diff --git a/doc/src/How2ReadData/How2ReadData.dlog b/doc/src/How2ReadData/How2ReadData.dlog new file mode 100644 index 000000000..915821c77 --- /dev/null +++ b/doc/src/How2ReadData/How2ReadData.dlog @@ -0,0 +1,22 @@ +translating doconce text in How2ReadData.do.txt to ipynb +*** replacing \bm{...} by \boldsymbol{...} (\bm is not supported by MathJax) +collected all required additional files in ipynb-How2ReadData-src.tar.gz which must be distributed with the notebook +*** warning: latex envir \begin{bmatrix} does not work well in Markdown. + Stick to \[ ... \], equation, equation*, align, or align* + environments in math environments. + +*** warning: latex envir \begin{bmatrix} does not work well in Markdown. + Stick to \[ ... \], equation, equation*, align, or align* + environments in math environments. + +*** warning: latex envir \begin{bmatrix} does not work well in Markdown. + Stick to \[ ... \], equation, equation*, align, or align* + environments in math environments. + +*** warning: latex envir \begin{cases} does not work well in Markdown. + Stick to \[ ... \], equation, equation*, align, or align* + environments in math environments. + +Failed to remove ans_at_end environment +Failed to remove sol_at_end environment +output in How2ReadData.ipynb diff --git a/doc/src/How2ReadData/How2ReadData.do.txt b/doc/src/How2ReadData/How2ReadData.do.txt index 863b19c39..a5d332464 100644 --- a/doc/src/How2ReadData/How2ReadData.do.txt +++ b/doc/src/How2ReadData/How2ReadData.do.txt @@ -1,10 +1,7 @@ -TITLE: Data Analysis and Machine Learning: Getting started, our first data and Machine Learning encounters -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 +======= Getting started, our first data and Machine Learning encounters ======= - -===== Introduction ===== +===== Introduction ===== Our emphasis throughout this series of lectures is on understanding the mathematical aspects of diff --git a/doc/src/How2ReadData/ipynb-How2ReadData-src.tar.gz b/doc/src/How2ReadData/ipynb-How2ReadData-src.tar.gz new file mode 100644 index 000000000..cb0f02a35 Binary files /dev/null and b/doc/src/How2ReadData/ipynb-How2ReadData-src.tar.gz differ diff --git a/doc/src/Introduction/Introduction.dlog b/doc/src/Introduction/Introduction.dlog new file mode 100644 index 000000000..383a05555 --- /dev/null +++ b/doc/src/Introduction/Introduction.dlog @@ -0,0 +1,10 @@ +translating doconce text in Introduction.do.txt to ipynb +*** replacing \bm{...} by \boldsymbol{...} (\bm is not supported by MathJax) +Failed to remove ans_at_end environment +Failed to remove sol_at_end environment +output in Introduction.ipynb +translating doconce text in Introduction.do.txt to ipynb +*** replacing \bm{...} by \boldsymbol{...} (\bm is not supported by MathJax) +Failed to remove ans_at_end environment +Failed to remove sol_at_end environment +output in Introduction.ipynb diff --git a/doc/src/Introduction/Introduction.do.txt b/doc/src/Introduction/Introduction.do.txt index ac5dd7c9e..2b7a5acd2 100644 --- a/doc/src/Introduction/Introduction.do.txt +++ b/doc/src/Introduction/Introduction.do.txt @@ -1,10 +1,4 @@ -TITLE: Introduction to Applied 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 - - - -===== Introduction ===== +======= Introduction to Applied Data Analysis and Machine Learning ======= During the last two decades there has been a swift and amazing development of Machine Learning techniques and algorithms that impact @@ -147,8 +141,6 @@ insights and knowledge we have about a specific system, play a central role when we develop a specific machine learning algorithm. -===== Machine Learning, an extremely rich field ===== - Machine learning is an extremely rich field, in spite of its young age. The increases we have seen during the last decades in computational capabilities have been followed by developments of @@ -164,8 +156,6 @@ communities of developers in the thousands or more. And the number of code developers and contributors keeps increasing. -===== A multidisciplinary approach ===== - Not all the algorithms and methods can be given a rigorous mathematical justification (for example decision trees and random forests), opening up thereby large rooms for experimenting and @@ -176,12 +166,6 @@ Carlo methods are central elements in a proper understanding of many of the 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 @@ -195,35 +179,24 @@ 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: -!bpop * 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 often 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. -!epop - - -===== Essential elements of ML ===== - The methods we cover have three main topics in common, irrespective of whether we deal with supervised or unsupervised learning. -!bpop + * The first ingredient is normally our data set (which can be subdivided into training, validation and test data). Many find the most difficult part of using Machine Learning to be the set up of your data in a meaningful way. * The second item is a model which is normally a function of some parameters. The model reflects our knowledge of the system (or lack thereof). As an example, if we know that our data show a behavior similar to what would be predicted by a polynomial, fitting our data to a polynomial of some degree would then determin our model. * The last ingredient is a so-called _cost/loss_ function (or error function) which allows us to present an estimate on how good our model is in reproducing the data it is supposed to train. -!epop - - -===== An optimization/minimization problem ===== - At the heart of basically all Machine Learning algorithms we will encounter so-called minimization or optimization algorithms. A large family of such methods are so-called _gradient methods_. @@ -279,10 +252,6 @@ is that if we are not specific about what we mean by a *correct* model, there could easily be many different models that fit the given data set *equally well*. - -===== What is a good model? Can we define it? ===== - - The central question is this: what leads us to say that a model is correct or optimal for a given data set? 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two decades there has been a swift and amazing\n", + "development of Machine Learning techniques and algorithms that impact\n", + "many areas in not only Science and Technology but also the Humanities,\n", + "Social Sciences, Medicine, Law, indeed, almost all possible\n", + "disciplines. The applications are incredibly many, from self-driving\n", + "cars to solving high-dimensional differential equations or complicated\n", + "quantum mechanical many-body problems. Machine Learning is perceived\n", + "by many as one of the main disruptive techniques nowadays. \n", + "\n", + "Statistics, Data science and Machine Learning form important\n", + "fields of research in modern science. They describe how to learn and\n", + "make predictions from data, as well as allowing us to extract\n", + "important correlations about physical process and the underlying laws\n", + "of motion in large data sets. The latter, big data sets, appear\n", + "frequently in essentially all disciplines, from the traditional\n", + "Science, Technology, Mathematics and Engineering fields to Life\n", + "Science, Law, education research, the Humanities and the Social\n", + "Sciences.\n", + "\n", + "It has become more\n", + "and more common to see research projects on big data in for example\n", + "the Social Sciences where extracting patterns from complicated survey\n", + "data is one of many research directions. Having a solid grasp of data\n", + "analysis and machine learning is thus becoming central to scientific\n", + "computing in many fields, and competences and skills within the fields\n", + "of machine learning and scientific computing are nowadays strongly\n", + "requested by many potential employers. The latter cannot be\n", + "overstated, familiarity with machine learning has almost become a\n", + "prerequisite for many of the most exciting employment opportunities,\n", + "whether they are in bioinformatics, life science, physics or finance,\n", + "in the private or the public sector. This author has had several\n", + "students or met students who have been hired recently based on their\n", + "skills and competences in scientific computing and data science, often\n", + "with marginal knowledge of machine learning.\n", + "\n", + "Machine learning is a subfield of computer science, and is closely\n", + "related to computational statistics. It evolved from the study of\n", + "pattern recognition in artificial intelligence (AI) research, and has\n", + "made contributions to AI tasks like computer vision, natural language\n", + "processing and speech recognition. Many of the methods we will study are also \n", + "strongly rooted in basic mathematics and physics research. \n", + "\n", + "Ideally, machine learning represents the science of giving computers\n", + "the ability to learn without being explicitly programmed. The idea is\n", + "that there exist generic algorithms which can be used to find patterns\n", + "in a broad class of data sets without having to write code\n", + "specifically for each problem. The algorithm will build its own logic\n", + "based on the data. You should however always keep in mind that\n", + "machines and algorithms are to a large extent developed by humans. The\n", + "insights and knowledge we have about a specific system, play a central\n", + "role when we develop a specific machine learning algorithm. \n", + "\n", + "Machine learning is an extremely rich field, in spite of its young\n", + "age. The increases we have seen during the last three decades in\n", + "computational capabilities have been followed by developments of\n", + "methods and techniques for analyzing and handling large date sets,\n", + "relying heavily on statistics, computer science and mathematics. The\n", + "field is rather new and developing rapidly. Popular software packages\n", + "written in Python for machine learning like\n", + "[Scikit-learn](http://scikit-learn.org/stable/),\n", + "[Tensorflow](https://www.tensorflow.org/),\n", + "[PyTorch](http://pytorch.org/) and [Keras](https://keras.io/), all\n", + "freely available at their respective GitHub sites, encompass\n", + "communities of developers in the thousands or more. And the number of\n", + "code developers and contributors keeps increasing. Not all the\n", + "algorithms and methods can be given a rigorous mathematical\n", + "justification, opening up thereby large rooms for experimenting and\n", + "trial and error and thereby exciting new developments. However, a\n", + "solid command of linear algebra, multivariate theory, probability\n", + "theory, statistical data analysis, understanding errors and Monte\n", + "Carlo methods are central elements in a proper understanding of many\n", + "of algorithms and methods we will discuss.\n", + "\n", + "\n", + "\n", + "## Learning outcomes\n", + "\n", + "These sets of lectures aim at giving you an overview of central aspects of\n", + "statistical data analysis as well as some of the central algorithms\n", + "used in machine learning. We will introduce a variety of central\n", + "algorithms and methods essential for studies of data analysis and\n", + "machine learning. \n", + "\n", + "Hands-on projects and experimenting with data and algorithms plays a central role in\n", + "these lectures, and our hope is, through the various\n", + "projects and exercises, to expose you to fundamental\n", + "research problems in these fields, with the aim to reproduce state of\n", + "the art scientific results. You will learn to develop and\n", + "structure codes for studying these systems, get acquainted with\n", + "computing facilities and learn to handle large scientific projects. A\n", + "good scientific and ethical conduct is emphasized throughout the\n", + "course. More specifically, you will\n", + "\n", + "1. Learn about basic data analysis, Bayesian statistics, Monte Carlo methods, data optimization and machine learning;\n", + "\n", + "2. Be capable of extending the acquired knowledge to other systems and cases;\n", + "\n", + "3. Have an understanding of central algorithms used in data analysis and machine learning;\n", + "\n", + "4. 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;\n", + "\n", + "5. Understand methods for regression and classification;\n", + "\n", + "6. Learn about neural network, genetic algorithms and Boltzmann machines;\n", + "\n", + "7. 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).\n", + "\n", + "There are several topics we will cover here, spanning from \n", + "statistical data analysis and its basic concepts such as expectation\n", + "values, variance, covariance, correlation functions and errors, via\n", + "well-known probability distribution functions like the uniform\n", + "distribution, the binomial distribution, the Poisson distribution and\n", + "simple and multivariate normal distributions to central elements of\n", + "Bayesian statistics and modeling. We will also remind the reader about\n", + "central elements from linear algebra and standard methods based on\n", + "linear algebra used to optimize (minimize) functions (the family of gradient descent methods)\n", + "and the Singular-value decomposition and\n", + "least square methods for parameterizing data.\n", + "\n", + "We will also cover Monte Carlo methods, Markov chains, well-known\n", + "algorithms for sampling stochastic events like the Metropolis-Hastings\n", + "and Gibbs sampling methods. An important aspect of all our\n", + "calculations is a proper estimation of errors. Here we will also\n", + "discuss famous resampling techniques like the blocking, the bootstrapping\n", + "and the jackknife methods and the infamous bias-variance tradeoff. \n", + "\n", + "The second part of the material covers several algorithms used in\n", + "machine learning.\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "## Machine Learning, a small (and probably biased) introduction\n", + "\n", + "\n", + "Ideally, machine learning represents the science of giving computers\n", + "the ability to learn without being explicitly programmed. The idea is\n", + "that there exist generic algorithms which can be used to find patterns\n", + "in a broad class of data sets without having to write code\n", + "specifically for each problem. The algorithm will build its own logic\n", + "based on the data. You should however always keep in mind that\n", + "machines and algorithms are to a large extent developed by humans. The\n", + "insights and knowledge we have about a specific system, play a central\n", + "role when we develop a specific machine learning algorithm. \n", + "\n", + "\n", + "Machine learning is an extremely rich field, in spite of its young\n", + "age. The increases we have seen during the last decades in\n", + "computational capabilities have been followed by developments of\n", + "methods and techniques for analyzing and handling large date sets,\n", + "relying heavily on statistics, computer science and mathematics. The\n", + "field is rather new and developing rapidly. Popular software libraries\n", + "written in Python for machine learning like\n", + "[Scikit-learn](http://scikit-learn.org/stable/),\n", + "[Tensorflow](https://www.tensorflow.org/),\n", + "[PyTorch](http://pytorch.org/) and [Keras](https://keras.io/), all\n", + "freely available at their respective GitHub sites, encompass\n", + "communities of developers in the thousands or more. And the number of\n", + "code developers and contributors keeps increasing.\n", + "\n", + "\n", + "Not all the\n", + "algorithms and methods can be given a rigorous mathematical\n", + "justification (for example decision trees and random forests), opening up thereby large rooms for experimenting and\n", + "trial and error and thereby exciting new developments. However, a\n", + "solid command of linear algebra, multivariate theory, probability\n", + "theory, statistical data analysis, understanding errors and Monte\n", + "Carlo methods are central elements in a proper understanding of many\n", + "of the algorithms and methods we will discuss.\n", + "\n", + "\n", + "The approaches to machine learning are many, but are often split into\n", + "two main categories. In *supervised learning* we know the answer to a\n", + "problem, and let the computer deduce the logic behind it. On the other\n", + "hand, *unsupervised learning* is a method for finding patterns and\n", + "relationship in data sets without any prior knowledge of the system.\n", + "Some authours also operate with a third category, namely\n", + "*reinforcement learning*. This is a paradigm of learning inspired by\n", + "behavioral psychology, where learning is achieved by trial-and-error,\n", + "solely from rewards and punishment.\n", + "\n", + "Another way to categorize machine learning tasks is to consider the\n", + "desired output of a system. Some of the most common tasks are:\n", + "\n", + "* 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 often supervised learning.\n", + "\n", + "* Regression: Finding a functional relationship between an input data set and a reference data set. The goal is to construct a function that maps input data to continuous output values.\n", + "\n", + "* Clustering: Data are divided into groups with certain common traits, without knowing the different groups beforehand. It is thus a form of unsupervised learning.\n", + "\n", + "The methods we cover have three main topics in common, irrespective of\n", + "whether we deal with supervised or unsupervised learning.\n", + "\n", + "* The first ingredient is normally our data set (which can be subdivided into training, validation and test data). Many find the most difficult part of using Machine Learning to be the set up of your data in a meaningful way. \n", + "\n", + "* The second item is a model which is normally a function of some parameters. The model reflects our knowledge of the system (or lack thereof). As an example, if we know that our data show a behavior similar to what would be predicted by a polynomial, fitting our data to a polynomial of some degree would then determin our model. \n", + "\n", + "* The last ingredient is a so-called **cost/loss** function (or error function) which allows us to present an estimate on how good our model is in reproducing the data it is supposed to train. \n", + "\n", + "At the heart of basically all Machine Learning algorithms we will encounter so-called minimization or optimization algorithms. A large family of such methods are so-called **gradient methods**.\n", + "\n", + "\n", + "## A Frequentist approach to data analysis\n", + "\n", + "When you hear phrases like **predictions and estimations** and\n", + "**correlations and causations**, what do you think of? May be you think\n", + "of the difference between classifying new data points and generating\n", + "new data points.\n", + "Or perhaps you consider that correlations represent some kind of symmetric statements like\n", + "if $A$ is correlated with $B$, then $B$ is correlated with\n", + "$A$. Causation on the other hand is directional, that is if $A$ causes $B$, $B$ does not\n", + "necessarily cause $A$.\n", + "\n", + "These concepts are in some sense the difference between machine\n", + "learning and statistics. In machine learning and prediction based\n", + "tasks, we are often interested in developing algorithms that are\n", + "capable of learning patterns from given data in an automated fashion,\n", + "and then using these learned patterns to make predictions or\n", + "assessments of newly given data. In many cases, our primary concern\n", + "is the quality of the predictions or assessments, and we are less\n", + "concerned about the underlying patterns that were learned in order\n", + "to make these predictions.\n", + "\n", + "In machine learning we normally use [a so-called frequentist approach](https://en.wikipedia.org/wiki/Frequentist_inference),\n", + "where the aim is to make predictions and find correlations. We focus\n", + "less on for example extracting a probability distribution function (PDF). The PDF can be\n", + "used in turn to make estimations and find causations such as given $A$\n", + "what is the likelihood of finding $B$.\n", + "\n", + "\n", + "\n", + "## What is a good model?\n", + "\n", + "In science and engineering we often end up in situations where we want to infer (or learn) a\n", + "quantitative model $M$ for a given set of sample points $\\boldsymbol{X} \\in [x_1, x_2,\\dots x_N]$.\n", + "\n", + "As we will see repeatedely in these lectures, we could try to fit these data points to a model given by a\n", + "straight line, or if we wish to be more sophisticated to a more complex\n", + "function.\n", + "\n", + "The reason for inferring such a model is that it\n", + "serves many useful purposes. On the one hand, the model can reveal information\n", + "encoded in the data or underlying mechanisms from which the data were generated. For instance, we could discover important\n", + "corelations that relate interesting physics interpretations.\n", + "\n", + "In addition, it can simplify the representation of the given data set and help\n", + "us in making predictions about future data samples.\n", + "\n", + "A first important consideration to keep in mind is that inferring the *correct* model\n", + "for a given data set is an elusive, if not impossible, task. The fundamental difficulty\n", + "is that if we are not specific about what we mean by a *correct* model, there\n", + "could easily be many different models that fit the given data set *equally well*.\n", + "\n", + "\n", + "The central question is this: what leads us to say that a model is correct or\n", + "optimal for a given data set? To make the model inference problem well posed, i.e.,\n", + "to guarantee that there is a unique optimal model for the given data, we need to\n", + "impose additional assumptions or restrictions on the class of models considered. To\n", + "this end, we should not be looking for just any model that can describe the data.\n", + "Instead, we should look for a **model** $M$ that is the best among a restricted class\n", + "of models. In addition, to make the model inference problem computationally\n", + "tractable, we need to specify how restricted the class of models needs to be. A\n", + "common strategy is to start \n", + "with the simplest possible class of models that is just necessary to describe the data\n", + "or solve the problem at hand. More precisely, the model class should be rich enough\n", + "to contain at least one model that can fit the data to a desired accuracy and yet be\n", + "restricted enough that it is relatively simple to find the best model for the given data.\n", + "\n", + "Thus, the most popular strategy is to start from the\n", + "simplest class of models and increase the complexity of the models only when the\n", + "simpler models become inadequate. For instance, if we work with a regression problem to fit a set of sample points, one\n", + "may first try the simplest class of models, namely linear models, followed obviously by more complex models.\n", + "\n", + "How to evaluate which model fits best the data is something we will come back to over and over again in these set of lectures.\n", + "\n", + "\n", + "\n", + "\n", + "## Choice of Programming Language\n", + "\n", + "Python plays nowadays a central role in the development of machine\n", + "learning techniques and tools for data analysis. In particular, seen\n", + "the wealth of machine learning and data analysis libraries written in\n", + "Python, easy to use libraries with immediate visualization(and not the\n", + "least impressive galleries of existing examples), the popularity of the\n", + "Jupyter notebook framework with the possibility to run **R** codes or\n", + "compiled programs written in C++, and much more made our choice of\n", + "programming language for this series of lectures easy. However,\n", + "since the focus here is not only on using existing Python libraries such\n", + "as **Scikit-Learn**, **Tensorflow** and **Pytorch**, but also on developing your own\n", + "algorithms and codes, we will as far as possible present many of these\n", + "algorithms either as a Python codes or C++ or Fortran (or other languages) codes. \n", + "\n", + "\n", + "\n", + "\n", + "## Data handling, machine learning and ethical aspects\n", + "\n", + "In most of the cases we will study, we will either generate the data\n", + "to analyze ourselves (both for supervised learning and unsupervised\n", + "learning) or we will recur again and again to data present in say\n", + "**Scikit-Learn** or **Tensorflow**. Many of the examples we end up\n", + "dealing with are from a privacy and data protection point of view,\n", + "rather inoccuous and boring results of numerical\n", + "calculations. However, this does not hinder us from developing a sound\n", + "ethical attitude to the data we use, how we analyze the data and how\n", + "we handle the data.\n", + "\n", + "The most immediate and simplest possible ethical aspects deal with our\n", + "approach to the scientific process. Nowadays, with version control\n", + "software like [Git](https://git-scm.com/) and various online\n", + "repositories like [Github](https://github.com/),\n", + "[Gitlab](https://about.gitlab.com/) etc, we can easily make our codes\n", + "and data sets we have used, freely and easily accessible to a wider\n", + "community. This helps us almost automagically in making our science\n", + "reproducible. The large open-source development communities involved\n", + "in say [Scikit-Learn](http://scikit-learn.org/stable/),\n", + "[Tensorflow](https://www.tensorflow.org/),\n", + "[PyTorch](http://pytorch.org/) and [Keras](https://keras.io/), are\n", + "all excellent examples of this. The codes can be tested and improved\n", + "upon continuosly, helping thereby our scientific community at large in\n", + "developing data analysis and machine learning tools. It is much\n", + "easier today to gain traction and acceptance for making your science\n", + "reproducible. From a societal stand, this is an important element\n", + "since many of the developers are employees of large public institutions like\n", + "universities and research labs. Our fellow taxpayers do deserve to get\n", + "something back for their bucks.\n", + "\n", + "However, this more mechanical aspect of the ethics of science (in\n", + "particular the reproducibility of scientific results) is something\n", + "which is obvious and everybody should do so as part of the dialectics of\n", + "science. The fact that many scientists are not willing to share their codes or \n", + "data is detrimental to the scientific discourse.\n", + "\n", + "Before we proceed, we should add a disclaimer. Even though\n", + "we may dream of computers developing some kind of higher learning\n", + "capabilities, at the end (even if the artificial intelligence\n", + "community keeps touting our ears full of fancy futuristic avenues), it is we, yes you reading these lines,\n", + "who end up constructing and instructing, via various algorithms, the\n", + "machine learning approaches. Self-driving cars for example, rely on sofisticated\n", + "programs which take into account all possible situations a car can\n", + "encounter. In addition, extensive usage of training data from GPS\n", + "information, maps etc, are typically fed into the software for\n", + "self-driving cars. Adding to this various sensors and cameras that\n", + "feed information to the programs, there are zillions of ethical issues\n", + "which arise from this.\n", + "\n", + "For self-driving cars, where basically many of the standard machine\n", + "learning algorithms discussed here enter into the codes, at a certain\n", + "stage we have to make choices. Yes, we , the lads and lasses who wrote\n", + "a program for a specific brand of a self-driving car. As an example,\n", + "all carmakers have as their utmost priority the security of the\n", + "driver and the accompanying passengers. A famous European carmaker, which is\n", + "one of the leaders in the market of self-driving cars, had **if**\n", + "statements of the following type: suppose there are two obstacles in\n", + "front of you and you cannot avoid to collide with one of them. One of\n", + "the obstacles is a monstertruck while the other one is a kindergarten\n", + "class trying to cross the road. The self-driving car algo would then\n", + "opt for the hitting the small folks instead of the monstertruck, since\n", + "the likelihood of surving a collision with our future citizens, is\n", + "much higher.\n", + "\n", + "This leads to serious ethical aspects. Why should we opt for such an\n", + "option? Who decides and who is entitled to make such choices? Keep in\n", + "mind that many of the algorithms you will encounter in this series of\n", + "lectures or hear about later, are indeed based on simple programming\n", + "instructions. And you are very likely to be one of the people who may\n", + "end up writing such a code. Thus, developing a sound ethical attitude\n", + "to what we do, an approach well beyond the simple mechanistic one of\n", + "making our science available and reproducible, is much needed. The\n", + "example of the self-driving cars is just one of infinitely many cases\n", + "where we have to make choices. When you analyze data on economic\n", + "inequalities, who guarantees that you are not weighting some data in a\n", + "particular way, perhaps because you dearly want a specific conclusion\n", + "which may support your political views? Or what about the recent\n", + "claims that a famous IT company like Apple has a sexist bias on the\n", + "their recently [launched credit card](https://qz.com/1748321/the-role-of-goldman-sachs-algorithms-in-the-apple-credit-card-scandal/)?\n", + "\n", + "We do not have the answers here, nor will we venture into a deeper\n", + "discussions of these aspects, but we want you think over these topics\n", + "in a more overarching way. A statistical data analysis with its dry\n", + "numbers and graphs meant to guide the eye, does not necessarily\n", + "reflect the truth, whatever that is. As a scientist, and after a\n", + "university education, you are supposedly a better citizen, with an\n", + "improved critical view and understanding of the scientific method, and\n", + "perhaps some deeper understanding of the ethics of science at\n", + "large. Use these insights. Be a critical citizen. You owe it to our\n", + "society." + ] + } + ], + "metadata": {}, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/doc/src/LectureNotes/_build/html/_sources/chapter2.ipynb b/doc/src/LectureNotes/_build/html/_sources/chapter2.ipynb new file mode 100644 index 000000000..51ce293e4 --- /dev/null +++ b/doc/src/LectureNotes/_build/html/_sources/chapter2.ipynb @@ -0,0 +1,2058 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Getting started, our first data and Machine Learning encounters\n", + "\n", + "## Introduction\n", + "\n", + "Our emphasis throughout this series of lectures \n", + "is on understanding the mathematical aspects of\n", + "different algorithms used in the fields of data analysis and machine learning. \n", + "\n", + "However, where possible we will emphasize the\n", + "importance of using available software. We start thus with a hands-on\n", + "and top-down approach to machine learning. The aim is thus to start with\n", + "relevant data or data we have produced \n", + "and use these to introduce statistical data analysis\n", + "concepts and machine learning algorithms before we delve into the\n", + "algorithms themselves. The examples we will use in the beginning, start with simple\n", + "polynomials with random noise added. We will use the Python\n", + "software package [Scikit-Learn](http://scikit-learn.org/stable/) and\n", + "introduce various machine learning algorithms to make fits of\n", + "the data and predictions. We move thereafter to more interesting\n", + "cases such as data from say experiments (below we will look at experimental nuclear binding energies as an example).\n", + "These are examples where we can easily set up the data and\n", + "then use machine learning algorithms included in for example\n", + "**Scikit-Learn**. \n", + "\n", + "These examples will serve us the purpose of getting\n", + "started. Furthermore, they allow us to catch more than two birds with\n", + "a stone. They will allow us to bring in some programming specific\n", + "topics and tools as well as showing the power of various Python \n", + "libraries for machine learning and statistical data analysis. \n", + "\n", + "Here, we will mainly focus on two\n", + "specific Python packages for Machine Learning, Scikit-Learn and\n", + "Tensorflow (see below for links etc). Moreover, the examples we\n", + "introduce will serve as inputs to many of our discussions later, as\n", + "well as allowing you to set up models and produce your own data and\n", + "get started with programming.\n", + "\n", + "\n", + "\n", + "## What is Machine Learning?\n", + "\n", + "Statistics, data science and machine learning form important fields of\n", + "research in modern science. They describe how to learn and make\n", + "predictions from data, as well as allowing us to extract important\n", + "correlations about physical process and the underlying laws of motion\n", + "in large data sets. The latter, big data sets, appear frequently in\n", + "essentially all disciplines, from the traditional Science, Technology,\n", + "Mathematics and Engineering fields to Life Science, Law, education\n", + "research, the Humanities and the Social Sciences. \n", + "\n", + "It has become more\n", + "and more common to see research projects on big data in for example\n", + "the Social Sciences where extracting patterns from complicated survey\n", + "data is one of many research directions. Having a solid grasp of data\n", + "analysis and machine learning is thus becoming central to scientific\n", + "computing in many fields, and competences and skills within the fields\n", + "of machine learning and scientific computing are nowadays strongly\n", + "requested by many potential employers. The latter cannot be\n", + "overstated, familiarity with machine learning has almost become a\n", + "prerequisite for many of the most exciting employment opportunities,\n", + "whether they are in bioinformatics, life science, physics or finance,\n", + "in the private or the public sector. This author has had several\n", + "students or met students who have been hired recently based on their\n", + "skills and competences in scientific computing and data science, often\n", + "with marginal knowledge of machine learning.\n", + "\n", + "Machine learning is a subfield of computer science, and is closely\n", + "related to computational statistics. It evolved from the study of\n", + "pattern recognition in artificial intelligence (AI) research, and has\n", + "made contributions to AI tasks like computer vision, natural language\n", + "processing and speech recognition. Many of the methods we will study are also \n", + "strongly rooted in basic mathematics and physics research. \n", + "\n", + "Ideally, machine learning represents the science of giving computers\n", + "the ability to learn without being explicitly programmed. The idea is\n", + "that there exist generic algorithms which can be used to find patterns\n", + "in a broad class of data sets without having to write code\n", + "specifically for each problem. The algorithm will build its own logic\n", + "based on the data. You should however always keep in mind that\n", + "machines and algorithms are to a large extent developed by humans. The\n", + "insights and knowledge we have about a specific system, play a central\n", + "role when we develop a specific machine learning algorithm. \n", + "\n", + "Machine learning is an extremely rich field, in spite of its young\n", + "age. The increases we have seen during the last three decades in\n", + "computational capabilities have been followed by developments of\n", + "methods and techniques for analyzing and handling large date sets,\n", + "relying heavily on statistics, computer science and mathematics. The\n", + "field is rather new and developing rapidly. Popular software packages\n", + "written in Python for machine learning like\n", + "[Scikit-learn](http://scikit-learn.org/stable/),\n", + "[Tensorflow](https://www.tensorflow.org/),\n", + "[PyTorch](http://pytorch.org/) and [Keras](https://keras.io/), all\n", + "freely available at their respective GitHub sites, encompass\n", + "communities of developers in the thousands or more. And the number of\n", + "code developers and contributors keeps increasing. Not all the\n", + "algorithms and methods can be given a rigorous mathematical\n", + "justification, opening up thereby large rooms for experimenting and\n", + "trial and error and thereby exciting new developments. However, a\n", + "solid command of linear algebra, multivariate theory, probability\n", + "theory, statistical data analysis, understanding errors and Monte\n", + "Carlo methods are central elements in a proper understanding of many\n", + "of algorithms and methods we will discuss.\n", + "\n", + "\n", + "\n", + "## Types of Machine Learning\n", + "\n", + "\n", + "The approaches to machine learning are many, but are often split into\n", + "two main categories. In *supervised learning* we know the answer to a\n", + "problem, and let the computer deduce the logic behind it. On the other\n", + "hand, *unsupervised learning* is a method for finding patterns and\n", + "relationship in data sets without any prior knowledge of the system.\n", + "Some authours also operate with a third category, namely\n", + "*reinforcement learning*. This is a paradigm of learning inspired by\n", + "behavioral psychology, where learning is achieved by trial-and-error,\n", + "solely from rewards and punishment.\n", + "\n", + "Another way to categorize machine learning tasks is to consider the\n", + "desired output of a system. Some of the most common tasks are:\n", + "\n", + " * 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.\n", + "\n", + " * Regression: Finding a functional relationship between an input data set and a reference data set. The goal is to construct a function that maps input data to continuous output values.\n", + "\n", + " * Clustering: Data are divided into groups with certain common traits, without knowing the different groups beforehand. It is thus a form of unsupervised learning.\n", + "\n", + "The methods we cover have three main topics in common, irrespective of\n", + "whether we deal with supervised or unsupervised learning. The first\n", + "ingredient is normally our data set (which can be subdivided into\n", + "training and test data), the second item is a model which is normally a\n", + "function of some parameters. The model reflects our knowledge of the system (or lack thereof). As an example, if we know that our data show a behavior similar to what would be predicted by a polynomial, fitting our data to a polynomial of some degree would then determin our model. \n", + "\n", + "The last ingredient is a so-called **cost**\n", + "function which allows us to present an estimate on how good our model\n", + "is in reproducing the data it is supposed to train. \n", + "At the heart of basically all ML algorithms there are so-called minimization algorithms, often we end up with various variants of **gradient** methods.\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "## Software and needed installations\n", + "\n", + "We will make extensive use of Python as programming language and its\n", + "myriad of available libraries. You will find\n", + "Jupyter notebooks invaluable in your work. You can run **R**\n", + "codes in the Jupyter/IPython notebooks, with the immediate benefit of\n", + "visualizing your data. You can also use compiled languages like C++,\n", + "Rust, Julia, Fortran etc if you prefer. The focus in these lectures will be\n", + "on Python.\n", + "\n", + "\n", + "If you have Python installed (we strongly recommend Python3) and you feel\n", + "pretty familiar with installing different packages, we recommend that\n", + "you install the following Python packages via **pip** as \n", + "\n", + "1. pip install numpy scipy matplotlib ipython scikit-learn mglearn sympy pandas pillow \n", + "\n", + "For Python3, replace **pip** with **pip3**.\n", + "\n", + "For OSX users we recommend, after having installed Xcode, to\n", + "install **brew**. Brew allows for a seamless installation of additional\n", + "software via for example \n", + "\n", + "1. brew install python3\n", + "\n", + "For Linux users, with its variety of distributions like for example the widely popular Ubuntu distribution,\n", + "you can use **pip** as well and simply install Python as \n", + "\n", + "1. sudo apt-get install python3 (or python for pyhton2.7)\n", + "\n", + "etc etc. \n", + "\n", + "\n", + "\n", + "## Python installers\n", + "\n", + "If you don't want to perform these operations separately and venture\n", + "into the hassle of exploring how to set up dependencies and paths, we\n", + "recommend two widely used distrubutions which set up all relevant\n", + "dependencies for Python, namely \n", + "\n", + "* [Anaconda](https://docs.anaconda.com/), \n", + "\n", + "which is an open source\n", + "distribution of the Python and R programming languages for large-scale\n", + "data processing, predictive analytics, and scientific computing, that\n", + "aims to simplify package management and deployment. Package versions\n", + "are managed by the package management system **conda**. \n", + "\n", + "* [Enthought canopy](https://www.enthought.com/product/canopy/) \n", + "\n", + "is a Python\n", + "distribution for scientific and analytic computing distribution and\n", + "analysis environment, available for free and under a commercial\n", + "license.\n", + "\n", + "Furthermore, [Google's Colab](https://colab.research.google.com/notebooks/welcome.ipynb) is a free Jupyter notebook environment that requires \n", + "no setup and runs entirely in the cloud. Try it out!\n", + "\n", + "## Useful Python libraries\n", + "Here we list several useful Python libraries we strongly recommend (if you use anaconda many of these are already there)\n", + "\n", + "* [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\n", + "\n", + "* [The pandas](https://pandas.pydata.org/) library provides high-performance, easy-to-use data structures and data analysis tools \n", + "\n", + "* [Xarray](http://xarray.pydata.org/en/stable/) is a Python package that makes working with labelled multi-dimensional arrays simple, efficient, and fun!\n", + "\n", + "* [Scipy](https://www.scipy.org/) (pronounced “Sigh Pie”) is a Python-based ecosystem of open-source software for mathematics, science, and engineering. \n", + "\n", + "* [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.\n", + "\n", + "* [Autograd](https://github.com/HIPS/autograd) can automatically differentiate native Python and Numpy code. It can handle a large subset of Python's features, including loops, ifs, recursion and closures, and it can even take derivatives of derivatives of derivatives\n", + "\n", + "* [SymPy](https://www.sympy.org/en/index.html) is a Python library for symbolic mathematics. \n", + "\n", + "* [scikit-learn](https://scikit-learn.org/stable/) has simple and efficient tools for machine learning, data mining and data analysis\n", + "\n", + "* [TensorFlow](https://www.tensorflow.org/) is a Python library for fast numerical computing created and released by Google\n", + "\n", + "* [Keras](https://keras.io/) is a high-level neural networks API, written in Python and capable of running on top of TensorFlow, CNTK, or Theano\n", + "\n", + "* And many more such as [pytorch](https://pytorch.org/), [Theano](https://pypi.org/project/Theano/) etc \n", + "\n", + "## Installing R, C++, cython or Julia\n", + "\n", + "You will also find it convenient to utilize **R**. We will mainly\n", + "use Python during our lectures and in various projects and exercises.\n", + "Those of you\n", + "already familiar with **R** should feel free to continue using **R**, keeping\n", + "however an eye on the parallel Python set ups. Similarly, if you are a\n", + "Python afecionado, feel free to explore **R** as well. Jupyter/Ipython\n", + "notebook allows you to run **R** codes interactively in your\n", + "browser. The software library **R** is really tailored for statistical data analysis\n", + "and allows for an easy usage of the tools and algorithms we will discuss in these\n", + "lectures.\n", + "\n", + "To install **R** with Jupyter notebook \n", + "[follow the link here](https://mpacer.org/maths/r-kernel-for-ipython-notebook)\n", + "\n", + "\n", + "\n", + "\n", + "## Installing R, C++, cython, Numba etc\n", + "\n", + "\n", + "For the C++ aficionados, Jupyter/IPython notebook allows you also to\n", + "install C++ and run codes written in this language interactively in\n", + "the browser. Since we will emphasize writing many of the algorithms\n", + "yourself, you can thus opt for either Python or C++ (or Fortran or other compiled languages) as programming\n", + "languages.\n", + "\n", + "To add more entropy, **cython** can also be used when running your\n", + "notebooks. It means that Python with the jupyter notebook\n", + "setup allows you to integrate widely popular softwares and tools for\n", + "scientific computing. Similarly, the \n", + "[Numba Python package](https://numba.pydata.org/) delivers increased performance\n", + "capabilities with minimal rewrites of your codes. With its\n", + "versatility, including symbolic operations, Python offers a unique\n", + "computational environment. Your jupyter notebook can easily be\n", + "converted into a nicely rendered **PDF** file or a Latex file for\n", + "further processing. For example, convert to latex as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + " pycod jupyter nbconvert filename.ipynb --to latex \n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "And to add more versatility, the Python package [SymPy](http://www.sympy.org/en/index.html) is a Python library for symbolic mathematics. It aims to become a full-featured computer algebra system (CAS) and is entirely written in Python. \n", + "\n", + "Finally, if you wish to use the light mark-up language \n", + "[doconce](https://github.com/hplgit/doconce) you can convert a standard ascii text file into various HTML \n", + "formats, ipython notebooks, latex files, pdf files etc with minimal edits. These lectures were generated using **doconce**.\n", + "\n", + "\n", + "\n", + "## Numpy examples and Important Matrix and vector handling packages\n", + "\n", + "There are several central software libraries for linear algebra and eigenvalue problems. Several of the more\n", + "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\n", + "software package LAPACK, which follows two other popular packages\n", + "developed in the 1970s, namely EISPACK and LINPACK. We describe them shortly here.\n", + "\n", + " * LINPACK: package for linear equations and least square problems.\n", + "\n", + " * LAPACK:package for solving symmetric, unsymmetric and generalized eigenvalue problems. From LAPACK's website it is possible to download for free all source codes from this library. Both C/C++ and Fortran versions are available.\n", + "\n", + " * BLAS (I, II and III): (Basic Linear Algebra Subprograms) are routines that provide standard building blocks for performing basic vector and matrix operations. Blas I is vector operations, II vector-matrix operations and III matrix-matrix operations. Highly parallelized and efficient codes, all available for download from .\n", + "\n", + "## Basic Matrix Features\n", + "\n", + "**Matrix properties reminder.**" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\mathbf{A} =\n", + " \\begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\\\\n", + " a_{21} & a_{22} & a_{23} & a_{24} \\\\\n", + " a_{31} & a_{32} & a_{33} & a_{34} \\\\\n", + " a_{41} & a_{42} & a_{43} & a_{44}\n", + " \\end{bmatrix}\\qquad\n", + "\\mathbf{I} =\n", + " \\begin{bmatrix} 1 & 0 & 0 & 0 \\\\\n", + " 0 & 1 & 0 & 0 \\\\\n", + " 0 & 0 & 1 & 0 \\\\\n", + " 0 & 0 & 0 & 1\n", + " \\end{bmatrix}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The inverse of a matrix is defined by" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\mathbf{A}^{-1} \\cdot \\mathbf{A} = I\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "
Relations Name matrix elements
$A = A^{T}$ symmetric $a_{ij} = a_{ji}$
$A = \\left (A^{T} \\right )^{-1}$ real orthogonal $\\sum_k a_{ik} a_{jk} = \\sum_k a_{ki} a_{kj} = \\delta_{ij}$
$A = A^{ * }$ real matrix $a_{ij} = a_{ij}^{ * }$
$A = A^{\\dagger}$ hermitian $a_{ij} = a_{ji}^{ * }$
$A = \\left (A^{\\dagger} \\right )^{-1}$ unitary $\\sum_k a_{ik} a_{jk}^{ * } = \\sum_k a_{ki}^{ * } a_{kj} = \\delta_{ij}$
\n", + "\n", + "\n", + "\n", + "\n", + "### Some famous Matrices\n", + "\n", + " * Diagonal if $a_{ij}=0$ for $i\\ne j$\n", + "\n", + " * Upper triangular if $a_{ij}=0$ for $i > j$\n", + "\n", + " * Lower triangular if $a_{ij}=0$ for $i < j$\n", + "\n", + " * Upper Hessenberg if $a_{ij}=0$ for $i > j+1$\n", + "\n", + " * Lower Hessenberg if $a_{ij}=0$ for $i < j+1$\n", + "\n", + " * Tridiagonal if $a_{ij}=0$ for $|i -j| > 1$\n", + "\n", + " * Lower banded with bandwidth $p$: $a_{ij}=0$ for $i > j+p$\n", + "\n", + " * Upper banded with bandwidth $p$: $a_{ij}=0$ for $i < j+p$\n", + "\n", + " * Banded, block upper triangular, block lower triangular....\n", + "\n", + "### More Basic Matrix Features\n", + "\n", + "**Some Equivalent Statements.**\n", + "\n", + "For an $N\\times N$ matrix $\\mathbf{A}$ the following properties are all equivalent\n", + "\n", + " * If the inverse of $\\mathbf{A}$ exists, $\\mathbf{A}$ is nonsingular.\n", + "\n", + " * The equation $\\mathbf{Ax}=0$ implies $\\mathbf{x}=0$.\n", + "\n", + " * The rows of $\\mathbf{A}$ form a basis of $R^N$.\n", + "\n", + " * The columns of $\\mathbf{A}$ form a basis of $R^N$.\n", + "\n", + " * $\\mathbf{A}$ is a product of elementary matrices.\n", + "\n", + " * $0$ is not eigenvalue of $\\mathbf{A}$.\n", + "\n", + "\n", + "\n", + "\n", + "## Numpy and arrays\n", + "[Numpy](http://www.numpy.org/) provides an easy way to handle arrays in Python. The standard way to import this library is as" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "import numpy as np" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Here follows a simple example where we set up an array of ten elements, all determined by random numbers drawn according to the normal distribution," + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "n = 10\n", + "x = np.random.normal(size=n)\n", + "print(x)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We defined a vector $x$ with $n=10$ elements with its values given by the Normal distribution $N(0,1)$.\n", + "Another alternative is to declare a vector as follows" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "import numpy as np\n", + "x = np.array([1, 2, 3])\n", + "print(x)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Here we have defined a vector with three elements, with $x_0=1$, $x_1=2$ and $x_2=3$. Note that both Python and C++\n", + "start numbering array elements from $0$ and on. This means that a vector with $n$ elements has a sequence of entities $x_0, x_1, x_2, \\dots, x_{n-1}$. We could also let (recommended) Numpy to compute the logarithms of a specific array as" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "import numpy as np\n", + "x = np.log(np.array([4, 7, 8]))\n", + "print(x)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "In the last example we used Numpy's unary function $np.log$. This function is\n", + "highly tuned to compute array elements since the code is vectorized\n", + "and does not require looping. We normaly recommend that you use the\n", + "Numpy intrinsic functions instead of the corresponding **log** function\n", + "from Python's **math** module. The looping is done explicitely by the\n", + "**np.log** function. The alternative, and slower way to compute the\n", + "logarithms of a vector would be to write" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "import numpy as np\n", + "from math import log\n", + "x = np.array([4, 7, 8])\n", + "for i in range(0, len(x)):\n", + " x[i] = log(x[i])\n", + "print(x)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We note that our code is much longer already and we need to import the **log** function from the **math** module. \n", + "The attentive reader will also notice that the output is $[1, 1, 2]$. Python interprets automagically our numbers as integers (like the **automatic** keyword in C++). To change this we could define our array elements to be double precision numbers as" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "import numpy as np\n", + "x = np.log(np.array([4, 7, 8], dtype = np.float64))\n", + "print(x)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "or simply write them as double precision numbers (Python uses 64 bits as default for floating point type variables), that is" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "import numpy as np\n", + "x = np.log(np.array([4.0, 7.0, 8.0])\n", + "print(x)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "To check the number of bytes (remember that one byte contains eight bits for double precision variables), you can use simple use the **itemsize** functionality (the array $x$ is actually an object which inherits the functionalities defined in Numpy) as" + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "import numpy as np\n", + "x = np.log(np.array([4.0, 7.0, 8.0])\n", + "print(x.itemsize)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Matrices in Python\n", + "\n", + "Having defined vectors, we are now ready to try out matrices. We can\n", + "define a $3 \\times 3 $ real matrix $\\hat{A}$ as (recall that we user\n", + "lowercase letters for vectors and uppercase letters for matrices)" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "import numpy as np\n", + "A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))\n", + "print(A)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "If we use the **shape** function we would get $(3, 3)$ as output, that is verifying that our matrix is a $3\\times 3$ matrix. We can slice the matrix and print for example the first column (Python organized matrix elements in a row-major order, see below) as" + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "import numpy as np\n", + "A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))\n", + "# print the first column, row-major order and elements start with 0\n", + "print(A[:,0])" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We can continue this was by printing out other columns or rows. The example here prints out the second column" + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "import numpy as np\n", + "A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))\n", + "# print the first column, row-major order and elements start with 0\n", + "print(A[1,:])" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Numpy contains many other functionalities that allow us to slice, subdivide etc etc arrays. We strongly recommend that you look up the [Numpy website for more details](http://www.numpy.org/). Useful functions when defining a matrix are the **np.zeros** function which declares a matrix of a given dimension and sets all elements to zero" + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "import numpy as np\n", + "n = 10\n", + "# define a matrix of dimension 10 x 10 and set all elements to zero\n", + "A = np.zeros( (n, n) )\n", + "print(A)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "or initializing all elements to" + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "import numpy as np\n", + "n = 10\n", + "# define a matrix of dimension 10 x 10 and set all elements to one\n", + "A = np.ones( (n, n) )\n", + "print(A)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "or as unitarily distributed random numbers (see the material on random number generators in the statistics part)" + ] + }, + { + "cell_type": "code", + "execution_count": 14, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "import numpy as np\n", + "n = 10\n", + "# define a matrix of dimension 10 x 10 and set all elements to random numbers with x \\in [0, 1]\n", + "A = np.random.rand(n, n)\n", + "print(A)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "As we will see throughout these lectures, there are several extremely useful functionalities in Numpy.\n", + "As an example, consider the discussion of the covariance matrix. Suppose we have defined three vectors\n", + "$\\hat{x}, \\hat{y}, \\hat{z}$ with $n$ elements each. The covariance matrix is defined as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\hat{\\Sigma} = \\begin{bmatrix} \\sigma_{xx} & \\sigma_{xy} & \\sigma_{xz} \\\\\n", + " \\sigma_{yx} & \\sigma_{yy} & \\sigma_{yz} \\\\\n", + " \\sigma_{zx} & \\sigma_{zy} & \\sigma_{zz} \n", + " \\end{bmatrix},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where for example" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\sigma_{xy} =\\frac{1}{n} \\sum_{i=0}^{n-1}(x_i- \\overline{x})(y_i- \\overline{y}).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The Numpy function **np.cov** calculates the covariance elements using the factor $1/(n-1)$ instead of $1/n$ since it assumes we do not have the exact mean values. \n", + "The following simple function uses the **np.vstack** function which takes each vector of dimension $1\\times n$ and produces a $3\\times n$ matrix $\\hat{W}$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\hat{W} = \\begin{bmatrix} x_0 & y_0 & z_0 \\\\\n", + " x_1 & y_1 & z_1 \\\\\n", + " x_2 & y_2 & z_2 \\\\\n", + " \\dots & \\dots & \\dots \\\\\n", + " x_{n-2} & y_{n-2} & z_{n-2} \\\\\n", + " x_{n-1} & y_{n-1} & z_{n-1}\n", + " \\end{bmatrix},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "which in turn is converted into into the $3\\times 3$ covariance matrix\n", + "$\\hat{\\Sigma}$ via the Numpy function **np.cov()**. We note that we can also calculate\n", + "the mean value of each set of samples $\\hat{x}$ etc using the Numpy\n", + "function **np.mean(x)**. We can also extract the eigenvalues of the\n", + "covariance matrix through the **np.linalg.eig()** function." + ] + }, + { + "cell_type": "code", + "execution_count": 15, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "# Importing various packages\n", + "import numpy as np\n", + "\n", + "n = 100\n", + "x = np.random.normal(size=n)\n", + "print(np.mean(x))\n", + "y = 4+3*x+np.random.normal(size=n)\n", + "print(np.mean(y))\n", + "z = x**3+np.random.normal(size=n)\n", + "print(np.mean(z))\n", + "W = np.vstack((x, y, z))\n", + "Sigma = np.cov(W)\n", + "print(Sigma)\n", + "Eigvals, Eigvecs = np.linalg.eig(Sigma)\n", + "print(Eigvals)" + ] + }, + { + "cell_type": "code", + "execution_count": 16, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "%matplotlib inline\n", + "\n", + "import numpy as np\n", + "import matplotlib.pyplot as plt\n", + "from scipy import sparse\n", + "eye = np.eye(4)\n", + "print(eye)\n", + "sparse_mtx = sparse.csr_matrix(eye)\n", + "print(sparse_mtx)\n", + "x = np.linspace(-10,10,100)\n", + "y = np.sin(x)\n", + "plt.plot(x,y,marker='x')\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Meet the Pandas\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "

