FYS-STK4155/doc/LectureNotes/book.do.txt

TITLE: Data Analysis and Machine Learning
AUTHOR: Morten Hjorth-Jensen {copyright, 1999-present|CC BY-NC} at Department of Physics, University of Oslo & Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
DATE: today

TOC: on


======= Introduction =======


During the last two decades there has been a swift and amazing
development of Machine Learning techniques and algorithms that impact
many areas in not only Science and Technology but also the Humanities,
Social Sciences, Medicine, Law, indeed, almost all possible
disciplines. The applications are incredibly many, from self-driving
cars to solving high-dimensional differential equations or complicated
quantum mechanical many-body problems. Machine Learning is perceived
by many as one of the main disruptive techniques nowadays. 

Statistics, Data science and Machine Learning form important
fields of research in modern science.  They describe how to learn and
make predictions from data, as well as allowing us to extract
important correlations about physical process and the underlying laws
of motion in large data sets. The latter, big data sets, appear
frequently in essentially all disciplines, from the traditional
Science, Technology, Mathematics and Engineering fields to Life
Science, Law, education research, the Humanities and the Social
Sciences.

It has become more
and more common to see research projects on big data in for example
the Social Sciences where extracting patterns from complicated survey
data is one of many research directions.  Having a solid grasp of data
analysis and machine learning is thus becoming central to scientific
computing in many fields, and competences and skills within the fields
of machine learning and scientific computing are nowadays strongly
requested by many potential employers. The latter cannot be
overstated, familiarity with machine learning has almost become a
prerequisite for many of the most exciting employment opportunities,
whether they are in bioinformatics, life science, physics or finance,
in the private or the public sector. This author has had several
students or met students who have been hired recently based on their
skills and competences in scientific computing and data science, often
with marginal knowledge of machine learning.

Machine learning is a subfield of computer science, and is closely
related to computational statistics.  It evolved from the study of
pattern recognition in artificial intelligence (AI) research, and has
made contributions to AI tasks like computer vision, natural language
processing and speech recognition. Many of the methods we will study are also 
strongly rooted in basic mathematics and physics research. 

Ideally, machine learning represents the science of giving computers
the ability to learn without being explicitly programmed.  The idea is
that there exist generic algorithms which can be used to find patterns
in a broad class of data sets without having to write code
specifically for each problem. The algorithm will build its own logic
based on the data.  You should however always keep in mind that
machines and algorithms are to a large extent developed by humans. The
insights and knowledge we have about a specific system, play a central
role when we develop a specific machine learning algorithm. 

Machine learning is an extremely rich field, in spite of its young
age. The increases we have seen during the last three decades in
computational capabilities have been followed by developments of
methods and techniques for analyzing and handling large date sets,
relying heavily on statistics, computer science and mathematics.  The
field is rather new and developing rapidly. Popular software packages
written in Python for machine learning like
"Scikit-learn":"http://scikit-learn.org/stable/",
"Tensorflow":"https://www.tensorflow.org/",
"PyTorch":"http://pytorch.org/" and "Keras":"https://keras.io/", all
freely available at their respective GitHub sites, encompass
communities of developers in the thousands or more. And the number of
code developers and contributors keeps increasing. Not all the
algorithms and methods can be given a rigorous mathematical
justification, opening up thereby large rooms for experimenting and
trial and error and thereby exciting new developments.  However, a
solid command of linear algebra, multivariate theory, probability
theory, statistical data analysis, understanding errors and Monte
Carlo methods are central elements in a proper understanding of many
of algorithms and methods we will discuss.


===== Learning outcomes =====

These sets of lectures aim at giving you an overview of central aspects of
statistical data analysis as well as some of the central algorithms
used in machine learning.  We will introduce a variety of central
algorithms and methods essential for studies of data analysis and
machine learning. 

Hands-on projects and experimenting with data and algorithms plays a central role in
these lectures, and our hope is, through the various
projects and exercises, to  expose you to fundamental
research problems in these fields, with the aim to reproduce state of
the art scientific results. You  will learn to develop and
structure codes for studying these systems, get acquainted with
computing facilities and learn to handle large scientific projects. A
good scientific and ethical conduct is emphasized throughout the
course. More specifically, you will

o Learn about basic data analysis, Bayesian statistics, Monte Carlo methods, data optimization and machine learning;
o Be capable of extending the acquired knowledge to other systems and cases;
o Have an understanding of central algorithms used in data analysis and machine learning;
o Gain knowledge of central aspects of Monte Carlo methods, Markov chains, Gibbs samplers and their possible applications, from numerical integration to simulation of stock markets;
o Understand methods for regression and classification;
o Learn about neural network, genetic algorithms and Boltzmann machines;
o 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++, in addition to a basic knowledge of linear algebra (typically taught during the first one or two years of undergraduate studies).

There are several topics we will cover here, spanning from 
statistical data analysis and its basic concepts such as expectation
values, variance, covariance, correlation functions and errors, via
well-known probability distribution functions like the uniform
distribution, the binomial distribution, the Poisson distribution and
simple and multivariate normal distributions to central elements of
Bayesian statistics and modeling. We will also remind the reader about
central elements from linear algebra and standard methods based on
linear algebra used to optimize (minimize) functions (the family of gradient descent methods)
and the Singular-value decomposition and
least square methods for parameterizing data.

We will also cover Monte Carlo methods, Markov chains, well-known
algorithms for sampling stochastic events like the Metropolis-Hastings
and Gibbs sampling methods. An important aspect of all our
calculations is a proper estimation of errors. Here we will also
discuss famous resampling techniques like the blocking, the bootstrapping
and the jackknife methods and the infamous bias-variance tradeoff. 

The second part of the material covers several algorithms used in
machine learning.






===== 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.


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 and test data), 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_
function which allows us to present an estimate on how good our model
is in reproducing the data it is supposed to train.  

Here we will build our machine learning approach on elements of the
statistical foundation discussed above, with elements from data
analysis, stochastic processes etc. We will discuss the following
machine learning algorithms

o Linear regression and its variants
o Decision tree algorithms, from single trees to random forests
o Bayesian statistics and regression
o Support vector machines and finally various variants of
o Artifical neural networks and deep learning, including convolutional neural networks and Bayesian neural networks
o Networks for unsupervised learning using for example reduced Boltzmann machines.



===== Choice of programming language =====

Python plays nowadays a central role in the development of machine
learning techniques and tools for data analysis. In particular, seen
the wealth of machine learning and data analysis libraries written in
Python, easy to use libraries with immediate visualization(and not the
least impressive galleries of existing examples), the popularity of the
Jupyter notebook framework with the possibility to run _R_ codes or
compiled programs written in C++, and much more made our choice of
programming language for this series of lectures easy. However,
since the focus here is not only on using existing Python libraries such
as _Scikit-Learn_ or _Tensorflow_, but also on developing your own
algorithms and codes, we will as far as possible present many of these
algorithms either as a Python codes or C++ or Fortran (or other languages) codes. 

The reason we also  focus on compiled languages like C++ (or
Fortran), is that Python is still notoriously slow when we do not
utilize highly streamlined computational libraries like
"Lapack":"http://www.netlib.org/lapack/" or other numerical libraries
written in compiled languages (many of these libraries are written in
Fortran).  Although a project like "Numba":"https://numba.pydata.org/"
holds great promise for speeding up the unrolling of lengthy loops, C++
and Fortran are presently still the performance winners. Numba gives
you potentially the power to speed up your applications with high
performance functions written directly in Python. In particular,
array-oriented and math-heavy Python code can achieve similar
performance to C, C++ and Fortran. However, even with these speed-ups,
for codes involving heavy Markov Chain Monte Carlo analyses and
optimizations of cost functions, C++/C or Fortran codes tend to
outperform Python codes.  

Presently thus, the community tends to let
code written in C++/C or Fortran do the heavy duty numerical
number crunching and leave the post-analysis of the data to the above
mentioned Python modules or software packages.  However, with the developments taking place in for example the Python community, and seen
the changes during the last decade, the above situation may change swiftly in the not too distant future. 

Many of the examples we discuss in this series of lectures come with
existing data files or provide code examples which produce the data to
be analyzed.  Most of the  applications we will discuss deal with
small data sets (less than a terabyte of information) and can easily
be analyzed and tested  on standard off the shelf laptops you find in general 
stores.

===== Data handling, machine learning  and ethical aspects =====

In most of the cases we will study, we will either generate the data
to analyze ourselves (both for supervised learning and unsupervised
learning) or we will recur again and again to data present in say
_Scikit-Learn_ or _Tensorflow_.  Many of the examples we end up
dealing with are from a privacy and data protection point of view,
rather inoccuous and boring results of numerical
calculations. However, this does not hinder us from developing a sound
ethical attitude to the data we use, how we analyze the data and how
we handle the data.

The most immediate and simplest possible ethical aspects deal with our
approach to the scientific process. Nowadays, with version control
software like "Git":"https://git-scm.com/" and various online
repositories like "Github":"https://github.com/",
"Gitlab":"https://about.gitlab.com/" etc, we can easily make our codes
and data sets we have used, freely and easily accessible to a wider
community. This helps us almost automagically in making our science
reproducible. The large open-source development communities involved
in say "Scikit-Learn":"http://scikit-learn.org/stable/",
"Tensorflow":"https://www.tensorflow.org/",
"PyTorch":"http://pytorch.org/" and "Keras":"https://keras.io/", are
all excellent examples of this. The codes can be tested and improved
upon continuosly, helping thereby our scientific community at large in
developing data analysis and machine learning tools.  It is much
easier today to gain traction and acceptance for making your science
reproducible. From a societal stand, this is an important element
since many of the developers are employees of large public institutions like
universities and research labs. Our fellow taxpayers do deserve to get
something back for their bucks.

However, this more mechanical aspect of the ethics of science (in
particular the reproducibility of scientific results) is something
which is obvious and everybody should do so as part of the dialectics of
science.  The fact that many scientists are not willing to share their codes or 
data is detrimental to the scientific discourse.

Before we proceed, we should add a disclaimer. Even though
we may dream of computers developing some kind of higher learning
capabilities, at the end (even if the artificial intelligence
community keeps touting our ears full of fancy futuristic avenues), it is we, yes you reading these lines,
who end up constructing and instructing, via various algorithms, the
machine learning approaches. Self-driving cars for example, rely on sofisticated
programs which take into account all possible situations a car can
encounter. In addition, extensive usage of training data from GPS
information, maps etc, are typically fed into the software for
self-driving cars. Adding to this various sensors and cameras that
feed information to the programs, there are zillions of ethical issues
which arise from this.

For self-driving cars, where basically many of the standard machine
learning algorithms discussed here enter into the codes, at a certain
stage we have to make choices. Yes, we , the lads and lasses who wrote
a program for a specific brand of a self-driving car.  As an example,
all carmakers have as their utmost priority the security of the
driver and the accompanying passengers. A famous European carmaker, which is
one of the leaders in the market of self-driving cars, had _if_
statements of the following type: suppose there are two obstacles in
front of you and you cannot avoid to collide with one of them. One of
the obstacles is a monstertruck while the other one is a kindergarten
class trying to cross the road. The self-driving car algo would then
opt for the hitting the small folks instead of the monstertruck, since
the likelihood of surving a collision with our future citizens, is
much higher.

This leads to serious ethical aspects. Why should we opt for such an
option? Who decides and who is entitled to make such choices? Keep in
mind that many of the algorithms you will encounter in this series of
lectures or hear about later, are indeed based on simple programming
instructions. And you are very likely to be one of the people who may
end up writing such a code. Thus, developing a sound ethical attitude
to what we do, an approach well beyond the simple mechanistic one of
making our science available and reproducible, is much needed. The
example of the self-driving cars is just one of infinitely many cases
where we have to make choices. When you analyze data on economic
inequalities, who guarantees that you are not weighting some data in a
particular way, perhaps because you dearly want a specific conclusion
which may support your political views? Or what about the recent
claims that a famous IT company like Apple has a sexist bias on the
their recently "launched credit card":"https://qz.com/1748321/the-role-of-goldman-sachs-algorithms-in-the-apple-credit-card-scandal/"?

We do not have the answers here, nor will we venture into a deeper
discussions of these aspects, but we want you think over these topics
in a more overarching way.  A statistical data analysis with its dry
numbers and graphs meant to guide the eye, does not necessarily
reflect the truth, whatever that is.  As a scientist, and after a
university education, you are supposedly a better citizen, with an
improved critical view and understanding of the scientific method, and
perhaps some deeper understanding of the ethics of science at
large. Use these insights. Be a critical citizen. You owe it to our
society.


======= Machine Learning Overview with Selectec Examples ======= 


===== Introduction =====

Our emphasis throughout this series of lectures  
is on understanding the mathematical aspects of
different algorithms used in the fields of data analysis and machine learning. 

However, where possible we will emphasize the
importance of using available software. We start thus with a hands-on
and top-down approach to machine learning. The aim is thus to start with
relevant data or data we have produced 
and use these to introduce statistical data analysis
concepts and machine learning algorithms before we delve into the
algorithms themselves. The examples we will use in the beginning, start with simple
polynomials with random noise added. We will use the Python
software package "Scikit-Learn":"http://scikit-learn.org/stable/" and
introduce various machine learning algorithms to make fits of
the data and predictions. We move thereafter to more interesting
cases such as data from say experiments (below we will look at experimental nuclear binding energies as an example).
These are examples where we can easily set up the data and
then use machine learning algorithms included in for example
_Scikit-Learn_. 

These examples will serve us the purpose of getting
started. Furthermore, they allow us to catch more than two birds with
a stone. They will allow us to bring in some programming specific
topics and tools as well as showing the power of various Python 
libraries for machine learning and statistical data analysis.  

Here, we will mainly focus on two
specific Python packages for Machine Learning, Scikit-Learn and
Tensorflow (see below for links etc).  Moreover, the examples we
introduce will serve as inputs to many of our discussions later, as
well as allowing you to set up models and produce your own data and
get started with programming.



===== What is Machine Learning?  =====

Statistics, data science and machine learning form important fields of
research in modern science.  They describe how to learn and make
predictions from data, as well as allowing us to extract important
correlations about physical process and the underlying laws of motion
in large data sets. The latter, big data sets, appear frequently in
essentially all disciplines, from the traditional Science, Technology,
Mathematics and Engineering fields to Life Science, Law, education
research, the Humanities and the Social Sciences. 

It has become more
and more common to see research projects on big data in for example
the Social Sciences where extracting patterns from complicated survey
data is one of many research directions.  Having a solid grasp of data
analysis and machine learning is thus becoming central to scientific
computing in many fields, and competences and skills within the fields
of machine learning and scientific computing are nowadays strongly
requested by many potential employers. The latter cannot be
overstated, familiarity with machine learning has almost become a
prerequisite for many of the most exciting employment opportunities,
whether they are in bioinformatics, life science, physics or finance,
in the private or the public sector. This author has had several
students or met students who have been hired recently based on their
skills and competences in scientific computing and data science, often
with marginal knowledge of machine learning.

Machine learning is a subfield of computer science, and is closely
related to computational statistics.  It evolved from the study of
pattern recognition in artificial intelligence (AI) research, and has
made contributions to AI tasks like computer vision, natural language
processing and speech recognition. Many of the methods we will study are also 
strongly rooted in basic mathematics and physics research. 

Ideally, machine learning represents the science of giving computers
the ability to learn without being explicitly programmed.  The idea is
that there exist generic algorithms which can be used to find patterns
in a broad class of data sets without having to write code
specifically for each problem. The algorithm will build its own logic
based on the data.  You should however always keep in mind that
machines and algorithms are to a large extent developed by humans. The
insights and knowledge we have about a specific system, play a central
role when we develop a specific machine learning algorithm. 

