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Week 34: Introduction to the course, Logistics and Practicalities

Morten Hjorth-Jensen, Department of Physics and Center for Computing in Science Education, University of Oslo, Norway

Date: Week 34, August 18-22, 2025

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.

  1. On Mondays we have a regular lecture which will be organized as a mix of active learning sessions and regular lectures. These lectures/active learning sessions start at 215pm and end at 4pm and serve the aims of giving an overview over various topics as well as solving specific problems. These lectures will also be recorded. Lectures can be attended in person or via zoom at https://uio.zoom.us/my/mortenhj

Videos and learning material with reading suggestions will be made available before each week starts.

Schedule first week

  • August 18: Lecture: Presentation of course, Linear regression, examples and theory

  • August 19: Introduction to software and repetition of Python Programming, linear algebra and basic elements of statistics. Please select group.

  • August 20: 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 (2.15pm-4pm)

  • 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>

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

Deadlines for projects (tentative)

  1. Project 1: October 6 (available September 1) graded with feedback)

  2. Project 2: November 3 (available October 6, graded with feedback)

  3. Project 3: December 8 (available November 3, 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.

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

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.

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.

Other courses on Data science and Machine Learning at UiO

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

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.

The plethora of machine learning algorithms/methods

  1. Deep learning: Neural Networks (NNs), Convolutional NNs, Recurrent NNs, Transformers, Boltzmann machines, autoencoders and variational autoencoders and generative adversarial networks and other generative models

  2. Bayesian statistics and Bayesian Machine Learning, Bayesian experimental design, Bayesian Regression models, Bayesian neural networks, Gaussian processes and much more

  3. Dimensionality reduction (Principal component analysis), Clustering Methods and more

  4. Ensemble Methods, Random forests, bagging and voting methods, gradient boosting approaches

  5. Linear and logistic regression, Kernel methods, support vector machines and more

  6. Reinforcement Learning; Transfer Learning and more

What Is Generative Modeling?

Generative modeling can be broadly defined as follows:

Generative modeling is a branch of machine learning that involves training a model to produce new data that is similar to a given dataset.

What does this mean in practice? Suppose we have a dataset containing photos of horses. We can train a generative model on this dataset to capture the rules that govern the complex relationships between pixels in images of horses. Then we can sample from this model to create novel, realistic images of horses that did not exist in the original dataset.

Example of generative modeling, taken from Generative Deep Learning by David Foster

Figure 1:

Generative Versus Discriminative Modeling

In order to truly understand what generative modeling aims to achieve and why this is important, it is useful to compare it to its counterpart, discriminative modeling. If you have studied machine learning, most problems you will have faced will have most likely been discriminative in nature.

Example of discriminative modeling, taken from Generative Deep Learning by David Foster

Figure 1:

Discriminative Modeling

When performing discriminative modeling, each observation in the training data has a label. For a binary classification problem such as our data could be labeled as ones and zeros. Our model then learns how to discriminate between these two groups and outputs the probability that a new observation has label 1 or 0

In contrast, generative modeling doesn’t require the dataset to be labeled because it concerns itself with generating entirely new data (for example an image), rather than trying to predict a label for say a given image.

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 \boldsymbol{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

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.

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

In [1]:
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,

In [3]:
n = 100
x = np.random.normal(size=n)
print(x)
[ 0.22889528  0.91634536  0.58026357 -0.20121369  1.63697022 -0.16620476
 -1.88888726 -2.65872949 -0.00563363  1.25894439 -0.07825328  1.43521368
  0.97253438 -1.64819989  0.98672233 -1.73087706 -0.99593933 -1.34926433
 -1.26715086 -0.95021858 -0.33244987  0.69501728 -0.48914472  1.87699249
 -0.91130987 -0.2889529  -0.74183199  0.74935175  1.34457198 -1.70306077
 -0.21142106  2.10793224 -0.40622194  2.34515157 -0.40761485  1.69110258
  0.72886914 -0.43700299 -1.14714793  0.15694733  0.66461823  0.89304652
  1.4543903  -0.37398506 -0.14684297 -0.57722604 -0.44036741 -0.40055163
  0.21510519 -0.53120826 -0.43941984  0.68555857  0.85074613 -0.8371588
  1.19968839  0.59476652  1.09647082  0.67324848  0.5365493  -0.11823534
  0.52454391  1.43031082  0.22692571 -0.87295854 -2.82106202 -0.27436094
  0.51321315  0.51606513 -1.56397437 -1.23442784  0.43472146  0.74179877
  0.73562128  1.00883303 -0.05851042 -1.19276526  1.16715641 -0.6401151
  0.81572064  1.0670422  -0.38749302 -1.67392967 -2.76996862  0.34870472
  1.3352494  -1.23489158 -1.70233983  0.62477383  0.14297718  0.23248733
  0.26787549  1.75420536 -1.22660411 -0.27597454  0.28073813  0.42879538
  1.64568937 -0.19205253  1.9311968  -0.26983672]

