Overview of first week#

1. The sessions on Tuesdays and Wednesdays last four hours for each group (four groups in total) and will include lectures in a flipped mode (promoting active learning) and work on exercices and projects.

2. The sessions will begin with lectures, discussions, questions and answers about the material to be covered every week. Videos and teaching material will be announced in due time.

3. There are four groups:

• Tuesdays 815am-12pm and 1215pm-4pm

• Wednesdays 815am-12pm and 1215pm-4pm.

4. On Mondays we have a regular lecture which will be organized as a mix of active learning sessions and regular lectures. These lectures/active learning sessions start at 1015am and end at 12pm and serve the aims of giving an overview over various topics as well as solving specific problems. These lectures will also be recorded.

• Link to recording of lecture

The labs are also available till 6pm Tuesdays and Wednesdays. Videos and learning material with reading suggestions will be made available before each week starts.

Schedule first week#

• August 19: Lecture: Presentation of course, Linear regression, examples and theory

• August 20: Introduction to software and repetition of Python Programming, linear algebra and basic elements of statistics. Please select group.

• August 23: Introduction to software and repetition of Python Programming, linear algebra and basic elements of statistics. Please select group.

Lectures and ComputerLab#

• Mondays: regular lectures/active learning sessions (10.15am-12pm)

• The sessions on Tuesdays and Wednesdays last four hours and will include partly lectures and discussions in the beginning.

• Weekly reading assignments and videos needed to solve projects and exercises.

• Weekly exercises. You can hand in exercises if you want and get an extra score, see below.

• Detailed lecture notes, exercises, all programs presented, projects etc can be found at the homepage of the course.

• Weekly plans and all other information are on the official website. This info will also be conveyed via weekly emails.

• No final exam, three projects that are graded and have to be approved.

Communication channels#

• Communications (email and more) via <canvas.uio.no>

• Discord channel at https://discord.gg/XBKjd4ccGq

Course Format#

• Three compulsory projects. Electronic reports only using Canvas to hand in projects and git as version control software and GitHub for repository (or GitLab) of all your material.

• Evaluation and grading: The three projects are graded and each counts 1/3 of the final mark. No final written or oral exam.

a. For the last project each group/participant submits a proposal or works with suggested (by us) proposals for the project.

b. If possible, we would like to organize the last project as a workshop where each group presents this to all other participants of the course

c. Based on feedback etc, each group finalizes the report and submits for grading.

• Python is the default programming language, but feel free to use C/C++, Julia and/or Fortran or other programming languages. All source codes discussed during the lectures can be found at the webpage and github address of the course.

Teachers#

• Morten Hjorth-Jensen, morten.hjorth-jensen@fys.uio.no

  • Phone: +47-48257387

  • Office: Department of Physics, University of Oslo, Eastern wing, room FØ470

  • Office hours: Anytime! Individual or group office hours can be arranged either in person or via zoom. Feel free to send an email for planning.

• Ida Torkjellsdatter Storehaug, i.t.storehaug@fys.uio.no

• Fahimeh Najafi, fahimeh.najafi@fys.uio.no

• Mia-Katrin Ose Kvalsund, m.k.o.kvalsund@fys.uio.no

• Karl Henrik Fredly, k.h.fredly@fys.uio.no

• Sigurd k. Huse, s.k.huse@fys.uio.no

• Odin Johansen, odin.johansen@fys.uio.no

Deadlines for projects (tentative)#

1. Project 1: October 7 (available September 2) graded with feedback)

2. Project 2: November 4 (available October 8, graded with feedback)

3. Project 3: December 9 (available November 5, graded with feedback)

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.

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.

Reading material#

The lecture notes are collected as a jupyter-book at https://compphysics.github.io/MachineLearning/doc/LectureNotes/_build/html/intro.html. The lecture notes can also be retrieved as a standard PDF file at 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.

