-1}\\ -\end{bmatrix} -\] -!et -and without loss of generality we rewrite again our equations as -!bt -\[ -\bm{y} = \bm{X}\bm{\beta}+\bm{\epsilon}. -\] -!et -The left-hand side of this equation is kwown. Our error vector $\bm{\epsilon}$ and the parameter vector $\bm{\beta}$ are our unknow quantities. How can we obtain the optimal set of $\beta_i$ values? -!eblock - - -!split -===== Optimizing our parameters ===== -!bblock -We have defined the matrix $\bm{X}$ via the equations -!bt -\begin{align*} -y_0&=\beta_0x_{00}+\beta_1x_{01}+\beta_2x_{02}+\dots+\beta_{n-1}x_{0n-1}+\epsilon_0\\ -y_1&=\beta_0x_{10}+\beta_1x_{11}+\beta_2x_{12}+\dots+\beta_{n-1}x_{1n-1}+\epsilon_1\\ -y_2&=\beta_0x_{20}+\beta_1x_{21}+\beta_2x_{22}+\dots+\beta_{n-1}x_{2n-1}+\epsilon_1\\ -\dots & \dots \\ -y_{i}&=\beta_0x_{i0}+\beta_1x_{i1}+\beta_2x_{i2}+\dots+\beta_{n-1}x_{in-1}+\epsilon_1\\ -\dots & \dots \\ -y_{n-1}&=\beta_0x_{n-1,0}+\beta_1x_{n-1,2}+\beta_2x_{n-1,2}+\dots+\beta_{n-1}x_{n-1,n-1}+\epsilon_{n-1}.\\ -\end{align*} -!et - -As we noted above, we stayed with a system with the design matrix - $\bm{X}\in {\mathbb{R}}^{n\times n}$, that is we have $p=n$. For reasons to come later (algorithmic arguments) we will hereafter define -our matrix as $\bm{X}\in {\mathbb{R}}^{n\times p}$, with the predictors refering to the column numbers and the entries $n$ being the row elements. - -!eblock - - -!split -===== Our model for the nuclear binding energies ===== - -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. - -We restate the parts of the code we are most interested in. -!bc pycod -# Common imports -import numpy as np -import pandas as pd -import matplotlib.pyplot as plt -from IPython.display import display -import os - -# Where to save the figures and data files -PROJECT_ROOT_DIR = "Results" -FIGURE_ID = "Results/FigureFiles" -DATA_ID = "DataFiles/" - -if not os.path.exists(PROJECT_ROOT_DIR): - os.mkdir(PROJECT_ROOT_DIR) - -if not os.path.exists(FIGURE_ID): - os.makedirs(FIGURE_ID) - -if not os.path.exists(DATA_ID): - os.makedirs(DATA_ID) - -def image_path(fig_id): - return os.path.join(FIGURE_ID, fig_id) - -def data_path(dat_id): - return os.path.join(DATA_ID, dat_id) - -def save_fig(fig_id): - plt.savefig(image_path(fig_id) + ".png", format='png') - -infile = open(data_path("MassEval2016.dat"),'r') - - -# Read the experimental data with Pandas -Masses = pd.read_fwf(infile, usecols=(2,3,4,6,11), - names=('N', 'Z', 'A', 'Element', 'Ebinding'), - widths=(1,3,5,5,5,1,3,4,1,13,11,11,9,1,2,11,9,1,3,1,12,11,1), - header=39, - index_col=False) - -# Extrapolated values are indicated by '#' in place of the decimal place, so -# the Ebinding column won't be numeric. Coerce to float and drop these entries. -Masses['Ebinding'] = pd.to_numeric(Masses['Ebinding'], errors='coerce') -Masses = Masses.dropna() -# Convert from keV to MeV. -Masses['Ebinding'] /= 1000 - -# Group the DataFrame by nucleon number, A. -Masses = Masses.groupby('A') -# Find the rows of the grouped DataFrame with the maximum binding energy. -Masses = Masses.apply(lambda t: t[t.Ebinding==t.Ebinding.max()]) -A = Masses['A'] -Z = Masses['Z'] -N = Masses['N'] -Element = Masses['Element'] -Energies = Masses['Ebinding'] - -# Now we set up the design matrix X -X = np.zeros((len(A),5)) -X[:,0] = 1 -X[:,1] = A -X[:,2] = A**(2.0/3.0) -X[:,3] = A**(-1.0/3.0) -X[:,4] = A**(-1.0) -# Then nice printout using pandas -DesignMatrix = pd.DataFrame(X) -DesignMatrix.index = A -DesignMatrix.columns = ['1', 'A', 'A^(2/3)', 'A^(-1/3)', '1/A'] -display(DesignMatrix) -!ec - -With $\bm{\beta}\in {\mathbb{R}}^{p\times 1}$, it means that we will hereafter write our equations for