Machine learning is an extremely rich field, in spite of its young
age. The increases we have seen during the last three decades in
computational capabilities have been followed by developments of
methods and techniques for analyzing and handling large date sets,
relying heavily on statistics, computer science and mathematics.  The
field is rather new and developing rapidly. Popular software packages
written in Python for machine learning like
"Scikit-learn":"http://scikit-learn.org/stable/",
"Tensorflow":"https://www.tensorflow.org/",
"PyTorch":"http://pytorch.org/" and "Keras":"https://keras.io/", all
freely available at their respective GitHub sites, encompass
communities of developers in the thousands or more. And the number of
code developers and contributors keeps increasing. Not all the
algorithms and methods can be given a rigorous mathematical
justification, opening up thereby large rooms for experimenting and
trial and error and thereby exciting new developments.  However, a
solid command of linear algebra, multivariate theory, probability
theory, statistical data analysis, understanding errors and Monte
Carlo methods are central elements in a proper understanding of many
of algorithms and methods we will discuss.



===== 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.


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 and test data), 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_
function which allows us to present an estimate on how good our model
is in reproducing the data it is supposed to train.  
At the heart of basically all ML algorithms there are so-called minimization algorithms, often we end up with various variants of _gradient_ methods.







===== Software and needed installations =====

We will make extensive use of Python as programming language and its
myriad of available libraries.  You will find
Jupyter notebooks invaluable in your work.  You can run _R_
codes in the Jupyter/IPython notebooks, with the immediate benefit of
visualizing your data. You can also use compiled languages like C++,
Rust, Julia, Fortran etc if you prefer. The focus in these lectures will be
on Python.


If you have Python installed (we strongly recommend Python3) and you feel
pretty familiar with installing different packages, we recommend that
you install the following Python packages via _pip_ as 

o pip install numpy scipy matplotlib ipython scikit-learn mglearn sympy pandas pillow 

For Python3, replace _pip_ with _pip3_.

For OSX users we recommend, after having installed Xcode, to
install _brew_. Brew allows for a seamless installation of additional
software via for example 

o brew install python3

For Linux users, with its variety of distributions like for example the widely popular Ubuntu distribution,
you can use _pip_ as well and simply install Python as 

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

etc etc. 



===== Python installers =====

If you don't want to perform these operations separately and venture
into the hassle of exploring how to set up dependencies and paths, we
recommend two widely used distrubutions which set up all relevant
dependencies for Python, namely 

* "Anaconda":"https://docs.anaconda.com/", 

which is an open source
distribution of the Python and R programming languages for large-scale
data processing, predictive analytics, and scientific computing, that
aims to simplify package management and deployment. Package versions
are managed by the package management system _conda_. 

* "Enthought canopy":"https://www.enthought.com/product/canopy/" 

is a Python
distribution for scientific and analytic computing distribution and
analysis environment, available for free and under a commercial
license.

Furthermore, "Google's Colab":"https://colab.research.google.com/notebooks/welcome.ipynb" is a free Jupyter notebook environment that requires 
no setup and runs entirely in the cloud. Try it out!

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

* "NumPy":"https://www.numpy.org/" is a highly popular library for large, multi-dimensional arrays and matrices, along with a large collection of high-level mathematical functions to operate on these arrays
* "The pandas":"https://pandas.pydata.org/" library provides high-performance, easy-to-use data structures and data analysis tools 
* "Xarray":"http://xarray.pydata.org/en/stable/" is a Python package that makes working with labelled multi-dimensional arrays simple, efficient, and fun!
* "Scipy":"https://www.scipy.org/" (pronounced “Sigh Pie”) is a Python-based ecosystem of open-source software for mathematics, science, and engineering. 
* "Matplotlib":"https://matplotlib.org/" is a Python 2D plotting library which produces publication quality figures in a variety of hardcopy formats and interactive environments across platforms.
* "Autograd":"https://github.com/HIPS/autograd" can automatically differentiate native Python and Numpy code. It can handle a large subset of Python's features, including loops, ifs, recursion and closures, and it can even take derivatives of derivatives of derivatives
* "SymPy":"https://www.sympy.org/en/index.html" is a Python library for symbolic mathematics. 
* "scikit-learn":"https://scikit-learn.org/stable/" has simple and efficient tools for machine learning, data mining and data analysis
* "TensorFlow":"https://www.tensorflow.org/" is a Python library for fast numerical computing created and released by Google
* "Keras":"https://keras.io/" is a high-level neural networks API, written in Python and capable of running on top of TensorFlow, CNTK, or Theano
* And many more such as "pytorch":"https://pytorch.org/",  "Theano":"https://pypi.org/project/Theano/" etc 

===== 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/Ipython
notebook allows you to run _R_ codes interactively in your
browser. The software library _R_ is really tailored  for statistical data analysis
and allows for an easy usage of the tools and algorithms we will discuss in these
lectures.

To install _R_ with Jupyter notebook 
"follow the link here":"https://mpacer.org/maths/r-kernel-for-ipython-notebook"




===== Installing R, C++, cython, Numba etc =====


For the C++ aficionados, Jupyter/IPython notebook allows you also to
install C++ and run codes written in this language interactively in
the browser. Since we will emphasize writing many of the algorithms
yourself, you can thus opt for either Python or C++ (or Fortran or other compiled languages) as programming
languages.

To add more entropy, _cython_ can also be used when running your
notebooks. It means that Python with the jupyter notebook
setup allows you to integrate widely popular softwares and tools for
scientific computing. Similarly, the 
"Numba Python package":"https://numba.pydata.org/" delivers increased performance
capabilities with minimal rewrites of your codes.  With its
versatility, including symbolic operations, Python offers a unique
computational environment. Your jupyter notebook can easily be
converted into a nicely rendered _PDF_ file or a Latex file for
further processing. For example, convert to latex as 

!bc 
pycod jupyter nbconvert filename.ipynb --to latex 
!ec

And to add more versatility, the Python package "SymPy":"http://www.sympy.org/en/index.html" is a Python library for symbolic mathematics. It aims to become a full-featured computer algebra system (CAS)  and is entirely written in Python. 

Finally, if you wish to use the light mark-up language 
"doconce":"https://github.com/hplgit/doconce" you can convert a standard ascii text file into various HTML 
formats, ipython notebooks, latex files, pdf files etc with minimal edits. These lectures were generated using _doconce_.



===== Numpy examples and Important Matrix and vector handling packages =====

There are several central software libraries for linear algebra and eigenvalue problems. Several of the more
popular ones have been wrapped into ofter software packages like those from the widely used text _Numerical Recipes_. The original source codes in many of the available packages are often taken from the widely used
software package LAPACK, which follows two other popular packages
developed in the 1970s, namely EISPACK and LINPACK.  We describe them shortly here.

  * LINPACK: package for linear equations and least square problems.
  * LAPACK:package for solving symmetric, unsymmetric and generalized eigenvalue problems. From LAPACK's website URL: "http://www.netlib.org" it is possible to download for free all source codes from this library. Both C/C++ and Fortran versions are available.
  * BLAS (I, II and III): (Basic Linear Algebra Subprograms) are routines that provide standard building blocks for performing basic vector and matrix operations. Blas I is vector operations, II vector-matrix operations and III matrix-matrix operations. Highly parallelized and efficient codes, all available for download from URL: "http://www.netlib.org".


===== Basic Matrix Features =====

!bblock Matrix properties reminder
!bt
\[
 \mathbf{A} =
      \begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\
                                 a_{21} & a_{22} & a_{23} & a_{24} \\
                                   a_{31} & a_{32} & a_{33} & a_{34} \\
                                  a_{41} & a_{42} & a_{43} & a_{44}
             \end{bmatrix}\qquad
\mathbf{I} =
      \begin{bmatrix} 1 & 0 & 0 & 0 \\
                                 0 & 1 & 0 & 0 \\
                                 0 & 0 & 1 & 0 \\
                                 0 & 0 & 0 & 1
             \end{bmatrix}
\]
!et



The inverse of a matrix is defined by

!bt
\[
\mathbf{A}^{-1} \cdot \mathbf{A} = I
\]
!et


|----------------------------------------------------------------------|
|       Relations      |       Name      | matrix elements             |
|----------------------------------------------------------------------|
| $A = A^{T}$         | symmetric       | $a_{ij} = a_{ji}$            |
| $A = \left (A^{T} \right )^{-1}$ | real orthogonal | $\sum_k a_{ik} a_{jk} = \sum_k a_{ki} a_{kj} = \delta_{ij}$ |
| $A = A^{ * }$          | real matrix     | $a_{ij} = a_{ij}^{ * }$       |
| $A = A^{\dagger}$     |  hermitian      | $a_{ij} = a_{ji}^{ * }$       |
| $A = \left (A^{\dagger} \right )^{-1}$ | unitary | $\sum_k a_{ik} a_{jk}^{ * } = \sum_k a_{ki}^{ * } a_{kj} = \delta_{ij}$ |
|----------------------------------------------------------------------|

!eblock


=== Some famous Matrices ===

  * Diagonal if $a_{ij}=0$ for $i\ne j$
  * Upper triangular if $a_{ij}=0$ for $i > j$
  * Lower triangular if $a_{ij}=0$ for $i < j$
  * Upper Hessenberg if $a_{ij}=0$ for $i > j+1$
  * Lower Hessenberg if $a_{ij}=0$ for $i < j+1$
  * Tridiagonal if $a_{ij}=0$ for $|i -j| > 1$
  * Lower banded with bandwidth $p$: $a_{ij}=0$ for $i > j+p$
  * Upper banded with bandwidth $p$: $a_{ij}=0$ for $i < j+p$
  * Banded, block upper triangular, block lower triangular....


=== More Basic Matrix Features ===

!bblock Some Equivalent Statements
For an $N\times N$ matrix  $\mathbf{A}$ the following properties are all equivalent

  * If the inverse of $\mathbf{A}$ exists, $\mathbf{A}$ is nonsingular.
  * The equation $\mathbf{Ax}=0$ implies $\mathbf{x}=0$.
  * The rows of $\mathbf{A}$ form a basis of $R^N$.
  * The columns of $\mathbf{A}$ form a basis of $R^N$.
  * $\mathbf{A}$ is a product of elementary matrices.
  * $0$ is not eigenvalue of $\mathbf{A}$.
!eblock


===== Numpy and arrays =====
"Numpy":"http://www.numpy.org/" provides an easy way to handle arrays in Python. The standard way to import this library is as

!bc pycod
import numpy as np
!ec
Here follows a simple example where we set up an array of ten elements, all determined by random numbers drawn according to the normal distribution,
!bc pycod
n = 10
x = np.random.normal(size=n)
print(x)
!ec
We defined a vector $x$ with $n=10$ elements with its values given by the Normal distribution $N(0,1)$.
Another alternative is to declare a vector as follows
!bc pycod
import numpy as np
x = np.array([1, 2, 3])
print(x)
!ec
Here we have defined a vector with three elements, with $x_0=1$, $x_1=2$ and $x_2=3$. Note that both Python and C++
start numbering array elements from $0$ and on. This means that a vector with $n$ elements has a sequence of entities $x_0, x_1, x_2, \dots, x_{n-1}$. We could also let (recommended) Numpy to compute the logarithms of a specific array as
!bc pycod
import numpy as np
x = np.log(np.array([4, 7, 8]))
print(x)
!ec

In the last example we used Numpy's unary function $np.log$. This function is
highly tuned to compute array elements since the code is vectorized
and does not require looping. We normaly recommend that you use the
Numpy intrinsic functions instead of the corresponding _log_ function
from Python's _math_ module. The looping is done explicitely by the
_np.log_ function. The alternative, and slower way to compute the
logarithms of a vector would be to write

!bc pycod
import numpy as np
from math import log
x = np.array([4, 7, 8])
for i in range(0, len(x)):
    x[i] = log(x[i])
print(x)
!ec
We note that our code is much longer already and we need to import the _log_ function from the _math_ module. 
The attentive reader will also notice that the output is $[1, 1, 2]$. Python interprets automagically our numbers as integers (like the _automatic_ keyword in C++). To change this we could define our array elements to be double precision numbers as
!bc pycod
import numpy as np
x = np.log(np.array([4, 7, 8], dtype = np.float64))
print(x)
!ec
or simply write them as double precision numbers (Python uses 64 bits as default for floating point type variables), that is
!bc pycod
import numpy as np
x = np.log(np.array([4.0, 7.0, 8.0])
print(x)
!ec
To check the number of bytes (remember that one byte contains eight bits for double precision variables), you can use simple use the _itemsize_ functionality (the array $x$ is actually an object which inherits the functionalities defined in Numpy) as 
!bc pycod
import numpy as np
x = np.log(np.array([4.0, 7.0, 8.0])
print(x.itemsize)
!ec


===== Matrices in Python =====

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

!bc pycod
import numpy as np
A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))
print(A)
!ec
If we use the _shape_ function we would get $(3, 3)$ as output, that is verifying that our matrix is a $3\times 3$ matrix. We can slice the matrix and print for example the first column (Python organized matrix elements in a row-major order, see below) as
!bc pycod
import numpy as np
A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))
# print the first column, row-major order and elements start with 0
print(A[:,0]) 
!ec
We can continue this was by printing out other columns or rows. The example here prints out the second column
!bc pycod
import numpy as np
A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))
# print the first column, row-major order and elements start with 0
print(A[1,:]) 
!ec 
Numpy contains many other functionalities that allow us to slice, subdivide etc etc arrays. We strongly recommend that you look up the "Numpy website for more details":"http://www.numpy.org/". Useful functions when defining a matrix are the _np.zeros_ function which declares a matrix of a given dimension and sets all elements to zero
!bc pycod
import numpy as np
n = 10
# define a matrix of dimension 10 x 10 and set all elements to zero
A = np.zeros( (n, n) )
print(A) 
!ec 
or initializing all elements to 
!bc pycod
import numpy as np
n = 10
# define a matrix of dimension 10 x 10 and set all elements to one
A = np.ones( (n, n) )
print(A) 
!ec 
or as unitarily distributed random numbers (see the material on random number generators in the statistics part)
!bc pycod
import numpy as np
n = 10
# define a matrix of dimension 10 x 10 and set all elements to random numbers with x \in [0, 1]
A = np.random.rand(n, n)
print(A) 
!ec 

As we will see throughout these lectures, there are several extremely useful functionalities in Numpy.
As an example, consider the discussion of the covariance matrix. Suppose we have defined three vectors
$\hat{x}, \hat{y}, \hat{z}$ with $n$ elements each. The covariance matrix is defined as 
!bt
\[
\hat{\Sigma} = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\
                              \sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\
                              \sigma_{zx} & \sigma_{zy} & \sigma_{zz} 
             \end{bmatrix},
\]
!et
where for example
!bt
\[
\sigma_{xy} =\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})(y_i- \overline{y}).
\]
!et
The Numpy function _np.cov_ calculates the covariance elements using the factor $1/(n-1)$ instead of $1/n$ since it assumes we do not have the exact mean values. 
The following simple function uses the _np.vstack_ function which takes each vector of dimension $1\times n$ and produces a $3\times n$ matrix $\hat{W}$
!bt
\[
\hat{W} = \begin{bmatrix} x_0 & y_0 & z_0 \\
                          x_1 & y_1 & z_1 \\
                          x_2 & y_2 & z_2 \\
                          \dots & \dots & \dots \\
                          x_{n-2} & y_{n-2} & z_{n-2} \\
                          x_{n-1} & y_{n-1} & z_{n-1}
             \end{bmatrix},
\]
!et

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

!bc pycod
# Importing various packages
import numpy as np

n = 100
x = np.random.normal(size=n)
print(np.mean(x))
y = 4+3*x+np.random.normal(size=n)
print(np.mean(y))
z = x**3+np.random.normal(size=n)
print(np.mean(z))
W = np.vstack((x, y, z))
Sigma = np.cov(W)
print(Sigma)
Eigvals, Eigvecs = np.linalg.eig(Sigma)
print(Eigvals)
!ec


!bc pycod
import numpy as np
import matplotlib.pyplot as plt
from scipy import sparse
eye = np.eye(4)
print(eye)
sparse_mtx = sparse.csr_matrix(eye)
print(sparse_mtx)
x = np.linspace(-10,10,100)
y = np.sin(x)
plt.plot(x,y,marker='x')
plt.show()
!ec


===== Meet the Pandas =====


FIGURE: [fig/pandas.jpg, width=600 frac=0.8]

Another useful Python package is
"pandas":"https://pandas.pydata.org/", which is an open source library
providing high-performance, easy-to-use data structures and data
analysis tools for Python. _pandas_ stands for panel data, a term borrowed from econometrics and is an efficient library for data analysis with an emphasis on tabular data.
_pandas_ has two major classes, the _DataFrame_ class with two-dimensional data objects and tabular data organized in columns and the class _Series_ with a focus on one-dimensional data objects. Both classes allow you to index data easily as we will see in the examples below. 
_pandas_ allows you also to perform mathematical operations on the data, spanning from simple reshapings of vectors and matrices to statistical operations. 