Main textbooks#

• Goodfellow, Bengio, and Courville (GBC), Deep Learning https://www.deeplearningbook.org/

• Sebastian Raschka, Yuxi Lie, and Vahid Mirjalili (RLM), Machine Learning with PyTorch and Scikit-Learn at https://www.packtpub.com/product/machine-learning-with-pytorch-and-scikit-learn/9781801819312, see also https://sebastianraschka.com/blog/2022/ml-pytorch-book.html

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.

Other popular texts#

Other texts.

• Christopher M. Bishop (CB), Pattern Recognition and Machine Learning

• Hastie, Tibshirani, and Friedman (HTF), The Elements of Statistical Learning, Springer.

• Aurelien Geron (AG), Hands‑On Machine Learning with Scikit‑Learn and TensorFlow, O’Reilly. 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

• David Foster (DF), Generative Deep Learning, https://www.oreilly.com/library/view/generative-deep-learning/9781098134174/

• Babcock and Gavras (BG), Generative AI with Python and TensorFlow, PacktPublishing/Hands-On-Generative-AI-with-Python-and-TensorFlow-2

Reading suggestions week 34#

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).

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.

Topics covered in this course: Statistical analysis and optimization of data#

The course has two central parts

1. Statistical analysis and optimization of data

2. 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.

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;

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;

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.

Extremely useful tools, strongly recommended#

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 https://computationalscienceuio.github.io/RefreshProgrammingSkills/intro.html

Other courses on Data science and Machine Learning at UiO#

• FYS5419 Quantum Computing and Quantum Machine Learning

• FYS5429 Advanced Machine Learning for the Physical Sciences

• STK2100 Machine learning and statistical methods for prediction and classification.

• IN3050/4050 Introduction to Artificial Intelligence and Machine Learning. Introductory course in machine learning and AI with an algorithmic approach.

• STK-INF3000/4000 Selected Topics in Data Science. The course provides insight into selected contemporary relevant topics within Data Science.

• IN4080 Natural Language Processing. Probabilistic and machine learning techniques applied to natural language processing.

Other courses on Data science and Machine Learning at UiO, contn#

• STK-IN4300 Statistical learning methods in Data Science. An advanced introduction to statistical and machine learning. For students with a good mathematics and statistics background.

• IN3310/4310 Deep Learnig for Image Analysis

• STK4051 Computational Statistics

• STK4021 Applied Bayesian Analysis and Numerical Methods

Learning outcomes#

• 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.

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.

Essential elements of ML#

The methods we cover have three main topics in common, irrespective of whether we deal with supervised or unsupervised learning.

• The first ingredient is normally our data set (which can be subdivided into training, 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.

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.

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, 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 $B$.

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 $\mathit{X}\in [x_1,x_2,\dots x_N]$.

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.

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.

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

1. 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

1. 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

1. sudo apt-get install python3 (or python for pyhton2.7)

etc etc.

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,

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

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 is a free Jupyter notebook environment that requires no setup and runs entirely in the cloud. Try it out!

Useful Python libraries#

Here we list several useful Python libraries we strongly recommend (if you use anaconda many of these are already there)

• NumPy 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 library provides high-performance, easy-to-use data structures and data analysis tools

• Xarray is a Python package that makes working with labelled multi-dimensional arrays simple, efficient, and fun!

• Scipy (pronounced “Sigh Pie”) is a Python-based ecosystem of open-source software for mathematics, science, and engineering.

• Matplotlib is a Python 2D plotting library which produces publication quality figures in a variety of hardcopy formats and interactive environments across platforms.

• 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 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 is a Python library for symbolic mathematics.

• scikit-learn has simple and efficient tools for machine learning, data mining and data analysis

• TensorFlow is a Python library for fast numerical computing created and released by Google

• Keras is a high-level neural networks API, written in Python and capable of running on top of TensorFlow, CNTK, or Theano

• Pytorch, highly recommened

• Theano and many other

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

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 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

    pycod jupyter nbconvert filename.ipynb --to latex 

And to add more versatility, the Python package SymPy 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.