\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "Another useful Python package is\n", + "[pandas](https://pandas.pydata.org/), which is an open source library\n", + "providing high-performance, easy-to-use data structures and data\n", + "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.\n", + "**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. \n", + "**pandas** allows you also to perform mathematical operations on the data, spanning from simple reshapings of vectors and matrices to statistical operations. \n", + "\n", + "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." + ] + }, + { + "cell_type": "code", + "execution_count": 17, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "import pandas as pd\n", + "from IPython.display import display\n", + "data = {'First Name': [\"Frodo\", \"Bilbo\", \"Aragorn II\", \"Samwise\"],\n", + " 'Last Name': [\"Baggins\", \"Baggins\",\"Elessar\",\"Gamgee\"],\n", + " 'Place of birth': [\"Shire\", \"Shire\", \"Eriador\", \"Shire\"],\n", + " 'Date of Birth T.A.': [2968, 2890, 2931, 2980]\n", + " }\n", + "data_pandas = pd.DataFrame(data)\n", + "display(data_pandas)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "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\n", + "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*.\n", + "Displaying these results, we see that the indices are given by the default numbers from zero to three.\n", + "**pandas** is extremely flexible and we can easily change the above indices by defining a new type of indexing as" + ] + }, + { + "cell_type": "code", + "execution_count": 18, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "data_pandas = pd.DataFrame(data,index=['Frodo','Bilbo','Aragorn','Sam'])\n", + "display(data_pandas)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Thereafter we display the content of the row which begins with the index **Aragorn**" + ] + }, + { + "cell_type": "code", + "execution_count": 19, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "display(data_pandas.loc['Aragorn'])" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We can easily append data to this, for example" + ] + }, + { + "cell_type": "code", + "execution_count": 20, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "new_hobbit = {'First Name': [\"Peregrin\"],\n", + " 'Last Name': [\"Took\"],\n", + " 'Place of birth': [\"Shire\"],\n", + " 'Date of Birth T.A.': [2990]\n", + " }\n", + "data_pandas=data_pandas.append(pd.DataFrame(new_hobbit, index=['Pippin']))\n", + "display(data_pandas)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "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 \n", + "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." + ] + }, + { + "cell_type": "code", + "execution_count": 21, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "import numpy as np\n", + "import pandas as pd\n", + "from IPython.display import display\n", + "np.random.seed(100)\n", + "# setting up a 10 x 5 matrix\n", + "rows = 10\n", + "cols = 5\n", + "a = np.random.randn(rows,cols)\n", + "df = pd.DataFrame(a)\n", + "display(df)\n", + "print(df.mean())\n", + "print(df.std())\n", + "display(df**2)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Thereafter we can select specific columns only and plot final results" + ] + }, + { + "cell_type": "code", + "execution_count": 22, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "df.columns = ['First', 'Second', 'Third', 'Fourth', 'Fifth']\n", + "df.index = np.arange(10)\n", + "\n", + "display(df)\n", + "print(df['Second'].mean() )\n", + "print(df.info())\n", + "print(df.describe())\n", + "\n", + "from pylab import plt, mpl\n", + "plt.style.use('seaborn')\n", + "mpl.rcParams['font.family'] = 'serif'\n", + "\n", + "df.cumsum().plot(lw=2.0, figsize=(10,6))\n", + "plt.show()\n", + "\n", + "\n", + "df.plot.bar(figsize=(10,6), rot=15)\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We can produce a $4\\times 4$ matrix" + ] + }, + { + "cell_type": "code", + "execution_count": 23, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "b = np.arange(16).reshape((4,4))\n", + "print(b)\n", + "df1 = pd.DataFrame(b)\n", + "print(df1)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and many other operations. \n", + "\n", + "The **Series** class is another important class included in\n", + "**pandas**. You can view it as a specialization of **DataFrame** but where\n", + "we have just a single column of data. It shares many of the same features as _DataFrame. As with **DataFrame**,\n", + "most operations are vectorized, achieving thereby a high performance when dealing with computations of arrays, in particular labeled arrays.\n", + "As we will see below it leads also to a very concice code close to the mathematical operations we may be interested in.\n", + "For multidimensional arrays, we recommend strongly [xarray](http://xarray.pydata.org/en/stable/). **xarray** has much of the same flexibility as **pandas**, but allows for the extension to higher dimensions than two. We will see examples later of the usage of both **pandas** and **xarray**. \n", + "\n", + "\n", + "\n", + "## Reading Data and fitting\n", + "\n", + "In order to study various Machine Learning algorithms, we need to\n", + "access data. Acccessing data is an essential step in all machine\n", + "learning algorithms. In particular, setting up the so-called **design\n", + "matrix** (to be defined below) is often the first element we need in\n", + "order to perform our calculations. To set up the design matrix means\n", + "reading (and later, when the calculations are done, writing) data\n", + "in various formats, The formats span from reading files from disk,\n", + "loading data from databases and interacting with online sources\n", + "like web application programming interfaces (APIs).\n", + "\n", + "In handling various input formats, as discussed above, we will mainly stay with **pandas**,\n", + "a Python package which allows us, in a seamless and painless way, to\n", + "deal with a multitude of formats, from standard **csv** (comma separated\n", + "values) files, via **excel**, **html** to **hdf5** formats. With **pandas**\n", + "and the **DataFrame** and **Series** functionalities we are able to convert text data\n", + "into the calculational formats we need for a specific algorithm. And our code is going to be \n", + "pretty close the basic mathematical expressions.\n", + "\n", + "Our first data set is going to be a classic from nuclear physics, namely all\n", + "available data on binding energies. Don't be intimidated if you are not familiar with nuclear physics. It serves simply as an example here of a data set. \n", + "\n", + "We will show some of the\n", + "strengths of packages like **Scikit-Learn** in fitting nuclear binding energies to\n", + "specific functions using linear regression first. Then, as a teaser, we will show you how \n", + "you can easily implement other algorithms like decision trees and random forests and neural networks.\n", + "\n", + "But before we really start with nuclear physics data, let's just look at some simpler polynomial fitting cases, such as,\n", + "(don't be offended) fitting straight lines!\n", + "\n", + "\n", + "### Simple linear regression model using **scikit-learn**\n", + "\n", + "We start with perhaps our simplest possible example, using **Scikit-Learn** to perform linear regression analysis on a data set produced by us. \n", + "\n", + "What follows is a simple Python code where we have defined a function\n", + "$y$ in terms of the variable $x$. Both are defined as vectors with $100$ entries. \n", + "The numbers in the vector $\\hat{x}$ are given\n", + "by random numbers generated with a uniform distribution with entries\n", + "$x_i \\in [0,1]$ (more about probability distribution functions\n", + "later). These values are then used to define a function $y(x)$\n", + "(tabulated again as a vector) with a linear dependence on $x$ plus a\n", + "random noise added via the normal distribution.\n", + "\n", + "\n", + "The Numpy functions are imported used the **import numpy as np**\n", + "statement and the random number generator for the uniform distribution\n", + "is called using the function **np.random.rand()**, where we specificy\n", + "that we want $100$ random variables. Using Numpy we define\n", + "automatically an array with the specified number of elements, $100$ in\n", + "our case. With the Numpy function **randn()** we can compute random\n", + "numbers with the normal distribution (mean value $\\mu$ equal to zero and\n", + "variance $\\sigma^2$ set to one) and produce the values of $y$ assuming a linear\n", + "dependence as function of $x$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "y = 2x+N(0,1),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $N(0,1)$ represents random numbers generated by the normal\n", + "distribution. From **Scikit-Learn** we import then the\n", + "**LinearRegression** functionality and make a prediction $\\tilde{y} =\n", + "\\alpha + \\beta x$ using the function **fit(x,y)**. We call the set of\n", + "data $(\\hat{x},\\hat{y})$ for our training data. The Python package\n", + "**scikit-learn** has also a functionality which extracts the above\n", + "fitting parameters $\\alpha$ and $\\beta$ (see below). Later we will\n", + "distinguish between training data and test data.\n", + "\n", + "For plotting we use the Python package\n", + "[matplotlib](https://matplotlib.org/) which produces publication\n", + "quality figures. Feel free to explore the extensive\n", + "[gallery](https://matplotlib.org/gallery/index.html) of examples. In\n", + "this example we plot our original values of $x$ and $y$ as well as the\n", + "prediction **ypredict** ($\\tilde{y}$), which attempts at fitting our\n", + "data with a straight line.\n", + "\n", + "The Python code follows here." + ] + }, + { + "cell_type": "code", + "execution_count": 24, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "# Importing various packages\n", + "import numpy as np\n", + "import matplotlib.pyplot as plt\n", + "from sklearn.linear_model import LinearRegression\n", + "\n", + "x = np.random.rand(100,1)\n", + "y = 2*x+np.random.randn(100,1)\n", + "linreg = LinearRegression()\n", + "linreg.fit(x,y)\n", + "xnew = np.array([[0],[1]])\n", + "ypredict = linreg.predict(xnew)\n", + "\n", + "plt.plot(xnew, ypredict, \"r-\")\n", + "plt.plot(x, y ,'ro')\n", + "plt.axis([0,1.0,0, 5.0])\n", + "plt.xlabel(r'$x$')\n", + "plt.ylabel(r'$y$')\n", + "plt.title(r'Simple Linear Regression')\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This example serves several aims. It allows us to demonstrate several\n", + "aspects of data analysis and later machine learning algorithms. The\n", + "immediate visualization shows that our linear fit is not\n", + "impressive. It goes through the data points, but there are many\n", + "outliers which are not reproduced by our linear regression. We could\n", + "now play around with this small program and change for example the\n", + "factor in front of $x$ and the normal distribution. Try to change the\n", + "function $y$ to" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "y = 10x+0.01 \\times N(0,1),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $x$ is defined as before. Does the fit look better? Indeed, by\n", + "reducing the role of the noise given by the normal distribution we see immediately that\n", + "our linear prediction seemingly reproduces better the training\n", + "set. However, this testing 'by the eye' is obviouly not satisfactory in the\n", + "long run. Here we have only defined the training data and our model, and \n", + "have not discussed a more rigorous approach to the **cost** function.\n", + "\n", + "We need more rigorous criteria in defining whether we have succeeded or\n", + "not in modeling our training data. You will be surprised to see that\n", + "many scientists seldomly venture beyond this 'by the eye' approach. A\n", + "standard approach for the *cost* function is the so-called $\\chi^2$\n", + "function (a variant of the mean-squared error (MSE))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\chi^2 = \\frac{1}{n}\n", + "\\sum_{i=0}^{n-1}\\frac{(y_i-\\tilde{y}_i)^2}{\\sigma_i^2},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $\\sigma_i^2$ is the variance (to be defined later) of the entry\n", + "$y_i$. We may not know the explicit value of $\\sigma_i^2$, it serves\n", + "however the aim of scaling the equations and make the cost function\n", + "dimensionless. \n", + "\n", + "Minimizing the cost function is a central aspect of\n", + "our discussions to come. Finding its minima as function of the model\n", + "parameters ($\\alpha$ and $\\beta$ in our case) will be a recurring\n", + "theme in these series of lectures. Essentially all machine learning\n", + "algorithms we will discuss center around the minimization of the\n", + "chosen cost function. This depends in turn on our specific\n", + "model for describing the data, a typical situation in supervised\n", + "learning. Automatizing the search for the minima of the cost function is a\n", + "central ingredient in all algorithms. Typical methods which are\n", + "employed are various variants of **gradient** methods. These will be\n", + "discussed in more detail later. Again, you'll be surprised to hear that\n", + "many practitioners minimize the above function ''by the eye', popularly dubbed as \n", + "'chi by the eye'. That is, change a parameter and see (visually and numerically) that \n", + "the $\\chi^2$ function becomes smaller. \n", + "\n", + "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 \n", + "the relative error (why would we prefer the MSE instead of the relative error?) as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\epsilon_{\\mathrm{relative}}= \\frac{\\vert \\hat{y} -\\hat{\\tilde{y}}\\vert}{\\vert \\hat{y}\\vert}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The squared cost function results in an arithmetic mean-unbiased\n", + "estimator, and the absolute-value cost function results in a\n", + "median-unbiased estimator (in the one-dimensional case, and a\n", + "geometric median-unbiased estimator for the multi-dimensional\n", + "case). The squared cost function has the disadvantage that it has the tendency\n", + "to be dominated by outliers.\n", + "\n", + "We can modify easily the above Python code and plot the relative error instead" + ] + }, + { + "cell_type": "code", + "execution_count": 25, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "import numpy as np\n", + "import matplotlib.pyplot as plt\n", + "from sklearn.linear_model import LinearRegression\n", + "\n", + "x = np.random.rand(100,1)\n", + "y = 5*x+0.01*np.random.randn(100,1)\n", + "linreg = LinearRegression()\n", + "linreg.fit(x,y)\n", + "ypredict = linreg.predict(x)\n", + "\n", + "plt.plot(x, np.abs(ypredict-y)/abs(y), \"ro\")\n", + "plt.axis([0,1.0,0.0, 0.5])\n", + "plt.xlabel(r'$x$')\n", + "plt.ylabel(r'$\\epsilon_{\\mathrm{relative}}$')\n", + "plt.title(r'Relative error')\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Depending on the parameter in front of the normal distribution, we may\n", + "have a small or larger relative error. Try to play around with\n", + "different training data sets and study (graphically) the value of the\n", + "relative error.\n", + "\n", + "As mentioned above, **Scikit-Learn** has an impressive functionality.\n", + "We can for example extract the values of $\\alpha$ and $\\beta$ and\n", + "their error estimates, or the variance and standard deviation and many\n", + "other properties from the statistical data analysis. \n", + "\n", + "Here we show an\n", + "example of the functionality of **Scikit-Learn**." + ] + }, + { + "cell_type": "code", + "execution_count": 26, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "import numpy as np \n", + "import matplotlib.pyplot as plt \n", + "from sklearn.linear_model import LinearRegression \n", + "from sklearn.metrics import mean_squared_error, r2_score, mean_squared_log_error, mean_absolute_error\n", + "\n", + "x = np.random.rand(100,1)\n", + "y = 2.0+ 5*x+0.5*np.random.randn(100,1)\n", + "linreg = LinearRegression()\n", + "linreg.fit(x,y)\n", + "ypredict = linreg.predict(x)\n", + "print('The intercept alpha: \\n', linreg.intercept_)\n", + "print('Coefficient beta : \\n', linreg.coef_)\n", + "# The mean squared error \n", + "print(\"Mean squared error: %.2f\" % mean_squared_error(y, ypredict))\n", + "# Explained variance score: 1 is perfect prediction \n", + "print('Variance score: %.2f' % r2_score(y, ypredict))\n", + "# Mean squared log error \n", + "print('Mean squared log error: %.2f' % mean_squared_log_error(y, ypredict) )\n", + "# Mean absolute error \n", + "print('Mean absolute error: %.2f' % mean_absolute_error(y, ypredict))\n", + "plt.plot(x, ypredict, \"r-\")\n", + "plt.plot(x, y ,'ro')\n", + "plt.axis([0.0,1.0,1.5, 7.0])\n", + "plt.xlabel(r'$x$')\n", + "plt.ylabel(r'$y$')\n", + "plt.title(r'Linear Regression fit ')\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The function **coef** gives us the parameter $\\beta$ of our fit while **intercept** yields \n", + "$\\alpha$. Depending on the constant in front of the normal distribution, we get values near or far from $alpha =2$ and $\\beta =5$. Try to play around with different parameters in front of the normal distribution. The function **meansquarederror** gives us the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error or loss defined as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "MSE(\\hat{y},\\hat{\\tilde{y}}) = \\frac{1}{n}\n", + "\\sum_{i=0}^{n-1}(y_i-\\tilde{y}_i)^2,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The smaller the value, the better the fit. Ideally we would like to\n", + "have an MSE equal zero. The attentive reader has probably recognized\n", + "this function as being similar to the $\\chi^2$ function defined above.\n", + "\n", + "The **r2score** function computes $R^2$, the coefficient of\n", + "determination. It provides a measure of how well future samples are\n", + "likely to be predicted by the model. Best possible score is 1.0 and it\n", + "can be negative (because the model can be arbitrarily worse). A\n", + "constant model that always predicts the expected value of $\\hat{y}$,\n", + "disregarding the input features, would get a $R^2$ score of $0.0$.\n", + "\n", + "If $\\tilde{\\hat{y}}_i$ is the predicted value of the $i-th$ sample and $y_i$ is the corresponding true value, then the score $R^2$ is defined as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "R^2(\\hat{y}, \\tilde{\\hat{y}}) = 1 - \\frac{\\sum_{i=0}^{n - 1} (y_i - \\tilde{y}_i)^2}{\\sum_{i=0}^{n - 1} (y_i - \\bar{y})^2},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where we have defined the mean value of $\\hat{y}$ as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\bar{y} = \\frac{1}{n} \\sum_{i=0}^{n - 1} y_i.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Another quantity taht we will meet again in our discussions of regression analysis is \n", + " the mean absolute error (MAE), a risk metric corresponding to the expected value of the absolute error loss or what we call the $l1$-norm loss. In our discussion above we presented the relative error.\n", + "The MAE is defined as follows" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\text{MAE}(\\hat{y}, \\hat{\\tilde{y}}) = \\frac{1}{n} \\sum_{i=0}^{n-1} \\left| y_i - \\tilde{y}_i \\right|.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We present the \n", + "squared logarithmic (quadratic) error" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\text{MSLE}(\\hat{y}, \\hat{\\tilde{y}}) = \\frac{1}{n} \\sum_{i=0}^{n - 1} (\\log_e (1 + y_i) - \\log_e (1 + \\tilde{y}_i) )^2,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $\\log_e (x)$ stands for the natural logarithm of $x$. This error\n", + "estimate is best to use when targets having exponential growth, such\n", + "as population counts, average sales of a commodity over a span of\n", + "years etc. \n", + "\n", + "\n", + "Finally, another cost function is the Huber cost function used in robust regression.\n", + "\n", + "The rationale behind this possible cost function is its reduced\n", + "sensitivity to outliers in the data set. In our discussions on\n", + "dimensionality reduction and normalization of data we will meet other\n", + "ways of dealing with outliers.\n", + "\n", + "The Huber cost function is defined as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "H_{\\delta}(a)={\\begin{cases}{\\frac {1}{2}}{a^{2}}&{\\text{for }}|a|\\leq \\delta ,\\\\\\delta (|a|-{\\frac {1}{2}}\\delta ),&{\\text{otherwise.}}\\end{cases}}}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Here $a=\\boldsymbol{y} - \\boldsymbol{\\tilde{y}}$.\n", + "We will discuss in more\n", + "detail these and other functions in the various lectures. We conclude this part with another example. Instead of \n", + "a linear $x$-dependence we study now a cubic polynomial and use the polynomial regression analysis tools of scikit-learn." + ] + }, + { + "cell_type": "code", + "execution_count": 27, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "import matplotlib.pyplot as plt\n", + "import numpy as np\n", + "import random\n", + "from sklearn.linear_model import Ridge\n", + "from sklearn.preprocessing import PolynomialFeatures\n", + "from sklearn.pipeline import make_pipeline\n", + "from sklearn.linear_model import LinearRegression\n", + "\n", + "x=np.linspace(0.02,0.98,200)\n", + "noise = np.asarray(random.sample((range(200)),200))\n", + "y=x**3*noise\n", + "yn=x**3*100\n", + "poly3 = PolynomialFeatures(degree=3)\n", + "X = poly3.fit_transform(x[:,np.newaxis])\n", + "clf3 = LinearRegression()\n", + "clf3.fit(X,y)\n", + "\n", + "Xplot=poly3.fit_transform(x[:,np.newaxis])\n", + "poly3_plot=plt.plot(x, clf3.predict(Xplot), label='Cubic Fit')\n", + "plt.plot(x,yn, color='red', label=\"True Cubic\")\n", + "plt.scatter(x, y, label='Data', color='orange', s=15)\n", + "plt.legend()\n", + "plt.show()\n", + "\n", + "def error(a):\n", + " for i in y:\n", + " err=(y-yn)/yn\n", + " return abs(np.sum(err))/len(err)\n", + "\n", + "print (error(y))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### To our real data: nuclear binding energies. Brief reminder on masses and binding energies\n", + "\n", + "Let us now dive into nuclear physics and remind ourselves briefly about some basic features about binding\n", + "energies. A basic quantity which can be measured for the ground\n", + "states of nuclei is the atomic mass $M(N, Z)$ of the neutral atom with\n", + "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). \n", + "\n", + "Atomic masses are usually tabulated in terms of the mass excess defined by" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\Delta M(N, Z) = M(N, Z) - uA,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $u$ is the Atomic Mass Unit" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "u = M(^{12}\\mathrm{C})/12 = 931.4940954(57) \\hspace{0.1cm} \\mathrm{MeV}/c^2.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The nucleon masses are" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "m_p = 1.00727646693(9)u,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "m_n = 939.56536(8)\\hspace{0.1cm} \\mathrm{MeV}/c^2 = 1.0086649156(6)u.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "In the [2016 mass evaluation of by W.J.Huang, G.Audi, M.Wang, F.G.Kondev, S.Naimi and X.Xu](http://nuclearmasses.org/resources_folder/Wang_2017_Chinese_Phys_C_41_030003.pdf)\n", + "there are data on masses and decays of 3437 nuclei.\n", + "\n", + "The nuclear binding energy is defined as the energy required to break\n", + "up a given nucleus into its constituent parts of $N$ neutrons and $Z$\n", + "protons. In terms of the atomic masses $M(N, Z)$ the binding energy is\n", + "defined by" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "BE(N, Z) = ZM_H c^2 + Nm_n c^2 - M(N, Z)c^2 ,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $M_H$ is the mass of the hydrogen atom and $m_n$ is the mass of the neutron.\n", + "In terms of the mass excess the binding energy is given by" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "BE(N, Z) = Z\\Delta_H c^2 + N\\Delta_n c^2 -\\Delta(N, Z)c^2 ,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $\\Delta_H c^2 = 7.2890$ MeV and $\\Delta_n c^2 = 8.0713$ MeV.\n", + "\n", + "\n", + "A popular and physically intuitive model which can be used to parametrize \n", + "the experimental binding energies as function of $A$, is the so-called \n", + "**liquid drop model**. The ansatz is based on the following expression" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "BE(N,Z) = a_1A-a_2A^{2/3}-a_3\\frac{Z^2}{A^{1/3}}-a_4\\frac{(N-Z)^2}{A},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $A$ stands for the number of nucleons and the $a_i$s are parameters which are determined by a fit \n", + "to the experimental data. \n", + "\n", + "\n", + "\n", + "\n", + "To arrive at the above expression we have assumed that we can make the following assumptions:\n", + "\n", + " * There is a volume term $a_1A$ proportional with the number of nucleons (the energy is also an extensive quantity). When an assembly of nucleons of the same size is packed together into the smallest volume, each interior nucleon has a certain number of other nucleons in contact with it. This contribution is proportional to the volume.\n", + "\n", + " * 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.\n", + "\n", + " * 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. \n", + "\n", + " * 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.\n", + "\n", + "We could also add a so-called pairing term, which is a correction term that\n", + "arises from the tendency of proton pairs and neutron pairs to\n", + "occur. An even number of particles is more stable than an odd number. \n", + "\n", + "\n", + "### Organizing our data\n", + "\n", + "Let us start with reading and organizing our data. \n", + "We start with the compilation of masses and binding energies from 2016.\n", + "After having downloaded this file to our own computer, we are now ready to read the file and start structuring our data.\n", + "\n", + "\n", + "We start with preparing folders for storing our calculations and the data file over masses and binding energies. We import also various modules that we will find useful in order to present various Machine Learning methods. Here we focus mainly on the functionality of **scikit-learn**." + ] + }, + { + "cell_type": "code", + "execution_count": 28, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "# Common imports\n", + "import numpy as np\n", + "import pandas as pd\n", + "import matplotlib.pyplot as plt\n", + "import sklearn.linear_model as skl\n", + "from sklearn.model_selection import train_test_split\n", + "from sklearn.metrics import mean_squared_error, r2_score, mean_absolute_error\n", + "import os\n", + "\n", + "# Where to save the figures and data files\n", + "PROJECT_ROOT_DIR = \"Results\"\n", + "FIGURE_ID = \"Results/FigureFiles\"\n", + "DATA_ID = \"DataFiles/\"\n", + "\n", + "if not os.path.exists(PROJECT_ROOT_DIR):\n", + " os.mkdir(PROJECT_ROOT_DIR)\n", + "\n", + "if not os.path.exists(FIGURE_ID):\n", + " os.makedirs(FIGURE_ID)\n", + "\n", + "if not os.path.exists(DATA_ID):\n", + " os.makedirs(DATA_ID)\n", + "\n", + "def image_path(fig_id):\n", + " return os.path.join(FIGURE_ID, fig_id)\n", + "\n", + "def data_path(dat_id):\n", + " return os.path.join(DATA_ID, dat_id)\n", + "\n", + "def save_fig(fig_id):\n", + " plt.savefig(image_path(fig_id) + \".png\", format='png')\n", + "\n", + "infile = open(data_path(\"MassEval2016.dat\"),'r')" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Before we proceed, we define also a function for making our plots. You can obviously avoid this and simply set up various **matplotlib** commands every time you need them. You may however find it convenient to collect all such commands in one function and simply call this function." + ] + }, + { + "cell_type": "code", + "execution_count": 29, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "from pylab import plt, mpl\n", + "plt.style.use('seaborn')\n", + "mpl.rcParams['font.family'] = 'serif'\n", + "\n", + "def MakePlot(x,y, styles, labels, axlabels):\n", + " plt.figure(figsize=(10,6))\n", + " for i in range(len(x)):\n", + " plt.plot(x[i], y[i], styles[i], label = labels[i])\n", + " plt.xlabel(axlabels[0])\n", + " plt.ylabel(axlabels[1])\n", + " plt.legend(loc=0)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Our next step is to read the data on experimental binding energies and\n", + "reorganize them as functions of the mass number $A$, the number of\n", + "protons $Z$ and neutrons $N$ using **pandas**. Before we do this it is\n", + "always useful (unless you have a binary file or other types of compressed\n", + "data) to actually open the file and simply take a look at it!\n", + "\n", + "\n", + "In particular, the program that outputs the final nuclear masses is written in Fortran with a specific format. It means that we need to figure out the format and which columns contain the data we are interested in. Pandas comes with a function that reads formatted output. After having admired the file, we are now ready to start massaging it with **pandas**. The file begins with some basic format information." + ] + }, + { + "cell_type": "code", + "execution_count": 30, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "\"\"\" \n", + "This is taken from the data file of the mass 2016 evaluation. \n", + "All files are 3436 lines long with 124 character per line. \n", + " Headers are 39 lines long. \n", + " col 1 : Fortran character control: 1 = page feed 0 = line feed \n", + " format : a1,i3,i5,i5,i5,1x,a3,a4,1x,f13.5,f11.5,f11.3,f9.3,1x,a2,f11.3,f9.3,1x,i3,1x,f12.5,f11.5 \n", + " These formats are reflected in the pandas widths variable below, see the statement \n", + " widths=(1,3,5,5,5,1,3,4,1,13,11,11,9,1,2,11,9,1,3,1,12,11,1), \n", + " Pandas has also a variable header, with length 39 in this case. \n", + "\"\"\"" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The data we are interested in are in columns 2, 3, 4 and 11, giving us\n", + "the number of neutrons, protons, mass numbers and binding energies,\n", + "respectively. We add also for the sake of completeness the element name. The data are in fixed-width formatted lines and we will\n", + "covert them into the **pandas** DataFrame structure." + ] + }, + { + "cell_type": "code", + "execution_count": 31, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "# Read the experimental data with Pandas\n", + "Masses = pd.read_fwf(infile, usecols=(2,3,4,6,11),\n", + " names=('N', 'Z', 'A', 'Element', 'Ebinding'),\n", + " widths=(1,3,5,5,5,1,3,4,1,13,11,11,9,1,2,11,9,1,3,1,12,11,1),\n", + " header=39,\n", + " index_col=False)\n", + "\n", + "# Extrapolated values are indicated by '#' in place of the decimal place, so\n", + "# the Ebinding column won't be numeric. Coerce to float and drop these entries.\n", + "Masses['Ebinding'] = pd.to_numeric(Masses['Ebinding'], errors='coerce')\n", + "Masses = Masses.dropna()\n", + "# Convert from keV to MeV.\n", + "Masses['Ebinding'] /= 1000\n", + "\n", + "# Group the DataFrame by nucleon number, A.\n", + "Masses = Masses.groupby('A')\n", + "# Find the rows of the grouped DataFrame with the maximum binding energy.\n", + "Masses = Masses.apply(lambda t: t[t.Ebinding==t.Ebinding.max()])" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We have now read in the data, grouped them according to the variables we are interested in. \n", + "We see how easy it is to reorganize the data using **pandas**. If we\n", + "were to do these operations in C/C++ or Fortran, we would have had to\n", + "write various functions/subroutines which perform the above\n", + "reorganizations for us. Having reorganized the data, we can now start\n", + "to make some simple fits using both the functionalities in **numpy** and\n", + "**Scikit-Learn** afterwards. \n", + "\n", + "Now we define five variables which contain\n", + "the number of nucleons $A$, the number of protons $Z$ and the number of neutrons $N$, the element name and finally the energies themselves." + ] + }, + { + "cell_type": "code", + "execution_count": 32, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "A = Masses['A']\n", + "Z = Masses['Z']\n", + "N = Masses['N']\n", + "Element = Masses['Element']\n", + "Energies = Masses['Ebinding']\n", + "print(Masses)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The next step, and we will define this mathematically later, is to set up the so-called **design matrix**. We will throughout call this matrix $\\boldsymbol{X}$.\n", + "It has dimensionality $p\\times n$, where $n$ is the number of data points and $p$ are the so-called predictors. In our case here they are given by the number of polynomials in $A$ we wish to include in the fit." + ] + }, + { + "cell_type": "code", + "execution_count": 33, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "# Now we set up the design matrix X\n", + "X = np.zeros((len(A),5))\n", + "X[:,0] = 1\n", + "X[:,1] = A\n", + "X[:,2] = A**(2.0/3.0)\n", + "X[:,3] = A**(-1.0/3.0)\n", + "X[:,4] = A**(-1.0)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "With **scikitlearn** we are now ready to use linear regression and fit our data." + ] + }, + { + "cell_type": "code", + "execution_count": 34, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "clf = skl.LinearRegression().fit(X, Energies)\n", + "fity = clf.predict(X)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Pretty simple! \n", + "Now we can print measures of how our fit is doing, the coefficients from the fits and plot the final fit together with our data." + ] + }, + { + "cell_type": "code", + "execution_count": 35, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "# The mean squared error \n", + "print(\"Mean squared error: %.2f\" % mean_squared_error(Energies, fity))\n", + "# Explained variance score: 1 is perfect prediction \n", + "print('Variance score: %.2f' % r2_score(Energies, fity))\n", + "# Mean absolute error \n", + "print('Mean absolute error: %.2f' % mean_absolute_error(Energies, fity))\n", + "print(clf.coef_, clf.intercept_)\n", + "\n", + "Masses['Eapprox'] = fity\n", + "# Generate a plot comparing the experimental with the fitted values values.\n", + "fig, ax = plt.subplots()\n", + "ax.set_xlabel(r'$A = N + Z$')\n", + "ax.set_ylabel(r'$E_\\mathrm{bind}\\,/\\mathrm{MeV}$')\n", + "ax.plot(Masses['A'], Masses['Ebinding'], alpha=0.7, lw=2,\n", + " label='Ame2016')\n", + "ax.plot(Masses['A'], Masses['Eapprox'], alpha=0.7, lw=2, c='m',\n", + " label='Fit')\n", + "ax.legend()\n", + "save_fig(\"Masses2016\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Seeing the wood for the trees\n", + "\n", + "As a teaser, let us now see how we can do this with decision trees using **scikit-learn**. Later we will switch to so-called **random forests**!" + ] + }, + { + "cell_type": "code", + "execution_count": 36, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "\n", + "#Decision Tree Regression\n", + "from sklearn.tree import DecisionTreeRegressor\n", + "regr_1=DecisionTreeRegressor(max_depth=5)\n", + "regr_2=DecisionTreeRegressor(max_depth=7)\n", + "regr_3=DecisionTreeRegressor(max_depth=9)\n", + "regr_1.fit(X, Energies)\n", + "regr_2.fit(X, Energies)\n", + "regr_3.fit(X, Energies)\n", + "\n", + "\n", + "y_1 = regr_1.predict(X)\n", + "y_2 = regr_2.predict(X)\n", + "y_3=regr_3.predict(X)\n", + "Masses['Eapprox'] = y_3\n", + "# Plot the results\n", + "plt.figure()\n", + "plt.plot(A, Energies, color=\"blue\", label=\"Data\", linewidth=2)\n", + "plt.plot(A, y_1, color=\"red\", label=\"max_depth=5\", linewidth=2)\n", + "plt.plot(A, y_2, color=\"green\", label=\"max_depth=7\", linewidth=2)\n", + "plt.plot(A, y_3, color=\"m\", label=\"max_depth=9\", linewidth=2)\n", + "\n", + "plt.xlabel(\"$A$\")\n", + "plt.ylabel(\"$E$[MeV]\")\n", + "plt.title(\"Decision Tree Regression\")\n", + "plt.legend()\n", + "save_fig(\"Masses2016Trees\")\n", + "plt.show()\n", + "print(Masses)\n", + "print(np.mean( (Energies-y_1)**2))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### And what about using neural networks?\n", + "\n", + "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) \n", + "functionality." + ] + }, + { + "cell_type": "code", + "execution_count": 37, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "from sklearn.neural_network import MLPRegressor\n", + "from sklearn.metrics import accuracy_score\n", + "import seaborn as sns\n", + "\n", + "X_train = X\n", + "Y_train = Energies\n", + "n_hidden_neurons = 100\n", + "epochs = 100\n", + "# store models for later use\n", + "eta_vals = np.logspace(-5, 1, 7)\n", + "lmbd_vals = np.logspace(-5, 1, 7)\n", + "# store the models for later use\n", + "DNN_scikit = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)\n", + "train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))\n", + "sns.set()\n", + "for i, eta in enumerate(eta_vals):\n", + " for j, lmbd in enumerate(lmbd_vals):\n", + " dnn = MLPRegressor(hidden_layer_sizes=(n_hidden_neurons), activation='logistic',\n", + " alpha=lmbd, learning_rate_init=eta, max_iter=epochs)\n", + " dnn.fit(X_train, Y_train)\n", + " DNN_scikit[i][j] = dnn\n", + " train_accuracy[i][j] = dnn.score(X_train, Y_train)\n", + "\n", + "fig, ax = plt.subplots(figsize = (10, 10))\n", + "sns.heatmap(train_accuracy, annot=True, ax=ax, cmap=\"viridis\")\n", + "ax.set_title(\"Training Accuracy\")\n", + "ax.set_ylabel(\"$\\eta$\")\n", + "ax.set_xlabel(\"$\\lambda$\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## A first summary\n", + "\n", + "The aim behind these introductory words was to present to you various\n", + "Python libraries and their functionalities, in particular libraries like\n", + "**numpy**, **pandas**, **xarray** and **matplotlib** and other that make our life much easier\n", + "in handling various data sets and visualizing data. \n", + "\n", + "Furthermore,\n", + "**Scikit-Learn** allows us with few lines of code to implement popular\n", + "Machine Learning algorithms for supervised learning. Later we will meet **Tensorflow**, a powerful library for deep learning. \n", + "Now it is time to dive more into the details of various methods. We will start with linear regression and try to take a deeper look at what it entails." + ] + } + ], + "metadata": {}, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/doc/src/LectureNotes/_build/html/_sources/chapter3.ipynb b/doc/src/LectureNotes/_build/html/_sources/chapter3.ipynb new file mode 100644 index 000000000..a8f98b7ba --- /dev/null +++ b/doc/src/LectureNotes/_build/html/_sources/chapter3.ipynb @@ -0,0 +1,5753 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Linear Regression and more Advanced Regression Analysis\n", + "\n", + "\n", + "## Why Linear Regression (aka Ordinary Least Squares and family)\n", + "\n", + "Fitting a continuous function with linear parameterization in terms of the parameters $\\boldsymbol{\\beta}$.