The following simple example shows how we can, in an easy way make tables of our data. Here we define a data set which includes names, place of birth and date of birth, and displays the data in an easy to read way. We will see repeated use of _pandas_, in particular in connection with classification of data. 

!bc pycod 
import pandas as pd
from IPython.display import display
data = {'First Name': ["Frodo", "Bilbo", "Aragorn II", "Samwise"],
        'Last Name': ["Baggins", "Baggins","Elessar","Gamgee"],
        'Place of birth': ["Shire", "Shire", "Eriador", "Shire"],
        'Date of Birth T.A.': [2968, 2890, 2931, 2980]
        }
data_pandas = pd.DataFrame(data)
display(data_pandas)
!ec

In the above we have imported _pandas_ with the shorthand _pd_, the latter has become the standard way we import _pandas_. We make then a list of various variables
and reorganize the aboves lists into a _DataFrame_ and then print out  a neat table with specific column labels as *Name*, *place of birth* and *date of birth*.
Displaying these results, we see that the indices are given by the default numbers from zero to three.
_pandas_ is extremely flexible and we can easily change the above indices by defining a new type of indexing as
!bc pycod
data_pandas = pd.DataFrame(data,index=['Frodo','Bilbo','Aragorn','Sam'])
display(data_pandas)
!ec
Thereafter we display the content of the row which begins with the index _Aragorn_
!bc pycod
display(data_pandas.loc['Aragorn'])
!ec

We can easily append data to this, for example
!bc pycod
new_hobbit = {'First Name': ["Peregrin"],
              'Last Name': ["Took"],
              'Place of birth': ["Shire"],
              'Date of Birth T.A.': [2990]
              }
data_pandas=data_pandas.append(pd.DataFrame(new_hobbit, index=['Pippin']))
display(data_pandas)
!ec


Here are other examples where we use the _DataFrame_ functionality to handle arrays, now with more interesting features for us, namely numbers. We set up a matrix 
of dimensionality $10\times 5$ and compute the mean value and standard deviation of each column. Similarly, we can perform mathematial operations like squaring the matrix elements and many other operations. 
!bc pycod
import numpy as np
import pandas as pd
from IPython.display import display
np.random.seed(100)
# setting up a 10 x 5 matrix
rows = 10
cols = 5
a = np.random.randn(rows,cols)
df = pd.DataFrame(a)
display(df)
print(df.mean())
print(df.std())
display(df**2)
!ec

Thereafter we can select specific columns only and plot final results
!bc pycod
df.columns = ['First', 'Second', 'Third', 'Fourth', 'Fifth']
df.index = np.arange(10)

display(df)
print(df['Second'].mean() )
print(df.info())
print(df.describe())

from pylab import plt, mpl
plt.style.use('seaborn')
mpl.rcParams['font.family'] = 'serif'

df.cumsum().plot(lw=2.0, figsize=(10,6))
plt.show()


df.plot.bar(figsize=(10,6), rot=15)
plt.show()
!ec
We can produce a $4\times 4$ matrix
!bc pycod
b = np.arange(16).reshape((4,4))
print(b)
df1 = pd.DataFrame(b)
print(df1)
!ec
and many other operations. 

The _Series_ class is another important class included in
_pandas_. You can view it as a specialization of _DataFrame_ but where
we have just a single column of data. It shares many of the same features as _DataFrame. As with _DataFrame_,
most operations are vectorized, achieving thereby a high performance when dealing with computations of arrays, in particular labeled arrays.
As we will see below it leads also to a very concice code close to the mathematical operations we may be interested in.
For multidimensional arrays, we recommend strongly "xarray":"http://xarray.pydata.org/en/stable/". _xarray_ has much of the same flexibility as _pandas_, but allows for the extension to higher dimensions than two. We will see examples later of the usage of both _pandas_ and _xarray_. 



===== Reading Data and fitting  =====

In order to study various Machine Learning algorithms, we need to
access data. Acccessing data is an essential step in all machine
learning algorithms. In particular, setting up the so-called _design
matrix_ (to be defined below) is often the first element we need in
order to perform our calculations. To set up the design matrix means
reading (and later, when the calculations are done, writing) data
in various formats, The formats span from reading files from disk,
loading data from databases and interacting with online sources
like web application programming interfaces (APIs).

In handling various input formats, as discussed above, we will mainly stay with _pandas_,
a Python package which allows us, in a seamless and painless way, to
deal with a multitude of formats, from standard _csv_ (comma separated
values) files, via _excel_, _html_ to _hdf5_ formats.  With _pandas_
and the _DataFrame_  and _Series_ functionalities we are able to convert text data
into the calculational formats we need for a specific algorithm. And our code is going to be 
pretty close the basic mathematical expressions.

Our first data set is going to be a classic from nuclear physics, namely all
available data on binding energies. Don't be intimidated if you are not familiar with nuclear physics. It serves simply as an example here of a data set. 

We will show some of the
strengths of packages like _Scikit-Learn_ in fitting nuclear binding energies to
specific functions using linear regression first. Then, as a teaser, we will show you how 
you can easily implement other algorithms like decision trees and random forests and neural networks.

But before we really start with nuclear physics data, let's just look at some simpler polynomial fitting cases, such as,
(don't be offended) fitting straight lines!


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


The Numpy functions are imported used the _import numpy as np_
statement and the random number generator for the uniform distribution
is called using the function _np.random.rand()_, where we specificy
that we want $100$ random variables.  Using Numpy we define
automatically an array with the specified number of elements, $100$ in
our case.  With the Numpy function _randn()_ we can compute random
numbers with the normal distribution (mean value $\mu$ equal to zero and
variance $\sigma^2$ set to one) and produce the values of $y$ assuming a linear
dependence as function of $x$

!bt
\[
y = 2x+N(0,1),
\]
!et

where $N(0,1)$ represents random numbers generated by the normal
distribution.  From _Scikit-Learn_ we import then the
_LinearRegression_ functionality and make a prediction $\tilde{y} =
\alpha + \beta x$ using the function _fit(x,y)_. We call the set of
data $(\hat{x},\hat{y})$ for our training data. The Python package
_scikit-learn_ has also a functionality which extracts the above
fitting parameters $\alpha$ and $\beta$ (see below). Later we will
distinguish between training data and test data.

For plotting we use the Python package
"matplotlib":"https://matplotlib.org/" which produces publication
quality figures. Feel free to explore the extensive
"gallery":"https://matplotlib.org/gallery/index.html" of examples. In
this example we plot our original values of $x$ and $y$ as well as the
prediction _ypredict_ ($\tilde{y}$), which attempts at fitting our
data with a straight line.

The Python code follows here.
!bc pycod
# Importing various packages
import numpy as np
import matplotlib.pyplot as plt
from sklearn.linear_model import LinearRegression

x = np.random.rand(100,1)
y = 2*x+np.random.randn(100,1)
linreg = LinearRegression()
linreg.fit(x,y)
xnew = np.array([[0],[1]])
ypredict = linreg.predict(xnew)

plt.plot(xnew, ypredict, "r-")
plt.plot(x, y ,'ro')
plt.axis([0,1.0,0, 5.0])
plt.xlabel(r'$x$')
plt.ylabel(r'$y$')
plt.title(r'Simple Linear Regression')
plt.show()
!ec

This example serves several aims. It allows us to demonstrate several
aspects of data analysis and later machine learning algorithms. The
immediate visualization shows that our linear fit is not
impressive. It goes through the data points, but there are many
outliers which are not reproduced by our linear regression.  We could
now play around with this small program and change for example the
factor in front of $x$ and the normal distribution.  Try to change the
function $y$ to

!bt
\[
y = 10x+0.01 \times N(0,1),
\]
!et

where $x$ is defined as before.  Does the fit look better? Indeed, by
reducing the role of the noise given by the normal distribution we see immediately that
our linear prediction seemingly reproduces better the training
set. However, this testing 'by the eye' is obviouly not satisfactory in the
long run. Here we have only defined the training data and our model, and 
have not discussed a more rigorous approach to the _cost_ function.

We need more rigorous criteria in defining whether we have succeeded or
not in modeling our training data.  You will be surprised to see that
many scientists seldomly venture beyond this 'by the eye' approach. A
standard approach for the *cost* function is the so-called $\chi^2$
function (a variant of the mean-squared error (MSE))

!bt 
\[ \chi^2 = \frac{1}{n}
\sum_{i=0}^{n-1}\frac{(y_i-\tilde{y}_i)^2}{\sigma_i^2}, 
\] 
!et 

where $\sigma_i^2$ is the variance (to be defined later) of the entry
$y_i$.  We may not know the explicit value of $\sigma_i^2$, it serves
however the aim of scaling the equations and make the cost function
dimensionless.  

Minimizing the cost function is a central aspect of
our discussions to come. Finding its minima as function of the model
parameters ($\alpha$ and $\beta$ in our case) will be a recurring
theme in these series of lectures. Essentially all machine learning
algorithms we will discuss center around the minimization of the
chosen cost function. This depends in turn on our specific
model for describing the data, a typical situation in supervised
learning. Automatizing the search for the minima of the cost function is a
central ingredient in all algorithms. Typical methods which are
employed are various variants of _gradient_ methods. These will be
discussed in more detail later. Again, you'll be surprised to hear that
many practitioners minimize the above function ''by the eye', popularly dubbed as 
'chi by the eye'. That is, change a parameter and see (visually and numerically) that 
the  $\chi^2$ function becomes smaller. 

There are many ways to define the cost function. A simpler approach is to look at the relative difference between the training data and the predicted data, that is we define 
the relative error (why would we prefer the MSE instead of the relative error?) as

!bt
\[
\epsilon_{\mathrm{relative}}= \frac{\vert \hat{y} -\hat{\tilde{y}}\vert}{\vert \hat{y}\vert}.
\]
!et

The squared cost function results in an arithmetic mean-unbiased
estimator, and the absolute-value cost function results in a
median-unbiased estimator (in the one-dimensional case, and a
geometric median-unbiased estimator for the multi-dimensional
case). The squared cost function has the disadvantage that it has the tendency
to be dominated by outliers.

We can modify easily the above Python code and plot the relative error instead
!bc pycod
import numpy as np
import matplotlib.pyplot as plt
from sklearn.linear_model import LinearRegression

x = np.random.rand(100,1)
y = 5*x+0.01*np.random.randn(100,1)
linreg = LinearRegression()
linreg.fit(x,y)
ypredict = linreg.predict(x)

plt.plot(x, np.abs(ypredict-y)/abs(y), "ro")
plt.axis([0,1.0,0.0, 0.5])
plt.xlabel(r'$x$')
plt.ylabel(r'$\epsilon_{\mathrm{relative}}$')
plt.title(r'Relative error')
plt.show()
!ec

Depending on the parameter in front of the normal distribution, we may
have a small or larger relative error. Try to play around with
different training data sets and study (graphically) the value of the
relative error.

As mentioned above, _Scikit-Learn_ has an impressive functionality.
We can for example extract the values of $\alpha$ and $\beta$ and
their error estimates, or the variance and standard deviation and many
other properties from the statistical data analysis. 

Here we show an
example of the functionality of _Scikit-Learn_.
!bc pycod 
import numpy as np 
import matplotlib.pyplot as plt 
from sklearn.linear_model import LinearRegression 
from sklearn.metrics import mean_squared_error, r2_score, mean_squared_log_error, mean_absolute_error

x = np.random.rand(100,1)
y = 2.0+ 5*x+0.5*np.random.randn(100,1)
linreg = LinearRegression()
linreg.fit(x,y)
ypredict = linreg.predict(x)
print('The intercept alpha: \n', linreg.intercept_)
print('Coefficient beta : \n', linreg.coef_)
# The mean squared error                               
print("Mean squared error: %.2f" % mean_squared_error(y, ypredict))
# Explained variance score: 1 is perfect prediction                                 
print('Variance score: %.2f' % r2_score(y, ypredict))
# Mean squared log error                                                        
print('Mean squared log error: %.2f' % mean_squared_log_error(y, ypredict) )
# Mean absolute error                                                           
print('Mean absolute error: %.2f' % mean_absolute_error(y, ypredict))
plt.plot(x, ypredict, "r-")
plt.plot(x, y ,'ro')
plt.axis([0.0,1.0,1.5, 7.0])
plt.xlabel(r'$x$')
plt.ylabel(r'$y$')
plt.title(r'Linear Regression fit ')
plt.show()

!ec
The function _coef_ gives us the parameter $\beta$ of our fit while _intercept_ yields 
$\alpha$. Depending on the constant in front of the normal distribution, we get values near or far from $alpha =2$ and $\beta =5$. Try to play around with different parameters in front of the normal distribution. The function _meansquarederror_ gives us the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error or loss defined as
!bt 
\[ MSE(\hat{y},\hat{\tilde{y}}) = \frac{1}{n}
\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2, 
\] 
!et 

The smaller the value, the better the fit. Ideally we would like to
have an MSE equal zero.  The attentive reader has probably recognized
this function as being similar to the $\chi^2$ function defined above.

The _r2score_ function computes $R^2$, the coefficient of
determination. It provides a measure of how well future samples are
likely to be predicted by the model. Best possible score is 1.0 and it
can be negative (because the model can be arbitrarily worse). A
constant model that always predicts the expected value of $\hat{y}$,
disregarding the input features, would get a $R^2$ score of $0.0$.

If $\tilde{\hat{y}}_i$ is the predicted value of the $i-th$ sample and $y_i$ is the corresponding true value, then the score $R^2$ is defined as
!bt
\[
R^2(\hat{y}, \tilde{\hat{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2},
\]
!et
where we have defined the mean value  of $\hat{y}$ as
!bt
\[
\bar{y} =  \frac{1}{n} \sum_{i=0}^{n - 1} y_i.
\]
!et
Another quantity taht we will meet again in our discussions of regression analysis is 
 the mean absolute error (MAE), a risk metric corresponding to the expected value of the absolute error loss or what we call the $l1$-norm loss. In our discussion above we presented the relative error.
The MAE is defined as follows
!bt
\[
\text{MAE}(\hat{y}, \hat{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n-1} \left| y_i - \tilde{y}_i \right|.
\]
!et
We present the 
squared logarithmic (quadratic) error
!bt
\[
\text{MSLE}(\hat{y}, \hat{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n - 1} (\log_e (1 + y_i) - \log_e (1 + \tilde{y}_i) )^2,
\]
!et

where $\log_e (x)$ stands for the natural logarithm of $x$. This error
estimate is best to use when targets having exponential growth, such
as population counts, average sales of a commodity over a span of
years etc. 