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 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 http://www.netlib.org.

Numpy and arrays#

Numpy provides an easy way to handle arrays in Python. The standard way to import this library is as

import numpy as np

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,

n = 10
x = np.random.normal(size=n)
print(x)

[ 2.05896588  0.43642532 -0.21096409  0.40779804 -1.13963886 -0.53677418
  0.83572684  0.58831595  0.30882611 -2.07060477]

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

import numpy as np
x = np.array([1, 2, 3])
print(x)

[1 2 3]

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

import numpy as np
x = np.log(np.array([4, 7, 8]))
print(x)

[1.38629436 1.94591015 2.07944154]

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

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)

[1 1 2]

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

import numpy as np
x = np.log(np.array([4, 7, 8], dtype = np.float64))
print(x)

[1.38629436 1.94591015 2.07944154]

or simply write them as double precision numbers (Python uses 64 bits as default for floating point type variables), that is

import numpy as np
x = np.log(np.array([4.0, 7.0, 8.0]))
print(x)

[1.38629436 1.94591015 2.07944154]

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

import numpy as np
x = np.log(np.array([4.0, 7.0, 8.0]))
print(x.itemsize)

8

Matrices in Python#

Having defined vectors, we are now ready to try out matrices. We can define a $3\times 3$ real matrix $\mathit{A}$ as (recall that we user lowercase letters for vectors and uppercase letters for matrices)

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)

[[1.38629436 1.94591015 2.07944154]
 [1.09861229 2.30258509 2.39789527]
 [1.38629436 1.60943791 1.94591015]]

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

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])

[1.38629436 1.09861229 1.38629436]

We can continue this was by printing out other columns or rows. The example here prints out the second column

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,:])

[1.09861229 2.30258509 2.39789527]

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. 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

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)

[[0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]]

or initializing all elements to

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)

[[1. 1. 1. 1. 1. 1. 1. 1. 1. 1.]
 [1. 1. 1. 1. 1. 1. 1. 1. 1. 1.]
 [1. 1. 1. 1. 1. 1. 1. 1. 1. 1.]
 [1. 1. 1. 1. 1. 1. 1. 1. 1. 1.]
 [1. 1. 1. 1. 1. 1. 1. 1. 1. 1.]
 [1. 1. 1. 1. 1. 1. 1. 1. 1. 1.]
 [1. 1. 1. 1. 1. 1. 1. 1. 1. 1.]
 [1. 1. 1. 1. 1. 1. 1. 1. 1. 1.]
 [1. 1. 1. 1. 1. 1. 1. 1. 1. 1.]
 [1. 1. 1. 1. 1. 1. 1. 1. 1. 1.]]

or as unitarily distributed random numbers (see the material on random number generators in the statistics part)

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)