the approximation as -!bt -\[ -\bm{\tilde{y}}= \bm{X}\bm{\beta}, -\] -!et -throughout these lectures. - - -!split -===== Optimizing our parameters, more details ===== -!bblock -With the above we use the design matrix to define the approximation $\bm{\tilde{y}}$ via the unknown quantity $\bm{\beta}$ as -!bt -\[ -\bm{\tilde{y}}= \bm{X}\bm{\beta}, -\] -!et -and in order to find the optimal parameters $\beta_i$ instead of solving the above linear algebra problem, we define a function which gives a measure of the spread between the values $y_i$ (which represent hopefully the exact values) and the parameterized values $\tilde{y}_i$, namely -!bt -\[ -C(\bm{\beta})=\frac{1}{n}\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2=\frac{1}{n}\left\{\left(\bm{y}-\bm{\tilde{y}}\right)^T\left(\bm{y}-\bm{\tilde{y}}\right)\right\}, -\] -!et -or using the matrix $\bm{X}$ and in a more compact matrix-vector notation as -!bt -\[ -C(\bm{\beta})=\frac{1}{n}\left\{\left(\bm{y}-\bm{X}\bm{\beta}\right)^T\left(\bm{y}-\bm{X}\bm{\beta}\right)\right\}. -\] -!et -This function is one possible way to define the so-called cost function. - - - -It is also common to define -the function $C$ as - -!bt -\[ -C(\bm{\beta})=\frac{1}{2n}\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2, -\] -!et -since when taking the first derivative with respect to the unknown parameters $\beta$, the factor of $2$ cancels out. -!eblock - - -!split -===== Interpretations and optimizing our parameters ===== -!bblock - -The function -!bt -\[ -C(\bm{\beta})=\frac{1}{n}\left\{\left(\bm{y}-\bm{X}\bm{\beta}\right)^T\left(\bm{y}-\bm{X}\bm{\beta}\right)\right\}, -\] -!et -can be linked to the variance of the quantity $y_i$ if we interpret the latter as the mean value. -When linking (see the discussion below) with the maximum likelihood approach below, we will indeed interpret $y_i$ as a mean value -!bt -\[ -y_{i}=\langle y_i \rangle = \beta_0x_{i,0}+\beta_1x_{i,1}+\beta_2x_{i,2}+\dots+\beta_{n-1}x_{i,n-1}+\epsilon_i, -\] -!et - -where $\langle y_i \rangle$ is the mean value. Keep in mind also that -till now we have treated $y_i$ as the exact value. Normally, the -response (dependent or outcome) variable $y_i$ the outcome of a -numerical experiment or another type of experiment and is thus only an -approximation to the true value. It is then always accompanied by an -error estimate, often limited to a statistical error estimate given by -the standard deviation discussed earlier. In the discussion here we -will treat $y_i$ as our exact value for the response variable. - -In order to find the parameters $\beta_i$ we will then minimize the spread of $C(\bm{\beta})$, that is we are going to solve the problem -!bt -\[ -{\displaystyle \min_{\bm{\beta}\in -{\mathbb{R}}^{p}}}\frac{1}{n}\left\{\left(\bm{y}-\bm{X}\bm{\beta}\right)^T\left(\bm{y}-\bm{X}\bm{\beta}\right)\right\}. -\] -!et -In practical terms it means we will require -!bt -\[ -\frac{\partial C(\bm{\beta})}{\partial \beta_j} = \frac{\partial }{\partial \beta_j}\left[ \frac{1}{n}\sum_{i=0}^{n-1}\left(y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}\right)^2\right]=0, -\] -!et -which results in -!bt -\[ -\frac{\partial C(\bm{\beta})}{\partial \beta_j} = -\frac{2}{n}\left[ \sum_{i=0}^{n-1}x_{ij}\left(y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}\right)\right]=0, -\] -!et -or in a matrix-vector form as -!bt -\[ -\frac{\partial C(\bm{\beta})}{\partial \bm{\beta}} = 0 = \bm{X}^T\left( \bm{y}-\bm{X}\bm{\beta}\right). -\] -!et - - -!eblock - - -!split -===== Interpretations and optimizing our parameters ===== -!bblock -We can rewrite -!bt -\[ -\frac{\partial C(\bm{\beta})}{\partial \bm{\beta}} = 0 = \bm{X}^T\left( \bm{y}-\bm{X}\bm{\beta}\right), -\] -!et -as -!bt -\[ -\bm{X}^T\bm{y} = \bm{X}^T\bm{X}\bm{\beta}, -\] -!et -and if the matrix $\bm{X}^T\bm{X}$ is invertible we have the solution -!bt -\[ -\bm{\beta} =\left(\bm{X}^T\bm{X}\right)^{-1}\bm{X}^T\bm{y}. -\] -!et - -We note also that since our design matrix is defined as $\bm{X}\in -{\mathbb{R}}^{n\times p}$, the product $\bm{X}^T\bm{X} \in -{\mathbb{R}}^{p\times p}$. In the above case we have that $p \ll n$, -in our case $p=5$ meaning that we end up with inverting a small -$5\times 5$ matrix. This is a rather common situation, in many cases we end up with low-dimensional -matrices to invert. The methods discussed here and for many other -supervised learning algorithms like classification with logistic -regression or support vector machines, exhibit dimensionalities which -allow for the usage of direct linear algebra methods such as _LU_ decomposition or _Singular Value Decomposition_ (SVD) for finding the inverse of the matrix -$\bm{X}^T\bm{X}$. -!eblock - -!bblock -_Small question_: Do you think the example we have at hand here (the nuclear binding energies) can lead to problems in inverting the matrix $\bm{X}^T\bm{X}$? What kind of problems can we expect? -!eblock - -!split -===== Interpretations and optimizing our parameters ===== -!bblock -The residuals $\bm{\epsilon}$ are in turn given by -!bt -\[ -\bm{\epsilon} = \bm{y}-\bm{\tilde{y}} = \bm{y}-\bm{X}\bm{\beta}, -\] -!et -and with -!bt -\[ -\bm{X}^T\left( \bm{y}-\bm{X}\bm{\beta}\right)= 0, -\] -!et -we have -!bt -\[ -\bm{X}^T\bm{\epsilon}=\bm{X}^T\left( \bm{y}-\bm{X}\bm{\beta}\right)= 0, -\] -!et -meaning that the solution for $\bm{\beta}$ is the one which minimizes the residuals. Later we will link this with the maximum likelihood approach. - -!eblock - - -Let us now return to our nuclear binding energies and simply code the above equations. - -!split -===== Own code for Ordinary Least Squares ===== - -It is rather straightforward to implement the matrix inversion and obtain the parameters $\bm{\beta}$. After having defined the matrix $\bm{X}$ we simply need to -write -!bc pycod -# matrix inversion to find beta -beta = np.linalg.inv(X.T.dot(X)).dot(X.T).dot(Energies) -# and then make the prediction -ytilde = X @ beta -!ec -Alternatively, you can use the least squares functionality in _Numpy_ as -!bc pycod -fit = np.linalg.lstsq(X, Energies, rcond =None)[0] -ytildenp = np.dot(fit,X.T) -!ec - -And finally we plot our fit with and compare with data -!bc pycod -Masses['Eapprox'] = ytilde -# Generate a plot comparing the experimental with the fitted values values. -fig, ax = plt.subplots() -ax.set_xlabel(r'$A = N + Z$') -ax.set_ylabel(r'$E_\mathrm{bind}\,/\mathrm{MeV}$') -ax.plot(Masses['A'], Masses['Ebinding'], alpha=0.7, lw=2, - label='Ame2016') -ax.plot(Masses['A'], Masses['Eapprox'], alpha=0.7, lw=2, c='m', - label='Fit') -ax.legend() -save_fig("Masses2016OLS") -plt.show() -!ec - -!split -===== Adding error analysis and training set up ===== - -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. -Since we are not using _Scikit-Learn_ here we can define our own $R2$ function as -!bc pycod -def R2(y_data, y_model): - return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2) -!ec -and we would be using it as -!bc pycod -print(R2(Energies,ytilde)) -!ec - -We can easily add our _MSE_ score as -!bc pycod -def MSE(y_data,y_model): - n = np.size(y_model) - return np.sum((y_data-y_model)**2)/n - -print(MSE(Energies,ytilde)) -!ec -and finally the relative error as -!bc pycod -def RelativeError(y_data,y_model): - return