\n", + "* Method of choice for fitting a continuous function!\n", + "\n", + "* Gives an excellent introduction to central Machine Learning features with **understandable pedagogical** links to other methods like **Neural Networks**, **Support Vector Machines** etc\n", + "\n", + "* Analytical expression for the fitting parameters $\\boldsymbol{\\beta}$\n", + "\n", + "* Analytical expressions for statistical propertiers like mean values, variances, confidence intervals and more\n", + "\n", + "* Analytical relation with probabilistic interpretations \n", + "\n", + "* Easy to introduce basic concepts like bias-variance tradeoff, cross-validation, resampling and regularization techniques and many other ML topics\n", + "\n", + "* Easy to code! And links well with classification problems and logistic regression and neural networks\n", + "\n", + "* Allows for **easy** hands-on understanding of gradient descent methods\n", + "\n", + "* and many more features\n", + "\n", + "For more discussions of Ridge and Lasso regression, [Wessel van Wieringen's](https://arxiv.org/abs/1509.09169) article is highly recommended.\n", + "Similarly, [Mehta et al's article](https://arxiv.org/abs/1803.08823) is also recommended.\n", + "\n", + "\n", + "## Regression analysis, overarching aims\n", + "\n", + "Regression modeling deals with the description of the sampling distribution of a given random variable $y$ and how it varies as function of another variable or a set of such variables $\\boldsymbol{x} =[x_0, x_1,\\dots, x_{n-1}]^T$. \n", + "The first variable is called the **dependent**, the **outcome** or the **response** variable while the set of variables $\\boldsymbol{x}$ is called the independent variable, or the predictor variable or the explanatory variable. \n", + "\n", + "A regression model aims at finding a likelihood function $p(\\boldsymbol{y}\\vert \\boldsymbol{x})$, that is the conditional distribution for $\\boldsymbol{y}$ with a given $\\boldsymbol{x}$. The estimation of $p(\\boldsymbol{y}\\vert \\boldsymbol{x})$ is made using a data set with \n", + "* $n$ cases $i = 0, 1, 2, \\dots, n-1$ \n", + "\n", + "* Response (target, dependent or outcome) variable $y_i$ with $i = 0, 1, 2, \\dots, n-1$ \n", + "\n", + "* $p$ so-called explanatory (independent or predictor) variables $\\boldsymbol{x}_i=[x_{i0}, x_{i1}, \\dots, x_{ip-1}]$ with $i = 0, 1, 2, \\dots, n-1$ and explanatory variables running from $0$ to $p-1$. See below for more explicit examples. \n", + "\n", + " The goal of the regression analysis is to extract/exploit relationship between $\\boldsymbol{y}$ and $\\boldsymbol{x}$ in or to infer causal dependencies, approximations to the likelihood functions, functional relationships and to make predictions, making fits and many other things.\n", + "\n", + "\n", + "\n", + "## Regression analysis, overarching aims II\n", + "\n", + "\n", + "Consider an experiment in which $p$ characteristics of $n$ samples are\n", + "measured. The data from this experiment, for various explanatory variables $p$ are normally represented by a matrix \n", + "$\\mathbf{X}$.\n", + "\n", + "The matrix $\\mathbf{X}$ is called the *design\n", + "matrix*. Additional information of the samples is available in the\n", + "form of $\\boldsymbol{y}$ (also as above). The variable $\\boldsymbol{y}$ is\n", + "generally referred to as the *response variable*. The aim of\n", + "regression analysis is to explain $\\boldsymbol{y}$ in terms of\n", + "$\\boldsymbol{X}$ through a functional relationship like $y_i =\n", + "f(\\mathbf{X}_{i,\\ast})$. When no prior knowledge on the form of\n", + "$f(\\cdot)$ is available, it is common to assume a linear relationship\n", + "between $\\boldsymbol{X}$ and $\\boldsymbol{y}$. This assumption gives rise to\n", + "the *linear regression model* where $\\boldsymbol{\\beta} = [\\beta_0, \\ldots,\n", + "\\beta_{p-1}]^{T}$ are the *regression parameters*. \n", + "\n", + "Linear regression gives us a set of analytical equations for the parameters $\\beta_j$.\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "## Examples\n", + "In order to understand the relation among the predictors $p$, the set of data $n$ and the target (outcome, output etc) $\\boldsymbol{y}$,\n", + "consider the model we discussed for describing nuclear binding energies. \n", + "\n", + "There we assumed that we could parametrize the data using a polynomial approximation based on the liquid drop model.\n", + "Assuming" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "BE(A) = a_0+a_1A+a_2A^{2/3}+a_3A^{-1/3}+a_4A^{-1},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "we have five predictors, that is the intercept, the $A$ dependent term, the $A^{2/3}$ term and the $A^{-1/3}$ and $A^{-1}$ terms.\n", + "This gives $p=0,1,2,3,4$. Furthermore we have $n$ entries for each predictor. It means that our design matrix is a \n", + "$p\\times n$ matrix $\\boldsymbol{X}$.\n", + "\n", + "Here the predictors are based on a model we have made. A popular data set which is widely encountered in ML applications is the\n", + "so-called [credit card default data from Taiwan](https://www.sciencedirect.com/science/article/pii/S0957417407006719?via%3Dihub). The data set contains data on $n=30000$ credit card holders with predictors like gender, marital status, age, profession, education, etc. In total there are $24$ such predictors or attributes leading to a design matrix of dimensionality $24 \\times 30000$. This is however a classification problem and we will come back to it when we discuss Logistic Regression.\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "## General linear models\n", + "Before we proceed let us study a case from linear algebra where we aim at fitting a set of data $\\boldsymbol{y}=[y_0,y_1,\\dots,y_{n-1}]$. We could think of these data as a result of an experiment or a complicated numerical experiment. These data are functions of a series of variables $\\boldsymbol{x}=[x_0,x_1,\\dots,x_{n-1}]$, that is $y_i = y(x_i)$ with $i=0,1,2,\\dots,n-1$. The variables $x_i$ could represent physical quantities like time, temperature, position etc. We assume that $y(x)$ is a smooth function. \n", + "\n", + "Since obtaining these data points may not be trivial, we want to use these data to fit a function which can allow us to make predictions for values of $y$ which are not in the present set. The perhaps simplest approach is to assume we can parametrize our function in terms of a polynomial of degree $n-1$ with $n$ points, that is" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "y=y(x) \\rightarrow y(x_i)=\\tilde{y}_i+\\epsilon_i=\\sum_{j=0}^{n-1} \\beta_j x_i^j+\\epsilon_i,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $\\epsilon_i$ is the error in our approximation.\n", + "\n", + "\n", + "\n", + "\n", + "## Rewriting the fitting procedure as a linear algebra problem\n", + "For every set of values $y_i,x_i$ we have thus the corresponding set of equations" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\begin{align*}\n", + "y_0&=\\beta_0+\\beta_1x_0^1+\\beta_2x_0^2+\\dots+\\beta_{n-1}x_0^{n-1}+\\epsilon_0\\\\\n", + "y_1&=\\beta_0+\\beta_1x_1^1+\\beta_2x_1^2+\\dots+\\beta_{n-1}x_1^{n-1}+\\epsilon_1\\\\\n", + "y_2&=\\beta_0+\\beta_1x_2^1+\\beta_2x_2^2+\\dots+\\beta_{n-1}x_2^{n-1}+\\epsilon_2\\\\\n", + "\\dots & \\dots \\\\\n", + "y_{n-1}&=\\beta_0+\\beta_1x_{n-1}^1+\\beta_2x_{n-1}^2+\\dots+\\beta_{n-1}x_{n-1}^{n-1}+\\epsilon_{n-1}.\\\\\n", + "\\end{align*}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Rewriting the fitting procedure as a linear algebra problem, more details\n", + "Defining the vectors" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{y} = [y_0,y_1, y_2,\\dots, y_{n-1}]^T,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{\\beta} = [\\beta_0,\\beta_1, \\beta_2,\\dots, \\beta_{n-1}]^T,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{\\epsilon} = [\\epsilon_0,\\epsilon_1, \\epsilon_2,\\dots, \\epsilon_{n-1}]^T,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and the design matrix" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{X}=\n", + "\\begin{bmatrix} \n", + "1& x_{0}^1 &x_{0}^2& \\dots & \\dots &x_{0}^{n-1}\\\\\n", + "1& x_{1}^1 &x_{1}^2& \\dots & \\dots &x_{1}^{n-1}\\\\\n", + "1& x_{2}^1 &x_{2}^2& \\dots & \\dots &x_{2}^{n-1}\\\\ \n", + "\\dots& \\dots &\\dots& \\dots & \\dots &\\dots\\\\\n", + "1& x_{n-1}^1 &x_{n-1}^2& \\dots & \\dots &x_{n-1}^{n-1}\\\\\n", + "\\end{bmatrix}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "we can rewrite our equations as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{y} = \\boldsymbol{X}\\boldsymbol{\\beta}+\\boldsymbol{\\epsilon}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The above design matrix is called a [Vandermonde matrix](https://en.wikipedia.org/wiki/Vandermonde_matrix).\n", + "\n", + "\n", + "\n", + "\n", + "## Generalizing the fitting procedure as a linear algebra problem\n", + "\n", + "We are obviously not limited to the above polynomial expansions. We\n", + "could replace the various powers of $x$ with elements of Fourier\n", + "series or instead of $x_i^j$ we could have $\\cos{(j x_i)}$ or $\\sin{(j\n", + "x_i)}$, or time series or other orthogonal functions. For every set\n", + "of values $y_i,x_i$ we can then generalize the equations to" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\begin{align*}\n", + "y_0&=\\beta_0x_{00}+\\beta_1x_{01}+\\beta_2x_{02}+\\dots+\\beta_{n-1}x_{0n-1}+\\epsilon_0\\\\\n", + "y_1&=\\beta_0x_{10}+\\beta_1x_{11}+\\beta_2x_{12}+\\dots+\\beta_{n-1}x_{1n-1}+\\epsilon_1\\\\\n", + "y_2&=\\beta_0x_{20}+\\beta_1x_{21}+\\beta_2x_{22}+\\dots+\\beta_{n-1}x_{2n-1}+\\epsilon_2\\\\\n", + "\\dots & \\dots \\\\\n", + "y_{i}&=\\beta_0x_{i0}+\\beta_1x_{i1}+\\beta_2x_{i2}+\\dots+\\beta_{n-1}x_{in-1}+\\epsilon_i\\\\\n", + "\\dots & \\dots \\\\\n", + "y_{n-1}&=\\beta_0x_{n-1,0}+\\beta_1x_{n-1,2}+\\beta_2x_{n-1,2}+\\dots+\\beta_{n-1}x_{n-1,n-1}+\\epsilon_{n-1}.\\\\\n", + "\\end{align*}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Note that we have $p=n$ here. The matrix is symmetric. This is generally not the case!**\n", + "\n", + "\n", + "\n", + "\n", + "## Generalizing the fitting procedure as a linear algebra problem\n", + "We redefine in turn the matrix $\\boldsymbol{X}$ as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{X}=\n", + "\\begin{bmatrix} \n", + "x_{00}& x_{01} &x_{02}& \\dots & \\dots &x_{0,n-1}\\\\\n", + "x_{10}& x_{11} &x_{12}& \\dots & \\dots &x_{1,n-1}\\\\\n", + "x_{20}& x_{21} &x_{22}& \\dots & \\dots &x_{2,n-1}\\\\ \n", + "\\dots& \\dots &\\dots& \\dots & \\dots &\\dots\\\\\n", + "x_{n-1,0}& x_{n-1,1} &x_{n-1,2}& \\dots & \\dots &x_{n-1,n-1}\\\\\n", + "\\end{bmatrix}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and without loss of generality we rewrite again our equations as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{y} = \\boldsymbol{X}\\boldsymbol{\\beta}+\\boldsymbol{\\epsilon}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The left-hand side of this equation is kwown. Our error vector $\\boldsymbol{\\epsilon}$ and the parameter vector $\\boldsymbol{\\beta}$ are our unknow quantities. How can we obtain the optimal set of $\\beta_i$ values?\n", + "\n", + "\n", + "\n", + "\n", + "## Optimizing our parameters\n", + "We have defined the matrix $\\boldsymbol{X}$ via the equations" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\begin{align*}\n", + "y_0&=\\beta_0x_{00}+\\beta_1x_{01}+\\beta_2x_{02}+\\dots+\\beta_{n-1}x_{0n-1}+\\epsilon_0\\\\\n", + "y_1&=\\beta_0x_{10}+\\beta_1x_{11}+\\beta_2x_{12}+\\dots+\\beta_{n-1}x_{1n-1}+\\epsilon_1\\\\\n", + "y_2&=\\beta_0x_{20}+\\beta_1x_{21}+\\beta_2x_{22}+\\dots+\\beta_{n-1}x_{2n-1}+\\epsilon_1\\\\\n", + "\\dots & \\dots \\\\\n", + "y_{i}&=\\beta_0x_{i0}+\\beta_1x_{i1}+\\beta_2x_{i2}+\\dots+\\beta_{n-1}x_{in-1}+\\epsilon_1\\\\\n", + "\\dots & \\dots \\\\\n", + "y_{n-1}&=\\beta_0x_{n-1,0}+\\beta_1x_{n-1,2}+\\beta_2x_{n-1,2}+\\dots+\\beta_{n-1}x_{n-1,n-1}+\\epsilon_{n-1}.\\\\\n", + "\\end{align*}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "As we noted above, we stayed with a system with the design matrix \n", + " $\\boldsymbol{X}\\in {\\mathbb{R}}^{n\\times n}$, that is we have $p=n$. For reasons to come later (algorithmic arguments) we will hereafter define \n", + "our matrix as $\\boldsymbol{X}\\in {\\mathbb{R}}^{n\\times p}$, with the predictors refering to the column numbers and the entries $n$ being the row elements.\n", + "\n", + "\n", + "\n", + "\n", + "## Our model for the nuclear binding energies\n", + "\n", + "In our [introductory notes](https://compphysics.github.io/MachineLearning/doc/pub/How2ReadData/html/How2ReadData.html) we looked at the so-called [liquid drop model](https://en.wikipedia.org/wiki/Semi-empirical_mass_formula). Let us remind ourselves about what we did by looking at the code.\n", + "\n", + "We restate the parts of the code we are most interested in." + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "%matplotlib inline\n", + "\n", + "# Common imports\n", + "import numpy as np\n", + "import pandas as pd\n", + "import matplotlib.pyplot as plt\n", + "from IPython.display import display\n", + "import os\n", + "\n", + "# Where to save the figures and data files\n", + "PROJECT_ROOT_DIR = \"Results\"\n", + "FIGURE_ID = \"Results/FigureFiles\"\n", + "DATA_ID = \"DataFiles/\"\n", + "\n", + "if not os.path.exists(PROJECT_ROOT_DIR):\n", + " os.mkdir(PROJECT_ROOT_DIR)\n", + "\n", + "if not os.path.exists(FIGURE_ID):\n", + " os.makedirs(FIGURE_ID)\n", + "\n", + "if not os.path.exists(DATA_ID):\n", + " os.makedirs(DATA_ID)\n", + "\n", + "def image_path(fig_id):\n", + " return os.path.join(FIGURE_ID, fig_id)\n", + "\n", + "def data_path(dat_id):\n", + " return os.path.join(DATA_ID, dat_id)\n", + "\n", + "def save_fig(fig_id):\n", + " plt.savefig(image_path(fig_id) + \".png\", format='png')\n", + "\n", + "infile = open(data_path(\"MassEval2016.dat\"),'r')\n", + "\n", + "\n", + "# Read the experimental data with Pandas\n", + "Masses = pd.read_fwf(infile, usecols=(2,3,4,6,11),\n", + " names=('N', 'Z', 'A', 'Element', 'Ebinding'),\n", + " widths=(1,3,5,5,5,1,3,4,1,13,11,11,9,1,2,11,9,1,3,1,12,11,1),\n", + " header=39,\n", + " index_col=False)\n", + "\n", + "# Extrapolated values are indicated by '#' in place of the decimal place, so\n", + "# the Ebinding column won't be numeric. Coerce to float and drop these entries.\n", + "Masses['Ebinding'] = pd.to_numeric(Masses['Ebinding'], errors='coerce')\n", + "Masses = Masses.dropna()\n", + "# Convert from keV to MeV.\n", + "Masses['Ebinding'] /= 1000\n", + "\n", + "# Group the DataFrame by nucleon number, A.\n", + "Masses = Masses.groupby('A')\n", + "# Find the rows of the grouped DataFrame with the maximum binding energy.\n", + "Masses = Masses.apply(lambda t: t[t.Ebinding==t.Ebinding.max()])\n", + "A = Masses['A']\n", + "Z = Masses['Z']\n", + "N = Masses['N']\n", + "Element = Masses['Element']\n", + "Energies = Masses['Ebinding']\n", + "\n", + "# Now we set up the design matrix X\n", + "X = np.zeros((len(A),5))\n", + "X[:,0] = 1\n", + "X[:,1] = A\n", + "X[:,2] = A**(2.0/3.0)\n", + "X[:,3] = A**(-1.0/3.0)\n", + "X[:,4] = A**(-1.0)\n", + "# Then nice printout using pandas\n", + "DesignMatrix = pd.DataFrame(X)\n", + "DesignMatrix.index = A\n", + "DesignMatrix.columns = ['1', 'A', 'A^(2/3)', 'A^(-1/3)', '1/A']\n", + "display(DesignMatrix)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "With $\\boldsymbol{\\beta}\\in {\\mathbb{R}}^{p\\times 1}$, it means that we will hereafter write our equations for the approximation as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{\\tilde{y}}= \\boldsymbol{X}\\boldsymbol{\\beta},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "throughout these lectures. \n", + "\n", + "\n", + "## Optimizing our parameters, more details\n", + "With the above we use the design matrix to define the approximation $\\boldsymbol{\\tilde{y}}$ via the unknown quantity $\\boldsymbol{\\beta}$ as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{\\tilde{y}}= \\boldsymbol{X}\\boldsymbol{\\beta},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and in order to find the optimal parameters $\\beta_i$ instead of solving the above linear algebra problem, we define a function which gives a measure of the spread between the values $y_i$ (which represent hopefully the exact values) and the parameterized values $\\tilde{y}_i$, namely" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "C(\\boldsymbol{\\beta})=\\frac{1}{n}\\sum_{i=0}^{n-1}\\left(y_i-\\tilde{y}_i\\right)^2=\\frac{1}{n}\\left\\{\\left(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}}\\right)^T\\left(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}}\\right)\\right\\},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "or using the matrix $\\boldsymbol{X}$ and in a more compact matrix-vector notation as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "C(\\boldsymbol{\\beta})=\\frac{1}{n}\\left\\{\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)^T\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)\\right\\}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This function is one possible way to define the so-called cost function.\n", + "\n", + "\n", + "\n", + "It is also common to define\n", + "the function $C$ as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "C(\\boldsymbol{\\beta})=\\frac{1}{2n}\\sum_{i=0}^{n-1}\\left(y_i-\\tilde{y}_i\\right)^2,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "since when taking the first derivative with respect to the unknown parameters $\\beta$, the factor of $2$ cancels out.\n", + "\n", + "\n", + "\n", + "\n", + "## Interpretations and optimizing our parameters\n", + "\n", + "The function" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "C(\\boldsymbol{\\beta})=\\frac{1}{n}\\left\\{\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)^T\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)\\right\\},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "can be linked to the variance of the quantity $y_i$ if we interpret the latter as the mean value. \n", + "When linking (see the discussion below) with the maximum likelihood approach below, we will indeed interpret $y_i$ as a mean value" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "y_{i}=\\langle y_i \\rangle = \\beta_0x_{i,0}+\\beta_1x_{i,1}+\\beta_2x_{i,2}+\\dots+\\beta_{n-1}x_{i,n-1}+\\epsilon_i,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $\\langle y_i \\rangle$ is the mean value. Keep in mind also that\n", + "till now we have treated $y_i$ as the exact value. Normally, the\n", + "response (dependent or outcome) variable $y_i$ the outcome of a\n", + "numerical experiment or another type of experiment and is thus only an\n", + "approximation to the true value. It is then always accompanied by an\n", + "error estimate, often limited to a statistical error estimate given by\n", + "the standard deviation discussed earlier. In the discussion here we\n", + "will treat $y_i$ as our exact value for the response variable.\n", + "\n", + "In order to find the parameters $\\beta_i$ we will then minimize the spread of $C(\\boldsymbol{\\beta})$, that is we are going to solve the problem" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "{\\displaystyle \\min_{\\boldsymbol{\\beta}\\in\n", + "{\\mathbb{R}}^{p}}}\\frac{1}{n}\\left\\{\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)^T\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)\\right\\}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "In practical terms it means we will require" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial C(\\boldsymbol{\\beta})}{\\partial \\beta_j} = \\frac{\\partial }{\\partial \\beta_j}\\left[ \\frac{1}{n}\\sum_{i=0}^{n-1}\\left(y_i-\\beta_0x_{i,0}-\\beta_1x_{i,1}-\\beta_2x_{i,2}-\\dots-\\beta_{n-1}x_{i,n-1}\\right)^2\\right]=0,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "which results in" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial C(\\boldsymbol{\\beta})}{\\partial \\beta_j} = -\\frac{2}{n}\\left[ \\sum_{i=0}^{n-1}x_{ij}\\left(y_i-\\beta_0x_{i,0}-\\beta_1x_{i,1}-\\beta_2x_{i,2}-\\dots-\\beta_{n-1}x_{i,n-1}\\right)\\right]=0,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "or in a matrix-vector form as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial C(\\boldsymbol{\\beta})}{\\partial \\boldsymbol{\\beta}} = 0 = \\boldsymbol{X}^T\\left( \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Interpretations and optimizing our parameters\n", + "We can rewrite" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial C(\\boldsymbol{\\beta})}{\\partial \\boldsymbol{\\beta}} = 0 = \\boldsymbol{X}^T\\left( \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{X}^T\\boldsymbol{y} = \\boldsymbol{X}^T\\boldsymbol{X}\\boldsymbol{\\beta},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and if the matrix $\\boldsymbol{X}^T\\boldsymbol{X}$ is invertible we have the solution" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{\\beta} =\\left(\\boldsymbol{X}^T\\boldsymbol{X}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We note also that since our design matrix is defined as $\\boldsymbol{X}\\in\n", + "{\\mathbb{R}}^{n\\times p}$, the product $\\boldsymbol{X}^T\\boldsymbol{X} \\in\n", + "{\\mathbb{R}}^{p\\times p}$. In the above case we have that $p \\ll n$,\n", + "in our case $p=5$ meaning that we end up with inverting a small\n", + "$5\\times 5$ matrix. This is a rather common situation, in many cases we end up with low-dimensional\n", + "matrices to invert. The methods discussed here and for many other\n", + "supervised learning algorithms like classification with logistic\n", + "regression or support vector machines, exhibit dimensionalities which\n", + "allow for the usage of direct linear algebra methods such as **LU** decomposition or **Singular Value Decomposition** (SVD) for finding the inverse of the matrix\n", + "$\\boldsymbol{X}^T\\boldsymbol{X}$.\n", + "\n", + "\n", + "\n", + "**Small question**: Do you think the example we have at hand here (the nuclear binding energies) can lead to problems in inverting the matrix $\\boldsymbol{X}^T\\boldsymbol{X}$? What kind of problems can we expect?\n", + "\n", + "\n", + "\n", + "## Some useful matrix and vector expressions\n", + "\n", + "The following matrix and vector relation will be useful here and for the rest of the course. Vectors are always written as boldfaced lower case letters and \n", + "matrices as upper case boldfaced letters." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "2\n", + "6\n", + " \n", + "<\n", + "<\n", + "<\n", + "!\n", + "!\n", + "M\n", + "A\n", + "T\n", + "H\n", + "_\n", + "B\n", + "L\n", + "O\n", + "C\n", + "K" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "2\n", + "7\n", + " \n", + "<\n", + "<\n", + "<\n", + "!\n", + "!\n", + "M\n", + "A\n", + "T\n", + "H\n", + "_\n", + "B\n", + "L\n", + "O\n", + "C\n", + "K" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "2\n", + "8\n", + " \n", + "<\n", + "<\n", + "<\n", + "!\n", + "!\n", + "M\n", + "A\n", + "T\n", + "H\n", + "_\n", + "B\n", + "L\n", + "O\n", + "C\n", + "K" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial \\log{\\vert\\boldsymbol{A}\\vert}}{\\partial \\boldsymbol{A}} = (\\boldsymbol{A}^{-1})^T.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Interpretations and optimizing our parameters\n", + "The residuals $\\boldsymbol{\\epsilon}$ are in turn given by" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{\\epsilon} = \\boldsymbol{y}-\\boldsymbol{\\tilde{y}} = \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and with" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{X}^T\\left( \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)= 0,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "we have" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{X}^T\\boldsymbol{\\epsilon}=\\boldsymbol{X}^T\\left( \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)= 0,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "meaning that the solution for $\\boldsymbol{\\beta}$ is the one which minimizes the residuals. Later we will link this with the maximum likelihood approach.\n", + "\n", + "\n", + "\n", + "\n", + "Let us now return to our nuclear binding energies and simply code the above equations. \n", + "\n", + "## Own code for Ordinary Least Squares\n", + "\n", + "It is rather straightforward to implement the matrix inversion and obtain the parameters $\\boldsymbol{\\beta}$. After having defined the matrix $\\boldsymbol{X}$ we simply need to \n", + "write" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "# matrix inversion to find beta\n", + "beta = np.linalg.inv(X.T.dot(X)).dot(X.T).dot(Energies)\n", + "# and then make the prediction\n", + "ytilde = X @ beta" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Alternatively, you can use the least squares functionality in **Numpy** as" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "fit = np.linalg.lstsq(X, Energies, rcond =None)[0]\n", + "ytildenp = np.dot(fit,X.T)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "And finally we plot our fit with and compare with data" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "Masses['Eapprox'] = ytilde\n", + "# Generate a plot comparing the experimental with the fitted values values.\n", + "fig, ax = plt.subplots()\n", + "ax.set_xlabel(r'$A = N + Z$')\n", + "ax.set_ylabel(r'$E_\\mathrm{bind}\\,/\\mathrm{MeV}$')\n", + "ax.plot(Masses['A'], Masses['Ebinding'], alpha=0.7, lw=2,\n", + " label='Ame2016')\n", + "ax.plot(Masses['A'], Masses['Eapprox'], alpha=0.7, lw=2, c='m',\n", + " label='Fit')\n", + "ax.legend()\n", + "save_fig(\"Masses2016OLS\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Adding error analysis and training set up\n", + "\n", + "We can easily test our fit by computing the $R2$ score that we discussed in connection with the functionality of **Scikit-Learn** in the introductory slides.\n", + "Since we are not using **Scikit-Learn** here we can define our own $R2$ function as" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "def R2(y_data, y_model):\n", + " return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and we would be using it as" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "print(R2(Energies,ytilde))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We can easily add our **MSE** score as" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "def MSE(y_data,y_model):\n", + " n = np.size(y_model)\n", + " return np.sum((y_data-y_model)**2)/n\n", + "\n", + "print(MSE(Energies,ytilde))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and finally the relative error as" + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "def RelativeError(y_data,y_model):\n", + " return abs((y_data-y_model)/y_data)\n", + "print(RelativeError(Energies, ytilde))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## The $\\chi^2$ function\n", + "\n", + "Normally, the response (dependent or outcome) variable $y_i$ is the\n", + "outcome of a numerical experiment or another type of experiment and is\n", + "thus only an approximation to the true value. It is then always\n", + "accompanied by an error estimate, often limited to a statistical error\n", + "estimate given by the standard deviation discussed earlier. In the\n", + "discussion here we will treat $y_i$ as our exact value for the\n", + "response variable.\n", + "\n", + "Introducing the standard deviation $\\sigma_i$ for each measurement\n", + "$y_i$, we define now the $\\chi^2$ function (omitting the $1/n$ term)\n", + "as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\chi^2(\\boldsymbol{\\beta})=\\frac{1}{n}\\sum_{i=0}^{n-1}\\frac{\\left(y_i-\\tilde{y}_i\\right)^2}{\\sigma_i^2}=\\frac{1}{n}\\left\\{\\left(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}}\\right)^T\\frac{1}{\\boldsymbol{\\Sigma^2}}\\left(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}}\\right)\\right\\},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where the matrix $\\boldsymbol{\\Sigma}$ is a diagonal matrix with $\\sigma_i$ as matrix elements.\n", + "\n", + "\n", + "\n", + "## The $\\chi^2$ function\n", + "\n", + "In order to find the parameters $\\beta_i$ we will then minimize the spread of $\\chi^2(\\boldsymbol{\\beta})$ by requiring" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial \\chi^2(\\boldsymbol{\\beta})}{\\partial \\beta_j} = \\frac{\\partial }{\\partial \\beta_j}\\left[ \\frac{1}{n}\\sum_{i=0}^{n-1}\\left(\\frac{y_i-\\beta_0x_{i,0}-\\beta_1x_{i,1}-\\beta_2x_{i,2}-\\dots-\\beta_{n-1}x_{i,n-1}}{\\sigma_i}\\right)^2\\right]=0,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "which results in" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial \\chi^2(\\boldsymbol{\\beta})}{\\partial \\beta_j} = -\\frac{2}{n}\\left[ \\sum_{i=0}^{n-1}\\frac{x_{ij}}{\\sigma_i}\\left(\\frac{y_i-\\beta_0x_{i,0}-\\beta_1x_{i,1}-\\beta_2x_{i,2}-\\dots-\\beta_{n-1}x_{i,n-1}}{\\sigma_i}\\right)\\right]=0,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "or in a matrix-vector form as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial \\chi^2(\\boldsymbol{\\beta})}{\\partial \\boldsymbol{\\beta}} = 0 = \\boldsymbol{A}^T\\left( \\boldsymbol{b}-\\boldsymbol{A}\\boldsymbol{\\beta}\\right).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where we have defined the matrix $\\boldsymbol{A} =\\boldsymbol{X}/\\boldsymbol{\\Sigma}$ with matrix elements $a_{ij} = x_{ij}/\\sigma_i$ and the vector $\\boldsymbol{b}$ with elements $b_i = y_i/\\sigma_i$.\n", + "\n", + "\n", + "\n", + "## The $\\chi^2$ function\n", + "\n", + "We can rewrite" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial \\chi^2(\\boldsymbol{\\beta})}{\\partial \\boldsymbol{\\beta}} = 0 = \\boldsymbol{A}^T\\left( \\boldsymbol{b}-\\boldsymbol{A}\\boldsymbol{\\beta}\\right),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{A}^T\\boldsymbol{b} = \\boldsymbol{A}^T\\boldsymbol{A}\\boldsymbol{\\beta},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and if the matrix $\\boldsymbol{A}^T\\boldsymbol{A}$ is invertible we have the solution" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{\\beta} =\\left(\\boldsymbol{A}^T\\boldsymbol{A}\\right)^{-1}\\boldsymbol{A}^T\\boldsymbol{b}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## The $\\chi^2$ function\n", + "\n", + "If we then introduce the matrix" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{H} = \\left(\\boldsymbol{A}^T\\boldsymbol{A}\\right)^{-1},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "we have then the following expression for the parameters $\\beta_j$ (the matrix elements of $\\boldsymbol{H}$ are $h_{ij}$)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\beta_j = \\sum_{k=0}^{p-1}h_{jk}\\sum_{i=0}^{n-1}\\frac{y_i}{\\sigma_i}\\frac{x_{ik}}{\\sigma_i} = \\sum_{k=0}^{p-1}h_{jk}\\sum_{i=0}^{n-1}b_ia_{ik}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We state without proof the expression for the uncertainty in the parameters $\\beta_j$ as (we leave this as an exercise)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\sigma^2(\\beta_j) = \\sum_{i=0}^{n-1}\\sigma_i^2\\left( \\frac{\\partial \\beta_j}{\\partial y_i}\\right)^2,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "resulting in" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\sigma^2(\\beta_j) = \\left(\\sum_{k=0}^{p-1}h_{jk}\\sum_{i=0}^{n-1}a_{ik}\\right)\\left(\\sum_{l=0}^{p-1}h_{jl}\\sum_{m=0}^{n-1}a_{ml}\\right) = h_{jj}!\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## The $\\chi^2$ function\n", + "The first step here is to approximate the function $y$ with a first-order polynomial, that is we write" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "y=y(x) \\rightarrow y(x_i) \\approx \\beta_0+\\beta_1 x_i.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "By computing the derivatives of $\\chi^2$ with respect to $\\beta_0$ and $\\beta_1$ show that these are given by" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial \\chi^2(\\boldsymbol{\\beta})}{\\partial \\beta_0} = -2\\left[ \\frac{1}{n}\\sum_{i=0}^{n-1}\\left(\\frac{y_i-\\beta_0-\\beta_1x_{i}}{\\sigma_i^2}\\right)\\right]=0,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial \\chi^2(\\boldsymbol{\\beta})}{\\partial \\beta_1} = -\\frac{2}{n}\\left[ \\sum_{i=0}^{n-1}x_i\\left(\\frac{y_i-\\beta_0-\\beta_1x_{i}}{\\sigma_i^2}\\right)\\right]=0.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## The $\\chi^2$ function\n", + "\n", + "For a linear fit (a first-order polynomial) we don't need to invert a matrix!! \n", + "Defining" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\gamma = \\sum_{i=0}^{n-1}\\frac{1}{\\sigma_i^2},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\gamma_x = \\sum_{i=0}^{n-1}\\frac{x_{i}}{\\sigma_i^2},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\gamma_y = \\sum_{i=0}^{n-1}\\left(\\frac{y_i}{\\sigma_i^2}\\right),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\gamma_{xx} = \\sum_{i=0}^{n-1}\\frac{x_ix_{i}}{\\sigma_i^2},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\gamma_{xy} = \\sum_{i=0}^{n-1}\\frac{y_ix_{i}}{\\sigma_i^2},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "we obtain" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\beta_0 = \\frac{\\gamma_{xx}\\gamma_y-\\gamma_x\\gamma_y}{\\gamma\\gamma_{xx}-\\gamma_x^2},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\beta_1 = \\frac{\\gamma_{xy}\\gamma-\\gamma_x\\gamma_y}{\\gamma\\gamma_{xx}-\\gamma_x^2}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This approach (different linear and non-linear regression) suffers\n", + "often from both being underdetermined and overdetermined in the\n", + "unknown coefficients $\\beta_i$. A better approach is to use the\n", + "Singular Value Decomposition (SVD) method discussed below. Or using\n", + "Lasso and Ridge regression. See below.\n", + "\n", + "\n", + "\n", + "\n", + "## Fitting an Equation of State for Dense Nuclear Matter\n", + "\n", + "Before we continue, let us introduce yet another example. We are going to fit the\n", + "nuclear equation of state using results from many-body calculations.\n", + "The equation of state we have made available here, as function of\n", + "density, has been derived using modern nucleon-nucleon potentials with\n", + "[the addition of three-body\n", + "forces](https://www.sciencedirect.com/science/article/pii/S0370157399001106). This\n", + "time the file is presented as a standard **csv** file.\n", + "\n", + "The beginning of the Python code here is similar to what you have seen\n", + "before, with the same initializations and declarations. We use also\n", + "**pandas** again, rather extensively in order to organize our data.\n", + "\n", + "The difference now is that we use **Scikit-Learn's** regression tools\n", + "instead of our own matrix inversion implementation. Furthermore, we\n", + "sneak in **Ridge** regression (to be discussed below) which includes a\n", + "hyperparameter $\\lambda$, also to be explained below.\n", + "\n", + "## The code" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "# Common imports\n", + "import os\n", + "import numpy as np\n", + "import pandas as pd\n", + "import matplotlib.pyplot as plt\n", + "import matplotlib.pyplot as plt\n", + "import sklearn.linear_model as skl\n", + "from sklearn.metrics import mean_squared_error, r2_score, mean_absolute_error\n", + "\n", + "# Where to save the figures and data files\n", + "PROJECT_ROOT_DIR = \"Results\"\n", + "FIGURE_ID = \"Results/FigureFiles\"\n", + "DATA_ID = \"DataFiles/\"\n", + "\n", + "if not os.path.exists(PROJECT_ROOT_DIR):\n", + " os.mkdir(PROJECT_ROOT_DIR)\n", + "\n", + "if not os.path.exists(FIGURE_ID):\n", + " os.makedirs(FIGURE_ID)\n", + "\n", + "if not os.path.exists(DATA_ID):\n", + " os.makedirs(DATA_ID)\n", + "\n", + "def image_path(fig_id):\n", + " return os.path.join(FIGURE_ID, fig_id)\n", + "\n", + "def data_path(dat_id):\n", + " return os.path.join(DATA_ID, dat_id)\n", + "\n", + "def save_fig(fig_id):\n", + " plt.savefig(image_path(fig_id) + \".png\", format='png')\n", + "\n", + "infile = open(data_path(\"EoS.csv\"),'r')\n", + "\n", + "# Read the EoS data as csv file and organize the data into two arrays with density and energies\n", + "EoS = pd.read_csv(infile, names=('Density', 'Energy'))\n", + "EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce')\n", + "EoS = EoS.dropna()\n", + "Energies = EoS['Energy']\n", + "Density = EoS['Density']\n", + "# The design matrix now as function of various polytrops\n", + "X = np.zeros((len(Density),4))\n", + "X[:,3] = Density**(4.0/3.0)\n", + "X[:,2] = Density\n", + "X[:,1] = Density**(2.0/3.0)\n", + "X[:,0] = 1\n", + "\n", + "# We use now Scikit-Learn's linear regressor and ridge regressor\n", + "# OLS part\n", + "clf = skl.LinearRegression().fit(X, Energies)\n", + "ytilde = clf.predict(X)\n", + "EoS['Eols'] = ytilde\n", + "# The mean squared error \n", + "print(\"Mean squared error: %.2f\" % mean_squared_error(Energies, ytilde))\n", + "# Explained variance score: 1 is perfect prediction \n", + "print('Variance score: %.2f' % r2_score(Energies, ytilde))\n", + "# Mean absolute error \n", + "print('Mean absolute error: %.2f' % mean_absolute_error(Energies, ytilde))\n", + "print(clf.coef_, clf.intercept_)\n", + "\n", + "# The Ridge regression with a hyperparameter lambda = 0.1\n", + "_lambda = 0.1\n", + "clf_ridge = skl.Ridge(alpha=_lambda).fit(X, Energies)\n", + "yridge = clf_ridge.predict(X)\n", + "EoS['Eridge'] = yridge\n", + "# The mean squared error \n", + "print(\"Mean squared error: %.2f\" % mean_squared_error(Energies, yridge))\n", + "# Explained variance score: 1 is perfect prediction \n", + "print('Variance score: %.2f' % r2_score(Energies, yridge))\n", + "# Mean absolute error \n", + "print('Mean absolute error: %.2f' % mean_absolute_error(Energies, yridge))\n", + "print(clf_ridge.coef_, clf_ridge.intercept_)\n", + "\n", + "fig, ax = plt.subplots()\n", + "ax.set_xlabel(r'$\\rho[\\mathrm{fm}^{-3}]$')\n", + "ax.set_ylabel(r'Energy per particle')\n", + "ax.plot(EoS['Density'], EoS['Energy'], alpha=0.7, lw=2,\n", + " label='Theoretical data')\n", + "ax.plot(EoS['Density'], EoS['Eols'], alpha=0.7, lw=2, c='m',\n", + " label='OLS')\n", + "ax.plot(EoS['Density'], EoS['Eridge'], alpha=0.7, lw=2, c='g',\n", + " label='Ridge $\\lambda = 0.1$')\n", + "ax.legend()\n", + "save_fig(\"EoSfitting\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The above simple polynomial in density $\\rho$ gives an excellent fit\n", + "to the data. \n", + "\n", + "We note also that there is a small deviation between the\n", + "standard OLS and the Ridge regression at higher densities. We discuss this in more detail\n", + "below.