[[4.62790954e-01 3.05355909e-01 4.36307837e-01 5.75663043e-01
  8.47455900e-01 8.88156953e-01 3.41214900e-01 4.80217592e-01
  3.51294625e-01 4.55894547e-01]
 [9.36928932e-01 1.34614286e-01 1.68706337e-01 5.12168283e-01
  1.82057187e-01 5.04384702e-01 3.90807328e-01 7.21325310e-01
  6.82139912e-04 7.75656839e-01]
 [7.39975586e-01 7.28371709e-02 4.47008569e-01 1.48689896e-01
  8.77801149e-01 8.77111122e-01 6.14302755e-01 3.05347541e-01
  5.22304438e-01 1.19480182e-01]
 [3.75927527e-01 1.79526117e-01 1.10245050e-01 2.14621443e-02
  2.18149821e-01 6.60754644e-01 7.14889592e-01 2.00365236e-01
  1.87081411e-01 8.98532247e-01]
 [9.76246245e-01 7.27600064e-02 6.40428736e-01 2.44633057e-01
  2.13315320e-01 9.04923596e-01 1.01278843e-01 1.60040487e-02
  9.57338433e-01 4.54609970e-01]
 [9.70716394e-01 8.19560587e-01 8.00696646e-01 8.50286966e-01
  6.01413231e-01 5.35401167e-01 5.09318595e-01 7.50871033e-01
  1.77320278e-01 1.41563755e-01]
 [5.58999323e-01 8.27823274e-02 2.45162718e-01 9.35805174e-01
  3.97967922e-01 6.41998971e-01 2.85999165e-01 9.42305845e-01
  6.51639460e-01 2.40037335e-01]
 [2.07463601e-01 6.36551310e-01 3.74118364e-01 9.48754846e-01
  6.84043753e-01 8.69630846e-01 2.90701292e-01 3.73794006e-01
  9.56898405e-01 7.53969725e-01]
 [8.69421667e-01 9.16907984e-01 2.60498453e-01 1.05394015e-01
  8.28209931e-01 2.18273724e-01 5.70725653e-01 5.90412777e-01
  3.15706255e-01 1.00031589e-01]
 [3.13129360e-02 9.74280470e-01 1.56927535e-01 2.05412497e-01
  3.29614274e-01 1.32235211e-01 9.66580589e-01 6.25369588e-01
  5.91506268e-01 3.63969272e-01]]

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 $\mathit{x},\mathit{y},\mathit{z}$ with $n$ elements each. The covariance matrix is defined as

where for example

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 $\mathit{W}$

which in turn is converted into into the $3\times 3$ covariance matrix $\mathbf{\Sigma }$ via the Numpy function np.cov(). We note that we can also calculate the mean value of each set of samples $\mathit{x}$ etc using the Numpy function np.mean(x). We can also extract the eigenvalues of the covariance matrix through the np.linalg.eig() function.

# 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)

-0.022796599742659046
4.026047728952776
-0.08626455960524361
[[ 1.07037904  3.23910863  3.46675556]
 [ 3.23910863 10.63080397 10.39964939]
 [ 3.46675556 10.39964939 18.1421739 ]]
[26.31853116  0.06730967  3.45751608]

%matplotlib inline

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()

[[1. 0. 0. 0.]
 [0. 1. 0. 0.]
 [0. 0. 1. 0.]
 [0. 0. 0. 1.]]
  (0, 0)	1.0
  (1, 1)	1.0
  (2, 2)	1.0
  (3, 3)	1.0

Meet the Pandas#

Figure 1:

Another useful Python package is pandas, 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.

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)

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

data_pandas = pd.DataFrame(data,index=['Frodo','Bilbo','Aragorn','Sam'])
display(data_pandas)

Thereafter we display the content of the row which begins with the index Aragorn

display(data_pandas.loc['Aragorn'])

First Name            Aragorn II
Last Name                Elessar
Place of birth           Eriador
Date of Birth T.A.          2931
Name: Aragorn, dtype: object

We can easily append data to this, for example

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)

---------------------------------------------------------------------------
AttributeError                            Traceback (most recent call last)
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23270/1326197715.py in ?()
----> 6 new_hobbit = {'First Name': ["Peregrin"],
      7               'Last Name': ["Took"],
      8               'Place of birth': ["Shire"],
      9               'Date of Birth T.A.': [2990]

~/miniforge3/envs/myenv/lib/python3.9/site-packages/pandas/core/generic.py in ?(self, name)
   6200             and name not in self._accessors
   6201             and self._info_axis._can_hold_identifiers_and_holds_name(name)
   6202         ):
   6203             return self[name]
-> 6204         return object.__getattribute__(self, name)

AttributeError: 'DataFrame' object has no attribute 'append'

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.