abs((y_data-y_model)/y_data) -print(RelativeError(Energies, ytilde)) -!ec - - - -!split -===== The $\chi^2$ function ===== -!bblock - -Normally, the response (dependent or outcome) variable $y_i$ is the -outcome of a numerical experiment or another type of experiment and is -thus only an approximation to the true value. It is then always -accompanied by an error estimate, often limited to a statistical error -estimate given by the standard deviation discussed earlier. In the -discussion here we will treat $y_i$ as our exact value for the -response variable. - -Introducing the standard deviation $\sigma_i$ for each measurement -$y_i$, we define now the $\chi^2$ function (omitting the $1/n$ term) -as - -!bt -\[ -\chi^2(\bm{\beta})=\frac{1}{n}\sum_{i=0}^{n-1}\frac{\left(y_i-\tilde{y}_i\right)^2}{\sigma_i^2}=\frac{1}{n}\left\{\left(\bm{y}-\bm{\tilde{y}}\right)^T\frac{1}{\bm{\Sigma^2}}\left(\bm{y}-\bm{\tilde{y}}\right)\right\}, -\] -!et -where the matrix $\bm{\Sigma}$ is a diagonal matrix with $\sigma_i$ as matrix elements. - -!eblock - -!split -===== The $\chi^2$ function ===== -!bblock - -In order to find the parameters $\beta_i$ we will then minimize the spread of $\chi^2(\bm{\beta})$ by requiring -!bt -\[ -\frac{\partial \chi^2(\bm{\beta})}{\partial \beta_j} = \frac{\partial }{\partial \beta_j}\left[ \frac{1}{n}\sum_{i=0}^{n-1}\left(\frac{y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}}{\sigma_i}\right)^2\right]=0, -\] -!et -which results in -!bt -\[ -\frac{\partial \chi^2(\bm{\beta})}{\partial \beta_j} = -\frac{2}{n}\left[ \sum_{i=0}^{n-1}\frac{x_{ij}}{\sigma_i}\left(\frac{y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}}{\sigma_i}\right)\right]=0, -\] -!et -or in a matrix-vector form as -!bt -\[ -\frac{\partial \chi^2(\bm{\beta})}{\partial \bm{\beta}} = 0 = \bm{A}^T\left( \bm{b}-\bm{A}\bm{\beta}\right). -\] -!et -where we have defined the matrix $\bm{A} =\bm{X}/\bm{\Sigma}$ with matrix elements $a_{ij} = x_{ij}/\sigma_i$ and the vector $\bm{b}$ with elements $b_i = y_i/\sigma_i$. -!eblock - -!split -===== The $\chi^2$ function ===== -!bblock - -We can rewrite -!bt -\[ -\frac{\partial \chi^2(\bm{\beta})}{\partial \bm{\beta}} = 0 = \bm{A}^T\left( \bm{b}-\bm{A}\bm{\beta}\right), -\] -!et -as -!bt -\[ -\bm{A}^T\bm{b} = \bm{A}^T\bm{A}\bm{\beta}, -\] -!et -and if the matrix $\bm{A}^T\bm{A}$ is invertible we have the solution -!bt -\[ -\bm{\beta} =\left(\bm{A}^T\bm{A}\right)^{-1}\bm{A}^T\bm{b}. -\] -!et -!eblock - -!split -===== The $\chi^2$ function ===== -!bblock - -If we then introduce the matrix -!bt -\[ -\bm{H} = \left(\bm{A}^T\bm{A}\right)^{-1}, -\] -!et -we have then the following expression for the parameters $\beta_j$ (the matrix elements of $\bm{H}$ are $h_{ij}$) -!bt -\[ -\beta_j = \sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}\frac{y_i}{\sigma_i}\frac{x_{ik}}{\sigma_i} = \sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}b_ia_{ik} -\] -!et -We state without proof the expression for the uncertainty in the parameters $\beta_j$ as (we leave this as an exercise) -!bt -\[ -\sigma^2(\beta_j) = \sum_{i=0}^{n-1}\sigma_i^2\left( \frac{\partial \beta_j}{\partial y_i}\right)^2, -\] -!et -resulting in -!bt -\[ -\sigma^2(\beta_j) = \left(\sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}a_{ik}\right)\left(\sum_{l=0}^{p-1}h_{jl}\sum_{m=0}^{n-1}a_{ml}\right) = h_{jj}! -\] -!et -!eblock - -!split -===== The $\chi^2$ function ===== -!bblock -The first step here is to approximate the function $y$ with a first-order polynomial, that is we write -!bt -\[ -y=y(x) \rightarrow y(x_i) \approx \beta_0+\beta_1 x_i. -\] -!et -By computing the derivatives of $\chi^2$ with respect to $\beta_0$ and $\beta_1$ show that these are given by -!bt -\[ -\frac{\partial \chi^2(\bm{\beta})}{\partial \beta_0} = -2\left[ \frac{1}{n}\sum_{i=0}^{n-1}\left(\frac{y_i-\beta_0-\beta_1x_{i}}{\sigma_i^2}\right)\right]=0, -\] -!et -and -!bt -\[ -\frac{\partial \chi^2(\bm{\beta})}{\partial \beta_1} = -\frac{2}{n}\left[ \sum_{i=0}^{n-1}x_i\left(\frac{y_i-\beta_0-\beta_1x_{i}}{\sigma_i^2}\right)\right]=0. -\] -!et -!eblock - -!split -===== The $\chi^2$ function ===== -!bblock - -For a linear fit (a first-order polynomial) we don't need to invert a matrix!! -Defining -!bt -\[ -\gamma = \sum_{i=0}^{n-1}\frac{1}{\sigma_i^2}, -\] -!et - -!bt -\[ -\gamma_x = \sum_{i=0}^{n-1}\frac{x_{i}}{\sigma_i^2}, -\] -!et - -!bt -\[ -\gamma_y = \sum_{i=0}^{n-1}\left(\frac{y_i}{\sigma_i^2}\right), -\] -!et - -!bt -\[ -\gamma_{xx} = \sum_{i=0}^{n-1}\frac{x_ix_{i}}{\sigma_i^2}, -\] -!et - -!bt -\[ -\gamma_{xy} = \sum_{i=0}^{n-1}\frac{y_ix_{i}}{\sigma_i^2}, -\] -!et - -we obtain - -!bt -\[ -\beta_0 = \frac{\gamma_{xx}\gamma_y-\gamma_x\gamma_y}{\gamma\gamma_{xx}-\gamma_x^2}, -\] -!et - -!bt -\[ -\beta_1 = \frac{\gamma_{xy}\gamma-\gamma_x\gamma_y}{\gamma\gamma_{xx}-\gamma_x^2}. -\] -!et - -This approach (different linear and non-linear regression) suffers -often from both being underdetermined and overdetermined in the -unknown coefficients $\beta_i$. A better approach is to use the -Singular Value Decomposition (SVD) method discussed next week. - -!eblock - - -!split -===== Fitting an Equation of State for Dense Nuclear Matter ===== - -Before we continue, let us introduce yet another example. We are going to fit the -nuclear equation of state using results from many-body calculations. -The equation of state we have made available here, as function of -density, has been derived using modern nucleon-nucleon potentials with -"the addition of three-body -forces":"https://www.sciencedirect.com/science/article/pii/S0370157399001106". This -time the file is presented as a standard _csv_ file. - -The beginning of the Python code here is similar to what you have seen -before, with the same initializations and declarations. We use also -_pandas_ again, rather extensively in order to organize our data. - -The difference now is that we use _Scikit-Learn's_ regression tools -instead of our own matrix inversion implementation. Furthermore, we -sneak in _Ridge_ regression (to be discussed below) which includes a -hyperparameter $\lambda$, also to be explained below. - -!split -===== The code ===== - -!bc pycod -# Common imports -import os -import numpy as np -import pandas as pd -import matplotlib.pyplot as plt -import matplotlib.pyplot as plt -import sklearn.linear_model as skl -from sklearn.metrics import mean_squared_error, r2_score, mean_absolute_error - -# Where to save the figures and data files -PROJECT_ROOT_DIR = "Results" -FIGURE_ID = "Results/FigureFiles" -DATA_ID = "DataFiles/" - -if not os.path.exists(PROJECT_ROOT_DIR): - os.mkdir(PROJECT_ROOT_DIR) - -if not os.path.exists(FIGURE_ID): - os.makedirs(FIGURE_ID) - -if not os.path.exists(DATA_ID): - os.makedirs(DATA_ID) - -def image_path(fig_id): - return os.path.join(FIGURE_ID, fig_id) - -def data_path(dat_id): - return os.path.join(DATA_ID, dat_id) - -def save_fig(fig_id): - plt.savefig(image_path(fig_id) + ".png", format='png') - -infile = open(data_path("EoS.csv"),'r') - -# Read the EoS data as csv file and organize the data into two arrays with density and energies -EoS = pd.read_csv(infile, names=('Density', 'Energy')) -EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce') -EoS = EoS.dropna() -Energies = EoS['Energy'] -Density = EoS['Density'] -# The design