\n", + "\n", + "\n", + "## Splitting our Data in Training and Test data\n", + "\n", + "It is normal in essentially all Machine Learning studies to split the\n", + "data in a training set and a test set (sometimes also an additional\n", + "validation set). **Scikit-Learn** has an own function for this. There\n", + "is no explicit recipe for how much data should be included as training\n", + "data and say test data. An accepted rule of thumb is to use\n", + "approximately $2/3$ to $4/5$ of the data as training data. We will\n", + "postpone a discussion of this splitting to the end of these notes and\n", + "our discussion of the so-called **bias-variance** tradeoff. Here we\n", + "limit ourselves to repeat the above equation of state fitting example\n", + "but now splitting the data into a training set and a test set." + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "import os\n", + "import numpy as np\n", + "import pandas as pd\n", + "import matplotlib.pyplot as plt\n", + "from sklearn.model_selection import train_test_split\n", + "# Where to save the figures and data files\n", + "PROJECT_ROOT_DIR = \"Results\"\n", + "FIGURE_ID = \"Results/FigureFiles\"\n", + "DATA_ID = \"DataFiles/\"\n", + "\n", + "if not os.path.exists(PROJECT_ROOT_DIR):\n", + " os.mkdir(PROJECT_ROOT_DIR)\n", + "\n", + "if not os.path.exists(FIGURE_ID):\n", + " os.makedirs(FIGURE_ID)\n", + "\n", + "if not os.path.exists(DATA_ID):\n", + " os.makedirs(DATA_ID)\n", + "\n", + "def image_path(fig_id):\n", + " return os.path.join(FIGURE_ID, fig_id)\n", + "\n", + "def data_path(dat_id):\n", + " return os.path.join(DATA_ID, dat_id)\n", + "\n", + "def save_fig(fig_id):\n", + " plt.savefig(image_path(fig_id) + \".png\", format='png')\n", + "\n", + "def R2(y_data, y_model):\n", + " return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)\n", + "def MSE(y_data,y_model):\n", + " n = np.size(y_model)\n", + " return np.sum((y_data-y_model)**2)/n\n", + "\n", + "infile = open(data_path(\"EoS.csv\"),'r')\n", + "\n", + "# Read the EoS data as csv file and organized into two arrays with density and energies\n", + "EoS = pd.read_csv(infile, names=('Density', 'Energy'))\n", + "EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce')\n", + "EoS = EoS.dropna()\n", + "Energies = EoS['Energy']\n", + "Density = EoS['Density']\n", + "# The design matrix now as function of various polytrops\n", + "X = np.zeros((len(Density),5))\n", + "X[:,0] = 1\n", + "X[:,1] = Density**(2.0/3.0)\n", + "X[:,2] = Density\n", + "X[:,3] = Density**(4.0/3.0)\n", + "X[:,4] = Density**(5.0/3.0)\n", + "# We split the data in test and training data\n", + "X_train, X_test, y_train, y_test = train_test_split(X, Energies, test_size=0.2)\n", + "# matrix inversion to find beta\n", + "beta = np.linalg.inv(X_train.T.dot(X_train)).dot(X_train.T).dot(y_train)\n", + "# and then make the prediction\n", + "ytilde = X_train @ beta\n", + "print(\"Training R2\")\n", + "print(R2(y_train,ytilde))\n", + "print(\"Training MSE\")\n", + "print(MSE(y_train,ytilde))\n", + "ypredict = X_test @ beta\n", + "print(\"Test R2\")\n", + "print(R2(y_test,ypredict))\n", + "print(\"Test MSE\")\n", + "print(MSE(y_test,ypredict))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "## The Boston housing data example\n", + "\n", + "The Boston housing \n", + "data set was originally a part of UCI Machine Learning Repository\n", + "and has been removed now. The data set is now included in **Scikit-Learn**'s \n", + "library. There are 506 samples and 13 feature (predictor) variables\n", + "in this data set. The objective is to predict the value of prices of\n", + "the house using the features (predictors) listed here.\n", + "\n", + "The features/predictors are\n", + "1. CRIM: Per capita crime rate by town\n", + "\n", + "2. ZN: Proportion of residential land zoned for lots over 25000 square feet\n", + "\n", + "3. INDUS: Proportion of non-retail business acres per town\n", + "\n", + "4. CHAS: Charles River dummy variable (= 1 if tract bounds river; 0 otherwise)\n", + "\n", + "5. NOX: Nitric oxide concentration (parts per 10 million)\n", + "\n", + "6. RM: Average number of rooms per dwelling\n", + "\n", + "7. AGE: Proportion of owner-occupied units built prior to 1940\n", + "\n", + "8. DIS: Weighted distances to five Boston employment centers\n", + "\n", + "9. RAD: Index of accessibility to radial highways\n", + "\n", + "10. TAX: Full-value property tax rate per USD10000\n", + "\n", + "11. B: $1000(Bk - 0.63)^2$, where $Bk$ is the proportion of [people of African American descent] by town\n", + "\n", + "12. LSTAT: Percentage of lower status of the population\n", + "\n", + "13. MEDV: Median value of owner-occupied homes in USD 1000s\n", + "\n", + "## Housing data, the code\n", + "We start by importing the libraries" + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "import numpy as np\n", + "import matplotlib.pyplot as plt \n", + "\n", + "import pandas as pd \n", + "import seaborn as sns" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and load the Boston Housing DataSet from **Scikit-Learn**" + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "from sklearn.datasets import load_boston\n", + "\n", + "boston_dataset = load_boston()\n", + "\n", + "# boston_dataset is a dictionary\n", + "# let's check what it contains\n", + "boston_dataset.keys()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Then we invoke Pandas" + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "boston = pd.DataFrame(boston_dataset.data, columns=boston_dataset.feature_names)\n", + "boston.head()\n", + "boston['MEDV'] = boston_dataset.target" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and preprocess the data" + ] + }, + { + "cell_type": "code", + "execution_count": 14, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "# check for missing values in all the columns\n", + "boston.isnull().sum()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We can then visualize the data" + ] + }, + { + "cell_type": "code", + "execution_count": 15, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "# set the size of the figure\n", + "sns.set(rc={'figure.figsize':(11.7,8.27)})\n", + "\n", + "# plot a histogram showing the distribution of the target values\n", + "sns.distplot(boston['MEDV'], bins=30)\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "It is now useful to look at the correlation matrix" + ] + }, + { + "cell_type": "code", + "execution_count": 16, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "# compute the pair wise correlation for all columns \n", + "correlation_matrix = boston.corr().round(2)\n", + "# use the heatmap function from seaborn to plot the correlation matrix\n", + "# annot = True to print the values inside the square\n", + "sns.heatmap(data=correlation_matrix, annot=True)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "From the above coorelation plot we can see that **MEDV** is strongly correlated to **LSTAT** and **RM**. We see also that **RAD** and **TAX** are stronly correlated, but we don't include this in our features together to avoid multi-colinearity" + ] + }, + { + "cell_type": "code", + "execution_count": 17, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "plt.figure(figsize=(20, 5))\n", + "\n", + "features = ['LSTAT', 'RM']\n", + "target = boston['MEDV']\n", + "\n", + "for i, col in enumerate(features):\n", + " plt.subplot(1, len(features) , i+1)\n", + " x = boston[col]\n", + " y = target\n", + " plt.scatter(x, y, marker='o')\n", + " plt.title(col)\n", + " plt.xlabel(col)\n", + " plt.ylabel('MEDV')" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Now we start training our model" + ] + }, + { + "cell_type": "code", + "execution_count": 18, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "X = pd.DataFrame(np.c_[boston['LSTAT'], boston['RM']], columns = ['LSTAT','RM'])\n", + "Y = boston['MEDV']" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We split the data into training and test sets" + ] + }, + { + "cell_type": "code", + "execution_count": 19, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "from sklearn.model_selection import train_test_split\n", + "\n", + "# splits the training and test data set in 80% : 20%\n", + "# assign random_state to any value.This ensures consistency.\n", + "X_train, X_test, Y_train, Y_test = train_test_split(X, Y, test_size = 0.2, random_state=5)\n", + "print(X_train.shape)\n", + "print(X_test.shape)\n", + "print(Y_train.shape)\n", + "print(Y_test.shape)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Then we use the linear regression functionality from **Scikit-Learn**" + ] + }, + { + "cell_type": "code", + "execution_count": 20, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "from sklearn.linear_model import LinearRegression\n", + "from sklearn.metrics import mean_squared_error, r2_score\n", + "\n", + "lin_model = LinearRegression()\n", + "lin_model.fit(X_train, Y_train)\n", + "\n", + "# model evaluation for training set\n", + "\n", + "y_train_predict = lin_model.predict(X_train)\n", + "rmse = (np.sqrt(mean_squared_error(Y_train, y_train_predict)))\n", + "r2 = r2_score(Y_train, y_train_predict)\n", + "\n", + "print(\"The model performance for training set\")\n", + "print(\"--------------------------------------\")\n", + "print('RMSE is {}'.format(rmse))\n", + "print('R2 score is {}'.format(r2))\n", + "print(\"\\n\")\n", + "\n", + "# model evaluation for testing set\n", + "\n", + "y_test_predict = lin_model.predict(X_test)\n", + "# root mean square error of the model\n", + "rmse = (np.sqrt(mean_squared_error(Y_test, y_test_predict)))\n", + "\n", + "# r-squared score of the model\n", + "r2 = r2_score(Y_test, y_test_predict)\n", + "\n", + "print(\"The model performance for testing set\")\n", + "print(\"--------------------------------------\")\n", + "print('RMSE is {}'.format(rmse))\n", + "print('R2 score is {}'.format(r2))" + ] + }, + { + "cell_type": "code", + "execution_count": 21, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "# plotting the y_test vs y_pred\n", + "# ideally should have been a straight line\n", + "plt.scatter(Y_test, y_test_predict)\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Reducing the number of degrees of freedom, overarching view\n", + "\n", + "Many Machine Learning problems involve thousands or even millions of\n", + "features for each training instance. Not only does this make training\n", + "extremely slow, it can also make it much harder to find a good\n", + "solution, as we will see. This problem is often referred to as the\n", + "curse of dimensionality. Fortunately, in real-world problems, it is\n", + "often possible to reduce the number of features considerably, turning\n", + "an intractable problem into a tractable one.\n", + "\n", + "Later we will discuss some of the most popular dimensionality reduction\n", + "techniques: the principal component analysis (PCA), Kernel PCA, and\n", + "Locally Linear Embedding (LLE). \n", + "\n", + "\n", + "Principal component analysis and its various variants deal with the\n", + "problem of fitting a low-dimensional [affine\n", + "subspace](https://en.wikipedia.org/wiki/Affine_space) to a set of of\n", + "data points in a high-dimensional space. With its family of methods it\n", + "is one of the most used tools in data modeling, compression and\n", + "visualization.\n", + "\n", + "\n", + "\n", + "\n", + "## Preprocessing our data\n", + "\n", + "Before we proceed however, we will discuss how to preprocess our\n", + "data. Till now and in connection with our previous examples we have\n", + "not met so many cases where we are too sensitive to the scaling of our\n", + "data. Normally the data may need a rescaling and/or may be sensitive\n", + "to extreme values. Scaling the data renders our inputs much more\n", + "suitable for the algorithms we want to employ.\n", + "\n", + "**Scikit-Learn** has several functions which allow us to rescale the\n", + "data, normally resulting in much better results in terms of various\n", + "accuracy scores. The **StandardScaler** function in **Scikit-Learn**\n", + "ensures that for each feature/predictor we study the mean value is\n", + "zero and the variance is one (every column in the design/feature\n", + "matrix). This scaling has the drawback that it does not ensure that\n", + "we have a particular maximum or minimum in our data set. Another\n", + "function included in **Scikit-Learn** is the **MinMaxScaler** which\n", + "ensures that all features are exactly between $0$ and $1$. The\n", + "\n", + "## More preprocessing\n", + "\n", + "\n", + "The **Normalizer** scales each data\n", + "point such that the feature vector has a euclidean length of one. In other words, it\n", + "projects a data point on the circle (or sphere in the case of higher dimensions) with a\n", + "radius of 1. This means every data point is scaled by a different number (by the\n", + "inverse of it’s length).\n", + "This normalization is often used when only the direction (or angle) of the data matters,\n", + "not the length of the feature vector.\n", + "\n", + "The **RobustScaler** works similarly to the StandardScaler in that it\n", + "ensures statistical properties for each feature that guarantee that\n", + "they are on the same scale. However, the RobustScaler uses the median\n", + "and quartiles, instead of mean and variance. This makes the\n", + "RobustScaler ignore data points that are very different from the rest\n", + "(like measurement errors). These odd data points are also called\n", + "outliers, and might often lead to trouble for other scaling\n", + "techniques.\n", + "\n", + "\n", + "\n", + "## Simple preprocessing examples, Franke function and regression" + ] + }, + { + "cell_type": "code", + "execution_count": 22, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "# Common imports\n", + "import os\n", + "import numpy as np\n", + "import pandas as pd\n", + "import matplotlib.pyplot as plt\n", + "import sklearn.linear_model as skl\n", + "from sklearn.metrics import mean_squared_error\n", + "from sklearn.model_selection import train_test_split\n", + "from sklearn.preprocessing import MinMaxScaler, StandardScaler, Normalizer\n", + "\n", + "# Where to save the figures and data files\n", + "PROJECT_ROOT_DIR = \"Results\"\n", + "FIGURE_ID = \"Results/FigureFiles\"\n", + "DATA_ID = \"DataFiles/\"\n", + "\n", + "if not os.path.exists(PROJECT_ROOT_DIR):\n", + " os.mkdir(PROJECT_ROOT_DIR)\n", + "\n", + "if not os.path.exists(FIGURE_ID):\n", + " os.makedirs(FIGURE_ID)\n", + "\n", + "if not os.path.exists(DATA_ID):\n", + " os.makedirs(DATA_ID)\n", + "\n", + "def image_path(fig_id):\n", + " return os.path.join(FIGURE_ID, fig_id)\n", + "\n", + "def data_path(dat_id):\n", + " return os.path.join(DATA_ID, dat_id)\n", + "\n", + "def save_fig(fig_id):\n", + " plt.savefig(image_path(fig_id) + \".png\", format='png')\n", + "\n", + "\n", + "def FrankeFunction(x,y):\n", + "\tterm1 = 0.75*np.exp(-(0.25*(9*x-2)**2) - 0.25*((9*y-2)**2))\n", + "\tterm2 = 0.75*np.exp(-((9*x+1)**2)/49.0 - 0.1*(9*y+1))\n", + "\tterm3 = 0.5*np.exp(-(9*x-7)**2/4.0 - 0.25*((9*y-3)**2))\n", + "\tterm4 = -0.2*np.exp(-(9*x-4)**2 - (9*y-7)**2)\n", + "\treturn term1 + term2 + term3 + term4\n", + "\n", + "\n", + "def create_X(x, y, n ):\n", + "\tif len(x.shape) > 1:\n", + "\t\tx = np.ravel(x)\n", + "\t\ty = np.ravel(y)\n", + "\n", + "\tN = len(x)\n", + "\tl = int((n+1)*(n+2)/2)\t\t# Number of elements in beta\n", + "\tX = np.ones((N,l))\n", + "\n", + "\tfor i in range(1,n+1):\n", + "\t\tq = int((i)*(i+1)/2)\n", + "\t\tfor k in range(i+1):\n", + "\t\t\tX[:,q+k] = (x**(i-k))*(y**k)\n", + "\n", + "\treturn X\n", + "\n", + "\n", + "# Making meshgrid of datapoints and compute Franke's function\n", + "n = 5\n", + "N = 1000\n", + "x = np.sort(np.random.uniform(0, 1, N))\n", + "y = np.sort(np.random.uniform(0, 1, N))\n", + "z = FrankeFunction(x, y)\n", + "X = create_X(x, y, n=n) \n", + "# split in training and test data\n", + "X_train, X_test, y_train, y_test = train_test_split(X,z,test_size=0.2)\n", + "\n", + "\n", + "clf = skl.LinearRegression().fit(X_train, y_train)\n", + "\n", + "# The mean squared error and R2 score\n", + "print(\"MSE before scaling: {:.2f}\".format(mean_squared_error(clf.predict(X_test), y_test)))\n", + "print(\"R2 score before scaling {:.2f}\".format(clf.score(X_test,y_test)))\n", + "\n", + "scaler = StandardScaler()\n", + "scaler.fit(X_train)\n", + "X_train_scaled = scaler.transform(X_train)\n", + "X_test_scaled = scaler.transform(X_test)\n", + "\n", + "print(\"Feature min values before scaling:\\n {}\".format(X_train.min(axis=0)))\n", + "print(\"Feature max values before scaling:\\n {}\".format(X_train.max(axis=0)))\n", + "\n", + "print(\"Feature min values after scaling:\\n {}\".format(X_train_scaled.min(axis=0)))\n", + "print(\"Feature max values after scaling:\\n {}\".format(X_train_scaled.max(axis=0)))\n", + "\n", + "clf = skl.LinearRegression().fit(X_train_scaled, y_train)\n", + "\n", + "\n", + "print(\"MSE after scaling: {:.2f}\".format(mean_squared_error(clf.predict(X_test_scaled), y_test)))\n", + "print(\"R2 score for scaled data: {:.2f}\".format(clf.score(X_test_scaled,y_test)))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## The singular value decomposition\n", + "\n", + "\n", + "The examples we have looked at so far are cases where we normally can\n", + "invert the matrix $\\boldsymbol{X}^T\\boldsymbol{X}$. Using a polynomial expansion as we\n", + "did both for the masses and the fitting of the equation of state,\n", + "leads to row vectors of the design matrix which are essentially\n", + "orthogonal due to the polynomial character of our model. Obtaining the inverse of the design matrix is then often done via a so-called LU, QR or Cholesky decomposition. \n", + "\n", + "\n", + "\n", + "This may\n", + "however not the be case in general and a standard matrix inversion\n", + "algorithm based on say LU, QR or Cholesky decomposition may lead to singularities. We will see examples of this below.\n", + "\n", + "There is however a way to partially circumvent this problem and also gain some insights about the ordinary least squares approach, and later shrinkage methods like Ridge and Lasso regressions. \n", + "\n", + "This is given by the **Singular Value Decomposition** algorithm, perhaps\n", + "the most powerful linear algebra algorithm. Let us look at a\n", + "different example where we may have problems with the standard matrix\n", + "inversion algorithm. Thereafter we dive into the math of the SVD.\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "## Linear Regression Problems\n", + "\n", + "One of the typical problems we encounter with linear regression, in particular \n", + "when the matrix $\\boldsymbol{X}$ (our so-called design matrix) is high-dimensional, \n", + "are problems with near singular or singular matrices. The column vectors of $\\boldsymbol{X}$ \n", + "may be linearly dependent, normally referred to as super-collinearity. \n", + "This means that the matrix may be rank deficient and it is basically impossible to \n", + "to model the data using linear regression. As an example, consider the matrix" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\begin{align*}\n", + "\\mathbf{X} & = \\left[\n", + "\\begin{array}{rrr}\n", + "1 & -1 & 2\n", + "\\\\\n", + "1 & 0 & 1\n", + "\\\\\n", + "1 & 2 & -1\n", + "\\\\\n", + "1 & 1 & 0\n", + "\\end{array} \\right]\n", + "\\end{align*}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The columns of $\\boldsymbol{X}$ are linearly dependent. We see this easily since the \n", + "the first column is the row-wise sum of the other two columns. The rank (more correct,\n", + "the column rank) of a matrix is the dimension of the space spanned by the\n", + "column vectors. Hence, the rank of $\\mathbf{X}$ is equal to the number\n", + "of linearly independent columns. In this particular case the matrix has rank 2.\n", + "\n", + "Super-collinearity of an $(n \\times p)$-dimensional design matrix $\\mathbf{X}$ implies\n", + "that the inverse of the matrix $\\boldsymbol{X}^T\\boldsymbol{X}$ (the matrix we need to invert to solve the linear regression equations) is non-invertible. If we have a square matrix that does not have an inverse, we say this matrix singular. The example here demonstrates this" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\begin{align*}\n", + "\\boldsymbol{X} & = \\left[\n", + "\\begin{array}{rr}\n", + "1 & -1\n", + "\\\\\n", + "1 & -1\n", + "\\end{array} \\right].\n", + "\\end{align*}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We see easily that $\\mbox{det}(\\boldsymbol{X}) = x_{11} x_{22} - x_{12} x_{21} = 1 \\times (-1) - 1 \\times (-1) = 0$. Hence, $\\mathbf{X}$ is singular and its inverse is undefined.\n", + "This is equivalent to saying that the matrix $\\boldsymbol{X}$ has at least an eigenvalue which is zero.\n", + "\n", + "\n", + "## Fixing the singularity\n", + "\n", + "If our design matrix $\\boldsymbol{X}$ which enters the linear regression problem" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + "\\boldsymbol{\\beta} = (\\boldsymbol{X}^{T} \\boldsymbol{X})^{-1} \\boldsymbol{X}^{T} \\boldsymbol{y},\n", + "\\label{_auto1} \\tag{1}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "has linearly dependent column vectors, we will not be able to compute the inverse\n", + "of $\\boldsymbol{X}^T\\boldsymbol{X}$ and we cannot find the parameters (estimators) $\\beta_i$. \n", + "The estimators are only well-defined if $(\\boldsymbol{X}^{T}\\boldsymbol{X})^{-1}$ exits. \n", + "This is more likely to happen when the matrix $\\boldsymbol{X}$ is high-dimensional. In this case it is likely to encounter a situation where \n", + "the regression parameters $\\beta_i$ cannot be estimated.\n", + "\n", + "A cheap *ad hoc* approach is simply to add a small diagonal component to the matrix to invert, that is we change" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{X}^{T} \\boldsymbol{X} \\rightarrow \\boldsymbol{X}^{T} \\boldsymbol{X}+\\lambda \\boldsymbol{I},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $\\boldsymbol{I}$ is the identity matrix. When we discuss **Ridge** regression this is actually what we end up evaluating. The parameter $\\lambda$ is called a hyperparameter. More about this later. \n", + "\n", + "\n", + "\n", + "## Basic math of the SVD\n", + "\n", + "\n", + "From standard linear algebra we know that a square matrix $\\boldsymbol{X}$ can be diagonalized if and only it is \n", + "a so-called [normal matrix](https://en.wikipedia.org/wiki/Normal_matrix), that is if $\\boldsymbol{X}\\in {\\mathbb{R}}^{n\\times n}$\n", + "we have $\\boldsymbol{X}\\boldsymbol{X}^T=\\boldsymbol{X}^T\\boldsymbol{X}$ or if $\\boldsymbol{X}\\in {\\mathbb{C}}^{n\\times n}$ we have $\\boldsymbol{X}\\boldsymbol{X}^{\\dagger}=\\boldsymbol{X}^{\\dagger}\\boldsymbol{X}$.\n", + "The matrix has then a set of eigenpairs" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "(\\lambda_1,\\boldsymbol{u}_1),\\dots, (\\lambda_n,\\boldsymbol{u}_n),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and the eigenvalues are given by the diagonal matrix" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{\\Sigma}=\\mathrm{Diag}(\\lambda_1, \\dots,\\lambda_n).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The matrix $\\boldsymbol{X}$ can be written in terms of an orthogonal/unitary transformation $\\boldsymbol{U}$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{X} = \\boldsymbol{U}\\boldsymbol{\\Sigma}\\boldsymbol{V}^T,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "with $\\boldsymbol{U}\\boldsymbol{U}^T=\\boldsymbol{I}$ or $\\boldsymbol{U}\\boldsymbol{U}^{\\dagger}=\\boldsymbol{I}$.\n", + "\n", + "Not all square matrices are diagonalizable. A matrix like the one discussed above" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{X} = \\begin{bmatrix} \n", + "1& -1 \\\\\n", + "1& -1\\\\\n", + "\\end{bmatrix}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "is not diagonalizable, it is a so-called [defective matrix](https://en.wikipedia.org/wiki/Defective_matrix). It is easy to see that the condition\n", + "$\\boldsymbol{X}\\boldsymbol{X}^T=\\boldsymbol{X}^T\\boldsymbol{X}$ is not fulfilled. \n", + "\n", + "\n", + "## The SVD, a Fantastic Algorithm\n", + "\n", + "\n", + "However, and this is the strength of the SVD algorithm, any general\n", + "matrix $\\boldsymbol{X}$ can be decomposed in terms of a diagonal matrix and\n", + "two orthogonal/unitary matrices. The [Singular Value Decompostion\n", + "(SVD) theorem](https://en.wikipedia.org/wiki/Singular_value_decomposition)\n", + "states that a general $m\\times n$ matrix $\\boldsymbol{X}$ can be written in\n", + "terms of a diagonal matrix $\\boldsymbol{\\Sigma}$ of dimensionality $m\\times n$\n", + "and two orthognal matrices $\\boldsymbol{U}$ and $\\boldsymbol{V}$, where the first has\n", + "dimensionality $m \\times m$ and the last dimensionality $n\\times n$.\n", + "We have then" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{X} = \\boldsymbol{U}\\boldsymbol{\\Sigma}\\boldsymbol{V}^T\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "As an example, the above defective matrix can be decomposed as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{X} = \\frac{1}{\\sqrt{2}}\\begin{bmatrix} 1& 1 \\\\ 1& -1\\\\ \\end{bmatrix} \\begin{bmatrix} 2& 0 \\\\ 0& 0\\\\ \\end{bmatrix} \\frac{1}{\\sqrt{2}}\\begin{bmatrix} 1& -1 \\\\ 1& 1\\\\ \\end{bmatrix}=\\boldsymbol{U}\\boldsymbol{\\Sigma}\\boldsymbol{V}^T,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "with eigenvalues $\\sigma_1=2$ and $\\sigma_2=0$. \n", + "The SVD exits always! \n", + "\n", + "The SVD\n", + "decomposition (singular values) gives eigenvalues \n", + "$\\sigma_i\\geq\\sigma_{i+1}$ for all $i$ and for dimensions larger than $i=p$, the\n", + "eigenvalues (singular values) are zero.\n", + "\n", + "In the general case, where our design matrix $\\boldsymbol{X}$ has dimension\n", + "$n\\times p$, the matrix is thus decomposed into an $n\\times n$\n", + "orthogonal matrix $\\boldsymbol{U}$, a $p\\times p$ orthogonal matrix $\\boldsymbol{V}$\n", + "and a diagonal matrix $\\boldsymbol{\\Sigma}$ with $r=\\mathrm{min}(n,p)$\n", + "singular values $\\sigma_i\\geq 0$ on the main diagonal and zeros filling\n", + "the rest of the matrix. There are at most $p$ singular values\n", + "assuming that $n > p$. In our regression examples for the nuclear\n", + "masses and the equation of state this is indeed the case, while for\n", + "the Ising model we have $p > n$. These are often cases that lead to\n", + "near singular or singular matrices.\n", + "\n", + "The columns of $\\boldsymbol{U}$ are called the left singular vectors while the columns of $\\boldsymbol{V}$ are the right singular vectors.\n", + "\n", + "## Economy-size SVD\n", + "\n", + "If we assume that $n > p$, then our matrix $\\boldsymbol{U}$ has dimension $n\n", + "\\times n$. The last $n-p$ columns of $\\boldsymbol{U}$ become however\n", + "irrelevant in our calculations since they are multiplied with the\n", + "zeros in $\\boldsymbol{\\Sigma}$.\n", + "\n", + "The economy-size decomposition removes extra rows or columns of zeros\n", + "from the diagonal matrix of singular values, $\\boldsymbol{\\Sigma}$, along with the columns\n", + "in either $\\boldsymbol{U}$ or $\\boldsymbol{V}$ that multiply those zeros in the expression. \n", + "Removing these zeros and columns can improve execution time\n", + "and reduce storage requirements without compromising the accuracy of\n", + "the decomposition.\n", + "\n", + "If $n > p$, we keep only the first $p$ columns of $\\boldsymbol{U}$ and $\\boldsymbol{\\Sigma}$ has dimension $p\\times p$. \n", + "If $p > n$, then only the first $n$ columns of $\\boldsymbol{V}$ are computed and $\\boldsymbol{\\Sigma}$ has dimension $n\\times n$.\n", + "The $n=p$ case is obvious, we retain the full SVD. \n", + "In general the economy-size SVD leads to less FLOPS and still conserving the desired accuracy.\n", + "\n", + "## Codes for the SVD" + ] + }, + { + "cell_type": "code", + "execution_count": 23, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "import numpy as np\n", + "# SVD inversion\n", + "def SVDinv(A):\n", + " ''' Takes as input a numpy matrix A and returns inv(A) based on singular value decomposition (SVD).\n", + " SVD is numerically more stable than the inversion algorithms provided by\n", + " numpy and scipy.linalg at the cost of being slower.\n", + " '''\n", + " U, s, VT = np.linalg.svd(A)\n", + "# print('test U')\n", + "# print( (np.transpose(U) @ U - U @np.transpose(U)))\n", + "# print('test VT')\n", + "# print( (np.transpose(VT) @ VT - VT @np.transpose(VT)))\n", + " print(U)\n", + " print(s)\n", + " print(VT)\n", + "\n", + " D = np.zeros((len(U),len(VT)))\n", + " for i in range(0,len(VT)):\n", + " D[i,i]=s[i]\n", + " UT = np.transpose(U); V = np.transpose(VT); invD = np.linalg.inv(D)\n", + " return np.matmul(V,np.matmul(invD,UT))\n", + "\n", + "\n", + "X = np.array([ [1.0, -1.0, 2.0], [1.0, 0.0, 1.0], [1.0, 2.0, -1.0], [1.0, 1.0, 0.0] ])\n", + "print(X)\n", + "A = np.transpose(X) @ X\n", + "print(A)\n", + "# Brute force inversion of super-collinear matrix\n", + "#B = np.linalg.inv(A)\n", + "#print(B)\n", + "C = SVDinv(A)\n", + "print(C)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The matrix $\\boldsymbol{X}$ has columns that are linearly dependent. The first\n", + "column is the row-wise sum of the other two columns. The rank of a\n", + "matrix (the column rank) is the dimension of space spanned by the\n", + "column vectors. The rank of the matrix is the number of linearly\n", + "independent columns, in this case just $2$. We see this from the\n", + "singular values when running the above code. Running the standard\n", + "inversion algorithm for matrix inversion with $\\boldsymbol{X}^T\\boldsymbol{X}$ results\n", + "in the program terminating due to a singular matrix.\n", + "\n", + "\n", + "\n", + "## Mathematical Properties\n", + "\n", + "There are several interesting mathematical properties which will be\n", + "relevant when we are going to discuss the differences between say\n", + "ordinary least squares (OLS) and **Ridge** regression.\n", + "\n", + "We have from OLS that the parameters of the linear approximation are given by" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{\\tilde{y}} = \\boldsymbol{X}\\boldsymbol{\\beta} = \\boldsymbol{X}\\left(\\boldsymbol{X}^T\\boldsymbol{X}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The matrix to invert can be rewritten in terms of our SVD decomposition as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{X}^T\\boldsymbol{X} = \\boldsymbol{V}\\boldsymbol{\\Sigma}^T\\boldsymbol{U}^T\\boldsymbol{U}\\boldsymbol{\\Sigma}\\boldsymbol{V}^T.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Using the orthogonality properties of $\\boldsymbol{U}$ we have" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{X}^T\\boldsymbol{X} = \\boldsymbol{V}\\boldsymbol{\\Sigma}^T\\boldsymbol{\\Sigma}\\boldsymbol{V}^T = \\boldsymbol{V}\\boldsymbol{D}\\boldsymbol{V}^T,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "with $\\boldsymbol{D}$ being a diagonal matrix with values along the diagonal given by the singular values squared. \n", + "\n", + "This means that" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "(\\boldsymbol{X}^T\\boldsymbol{X})\\boldsymbol{V} = \\boldsymbol{V}\\boldsymbol{D},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "that is the eigenvectors of $(\\boldsymbol{X}^T\\boldsymbol{X})$ are given by the columns of the right singular matrix of $\\boldsymbol{X}$ and the eigenvalues are the squared singular values. It is easy to show (show this) that" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "(\\boldsymbol{X}\\boldsymbol{X}^T)\\boldsymbol{U} = \\boldsymbol{U}\\boldsymbol{D},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "that is, the eigenvectors of $(\\boldsymbol{X}\\boldsymbol{X})^T$ are the columns of the left singular matrix and the eigenvalues are the same. \n", + "\n", + "Going back to our OLS equation we have" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{X}\\boldsymbol{\\beta} = \\boldsymbol{X}\\left(\\boldsymbol{V}\\boldsymbol{D}\\boldsymbol{V}^T \\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y}=\\boldsymbol{U\\Sigma V^T}\\left(\\boldsymbol{V}\\boldsymbol{D}\\boldsymbol{V}^T \\right)^{-1}(\\boldsymbol{U\\Sigma V^T})^T\\boldsymbol{y}=\\boldsymbol{U}\\boldsymbol{U}^T\\boldsymbol{y}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We will come back to this expression when we discuss Ridge regression. \n", + "\n", + "\n", + "$$ \\tilde{y}^{OLS}=\\boldsymbol{X}\\hat{\\beta}^{OLS}=\\sum_{j=1}^p \\boldsymbol{u}_j\\boldsymbol{u}_j^T\\boldsymbol{y}$$ and for Ridge we have \n", + "\n", + "$$ \\tilde{y}^{Ridge}=\\boldsymbol{X}\\hat{\\beta}^{Ridge}=\\sum_{j=1}^p \\boldsymbol{u}_j\\frac{\\sigma_j^2}{\\sigma_j^2+\\lambda}\\boldsymbol{u}_j^T\\boldsymbol{y}$$ . \n", + "\n", + "It is indeed the economy-sized SVD, note the summation runs up tp $$p$$ only and not $$n$$. \n", + "\n", + "Here we have that $$\\boldsymbol{X} = \\boldsymbol{U}\\boldsymbol{\\Sigma}\\boldsymbol{V}^T$$, with $$\\Sigma$$ being an $$ n\\times p$$ matrix and $$\\boldsymbol{V}$$ being a $$ p\\times p$$ matrix. We also have assumed here that $$ n > p$$. \n", + "\n", + "\n", + "## Ridge and LASSO Regression\n", + "\n", + "Let us remind ourselves about the expression for the standard Mean Squared Error (MSE) which we used to define our cost function and the equations for the ordinary least squares (OLS) method, that is \n", + "our optimization problem is" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "{\\displaystyle \\min_{\\boldsymbol{\\beta}\\in {\\mathbb{R}}^{p}}}\\frac{1}{n}\\left\\{\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)^T\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)\\right\\}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "or we can state it as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "{\\displaystyle \\min_{\\boldsymbol{\\beta}\\in\n", + "{\\mathbb{R}}^{p}}}\\frac{1}{n}\\sum_{i=0}^{n-1}\\left(y_i-\\tilde{y}_i\\right)^2=\\frac{1}{n}\\vert\\vert \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\vert\\vert_2^2,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where we have used the definition of a norm-2 vector, that is" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\vert\\vert \\boldsymbol{x}\\vert\\vert_2 = \\sqrt{\\sum_i x_i^2}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "By minimizing the above equation with respect to the parameters\n", + "$\\boldsymbol{\\beta}$ we could then obtain an analytical expression for the\n", + "parameters $\\boldsymbol{\\beta}$. We can add a regularization parameter $\\lambda$ by\n", + "defining a new cost function to be optimized, that is" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "{\\displaystyle \\min_{\\boldsymbol{\\beta}\\in\n", + "{\\mathbb{R}}^{p}}}\\frac{1}{n}\\vert\\vert \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\vert\\vert_2^2+\\lambda\\vert\\vert \\boldsymbol{\\beta}\\vert\\vert_2^2\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "which leads to the Ridge regression minimization problem where we\n", + "require that $\\vert\\vert \\boldsymbol{\\beta}\\vert\\vert_2^2\\le t$, where $t$ is\n", + "a finite number larger than zero. By defining" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "C(\\boldsymbol{X},\\boldsymbol{\\beta})=\\frac{1}{n}\\vert\\vert \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\vert\\vert_2^2+\\lambda\\vert\\vert \\boldsymbol{\\beta}\\vert\\vert_1,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "we have a new optimization equation" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "{\\displaystyle \\min_{\\boldsymbol{\\beta}\\in\n", + "{\\mathbb{R}}^{p}}}\\frac{1}{n}\\vert\\vert \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\vert\\vert_2^2+\\lambda\\vert\\vert \\boldsymbol{\\beta}\\vert\\vert_1\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "which leads to Lasso regression. Lasso stands for least absolute shrinkage and selection operator. \n", + "\n", + "Here we have defined the norm-1 as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\vert\\vert \\boldsymbol{x}\\vert\\vert_1 = \\sum_i \\vert x_i\\vert.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## More on Ridge Regression\n", + "\n", + "Using the matrix-vector expression for Ridge regression," + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "C(\\boldsymbol{X},\\boldsymbol{\\beta})=\\frac{1}{n}\\left\\{(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta})^T(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta})\\right\\}+\\lambda\\boldsymbol{\\beta}^T\\boldsymbol{\\beta},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "by taking the derivatives with respect to $\\boldsymbol{\\beta}$ we obtain then\n", + "a slightly modified matrix inversion problem which for finite values\n", + "of $\\lambda$ does not suffer from singularity problems. We obtain" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{\\beta}^{\\mathrm{Ridge}} = \\left(\\boldsymbol{X}^T\\boldsymbol{X}+\\lambda\\boldsymbol{I}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "with $\\boldsymbol{I}$ being a $p\\times p$ identity matrix with the constraint that" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\sum_{i=0}^{p-1} \\beta_i^2 \\leq t,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "with $t$ a finite positive number. \n", + "\n", + "We see that Ridge regression is nothing but the standard\n", + "OLS with a modified diagonal term added to $\\boldsymbol{X}^T\\boldsymbol{X}$. The\n", + "consequences, in particular for our discussion of the bias-variance tradeoff \n", + "are rather interesting.\n", + "\n", + "Furthermore, if we use the result above in terms of the SVD decomposition (our analysis was done for the OLS method), we had" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "(\\boldsymbol{X}\\boldsymbol{X}^T)\\boldsymbol{U} = \\boldsymbol{U}\\boldsymbol{D}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We can analyse the OLS solutions in terms of the eigenvectors (the columns) of the right singular value matrix $\\boldsymbol{U}$ as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{X}\\boldsymbol{\\beta} = \\boldsymbol{X}\\left(\\boldsymbol{V}\\boldsymbol{D}\\boldsymbol{V}^T \\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y}=\\boldsymbol{U\\Sigma V^T}\\left(\\boldsymbol{V}\\boldsymbol{D}\\boldsymbol{V}^T \\right)^{-1}(\\boldsymbol{U\\Sigma V^T})^T\\boldsymbol{y}=\\boldsymbol{U}\\boldsymbol{U}^T\\boldsymbol{y}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "For Ridge regression this becomes" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{X}\\boldsymbol{\\beta}^{\\mathrm{Ridge}} = \\boldsymbol{U\\Sigma V^T}\\left(\\boldsymbol{V}\\boldsymbol{D}\\boldsymbol{V}^T+\\lambda\\boldsymbol{I} \\right)^{-1}(\\boldsymbol{U\\Sigma V^T})^T\\boldsymbol{y}=\\sum_{j=0}^{p-1}\\boldsymbol{u}_j\\boldsymbol{u}_j^T\\frac{\\sigma_j^2}{\\sigma_j^2+\\lambda}\\boldsymbol{y},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "with the vectors $\\boldsymbol{u}_j$ being the columns of $\\boldsymbol{U}$. \n", + "\n", + "## Interpreting the Ridge results\n", + "\n", + "Since $\\lambda \\geq 0$, it means that compared to OLS, we have" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\sigma_j^2}{\\sigma_j^2+\\lambda} \\leq 1.