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)

Thereafter we can select specific columns only and plot final results

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()

We can produce a $4\times 4$ matrix

b = np.arange(16).reshape((4,4))
print(b)
df1 = pd.DataFrame(b)
print(df1)

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. 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.

Pandas AI#

Try out Pandas AI

# 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()

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()

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()

# 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')

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)

"""                                                                                                                         
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.                                                          
"""

# 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']
print(Masses)

# 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)

clf = skl.LinearRegression().fit(X, Energies)
fity = clf.predict(X)

# 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()

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)

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.

Why Linear Regression (aka Ordinary Least Squares and family)#

Fitting a continuous function with linear parameterization in terms of the parameters $\mathit{\beta }$.

• 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 $\mathit{\beta }$

• 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 article is highly recommended. Similarly, Mehta et al’s article is also recommended.

Regression analysis, overarching aims#

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 $\mathit{x}=[x_0,x_1,\dots ,x_{n-1}{]}^T$. The first variable is called the dependent, the outcome or the response variable while the set of variables $\mathit{x}$ 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 $p(\mathit{y}|\mathit{x})$ or in the more traditional sense a function $\mathit{y}(\mathit{x})$, that is the conditional distribution for $\mathit{y}$ with a given $\mathit{x}$. The estimation of $p(\mathit{y}|\mathit{x})$ 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 ${\mathit{x}}_i=[x_{i0},x_{i1},\dots ,x_{ip-1}]$ 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 $\mathit{y}$ and $\mathit{x}$ in order to infer specific dependencies, approximations to the likelihood functions, functional relationships and to make predictions, making fits and many other things.

Regression analysis, overarching aims II#

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 $\mathit{y}$ (also as above). The variable $\mathit{y}$ is generally referred to as the response variable. The aim of regression analysis is to explain $\mathit{y}$ in terms of $\mathit{X}$ 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 $\mathit{X}$ and $\mathit{y}$. This assumption gives rise to the linear regression model where $\mathit{\beta }=[{\beta }_0,\dots ,{\beta }_{p-1}{]}^T$ are the regression parameters.

Linear regression gives us a set of analytical equations for the parameters ${\beta }_j$.

Examples#

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) $\mathit{y}$, 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

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 $\mathit{X}$.

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. 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.

General linear models and linear algebra#

Before we proceed let us study a case where we aim at fitting a set of data $\mathit{y}=[y_0,y_1,\dots ,y_{n-1}]$. 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 $\mathit{x}=[x_0,x_1,\dots ,x_{n-1}]$, 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

where ${ϵ}_i$ is the error in our approximation.

Rewriting the fitting procedure as a linear algebra problem#

For every set of values $y_i,x_i$ we have thus the corresponding set of equations

Rewriting the fitting procedure as a linear algebra problem, more details#

Defining the vectors

and

and

and the design matrix

we can rewrite our equations as

The above design matrix is called a Vandermonde matrix.

Generalizing the fitting procedure as a linear algebra problem#

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

Note that we have $p=n$ here. The matrix is symmetric. This is generally not the case!

Generalizing the fitting procedure as a linear algebra problem#

We redefine in turn the matrix $\mathit{X}$ as

and without loss of generality we rewrite again our equations as

The left-hand side of this equation is kwown. Our error vector $\mathit{ϵ}$ and the parameter vector $\mathit{\beta }$ are our unknow quantities. How can we obtain the optimal set of ${\beta }_i$ values?

Optimizing our parameters#

We have defined the matrix $\mathit{X}$ via the equations

As we noted above, we stayed with a system with the design matrix $\mathit{X}\in {\mathbb{R}}^{n\times n}$, that is we have $p=n$. For reasons to come later (algorithmic arguments) we will hereafter define our matrix as $\mathit{X}\in {\mathbb{R}}^{n\times p}$, with the predictors refering to the column numbers and the entries $n$ being the row elements.

Our model for the nuclear binding energies#

In our introductory notes we looked at the so-called liquid drop model. 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.

# 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)