matrix now as function of various polytrops -X = np.zeros((len(Density),4)) -X[:,3] = Density**(4.0/3.0) -X[:,2] = Density -X[:,1] = Density**(2.0/3.0) -X[:,0] = 1 - -# We use now Scikit-Learn's linear regressor and ridge regressor -# OLS part -clf = skl.LinearRegression().fit(X, Energies) -ytilde = clf.predict(X) -EoS['Eols'] = ytilde -# The mean squared error -print("Mean squared error: %.2f" % mean_squared_error(Energies, ytilde)) -# Explained variance score: 1 is perfect prediction -print('Variance score: %.2f' % r2_score(Energies, ytilde)) -# Mean absolute error -print('Mean absolute error: %.2f' % mean_absolute_error(Energies, ytilde)) -print(clf.coef_, clf.intercept_) - -# The Ridge regression with a hyperparameter lambda = 0.1 -_lambda = 0.1 -clf_ridge = skl.Ridge(alpha=_lambda).fit(X, Energies) -yridge = clf_ridge.predict(X) -EoS['Eridge'] = yridge -# The mean squared error -print("Mean squared error: %.2f" % mean_squared_error(Energies, yridge)) -# Explained variance score: 1 is perfect prediction -print('Variance score: %.2f' % r2_score(Energies, yridge)) -# Mean absolute error -print('Mean absolute error: %.2f' % mean_absolute_error(Energies, yridge)) -print(clf_ridge.coef_, clf_ridge.intercept_) - -fig, ax = plt.subplots() -ax.set_xlabel(r'$\rho[\mathrm{fm}^{-3}]$') -ax.set_ylabel(r'Energy per particle') -ax.plot(EoS['Density'], EoS['Energy'], alpha=0.7, lw=2, - label='Theoretical data') -ax.plot(EoS['Density'], EoS['Eols'], alpha=0.7, lw=2, c='m', - label='OLS') -ax.plot(EoS['Density'], EoS['Eridge'], alpha=0.7, lw=2, c='g', - label='Ridge $\lambda = 0.1$') -ax.legend() -save_fig("EoSfitting") -plt.show() -!ec - -The above simple polynomial in density $\rho$ gives an excellent fit -to the data. - -We note also that there is a small deviation between the -standard OLS and the Ridge regression at higher densities. We discuss this in more detail -below. - - -!split -===== Splitting our Data in Training and Test data ===== - -It is normal in essentially all Machine Learning studies to split the -data in a training set and a test set (sometimes also an additional -validation set). _Scikit-Learn_ has an own function for this. There -is no explicit recipe for how much data should be included as training -data and say test data. An accepted rule of thumb is to use -approximately $2/3$ to $4/5$ of the data as training data. We will -postpone a discussion of this splitting to the end of these notes and -our discussion of the so-called _bias-variance_ tradeoff. Here we -limit ourselves to repeat the above equation of state fitting example -but now splitting the data into a training set and a test set. - -!bc pycod -import os -import numpy as np -import pandas as pd -import matplotlib.pyplot as plt -from sklearn.model_selection import train_test_split -# Where to save the figures and data files -PROJECT_ROOT_DIR = "Results" -FIGURE_ID = "Results/FigureFiles" -DATA_ID = "DataFiles/" - -if not os.path.exists(PROJECT_ROOT_DIR): - os.mkdir(PROJECT_ROOT_DIR) - -if not os.path.exists(FIGURE_ID): - os.makedirs(FIGURE_ID) - -if not os.path.exists(DATA_ID): - os.makedirs(DATA_ID) - -def image_path(fig_id): - return os.path.join(FIGURE_ID, fig_id) - -def data_path(dat_id): - return os.path.join(DATA_ID, dat_id) - -def save_fig(fig_id): - plt.savefig(image_path(fig_id) + ".png", format='png') - -def R2(y_data, y_model): - return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2) -def MSE(y_data,y_model): - n = np.size(y_model) - return np.sum((y_data-y_model)**2)/n - -infile = open(data_path("EoS.csv"),'r') - -# Read the EoS data as csv file and organized into two arrays with density