Finally, another cost function is the Huber cost function used in robust regression.
It is less sensitive to outliers in data than the squared error cost function.
A variant for classification is also sometimes used, a quantity we will meet later. 
!bt
\[
L_{\delta }(a)={\begin{cases}{\frac {1}{2}}{a^{2}}&{\text{for }}|a|\leq \delta ,\\\delta (|a|-{\frac {1}{2}}\delta ),&{\text{otherwise.}}\end{cases}}}L_{\delta }(a)={\begin{cases}{\frac  {1}{2}}{a^{2}}&{\text{for }}|a|\leq \delta ,\\\delta (|a|-{\frac  {1}{2}}\delta ),&{\text{otherwise.}}\end{cases}}
\]
!et

We will discuss in more
detail these and other functions in the various lectures.  We conclude this part with another example. Instead of 
a linear $x$-dependence we study now a cubic polynomial and use the polynomial regression analysis tools of scikit-learn. 

!bc pycod
import matplotlib.pyplot as plt
import numpy as np
import random
from sklearn.linear_model import Ridge
from sklearn.preprocessing import PolynomialFeatures
from sklearn.pipeline import make_pipeline
from sklearn.linear_model import LinearRegression

x=np.linspace(0.02,0.98,200)
noise = np.asarray(random.sample((range(200)),200))
y=x**3*noise
yn=x**3*100
poly3 = PolynomialFeatures(degree=3)
X = poly3.fit_transform(x[:,np.newaxis])
clf3 = LinearRegression()
clf3.fit(X,y)

Xplot=poly3.fit_transform(x[:,np.newaxis])
poly3_plot=plt.plot(x, clf3.predict(Xplot), label='Cubic Fit')
plt.plot(x,yn, color='red', label="True Cubic")
plt.scatter(x, y, label='Data', color='orange', s=15)
plt.legend()
plt.show()

def error(a):
    for i in y:
        err=(y-yn)/yn
    return abs(np.sum(err))/len(err)

print (error(y))
!ec




=== To our real data: nuclear binding energies. Brief reminder on masses and binding energies ===

Let us now dive into  nuclear physics and remind ourselves briefly about some basic features about binding
energies.  A basic quantity which can be measured for the ground
states of nuclei is the atomic mass $M(N, Z)$ of the neutral atom with
atomic mass number $A$ and charge $Z$. The number of neutrons is $N$. There are indeed several sophisticated experiments worldwide which allow us to measure this quantity to high precision (parts per million even). 

Atomic masses are usually tabulated in terms of the mass excess defined by
!bt
\[
\Delta M(N, Z) =  M(N, Z) - uA,
\]
!et
where $u$ is the Atomic Mass Unit 
!bt
\[
u = M(^{12}\mathrm{C})/12 = 931.4940954(57) \hspace{0.1cm} \mathrm{MeV}/c^2.
\]
!et
The nucleon masses are
!bt 
\[
m_p =  1.00727646693(9)u,
\] 
!et
and
!bt 
\[
m_n = 939.56536(8)\hspace{0.1cm} \mathrm{MeV}/c^2 = 1.0086649156(6)u.
\]
!et

In the "2016 mass evaluation of by W.J.Huang, G.Audi, M.Wang, F.G.Kondev, S.Naimi and X.Xu":"http://nuclearmasses.org/resources_folder/Wang_2017_Chinese_Phys_C_41_030003.pdf"
there are data on masses and decays of 3437 nuclei.

The nuclear binding energy is defined as the energy required to break
up a given nucleus into its constituent parts of $N$ neutrons and $Z$
protons. In terms of the atomic masses $M(N, Z)$ the binding energy is
defined by


!bt
\[
BE(N, Z) = ZM_H c^2 + Nm_n c^2 - M(N, Z)c^2 ,
\]
!et
where $M_H$ is the mass of the hydrogen atom and $m_n$ is the mass of the neutron.
In terms of the mass excess the binding energy is given by
!bt
\[
BE(N, Z) = Z\Delta_H c^2 + N\Delta_n c^2 -\Delta(N, Z)c^2 ,
\]
!et
where $\Delta_H c^2 = 7.2890$ MeV and $\Delta_n c^2 = 8.0713$ MeV.


A popular and physically intuitive model which can be used to parametrize 
the experimental binding energies as function of $A$, is the so-called 
_liquid drop model_. The ansatz is based on the following expression

!bt
\[ 
BE(N,Z) = a_1A-a_2A^{2/3}-a_3\frac{Z^2}{A^{1/3}}-a_4\frac{(N-Z)^2}{A},
\]
!et

where $A$ stands for the number of nucleons and the $a_i$s are parameters which are determined by a fit 
to the experimental data.  




To arrive at the above expression we have assumed that we can make the following assumptions:

 * There is a volume term $a_1A$ proportional with the number of nucleons (the energy is also an extensive quantity). When an assembly of nucleons of the same size is packed together into the smallest volume, each interior nucleon has a certain number of other nucleons in contact with it. This contribution is proportional to the volume.

 * There is a surface energy term $a_2A^{2/3}$. The assumption here is that a nucleon at the surface of a nucleus interacts with fewer other nucleons than one in the interior of the nucleus and hence its binding energy is less. This surface energy term takes that into account and is therefore negative and is proportional to the surface area.


 * There is a Coulomb energy term $a_3\frac{Z^2}{A^{1/3}}$. The electric repulsion between each pair of protons in a nucleus yields less binding. 

 * There is an asymmetry term $a_4\frac{(N-Z)^2}{A}$. This term is associated with the Pauli exclusion principle and reflects the fact that the proton-neutron interaction is more attractive on the average than the neutron-neutron and proton-proton interactions.

We could also add a so-called pairing term, which is a correction term that
arises from the tendency of proton pairs and neutron pairs to
occur. An even number of particles is more stable than an odd number. 


=== Organizing our data ===

Let us start with reading and organizing our data. 
We start with the compilation of masses and binding energies from 2016.
After having downloaded this file to our own computer, we are now ready to read the file and start structuring our data.


We start with preparing folders for storing our calculations and the data file over masses and binding energies. We import also various modules that we will find useful in order to present various Machine Learning methods. Here we focus mainly on the functionality of _scikit-learn_.
!bc pycod
# Common imports
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import sklearn.linear_model as skl
from sklearn.model_selection import train_test_split
from sklearn.metrics import mean_squared_error, r2_score, mean_absolute_error
import os

# Where to save the figures and data files
PROJECT_ROOT_DIR = "Results"
FIGURE_ID = "Results/FigureFiles"
DATA_ID = "DataFiles/"

if not os.path.exists(PROJECT_ROOT_DIR):
    os.mkdir(PROJECT_ROOT_DIR)

if not os.path.exists(FIGURE_ID):
    os.makedirs(FIGURE_ID)

if not os.path.exists(DATA_ID):
    os.makedirs(DATA_ID)

def image_path(fig_id):
    return os.path.join(FIGURE_ID, fig_id)

def data_path(dat_id):
    return os.path.join(DATA_ID, dat_id)

def save_fig(fig_id):
    plt.savefig(image_path(fig_id) + ".png", format='png')

infile = open(data_path("MassEval2016.dat"),'r')
!ec


Before we proceed, we define also a function for making our plots. You can obviously avoid this and simply set up various _matplotlib_ commands every time you need them. You may however find it convenient to collect all such commands in one function and simply call this function. 
!bc pycod 
from pylab import plt, mpl
plt.style.use('seaborn')
mpl.rcParams['font.family'] = 'serif'

def MakePlot(x,y, styles, labels, axlabels):
    plt.figure(figsize=(10,6))
    for i in range(len(x)):
        plt.plot(x[i], y[i], styles[i], label = labels[i])
        plt.xlabel(axlabels[0])
        plt.ylabel(axlabels[1])
    plt.legend(loc=0)
!ec

Our next step is to read the data on experimental binding energies and
reorganize them as functions of the mass number $A$, the number of
protons $Z$ and neutrons $N$ using _pandas_.  Before we do this it is
always useful (unless you have a binary file or other types of compressed
data) to actually open the file and simply take a look at it!


In particular, the program that outputs the final nuclear masses is written in Fortran with a specific format. It means that we need to figure out the format and which columns contain the data we are interested in. Pandas comes with a function that reads formatted output. After having admired the file, we are now ready to start massaging it with _pandas_. The file begins with some basic format information.
!bc pycod 
"""                                                                                                                         
This is taken from the data file of the mass 2016 evaluation.                                                               
All files are 3436 lines long with 124 character per line.                                                                  
       Headers are 39 lines long.                                                                                           
   col 1     :  Fortran character control: 1 = page feed  0 = line feed                                                     
   format    :  a1,i3,i5,i5,i5,1x,a3,a4,1x,f13.5,f11.5,f11.3,f9.3,1x,a2,f11.3,f9.3,1x,i3,1x,f12.5,f11.5                     
   These formats are reflected in the pandas widths variable below, see the statement                                       
   widths=(1,3,5,5,5,1,3,4,1,13,11,11,9,1,2,11,9,1,3,1,12,11,1),                                                            
   Pandas has also a variable header, with length 39 in this case.                                                          
"""
!ec

The data we are interested in are in columns 2, 3, 4 and 11, giving us
the number of neutrons, protons, mass numbers and binding energies,
respectively. We add also for the sake of completeness the element name. The data are in fixed-width formatted lines and we will
covert them into the _pandas_ DataFrame structure.

!bc pycod 
# Read the experimental data with Pandas
Masses = pd.read_fwf(infile, usecols=(2,3,4,6,11),
              names=('N', 'Z', 'A', 'Element', 'Ebinding'),
              widths=(1,3,5,5,5,1,3,4,1,13,11,11,9,1,2,11,9,1,3,1,12,11,1),
              header=39,
              index_col=False)

# Extrapolated values are indicated by '#' in place of the decimal place, so
# the Ebinding column won't be numeric. Coerce to float and drop these entries.
Masses['Ebinding'] = pd.to_numeric(Masses['Ebinding'], errors='coerce')
Masses = Masses.dropna()
# Convert from keV to MeV.
Masses['Ebinding'] /= 1000

# Group the DataFrame by nucleon number, A.
Masses = Masses.groupby('A')
# Find the rows of the grouped DataFrame with the maximum binding energy.
Masses = Masses.apply(lambda t: t[t.Ebinding==t.Ebinding.max()])
!ec

We have now read in the data, grouped them according to the variables we are interested in. 
We see how easy it is to reorganize the data using _pandas_. If we
were to do these operations in C/C++ or Fortran, we would have had to
write various functions/subroutines which perform the above
reorganizations for us.  Having reorganized the data, we can now start
to make some simple fits using both the functionalities in _numpy_ and
_Scikit-Learn_ afterwards. 

Now we define five variables which contain
the number of nucleons $A$, the number of protons $Z$ and the number of neutrons $N$, the element name and finally the energies themselves.
!bc pycod 
A = Masses['A']
Z = Masses['Z']
N = Masses['N']
Element = Masses['Element']
Energies = Masses['Ebinding']
print(Masses)
!ec
The next step, and we will define this mathematically later, is to set up the so-called _design matrix_. We will throughout call this matrix $\bm{X}$.
It has dimensionality $p\times n$, where $n$ is the number of data points and $p$ are the so-called predictors. In our case here they are given by the number of polynomials in $A$ we wish to include in the fit. 
!bc pycod
# Now we set up the design matrix X
X = np.zeros((len(A),5))
X[:,0] = 1
X[:,1] = A
X[:,2] = A**(2.0/3.0)
X[:,3] = A**(-1.0/3.0)
X[:,4] = A**(-1.0)
!ec
With _scikitlearn_ we are now ready to use linear regression and fit our data.
!bc pycod 
clf = skl.LinearRegression().fit(X, Energies)
fity = clf.predict(X)
!ec
Pretty simple!  
Now we can print measures of how our fit is doing, the coefficients from the fits and plot the final fit together with our data.
!bc pycod
# The mean squared error                               
print("Mean squared error: %.2f" % mean_squared_error(Energies, fity))
# Explained variance score: 1 is perfect prediction                                 
print('Variance score: %.2f' % r2_score(Energies, fity))
# Mean absolute error                                                           
print('Mean absolute error: %.2f' % mean_absolute_error(Energies, fity))
print(clf.coef_, clf.intercept_)

Masses['Eapprox']  = fity
# Generate a plot comparing the experimental with the fitted values values.
fig, ax = plt.subplots()
ax.set_xlabel(r'$A = N + Z$')
ax.set_ylabel(r'$E_\mathrm{bind}\,/\mathrm{MeV}$')
ax.plot(Masses['A'], Masses['Ebinding'], alpha=0.7, lw=2,
            label='Ame2016')
ax.plot(Masses['A'], Masses['Eapprox'], alpha=0.7, lw=2, c='m',
            label='Fit')
ax.legend()
save_fig("Masses2016")
plt.show()
!ec


=== Seeing the wood for the trees ===

As a teaser, let us now see how we can do this with decision trees using _scikit-learn_. Later we will switch to so-called _random forests_!


!bc pycod

#Decision Tree Regression
from sklearn.tree import DecisionTreeRegressor
regr_1=DecisionTreeRegressor(max_depth=5)
regr_2=DecisionTreeRegressor(max_depth=7)
regr_3=DecisionTreeRegressor(max_depth=9)
regr_1.fit(X, Energies)
regr_2.fit(X, Energies)
regr_3.fit(X, Energies)


y_1 = regr_1.predict(X)
y_2 = regr_2.predict(X)
y_3=regr_3.predict(X)
Masses['Eapprox'] = y_3
# Plot the results
plt.figure()
plt.plot(A, Energies, color="blue", label="Data", linewidth=2)
plt.plot(A, y_1, color="red", label="max_depth=5", linewidth=2)
plt.plot(A, y_2, color="green", label="max_depth=7", linewidth=2)
plt.plot(A, y_3, color="m", label="max_depth=9", linewidth=2)

plt.xlabel("$A$")
plt.ylabel("$E$[MeV]")
plt.title("Decision Tree Regression")
plt.legend()
save_fig("Masses2016Trees")
plt.show()
print(Masses)
print(np.mean( (Energies-y_1)**2))
!ec


=== And what about using neural networks? ===
The _seaborn_ package allows us to visualize data in an efficient way. Note that we use _scikit-learn_'s multi-layer perceptron (or feed forward neural network) 
functionality.
!bc pycod
from sklearn.neural_network import MLPRegressor
from sklearn.metrics import accuracy_score
import seaborn as sns

X_train = X
Y_train = Energies
n_hidden_neurons = 100
epochs = 100
# store models for later use
eta_vals = np.logspace(-5, 1, 7)
lmbd_vals = np.logspace(-5, 1, 7)
# 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='logistic',
                            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)

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()



!ec


===== A first summary =====

The aim behind these introductory words was to present to you various
Python libraries and their functionalities, in particular libraries like
_numpy_, _pandas_, _xarray_ and _matplotlib_ and other that make our life much easier
in handling various data sets and visualizing data. 

Furthermore,
_Scikit-Learn_ allows us with few lines of code to implement popular
Machine Learning algorithms for supervised learning. Later we will meet _Tensorflow_, a powerful library for deep learning. 
Now it is time to dive more into the details of various methods. We will start with linear regression and try to take a deeper look at what it entails.


=======  Review of Basic Statistics =======


===== Domains and probabilities  =====
!bblock
Consider the following simple example, namely the tossing of two dice, resulting in  the following possible values
!bt
\begin{equation*}
\{2,3,4,5,6,7,8,9,10,11,12\}. 
\end{equation*}
!et
These values are called the *domain*. 
To this domain we have the corresponding *probabilities*
!bt
\begin{equation*}
\{1/36,2/36/,3/36,4/36,5/36,6/36,5/36,4/36,3/36,2/36,1/36\}.
\end{equation*}
!et
!eblock


===== Tossing  the dice =====
!bblock
The numbers in the domain are the outcomes of the physical process of tossing say two dice.
We cannot tell beforehand whether the outcome is 3 or 5 or any other number in this domain.
This defines the randomness of the outcome, or unexpectedness or any other synonimous word which
encompasses the uncertitude of the final outcome. 