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Ridge regression finds the coordinates of $\\boldsymbol{y}$ with respect to the\n", + "orthonormal basis $\\boldsymbol{U}$, it then shrinks the coordinates by\n", + "$\\frac{\\sigma_j^2}{\\sigma_j^2+\\lambda}$. Recall that the SVD has\n", + "eigenvalues ordered in a descending way, that is $\\sigma_i \\geq\n", + "\\sigma_{i+1}$.\n", + "\n", + "For small eigenvalues $\\sigma_i$ it means that their contributions become less important, a fact which can be used to reduce the number of degrees of freedom.\n", + "Actually, calculating the variance of $\\boldsymbol{X}\\boldsymbol{v}_j$ shows that this quantity is equal to $\\sigma_j^2/n$.\n", + "With a parameter $\\lambda$ we can thus shrink the role of specific parameters. \n", + "\n", + "\n", + "## More interpretations\n", + "\n", + "For the sake of simplicity, let us assume that the design matrix is orthonormal, that is" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{X}^T\\boldsymbol{X}=(\\boldsymbol{X}^T\\boldsymbol{X})^{-1} =\\boldsymbol{I}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "In this case the standard OLS results in" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{\\beta}^{\\mathrm{OLS}} = \\boldsymbol{X}^T\\boldsymbol{y}=\\sum_{i=0}^{p-1}\\boldsymbol{u}_j\\boldsymbol{u}_j^T\\boldsymbol{y},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{\\beta}^{\\mathrm{Ridge}} = \\left(\\boldsymbol{I}+\\lambda\\boldsymbol{I}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y}=\\left(1+\\lambda\\right)^{-1}\\boldsymbol{\\beta}^{\\mathrm{OLS}},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "that is the Ridge estimator scales the OLS estimator by the inverse of a factor $1+\\lambda$, and\n", + "the Ridge estimator converges to zero when the hyperparameter goes to\n", + "infinity.\n", + "\n", + "We will come back to more interpreations after we have gone through some of the statistical analysis part. \n", + "\n", + "For more discussions of Ridge and Lasso regression, [Wessel van Wieringen's](https://arxiv.org/abs/1509.09169) article is highly recommended.\n", + "Similarly, [Mehta et al's article](https://arxiv.org/abs/1803.08823) is also recommended.\n", + "\n", + "\n", + "\n", + "## A better understanding of regularization\n", + "\n", + "The parameter $\\lambda$ that we have introduced in the Ridge (and\n", + "Lasso as well) regression is often called a regularization parameter\n", + "or shrinkage parameter. It is common to call it a hyperparameter. What does it mean mathemtically?\n", + "\n", + "Here we will first look at how to analyze the difference between the\n", + "standard OLS equations and the Ridge expressions in terms of a linear\n", + "algebra analysis using the SVD algorithm. Thereafter, we will link\n", + "(see the material on the bias-variance tradeoff below) these\n", + "observation to the statisical analysis of the results. In particular\n", + "we consider how the variance of the parameters $\\boldsymbol{\\beta}$ is\n", + "affected by changing the parameter $\\lambda$.\n", + "\n", + "## Decomposing the OLS and Ridge expressions\n", + "\n", + "We have our design matrix\n", + " $\\boldsymbol{X}\\in {\\mathbb{R}}^{n\\times p}$. With the SVD we decompose it as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{X} = \\boldsymbol{U\\Sigma V^T},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "with $\\boldsymbol{U}\\in {\\mathbb{R}}^{n\\times n}$, $\\boldsymbol{\\Sigma}\\in {\\mathbb{R}}^{n\\times p}$\n", + "and $\\boldsymbol{V}\\in {\\mathbb{R}}^{p\\times p}$.\n", + "\n", + "The matrices $\\boldsymbol{U}$ and $\\boldsymbol{V}$ are unitary/orthonormal matrices, that is in case the matrices are real we have $\\boldsymbol{U}^T\\boldsymbol{U}=\\boldsymbol{U}\\boldsymbol{U}^T=\\boldsymbol{I}$ and $\\boldsymbol{V}^T\\boldsymbol{V}=\\boldsymbol{V}\\boldsymbol{V}^T=\\boldsymbol{I}$.\n", + "\n", + "\n", + "\n", + "## Introducing the Covariance and Correlation functions\n", + "\n", + "Before we discuss the link between for example Ridge regression and the singular value decomposition, we need to remind ourselves about\n", + "the definition of the covariance and the correlation function. These are quantities \n", + "\n", + "Suppose we have defined two vectors\n", + "$\\hat{x}$ and $\\hat{y}$ with $n$ elements each. The covariance matrix $\\boldsymbol{C}$ is defined as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{C}[\\boldsymbol{x},\\boldsymbol{y}] = \\begin{bmatrix} \\mathrm{cov}[\\boldsymbol{x},\\boldsymbol{x}] & \\mathrm{cov}[\\boldsymbol{x},\\boldsymbol{y}] \\\\\n", + " \\mathrm{cov}[\\boldsymbol{y},\\boldsymbol{x}] & \\mathrm{cov}[\\boldsymbol{y},\\boldsymbol{y}] \\\\\n", + " \\end{bmatrix},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where for example" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\mathrm{cov}[\\boldsymbol{x},\\boldsymbol{y}] =\\frac{1}{n} \\sum_{i=0}^{n-1}(x_i- \\overline{x})(y_i- \\overline{y}).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "With this definition and recalling that the variance is defined as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\mathrm{var}[\\boldsymbol{x}]=\\frac{1}{n} \\sum_{i=0}^{n-1}(x_i- \\overline{x})^2,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "we can rewrite the covariance matrix as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{C}[\\boldsymbol{x},\\boldsymbol{y}] = \\begin{bmatrix} \\mathrm{var}[\\boldsymbol{x}] & \\mathrm{cov}[\\boldsymbol{x},\\boldsymbol{y}] \\\\\n", + " \\mathrm{cov}[\\boldsymbol{x},\\boldsymbol{y}] & \\mathrm{var}[\\boldsymbol{y}] \\\\\n", + " \\end{bmatrix}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The covariance takes values between zero and infinity and may thus\n", + "lead to problems with loss of numerical precision for particularly\n", + "large values. It is common to scale the covariance matrix by\n", + "introducing instead the correlation matrix defined via the so-called\n", + "correlation function" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\mathrm{corr}[\\boldsymbol{x},\\boldsymbol{y}]=\\frac{\\mathrm{cov}[\\boldsymbol{x},\\boldsymbol{y}]}{\\sqrt{\\mathrm{var}[\\boldsymbol{x}] \\mathrm{var}[\\boldsymbol{y}]}}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The correlation function is then given by values $\\mathrm{corr}[\\boldsymbol{x},\\boldsymbol{y}]\n", + "\\in [-1,1]$. This avoids eventual problems with too large values. We\n", + "can then define the correlation matrix for the two vectors $\\boldsymbol{x}$\n", + "and $\\boldsymbol{y}$ as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{K}[\\boldsymbol{x},\\boldsymbol{y}] = \\begin{bmatrix} 1 & \\mathrm{corr}[\\boldsymbol{x},\\boldsymbol{y}] \\\\\n", + " \\mathrm{corr}[\\boldsymbol{y},\\boldsymbol{x}] & 1 \\\\\n", + " \\end{bmatrix},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "In the above example this is the function we constructed using **pandas**.\n", + "\n", + "## Correlation Function and Design/Feature Matrix\n", + "\n", + "In our derivation of the various regression algorithms like **Ordinary Least Squares** or **Ridge regression**\n", + "we defined the design/feature matrix $\\boldsymbol{X}$ as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{X}=\\begin{bmatrix}\n", + "x_{0,0} & x_{0,1} & x_{0,2}& \\dots & \\dots x_{0,p-1}\\\\\n", + "x_{1,0} & x_{1,1} & x_{1,2}& \\dots & \\dots x_{1,p-1}\\\\\n", + "x_{2,0} & x_{2,1} & x_{2,2}& \\dots & \\dots x_{2,p-1}\\\\\n", + "\\dots & \\dots & \\dots & \\dots \\dots & \\dots \\\\\n", + "x_{n-2,0} & x_{n-2,1} & x_{n-2,2}& \\dots & \\dots x_{n-2,p-1}\\\\\n", + "x_{n-1,0} & x_{n-1,1} & x_{n-1,2}& \\dots & \\dots x_{n-1,p-1}\\\\\n", + "\\end{bmatrix},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "with $\\boldsymbol{X}\\in {\\mathbb{R}}^{n\\times p}$, with the predictors/features $p$ refering to the column numbers and the\n", + "entries $n$ being the row elements.\n", + "We can rewrite the design/feature matrix in terms of its column vectors as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{X}=\\begin{bmatrix} \\boldsymbol{x}_0 & \\boldsymbol{x}_1 & \\boldsymbol{x}_2 & \\dots & \\dots & \\boldsymbol{x}_{p-1}\\end{bmatrix},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "with a given vector" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{x}_i^T = \\begin{bmatrix}x_{0,i} & x_{1,i} & x_{2,i}& \\dots & \\dots x_{n-1,i}\\end{bmatrix}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "With these definitions, we can now rewrite our $2\\times 2$\n", + "correaltion/covariance matrix in terms of a moe general design/feature\n", + "matrix $\\boldsymbol{X}\\in {\\mathbb{R}}^{n\\times p}$. This leads to a $p\\times p$\n", + "covariance matrix for the vectors $\\boldsymbol{x}_i$ with $i=0,1,\\dots,p-1$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{C}[\\boldsymbol{x}] = \\begin{bmatrix}\n", + "\\mathrm{var}[\\boldsymbol{x}_0] & \\mathrm{cov}[\\boldsymbol{x}_0,\\boldsymbol{x}_1] & \\mathrm{cov}[\\boldsymbol{x}_0,\\boldsymbol{x}_2] & \\dots & \\dots & \\mathrm{cov}[\\boldsymbol{x}_0,\\boldsymbol{x}_{p-1}]\\\\\n", + "\\mathrm{cov}[\\boldsymbol{x}_1,\\boldsymbol{x}_0] & \\mathrm{var}[\\boldsymbol{x}_1] & \\mathrm{cov}[\\boldsymbol{x}_1,\\boldsymbol{x}_2] & \\dots & \\dots & \\mathrm{cov}[\\boldsymbol{x}_1,\\boldsymbol{x}_{p-1}]\\\\\n", + "\\mathrm{cov}[\\boldsymbol{x}_2,\\boldsymbol{x}_0] & \\mathrm{cov}[\\boldsymbol{x}_2,\\boldsymbol{x}_1] & \\mathrm{var}[\\boldsymbol{x}_2] & \\dots & \\dots & \\mathrm{cov}[\\boldsymbol{x}_2,\\boldsymbol{x}_{p-1}]\\\\\n", + "\\dots & \\dots & \\dots & \\dots & \\dots & \\dots \\\\\n", + "\\dots & \\dots & \\dots & \\dots & \\dots & \\dots \\\\\n", + "\\mathrm{cov}[\\boldsymbol{x}_{p-1},\\boldsymbol{x}_0] & \\mathrm{cov}[\\boldsymbol{x}_{p-1},\\boldsymbol{x}_1] & \\mathrm{cov}[\\boldsymbol{x}_{p-1},\\boldsymbol{x}_{2}] & \\dots & \\dots & \\mathrm{var}[\\boldsymbol{x}_{p-1}]\\\\\n", + "\\end{bmatrix},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and the correlation matrix" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{K}[\\boldsymbol{x}] = \\begin{bmatrix}\n", + "1 & \\mathrm{corr}[\\boldsymbol{x}_0,\\boldsymbol{x}_1] & \\mathrm{corr}[\\boldsymbol{x}_0,\\boldsymbol{x}_2] & \\dots & \\dots & \\mathrm{corr}[\\boldsymbol{x}_0,\\boldsymbol{x}_{p-1}]\\\\\n", + "\\mathrm{corr}[\\boldsymbol{x}_1,\\boldsymbol{x}_0] & 1 & \\mathrm{corr}[\\boldsymbol{x}_1,\\boldsymbol{x}_2] & \\dots & \\dots & \\mathrm{corr}[\\boldsymbol{x}_1,\\boldsymbol{x}_{p-1}]\\\\\n", + "\\mathrm{corr}[\\boldsymbol{x}_2,\\boldsymbol{x}_0] & \\mathrm{corr}[\\boldsymbol{x}_2,\\boldsymbol{x}_1] & 1 & \\dots & \\dots & \\mathrm{corr}[\\boldsymbol{x}_2,\\boldsymbol{x}_{p-1}]\\\\\n", + "\\dots & \\dots & \\dots & \\dots & \\dots & \\dots \\\\\n", + "\\dots & \\dots & \\dots & \\dots & \\dots & \\dots \\\\\n", + "\\mathrm{corr}[\\boldsymbol{x}_{p-1},\\boldsymbol{x}_0] & \\mathrm{corr}[\\boldsymbol{x}_{p-1},\\boldsymbol{x}_1] & \\mathrm{corr}[\\boldsymbol{x}_{p-1},\\boldsymbol{x}_{2}] & \\dots & \\dots & 1\\\\\n", + "\\end{bmatrix},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Covariance Matrix Examples\n", + "\n", + "\n", + "The Numpy function **np.cov** calculates the covariance elements using\n", + "the factor $1/(n-1)$ instead of $1/n$ since it assumes we do not have\n", + "the exact mean values. The following simple function uses the\n", + "**np.vstack** function which takes each vector of dimension $1\\times n$\n", + "and produces a $2\\times n$ matrix $\\boldsymbol{W}$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{W} = \\begin{bmatrix} x_0 & y_0 \\\\\n", + " x_1 & y_1 \\\\\n", + " x_2 & y_2\\\\\n", + " \\dots & \\dots \\\\\n", + " x_{n-2} & y_{n-2}\\\\\n", + " x_{n-1} & y_{n-1} & \n", + " \\end{bmatrix},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "which in turn is converted into into the $2\\times 2$ covariance matrix\n", + "$\\boldsymbol{C}$ via the Numpy function **np.cov()**. We note that we can also calculate\n", + "the mean value of each set of samples $\\boldsymbol{x}$ etc using the Numpy\n", + "function **np.mean(x)**. We can also extract the eigenvalues of the\n", + "covariance matrix through the **np.linalg.eig()** function." + ] + }, + { + "cell_type": "code", + "execution_count": 24, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "# Importing various packages\n", + "import numpy as np\n", + "n = 100\n", + "x = np.random.normal(size=n)\n", + "print(np.mean(x))\n", + "y = 4+3*x+np.random.normal(size=n)\n", + "print(np.mean(y))\n", + "W = np.vstack((x, y))\n", + "C = np.cov(W)\n", + "print(C)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Correlation Matrix\n", + "\n", + "The previous example can be converted into the correlation matrix by\n", + "simply scaling the matrix elements with the variances. We should also\n", + "subtract the mean values for each column. This leads to the following\n", + "code which sets up the correlations matrix for the previous example in\n", + "a more brute force way. Here we scale the mean values for each column of the design matrix, calculate the relevant mean values and variances and then finally set up the $2\\times 2$ correlation matrix (since we have only two vectors)." + ] + }, + { + "cell_type": "code", + "execution_count": 25, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "import numpy as np\n", + "n = 100\n", + "# define two vectors \n", + "x = np.random.random(size=n)\n", + "y = 4+3*x+np.random.normal(size=n)\n", + "#scaling the x and y vectors \n", + "x = x - np.mean(x)\n", + "y = y - np.mean(y)\n", + "variance_x = np.sum(x@x)/n\n", + "variance_y = np.sum(y@y)/n\n", + "print(variance_x)\n", + "print(variance_y)\n", + "cov_xy = np.sum(x@y)/n\n", + "cov_xx = np.sum(x@x)/n\n", + "cov_yy = np.sum(y@y)/n\n", + "C = np.zeros((2,2))\n", + "C[0,0]= cov_xx/variance_x\n", + "C[1,1]= cov_yy/variance_y\n", + "C[0,1]= cov_xy/np.sqrt(variance_y*variance_x)\n", + "C[1,0]= C[0,1]\n", + "print(C)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We see that the matrix elements along the diagonal are one as they\n", + "should be and that the matrix is symmetric. Furthermore, diagonalizing\n", + "this matrix we easily see that it is a positive definite matrix.\n", + "\n", + "The above procedure with **numpy** can be made more compact if we use **pandas**.\n", + "\n", + "## Correlation Matrix with Pandas\n", + "\n", + "We whow here how we can set up the correlation matrix using **pandas**, as done in this simple code" + ] + }, + { + "cell_type": "code", + "execution_count": 26, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "import numpy as np\n", + "import pandas as pd\n", + "n = 10\n", + "x = np.random.normal(size=n)\n", + "x = x - np.mean(x)\n", + "y = 4+3*x+np.random.normal(size=n)\n", + "y = y - np.mean(y)\n", + "X = (np.vstack((x, y))).T\n", + "print(X)\n", + "Xpd = pd.DataFrame(X)\n", + "print(Xpd)\n", + "correlation_matrix = Xpd.corr()\n", + "print(correlation_matrix)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We expand this model to the Franke function discussed above.\n", + "\n", + "## Correlation Matrix with Pandas and the Franke function" + ] + }, + { + "cell_type": "code", + "execution_count": 27, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "# Common imports\n", + "import numpy as np\n", + "import pandas as pd\n", + "\n", + "\n", + "def FrankeFunction(x,y):\n", + "\tterm1 = 0.75*np.exp(-(0.25*(9*x-2)**2) - 0.25*((9*y-2)**2))\n", + "\tterm2 = 0.75*np.exp(-((9*x+1)**2)/49.0 - 0.1*(9*y+1))\n", + "\tterm3 = 0.5*np.exp(-(9*x-7)**2/4.0 - 0.25*((9*y-3)**2))\n", + "\tterm4 = -0.2*np.exp(-(9*x-4)**2 - (9*y-7)**2)\n", + "\treturn term1 + term2 + term3 + term4\n", + "\n", + "\n", + "def create_X(x, y, n ):\n", + "\tif len(x.shape) > 1:\n", + "\t\tx = np.ravel(x)\n", + "\t\ty = np.ravel(y)\n", + "\n", + "\tN = len(x)\n", + "\tl = int((n+1)*(n+2)/2)\t\t# Number of elements in beta\n", + "\tX = np.ones((N,l))\n", + "\n", + "\tfor i in range(1,n+1):\n", + "\t\tq = int((i)*(i+1)/2)\n", + "\t\tfor k in range(i+1):\n", + "\t\t\tX[:,q+k] = (x**(i-k))*(y**k)\n", + "\n", + "\treturn X\n", + "\n", + "\n", + "# Making meshgrid of datapoints and compute Franke's function\n", + "n = 4\n", + "N = 100\n", + "x = np.sort(np.random.uniform(0, 1, N))\n", + "y = np.sort(np.random.uniform(0, 1, N))\n", + "z = FrankeFunction(x, y)\n", + "X = create_X(x, y, n=n) \n", + "\n", + "Xpd = pd.DataFrame(X)\n", + "# subtract the mean values and set up the covariance matrix\n", + "Xpd = Xpd - Xpd.mean()\n", + "covariance_matrix = Xpd.cov()\n", + "print(covariance_matrix)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We note here that the covariance is zero for the first rows and\n", + "columns since all matrix elements in the design matrix were set to one\n", + "(we are fitting the function in terms of a polynomial of degree $n$).\n", + "\n", + "This means that the variance for these elements will be zero and will\n", + "cause problems when we set up the correlation matrix. We can simply\n", + "drop these elements and construct a correlation\n", + "matrix without these elements. \n", + "\n", + "\n", + "## Rewriting the Covariance and/or Correlation Matrix\n", + "\n", + "We can rewrite the covariance matrix in a more compact form in terms of the design/feature matrix $\\boldsymbol{X}$ as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{C}[\\boldsymbol{x}] = \\frac{1}{n}\\boldsymbol{X}^T\\boldsymbol{X}= \\mathbb{E}[\\boldsymbol{X}^T\\boldsymbol{X}].\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "To see this let us simply look at a design matrix $\\boldsymbol{X}\\in {\\mathbb{R}}^{2\\times 2}$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{X}=\\begin{bmatrix}\n", + "x_{00} & x_{01}\\\\\n", + "x_{10} & x_{11}\\\\\n", + "\\end{bmatrix}=\\begin{bmatrix}\n", + "\\boldsymbol{x}_{0} & \\boldsymbol{x}_{1}\\\\\n", + "\\end{bmatrix}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "If we then compute the expectation value" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\mathbb{E}[\\boldsymbol{X}^T\\boldsymbol{X}] = \\frac{1}{n}\\boldsymbol{X}^T\\boldsymbol{X}=\\begin{bmatrix}\n", + "x_{00}^2+x_{01}^2 & x_{00}x_{10}+x_{01}x_{11}\\\\\n", + "x_{10}x_{00}+x_{11}x_{01} & x_{10}^2+x_{11}^2\\\\\n", + "\\end{bmatrix},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "which is just" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{C}[\\boldsymbol{x}_0,\\boldsymbol{x}_1] = \\boldsymbol{C}[\\boldsymbol{x}]=\\begin{bmatrix} \\mathrm{var}[\\boldsymbol{x}_0] & \\mathrm{cov}[\\boldsymbol{x}_0,\\boldsymbol{x}_1] \\\\\n", + " \\mathrm{cov}[\\boldsymbol{x}_1,\\boldsymbol{x}_0] & \\mathrm{var}[\\boldsymbol{x}_1] \\\\\n", + " \\end{bmatrix},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where we wrote $$\\boldsymbol{C}[\\boldsymbol{x}_0,\\boldsymbol{x}_1] = \\boldsymbol{C}[\\boldsymbol{x}]$$ to indicate that this the covariance of the vectors $\\boldsymbol{x}$ of the design/feature matrix $\\boldsymbol{X}$.\n", + "\n", + "It is easy to generalize this to a matrix $\\boldsymbol{X}\\in {\\mathbb{R}}^{n\\times p}$.\n", + "\n", + "\n", + "## Linking with SVD\n", + "\n", + "See lecture september 11. More text to be added here soon.\n", + "\n", + "\n", + "\n", + "\n", + "## Where are we going?\n", + "\n", + "Before we proceed, we need to rethink what we have been doing. In our\n", + "eager to fit the data, we have omitted several important elements in\n", + "our regression analysis. In what follows we will\n", + "1. look at statistical properties, including a discussion of mean values, variance and the so-called bias-variance tradeoff\n", + "\n", + "2. introduce resampling techniques like cross-validation, bootstrapping and jackknife and more\n", + "\n", + "This will allow us to link the standard linear algebra methods we have discussed above to a statistical interpretation of the methods. \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "## Resampling methods\n", + "Resampling methods are an indispensable tool in modern\n", + "statistics. They involve repeatedly drawing samples from a training\n", + "set and refitting a model of interest on each sample in order to\n", + "obtain additional information about the fitted model. For example, in\n", + "order to estimate the variability of a linear regression fit, we can\n", + "repeatedly draw different samples from the training data, fit a linear\n", + "regression to each new sample, and then examine the extent to which\n", + "the resulting fits differ. Such an approach may allow us to obtain\n", + "information that would not be available from fitting the model only\n", + "once using the original training sample.\n", + "\n", + "Two resampling methods are often used in Machine Learning analyses,\n", + "1. The **bootstrap method**\n", + "\n", + "2. and **Cross-Validation**\n", + "\n", + "In addition there are several other methods such as the Jackknife and the Blocking methods. We will discuss in particular\n", + "cross-validation and the bootstrap method.\n", + "\n", + "\n", + "\n", + "\n", + "## Resampling approaches can be computationally expensive\n", + "\n", + "Resampling approaches can be computationally expensive, because they\n", + "involve fitting the same statistical method multiple times using\n", + "different subsets of the training data. However, due to recent\n", + "advances in computing power, the computational requirements of\n", + "resampling methods generally are not prohibitive. In this chapter, we\n", + "discuss two of the most commonly used resampling methods,\n", + "cross-validation and the bootstrap. Both methods are important tools\n", + "in the practical application of many statistical learning\n", + "procedures. For example, cross-validation can be used to estimate the\n", + "test error associated with a given statistical learning method in\n", + "order to evaluate its performance, or to select the appropriate level\n", + "of flexibility. The process of evaluating a model’s performance is\n", + "known as model assessment, whereas the process of selecting the proper\n", + "level of flexibility for a model is known as model selection. The\n", + "bootstrap is widely used.\n", + "\n", + "\n", + "\n", + "## Why resampling methods ?\n", + "**Statistical analysis.**\n", + "\n", + "\n", + "* Our simulations can be treated as *computer experiments*. This is particularly the case for Monte Carlo methods\n", + "\n", + "* The results can be analysed with the same statistical tools as we would use analysing experimental data.\n", + "\n", + "* As in all experiments, we are looking for expectation values and an estimate of how accurate they are, i.e., possible sources for errors.\n", + "\n", + " \n", + "\n", + "## Statistical analysis\n", + "\n", + "* As in other experiments, many numerical experiments have two classes of errors:\n", + "\n", + " * Statistical errors\n", + "\n", + " * Systematical errors\n", + "\n", + "\n", + "* Statistical errors can be estimated using standard tools from statistics\n", + "\n", + "* Systematical errors are method specific and must be treated differently from case to case.\n", + "\n", + " \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "## Linking the regression analysis with a statistical interpretation\n", + "\n", + "\n", + "The\n", + "advantage of doing linear regression is that we actually end up with\n", + "analytical expressions for several statistical quantities. \n", + "Standard least squares and Ridge regression allow us to\n", + "derive quantities like the variance and other expectation values in a\n", + "rather straightforward way.\n", + "\n", + "\n", + "It is assumed that $\\varepsilon_i\n", + "\\sim \\mathcal{N}(0, \\sigma^2)$ and the $\\varepsilon_{i}$ are\n", + "independent, i.e.:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\begin{align*} \n", + "\\mbox{Cov}(\\varepsilon_{i_1},\n", + "\\varepsilon_{i_2}) & = \\left\\{ \\begin{array}{lcc} \\sigma^2 & \\mbox{if}\n", + "& i_1 = i_2, \\\\ 0 & \\mbox{if} & i_1 \\not= i_2. \\end{array} \\right.\n", + "\\end{align*}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The randomness of $\\varepsilon_i$ implies that\n", + "$\\mathbf{y}_i$ is also a random variable. In particular,\n", + "$\\mathbf{y}_i$ is normally distributed, because $\\varepsilon_i \\sim\n", + "\\mathcal{N}(0, \\sigma^2)$ and $\\mathbf{X}_{i,\\ast} \\, \\boldsymbol{\\beta}$ is a\n", + "non-random scalar. To specify the parameters of the distribution of\n", + "$\\mathbf{y}_i$ we need to calculate its first two moments. \n", + "\n", + "Recall that $\\boldsymbol{X}$ is a matrix of dimensionality $n\\times p$. The\n", + "notation above $\\mathbf{X}_{i,\\ast}$ means that we are looking at the\n", + "row number $i$ and perform a sum over all values $p$.\n", + "\n", + "\n", + "## Assumptions made\n", + "\n", + "The assumption we have made here can be summarized as (and this is going to be useful when we discuss the bias-variance trade off)\n", + "that there exists a function $f(\\boldsymbol{x})$ and a normal distributed error $\\boldsymbol{\\varepsilon}\\sim \\mathcal{N}(0, \\sigma^2)$\n", + "which describe our data" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{y} = f(\\boldsymbol{x})+\\boldsymbol{\\varepsilon}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We approximate this function with our model from the solution of the linear regression equations, that is our\n", + "function $f$ is approximated by $\\boldsymbol{\\tilde{y}}$ where we want to minimize $(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2$, our MSE, with" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{\\tilde{y}} = \\boldsymbol{X}\\boldsymbol{\\beta}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Expectation value and variance\n", + "\n", + "We can calculate the expectation value of $\\boldsymbol{y}$ for a given element $i$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\begin{align*} \n", + "\\mathbb{E}(y_i) & =\n", + "\\mathbb{E}(\\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta}) + \\mathbb{E}(\\varepsilon_i)\n", + "\\, \\, \\, = \\, \\, \\, \\mathbf{X}_{i, \\ast} \\, \\beta, \n", + "\\end{align*}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "while\n", + "its variance is" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\begin{align*} \\mbox{Var}(y_i) & = \\mathbb{E} \\{ [y_i\n", + "- \\mathbb{E}(y_i)]^2 \\} \\, \\, \\, = \\, \\, \\, \\mathbb{E} ( y_i^2 ) -\n", + "[\\mathbb{E}(y_i)]^2 \\\\ & = \\mathbb{E} [ ( \\mathbf{X}_{i, \\ast} \\,\n", + "\\beta + \\varepsilon_i )^2] - ( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta})^2 \\\\ &\n", + "= \\mathbb{E} [ ( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta})^2 + 2 \\varepsilon_i\n", + "\\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta} + \\varepsilon_i^2 ] - ( \\mathbf{X}_{i,\n", + "\\ast} \\, \\beta)^2 \\\\ & = ( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta})^2 + 2\n", + "\\mathbb{E}(\\varepsilon_i) \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta} +\n", + "\\mathbb{E}(\\varepsilon_i^2 ) - ( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta})^2 \n", + "\\\\ & = \\mathbb{E}(\\varepsilon_i^2 ) \\, \\, \\, = \\, \\, \\,\n", + "\\mbox{Var}(\\varepsilon_i) \\, \\, \\, = \\, \\, \\, \\sigma^2. \n", + "\\end{align*}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Hence, $y_i \\sim \\mathcal{N}( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta}, \\sigma^2)$, that is $\\boldsymbol{y}$ follows a normal distribution with \n", + "mean value $\\boldsymbol{X}\\boldsymbol{\\beta}$ and variance $\\sigma^2$ (not be confused with the singular values of the SVD). \n", + "\n", + "## Expectation value and variance for $\\boldsymbol{\\beta}$\n", + "\n", + "With the OLS expressions for the parameters $\\boldsymbol{\\beta}$ we can evaluate the expectation value" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\mathbb{E}(\\boldsymbol{\\beta}) = \\mathbb{E}[ (\\mathbf{X}^{\\top} \\mathbf{X})^{-1}\\mathbf{X}^{T} \\mathbf{Y}]=(\\mathbf{X}^{T} \\mathbf{X})^{-1}\\mathbf{X}^{T} \\mathbb{E}[ \\mathbf{Y}]=(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\mathbf{X}^{T}\\mathbf{X}\\boldsymbol{\\beta}=\\boldsymbol{\\beta}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This means that the estimator of the regression parameters is unbiased.\n", + "\n", + "We can also calculate the variance\n", + "\n", + "The variance of $\\boldsymbol{\\beta}$ is" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\begin{eqnarray*}\n", + "\\mbox{Var}(\\boldsymbol{\\beta}) & = & \\mathbb{E} \\{ [\\boldsymbol{\\beta} - \\mathbb{E}(\\boldsymbol{\\beta})] [\\boldsymbol{\\beta} - \\mathbb{E}(\\boldsymbol{\\beta})]^{T} \\}\n", + "\\\\\n", + "& = & \\mathbb{E} \\{ [(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\mathbf{Y} - \\boldsymbol{\\beta}] \\, [(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\mathbf{Y} - \\boldsymbol{\\beta}]^{T} \\}\n", + "\\\\\n", + "% & = & \\mathbb{E} \\{ [(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\mathbf{Y}] \\, [(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\mathbf{Y}]^{T} \\} - \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T}\n", + "% \\\\\n", + "% & = & \\mathbb{E} \\{ (\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\mathbf{Y} \\, \\mathbf{Y}^{T} \\, \\mathbf{X} \\, (\\mathbf{X}^{T} \\mathbf{X})^{-1} \\} - \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T}\n", + "% \\\\\n", + "& = & (\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\, \\mathbb{E} \\{ \\mathbf{Y} \\, \\mathbf{Y}^{T} \\} \\, \\mathbf{X} \\, (\\mathbf{X}^{T} \\mathbf{X})^{-1} - \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T}\n", + "\\\\\n", + "& = & (\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\, \\{ \\mathbf{X} \\, \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T} \\, \\mathbf{X}^{T} + \\sigma^2 \\} \\, \\mathbf{X} \\, (\\mathbf{X}^{T} \\mathbf{X})^{-1} - \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T}\n", + "% \\\\\n", + "% & = & (\\mathbf{X}^T \\mathbf{X})^{-1} \\, \\mathbf{X}^T \\, \\mathbf{X} \\, \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^T \\, \\mathbf{X}^T \\, \\mathbf{X} \\, (\\mathbf{X}^T % \\mathbf{X})^{-1}\n", + "% \\\\\n", + "% & & + \\, \\, \\sigma^2 \\, (\\mathbf{X}^T \\mathbf{X})^{-1} \\, \\mathbf{X}^T \\, \\mathbf{X} \\, (\\mathbf{X}^T \\mathbf{X})^{-1} - \\boldsymbol{\\beta} \\boldsymbol{\\beta}^T\n", + "\\\\\n", + "& = & \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T} + \\sigma^2 \\, (\\mathbf{X}^{T} \\mathbf{X})^{-1} - \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T}\n", + "\\, \\, \\, = \\, \\, \\, \\sigma^2 \\, (\\mathbf{X}^{T} \\mathbf{X})^{-1},\n", + "\\end{eqnarray*}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where we have used that $\\mathbb{E} (\\mathbf{Y} \\mathbf{Y}^{T}) =\n", + "\\mathbf{X} \\, \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T} \\, \\mathbf{X}^{T} +\n", + "\\sigma^2 \\, \\mathbf{I}_{nn}$. From $\\mbox{Var}(\\boldsymbol{\\beta}) = \\sigma^2\n", + "\\, (\\mathbf{X}^{T} \\mathbf{X})^{-1}$, one obtains an estimate of the\n", + "variance of the estimate of the $j$-th regression coefficient:\n", + "$\\boldsymbol{\\sigma}^2 (\\boldsymbol{\\beta}_j ) = \\boldsymbol{\\sigma}^2 \\sqrt{\n", + "[(\\mathbf{X}^{T} \\mathbf{X})^{-1}]_{jj} }$. This may be used to\n", + "construct a confidence interval for the estimates.\n", + "\n", + "\n", + "In a similar way, we can obtain analytical expressions for say the\n", + "expectation values of the parameters $\\boldsymbol{\\beta}$ and their variance\n", + "when we employ Ridge regression, allowing us again to define a confidence interval. \n", + "\n", + "It is rather straightforward to show that" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\mathbb{E} \\big[ \\boldsymbol{\\beta}^{\\mathrm{Ridge}} \\big]=(\\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I}_{pp})^{-1} (\\mathbf{X}^{\\top} \\mathbf{X})\\boldsymbol{\\beta}^{\\mathrm{OLS}}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We see clearly that \n", + "$\\mathbb{E} \\big[ \\boldsymbol{\\beta}^{\\mathrm{Ridge}} \\big] \\not= \\boldsymbol{\\beta}^{\\mathrm{OLS}}$ for any $\\lambda > 0$. We say then that the ridge estimator is biased.\n", + "\n", + "We can also compute the variance as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\mbox{Var}[\\boldsymbol{\\beta}^{\\mathrm{Ridge}}]=\\sigma^2[ \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1} \\mathbf{X}^{T} \\mathbf{X} \\{ [ \\mathbf{X}^{\\top} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}\\}^{T},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and it is easy to see that if the parameter $\\lambda$ goes to infinity then the variance of Ridge parameters $\\boldsymbol{\\beta}$ goes to zero. \n", + "\n", + "With this, we can compute the difference" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\mbox{Var}[\\boldsymbol{\\beta}^{\\mathrm{OLS}}]-\\mbox{Var}(\\boldsymbol{\\beta}^{\\mathrm{Ridge}})=\\sigma^2 [ \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}[ 2\\lambda\\mathbf{I} + \\lambda^2 (\\mathbf{X}^{T} \\mathbf{X})^{-1} ] \\{ [ \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}\\}^{T}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The difference is non-negative definite since each component of the\n", + "matrix product is non-negative definite. \n", + "This means the variance we obtain with the standard OLS will always for $\\lambda > 0$ be larger than the variance of $\\boldsymbol{\\beta}$ obtained with the Ridge estimator. This has interesting consequences when we discuss the so-called bias-variance trade-off below. \n", + "\n", + "\n", + "## Resampling methods\n", + "\n", + "With all these analytical equations for both the OLS and Ridge\n", + "regression, we will now outline how to assess a given model. This will\n", + "lead us to a discussion of the so-called bias-variance tradeoff (see\n", + "below) and so-called resampling methods.\n", + "\n", + "One of the quantities we have discussed as a way to measure errors is\n", + "the mean-squared error (MSE), mainly used for fitting of continuous\n", + "functions. Another choice is the absolute error.\n", + "\n", + "In the discussions below we will focus on the MSE and in particular since we will split the data into test and training data,\n", + "we discuss the\n", + "1. prediction error or simply the **test error** $\\mathrm{Err_{Test}}$, where we have a fixed training set and the test error is the MSE arising from the data reserved for testing. We discuss also the \n", + "\n", + "2. training error $\\mathrm{Err_{Train}}$, which is the average loss over the training data.\n", + "\n", + "As our model becomes more and more complex, more of the training data tends to used. The training may thence adapt to more complicated structures in the data. This may lead to a decrease in the bias (see below for code example) and a slight increase of the variance for the test error.\n", + "For a certain level of complexity the test error will reach minimum, before starting to increase again. The\n", + "training error reaches a saturation.\n", + "\n", + "\n", + "\n", + "\n", + "## Resampling methods: Jackknife and Bootstrap\n", + "\n", + "Two famous\n", + "resampling methods are the **independent bootstrap** and **the jackknife**. \n", + "\n", + "The jackknife is a special case of the independent bootstrap. Still, the jackknife was made\n", + "popular prior to the independent bootstrap. And as the popularity of\n", + "the independent bootstrap soared, new variants, such as **the dependent bootstrap**.\n", + "\n", + "The Jackknife and independent bootstrap work for\n", + "independent, identically distributed random variables.\n", + "If these conditions are not\n", + "satisfied, the methods will fail. Yet, it should be said that if the data are\n", + "independent, identically distributed, and we only want to estimate the\n", + "variance of $\\overline{X}$ (which often is the case), then there is no\n", + "need for bootstrapping. \n", + "\n", + "## Resampling methods: Jackknife\n", + "\n", + "The Jackknife works by making many replicas of the estimator $\\widehat{\\theta}$. \n", + "The jackknife is a resampling method where we systematically leave out one observation from the vector of observed values $\\boldsymbol{x} = (x_1,x_2,\\cdots,X_n)$. \n", + "Let $\\boldsymbol{x}_i$ denote the vector" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{x}_i = (x_1,x_2,\\cdots,x_{i-1},x_{i+1},\\cdots,x_n),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "which equals the vector $\\boldsymbol{x}$ with the exception that observation\n", + "number $i$ is left out. Using this notation, define\n", + "$\\widehat{\\theta}_i$ to be the estimator\n", + "$\\widehat{\\theta}$ computed using $\\vec{X}_i$. \n", + "\n", + "\n", + "## Jackknife code example" + ] + }, + { + "cell_type": "code", + "execution_count": 28, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "from numpy import *\n", + "from numpy.random import randint, randn\n", + "from time import time\n", + "\n", + "def jackknife(data, stat):\n", + " n = len(data);t = zeros(n); inds = arange(n); t0 = time()\n", + " ## 'jackknifing' by leaving out an observation for each i \n", + " for i in range(n):\n", + " t[i] = stat(delete(data,i) )\n", + "\n", + " # analysis \n", + " print(\"Runtime: %g sec\" % (time()-t0)); print(\"Jackknife Statistics :\")\n", + " print(\"original bias std. error\")\n", + " print(\"%8g %14g %15g\" % (stat(data),(n-1)*mean(t)/n, (n*var(t))**.5))\n", + "\n", + " return t\n", + "\n", + "\n", + "# Returns mean of data samples \n", + "def stat(data):\n", + " return mean(data)\n", + "\n", + "\n", + "mu, sigma = 100, 15\n", + "datapoints = 10000\n", + "x = mu + sigma*random.randn(datapoints)\n", + "# jackknife returns the data sample \n", + "t = jackknife(x, stat)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Resampling methods: Bootstrap\n", + "Bootstrapping is a nonparametric approach to statistical inference\n", + "that substitutes computation for more traditional distributional\n", + "assumptions and asymptotic results. Bootstrapping offers a number of\n", + "advantages: \n", + "1. The bootstrap is quite general, although there are some cases in which it fails. \n", + "\n", + "2. Because it does not require distributional assumptions (such as normally distributed errors), the bootstrap can provide more accurate inferences when the data are not well behaved or when the sample size is small. \n", + "\n", + "3. It is possible to apply the bootstrap to statistics with sampling distributions that are difficult to derive, even asymptotically. \n", + "\n", + "4. It is relatively simple to apply the bootstrap to complex data-collection plans (such as stratified and clustered samples).