and energies -EoS = pd.read_csv(infile, names=('Density', 'Energy')) -EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce') -EoS = EoS.dropna() -Energies = EoS['Energy'] -Density = EoS['Density'] -# The design matrix now as function of various polytrops -X = np.zeros((len(Density),5)) -X[:,0] = 1 -X[:,1] = Density**(2.0/3.0) -X[:,2] = Density -X[:,3] = Density**(4.0/3.0) -X[:,4] = Density**(5.0/3.0) -# We split the data in test and training data -X_train, X_test, y_train, y_test = train_test_split(X, Energies, test_size=0.2) -# matrix inversion to find beta -beta = np.linalg.inv(X_train.T.dot(X_train)).dot(X_train.T).dot(y_train) -# and then make the prediction -ytilde = X_train @ beta -print("Training R2") -print(R2(y_train,ytilde)) -print("Training MSE") -print(MSE(y_train,ytilde)) -ypredict = X_test @ beta -print("Test R2") -print(R2(y_test,ypredict)) -print("Test MSE") -print(MSE(y_test,ypredict)) -!ec - - - - - -!split -===== Exercises ===== - -Here are three possible exercises for week 34 - - -===== Exercise: Setting up various Python environments ===== - -The first exercise here is of a mere technical art. We want you to have -* git as a version control software and to establish a user account on a provider like GitHub. Other providers like GitLab etc are equally fine. You can also use the University of Oslo "GitHub facilities":"https://www.uio.no/tjenester/it/maskin/filer/versjonskontroll/github.html". -* Install various Python packages - -We will make extensive use of Python as programming language and its -myriad of available libraries. You will find -IPython/Jupyter notebooks invaluable in your work. You can run _R_ -codes in the Jupyter/IPython notebooks, with the immediate benefit of -visualizing your data. You can also use compiled languages like C++, -Rust, Fortran etc if you prefer. The focus in these lectures will be -on Python. - -If you have Python installed (we recommend Python3) and you feel -pretty familiar with installing different packages, we recommend that -you install the following Python packages via _pip_ as - -o pip install numpy scipy matplotlib ipython scikit-learn sympy pandas pillow - -For _Tensorflow_, we recommend following the instructions in the text of -"Aurelien Geron, Hands‑On Machine Learning with Scikit‑Learn and TensorFlow, O'Reilly":"http://shop.oreilly.com/product/0636920052289.do" - -We will come back to _tensorflow_ later. - -For Python3, replace _pip_ with _pip3_. - -For OSX users we recommend, after having installed Xcode, to -install _brew_. Brew allows for a seamless installation of additional -software via for example - -o brew install python3 - -For Linux users, with its variety of distributions like for example the widely popular Ubuntu distribution, -you can use _pip_ as well and simply install Python as - -o sudo apt-get install python3 (or python for Python2.7) - -If you don't want to perform these operations separately and venture -into the hassle of exploring how to set up dependencies and paths, we -recommend two widely used distrubutions which set up all relevant -dependencies for Python, namely - -* "Anaconda":"https://docs.anaconda.com/", - -which is an open source -distribution of the Python and R programming languages for large-scale -data processing, predictive analytics, and scientific computing, that -aims to simplify package management and deployment. Package versions -are managed by the package management system _conda_. - -* "Enthought canopy":"https://www.enthought.com/product/canopy/" - -is a Python -distribution for scientific and analytic computing distribution and -analysis environment, available for