The only thing we can tell beforehand
is that say the outcome 2 has a certain probability.  
If our favorite hobby is to  spend an hour every evening throwing dice and 
registering the sequence of outcomes, we will note that the numbers in the above domain
!bt
\begin{equation*}
\{2,3,4,5,6,7,8,9,10,11,12\},
\end{equation*}
!et 
appear in a random order. After 11 throws the results may look like

!bt
\begin{equation*}
\{10,8,6,3,6,9,11,8,12,4,5\}. 
\end{equation*}
!et
!eblock


===== Stochastic variables =====
!bblock

_Random variables are characterized by a domain which contains all possible values that the random value may take. This domain has a corresponding probability distribution function(PDF)_.
!eblock


===== Stochastic variables and the main concepts, the discrete case =====
!bblock
There are two main concepts associated with a stochastic variable. The
*domain* is the set $\mathbb D = \{x\}$ of all accessible values
the variable can assume, so that $X \in \mathbb D$. An example of a
discrete domain is the set of six different numbers that we may get by
throwing of a dice, $x\in\{1,\,2,\,3,\,4,\,5,\,6\}$.

The *probability distribution function (PDF)* is a function
$p(x)$ on the domain which, in the discrete case, gives us the
probability or relative frequency with which these values of $X$
occur
!bt
\begin{equation*}
p(x) = \mathrm{Prob}(X=x).
\end{equation*}
!et
!eblock



===== Stochastic variables and the main concepts, the continuous case =====
!bblock
In the continuous case, the PDF does not directly depict the
actual probability. Instead we define the probability for the
stochastic variable to assume any value on an infinitesimal interval
around $x$ to be $p(x)dx$. The continuous function $p(x)$ then gives us
the *density* of the probability rather than the probability
itself. The probability for a stochastic variable to assume any value
on a non-infinitesimal interval $[a,\,b]$ is then just the integral

!bt
\begin{equation*}
\mathrm{Prob}(a\leq X\leq b) = \int_a^b p(x)dx.
\end{equation*}
!et
Qualitatively speaking, a stochastic variable represents the values of
numbers chosen as if by chance from some specified PDF so that the
selection of a large set of these numbers reproduces this PDF.
!eblock


===== The cumulative probability =====
!bblock
Of interest to us is the *cumulative probability
distribution function* (_CDF_), $P(x)$, which is just the probability
for a stochastic variable $X$ to assume any value less than $x$
!bt
\begin{equation*}
P(x)=\mathrm{Prob(}X\leq x\mathrm{)} =
\int_{-\infty}^x p(x^{\prime})dx^{\prime}.
\end{equation*}
!et
The relation between a CDF and its corresponding PDF is then

!bt
\begin{equation*}
p(x) = \frac{d}{dx}P(x).
\end{equation*}
!et
!eblock


===== Properties of PDFs =====
!bblock

There are two properties that all PDFs must satisfy. The first one is
positivity (assuming that the PDF is normalized)

!bt
\begin{equation*}
0 \leq p(x) \leq 1.
\end{equation*}
!et
Naturally, it would be nonsensical for any of the values of the domain
to occur with a probability greater than $1$ or less than $0$. Also,
the PDF must be normalized. That is, all the probabilities must add up
to unity.  The probability of ``anything'' to happen is always unity. For
both discrete and continuous PDFs, this condition is
!bt
\begin{align*}
\sum_{x_i\in\mathbb D} p(x_i) & =  1,\\
\int_{x\in\mathbb D} p(x)\,dx & =  1.
\end{align*}
!et
!eblock


===== Important distributions, the uniform distribution =====
!bblock
The first one
is the most basic PDF; namely the uniform distribution
!bt
\begin{equation}
p(x) = \frac{1}{b-a}\theta(x-a)\theta(b-x).
label{eq:unifromPDF}
\end{equation}
!et
For $a=0$ and $b=1$ we have 
!bt
\[
\begin{array}{ll}
p(x)dx = dx & \in [0,1].
\end{array}
\]
!et
The latter distribution is used to generate random numbers. For other PDFs, one needs normally a mapping from this distribution to say for example the exponential distribution. 
!eblock


===== Gaussian distribution =====
!bblock
The second one is the Gaussian Distribution
!bt
\begin{equation*}
p(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp{(-\frac{(x-\mu)^2}{2\sigma^2})},
\end{equation*}
!et
with mean value $\mu$ and standard deviation $\sigma$. If $\mu=0$ and $\sigma=1$, it is normally called the _standard normal distribution_
!bt
\begin{equation*}
p(x) = \frac{1}{\sqrt{2\pi}} \exp{(-\frac{x^2}{2})},
\end{equation*}
!et

The following simple Python code plots the above distribution for different values of $\mu$ and $\sigma$. 
!bc pyscpro
import numpy as np
from math import acos, exp, sqrt
from  matplotlib import pyplot as plt
from matplotlib import rc, rcParams
import matplotlib.units as units
import matplotlib.ticker as ticker
rc('text',usetex=True)
rc('font',**{'family':'serif','serif':['Gaussian distribution']})
font = {'family' : 'serif',
        'color'  : 'darkred',
        'weight' : 'normal',
        'size'   : 16,
        }
pi = acos(-1.0)
mu0 = 0.0
sigma0 = 1.0
mu1= 1.0
sigma1 = 2.0
mu2 = 2.0
sigma2 = 4.0

x = np.linspace(-20.0, 20.0)
v0 = np.exp(-(x*x-2*x*mu0+mu0*mu0)/(2*sigma0*sigma0))/sqrt(2*pi*sigma0*sigma0)
v1 = np.exp(-(x*x-2*x*mu1+mu1*mu1)/(2*sigma1*sigma1))/sqrt(2*pi*sigma1*sigma1)
v2 = np.exp(-(x*x-2*x*mu2+mu2*mu2)/(2*sigma2*sigma2))/sqrt(2*pi*sigma2*sigma2)
plt.plot(x, v0, 'b-', x, v1, 'r-', x, v2, 'g-')
plt.title(r'{\bf Gaussian distributions}', fontsize=20)
plt.text(-19, 0.3, r'Parameters: $\mu = 0$, $\sigma = 1$', fontdict=font)
plt.text(-19, 0.18, r'Parameters: $\mu = 1$, $\sigma = 2$', fontdict=font)
plt.text(-19, 0.08, r'Parameters: $\mu = 2$, $\sigma = 4$', fontdict=font)
plt.xlabel(r'$x$',fontsize=20)
plt.ylabel(r'$p(x)$ [MeV]',fontsize=20)

# Tweak spacing to prevent clipping of ylabel                                                                       
plt.subplots_adjust(left=0.15)
plt.savefig('gaussian.pdf', format='pdf')
plt.show()
!ec
!eblock



===== Exponential distribution =====
!bblock
Another important distribution in science is the exponential distribution
!bt
\begin{equation*}
p(x) = \alpha\exp{-(\alpha x)}.
\end{equation*}
!et
!eblock


===== Expectation values =====
!bblock
Let $h(x)$ be an arbitrary continuous function on the domain of the stochastic
variable $X$ whose PDF is $p(x)$. We define the *expectation value*
of $h$ with respect to $p$ as follows

!bt
\begin{equation}
\langle h \rangle_X \equiv \int\! h(x)p(x)\,dx
label{eq:expectation_value_of_h_wrt_p}
\end{equation}
!et
Whenever the PDF is known implicitly, like in this case, we will drop
the index $X$ for clarity.  
A particularly useful class of special expectation values are the
*moments*. The $n$-th moment of the PDF $p$ is defined as
follows
!bt
\begin{equation*}
\langle x^n \rangle \equiv \int\! x^n p(x)\,dx
\end{equation*}
!et
!eblock


===== Stochastic variables and the main concepts, mean values =====
!bblock
The zero-th moment $\langle 1\rangle$ is just the normalization condition of
$p$. The first moment, $\langle x\rangle$, is called the *mean* of $p$
and often denoted by the letter $\mu$
!bt
\begin{equation*}
\langle x\rangle  = \mu \equiv \int x p(x)dx,
\end{equation*}
!et
for a continuous distribution and 
!bt
\begin{equation*}
\langle x\rangle  = \mu \equiv \sum_{i=1}^N x_i p(x_i),
\end{equation*}
!et
for a discrete distribution. 
Qualitatively it represents the centroid or the average value of the
PDF and is therefore simply called the expectation value of $p(x)$.
!eblock


===== Stochastic variables and the main concepts, central moments, the variance =====
!bblock

A special version of the moments is the set of *central moments*, the n-th central moment defined as
!bt
\begin{equation*}
\langle (x-\langle x\rangle )^n\rangle  \equiv \int\! (x-\langle x\rangle)^n p(x)\,dx
\end{equation*}
!et
The zero-th and first central moments are both trivial, equal $1$ and
$0$, respectively. But the second central moment, known as the
*variance* of $p$, is of particular interest. For the stochastic
variable $X$, the variance is denoted as $\sigma^2_X$ or $\mathrm{Var}(X)$
!bt
\begin{align*}
\sigma^2_X &=\mathrm{Var}(X) =  \langle (x-\langle x\rangle)^2\rangle  =
\int (x-\langle x\rangle)^2 p(x)dx\\
& =  \int\left(x^2 - 2 x \langle x\rangle^{2} +\langle x\rangle^2\right)p(x)dx\\
& =  \langle x^2\rangle - 2 \langle x\rangle\langle x\rangle + \langle x\rangle^2\\
& =  \langle x^2 \rangle - \langle x\rangle^2
\end{align*}
!et
The square root of the variance, $\sigma =\sqrt{\langle (x-\langle x\rangle)^2\rangle}$ is called the 
_standard deviation_ of $p$. It is the RMS (root-mean-square)
value of the deviation of the PDF from its mean value, interpreted
qualitatively as the ``spread'' of $p$ around its mean.
!eblock




===== Probability Distribution Functions =====
!bblock

The following table collects properties of probability distribution functions.
In our notation we reserve the label $p(x)$ for the probability of a certain event,
while $P(x)$ is the cumulative probability. 


|--------------------------------------------------------------------------------------------------------------------------------------|
|                                            | Discrete PDF                               | Continuous PDF                             |
|---------------------l-------------------------------------------c-------------------------------------------c------------------------|
| Domain                                     | $\left\{x_1, x_2, x_3, \dots, x_N\right\}$ | $[a,b]$                                    |
| Probability                                | $p(x_i)$                                   | $p(x)dx$                                   |
| Cumulative                                 | $P_i=\sum_{l=1}^ip(x_l)$                   | $P(x)=\int_a^xp(t)dt$                      |
| Positivity                                 | $0 \le p(x_i) \le 1$                      | $p(x) \ge 0$                              |
| Positivity                                 | $0 \le P_i \le 1$                         | $0 \le P(x) \le 1$                        |
| Monotonic                                  | $P_i \ge P_j$ if $x_i \ge x_j$             | $P(x_i) \ge P(x_j)$ if $x_i \ge x_j$       |
| Normalization                              | $P_N=1$                                    | $P(b)=1$                                   |
|--------------------------------------------------------------------------------------------------------------------------------------|

!eblock
     


===== Probability Distribution Functions =====
!bblock
With a PDF we can compute expectation values of selected quantities such as

!bt
\begin{equation*}
    \langle x^k\rangle=\sum_{i=1}^{N}x_i^kp(x_i),
\end{equation*}
!et
if we have a discrete PDF or

!bt
\begin{equation*}
    \langle x^k\rangle=\int_a^b x^kp(x)dx,
\end{equation*}
!et
in the case of a continuous PDF. We have already defined the mean value $\mu$
and the variance $\sigma^2$. 
!eblock


===== The three famous Probability Distribution Functions =====
!bblock

There are at least three PDFs which one may encounter. These are the

_Uniform distribution_  
!bt 
\begin{equation*}   
p(x)=\frac{1}{b-a}\Theta(x-a)\Theta(b-x),
\end{equation*}
!et
yielding probabilities different from zero in the interval $[a,b]$.

_The exponential distribution_
!bt 
\begin{equation*}   
p(x)=\alpha \exp{(-\alpha x)},
\end{equation*}
!et
yielding probabilities different from zero in the interval $[0,\infty)$ and with mean value
!bt
\begin{equation*} 
\mu = \int_0^{\infty}xp(x)dx=\int_0^{\infty}x\alpha \exp{(-\alpha x)}dx=\frac{1}{\alpha},
\end{equation*}
!et
!eblock
with variance
!bt
\begin{equation*}
\sigma^2=\int_0^{\infty}x^2p(x)dx-\mu^2 = \frac{1}{\alpha^2}.
\end{equation*}
!et 


===== Probability Distribution Functions, the normal distribution =====
!bblock
Finally, we have the so-called univariate normal  distribution, or just the _normal distribution_
!bt 
\begin{equation*}    
p(x)=\frac{1}{b\sqrt{2\pi}}\exp{\left(-\frac{(x-a)^2}{2b^2}\right)}
\end{equation*}
!et
with probabilities different from zero in the interval $(-\infty,\infty)$.
The integral $\int_{-\infty}^{\infty}\exp{\left(-(x^2\right)}dx$ appears in many calculations, its value
is $\sqrt{\pi}$,  a result we will need when we compute the mean value and the variance.
The mean value is
!bt
\begin{equation*}  
 \mu = \int_0^{\infty}xp(x)dx=\frac{1}{b\sqrt{2\pi}}\int_{-\infty}^{\infty}x \exp{\left(-\frac{(x-a)^2}{2b^2}\right)}dx,
\end{equation*}
!et
which becomes with a suitable change of variables
!bt
\begin{equation*}  
 \mu =\frac{1}{b\sqrt{2\pi}}\int_{-\infty}^{\infty}b\sqrt{2}(a+b\sqrt{2}y)\exp{-y^2}dy=a.
\end{equation*}
!et
!eblock


===== Probability Distribution Functions, the normal distribution =====
!bblock
Similarly, the variance becomes
!bt
\begin{equation*}  
 \sigma^2 = \frac{1}{b\sqrt{2\pi}}\int_{-\infty}^{\infty}(x-\mu)^2 \exp{\left(-\frac{(x-a)^2}{2b^2}\right)}dx,
\end{equation*}
!et
and inserting the mean value and performing a variable change we obtain

!bt
\begin{equation*}  
 \sigma^2 = \frac{1}{b\sqrt{2\pi}}\int_{-\infty}^{\infty}b\sqrt{2}(b\sqrt{2}y)^2\exp{\left(-y^2\right)}dy=
\frac{2b^2}{\sqrt{\pi}}\int_{-\infty}^{\infty}y^2\exp{\left(-y^2\right)}dy,
\end{equation*}
!et
and performing a final integration by parts we obtain the well-known result $\sigma^2=b^2$.
It is useful to introduce the standard normal distribution as well, defined by $\mu=a=0$, viz. a distribution
centered around zero and with a variance $\sigma^2=1$, leading to

!bt
\begin{equation}
   p(x)=\frac{1}{\sqrt{2\pi}}\exp{\left(-\frac{x^2}{2}\right)}.
\end{equation}
!et
!eblock


===== Probability Distribution Functions, the cumulative distribution =====
!bblock

The exponential and uniform distributions have simple cumulative functions,
whereas the normal distribution does not, being proportional to the so-called
error function $erf(x)$, given by

!bt
\begin{equation*} 
P(x) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^x\exp{\left(-\frac{t^2}{2}\right)}dt,
\end{equation*}
!et
which is difficult to evaluate in a quick way. 
!eblock



===== Probability Distribution Functions, other important distribution =====
!bblock

Some other PDFs which one encounters often in the natural sciences are the binomial distribution
!bt
\begin{equation*}
   p(x) = \left(\begin{array}{c} n \\ x\end{array}\right)y^x(1-y)^{n-x} \hspace{0.5cm}x=0,1,\dots,n,
\end{equation*}
!et
where $y$ is the probability for a specific event, such as the tossing of a coin or moving left or right
in case of a random walker. Note that $x$ is a discrete stochastic variable. 
 