\n", + "\n", + "\n", + "\n", + "\n", + "## Resampling methods: Bootstrap background\n", + "\n", + "Since $\\widehat{\\theta} = \\widehat{\\theta}(\\boldsymbol{X})$ is a function of random variables,\n", + "$\\widehat{\\theta}$ itself must be a random variable. Thus it has\n", + "a pdf, call this function $p(\\boldsymbol{t})$. The aim of the bootstrap is to\n", + "estimate $p(\\boldsymbol{t})$ by the relative frequency of\n", + "$\\widehat{\\theta}$. You can think of this as using a histogram\n", + "in the place of $p(\\boldsymbol{t})$. If the relative frequency closely\n", + "resembles $p(\\vec{t})$, then using numerics, it is straight forward to\n", + "estimate all the interesting parameters of $p(\\boldsymbol{t})$ using point\n", + "estimators. \n", + "\n", + "\n", + "## Resampling methods: More Bootstrap background\n", + "\n", + "In the case that $\\widehat{\\theta}$ has\n", + "more than one component, and the components are independent, we use the\n", + "same estimator on each component separately. If the probability\n", + "density function of $X_i$, $p(x)$, had been known, then it would have\n", + "been straight forward to do this by: \n", + "1. Drawing lots of numbers from $p(x)$, suppose we call one such set of numbers $(X_1^*, X_2^*, \\cdots, X_n^*)$. \n", + "\n", + "2. Then using these numbers, we could compute a replica of $\\widehat{\\theta}$ called $\\widehat{\\theta}^*$. \n", + "\n", + "By repeated use of (1) and (2), many\n", + "estimates of $\\widehat{\\theta}$ could have been obtained. The\n", + "idea is to use the relative frequency of $\\widehat{\\theta}^*$\n", + "(think of a histogram) as an estimate of $p(\\boldsymbol{t})$.\n", + "\n", + "## Resampling methods: Bootstrap approach\n", + "\n", + "But\n", + "unless there is enough information available about the process that\n", + "generated $X_1,X_2,\\cdots,X_n$, $p(x)$ is in general\n", + "unknown. Therefore, [Efron in 1979](https://projecteuclid.org/euclid.aos/1176344552) asked the\n", + "question: What if we replace $p(x)$ by the relative frequency\n", + "of the observation $X_i$; if we draw observations in accordance with\n", + "the relative frequency of the observations, will we obtain the same\n", + "result in some asymptotic sense? The answer is yes.\n", + "\n", + "\n", + "Instead of generating the histogram for the relative\n", + "frequency of the observation $X_i$, just draw the values\n", + "$(X_1^*,X_2^*,\\cdots,X_n^*)$ with replacement from the vector\n", + "$\\boldsymbol{X}$. \n", + "\n", + "## Resampling methods: Bootstrap steps\n", + "\n", + "The independent bootstrap works like this: \n", + "\n", + "1. Draw with replacement $n$ numbers for the observed variables $\\boldsymbol{x} = (x_1,x_2,\\cdots,x_n)$. \n", + "\n", + "2. Define a vector $\\boldsymbol{x}^*$ containing the values which were drawn from $\\boldsymbol{x}$. \n", + "\n", + "3. Using the vector $\\boldsymbol{x}^*$ compute $\\widehat{\\theta}^*$ by evaluating $\\widehat \\theta$ under the observations $\\boldsymbol{x}^*$. \n", + "\n", + "4. Repeat this process $k$ times. \n", + "\n", + "When you are done, you can draw a histogram of the relative frequency\n", + "of $\\widehat \\theta^*$. This is your estimate of the probability\n", + "distribution $p(t)$. Using this probability distribution you can\n", + "estimate any statistics thereof. In principle you never draw the\n", + "histogram of the relative frequency of $\\widehat{\\theta}^*$. Instead\n", + "you use the estimators corresponding to the statistic of interest. For\n", + "example, if you are interested in estimating the variance of $\\widehat\n", + "\\theta$, apply the etsimator $\\widehat \\sigma^2$ to the values\n", + "$\\widehat \\theta ^*$.\n", + "\n", + "\n", + "## Code example for the Bootstrap method\n", + "\n", + "The following code starts with a Gaussian distribution with mean value\n", + "$\\mu =100$ and variance $\\sigma=15$. We use this to generate the data\n", + "used in the bootstrap analysis. The bootstrap analysis returns a data\n", + "set after a given number of bootstrap operations (as many as we have\n", + "data points). This data set consists of estimated mean values for each\n", + "bootstrap operation. The histogram generated by the bootstrap method\n", + "shows that the distribution for these mean values is also a Gaussian,\n", + "centered around the mean value $\\mu=100$ but with standard deviation\n", + "$\\sigma/\\sqrt{n}$, where $n$ is the number of bootstrap samples (in\n", + "this case the same as the number of original data points). The value\n", + "of the standard deviation is what we expect from the central limit\n", + "theorem." + ] + }, + { + "cell_type": "code", + "execution_count": 29, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "from numpy import *\n", + "from numpy.random import randint, randn\n", + "from time import time\n", + "import matplotlib.mlab as mlab\n", + "import matplotlib.pyplot as plt\n", + "\n", + "# Returns mean of bootstrap samples \n", + "def stat(data):\n", + " return mean(data)\n", + "\n", + "# Bootstrap algorithm\n", + "def bootstrap(data, statistic, R):\n", + " t = zeros(R); n = len(data); inds = arange(n); t0 = time()\n", + " # non-parametric bootstrap \n", + " for i in range(R):\n", + " t[i] = statistic(data[randint(0,n,n)])\n", + "\n", + " # analysis \n", + " print(\"Runtime: %g sec\" % (time()-t0)); print(\"Bootstrap Statistics :\")\n", + " print(\"original bias std. error\")\n", + " print(\"%8g %8g %14g %15g\" % (statistic(data), std(data),mean(t),std(t)))\n", + " return t\n", + "\n", + "\n", + "mu, sigma = 100, 15\n", + "datapoints = 10000\n", + "x = mu + sigma*random.randn(datapoints)\n", + "# bootstrap returns the data sample \n", + "t = bootstrap(x, stat, datapoints)\n", + "# the histogram of the bootstrapped data \n", + "n, binsboot, patches = plt.hist(t, 50, normed=1, facecolor='red', alpha=0.75)\n", + "\n", + "# add a 'best fit' line \n", + "y = mlab.normpdf( binsboot, mean(t), std(t))\n", + "lt = plt.plot(binsboot, y, 'r--', linewidth=1)\n", + "plt.xlabel('Smarts')\n", + "plt.ylabel('Probability')\n", + "plt.axis([99.5, 100.6, 0, 3.0])\n", + "plt.grid(True)\n", + "\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "## Various steps in cross-validation\n", + "\n", + "When the repetitive splitting of the data set is done randomly,\n", + "samples may accidently end up in a fast majority of the splits in\n", + "either training or test set. Such samples may have an unbalanced\n", + "influence on either model building or prediction evaluation. To avoid\n", + "this $k$-fold cross-validation structures the data splitting. The\n", + "samples are divided into $k$ more or less equally sized exhaustive and\n", + "mutually exclusive subsets. In turn (at each split) one of these\n", + "subsets plays the role of the test set while the union of the\n", + "remaining subsets constitutes the training set. Such a splitting\n", + "warrants a balanced representation of each sample in both training and\n", + "test set over the splits. Still the division into the $k$ subsets\n", + "involves a degree of randomness. This may be fully excluded when\n", + "choosing $k=n$. This particular case is referred to as leave-one-out\n", + "cross-validation (LOOCV). \n", + "\n", + "\n", + "## How to set up the cross-validation for Ridge and/or Lasso\n", + "\n", + "* Define a range of interest for the penalty parameter.\n", + "\n", + "* Divide the data set into training and test set comprising samples $\\{1, \\ldots, n\\} \\setminus i$ and $\\{ i \\}$, respectively.\n", + "\n", + "* Fit the linear regression model by means of ridge estimation for each $\\lambda$ in the grid using the training set, and the corresponding estimate of the error variance $\\boldsymbol{\\sigma}_{-i}^2(\\lambda)$, as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\begin{align*}\n", + "\\boldsymbol{\\beta}_{-i}(\\lambda) & = ( \\boldsymbol{X}_{-i, \\ast}^{T}\n", + "\\boldsymbol{X}_{-i, \\ast} + \\lambda \\boldsymbol{I}_{pp})^{-1}\n", + "\\boldsymbol{X}_{-i, \\ast}^{T} \\boldsymbol{y}_{-i}\n", + "\\end{align*}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "* Evaluate the prediction performance of these models on the test set by $\\log\\{L[y_i, \\boldsymbol{X}_{i, \\ast}; \\boldsymbol{\\beta}_{-i}(\\lambda), \\boldsymbol{\\sigma}_{-i}^2(\\lambda)]\\}$. Or, by the prediction error $|y_i - \\boldsymbol{X}_{i, \\ast} \\boldsymbol{\\beta}_{-i}(\\lambda)|$, the relative error, the error squared or the R2 score function.\n", + "\n", + "* Repeat the first three steps such that each sample plays the role of the test set once.\n", + "\n", + "* Average the prediction performances of the test sets at each grid point of the penalty bias/parameter. It is an estimate of the prediction performance of the model corresponding to this value of the penalty parameter on novel data. It is defined as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\begin{align*}\n", + "\\frac{1}{n} \\sum_{i = 1}^n \\log\\{L[y_i, \\mathbf{X}_{i, \\ast}; \\boldsymbol{\\beta}_{-i}(\\lambda), \\boldsymbol{\\sigma}_{-i}^2(\\lambda)]\\}.\n", + "\\end{align*}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Cross-validation in brief\n", + "\n", + "For the various values of $k$\n", + "\n", + "1. shuffle the dataset randomly.\n", + "\n", + "2. Split the dataset into $k$ groups.\n", + "\n", + "3. For each unique group:\n", + "\n", + "a. Decide which group to use as set for test data\n", + "\n", + "b. Take the remaining groups as a training data set\n", + "\n", + "c. Fit a model on the training set and evaluate it on the test set\n", + "\n", + "d. Retain the evaluation score and discard the model\n", + "\n", + "\n", + "5. Summarize the model using the sample of model evaluation scores\n", + "\n", + "## Code Example for Cross-validation and $k$-fold Cross-validation\n", + "\n", + "The code here uses Ridge regression with cross-validation (CV) resampling and $k$-fold CV in order to fit a specific polynomial." + ] + }, + { + "cell_type": "code", + "execution_count": 30, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "import numpy as np\n", + "import matplotlib.pyplot as plt\n", + "from sklearn.model_selection import KFold\n", + "from sklearn.linear_model import Ridge\n", + "from sklearn.model_selection import cross_val_score\n", + "from sklearn.preprocessing import PolynomialFeatures\n", + "\n", + "# A seed just to ensure that the random numbers are the same for every run.\n", + "# Useful for eventual debugging.\n", + "np.random.seed(3155)\n", + "\n", + "# Generate the data.\n", + "nsamples = 100\n", + "x = np.random.randn(nsamples)\n", + "y = 3*x**2 + np.random.randn(nsamples)\n", + "\n", + "## Cross-validation on Ridge regression using KFold only\n", + "\n", + "# Decide degree on polynomial to fit\n", + "poly = PolynomialFeatures(degree = 6)\n", + "\n", + "# Decide which values of lambda to use\n", + "nlambdas = 500\n", + "lambdas = np.logspace(-3, 5, nlambdas)\n", + "\n", + "# Initialize a KFold instance\n", + "k = 5\n", + "kfold = KFold(n_splits = k)\n", + "\n", + "# Perform the cross-validation to estimate MSE\n", + "scores_KFold = np.zeros((nlambdas, k))\n", + "\n", + "i = 0\n", + "for lmb in lambdas:\n", + " ridge = Ridge(alpha = lmb)\n", + " j = 0\n", + " for train_inds, test_inds in kfold.split(x):\n", + " xtrain = x[train_inds]\n", + " ytrain = y[train_inds]\n", + "\n", + " xtest = x[test_inds]\n", + " ytest = y[test_inds]\n", + "\n", + " Xtrain = poly.fit_transform(xtrain[:, np.newaxis])\n", + " ridge.fit(Xtrain, ytrain[:, np.newaxis])\n", + "\n", + " Xtest = poly.fit_transform(xtest[:, np.newaxis])\n", + " ypred = ridge.predict(Xtest)\n", + "\n", + " scores_KFold[i,j] = np.sum((ypred - ytest[:, np.newaxis])**2)/np.size(ypred)\n", + "\n", + " j += 1\n", + " i += 1\n", + "\n", + "\n", + "estimated_mse_KFold = np.mean(scores_KFold, axis = 1)\n", + "\n", + "## Cross-validation using cross_val_score from sklearn along with KFold\n", + "\n", + "# kfold is an instance initialized above as:\n", + "# kfold = KFold(n_splits = k)\n", + "\n", + "estimated_mse_sklearn = np.zeros(nlambdas)\n", + "i = 0\n", + "for lmb in lambdas:\n", + " ridge = Ridge(alpha = lmb)\n", + "\n", + " X = poly.fit_transform(x[:, np.newaxis])\n", + " estimated_mse_folds = cross_val_score(ridge, X, y[:, np.newaxis], scoring='neg_mean_squared_error', cv=kfold)\n", + "\n", + " # cross_val_score return an array containing the estimated negative mse for every fold.\n", + " # we have to the the mean of every array in order to get an estimate of the mse of the model\n", + " estimated_mse_sklearn[i] = np.mean(-estimated_mse_folds)\n", + "\n", + " i += 1\n", + "\n", + "## Plot and compare the slightly different ways to perform cross-validation\n", + "\n", + "plt.figure()\n", + "\n", + "plt.plot(np.log10(lambdas), estimated_mse_sklearn, label = 'cross_val_score')\n", + "plt.plot(np.log10(lambdas), estimated_mse_KFold, 'r--', label = 'KFold')\n", + "\n", + "plt.xlabel('log10(lambda)')\n", + "plt.ylabel('mse')\n", + "\n", + "plt.legend()\n", + "\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## The bias-variance tradeoff\n", + "\n", + "\n", + "We will discuss the bias-variance tradeoff in the context of\n", + "continuous predictions such as regression. However, many of the\n", + "intuitions and ideas discussed here also carry over to classification\n", + "tasks. Consider a dataset $\\mathcal{L}$ consisting of the data\n", + "$\\mathbf{X}_\\mathcal{L}=\\{(y_j, \\boldsymbol{x}_j), j=0\\ldots n-1\\}$. \n", + "\n", + "Let us assume that the true data is generated from a noisy model" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{y}=f(\\boldsymbol{x}) + \\boldsymbol{\\epsilon}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $\\epsilon$ is normally distributed with mean zero and standard deviation $\\sigma^2$.\n", + "\n", + "In our derivation of the ordinary least squares method we defined then\n", + "an approximation to the function $f$ in terms of the parameters\n", + "$\\boldsymbol{\\beta}$ and the design matrix $\\boldsymbol{X}$ which embody our model,\n", + "that is $\\boldsymbol{\\tilde{y}}=\\boldsymbol{X}\\boldsymbol{\\beta}$. \n", + "\n", + "Thereafter we found the parameters $\\boldsymbol{\\beta}$ by optimizing the means squared error via the so-called cost function" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "C(\\boldsymbol{X},\\boldsymbol{\\beta}) =\\frac{1}{n}\\sum_{i=0}^{n-1}(y_i-\\tilde{y}_i)^2=\\mathbb{E}\\left[(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2\\right].\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We can rewrite this as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\mathbb{E}\\left[(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2\\right]=\\frac{1}{n}\\sum_i(f_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2+\\frac{1}{n}\\sum_i(\\tilde{y}_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2+\\sigma^2.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The three terms represent the square of the bias of the learning\n", + "method, which can be thought of as the error caused by the simplifying\n", + "assumptions built into the method. The second term represents the\n", + "variance of the chosen model and finally the last terms is variance of\n", + "the error $\\boldsymbol{\\epsilon}$.\n", + "\n", + "To derive this equation, we need to recall that the variance of $\\boldsymbol{y}$ and $\\boldsymbol{\\epsilon}$ are both equal to $\\sigma^2$. The mean value of $\\boldsymbol{\\epsilon}$ is by definition equal to zero. Furthermore, the function $f$ is not a stochastics variable, idem for $\\boldsymbol{\\tilde{y}}$.\n", + "We use a more compact notation in terms of the expectation value" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\mathbb{E}\\left[(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2\\right]=\\mathbb{E}\\left[(\\boldsymbol{f}+\\boldsymbol{\\epsilon}-\\boldsymbol{\\tilde{y}})^2\\right],\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and adding and subtracting $\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right]$ we get" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\mathbb{E}\\left[(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2\\right]=\\mathbb{E}\\left[(\\boldsymbol{f}+\\boldsymbol{\\epsilon}-\\boldsymbol{\\tilde{y}}+\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right]-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2\\right],\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "which, using the abovementioned expectation values can be rewritten as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\mathbb{E}\\left[(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2\\right]=\\mathbb{E}\\left[(\\boldsymbol{y}-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2\\right]+\\mathrm{Var}\\left[\\boldsymbol{\\tilde{y}}\\right]+\\sigma^2,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "that is the rewriting in terms of the so-called bias, the variance of the model $\\boldsymbol{\\tilde{y}}$ and the variance of $\\boldsymbol{\\epsilon}$.\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "## Example code for Bias-Variance tradeoff" + ] + }, + { + "cell_type": "code", + "execution_count": 31, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "import matplotlib.pyplot as plt\n", + "import numpy as np\n", + "from sklearn.linear_model import LinearRegression, Ridge, Lasso\n", + "from sklearn.preprocessing import PolynomialFeatures\n", + "from sklearn.model_selection import train_test_split\n", + "from sklearn.pipeline import make_pipeline\n", + "from sklearn.utils import resample\n", + "\n", + "np.random.seed(2018)\n", + "\n", + "n = 500\n", + "n_boostraps = 100\n", + "degree = 18 # A quite high value, just to show.\n", + "noise = 0.1\n", + "\n", + "# Make data set.\n", + "x = np.linspace(-1, 3, n).reshape(-1, 1)\n", + "y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2) + np.random.normal(0, 0.1, x.shape)\n", + "\n", + "# Hold out some test data that is never used in training.\n", + "x_train, x_test, y_train, y_test = train_test_split(x, y, test_size=0.2)\n", + "\n", + "# Combine x transformation and model into one operation.\n", + "# Not neccesary, but convenient.\n", + "model = make_pipeline(PolynomialFeatures(degree=degree), LinearRegression(fit_intercept=False))\n", + "\n", + "# The following (m x n_bootstraps) matrix holds the column vectors y_pred\n", + "# for each bootstrap iteration.\n", + "y_pred = np.empty((y_test.shape[0], n_boostraps))\n", + "for i in range(n_boostraps):\n", + " x_, y_ = resample(x_train, y_train)\n", + "\n", + " # Evaluate the new model on the same test data each time.\n", + " y_pred[:, i] = model.fit(x_, y_).predict(x_test).ravel()\n", + "\n", + "# Note: Expectations and variances taken w.r.t. different training\n", + "# data sets, hence the axis=1. Subsequent means are taken across the test data\n", + "# set in order to obtain a total value, but before this we have error/bias/variance\n", + "# calculated per data point in the test set.\n", + "# Note 2: The use of keepdims=True is important in the calculation of bias as this \n", + "# maintains the column vector form. Dropping this yields very unexpected results.\n", + "error = np.mean( np.mean((y_test - y_pred)**2, axis=1, keepdims=True) )\n", + "bias = np.mean( (y_test - np.mean(y_pred, axis=1, keepdims=True))**2 )\n", + "variance = np.mean( np.var(y_pred, axis=1, keepdims=True) )\n", + "print('Error:', error)\n", + "print('Bias^2:', bias)\n", + "print('Var:', variance)\n", + "print('{} >= {} + {} = {}'.format(error, bias, variance, bias+variance))\n", + "\n", + "plt.plot(x[::5, :], y[::5, :], label='f(x)')\n", + "plt.scatter(x_test, y_test, label='Data points')\n", + "plt.scatter(x_test, np.mean(y_pred, axis=1), label='Pred')\n", + "plt.legend()\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Understanding what happens" + ] + }, + { + "cell_type": "code", + "execution_count": 32, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "import matplotlib.pyplot as plt\n", + "import numpy as np\n", + "from sklearn.linear_model import LinearRegression, Ridge, Lasso\n", + "from sklearn.preprocessing import PolynomialFeatures\n", + "from sklearn.model_selection import train_test_split\n", + "from sklearn.pipeline import make_pipeline\n", + "from sklearn.utils import resample\n", + "\n", + "np.random.seed(2018)\n", + "\n", + "n = 40\n", + "n_boostraps = 100\n", + "maxdegree = 14\n", + "\n", + "\n", + "# Make data set.\n", + "x = np.linspace(-3, 3, n).reshape(-1, 1)\n", + "y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)\n", + "error = np.zeros(maxdegree)\n", + "bias = np.zeros(maxdegree)\n", + "variance = np.zeros(maxdegree)\n", + "polydegree = np.zeros(maxdegree)\n", + "x_train, x_test, y_train, y_test = train_test_split(x, y, test_size=0.2)\n", + "\n", + "for degree in range(maxdegree):\n", + " model = make_pipeline(PolynomialFeatures(degree=degree), LinearRegression(fit_intercept=False))\n", + " y_pred = np.empty((y_test.shape[0], n_boostraps))\n", + " for i in range(n_boostraps):\n", + " x_, y_ = resample(x_train, y_train)\n", + " y_pred[:, i] = model.fit(x_, y_).predict(x_test).ravel()\n", + "\n", + " polydegree[degree] = degree\n", + " error[degree] = np.mean( np.mean((y_test - y_pred)**2, axis=1, keepdims=True) )\n", + " bias[degree] = np.mean( (y_test - np.mean(y_pred, axis=1, keepdims=True))**2 )\n", + " variance[degree] = np.mean( np.var(y_pred, axis=1, keepdims=True) )\n", + " print('Polynomial degree:', degree)\n", + " print('Error:', error[degree])\n", + " print('Bias^2:', bias[degree])\n", + " print('Var:', variance[degree])\n", + " print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))\n", + "\n", + "plt.plot(polydegree, error, label='Error')\n", + "plt.plot(polydegree, bias, label='bias')\n", + "plt.plot(polydegree, variance, label='Variance')\n", + "plt.legend()\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "## Summing up\n", + "\n", + "\n", + "\n", + "\n", + "The bias-variance tradeoff summarizes the fundamental tension in\n", + "machine learning, particularly supervised learning, between the\n", + "complexity of a model and the amount of training data needed to train\n", + "it. Since data is often limited, in practice it is often useful to\n", + "use a less-complex model with higher bias, that is a model whose asymptotic\n", + "performance is worse than another model because it is easier to\n", + "train and less sensitive to sampling noise arising from having a\n", + "finite-sized training dataset (smaller variance). \n", + "\n", + "\n", + "\n", + "The above equations tell us that in\n", + "order to minimize the expected test error, we need to select a\n", + "statistical learning method that simultaneously achieves low variance\n", + "and low bias. Note that variance is inherently a nonnegative quantity,\n", + "and squared bias is also nonnegative. Hence, we see that the expected\n", + "test MSE can never lie below $Var(\\epsilon)$, the irreducible error.\n", + "\n", + "\n", + "What do we mean by the variance and bias of a statistical learning\n", + "method? The variance refers to the amount by which our model would change if we\n", + "estimated it using a different training data set. Since the training\n", + "data are used to fit the statistical learning method, different\n", + "training data sets will result in a different estimate. But ideally the\n", + "estimate for our model should not vary too much between training\n", + "sets. However, if a method has high variance then small changes in\n", + "the training data can result in large changes in the model. In general, more\n", + "flexible statistical methods have higher variance.\n", + "\n", + "\n", + "You may also find this recent [article](https://www.pnas.org/content/116/32/15849) of interest.\n", + "\n", + "## Another Example from Scikit-Learn's Repository" + ] + }, + { + "cell_type": "code", + "execution_count": 33, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "\"\"\"\n", + "============================\n", + "Underfitting vs. Overfitting\n", + "============================\n", + "\n", + "This example demonstrates the problems of underfitting and overfitting and\n", + "how we can use linear regression with polynomial features to approximate\n", + "nonlinear functions. The plot shows the function that we want to approximate,\n", + "which is a part of the cosine function. In addition, the samples from the\n", + "real function and the approximations of different models are displayed. The\n", + "models have polynomial features of different degrees. We can see that a\n", + "linear function (polynomial with degree 1) is not sufficient to fit the\n", + "training samples. This is called **underfitting**. A polynomial of degree 4\n", + "approximates the true function almost perfectly. However, for higher degrees\n", + "the model will **overfit** the training data, i.e. it learns the noise of the\n", + "training data.\n", + "We evaluate quantitatively **overfitting** / **underfitting** by using\n", + "cross-validation. We calculate the mean squared error (MSE) on the validation\n", + "set, the higher, the less likely the model generalizes correctly from the\n", + "training data.\n", + "\"\"\"\n", + "\n", + "print(__doc__)\n", + "\n", + "import numpy as np\n", + "import matplotlib.pyplot as plt\n", + "from sklearn.pipeline import Pipeline\n", + "from sklearn.preprocessing import PolynomialFeatures\n", + "from sklearn.linear_model import LinearRegression\n", + "from sklearn.model_selection import cross_val_score\n", + "\n", + "\n", + "def true_fun(X):\n", + " return np.cos(1.5 * np.pi * X)\n", + "\n", + "np.random.seed(0)\n", + "\n", + "n_samples = 30\n", + "degrees = [1, 4, 15]\n", + "\n", + "X = np.sort(np.random.rand(n_samples))\n", + "y = true_fun(X) + np.random.randn(n_samples) * 0.1\n", + "\n", + "plt.figure(figsize=(14, 5))\n", + "for i in range(len(degrees)):\n", + " ax = plt.subplot(1, len(degrees), i + 1)\n", + " plt.setp(ax, xticks=(), yticks=())\n", + "\n", + " polynomial_features = PolynomialFeatures(degree=degrees[i],\n", + " include_bias=False)\n", + " linear_regression = LinearRegression()\n", + " pipeline = Pipeline([(\"polynomial_features\", polynomial_features),\n", + " (\"linear_regression\", linear_regression)])\n", + " pipeline.fit(X[:, np.newaxis], y)\n", + "\n", + " # Evaluate the models using crossvalidation\n", + " scores = cross_val_score(pipeline, X[:, np.newaxis], y,\n", + " scoring=\"neg_mean_squared_error\", cv=10)\n", + "\n", + " X_test = np.linspace(0, 1, 100)\n", + " plt.plot(X_test, pipeline.predict(X_test[:, np.newaxis]), label=\"Model\")\n", + " plt.plot(X_test, true_fun(X_test), label=\"True function\")\n", + " plt.scatter(X, y, edgecolor='b', s=20, label=\"Samples\")\n", + " plt.xlabel(\"x\")\n", + " plt.ylabel(\"y\")\n", + " plt.xlim((0, 1))\n", + " plt.ylim((-2, 2))\n", + " plt.legend(loc=\"best\")\n", + " plt.title(\"Degree {}\\nMSE = {:.2e}(+/- {:.2e})\".format(\n", + " degrees[i], -scores.mean(), scores.std()))\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## More examples on bootstrap and cross-validation and errors" + ] + }, + { + "cell_type": "code", + "execution_count": 34, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "# Common imports\n", + "import os\n", + "import numpy as np\n", + "import pandas as pd\n", + "import matplotlib.pyplot as plt\n", + "from sklearn.linear_model import LinearRegression, Ridge, Lasso\n", + "from sklearn.model_selection import train_test_split\n", + "from sklearn.utils import resample\n", + "from sklearn.metrics import mean_squared_error\n", + "# Where to save the figures and data files\n", + "PROJECT_ROOT_DIR = \"Results\"\n", + "FIGURE_ID = \"Results/FigureFiles\"\n", + "DATA_ID = \"DataFiles/\"\n", + "\n", + "if not os.path.exists(PROJECT_ROOT_DIR):\n", + " os.mkdir(PROJECT_ROOT_DIR)\n", + "\n", + "if not os.path.exists(FIGURE_ID):\n", + " os.makedirs(FIGURE_ID)\n", + "\n", + "if not os.path.exists(DATA_ID):\n", + " os.makedirs(DATA_ID)\n", + "\n", + "def image_path(fig_id):\n", + " return os.path.join(FIGURE_ID, fig_id)\n", + "\n", + "def data_path(dat_id):\n", + " return os.path.join(DATA_ID, dat_id)\n", + "\n", + "def save_fig(fig_id):\n", + " plt.savefig(image_path(fig_id) + \".png\", format='png')\n", + "\n", + "infile = open(data_path(\"EoS.csv\"),'r')\n", + "\n", + "# Read the EoS data as csv file and organize the data into two arrays with density and energies\n", + "EoS = pd.read_csv(infile, names=('Density', 'Energy'))\n", + "EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce')\n", + "EoS = EoS.dropna()\n", + "Energies = EoS['Energy']\n", + "Density = EoS['Density']\n", + "# The design matrix now as function of various polytrops\n", + "\n", + "Maxpolydegree = 30\n", + "X = np.zeros((len(Density),Maxpolydegree))\n", + "X[:,0] = 1.0\n", + "testerror = np.zeros(Maxpolydegree)\n", + "trainingerror = np.zeros(Maxpolydegree)\n", + "polynomial = np.zeros(Maxpolydegree)\n", + "\n", + "trials = 100\n", + "for polydegree in range(1, Maxpolydegree):\n", + " polynomial[polydegree] = polydegree\n", + " for degree in range(polydegree):\n", + " X[:,degree] = Density**(degree/3.0)\n", + "\n", + "# loop over trials in order to estimate the expectation value of the MSE\n", + " testerror[polydegree] = 0.0\n", + " trainingerror[polydegree] = 0.0\n", + " for samples in range(trials):\n", + " x_train, x_test, y_train, y_test = train_test_split(X, Energies, test_size=0.2)\n", + " model = LinearRegression(fit_intercept=True).fit(x_train, y_train)\n", + " ypred = model.predict(x_train)\n", + " ytilde = model.predict(x_test)\n", + " testerror[polydegree] += mean_squared_error(y_test, ytilde)\n", + " trainingerror[polydegree] += mean_squared_error(y_train, ypred) \n", + "\n", + " testerror[polydegree] /= trials\n", + " trainingerror[polydegree] /= trials\n", + " print(\"Degree of polynomial: %3d\"% polynomial[polydegree])\n", + " print(\"Mean squared error on training data: %.8f\" % trainingerror[polydegree])\n", + " print(\"Mean squared error on test data: %.8f\" % testerror[polydegree])\n", + "\n", + "plt.plot(polynomial, np.log10(trainingerror), label='Training Error')\n", + "plt.plot(polynomial, np.log10(testerror), label='Test Error')\n", + "plt.xlabel('Polynomial degree')\n", + "plt.ylabel('log10[MSE]')\n", + "plt.legend()\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "## The same example but now with cross-validation" + ] + }, + { + "cell_type": "code", + "execution_count": 35, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "# Common imports\n", + "import os\n", + "import numpy as np\n", + "import pandas as pd\n", + "import matplotlib.pyplot as plt\n", + "from sklearn.linear_model import LinearRegression, Ridge, Lasso\n", + "from sklearn.metrics import mean_squared_error\n", + "from sklearn.model_selection import KFold\n", + "from sklearn.model_selection import cross_val_score\n", + "\n", + "\n", + "# Where to save the figures and data files\n", + "PROJECT_ROOT_DIR = \"Results\"\n", + "FIGURE_ID = \"Results/FigureFiles\"\n", + "DATA_ID = \"DataFiles/\"\n", + "\n", + "if not os.path.exists(PROJECT_ROOT_DIR):\n", + " os.mkdir(PROJECT_ROOT_DIR)\n", + "\n", + "if not os.path.exists(FIGURE_ID):\n", + " os.makedirs(FIGURE_ID)\n", + "\n", + "if not os.path.exists(DATA_ID):\n", + " os.makedirs(DATA_ID)\n", + "\n", + "def image_path(fig_id):\n", + " return os.path.join(FIGURE_ID, fig_id)\n", + "\n", + "def data_path(dat_id):\n", + " return os.path.join(DATA_ID, dat_id)\n", + "\n", + "def save_fig(fig_id):\n", + " plt.savefig(image_path(fig_id) + \".png\", format='png')\n", + "\n", + "infile = open(data_path(\"EoS.csv\"),'r')\n", + "\n", + "# Read the EoS data as csv file and organize the data into two arrays with density and energies\n", + "EoS = pd.read_csv(infile, names=('Density', 'Energy'))\n", + "EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce')\n", + "EoS = EoS.dropna()\n", + "Energies = EoS['Energy']\n", + "Density = EoS['Density']\n", + "# The design matrix now as function of various polytrops\n", + "\n", + "Maxpolydegree = 30\n", + "X = np.zeros((len(Density),Maxpolydegree))\n", + "X[:,0] = 1.0\n", + "estimated_mse_sklearn = np.zeros(Maxpolydegree)\n", + "polynomial = np.zeros(Maxpolydegree)\n", + "k =5\n", + "kfold = KFold(n_splits = k)\n", + "\n", + "for polydegree in range(1, Maxpolydegree):\n", + " polynomial[polydegree] = polydegree\n", + " for degree in range(polydegree):\n", + " X[:,degree] = Density**(degree/3.0)\n", + " OLS = LinearRegression()\n", + "# loop over trials in order to estimate the expectation value of the MSE\n", + " estimated_mse_folds = cross_val_score(OLS, X, Energies, scoring='neg_mean_squared_error', cv=kfold)\n", + "#[:, np.newaxis]\n", + " estimated_mse_sklearn[polydegree] = np.mean(-estimated_mse_folds)\n", + "\n", + "plt.plot(polynomial, np.log10(estimated_mse_sklearn), label='Test Error')\n", + "plt.xlabel('Polynomial degree')\n", + "plt.ylabel('log10[MSE]')\n", + "plt.legend()\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Cross-validation with Ridge" + ] + }, + { + "cell_type": "code", + "execution_count": 36, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "import numpy as np\n", + "import matplotlib.pyplot as plt\n", + "from sklearn.model_selection import KFold\n", + "from sklearn.linear_model import Ridge\n", + "from sklearn.model_selection import cross_val_score\n", + "from sklearn.preprocessing import PolynomialFeatures\n", + "\n", + "# A seed just to ensure that the random numbers are the same for every run.\n", + "np.random.seed(3155)\n", + "# Generate the data.\n", + "n = 100\n", + "x = np.linspace(-3, 3, n).reshape(-1, 1)\n", + "y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)\n", + "# Decide degree on polynomial to fit\n", + "poly = PolynomialFeatures(degree = 10)\n", + "\n", + "# Decide which values of lambda to use\n", + "nlambdas = 500\n", + "lambdas = np.logspace(-3, 5, nlambdas)\n", + "# Initialize a KFold instance\n", + "k = 5\n", + "kfold = KFold(n_splits = k)\n", + "estimated_mse_sklearn = np.zeros(nlambdas)\n", + "i = 0\n", + "for lmb in lambdas:\n", + " ridge = Ridge(alpha = lmb)\n", + " estimated_mse_folds = cross_val_score(ridge, x, y, scoring='neg_mean_squared_error', cv=kfold)\n", + " estimated_mse_sklearn[i] = np.mean(-estimated_mse_folds)\n", + " i += 1\n", + "plt.figure()\n", + "plt.plot(np.log10(lambdas), estimated_mse_sklearn, label = 'cross_val_score')\n", + "plt.xlabel('log10(lambda)')\n", + "plt.ylabel('MSE')\n", + "plt.legend()\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## The Ising model\n", + "\n", + "The one-dimensional Ising model with nearest neighbor interaction, no\n", + "external field and a constant coupling constant $J$ is given by" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + " H = -J \\sum_{k}^L s_k s_{k + 1},\n", + "\\label{_auto2} \\tag{2}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $s_i \\in \\{-1, 1\\}$ and $s_{N + 1} = s_1$. The number of spins\n", + "in the system is determined by $L$. For the one-dimensional system\n", + "there is no phase transition.\n", + "\n", + "We will look at a system of $L = 40$ spins with a coupling constant of\n", + "$J = 1$. To get enough training data we will generate 10000 states\n", + "with their respective energies." + ] + }, + { + "cell_type": "code", + "execution_count": 37, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "import numpy as np\n", + "import matplotlib.pyplot as plt\n", + "from mpl_toolkits.axes_grid1 import make_axes_locatable\n", + "import seaborn as sns\n", + "import scipy.linalg as scl\n", + "from sklearn.model_selection import train_test_split\n", + "import tqdm\n", + "sns.set(color_codes=True)\n", + "cmap_args=dict(vmin=-1., vmax=1., cmap='seismic')\n", + "\n", + "L = 40\n", + "n = int(1e4)\n", + "\n", + "spins = np.random.choice([-1, 1], size=(n, L))\n", + "J = 1.0\n", + "\n", + "energies = np.zeros(n)\n", + "\n", + "for i in range(n):\n", + " energies[i] = - J * np.dot(spins[i], np.roll(spins[i], 1))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Here we use ordinary least squares\n", + "regression to predict the energy for the nearest neighbor\n", + "one-dimensional Ising model on a ring, i.e., the endpoints wrap\n", + "around. We will use linear regression to fit a value for\n", + "the coupling constant to achieve this.\n", + "\n", + "## Reformulating the problem to suit regression\n", + "\n", + "A more general form for the one-dimensional Ising model is" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + " H = - \\sum_j^L \\sum_k^L s_j s_k J_{jk}.\n", + "\\label{_auto3} \\tag{3}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Here we allow for interactions beyond the nearest neighbors and a state dependent\n", + "coupling constant. This latter expression can be formulated as\n", + "a matrix-product" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + " \\boldsymbol{H} = \\boldsymbol{X} J,\n", + "\\label{_auto4} \\tag{4}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $X_{jk} = s_j s_k$ and $J$ is a matrix which consists of the\n", + "elements $-J_{jk}$. This form of writing the energy fits perfectly\n", + "with the form utilized in linear regression, that is" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + " \\boldsymbol{y} = \\boldsymbol{X}\\boldsymbol{\\beta} + \\boldsymbol{\\epsilon},\n", + "\\label{_auto5} \\tag{5}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We split the data in training and test data as discussed in the previous example" + ] + }, + { + "cell_type": "code", + "execution_count": 38, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "X = np.zeros((n, L ** 2))\n", + "for i in range(n):\n", + " X[i] = np.outer(spins[i], spins[i]).ravel()\n", + "y = energies\n", + "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Linear regression\n", + "\n", + "In the ordinary least squares method we choose the cost function" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + " C(\\boldsymbol{X}, \\boldsymbol{\\beta})= \\frac{1}{n}\\left\\{(\\boldsymbol{X}\\boldsymbol{\\beta} - \\boldsymbol{y})^T(\\boldsymbol{X}\\boldsymbol{\\beta} - \\boldsymbol{y})\\right\\}.\n", + "\\label{_auto6} \\tag{6}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We then find the extremal point of $C$ by taking the derivative with respect to $\\boldsymbol{\\beta}$ as discussed above.