free and under a commercial -license. - -We recommend using _Anaconda_ if you are not too familiar with setting paths in a terminal environment. - - - - -===== Exercise: making your own data and exploring scikit-learn ===== - - -We will generate our own dataset for a function $y(x)$ where $x \in [0,1]$ and defined by random numbers computed with the uniform distribution. The function $y$ is a quadratic polynomial in $x$ with added stochastic noise according to the normal distribution $\cal {N}(0,1)$. -The following simple Python instructions define our $x$ and $y$ values (with 100 data points). -!bc pycod -x = np.random.rand(100,1) -y = 2.0+5*x*x+0.1*np.random.randn(100,1) -!ec - -o Write your own code (following the examples under the "regression notes":"https://compphysics.github.io/MachineLearning/doc/LectureNotes/_build/html/chapter1.html") for computing the parametrization of the data set fitting a second-order polynomial. -o Use thereafter _scikit-learn_ (see again the examples in the regression slides) and compare with your own code. -o Using scikit-learn, compute also the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error defined as -!bt -\[ MSE(\bm{y},\bm{\tilde{y}}) = \frac{1}{n} -\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2, -\] -!et -and the $R^2$ score function. -If $\tilde{\bm{y}}_i$ 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 -!bt -\[ -R^2(\bm{y}, \tilde{\bm{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2}, -\] -!et -where we have defined the mean value of $\bm{y}$ as -!bt -\[ -\bar{y} = \frac{1}{n} \sum_{i=0}^{n - 1} y_i. -\] -!et -You can use the functionality included in scikit-learn. If you feel for it, you can use your own program and define functions which compute the above two functions. -Discuss the meaning of these results. Try also to vary the coefficient in front of the added stochastic noise term and discuss the quality of the fits. - -===== Exercise: Split data in test and training data ===== - -In this exercise we want you to to compute the MSE for the training -data and the test data as function of the complexity of a polynomial, -that is the degree of a given polynomial. - -The aim is to reproduce Figure 2.11 of "Hastie et al":"https://github.com/CompPhysics/MLErasmus/blob/master/doc/Textbooks/elementsstat.pdf". - -Our data is defined by $x\in [-3,3]$ with a total of for example $n=100$ data points. You should try to vary the number of data points $n$ in your analysis. -!bc pycod -np.random.seed() -n = 100 -# Make data set. -x = np.linspace(-3, 3, n).reshape(-1, 1) -y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape) -!ec -where $y$ is the function we want to fit with a given polynomial. -!bsubex -Write a first code which sets up a design matrix $X$ defined by a fifth-order polynomial and split your data set in training and test data. -!esubex - -!bsubex -Write thereafter (using either _scikit-learn_ or your matrix inversion code using for example _numpy_) -and perform an ordinary least squares fitting and compute the mean squared error for the training data and the test data. These calculations should apply to a model given by a fifth-order polynomial. -!esubex - -!bsubex -Add now a model which allows you to make polynomials up to degree $15$. Perform a standard OLS fitting of the training data and compute the MSE for the training and test data and plot both test and training data MSE as functions of the polynomial degree. Compare what you see with Figure 2.11 of Hastie et al. Comment your results. For which polynomial degree do you find an optimal MSE (smallest value)? -!esubex - - - - - - - - - - - - - - -