The sequence of binomial trials is characterized by the following definitions

  * Every experiment is thought to consist of $N$ independent trials.

  * In every independent trial one registers if a specific situation happens or not, such as the  jump to the left or right of a random walker.

  * The probability for every outcome in a single trial has the same value, for example the outcome of tossing (either heads or tails) a coin is always $1/2$.
!eblock


===== Probability Distribution Functions, the binomial distribution =====
!bblock

In order to compute the mean and variance we need to recall Newton's binomial
formula
!bt
\begin{equation*}
   (a+b)^m=\sum_{n=0}^m \left(\begin{array}{c} m \\ n\end{array}\right)a^nb^{m-n},
\end{equation*}
!et
which can be used to show that

!bt
\begin{equation*}
\sum_{x=0}^n\left(\begin{array}{c} n \\ x\end{array}\right)y^x(1-y)^{n-x} = (y+1-y)^n = 1,
\end{equation*}
!et
the PDF is normalized to one. 
The mean value is
!bt
\begin{equation*}
\mu = \sum_{x=0}^n x\left(\begin{array}{c} n \\ x\end{array}\right)y^x(1-y)^{n-x} =
\sum_{x=0}^n x\frac{n!}{x!(n-x)!}y^x(1-y)^{n-x}, 
\end{equation*}
!et
resulting in
!bt
\begin{equation*}
\mu = 
\sum_{x=0}^n x\frac{(n-1)!}{(x-1)!(n-1-(x-1))!}y^{x-1}(1-y)^{n-1-(x-1)},
\end{equation*}
!et
which we rewrite as

!bt
\begin{equation*}
\mu=ny\sum_{\nu=0}^n\left(\begin{array}{c} n-1 \\ \nu\end{array}\right)y^{\nu}(1-y)^{n-1-\nu} =ny(y+1-y)^{n-1}=ny. 
\end{equation*}
!et
!eblock
The variance is slightly trickier to get. It reads $\sigma^2=ny(1-y)$. 


===== Probability Distribution Functions, Poisson's  distribution =====
!bblock

Another important distribution with discrete stochastic variables $x$ is  
the Poisson model, which resembles the exponential distribution and reads
!bt
\begin{equation*}
    p(x) = \frac{\lambda^x}{x!} e^{-\lambda} \hspace{0.5cm}x=0,1,\dots,;\lambda > 0.
\end{equation*}
!et
In this case both the mean value and the variance are easier to calculate,

!bt
\begin{equation*}
\mu = \sum_{x=0}^{\infty} x \frac{\lambda^x}{x!} e^{-\lambda} = \lambda e^{-\lambda}\sum_{x=1}^{\infty}
\frac{\lambda^{x-1}}{(x-1)!}=\lambda,
\end{equation*}
!et
and the variance is $\sigma^2=\lambda$. 
!eblock




===== Probability Distribution Functions, Poisson's  distribution =====
!bblock
An example of applications of the Poisson distribution could be the counting
of the number of $\alpha$-particles emitted from a radioactive source in a given time interval.
In the limit of $n\rightarrow \infty$ and for small probabilities $y$, the binomial distribution
approaches the Poisson distribution. Setting $\lambda = ny$, with $y$ the probability for an event in
the binomial distribution we can show that

!bt
\begin{equation*} 
\lim_{n\rightarrow \infty}\left(\begin{array}{c} n \\ x\end{array}\right)y^x(1-y)^{n-x} e^{-\lambda}=\sum_{x=1}^{\infty}\frac{\lambda^x}{x!} e^{-\lambda}.
\end{equation*}
!et
!eblock



===== Meet the  covariance! =====
!bblock
An important quantity in a statistical analysis is the so-called covariance. 

Consider the set $\{X_i\}$ of $n$
stochastic variables (not necessarily uncorrelated) with the
multivariate PDF $P(x_1,\dots,x_n)$. The *covariance* of two
of the stochastic variables, $X_i$ and $X_j$, is defined as follows

!bt
\begin{align}
\mathrm{Cov}(X_i,\,X_j) & = \langle (x_i-\langle x_i\rangle)(x_j-\langle x_j\rangle)\rangle \\
&=\int\cdots\int (x_i-\langle x_i\rangle)(x_j-\langle x_j\rangle)P(x_1,\dots,x_n)\,dx_1\dots dx_n,
label{eq:def_covariance}
\end{align}
!et
with
!bt
\begin{equation*}
\langle x_i\rangle =
\int\cdots\int x_i P(x_1,\dots,x_n)\,dx_1\dots dx_n.
\end{equation*}
!et
!eblock




===== Meet the  covariance in matrix disguise =====
!bblock
If we consider the above covariance as a matrix 
!bt
\[
C_{ij} =\mathrm{Cov}(X_i,\,X_j), 
\]
!et
then the diagonal elements are just the familiar
variances, $C_{ii} = \mathrm{Cov}(X_i,\,X_i) = \mathrm{Var}(X_i)$. It turns out that
all the off-diagonal elements are zero if the stochastic variables are
uncorrelated. 
!eblock


=====  Covariance =====
!bc pycod
# Importing various packages
from math import exp, sqrt
from random import random, seed
import numpy as np
import matplotlib.pyplot as plt

def covariance(x, y, n):
    sum = 0.0
    mean_x = np.mean(x)
    mean_y = np.mean(y)
    for i in range(0, n):
        sum += (x[(i)]-mean_x)*(y[i]-mean_y)
    return  sum/n

n = 10

x=np.random.normal(size=n)
y = 4+3*x+np.random.normal(size=n)
covxy = covariance(x,y,n)
print(covxy)
z = np.vstack((x, y))
c = np.cov(z.T)

print(c)

!ec




===== Meet the  covariance, uncorrelated events =====
!bblock

Consider the stochastic variables $X_i$ and $X_j$, ($i\neq j$). We have
!bt
\begin{align*}
Cov(X_i,\,X_j) &= \langle (x_i-\langle x_i\rangle)(x_j-\langle x_j\rangle)\rangle\\
&=\langle x_i x_j - x_i\langle x_j\rangle - \langle x_i\rangle x_j + \langle x_i\rangle\langle x_j\rangle\rangle\\
&=\langle x_i x_j\rangle - \langle x_i\langle x_j\rangle\rangle - \langle \langle x_i\rangle x_j \rangle +
\langle \langle x_i\rangle\langle x_j\rangle\rangle \\
&=\langle x_i x_j\rangle - \langle x_i\rangle\langle x_j\rangle - \langle x_i\rangle\langle x_j\rangle +
\langle x_i\rangle\langle x_j\rangle \\
&=\langle x_i x_j\rangle - \langle x_i\rangle\langle x_j\rangle
\end{align*}
!et
If $X_i$ and $X_j$ are independent (assuming $i \neq j$), we have that 
!bt
\[
\langle x_i x_j\rangle = \langle x_i\rangle\langle x_j\rangle,
\]
!et
leading to 
!bt
\[
Cov(X_i, X_j) = 0 \hspace{0.1cm} (i\neq j).
\]
!et
!eblock




===== Numerical experiments and the covariance =====
!bblock

Now that we have constructed an idealized mathematical framework, let
us try to apply it to empirical observations. Examples of relevant
physical phenomena may be spontaneous decays of nuclei, or a purely
mathematical set of numbers produced by some deterministic
mechanism. It is the latter we will deal with, using so-called pseudo-random
number generators.  In general our observations will contain only a limited set of
observables. We remind the reader that
a *stochastic process* is a process that produces sequentially a
chain of values
!bt
\begin{equation*}
\{x_1, x_2,\dots\,x_k,\dots\}.
\end{equation*}
!et
!eblock



===== Numerical experiments and the covariance =====
!bblock
We will call these
values our *measurements* and the entire set as our measured
*sample*.  The action of measuring all the elements of a sample
we will call a stochastic *experiment* (since, operationally,
they are often associated with results of empirical observation of
some physical or mathematical phenomena; precisely an experiment). We
assume that these values are distributed according to some 
PDF $p_X^{\phantom X}(x)$, where $X$ is just the formal symbol for the
stochastic variable whose PDF is $p_X^{\phantom X}(x)$. Instead of
trying to determine the full distribution $p$ we are often only
interested in finding the few lowest moments, like the mean
$\mu_X^{\phantom X}$ and the variance $\sigma_X^{\phantom X}$.
!eblock



===== Numerical experiments and the covariance, actual situations =====
!bblock
In practical situations however, a sample is always of finite size. Let that
size be $n$. The expectation value of a sample $\alpha$, the _sample mean_, is then defined as follows
!bt
\begin{equation*}
\langle x_{\alpha} \rangle \equiv \frac{1}{n}\sum_{k=1}^n x_{\alpha,k}.
\end{equation*}
!et
The *sample variance* is:
!bt
\begin{equation*}
\mathrm{Var}(x) \equiv \frac{1}{n}\sum_{k=1}^n (x_{\alpha,k} - \langle x_{\alpha} \rangle)^2,
\end{equation*}
!et
with its square root being the *standard deviation of the sample*. 
!eblock



===== Numerical experiments and the covariance, our observables =====
!bblock
You can think of the above observables as a set of quantities which define
a given experiment. This experiment is then repeated several times, say $m$ times.
The total average is then
!bt
\begin{equation}
\langle X_m \rangle= \frac{1}{m}\sum_{\alpha=1}^mx_{\alpha}=\frac{1}{mn}\sum_{\alpha, k} x_{\alpha,k},
label{eq:exptmean}
\end{equation}
!et
where the last sums end at $m$ and $n$.
The total variance is
!bt
\begin{equation*}
\sigma^2_m= \frac{1}{mn^2}\sum_{\alpha=1}^m(\langle x_{\alpha} \rangle-\langle X_m \rangle)^2,
\end{equation*}
!et
which we rewrite as
!bt
\begin{equation}
\sigma^2_m=\frac{1}{m}\sum_{\alpha=1}^m\sum_{kl=1}^n (x_{\alpha,k}-\langle X_m \rangle)(x_{\alpha,l}-\langle X_m \rangle).
label{eq:exptvariance}
\end{equation}
!et
!eblock


===== Numerical experiments and the covariance, the sample variance =====
!bblock

We define also the sample variance $\sigma^2$ of all $mn$ individual experiments as
!bt
\begin{equation}
\sigma^2=\frac{1}{mn}\sum_{\alpha=1}^m\sum_{k=1}^n (x_{\alpha,k}-\langle X_m \rangle)^2.
label{eq:sampleexptvariance}
\end{equation}
!et



These quantities, being known experimental values or the results from our calculations, 
may differ, in some cases
significantly,  from the similarly named
exact values for the mean value $\mu_X$, the variance $\mathrm{Var}(X)$
and the covariance $\mathrm{Cov}(X,Y)$. 
!eblock


===== Numerical experiments and the covariance, central limit theorem =====
!bblock

The central limit theorem states that the PDF $\tilde{p}(z)$ of
the average of $m$ random values corresponding to a PDF $p(x)$ 
is a normal distribution whose mean is the 
mean value of the PDF $p(x)$ and whose variance is the variance
of the PDF $p(x)$ divided by $m$, the number of values used to compute $z$.

The central limit theorem leads then to the well-known expression for the
standard deviation, given by
!bt
\begin{equation*}
    \sigma_m=
\frac{\sigma}{\sqrt{m}}.
\end{equation*}
!et

In many cases the above estimate for the standard deviation, in particular if correlations are strong, may be too simplistic.  We need therefore a more precise defintion of the error and the variance in our results.
!eblock


===== Definition of Correlation Functions and Standard Deviation =====
!bblock
Our estimate of the true average $\mu_{X}$ is the sample mean $\langle X_m \rangle$

!bt
\begin{equation*}
\mu_{X}^{\phantom X} \approx X_m=\frac{1}{mn}\sum_{\alpha=1}^m\sum_{k=1}^n x_{\alpha,k}.
\end{equation*}
!et


We can then use Eq. (ref{eq:exptvariance})
!bt
\begin{equation*}
\sigma^2_m=\frac{1}{mn^2}\sum_{\alpha=1}^m\sum_{kl=1}^n (x_{\alpha,k}-\langle X_m \rangle)(x_{\alpha,l}-\langle X_m \rangle),
\end{equation*}
!et
and rewrite it as
!bt
\begin{equation*}
\sigma^2_m=\frac{\sigma^2}{n}+\frac{2}{mn^2}\sum_{\alpha=1}^m\sum_{k<l}^n (x_{\alpha,k}-\langle X_m \rangle)(x_{\alpha,l}-\langle X_m \rangle),
\end{equation*}
!et
where the first term is the sample variance of all $mn$ experiments divided by $n$
and the last term is nothing but the covariance which arises when $k\ne l$. 
!eblock






===== Definition of Correlation Functions and Standard Deviation =====
!bblock
Our estimate of the true average $\mu_{X}$ is the sample mean $\langle X_m \rangle$

If the 
observables are uncorrelated, then the covariance is zero and we obtain a total variance
which agrees with the central limit theorem. Correlations may often be present in our data set, resulting in a non-zero covariance.  The first term is normally called the uncorrelated 
contribution.
Computationally the uncorrelated first term is much easier to treat
efficiently than the second.
We just accumulate separately the values $x^2$ and $x$ for every
measurement $x$ we receive. The correlation term, though, has to be
calculated at the end of the experiment since we need all the
measurements to calculate the cross terms. Therefore, all measurements
have to be stored throughout the experiment.
!eblock


===== Definition of Correlation Functions and Standard Deviation =====
!bblock

Let us analyze the problem by splitting up the correlation term into
partial sums of the form

!bt
\begin{equation*}
f_d = \frac{1}{nm}\sum_{\alpha=1}^m\sum_{k=1}^{n-d}(x_{\alpha,k}-\langle X_m \rangle)(x_{\alpha,k+d}-\langle X_m \rangle),
\end{equation*}
!et
The correlation term of the total variance can now be rewritten in terms of
$f_d$

!bt
\begin{equation*}
\frac{2}{mn^2}\sum_{\alpha=1}^m\sum_{k<l}^n (x_{\alpha,k}-\langle X_m \rangle)(x_{\alpha,l}-\langle X_m \rangle)=
\frac{2}{n}\sum_{d=1}^{n-1} f_d
\end{equation*}
!et
!eblock



===== Definition of Correlation Functions and Standard Deviation =====
!bblock
The value of $f_d$ reflects the correlation between measurements
separated by the distance $d$ in the samples.  Notice that for
$d=0$, $f$ is just the sample variance, $\sigma^2$. If we divide $f_d$
by $\sigma^2$, we arrive at the so called _autocorrelation function_

!bt
\begin{equation}
\kappa_d = \frac{f_d}{\sigma^2}
label{eq:autocorrelformal}
\end{equation}
!et
which gives us a useful measure of the correlation pair correlation
starting always at $1$ for $d=0$.
!eblock


===== Definition of Correlation Functions and Standard Deviation, sample variance =====
!bblock

The sample variance of the $mn$ experiments can now be
written in terms of the autocorrelation function

!bt
\begin{equation}
\sigma_m^2=\frac{\sigma^2}{n}+\frac{2}{n}\cdot\sigma^2\sum_{d=1}^{n-1}
\frac{f_d}{\sigma^2}=\left(1+2\sum_{d=1}^{n-1}\kappa_d\right)\frac{1}{n}\sigma^2=\frac{\tau}{n}\cdot\sigma^2
label{eq:error_estimate_corr_time}
\end{equation}
!et
and we see that $\sigma_m$ can be expressed in terms of the
uncorrelated sample variance times a correction factor $\tau$ which
accounts for the correlation between measurements. We call this
correction factor the *autocorrelation time*

!bt
\begin{equation}
\tau = 1+2\sum_{d=1}^{n-1}\kappa_d
label{eq:autocorrelation_time}
\end{equation}
!et
# It is closely related to the area under the graph of the
# autocorrelation function. 
For a correlation free experiment, $\tau$
equals 1. 
!eblock




===== Definition of Correlation Functions and Standard Deviation =====
!bblock
From the point of view of
Eq. (ref{eq:error_estimate_corr_time}) we can interpret a sequential
correlation as an effective reduction of the number of measurements by
a factor $\tau$. The effective number of measurements becomes
!bt
\begin{equation*}
n_\mathrm{eff} = \frac{n}{\tau}
\end{equation*}
!et
To neglect the autocorrelation time $\tau$ will always cause our
simple uncorrelated estimate of $\sigma_m^2\approx \sigma^2/n$ to
be less than the true sample error. The estimate of the error will be
too ``good''. On the other hand, the calculation of the full
autocorrelation time poses an efficiency problem if the set of
measurements is very large.  The solution to this problem is given by 
more practically oriented methods like the blocking technique.
#add ref here to flybjerg
!eblock


 
===== Code to compute the Covariance matrix and the Covariance =====
!bc pycod
# Importing various packages
from math import exp, sqrt
from random import random, seed
import numpy as np
import matplotlib.pyplot as plt

# Sample covariance, note the factor 1/(n-1)
def covariance(x, y, n):
    sum = 0.0
    mean_x = np.mean(x)
    mean_y = np.mean(y)
    for i in range(0, n):
        sum += (x[(i)]-mean_x)*(y[i]-mean_y)
    return  sum/(n-1.)