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

In [5]:
import numpy as np
x = np.array([1, 2, 4])
print(x)
[1 2 4]

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

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

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

In [5]:
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)

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

In [6]:
import numpy as np
x = np.log(np.array([4, 7, 8], dtype = np.float64))
print(x)

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

In [7]:
import numpy as np
x = np.log(np.array([4.0, 7.0, 8.0]))
print(x)

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

In [8]:
import numpy as np
x = np.log(np.array([4.0, 7.0, 8.0]))
print(x.itemsize)

Matrices in Python

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

In [9]:
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)

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

In [10]:
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])

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

In [11]:
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,:])

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

In [12]:
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)

or initializing all elements to

In [13]:
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)

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

In [14]:
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)

As we will see throughout these lectures, there are several extremely useful functionalities in Numpy. As an example, consider the discussion of the covariance matrix. Suppose we have defined three vectors \boldsymbol{x}, \boldsymbol{y}, \boldsymbol{z} with n elements each. The covariance matrix is defined as


\boldsymbol{\Sigma} = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\
                              \sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\
                              \sigma_{zx} & \sigma_{zy} & \sigma_{zz} 
             \end{bmatrix},

where for example


\sigma_{xy} =\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})(y_i- \overline{y}).

The Numpy function 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 \boldsymbol{W}


\boldsymbol{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},

which in turn is converted into into the 3\times 3 covariance matrix \boldsymbol{\Sigma} via the Numpy function np.cov(). We note that we can also calculate the mean value of each set of samples \boldsymbol{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.

In [15]:
# 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)
In [16]:
%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()

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.

In [1]:
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)
First Name Last Name Place of birth Date of Birth T.A.
0 Frodo Baggins Shire 2968
1 Bilbo Baggins Shire 2890
2 Aragorn II Elessar Eriador 2931
3 Samwise Gamgee Shire 2980

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

In [18]:
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

In [19]:
display(data_pandas.loc['Aragorn'])

We can easily append data to this, for example

In [20]:
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)

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.

In [2]:
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)
0 1 2 3 4
0 -1.749765 0.342680 1.153036 -0.252436 0.981321
1 0.514219 0.221180 -1.070043 -0.189496 0.255001
2 -0.458027 0.435163 -0.583595 0.816847 0.672721
3 -0.104411 -0.531280 1.029733 -0.438136 -1.118318
4 1.618982 1.541605 -0.251879 -0.842436 0.184519
5 0.937082 0.731000 1.361556 -0.326238 0.055676
6 0.222400 -1.443217 -0.756352 0.816454 0.750445
7 -0.455947 1.189622 -1.690617 -1.356399 -1.232435
8 -0.544439 -0.668172 0.007315 -0.612939 1.299748
9 -1.733096 -0.983310 0.357508 -1.613579 1.470714 
0   -0.175300
1    0.083527
2   -0.044334
3   -0.399836
4    0.331939
dtype: float64
0    1.069584
1    0.965548
2    1.018232
3    0.793167
4    0.918992
dtype: float64
0 1 2 3 4
0 3.061679 0.117430 1.329492 0.063724 0.962990
1 0.264421 0.048920 1.144993 0.035909 0.065026
2 0.209789 0.189367 0.340583 0.667239 0.452553
3 0.010902 0.282259 1.060349 0.191963 1.250636
4 2.621102 2.376547 0.063443 0.709698 0.034047
5 0.878123 0.534362 1.853835 0.106431 0.003100
6 0.049462 2.082875 0.572069 0.666597 0.563167
7 0.207888 1.415201 2.858185 1.839818 1.518895
8 0.296414 0.446453 0.000054 0.375694 1.689345
9 3.003620 0.966899 0.127812 2.603636 2.162999

Thereafter we can select specific columns only and plot final results

In [6]:
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()
---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
Cell In[6], line 1
----> 1 df.columns = ['First', 'Second', 'Third', 'Fourth', 'Fifth']
      2 df.index = np.arange(10)
      4 display(df)

NameError: name 'df' is not defined

We can produce a 4\times 4 matrix

In [23]:
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

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 \boldsymbol{x} are given by random numbers generated with a uniform distribution with entries x_i \in [0,1] (more about probability distribution functions later). These values are then used to define a function y(x) (tabulated again as a vector) with a linear dependence on x plus a random noise added via the normal distribution.

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


y = 2x+N(0,1),

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 (\boldsymbol{x},\boldsymbol{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 which produces publication quality figures. Feel free to explore the extensive gallery 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.

In [24]:
# 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()
Warning:
Output truncated. This notebook contains too many cells to display efficiently.
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