\n", + "This yields the expression for $\\boldsymbol{\\beta}$ to be" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{\\beta} = \\frac{\\boldsymbol{X}^T \\boldsymbol{y}}{\\boldsymbol{X}^T \\boldsymbol{X}},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "which immediately imposes some requirements on $\\boldsymbol{X}$ as there must exist\n", + "an inverse of $\\boldsymbol{X}^T \\boldsymbol{X}$. If the expression we are modeling contains an\n", + "intercept, i.e., a constant term, we must make sure that the\n", + "first column of $\\boldsymbol{X}$ consists of $1$. We do this here" + ] + }, + { + "cell_type": "code", + "execution_count": 39, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "X_train_own = np.concatenate(\n", + " (np.ones(len(X_train))[:, np.newaxis], X_train),\n", + " axis=1\n", + ")\n", + "X_test_own = np.concatenate(\n", + " (np.ones(len(X_test))[:, np.newaxis], X_test),\n", + " axis=1\n", + ")" + ] + }, + { + "cell_type": "code", + "execution_count": 40, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "def ols_inv(x: np.ndarray, y: np.ndarray) -> np.ndarray:\n", + " return scl.inv(x.T @ x) @ (x.T @ y)\n", + "beta = ols_inv(X_train_own, y_train)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Singular Value decomposition\n", + "\n", + "Doing the inversion directly turns out to be a bad idea since the matrix\n", + "$\\boldsymbol{X}^T\\boldsymbol{X}$ is singular. An alternative approach is to use the **singular\n", + "value decomposition**. Using the definition of the Moore-Penrose\n", + "pseudoinverse we can write the equation for $\\boldsymbol{\\beta}$ as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{\\beta} = \\boldsymbol{X}^{+}\\boldsymbol{y},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where the pseudoinverse of $\\boldsymbol{X}$ is given by" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{X}^{+} = \\frac{\\boldsymbol{X}^T}{\\boldsymbol{X}^T\\boldsymbol{X}}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Using singular value decomposition we can decompose the matrix $\\boldsymbol{X} = \\boldsymbol{U}\\boldsymbol{\\Sigma} \\boldsymbol{V}^T$,\n", + "where $\\boldsymbol{U}$ and $\\boldsymbol{V}$ are orthogonal(unitary) matrices and $\\boldsymbol{\\Sigma}$ contains the singular values (more details below).\n", + "where $X^{+} = V\\Sigma^{+} U^T$. This reduces the equation for\n", + "$\\omega$ to" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + " \\boldsymbol{\\beta} = \\boldsymbol{V}\\boldsymbol{\\Sigma}^{+} \\boldsymbol{U}^T \\boldsymbol{y}.\n", + "\\label{_auto7} \\tag{7}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Note that solving this equation by actually doing the pseudoinverse\n", + "(which is what we will do) is not a good idea as this operation scales\n", + "as $\\mathcal{O}(n^3)$, where $n$ is the number of elements in a\n", + "general matrix. Instead, doing $QR$-factorization and solving the\n", + "linear system as an equation would reduce this down to\n", + "$\\mathcal{O}(n^2)$ operations." + ] + }, + { + "cell_type": "code", + "execution_count": 41, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "def ols_svd(x: np.ndarray, y: np.ndarray) -> np.ndarray:\n", + " u, s, v = scl.svd(x)\n", + " return v.T @ scl.pinv(scl.diagsvd(s, u.shape[0], v.shape[0])) @ u.T @ y" + ] + }, + { + "cell_type": "code", + "execution_count": 42, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "beta = ols_svd(X_train_own,y_train)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "When extracting the $J$-matrix we need to make sure that we remove the intercept, as is done here" + ] + }, + { + "cell_type": "code", + "execution_count": 43, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "J = beta[1:].reshape(L, L)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "A way of looking at the coefficients in $J$ is to plot the matrices as images." + ] + }, + { + "cell_type": "code", + "execution_count": 44, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "fig = plt.figure(figsize=(20, 14))\n", + "im = plt.imshow(J, **cmap_args)\n", + "plt.title(\"OLS\", fontsize=18)\n", + "plt.xticks(fontsize=18)\n", + "plt.yticks(fontsize=18)\n", + "cb = fig.colorbar(im)\n", + "cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "It is interesting to note that OLS\n", + "considers both $J_{j, j + 1} = -0.5$ and $J_{j, j - 1} = -0.5$ as\n", + "valid matrix elements for $J$.\n", + "In our discussion below on hyperparameters and Ridge and Lasso regression we will see that\n", + "this problem can be removed, partly and only with Lasso regression. \n", + "\n", + "In this case our matrix inversion was actually possible. The obvious question now is what is the mathematics behind the SVD?\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "## The one-dimensional Ising model\n", + "\n", + "Let us bring back the Ising model again, but now with an additional\n", + "focus on Ridge and Lasso regression as well. We repeat some of the\n", + "basic parts of the Ising model and the setup of the training and test\n", + "data. The one-dimensional Ising model with nearest neighbor\n", + "interaction, no external field and a constant coupling constant $J$ is\n", + "given by" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + " H = -J \\sum_{k}^L s_k s_{k + 1},\n", + "\\label{_auto8} \\tag{8}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $s_i \\in \\{-1, 1\\}$ and $s_{N + 1} = s_1$. The number of spins in the system is determined by $L$. For the one-dimensional system there is no phase transition.\n", + "\n", + "We will look at a system of $L = 40$ spins with a coupling constant of $J = 1$. To get enough training data we will generate 10000 states with their respective energies." + ] + }, + { + "cell_type": "code", + "execution_count": 45, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "import numpy as np\n", + "import matplotlib.pyplot as plt\n", + "from mpl_toolkits.axes_grid1 import make_axes_locatable\n", + "import seaborn as sns\n", + "import scipy.linalg as scl\n", + "from sklearn.model_selection import train_test_split\n", + "import sklearn.linear_model as skl\n", + "import tqdm\n", + "sns.set(color_codes=True)\n", + "cmap_args=dict(vmin=-1., vmax=1., cmap='seismic')\n", + "\n", + "L = 40\n", + "n = int(1e4)\n", + "\n", + "spins = np.random.choice([-1, 1], size=(n, L))\n", + "J = 1.0\n", + "\n", + "energies = np.zeros(n)\n", + "\n", + "for i in range(n):\n", + " energies[i] = - J * np.dot(spins[i], np.roll(spins[i], 1))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "A more general form for the one-dimensional Ising model is" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + " H = - \\sum_j^L \\sum_k^L s_j s_k J_{jk}.\n", + "\\label{_auto9} \\tag{9}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Here we allow for interactions beyond the nearest neighbors and a more\n", + "adaptive coupling matrix. This latter expression can be formulated as\n", + "a matrix-product on the form" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + " H = X J,\n", + "\\label{_auto10} \\tag{10}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $X_{jk} = s_j s_k$ and $J$ is the matrix consisting of the\n", + "elements $-J_{jk}$. This form of writing the energy fits perfectly\n", + "with the form utilized in linear regression, viz." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + " \\boldsymbol{y} = \\boldsymbol{X}\\boldsymbol{\\beta} + \\boldsymbol{\\epsilon}.\n", + "\\label{_auto11} \\tag{11}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We organize the data as we did above" + ] + }, + { + "cell_type": "code", + "execution_count": 46, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "X = np.zeros((n, L ** 2))\n", + "for i in range(n):\n", + " X[i] = np.outer(spins[i], spins[i]).ravel()\n", + "y = energies\n", + "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.96)\n", + "\n", + "X_train_own = np.concatenate(\n", + " (np.ones(len(X_train))[:, np.newaxis], X_train),\n", + " axis=1\n", + ")\n", + "\n", + "X_test_own = np.concatenate(\n", + " (np.ones(len(X_test))[:, np.newaxis], X_test),\n", + " axis=1\n", + ")" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We will do all fitting with **Scikit-Learn**," + ] + }, + { + "cell_type": "code", + "execution_count": 47, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "clf = skl.LinearRegression().fit(X_train, y_train)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "When extracting the $J$-matrix we make sure to remove the intercept" + ] + }, + { + "cell_type": "code", + "execution_count": 48, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "J_sk = clf.coef_.reshape(L, L)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "And then we plot the results" + ] + }, + { + "cell_type": "code", + "execution_count": 49, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "fig = plt.figure(figsize=(20, 14))\n", + "im = plt.imshow(J_sk, **cmap_args)\n", + "plt.title(\"LinearRegression from Scikit-learn\", fontsize=18)\n", + "plt.xticks(fontsize=18)\n", + "plt.yticks(fontsize=18)\n", + "cb = fig.colorbar(im)\n", + "cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The results perfectly with our previous discussion where we used our own code.\n", + "\n", + "## Ridge regression\n", + "\n", + "Having explored the ordinary least squares we move on to ridge\n", + "regression. In ridge regression we include a **regularizer**. This\n", + "involves a new cost function which leads to a new estimate for the\n", + "weights $\\boldsymbol{\\beta}$. This results in a penalized regression problem. The\n", + "cost function is given by" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "1\n", + "3\n", + "6\n", + " \n", + "<\n", + "<\n", + "<\n", + "!\n", + "!\n", + "M\n", + "A\n", + "T\n", + "H\n", + "_\n", + "B\n", + "L\n", + "O\n", + "C\n", + "K" + ] + }, + { + "cell_type": "code", + "execution_count": 50, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "_lambda = 0.1\n", + "clf_ridge = skl.Ridge(alpha=_lambda).fit(X_train, y_train)\n", + "J_ridge_sk = clf_ridge.coef_.reshape(L, L)\n", + "fig = plt.figure(figsize=(20, 14))\n", + "im = plt.imshow(J_ridge_sk, **cmap_args)\n", + "plt.title(\"Ridge from Scikit-learn\", fontsize=18)\n", + "plt.xticks(fontsize=18)\n", + "plt.yticks(fontsize=18)\n", + "cb = fig.colorbar(im)\n", + "cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)\n", + "\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## LASSO regression\n", + "\n", + "In the **Least Absolute Shrinkage and Selection Operator** (LASSO)-method we get a third cost function." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + " C(\\boldsymbol{X}, \\boldsymbol{\\beta}; \\lambda) = (\\boldsymbol{X}\\boldsymbol{\\beta} - \\boldsymbol{y})^T(\\boldsymbol{X}\\boldsymbol{\\beta} - \\boldsymbol{y}) + \\lambda \\sqrt{\\boldsymbol{\\beta}^T\\boldsymbol{\\beta}}.\n", + "\\label{_auto13} \\tag{13}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Finding the extremal point of this cost function is not so straight-forward as in least squares and ridge. We will therefore rely solely on the function ``Lasso`` from **Scikit-Learn**." + ] + }, + { + "cell_type": "code", + "execution_count": 51, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "clf_lasso = skl.Lasso(alpha=_lambda).fit(X_train, y_train)\n", + "J_lasso_sk = clf_lasso.coef_.reshape(L, L)\n", + "fig = plt.figure(figsize=(20, 14))\n", + "im = plt.imshow(J_lasso_sk, **cmap_args)\n", + "plt.title(\"Lasso from Scikit-learn\", fontsize=18)\n", + "plt.xticks(fontsize=18)\n", + "plt.yticks(fontsize=18)\n", + "cb = fig.colorbar(im)\n", + "cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)\n", + "\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "It is quite striking how LASSO breaks the symmetry of the coupling\n", + "constant as opposed to ridge and OLS. We get a sparse solution with\n", + "$J_{j, j + 1} = -1$.\n", + "\n", + "\n", + "\n", + "## Performance as function of the regularization parameter\n", + "\n", + "We see how the different models perform for a different set of values for $\\lambda$." + ] + }, + { + "cell_type": "code", + "execution_count": 52, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "lambdas = np.logspace(-4, 5, 10)\n", + "\n", + "train_errors = {\n", + " \"ols_sk\": np.zeros(lambdas.size),\n", + " \"ridge_sk\": np.zeros(lambdas.size),\n", + " \"lasso_sk\": np.zeros(lambdas.size)\n", + "}\n", + "\n", + "test_errors = {\n", + " \"ols_sk\": np.zeros(lambdas.size),\n", + " \"ridge_sk\": np.zeros(lambdas.size),\n", + " \"lasso_sk\": np.zeros(lambdas.size)\n", + "}\n", + "\n", + "plot_counter = 1\n", + "\n", + "fig = plt.figure(figsize=(32, 54))\n", + "\n", + "for i, _lambda in enumerate(tqdm.tqdm(lambdas)):\n", + " for key, method in zip(\n", + " [\"ols_sk\", \"ridge_sk\", \"lasso_sk\"],\n", + " [skl.LinearRegression(), skl.Ridge(alpha=_lambda), skl.Lasso(alpha=_lambda)]\n", + " ):\n", + " method = method.fit(X_train, y_train)\n", + "\n", + " train_errors[key][i] = method.score(X_train, y_train)\n", + " test_errors[key][i] = method.score(X_test, y_test)\n", + "\n", + " omega = method.coef_.reshape(L, L)\n", + "\n", + " plt.subplot(10, 5, plot_counter)\n", + " plt.imshow(omega, **cmap_args)\n", + " plt.title(r\"%s, $\\lambda = %.4f$\" % (key, _lambda))\n", + " plot_counter += 1\n", + "\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We see that LASSO reaches a good solution for low\n", + "values of $\\lambda$, but will \"wither\" when we increase $\\lambda$ too\n", + "much. Ridge is more stable over a larger range of values for\n", + "$\\lambda$, but eventually also fades away.\n", + "\n", + "## Finding the optimal value of $\\lambda$\n", + "\n", + "To determine which value of $\\lambda$ is best we plot the accuracy of\n", + "the models when predicting the training and the testing set. We expect\n", + "the accuracy of the training set to be quite good, but if the accuracy\n", + "of the testing set is much lower this tells us that we might be\n", + "subject to an overfit model. The ideal scenario is an accuracy on the\n", + "testing set that is close to the accuracy of the training set." + ] + }, + { + "cell_type": "code", + "execution_count": 53, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "fig = plt.figure(figsize=(20, 14))\n", + "\n", + "colors = {\n", + " \"ols_sk\": \"r\",\n", + " \"ridge_sk\": \"y\",\n", + " \"lasso_sk\": \"c\"\n", + "}\n", + "\n", + "for key in train_errors:\n", + " plt.semilogx(\n", + " lambdas,\n", + " train_errors[key],\n", + " colors[key],\n", + " label=\"Train {0}\".format(key),\n", + " linewidth=4.0\n", + " )\n", + "\n", + "for key in test_errors:\n", + " plt.semilogx(\n", + " lambdas,\n", + " test_errors[key],\n", + " colors[key] + \"--\",\n", + " label=\"Test {0}\".format(key),\n", + " linewidth=4.0\n", + " )\n", + "plt.legend(loc=\"best\", fontsize=18)\n", + "plt.xlabel(r\"$\\lambda$\", fontsize=18)\n", + "plt.ylabel(r\"$R^2$\", fontsize=18)\n", + "plt.tick_params(labelsize=18)\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "From the above figure we can see that LASSO with $\\lambda = 10^{-2}$\n", + "achieves a very good accuracy on the test set. This by far surpasses the\n", + "other models for all values of $\\lambda$." + ] + } + ], + "metadata": {}, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/doc/src/LectureNotes/chapter5.ipynb b/doc/src/LectureNotes/_build/html/_sources/chapter4.ipynb similarity index 69% rename from doc/src/LectureNotes/chapter5.ipynb rename to doc/src/LectureNotes/_build/html/_sources/chapter4.ipynb index 4da84cb3d..8372ef384 100644 --- a/doc/src/LectureNotes/chapter5.ipynb +++ b/doc/src/LectureNotes/_build/html/_sources/chapter4.ipynb @@ -4,22 +4,9 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "\n", - "# Data Analysis and Machine Learning: Logistic Regression\n", - "\n", - " \n", - "**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University\n", - "\n", - "Date: **Oct 17, 2019**\n", - "\n", - "Copyright 1999-2019, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Logistic Regression\n", + "# Logistic Regression\n", "\n", + "## Introduction\n", "In linear regression our main interest was centered on learning the\n", "coefficients of a functional fit (say a polynomial) in order to be\n", "able to predict the response of a continuous variable on some unseen\n", @@ -49,8 +36,6 @@ "binary outcome, true or false, positive or negative, success or\n", "failure etc.\n", "\n", - "## Optimization and Deep learning\n", - "\n", "Logistic regression will also serve as our stepping stone towards\n", "neural network algorithms and supervised deep learning. For logistic\n", "learning, the minimization of the cost function leads to a non-linear\n", @@ -66,7 +51,7 @@ "models, as we will see later.\n", "\n", "\n", - "\n", + "\n", "## Basics\n", "\n", "We consider the case where the dependent variables, also called the\n", @@ -98,8 +83,6 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "## Linear classifier\n", - "\n", "Before moving to the logistic model, let us try to use our linear\n", "regression model to classify these two outcomes. We could for example\n", "fit a linear model to the default case if $y_i > 0.5$ and the no\n", @@ -131,7 +114,6 @@ "where $\\hat{y}$ is a vector representing the possible outcomes, $\\hat{X}$ is our\n", "$n\\times p$ design matrix and $\\hat{\\beta}$ represents our estimators/predictors.\n", "\n", - "## Some selected properties\n", "\n", "The main problem with our function is that it takes values on the\n", "entire real axis. In the case of logistic regression, however, the\n", @@ -149,6 +131,7 @@ "the probability of a given category. This leads us to the logistic function.\n", "\n", "\n", + "\n", "## The logistic function\n", "\n", "The perceptron is an example of a ``hard classification\" model. We\n", @@ -178,8 +161,6 @@ "source": [ "Note that $1-p(t)= p(-t)$.\n", "\n", - "## Examples of likelihood functions used in logistic regression and nueral networks\n", - "\n", "\n", "The following code plots the logistic function, the step function and other functions we will encounter from here and on." ] @@ -187,7 +168,9 @@ { "cell_type": "code", "execution_count": 1, - "metadata": {}, + "metadata": { + "collapsed": false + }, "outputs": [], "source": [ "%matplotlib inline\n", @@ -291,7 +274,6 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "\n", "## Maximum likelihood\n", "\n", "In order to define the total likelihood for all possible outcomes from a \n", @@ -333,8 +315,6 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "## The cost function rewritten\n", - "\n", "Reordering the logarithms, we can rewrite the **cost/loss** function as" ] }, @@ -371,7 +351,6 @@ "This equation is known in statistics as the **cross entropy**. Finally, we note that just as in linear regression, \n", "in practice we often supplement the cross-entropy with additional regularization terms, usually $L_1$ and $L_2$ regularization as we did for Ridge and Lasso regression.\n", "\n", - "## Minimizing the cross entropy\n", "\n", "The cross entropy is a convex function of the weights $\\hat{\\beta}$ and,\n", "therefore, any local minimizer is a global minimizer. \n", @@ -410,8 +389,6 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "## A more compact expression\n", - "\n", "Let us now define a vector $\\hat{y}$ with $n$ elements $y_i$, an\n", "$n\\times p$ matrix $\\hat{X}$ which contains the $x_i$ values and a\n", "vector $\\hat{p}$ of fitted probabilities $p(y_i\\vert x_i,\\hat{\\beta})$. We can rewrite in a more compact form the first\n", @@ -448,8 +425,6 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "## Extending to more predictors\n", - "\n", "Within a binary classification problem, we can easily expand our model to include multiple predictors. Our ratio between likelihoods is then with $p$ predictors" ] }, @@ -547,7 +522,6 @@ "**logit** transformations.\n", "\n", "\n", - "## More classes\n", "\n", "In our discussion of neural networks we will encounter the above again\n", "in terms of a slightly modified function, the so-called **Softmax** function.\n", @@ -599,215 +573,11 @@ "To find the optimal parameters we would typically use a gradient\n", "descent method. Newton's method and gradient descent methods are\n", "discussed in the material on [optimization\n", - "methods](https://compphysics.github.io/MachineLearning/doc/pub/Splines/html/Splines-bs.html).\n", - "\n", - "\n", - "\n", - "\n", - "## A simple classification problem" - ] - }, - { - "cell_type": "code", - "execution_count": 2, - "metadata": {}, - "outputs": [], - "source": [ - "import numpy as np\n", - "from sklearn import datasets, linear_model\n", - "import matplotlib.pyplot as plt\n", - "\n", - "\n", - "def generate_data():\n", - " np.random.seed(0)\n", - " X, y = datasets.make_moons(200, noise=0.20)\n", - " return X, y\n", - "\n", - "\n", - "def visualize(X, y, clf):\n", - " plot_decision_boundary(lambda x: clf.predict(x), X, y)\n", - "\n", - "def plot_decision_boundary(pred_func, X, y):\n", - " # Set min and max values and give it some padding\n", - " x_min, x_max = X[:, 0].min() - .5, X[:, 0].max() + .5\n", - " y_min, y_max = X[:, 1].min() - .5, X[:, 1].max() + .5\n", - " h = 0.01\n", - " # Generate a grid of points with distance h between them\n", - " xx, yy = np.meshgrid(np.arange(x_min, x_max, h), np.arange(y_min, y_max, h))\n", - " # Predict the function value for the whole gid\n", - " Z = pred_func(np.c_[xx.ravel(), yy.ravel()])\n", - " Z = Z.reshape(xx.shape)\n", - " # Plot the contour and training examples\n", - " plt.contourf(xx, yy, Z, cmap=plt.cm.Spectral)\n", - " plt.scatter(X[:, 0], X[:, 1], c=y, cmap=plt.cm.Spectral)\n", - " plt.show()\n", - "\n", - "\n", - "def classify(X, y):\n", - " clf = linear_model.LogisticRegressionCV()\n", - " clf.fit(X, y)\n", - " return clf\n", - "\n", - "\n", - "def main():\n", - " X, y = generate_data()\n", - " # visualize(X, y)\n", - " clf = classify(X, y)\n", - " visualize(X, y, clf)\n", - "\n", - "if __name__ == \"__main__\":\n", - " main()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## The Credit Card example\n", - "Here we use the the [credit card data](https://archive.ics.uci.edu/ml/datasets/default+of+credit+card+clients). \n", - "The data are from an extensive database from Taiwan and include more than ten predictors.\n", - "\n", - "For categorical data -Scikit-Learn- provides a so-called **one-hot encoder**.\n", - "This is called one-hot\n", - "encoding, because only one attribute will be equal to 1 (hot), while the others will be 0 (cold).\n", - "**Scikit-Learn** provides a OneHotEncoder encoder to convert integer categorical values into one-hot" - ] - }, - { - "cell_type": "code", - "execution_count": 3, - "metadata": {}, - "outputs": [], - "source": [ - "from sklearn.preprocessing import OneHotEncoder\n", - "encoder = OneHotEncoder()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## How to read the Credit Card data" - ] - }, - { - "cell_type": "code", - "execution_count": 4, - "metadata": {}, - "outputs": [], - "source": [ - "import pandas as pd\n", - "import os\n", - "import numpy as np\n", - "\n", - "\n", - "from sklearn.model_selection import train_test_split\n", - "from sklearn.preprocessing import OneHotEncoder\n", - "from sklearn.compose import ColumnTransformer\n", - "from sklearn.preprocessing import StandardScaler, OneHotEncoder\n", - "from sklearn.metrics import confusion_matrix, accuracy_score, roc_auc_score\n", - "\n", - "# Trying to set the seed\n", - "np.random.seed(0)\n", - "import random\n", - "random.seed(0)\n", - "\n", - "# Reading file into data frame\n", - "cwd = os.getcwd()\n", - "filename = cwd + '/default of credit card clients.xls'\n", - "nanDict = {}\n", - "df = pd.read_excel(filename, header=1, skiprows=0, index_col=0, na_values=nanDict)\n", - "\n", - "df.rename(index=str, columns={\"default payment next month\": \"defaultPaymentNextMonth\"}, inplace=True)\n", - "\n", - "# Features and targets \n", - "X = df.loc[:, df.columns != 'defaultPaymentNextMonth'].values\n", - "y = df.loc[:, df.columns == 'defaultPaymentNextMonth'].values\n", - "\n", - "# Categorical variables to one-hot's\n", - "onehotencoder = OneHotEncoder(categories=\"auto\")\n", - "\n", - "X = ColumnTransformer(\n", - " [(\"\", onehotencoder, [3]),],\n", - " remainder=\"passthrough\"\n", - ").fit_transform(X)\n", - "\n", - "y.shape\n", - "\n", - "# Train-test split\n", - "trainingShare = 0.5 \n", - "seed = 1\n", - "XTrain, XTest, yTrain, yTest=train_test_split(X, y, train_size=trainingShare, \\\n", - " test_size = 1-trainingShare,\n", - " random_state=seed)\n", - "\n", - "# Input Scaling\n", - "sc = StandardScaler()\n", - "XTrain = sc.fit_transform(XTrain)\n", - "XTest = sc.transform(XTest)\n", - "\n", - "# One-hot's of the target vector\n", - "Y_train_onehot, Y_test_onehot = onehotencoder.fit_transform(yTrain), onehotencoder.fit_transform(yTest)\n", - "\n", - "# Remove instances with zeros only for past bill statements or paid amounts\n", - "'''\n", - "df = df.drop(df[(df.BILL_AMT1 == 0) &\n", - " (df.BILL_AMT2 == 0) &\n", - " (df.BILL_AMT3 == 0) &\n", - " (df.BILL_AMT4 == 0) &\n", - " (df.BILL_AMT5 == 0) &\n", - " (df.BILL_AMT6 == 0) &\n", - " (df.PAY_AMT1 == 0) &\n", - " (df.PAY_AMT2 == 0) &\n", - " (df.PAY_AMT3 == 0) &\n", - " (df.PAY_AMT4 == 0) &\n", - " (df.PAY_AMT5 == 0) &\n", - " (df.PAY_AMT6 == 0)].index)\n", - "'''\n", - "df = df.drop(df[(df.BILL_AMT1 == 0) &\n", - " (df.BILL_AMT2 == 0) &\n", - " (df.BILL_AMT3 == 0) &\n", - " (df.BILL_AMT4 == 0) &\n", - " (df.BILL_AMT5 == 0) &\n", - " (df.BILL_AMT6 == 0)].index)\n", - "\n", - "df = df.drop(df[(df.PAY_AMT1 == 0) &\n", - " (df.PAY_AMT2 == 0) &\n", - " (df.PAY_AMT3 == 0) &\n", - " (df.PAY_AMT4 == 0) &\n", - " (df.PAY_AMT5 == 0) &\n", - " (df.PAY_AMT6 == 0)].index)\n", - "\n", - "from sklearn.linear_model import LogisticRegression\n", - "from sklearn.model_selection import GridSearchCV\n", - "\n", - "lambdas=np.logspace(-5,7,13)\n", - "parameters = [{'C': 1./lambdas, \"solver\":[\"lbfgs\"]}]#*len(parameters)}]\n", - "scoring = ['accuracy', 'roc_auc']\n", - "logReg = LogisticRegression()\n", - "gridSearch = GridSearchCV(logReg, parameters, cv=5, scoring=scoring, refit='roc_auc')" + "methods](https://compphysics.github.io/MachineLearning/doc/pub/Splines/html/Splines-bs.html)." ] } ], - "metadata": { - "kernelspec": { - "display_name": "Python 3", - "language": "python", - "name": "python3" - }, - "language_info": { - "codemirror_mode": { - "name": "ipython", - "version": 3 - }, - "file_extension": ".py", - "mimetype": "text/x-python", - "name": "python", - "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.7.4" - } - }, + "metadata": {}, "nbformat": 4, - "nbformat_minor": 2 + "nbformat_minor": 4 } diff --git a/doc/src/LectureNotes/_build/html/_sources/content.md b/doc/src/LectureNotes/_build/html/_sources/content.md deleted file mode 100644 index 0f6aca77a..000000000 --- a/doc/src/LectureNotes/_build/html/_sources/content.md +++ /dev/null @@ -1,5 +0,0 @@ -Content in Jupyter Book -======================= - -There are many ways to write content in Jupyter Book. This short section -covers a few tips for how to do so. diff --git a/doc/src/LectureNotes/_build/html/_sources/intro.md b/doc/src/LectureNotes/_build/html/_sources/intro.md deleted file mode 100644 index 19ce5e005..000000000 --- a/doc/src/LectureNotes/_build/html/_sources/intro.md +++ /dev/null @@ -1,7 +0,0 @@ -Welcome to your Jupyter Book -============================ - -This is a small sample book to give you a feel for how book content is -structured. - -Check out the content pages bundled with this sample book to get started. diff --git a/doc/src/LectureNotes/_build/html/_sources/markdown.md b/doc/src/LectureNotes/_build/html/_sources/markdown.md deleted file mode 100644 index c186497d2..000000000 --- a/doc/src/LectureNotes/_build/html/_sources/markdown.md +++ /dev/null @@ -1,125 +0,0 @@ -# Markdown Files - -Whether you write your book's content in Jupyter Notebooks (`.ipynb`) or -in regular markdown files (`.md`), you'll write in the same flavor of markdown -called **MyST Markdown**. - -## What is MyST? - -MyST stands for "Markedly Structured Text". It -is a slight variation on a flavor of markdown called "CommonMark" markdown, -with small syntax extensions to allow you to write **roles** and **directives** -in the Sphinx ecosystem. - -## What are roles and directives? - -Roles and directives are two of the most powerful tools in Jupyter Book. They -are kind of like functions, but written in a markup language. They both -serve a similar purpose, but **roles are written in one line**, whereas -**directives span many lines**. They both accept different kinds of inputs, -and what they do with those inputs depends on the specific role or directive -that is being called. - -### Using a directive - -At its simplest, you can insert a directive into your book's content like so: - -```` -```{mydirectivename} -My directive content -``` -```` - -This will only work if a directive with name `mydirectivename` already exists -(which it doesn't). There are many pre-defined directives associated with -Jupyter Book. For example, to insert a note box into your content, you can -use the following directive: - -```` -```{note} -Here is a note -``` -```` - -This results in: - -```{note} -Here is a note -``` - -In your built book. - -For more information on writing directives, see the -[MyST documentation](https://myst-parser.readthedocs.io/). - - -### Using a role - -Roles are very similar to directives, but they are less-complex and written -entirely on one line. You can insert a role into your book's content with -this pattern: - -``` -Some content {rolename}`and here is my role's content!` -``` - -Again, roles will only work if `rolename` is a valid role's name. For example, -the `doc` role can be used to refer to another page in your book. You can -refer directly to another page by its relative path. For example, the -role syntax `` {doc}`intro` `` will result in: {doc}`intro`. - -For more information on writing roles, see the -[MyST documentation](https://myst-parser.readthedocs.io/). - - -### Adding a citation - -You can also cite references that are stored in a `bibtex` file. For example, -the following syntax: `` {cite}`holdgraf_evidence_2014` `` will render like -this: {cite}`holdgraf_evidence_2014`. - -Moreoever, you can insert a bibliography into your page with this syntax: -The `{bibliography}` directive must be used for all the `{cite}` roles to -render properly. -For example, if the references for your book are stored in `references.bib`, -then the bibliography is inserted with: - -```` -```{bibliography} references.bib -``` -```` - -Resulting in a rendered bibliography that looks like: - -```{bibliography} references.bib -``` - - -### Executing code in your markdown files - -If you'd like to include computational content inside these markdown files, -you can use MyST Markdown to define cells that will be executed when your -book is built. Jupyter Book uses *jupytext* to do this. - -First, add Jupytext metadata to the file. For example, to add Jupytext metadata -to this markdown page, run this command: - -``` -jupyter-book myst init markdown.md -``` - -Once a markdown file has Jupytext metadata in it, you can add the following -directive to run the code at build time: - -```` -```{code-cell} -print("Here is some code to execute") -``` -```` - -When your book is built, the contents of any `{code-cell}` blocks will be -executed with your default Jupyter kernel, and their outputs will be displayed -in-line with the rest of your content. - -For more information about executing computational content with Jupyter Book, -see [The MyST-NB documentation](https://myst-nb.readthedocs.io/). diff --git a/doc/src/LectureNotes/_build/html/_sources/notebooks.ipynb b/doc/src/LectureNotes/_build/html/_sources/notebooks.ipynb deleted file mode 100644 index b33ee373d..000000000 --- a/doc/src/LectureNotes/_build/html/_sources/notebooks.ipynb +++ /dev/null @@ -1,122 +0,0 @@ -{ - "cells": [ - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "# Content with notebooks\n", - "\n", - "You can also create content with Jupyter Notebooks. This means that you can include\n", - "code blocks and their outputs in your book.\n", - "\n", - "## Markdown + notebooks\n", - "\n", - "As it is markdown, you can embed images, HTML, etc into your posts!\n", - "\n", - "![](https://myst-parser.readthedocs.io/en/latest/_static/logo.png)\n", - "\n", - "You an also $add_{math}$ and\n", - "\n", - "$$\n", - "math^{blocks}\n", - "$$\n", - "\n", - "or\n", - "\n", - "$$\n", - "\\begin{aligned}\n", - "\\mbox{mean} la_{tex} \\\\ \\\\\n", - "math blocks\n", - "\\end{aligned}\n", - "$$\n", - "\n", - "But make sure you \\$Escape \\$your \\$dollar signs \\$you want to keep!\n", - "\n", - "## MyST markdown\n", - "\n", - "MyST markdown works in Jupyter Notebooks as well. For more information about MyST markdown, check\n", - "out [the MyST guide in Jupyter Book](https://jupyterbook.org/content/myst.html),\n", - "or see [the MyST markdown documentation](https://myst-parser.readthedocs.io/en/latest/).\n", - "\n", - "## Code blocks and outputs\n", - "\n", - "Jupyter Book will also embed your code blocks and output in your book.\n", - "For example, here's some sample Matplotlib code:" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [ - "from matplotlib import rcParams, cycler\n", - "import matplotlib.pyplot as plt\n", - "import numpy as np\n", - "plt.ion()" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [ - "# Fixing random state for reproducibility\n", - "np.random.seed(19680801)\n", - "\n", - "N = 10\n", - "data = [np.logspace(0, 1, 100) + np.random.randn(100) + ii for ii in range(N)]\n", - "data = np.array(data).T\n", - "cmap = plt.cm.coolwarm\n", - "rcParams['axes.prop_cycle'] = cycler(color=cmap(np.linspace(0, 1, N)))\n", - "\n", - "\n", - "from matplotlib.lines import Line2D\n", - "custom_lines = [Line2D([0], [0], color=cmap(0.), lw=4),\n", - " Line2D([0], [0], color=cmap(.5), lw=4),\n", - " Line2D([0], [0], color=cmap(1.), lw=4)]\n", - "\n", - "fig, ax = plt.subplots(figsize=(10, 5))\n", - "lines = ax.plot(data)\n", - "ax.legend(custom_lines, ['Cold', 'Medium', 'Hot']);" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "There is a lot more that you can do with outputs (such as including interactive outputs)\n", - "with your book. For more information about this, see [the Jupyter Book documentation](https://jupyterbook.org)" - ] - } - ], - "metadata": { - "kernelspec": { - "display_name": "Python 3", - "language": "python", - "name": "python3" - }, - "language_info": { - "codemirror_mode": { - "name": "ipython", - "version": 3 - }, - "file_extension": ".py", - "mimetype": "text/x-python", - "name": "python", - "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.8.0" - }, - "widgets": { - "application/vnd.jupyter.widget-state+json": { - "state": {}, - "version_major": 2, - "version_minor": 0 - } - } - }, - "nbformat": 4, - "nbformat_minor": 4 -} diff --git a/doc/src/LectureNotes/_build/html/_static/Picture.png b/doc/src/LectureNotes/_build/html/_static/Picture.png new file mode 100644 index 000000000..2d139ee9b Binary files /dev/null and b/doc/src/LectureNotes/_build/html/_static/Picture.png differ diff --git a/doc/src/LectureNotes/_build/html/_static/logo.png b/doc/src/LectureNotes/_build/html/_static/logo.png deleted file mode 100644 index 06d56f40c..000000000 Binary files a/doc/src/LectureNotes/_build/html/_static/logo.png and /dev/null differ diff --git a/doc/src/LectureNotes/_build/html/_static/panels-main.c949a650a448cc0ae9fd3441c0e17fb0.css b/doc/src/LectureNotes/_build/html/_static/panels-main.c949a650a448cc0ae9fd3441c0e17fb0.css new file mode 100644 index 000000000..fc14abc85 --- /dev/null +++ b/doc/src/LectureNotes/_build/html/_static/panels-main.c949a650a448cc0ae9fd3441c0e17fb0.css @@ -0,0 +1 @@ +details.dropdown .summary-title{padding-right:3em !important;-moz-user-select:none;-ms-user-select:none;-webkit-user-select:none;user-select:none}details.dropdown:hover{cursor:pointer}details.dropdown .summary-content{cursor:default}details.dropdown summary{list-style:none;padding:1em}details.dropdown summary .octicon.no-title{vertical-align:middle}details.dropdown[open] summary .octicon.no-title{visibility:hidden}details.dropdown summary::-webkit-details-marker{display:none}details.dropdown summary:focus{outline:none}details.dropdown summary:hover .summary-up svg,details.dropdown summary:hover .summary-down svg{opacity:1}details.dropdown .summary-up svg,details.dropdown .summary-down svg{display:block;opacity:.6}details.dropdown .summary-up,details.dropdown .summary-down{pointer-events:none;position:absolute;right:1em;top:.75em}details.dropdown[open] .summary-down{visibility:hidden}details.dropdown:not([open]) .summary-up{visibility:hidden}details.dropdown.fade-in[open] summary~*{-moz-animation:panels-fade-in .5s ease-in-out;-webkit-animation:panels-fade-in .5s ease-in-out;animation:panels-fade-in .5s ease-in-out}details.dropdown.fade-in-slide-down[open] summary~*{-moz-animation:panels-fade-in .5s ease-in-out, panels-slide-down .5s ease-in-out;-webkit-animation:panels-fade-in .5s ease-in-out, panels-slide-down .5s ease-in-out;animation:panels-fade-in .5s ease-in-out, panels-slide-down .5s ease-in-out}@keyframes panels-fade-in{0%{opacity:0}100%{opacity:1}}@keyframes panels-slide-down{0%{transform:translate(0, -10px)}100%{transform:translate(0, 0)}}.octicon{display:inline-block;fill:currentColor;vertical-align:text-top}.tabbed-content{box-shadow:0 -.0625rem var(--tabs-color-overline),0 .0625rem var(--tabs-color-underline);display:none;order:99;padding-bottom:.75rem;padding-top:.75rem;width:100%}.tabbed-content>:first-child{margin-top:0 !important}.tabbed-content>:last-child{margin-bottom:0 !important}.tabbed-content>.tabbed-set{margin:0}.tabbed-set{border-radius:.125rem;display:flex;flex-wrap:wrap;margin:1em 0;position:relative}.tabbed-set>input{opacity:0;position:absolute}.tabbed-set>input:checked+label{border-color:var(--tabs-color-label-active);color:var(--tabs-color-label-active)}.tabbed-set>input:checked+label+.tabbed-content{display:block}.tabbed-set>input:focus+label{outline-style:auto}.tabbed-set>input:not(.focus-visible)+label{outline:none;-webkit-tap-highlight-color:transparent}.tabbed-set>label{border-bottom:.125rem solid transparent;color:var(--tabs-color-label-inactive);cursor:pointer;font-size:var(--tabs-size-label);font-weight:700;padding:1em 1.25em .5em;transition:color 250ms;width:auto;z-index:1}html .tabbed-set>label:hover{color:var(--tabs-color-label-active)} diff --git a/doc/src/LectureNotes/_build/html/_static/panels-variables.06eb56fa6e07937060861dad626602ad.css b/doc/src/LectureNotes/_build/html/_static/panels-variables.06eb56fa6e07937060861dad626602ad.css new file mode 100644 index 000000000..adc616622 --- /dev/null +++ b/doc/src/LectureNotes/_build/html/_static/panels-variables.06eb56fa6e07937060861dad626602ad.css @@ -0,0 +1,7 @@ +:root { +--tabs-color-label-active: hsla(231, 99%, 66%, 1); 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Introduction to Applied Data Analysis and Machine Learning