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))
covxx = covariance(x,x,n)
covyy = covariance(y,y,n)
covzz = covariance(z,z,n)
covxy = covariance(x,y,n)
covxz = covariance(x,z,n)
covyz = covariance(y,z,n)
print(covxx,covyy, covzz)
print(covxy,covxz, covyz)
w = np.vstack((x, y, z))
#print(w)
c = np.cov(w)
print(c)
#eigen = np.zeros(n)
Eigvals, Eigvecs = np.linalg.eig(c)
print(Eigvals)
!ec


======= Random Numbers =======
!bblock

Uniform deviates are just random numbers that lie within a specified range
(typically 0 to 1), with any one number in the range just as likely as any other. They
are, in other words, what you probably think random numbers are. However,
we want to distinguish uniform deviates from other sorts of random numbers, for
example numbers drawn from a normal (Gaussian) distribution of specified mean
and standard deviation. These other sorts of deviates are almost always generated by
performing appropriate operations on one or more uniform deviates, as we will see
in subsequent sections. So, a reliable source of random uniform deviates, the subject
of this section, is an essential building block for any sort of stochastic modeling
or Monte Carlo computer work.
!eblock



======= Random Numbers, better name: pseudo random numbers  =======
!bblock

A disclaimer is however appropriate. It should be fairly obvious that 
something as deterministic as a computer cannot generate purely random numbers.

Numbers generated by any of the standard algorithms are in reality pseudo random
numbers, hopefully abiding to the following criteria:

  * they produce a uniform distribution in the interval [0,1].

  * correlations between random numbers are negligible

  * the period before the same sequence of random numbers is repeated   is as large as possible and finally

  * the algorithm should be fast.
!eblock



======= Random number generator RNG  =======
!bblock
 The most common random number generators are based on so-called
Linear congruential relations of the type

!bt
\begin{equation*}
  N_i=(aN_{i-1}+c) \mathrm{MOD} (M),
\end{equation*}
!et
which yield a number in the interval [0,1] through

!bt
\begin{equation*}
  x_i=N_i/M
\end{equation*}
!et

The number 
$M$ is called the period and it should be as large as possible 
 and 
$N_0$ is the starting value, or seed. The function $\mathrm{MOD}$ means the remainder,
that is if we were to evaluate $(13)\mathrm{MOD}(9)$, the outcome is the remainder
of the division $13/9$, namely $4$.
!eblock


======= Random number generator RNG and periodic outputs =======
!bblock

The problem with such generators is that their outputs are periodic;
they 
will start to repeat themselves with a period that is at most $M$. If however
the parameters $a$ and $c$ are badly chosen, the period may be even shorter.

Consider the following example

!bt
\begin{equation*}
  N_i=(6N_{i-1}+7) \mathrm{MOD} (5),
\end{equation*}
!et
with a seed $N_0=2$. This generator produces the sequence
$4,1,3,0,2,4,1,3,0,2,...\dots$, i.e., a sequence with period $5$.
However, increasing $M$ may not guarantee a larger period as the following
example shows

!bt
\begin{equation*}
  N_i=(27N_{i-1}+11) \mathrm{MOD} (54),
\end{equation*}
!et
which still, with $N_0=2$, results in $11,38,11,38,11,38,\dots$, a period of
just $2$.
!eblock


======= Random number generator RNG and its period  =======
!bblock
Typical periods for the random generators provided in the program library 
are of the order of $\sim 10^9$ or larger. Other random number generators which have
become increasingly popular are so-called shift-register generators.
In these generators each successive number depends on many preceding
values (rather than the last values as in the linear congruential
generator).
For example, you could make a shift register generator whose $l$th 
number is the sum of the $l-i$th and $l-j$th values with modulo $M$,
!bt
\begin{equation*}
   N_l=(aN_{l-i}+cN_{l-j})\mathrm{MOD}(M).
\end{equation*}
!et
!eblock


======= Random number generator RNG, other examples =======
!bblock
Such a generator again produces a sequence of pseudorandom numbers
but this time with a period much larger than $M$.
It is also possible to construct more elaborate algorithms by including
more than two past terms in the sum of each iteration.
One example is the generator of "Marsaglia and Zaman":"http://dl.acm.org/citation.cfm?id=187154"
which consists of two congruential relations

!bt
\begin{equation}
   N_l=(N_{l-3}-N_{l-1})\mathrm{MOD}(2^{31}-69),
   label{eq:mz1}
\end{equation}
!et
followed by
!bt
\begin{equation}
   N_l=(69069N_{l-1}+1013904243)\mathrm{MOD}(2^{32}),
   label{eq:mz2}
\end{equation}
!et
which according to the authors has a period larger than $2^{94}$.
!eblock


======= Random number generator RNG, other examples =======
!bblock
Instead of  using modular addition, we could use the bitwise
exclusive-OR ($\oplus$) operation so that

!bt
\begin{equation*}
   N_l=(N_{l-i})\oplus (N_{l-j})
\end{equation*}
!et
where the bitwise action of $\oplus$ means that if $N_{l-i}=N_{l-j}$ the result is
$0$ whereas if $N_{l-i}\ne N_{l-j}$ the result is
$1$. As an example, consider the case where  $N_{l-i}=6$ and $N_{l-j}=11$. The first
one has a bit representation (using 4 bits only) which reads $0110$ whereas the 
second number is $1011$. Employing the $\oplus$ operator yields 
$1101$, or $2^3+2^2+2^0=13$.

In Fortran90, the bitwise $\oplus$ operation is coded through the intrinsic
function $\mathrm{IEOR}(m,n)$ where $m$ and $n$ are the input numbers, while in $C$
it is given by $m\wedge n$. 
!eblock



======= Random number generator RNG, RAN0 =======
!bblock

We show here how the linear congruential algorithm can be implemented, namely
!bt
\begin{equation*}
  N_i=(aN_{i-1}) \mathrm{MOD} (M).
\end{equation*}
!et
However, since $a$ and $N_{i-1}$ are integers and their multiplication 
could become greater than the standard 32 bit integer, there is a trick via 
Schrage's algorithm which approximates the multiplication
of large integers through the factorization
!bt
\begin{equation*}
  M=aq+r,
\end{equation*}
!et
where we have defined

!bt
\begin{equation*}
   q=[M/a],
\end{equation*}
!et
and
!bt
\begin{equation*}
  r = M\hspace{0.1cm}\mathrm{MOD} \hspace{0.1cm}a.
\end{equation*}
!et
where the brackets denote integer division. In the code below the numbers 
$q$ and $r$ are chosen so that $r < q$.
!eblock




======= Random number generator RNG, RAN0 =======
!bblock

To see how this works we note first that
!bt
\begin{equation}
(aN_{i-1}) \mathrm{MOD} (M)= (aN_{i-1}-[N_{i-1}/q]M)\mathrm{MOD} (M),
label{eq:rntrick1}
\end{equation}
!et
since we can add or subtract any integer multiple of $M$ from $aN_{i-1}$.
The last term $[N_{i-1}/q]M\mathrm{MOD}(M)$ is zero since the integer division 
$[N_{i-1}/q]$ just yields a constant which is multiplied with $M$. 
!eblock



======= Random number generator RNG, RAN0 =======
!bblock
We can now rewrite Eq. (ref{eq:rntrick1}) as

!bt
\begin{equation}
(aN_{i-1}) \mathrm{MOD} (M)= (aN_{i-1}-[N_{i-1}/q](aq+r))\mathrm{MOD} (M),
label{eq:rntrick2}
\end{equation}
!et 
which results
in

!bt
\begin{equation}
(aN_{i-1}) \mathrm{MOD} (M)= \left(a(N_{i-1}-[N_{i-1}/q]q)-[N_{i-1}/q]r)\right)\mathrm{MOD} (M),
label{eq:rntrick3}
\end{equation}
!et 
yielding
!bt
\begin{equation}
(aN_{i-1}) \mathrm{MOD} (M)= \left(a(N_{i-1}\mathrm{MOD} (q)) -[N_{i-1}/q]r)\right)\mathrm{MOD} (M).
label{eq:rntrick4}
\end{equation}
!et 
!eblock



======= Random number generator RNG, RAN0 =======
!bblock
The term $[N_{i-1}/q]r$ is always smaller or equal $N_{i-1}(r/q)$ and with $r < q$ we obtain always a 
number smaller than $N_{i-1}$, which is smaller than $M$. 
And since the number $N_{i-1}\mathrm{MOD} (q)$ is between zero and $q-1$ then
$a(N_{i-1}\mathrm{MOD} (q))< aq$. Combined with our definition of $q=[M/a]$ ensures that 
this term is also smaller than $M$ meaning that both terms fit into a
32-bit signed integer. None of these two terms can be negative, but their difference could.
The algorithm below adds $M$ if their difference is negative.
Note that the program uses the bitwise $\oplus$ operator to generate
the starting point for each generation of a random number. The period
of $ran0$ is $\sim 2.1\times 10^{9}$. A special feature of this
algorithm is that is should never be called with the initial seed 
set to $0$. 
!eblock



======= Random number generator RNG, RAN0 code =======
!bblock

!bc cppcod
    /*
     ** The function
     **           ran0()
     ** is an "Minimal" random number generator of Park and Miller
     ** Set or reset the input value
     ** idum to any integer value (except the unlikely value MASK)
     ** to initialize the sequence; idum must not be altered between
     ** calls for sucessive deviates in a sequence.
     ** The function returns a uniform deviate between 0.0 and 1.0.
     */
double ran0(long &idum)
{
   const int a = 16807, m = 2147483647, q = 127773;
   const int r = 2836, MASK = 123459876;
   const double am = 1./m;
   long     k;
   double   ans;
   idum ^= MASK;
   k = (*idum)/q;
   idum = a*(idum - k*q) - r*k;
   // add m if negative difference
   if(idum < 0) idum += m;
   ans=am*(idum);
   idum ^= MASK;
   return ans;
} // End: function ran0() 
!ec
!eblock


===== Properties of Selected Random Number Generators =====
!bblock

As mentioned previously, the underlying PDF for the generation of
random numbers is the uniform distribution, meaning that the 
probability for finding a number $x$ in the interval [0,1] is $p(x)=1$.

A random number generator should produce numbers which are uniformly distributed
in this interval. The table  shows the distribution of $N=10000$ random
numbers generated by the functions in the program library.
We note in this table that the number of points in the various
intervals $0.0-0.1$, $0.1-0.2$ etc are fairly close to $1000$, with some minor
deviations. 

Two additional measures are the standard deviation $\sigma$ and the mean
$\mu=\langle x\rangle$.
!eblock


===== Properties of Selected Random Number Generators =====
!bblock
For the uniform distribution, the mean value $\mu$ is then

!bt
\begin{equation*}
  \mu=\langle x\rangle=\frac{1}{2}
\end{equation*}
!et
while the standard deviation is

!bt
\begin{equation*}
   \sigma=\sqrt{\langle x^2\rangle-\mu^2}=\frac{1}{\sqrt{12}}=0.2886.
\end{equation*}
!et
!eblock


===== Properties of Selected Random Number Generators =====
!bblock
The various random number generators produce results which agree rather well with
these limiting values. 

|------------------------------------------------------|
| $x$-bin  | ran0     | ran1     | ran2     | ran3     |
|----c---------r---------r---------r---------r---------|
| 0.0-0.1  | 1013     | 991      | 938      | 1047     |
| 0.1-0.2  | 1002     | 1009     | 1040     | 1030     |
| 0.2-0.3  | 989      | 999      | 1030     | 993      |
| 0.3-0.4  | 939      | 960      | 1023     | 937      |
| 0.4-0.5  | 1038     | 1001     | 1002     | 992      |
| 0.5-0.6  | 1037     | 1047     | 1009     | 1009     |
| 0.6-0.7  | 1005     | 989      | 1003     | 989      |
| 0.7-0.8  | 986      | 962      | 985      | 954      |
| 0.8-0.9  | 1000     | 1027     | 1009     | 1023     |
| 0.9-1.0  | 991      | 1015     | 961      | 1026     |
| $\mu$    | 0.4997   | 0.5018   | 0.4992   | 0.4990   |
| $\sigma$ | 0.2882   | 0.2892   | 0.2861   | 0.2915   |
|------------------------------------------------------|

!eblock


===== Simple demonstration of RNGs using python =====
!bblock
The following simple Python code plots the distribution of the produced random numbers using the linear congruential RNG employed by Python. The trend displayed in the previous table is seen rather clearly.
!bc pyscpro
#!/usr/bin/env python
import numpy as np
import matplotlib.mlab as mlab
import matplotlib.pyplot as plt
import random

# initialize the rng with a seed
random.seed() 
counts = 10000
values = np.zeros(counts)   
for i in range (1, counts, 1):
    values[i] = random.random()

# the histogram of the data
n, bins, patches = plt.hist(values, 10, facecolor='green')

plt.xlabel('$x$')
plt.ylabel('Number of counts')
plt.title(r'Test of uniform distribution')
plt.axis([0, 1, 0, 1100])
plt.grid(True)
plt.show()
!ec
!eblock



===== Properties of Selected Random Number Generators =====
!bblock
Since our random numbers, which are typically generated via a linear congruential algorithm,
are never fully independent, we can then define 
an important test which measures the degree of correlation, namely the  so-called  
auto-correlation function defined previously, see again Eq. (ref{eq:autocorrelformal}).
We rewrite it here as
!bt
\begin{equation*}
    C_k=\frac{f_d}
             {\sigma^2},
\end{equation*}
!et
with $C_0=1$. Recall that 
$\sigma^2=\langle x_i^2\rangle-\langle x_i\rangle^2$ and that
!bt
\begin{equation*}
f_d = \frac{1}{nm}\sum_{\alpha=1}^m\sum_{k=1}^{n-d}(x_{\alpha,k}-\langle X_m \rangle)(x_{\alpha,k+d}-\langle X_m \rangle),
\end{equation*}
!et


The non-vanishing of $C_k$ for $k\ne 0$ means that the random
numbers are not independent. The independence of the random numbers is crucial 
in the evaluation of other expectation values. If they are not independent, our
assumption for approximating $\sigma_N$ is no longer valid.

!eblock



===== Autocorrelation function =====
This program computes the autocorrelation function as discussed in the equation on the previous slide for random numbers generated with the normal distribution $N(0,1)$.
!bc pycod
# Importing various packages
from math import exp, sqrt
from random import random, seed
import numpy as np
import matplotlib.pyplot as plt

def autocovariance(x, n, k, mean_x):
    sum = 0.0
    for i in range(0, n-k):
        sum += (x[(i+k)]-mean_x)*(x[i]-mean_x)
    return  sum/n

n = 1000
x=np.random.normal(size=n)
autocor = np.zeros(n)
figaxis = np.zeros(n)
mean_x=np.mean(x)
var_x = np.var(x)
print(mean_x, var_x)
for i in range (0, n):
    figaxis[i] = i
    autocor[i]=(autocovariance(x, n, i, mean_x))/var_x    

plt.plot(figaxis, autocor, "r-")
plt.axis([0,n,-0.1, 1.0])
plt.xlabel(r'$i$')
plt.ylabel(r'$\gamma_i$')
plt.title(r'Autocorrelation function')
plt.show()
!ec
As can be seen from the plot, the first point gives back the variance and a value of one. 
For the remaining values we notice that there are still non-zero values for the auto-correlation function.

















===== Correlation function and which random number generators should I use =====
!bblock
The program here computes the correlation function for one of the standard functions included with the c++ compiler. 
!bc cppcod
//  This function computes the autocorrelation function for 
//  the standard c++ random number generator

#include <fstream>
#include <iomanip>
#include <iostream>
#include <cmath>
using namespace std;
// output file as global variable
ofstream ofile;  

//     Main function begins here     
int main(int argc, char* argv[])
{
     int n;
     char *outfilename;

     cin >> n;
     double MCint = 0.;      double MCintsqr2=0.;
     double invers_period = 1./RAND_MAX; // initialise the random number generator
     srand(time(NULL));  // This produces the so-called seed in MC jargon
     // Compute the variance and the mean value of the uniform distribution
     // Compute also the specific values x for each cycle in order to be able to
     // the covariance and the correlation function  
     // Read in output file, abort if there are too few command-line arguments
     if( argc <= 2 ){
       cout << "Bad Usage: " << argv[0] << 
	 " read also output file and number of cycles on same line" << endl;
       exit(1);
     }
     else{
       outfilename=argv[1];
     }
     ofile.open(outfilename); 
     // Get  the number of Monte-Carlo samples
     n = atoi(argv[2]);
     double *X;  
     X = new double[n];
     for (int i = 0;  i < n; i++){
           double x = double(rand())*invers_period; 
           X[i] = x;
           MCint += x;
           MCintsqr2 += x*x;
     }
     double Mean = MCint/((double) n );
     MCintsqr2 = MCintsqr2/((double) n );
     double STDev = sqrt(MCintsqr2-Mean*Mean);
     double Variance = MCintsqr2-Mean*Mean;
//   Write mean value and standard deviation 
     cout << " Standard deviation= " << STDev << " Integral = " << Mean << endl;

     // Now we compute the autocorrelation function
     double *autocor;  autocor = new double[n];
     for (int j = 0; j < n; j++){
       double sum = 0.0;
       for (int k = 0; k < (n-j); k++){
	 sum  += (X[k]-Mean)*(X[k+j]-Mean); 
       }
       autocor[j] = sum/Variance/((double) n );
       ofile << setiosflags(ios::showpoint | ios::uppercase);
       ofile << setw(15) << setprecision(8) << j;
       ofile << setw(15) << setprecision(8) << autocor[j] << endl;
     }
     ofile.close();  // close output file
     return 0;
}  // end of main program 
!ec
!eblock




======= Which RNG should I use?  =======
!bblock
* C++ has a class called _random_. The "random class":"http://www.cplusplus.com/reference/random/" contains a large selection of RNGs and is highly recommended. Some of these RNGs have very large periods making it thereby very safe to use these RNGs in case one is performing large calculations. In particular, the "Mersenne twister random number engine":"http://www.cplusplus.com/reference/random/mersenne_twister_engine/" has a period of $2^{19937}$. 
* Add RNGs in Python
 
!eblock



===== How to use the Mersenne generator =====
!bblock
The following part of a c++ code (from project 4) sets up the uniform distribution for $x\in [0,1]$. 
!bc cppcod
/*

//  You need this 
#include <random>

// Initialize the seed and call the Mersienne algo
std::random_device rd;
std::mt19937_64 gen(rd());
// Set up the uniform distribution for x \in [[0, 1]
std::uniform_real_distribution<double> RandomNumberGenerator(0.0,1.0);

// Now use the RNG
int ix = (int) (RandomNumberGenerator(gen)*NSpins);
!ec
!eblock





===== Why blocking? ===== 
!bblock Statistical analysis
    * Monte Carlo simulations can be treated as *computer experiments*
    * The results can be analysed with the same statistical tools as we would use analysing experimental data.
    * As in all experiments, we are looking for expectation values and an estimate of how accurate they are, i.e., possible sources for errors.

A very good article which explains blocking is H. Flyvbjerg and H. G. Petersen, *Error estimates on averages of correlated data*,  "Journal of Chemical Physics 91, 461-466 (1989)":"http://scitation.aip.org/content/aip/journal/jcp/91/1/10.1063/1.457480".

!eblock    




===== Why blocking? ===== 
!bblock Statistical analysis
    * As in other experiments, Monte Carlo experiments have two classes of errors:
      * Statistical errors
      * Systematical errors
    * Statistical errors can be estimated using standard tools from statistics
    * Systematical errors are method specific and must be treated differently from case to case. (In VMC a common source is the step length or time step in importance sampling)
!eblock    



===== Code to demonstrate the calculation of the autocorrelation function =====
The following code computes the autocorrelation function, the covariance and the standard deviation
for standard RNG. 
The "following  file":"https://github.com/CompPhysics/ComputationalPhysics2/tree/gh-pages/doc/Programs/LecturePrograms/programs/Blocking/autocorrelation.cpp" gives the code.
!bc cppcod
//  This function computes the autocorrelation function for 
//  the Mersenne random number generator with a uniform distribution
#include <iostream>
#include <fstream>
#include <iomanip>
#include <cstdlib>
#include <random>
#include <armadillo>
#include <string>
#include <cmath>
using namespace  std;
using namespace arma;
// output file
ofstream ofile;

//     Main function begins here     
int main(int argc, char* argv[])
{
  int MonteCarloCycles;
  string filename;
  if (argc > 1) {
    filename=argv[1];
    MonteCarloCycles = atoi(argv[2]);
    string fileout = filename;
    string argument = to_string(MonteCarloCycles);
    fileout.append(argument);
    ofile.open(fileout);
  }

  // Compute the variance and the mean value of the uniform distribution
  // Compute also the specific values x for each cycle in order to be able to
  // compute the covariance and the correlation function  

  vec X  = zeros<vec>(MonteCarloCycles);
  double MCint = 0.;      double MCintsqr2=0.;
  std::random_device rd;
  std::mt19937_64 gen(rd());
  // Set up the uniform distribution for x \in [[0, 1]
  std::uniform_real_distribution<double> RandomNumberGenerator(0.0,1.0);
  for (int i = 0;  i < MonteCarloCycles; i++){
    double x =   RandomNumberGenerator(gen); 
    X(i) = x;
    MCint += x;
    MCintsqr2 += x*x;
  }
  double Mean = MCint/((double) MonteCarloCycles );
  MCintsqr2 = MCintsqr2/((double) MonteCarloCycles );
  double STDev = sqrt(MCintsqr2-Mean*Mean);
  double Variance = MCintsqr2-Mean*Mean;
  //   Write mean value and variance
  cout << " Sample variance= " << Variance  << " Mean value = " << Mean << endl;
  // Now we compute the autocorrelation function
  vec autocorrelation = zeros<vec>(MonteCarloCycles);
  for (int j = 0; j < MonteCarloCycles; j++){
    double sum = 0.0;
    for (int k = 0; k < (MonteCarloCycles-j); k++){
      sum  += (X(k)-Mean)*(X(k+j)-Mean); 
    }
    autocorrelation(j) = sum/Variance/((double) MonteCarloCycles );
    ofile << setiosflags(ios::showpoint | ios::uppercase);
    ofile << setw(15) << setprecision(8) << j;
    ofile << setw(15) << setprecision(8) << autocorrelation(j) << endl;
  }
  // Now compute the exact covariance using the autocorrelation function
  double Covariance = 0.0;
  for (int j = 0; j < MonteCarloCycles; j++){
    Covariance  += autocorrelation(j);
  }
  Covariance *=  2.0/((double) MonteCarloCycles);
  // Compute now the total variance, including the covariance, and obtain the standard deviation
  double TotalVariance = (Variance/((double) MonteCarloCycles ))+Covariance;
  cout << "Covariance =" << Covariance << "Totalvariance= " << TotalVariance << "Sample Variance/n= " << (Variance/((double) MonteCarloCycles )) << endl;
  cout << " STD from sample variance= " << sqrt(Variance/((double) MonteCarloCycles )) << " STD with covariance = " << sqrt(TotalVariance) << endl;

  ofile.close();  // close output file
  return 0;
}  // end of main program 


!ec



===== What is blocking? =====
!bblock Blocking
    * Say that we have a set of samples from a Monte Carlo experiment
    * Assuming (wrongly) that our samples are uncorrelated our best estimate of the standard deviation of the mean $\langle \mathbf{M}\rangle$ is given by
!bt   
\[
\sigma=\sqrt{\frac{1}{n}\left(\langle \mathbf{M}^2\rangle-\langle \mathbf{M}\rangle^2\right)} 
\]
!et
    * If the samples are correlated we can rewrite our results to show  that
!bt      
\[
\sigma=\sqrt{\frac{1+2\tau/\Delta t}{n}\left(\langle \mathbf{M}^2\rangle-\langle \mathbf{M}\rangle^2\right)}
\]
!et
      where $\tau$ is the correlation time (the time between a sample and the next uncorrelated sample) and $\Delta t$ is time between each sample
!eblock    


===== What is blocking? =====
!bblock Blocking 
    * If $\Delta t\gg\tau$ our first estimate of $\sigma$ still holds
    * Much more common that $\Delta t<\tau$
    * In the method of data blocking we divide the sequence of samples into blocks
    * We then take the mean $\langle \mathbf{M}_i\rangle$ of block $i=1\ldots n_{blocks}$ to calculate the total mean and variance
    * The size of each block must be so large that sample $j$ of block $i$ is not correlated with sample $j$ of block $i+1$
    * The correlation time $\tau$ would be a good choice
!eblock


===== What is blocking? =====
!bblock Blocking
    * Problem: We don't know $\tau$ or it is too expensive to compute
    * Solution: Make a plot of std. dev. as a function of blocksize
    * The estimate of std. dev. of correlated data is too low $\to$ the error will increase with increasing block size until the blocks are uncorrelated, where we reach a plateau
    * When the std. dev. stops increasing the blocks are uncorrelated
!eblock


===== Implementation =====
!bblock 
    * Do a Monte Carlo simulation, storing all samples to file
    * Do the statistical analysis on this file, independently of your Monte Carlo program
    * Read the file into an array
    * Loop over various block sizes
    * For each block size $n_b$, loop over the array in steps of $n_b$ taking the mean of elements $i n_b,\ldots,(i+1) n_b$
    * Take the mean and variance of the resulting array
    * Write the results for each block size to file for later
      analysis
!eblock






===== Actual implementation with code, main function =====
When the file gets large, it can be useful to write your data in binary mode instead of ascii characters.
The "following python file":"https://github.com/CompPhysics/MachineLearning/blob/master/doc/Programs/Sampling/analysis.py"   reads data from file with the output from every Monte Carlo cycle.
!bc pycod
# Blocking
    @timeFunction
    def blocking(self, blockSizeMax = 500):
        blockSizeMin = 1

        self.blockSizes = []
        self.meanVec = []
        self.varVec = []

        for i in range(blockSizeMin, blockSizeMax):
            if(len(self.data) % i != 0):
                pass#continue
            blockSize = i
            meanTempVec = []
            varTempVec = []
            startPoint = 0
            endPoint = blockSize

            while endPoint <= len(self.data):
                meanTempVec.append(np.average(self.data[startPoint:endPoint]))
                startPoint = endPoint
                endPoint += blockSize
            mean, var = np.average(meanTempVec), np.var(meanTempVec)/len(meanTempVec)
            self.meanVec.append(mean)
            self.varVec.append(var)
            self.blockSizes.append(blockSize)

        self.blockingAvg = np.average(self.meanVec[-200:])
        self.blockingVar = (np.average(self.varVec[-200:]))
        self.blockingStd = np.sqrt(self.blockingVar)

!ec





===== The Bootstrap method =====

The Bootstrap  resampling method is also very popular. It is very simple:

o Start with your sample of measurements and compute the sample variance and the mean values
o Then start again but pick in a random way the numbers in the sample and recalculate the mean and the sample variance.
o Repeat this $K$ times.

It can be shown, see the article by "Efron":"https://projecteuclid.org/download/pdf_1/euclid.aos/1176344552"
that it produces the correct standard deviation.

This method is very useful for small ensembles of data points.  

 
===== Bootstrapping =====
Given a set of $N$ data, assume that we are interested in some 
observable $\theta$ which may be estimated from that set. This observable can also be for example the result of a fit based on all $N$ raw data. 
Let us call the value of the observable obtained from the original 
data set $\hat{\theta}$. One recreates from the sample repeatedly 
other samples by choosing randomly $N$ data out of the original set. 
This costs essentially nothing, since we just recycle the original data set for the building of new sets. 

 
===== Bootstrapping, recipe =====
Let us assume we have done this $K$ times and thus have $K$ sets of $N$ 
data values each. 
Of course some values will enter more than once in the new sets. For each of these sets one computes the observable $\theta$ resulting in values $\theta_k$ with $k = 1,...,K$. Then one determines
!bt
\[
\tilde{\theta} = \frac{1}{K} \sum_{k=1}^K \theta_k,
\]
!et
and
!bt
\[
sigma^2_{\tilde{\theta}} = \frac{1}{K} \sum_{k=1}^K \left(\theta_k-\tilde{\theta}\right)^2.
\]
!et

These are estimators for $\angle\theta\rangle$ and its variance. They are not unbiased and therefore 
$\tilde{\theta}\neq\hat{\theta}$  for finite K. 

The difference is called bias and gives an idea on how far away the result may be from 
the true $\angle\theta\rangle$. As final result for the observable one quotes $\angle\theta\rangle = \tilde{\theta} \pm \sigma_{\tilde{\theta}}$ .


 
===== Bootstrapping, "code":"https://github.com/CompPhysics/MachineLearning/blob/master/doc/Programs/Sampling/analysis.py" =====
!bc
# Bootstrap
    @timeFunction
    def bootstrap(self, nBoots = 1000):
        bootVec = np.zeros(nBoots)
        for k in range(0,nBoots):
            bootVec[k] = np.average(np.random.choice(self.data, len(self.data)))
        self.bootAvg = np.average(bootVec)
        self.bootVar = np.var(bootVec)
        self.bootStd = np.std(bootVec)
!ec

 
===== Jackknife, "code":"https://github.com/CompPhysics/MachineLearning/blob/master/doc/Programs/Sampling/analysis.py" =====
!bc
# Jackknife
    @timeFunction
    def jackknife(self):
        jackknVec = np.zeros(len(self.data))
        for k in range(0,len(self.data)):
            jackknVec[k] = np.average(np.delete(self.data, k))
        self.jackknAvg = self.avg - (len(self.data) - 1) * (np.average(jackknVec) - self.avg)
        self.jackknVar = float(len(self.data) - 1) * np.var(jackknVec)
        self.jackknStd = np.sqrt(self.jackknVar)
!ec