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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, +Tensorflow, +PyTorch and Keras, 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

+
    +

  1. Learn about basic data analysis, Bayesian statistics, Monte Carlo methods, data optimization and machine learning;

    2. +

  2. Be capable of extending the acquired knowledge to other systems and cases;

    3. +

  3. Have an understanding of central algorithms used in data analysis and machine learning;

    4. +

  4. 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;

    5. +

  5. Understand methods for regression and classification;

    6. +

  6. Learn about neural network, genetic algorithms and Boltzmann machines;

    7. +

  7. 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).

    8. +
+

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.

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+

Machine Learning, a small (and probably biased) introduction

+

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 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 libraries +written in Python for machine learning like +Scikit-learn, +Tensorflow, +PyTorch and Keras, 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 (for example decision trees and random forests), 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 the algorithms and methods we will discuss.

+

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:

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  • 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 often 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.

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

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  • The first ingredient is normally our data set (which can be subdivided into training, validation and test data). Many find the most difficult part of using Machine Learning to be the set up of your data in a meaningful way.

    • +

  • The second item is a model which is normally a function of some parameters. The model reflects our knowledge of the system (or lack thereof). As an example, if we know that our data show a behavior similar to what would be predicted by a polynomial, fitting our data to a polynomial of some degree would then determin our model.

    • +

  • The last ingredient is a so-called cost/loss function (or error function) which allows us to present an estimate on how good our model is in reproducing the data it is supposed to train.

    • +
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At the heart of basically all Machine Learning algorithms we will encounter so-called minimization or optimization algorithms. A large family of such methods are so-called gradient methods.

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+

A Frequentist approach to data analysis

+

When you hear phrases like predictions and estimations and +correlations and causations, what do you think of? May be you think +of the difference between classifying new data points and generating +new data points. +Or perhaps you consider that correlations represent some kind of symmetric statements like +if \(A\) is correlated with \(B\), then \(B\) is correlated with +\(A\). Causation on the other hand is directional, that is if \(A\) causes \(B\), \(B\) does not +necessarily cause \(A\).

+

These concepts are in some sense the difference between machine +learning and statistics. In machine learning and prediction based +tasks, we are often interested in developing algorithms that are +capable of learning patterns from given data in an automated fashion, +and then using these learned patterns to make predictions or +assessments of newly given data. In many cases, our primary concern +is the quality of the predictions or assessments, and we are less +concerned about the underlying patterns that were learned in order +to make these predictions.

+

In machine learning we normally use a so-called frequentist approach, +where the aim is to make predictions and find correlations. We focus +less on for example extracting a probability distribution function (PDF). The PDF can be +used in turn to make estimations and find causations such as given \(A\) +what is the likelihood of finding \(B\).

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+

What is a good model?

+

In science and engineering we often end up in situations where we want to infer (or learn) a +quantitative model \(M\) for a given set of sample points \(\boldsymbol{X} \in [x_1, x_2,\dots x_N]\).

+

As we will see repeatedely in these lectures, we could try to fit these data points to a model given by a +straight line, or if we wish to be more sophisticated to a more complex +function.

+

The reason for inferring such a model is that it +serves many useful purposes. On the one hand, the model can reveal information +encoded in the data or underlying mechanisms from which the data were generated. For instance, we could discover important +corelations that relate interesting physics interpretations.

+

In addition, it can simplify the representation of the given data set and help +us in making predictions about future data samples.

+

A first important consideration to keep in mind is that inferring the correct model +for a given data set is an elusive, if not impossible, task. The fundamental difficulty +is that if we are not specific about what we mean by a correct model, there +could easily be many different models that fit the given data set equally well.

+

The central question is this: what leads us to say that a model is correct or +optimal for a given data set? To make the model inference problem well posed, i.e., +to guarantee that there is a unique optimal model for the given data, we need to +impose additional assumptions or restrictions on the class of models considered. To +this end, we should not be looking for just any model that can describe the data. +Instead, we should look for a model \(M\) that is the best among a restricted class +of models. In addition, to make the model inference problem computationally +tractable, we need to specify how restricted the class of models needs to be. A +common strategy is to start +with the simplest possible class of models that is just necessary to describe the data +or solve the problem at hand. More precisely, the model class should be rich enough +to contain at least one model that can fit the data to a desired accuracy and yet be +restricted enough that it is relatively simple to find the best model for the given data.

+

Thus, the most popular strategy is to start from the +simplest class of models and increase the complexity of the models only when the +simpler models become inadequate. For instance, if we work with a regression problem to fit a set of sample points, one +may first try the simplest class of models, namely linear models, followed obviously by more complex models.

+

How to evaluate which model fits best the data is something we will come back to over and over again in these set of lectures.

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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, Tensorflow and Pytorch, 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.

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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 and various online +repositories like Github, +Gitlab 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, +Tensorflow, +PyTorch and Keras, 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?

+

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.

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+ + 1. Getting started, our first data and Machine Learning encounters + +
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+ + By Morten Hjorth-Jensen
+ + © Copyright 2020.
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1. Getting started, our first data and Machine Learning encounters

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1.1. Introduction

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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 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.

+
+
+

1.2. 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, +Tensorflow, +PyTorch and Keras, 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.

+
+
+

1.3. 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.

+
+
+

1.4. Software and needed installations

+

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

+

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

+
    +

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

    2. +
+

For Python3, replace pip with pip3.

+

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

+
    +

  1. brew install python3

    2. +
+

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

+
    +

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

    2. +
+

etc etc.

+
+
+

1.5. Python installers

+

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

+ +

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

+ +

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

+

Furthermore, Google’s Colab is a free Jupyter notebook environment that requires +no setup and runs entirely in the cloud. Try it out!

+
+
+

1.6. Useful Python libraries

+

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

+
    +

  • NumPy is a highly popular library for large, multi-dimensional arrays and matrices, along with a large collection of high-level mathematical functions to operate on these arrays

    • +

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

    • +

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

    • +

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

    • +

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

    • +

  • Autograd can automatically differentiate native Python and Numpy code. It can handle a large subset of Python’s features, including loops, ifs, recursion and closures, and it can even take derivatives of derivatives of derivatives

    • +

  • SymPy is a Python library for symbolic mathematics.

    • +

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

    • +

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

    • +

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

    • +

  • And many more such as pytorch, Theano etc

    • +
+
+
+

1.7. 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

+
+
+

1.8. Installing R, C++, cython, Numba etc

+

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

+

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

+
    pycod jupyter nbconvert filename.ipynb --to latex 
+
+
+

And to add more versatility, the Python package SymPy is a Python library for symbolic mathematics. It aims to become a full-featured computer algebra system (CAS) and is entirely written in Python.

+

Finally, if you wish to use the light mark-up language +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.

+
+
+

1.9. Numpy examples and Important Matrix and vector handling packages

+

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

+
    +

  • LINPACK: package for linear equations and least square problems.

    • +

  • LAPACK:package for solving symmetric, unsymmetric and generalized eigenvalue problems. From LAPACK’s website http://www.netlib.org it is possible to download for free all source codes from this library. Both C/C++ and Fortran versions are available.

    • +

  • BLAS (I, II and III): (Basic Linear Algebra Subprograms) are routines that provide standard building blocks for performing basic vector and matrix operations. Blas I is vector operations, II vector-matrix operations and III matrix-matrix operations. Highly parallelized and efficient codes, all available for download from http://www.netlib.org.

    • +
+
+
+

1.10. Basic Matrix Features

+

Matrix properties reminder.

+
+\[\begin{split} +\mathbf{A} = + \begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ + a_{21} & a_{22} & a_{23} & a_{24} \\ + a_{31} & a_{32} & a_{33} & a_{34} \\ + a_{41} & a_{42} & a_{43} & a_{44} + \end{bmatrix}\qquad +\mathbf{I} = + \begin{bmatrix} 1 & 0 & 0 & 0 \\ + 0 & 1 & 0 & 0 \\ + 0 & 0 & 1 & 0 \\ + 0 & 0 & 0 & 1 + \end{bmatrix} +\end{split}\]
+

The inverse of a matrix is defined by

+
+\[ +\mathbf{A}^{-1} \cdot \mathbf{A} = I +\]
+ + + + + + + + + + + +
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 $\sum_k a_{ik} a_{jk}^{ * } = \sum_k a_{ki}^{ * } a_{kj} = \delta_{ij}$
+
+

1.10.1. 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….

    • +
+
+
+

1.10.2. More Basic Matrix Features

+

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 \(\mathbf{A}\).

    • +
+
+
+
+

1.11. Numpy and arrays

+

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

+
+
+
import numpy as np
+
+
+
+
+

Here follows a simple example where we set up an array of ten elements, all determined by random numbers drawn according to the normal distribution,

+
+
+
n = 10
+x = np.random.normal(size=n)
+print(x)
+
+
+
+
+
[-0.471679   -0.31070947 -1.01764096  0.40557878 -0.35432968  0.88192231
+ -0.14344427 -1.6439498   0.83757805 -0.49964924]
+
+
+
+
+

We defined a vector \(x\) with \(n=10\) elements with its values given by the Normal distribution \(N(0,1)\). +Another alternative is to declare a vector as follows

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

Here we have defined a vector with three elements, with \(x_0=1\), \(x_1=2\) and \(x_2=3\). Note that both Python and C++ +start numbering array elements from \(0\) and on. This means that a vector with \(n\) elements has a sequence of entities \(x_0, x_1, x_2, \dots, x_{n-1}\). We could also let (recommended) Numpy to compute the logarithms of a specific array as

+
+
+
import numpy as np
+x = np.log(np.array([4, 7, 8]))
+print(x)
+
+
+
+
+
[1.38629436 1.94591015 2.07944154]
+
+
+
+
+

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

+
+
+
import numpy as np
+from math import log
+x = np.array([4, 7, 8])
+for i in range(0, len(x)):
+    x[i] = log(x[i])
+print(x)
+
+
+
+
+
[1 1 2]
+
+
+
+
+

We note that our code is much longer already and we need to import the log function from the math module. +The attentive reader will also notice that the output is \([1, 1, 2]\). Python interprets automagically our numbers as integers (like the automatic keyword in C++). To change this we could define our array elements to be double precision numbers as

+
+
+
import numpy as np
+x = np.log(np.array([4, 7, 8], dtype = np.float64))
+print(x)
+
+
+
+
+
[1.38629436 1.94591015 2.07944154]
+
+
+
+
+

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

+
+
+
import numpy as np
+x = np.log(np.array([4.0, 7.0, 8.0])
+print(x)
+
+
+
+
+
  File "<ipython-input-7-f6d7a289d493>", line 3
+    print(x)
+    ^
+SyntaxError: invalid syntax
+
+
+
+
+

To check the number of bytes (remember that one byte contains eight bits for double precision variables), you can use simple use the itemsize functionality (the array \(x\) is actually an object which inherits the functionalities defined in Numpy) as

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

1.12. 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)

+
+
+
import numpy as np
+A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))
+print(A)
+
+
+
+
+

If we use the shape function we would get \((3, 3)\) as output, that is verifying that our matrix is a \(3\times 3\) matrix. We can slice the matrix and print for example the first column (Python organized matrix elements in a row-major order, see below) as

+
+
+
import numpy as np
+A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))
+# print the first column, row-major order and elements start with 0
+print(A[:,0])
+
+
+
+
+

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

+
+
+
import numpy as np
+A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))
+# print the first column, row-major order and elements start with 0
+print(A[1,:])
+
+
+
+
+

Numpy contains many other functionalities that allow us to slice, subdivide etc etc arrays. We strongly recommend that you look up the Numpy website for more details. Useful functions when defining a matrix are the np.zeros function which declares a matrix of a given dimension and sets all elements to zero

+
+
+
import numpy as np
+n = 10
+# define a matrix of dimension 10 x 10 and set all elements to zero
+A = np.zeros( (n, n) )
+print(A)
+
+
+
+
+

or initializing all elements to

+
+
+
import numpy as np
+n = 10
+# define a matrix of dimension 10 x 10 and set all elements to one
+A = np.ones( (n, n) )
+print(A)
+
+
+
+
+

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

+
+
+
import numpy as np
+n = 10
+# define a matrix of dimension 10 x 10 and set all elements to random numbers with x \in [0, 1]
+A = np.random.rand(n, n)
+print(A)
+
+
+
+
+

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

+
+\[\begin{split} +\hat{\Sigma} = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\ + \sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\ + \sigma_{zx} & \sigma_{zy} & \sigma_{zz} + \end{bmatrix}, +\end{split}\]
+

where for example

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

The Numpy function np.cov calculates the covariance elements using the factor \(1/(n-1)\) instead of \(1/n\) since it assumes we do not have the exact mean values. +The following simple function uses the np.vstack function which takes each vector of dimension \(1\times n\) and produces a \(3\times n\) matrix \(\hat{W}\)

+
+\[\begin{split} +\hat{W} = \begin{bmatrix} x_0 & y_0 & z_0 \\ + x_1 & y_1 & z_1 \\ + x_2 & y_2 & z_2 \\ + \dots & \dots & \dots \\ + x_{n-2} & y_{n-2} & z_{n-2} \\ + x_{n-1} & y_{n-1} & z_{n-1} + \end{bmatrix}, +\end{split}\]
+

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.

+
+
+
# 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)
+
+
+
+
+
+
+
%matplotlib inline
+
+import numpy as np
+import matplotlib.pyplot as plt
+from scipy import sparse
+eye = np.eye(4)
+print(eye)
+sparse_mtx = sparse.csr_matrix(eye)
+print(sparse_mtx)
+x = np.linspace(-10,10,100)
+y = np.sin(x)
+plt.plot(x,y,marker='x')
+plt.show()
+
+
+
+
+
+
+

1.13. Meet the Pandas

+ + +

+ + +

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

+

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

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

In the above we have imported pandas with the shorthand pd, the latter has become the standard way we import pandas. We make then a list of various variables +and reorganize the aboves lists into a DataFrame and then print out a neat table with specific column labels as Name, place of birth and date of birth. +Displaying these results, we see that the indices are given by the default numbers from zero to three. +pandas is extremely flexible and we can easily change the above indices by defining a new type of indexing as

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

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

+
+
+
display(data_pandas.loc['Aragorn'])
+
+
+
+
+

We can easily append data to this, for example

+
+
+
new_hobbit = {'First Name': ["Peregrin"],
+              'Last Name': ["Took"],
+              'Place of birth': ["Shire"],
+              'Date of Birth T.A.': [2990]
+              }
+data_pandas=data_pandas.append(pd.DataFrame(new_hobbit, index=['Pippin']))
+display(data_pandas)
+
+
+
+
+

Here are other examples where we use the DataFrame functionality to handle arrays, now with more interesting features for us, namely numbers. We set up a matrix +of dimensionality \(10\times 5\) and compute the mean value and standard deviation of each column. Similarly, we can perform mathematial operations like squaring the matrix elements and many other operations.

+
+
+
import numpy as np
+import pandas as pd
+from IPython.display import display
+np.random.seed(100)
+# setting up a 10 x 5 matrix
+rows = 10
+cols = 5
+a = np.random.randn(rows,cols)
+df = pd.DataFrame(a)
+display(df)
+print(df.mean())
+print(df.std())
+display(df**2)
+
+
+
+
+

Thereafter we can select specific columns only and plot final results

+
+
+
df.columns = ['First', 'Second', 'Third', 'Fourth', 'Fifth']
+df.index = np.arange(10)
+
+display(df)
+print(df['Second'].mean() )
+print(df.info())
+print(df.describe())
+
+from pylab import plt, mpl
+plt.style.use('seaborn')
+mpl.rcParams['font.family'] = 'serif'
+
+df.cumsum().plot(lw=2.0, figsize=(10,6))
+plt.show()
+
+
+df.plot.bar(figsize=(10,6), rot=15)
+plt.show()
+
+
+
+
+

We can produce a \(4\times 4\) matrix

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

and many other operations.

+

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

+
+
+

1.14. 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!

+
+

1.14.1. 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 \(x\)

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

where \(N(0,1)\) represents random numbers generated by the normal +distribution. From Scikit-Learn we import then the +LinearRegression functionality and make a prediction \(\tilde{y} = +\alpha + \beta x\) using the function fit(x,y). We call the set of +data \((\hat{x},\hat{y})\) 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 which produces publication +quality figures. Feel free to explore the extensive +gallery of examples. In +this example we plot our original values of \(x\) and \(y\) as well as the +prediction ypredict (\(\tilde{y}\)), which attempts at fitting our +data with a straight line.

+

The Python code follows here.

+
+
+
# 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()
+
+
+
+
+

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

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

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))

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

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

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

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

+
+
+
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()
+
+
+
+
+

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.

+
+
+
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()
+
+
+
+
+

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

+
+\[ +MSE(\hat{y},\hat{\tilde{y}}) = \frac{1}{n} +\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2, +\]
+

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

+
+\[ +R^2(\hat{y}, \tilde{\hat{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2}, +\]
+

where we have defined the mean value of \(\hat{y}\) as

+
+\[ +\bar{y} = \frac{1}{n} \sum_{i=0}^{n - 1} y_i. +\]
+

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

+
+\[ +\text{MAE}(\hat{y}, \hat{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n-1} \left| y_i - \tilde{y}_i \right|. +\]
+

We present the +squared logarithmic (quadratic) error

+
+\[ +\text{MSLE}(\hat{y}, \hat{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n - 1} (\log_e (1 + y_i) - \log_e (1 + \tilde{y}_i) )^2, +\]
+

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

+

Finally, another cost function is the Huber cost function used in robust regression.

+

The rationale behind this possible cost function is its reduced +sensitivity to outliers in the data set. In our discussions on +dimensionality reduction and normalization of data we will meet other +ways of dealing with outliers.

+

The Huber cost function is defined as

+
+\[\begin{split} +H_{\delta}(a)={\begin{cases}{\frac {1}{2}}{a^{2}}&{\text{for }}|a|\leq \delta ,\\\delta (|a|-{\frac {1}{2}}\delta ),&{\text{otherwise.}}\end{cases}}}. +\end{split}\]
+

Here \(a=\boldsymbol{y} - \boldsymbol{\tilde{y}}\). +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.

+
+
+
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))
+
+
+
+
+
+
+

1.14.2. 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

+
+\[ +\Delta M(N, Z) = M(N, Z) - uA, +\]
+

where \(u\) is the Atomic Mass Unit

+
+\[ +u = M(^{12}\mathrm{C})/12 = 931.4940954(57) \hspace{0.1cm} \mathrm{MeV}/c^2. +\]
+

The nucleon masses are

+
+\[ +m_p = 1.00727646693(9)u, +\]
+

and

+
+\[ +m_n = 939.56536(8)\hspace{0.1cm} \mathrm{MeV}/c^2 = 1.0086649156(6)u. +\]
+

In the 2016 mass evaluation of by W.J.Huang, G.Audi, M.Wang, F.G.Kondev, S.Naimi and X.Xu +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

+
+\[ +BE(N, Z) = ZM_H c^2 + Nm_n c^2 - M(N, Z)c^2 , +\]
+

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

+
+\[ +BE(N, Z) = Z\Delta_H c^2 + N\Delta_n c^2 -\Delta(N, Z)c^2 , +\]
+

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

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

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.

+
+
+

1.14.3. 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.

+
+
+
# 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')
+
+
+
+
+

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.

+
+
+
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)
+
+
+
+
+

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.

+
+
+
"""                                                                                                                         
+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.                                                          
+"""
+
+
+
+
+

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.

+
+
+
# 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()])
+
+
+
+
+

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.

+
+
+
A = Masses['A']
+Z = Masses['Z']
+N = Masses['N']
+Element = Masses['Element']
+Energies = Masses['Ebinding']
+print(Masses)
+
+
+
+
+

The next step, and we will define this mathematically later, is to set up the so-called design matrix. We will throughout call this matrix \(\boldsymbol{X}\). +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.

+
+
+
# 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)
+
+
+
+
+

With scikitlearn we are now ready to use linear regression and fit our data.

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

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.

+
+
+
# 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()
+
+
+
+
+
+
+

1.14.4. 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!

+
+
+
#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))
+
+
+
+
+
+
+

1.14.5. 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.

+
+
+
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()
+
+
+
+
+
+
+
+

1.15. 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.

+
+
+ + + + +
+ +
+
+ + +
+ + Introduction to Applied Data Analysis and Machine Learning + 2. Linear Regression and more Advanced Regression Analysis + +
+
+
+

+ + By Morten Hjorth-Jensen
+ + © Copyright 2020.
+

+
+
+
+ + +
+
+ + + + + \ No newline at end of file diff --git a/doc/src/LectureNotes/_build/html/chapter3.html b/doc/src/LectureNotes/_build/html/chapter3.html new file mode 100644 index 000000000..bbf3adbfc --- /dev/null +++ b/doc/src/LectureNotes/_build/html/chapter3.html @@ -0,0 +1,4760 @@ + + + + + + + + + 2. Linear Regression and more Advanced Regression Analysis — Applied Machine Learning and Data Analysis + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + + + + + + + +
+ +
+
+
+
+ + + +
+ + + +
+ + + + + + +
+ +
+ + +
+ +
+ + + +
+
+ + + + + + + +
+ + +
+
+ Contents +
+ +
+
+
+
+ +
+ +
+

2. Linear Regression and more Advanced Regression Analysis

+
+

2.1. Why Linear Regression (aka Ordinary Least Squares and family)

+

Fitting a continuous function with linear parameterization in terms of the parameters \(\boldsymbol{\beta}\).

+
    +

  • Method of choice for fitting a continuous function!

    • +

  • Gives an excellent introduction to central Machine Learning features with understandable pedagogical links to other methods like Neural Networks, Support Vector Machines etc

    • +

  • Analytical expression for the fitting parameters \(\boldsymbol{\beta}\)

    • +

  • Analytical expressions for statistical propertiers like mean values, variances, confidence intervals and more

    • +

  • Analytical relation with probabilistic interpretations

    • +

  • Easy to introduce basic concepts like bias-variance tradeoff, cross-validation, resampling and regularization techniques and many other ML topics

    • +

  • Easy to code! And links well with classification problems and logistic regression and neural networks

    • +

  • Allows for easy hands-on understanding of gradient descent methods

    • +

  • and many more features

    • +
+

For more discussions of Ridge and Lasso regression, Wessel van Wieringen’s article is highly recommended. +Similarly, Mehta et al’s article is also recommended.

+
+
+

2.2. Regression analysis, overarching aims

+

Regression modeling deals with the description of the sampling distribution of a given random variable \(y\) and how it varies as function of another variable or a set of such variables \(\boldsymbol{x} =[x_0, x_1,\dots, x_{n-1}]^T\). +The first variable is called the dependent, the outcome or the response variable while the set of variables \(\boldsymbol{x}\) is called the independent variable, or the predictor variable or the explanatory variable.

+

A regression model aims at finding a likelihood function \(p(\boldsymbol{y}\vert \boldsymbol{x})\), that is the conditional distribution for \(\boldsymbol{y}\) with a given \(\boldsymbol{x}\). The estimation of \(p(\boldsymbol{y}\vert \boldsymbol{x})\) is made using a data set with

+
    +

  • \(n\) cases \(i = 0, 1, 2, \dots, n-1\)

    • +

  • Response (target, dependent or outcome) variable \(y_i\) with \(i = 0, 1, 2, \dots, n-1\)

    • +

  • \(p\) so-called explanatory (independent or predictor) variables \(\boldsymbol{x}_i=[x_{i0}, x_{i1}, \dots, x_{ip-1}]\) with \(i = 0, 1, 2, \dots, n-1\) and explanatory variables running from \(0\) to \(p-1\). See below for more explicit examples.

    • +
+

The goal of the regression analysis is to extract/exploit relationship between \(\boldsymbol{y}\) and \(\boldsymbol{x}\) in or to infer causal dependencies, approximations to the likelihood functions, functional relationships and to make predictions, making fits and many other things.

+
+
+

2.3. Regression analysis, overarching aims II

+

Consider an experiment in which \(p\) characteristics of \(n\) samples are +measured. The data from this experiment, for various explanatory variables \(p\) are normally represented by a matrix
+\(\mathbf{X}\).

+

The matrix \(\mathbf{X}\) is called the design +matrix. Additional information of the samples is available in the +form of \(\boldsymbol{y}\) (also as above). The variable \(\boldsymbol{y}\) is +generally referred to as the response variable. The aim of +regression analysis is to explain \(\boldsymbol{y}\) in terms of +\(\boldsymbol{X}\) through a functional relationship like \(y_i = +f(\mathbf{X}_{i,\ast})\). When no prior knowledge on the form of +\(f(\cdot)\) is available, it is common to assume a linear relationship +between \(\boldsymbol{X}\) and \(\boldsymbol{y}\). This assumption gives rise to +the linear regression model where \(\boldsymbol{\beta} = [\beta_0, \ldots, +\beta_{p-1}]^{T}\) are the regression parameters.

+

Linear regression gives us a set of analytical equations for the parameters \(\beta_j\).

+
+
+

2.4. Examples

+

In order to understand the relation among the predictors \(p\), the set of data \(n\) and the target (outcome, output etc) \(\boldsymbol{y}\), +consider the model we discussed for describing nuclear binding energies.

+

There we assumed that we could parametrize the data using a polynomial approximation based on the liquid drop model. +Assuming

+
+\[ +BE(A) = a_0+a_1A+a_2A^{2/3}+a_3A^{-1/3}+a_4A^{-1}, +\]
+

we have five predictors, that is the intercept, the \(A\) dependent term, the \(A^{2/3}\) term and the \(A^{-1/3}\) and \(A^{-1}\) terms. +This gives \(p=0,1,2,3,4\). Furthermore we have \(n\) entries for each predictor. It means that our design matrix is a +\(p\times n\) matrix \(\boldsymbol{X}\).

+

Here the predictors are based on a model we have made. A popular data set which is widely encountered in ML applications is the +so-called credit card default data from Taiwan. The data set contains data on \(n=30000\) credit card holders with predictors like gender, marital status, age, profession, education, etc. In total there are \(24\) such predictors or attributes leading to a design matrix of dimensionality \(24 \times 30000\). This is however a classification problem and we will come back to it when we discuss Logistic Regression.

+
+
+

2.5. General linear models

+

Before we proceed let us study a case from linear algebra where we aim at fitting a set of data \(\boldsymbol{y}=[y_0,y_1,\dots,y_{n-1}]\). We could think of these data as a result of an experiment or a complicated numerical experiment. These data are functions of a series of variables \(\boldsymbol{x}=[x_0,x_1,\dots,x_{n-1}]\), that is \(y_i = y(x_i)\) with \(i=0,1,2,\dots,n-1\). The variables \(x_i\) could represent physical quantities like time, temperature, position etc. We assume that \(y(x)\) is a smooth function.

+

Since obtaining these data points may not be trivial, we want to use these data to fit a function which can allow us to make predictions for values of \(y\) which are not in the present set. The perhaps simplest approach is to assume we can parametrize our function in terms of a polynomial of degree \(n-1\) with \(n\) points, that is

+
+\[ +y=y(x) \rightarrow y(x_i)=\tilde{y}_i+\epsilon_i=\sum_{j=0}^{n-1} \beta_j x_i^j+\epsilon_i, +\]
+

where \(\epsilon_i\) is the error in our approximation.

+
+
+

2.6. Rewriting the fitting procedure as a linear algebra problem

+

For every set of values \(y_i,x_i\) we have thus the corresponding set of equations

+
+\[\begin{split} +\begin{align*} +y_0&=\beta_0+\beta_1x_0^1+\beta_2x_0^2+\dots+\beta_{n-1}x_0^{n-1}+\epsilon_0\\ +y_1&=\beta_0+\beta_1x_1^1+\beta_2x_1^2+\dots+\beta_{n-1}x_1^{n-1}+\epsilon_1\\ +y_2&=\beta_0+\beta_1x_2^1+\beta_2x_2^2+\dots+\beta_{n-1}x_2^{n-1}+\epsilon_2\\ +\dots & \dots \\ +y_{n-1}&=\beta_0+\beta_1x_{n-1}^1+\beta_2x_{n-1}^2+\dots+\beta_{n-1}x_{n-1}^{n-1}+\epsilon_{n-1}.\\ +\end{align*} +\end{split}\]
+
+
+

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

+

Defining the vectors

+
+\[ +\boldsymbol{y} = [y_0,y_1, y_2,\dots, y_{n-1}]^T, +\]
+

and

+
+\[ +\boldsymbol{\beta} = [\beta_0,\beta_1, \beta_2,\dots, \beta_{n-1}]^T, +\]
+

and

+
+\[ +\boldsymbol{\epsilon} = [\epsilon_0,\epsilon_1, \epsilon_2,\dots, \epsilon_{n-1}]^T, +\]
+

and the design matrix

+
+\[\begin{split} +\boldsymbol{X}= +\begin{bmatrix} +1& x_{0}^1 &x_{0}^2& \dots & \dots &x_{0}^{n-1}\\ +1& x_{1}^1 &x_{1}^2& \dots & \dots &x_{1}^{n-1}\\ +1& x_{2}^1 &x_{2}^2& \dots & \dots &x_{2}^{n-1}\\ +\dots& \dots &\dots& \dots & \dots &\dots\\ +1& x_{n-1}^1 &x_{n-1}^2& \dots & \dots &x_{n-1}^{n-1}\\ +\end{bmatrix} +\end{split}\]
+

we can rewrite our equations as

+
+\[ +\boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta}+\boldsymbol{\epsilon}. +\]
+

The above design matrix is called a Vandermonde matrix.

+
+
+

2.8. Generalizing the fitting procedure as a linear algebra problem

+

We are obviously not limited to the above polynomial expansions. We +could replace the various powers of \(x\) with elements of Fourier +series or instead of \(x_i^j\) we could have \(\cos{(j x_i)}\) or \(\sin{(j +x_i)}\), or time series or other orthogonal functions. For every set +of values \(y_i,x_i\) we can then generalize the equations to

+
+\[\begin{split} +\begin{align*} +y_0&=\beta_0x_{00}+\beta_1x_{01}+\beta_2x_{02}+\dots+\beta_{n-1}x_{0n-1}+\epsilon_0\\ +y_1&=\beta_0x_{10}+\beta_1x_{11}+\beta_2x_{12}+\dots+\beta_{n-1}x_{1n-1}+\epsilon_1\\ +y_2&=\beta_0x_{20}+\beta_1x_{21}+\beta_2x_{22}+\dots+\beta_{n-1}x_{2n-1}+\epsilon_2\\ +\dots & \dots \\ +y_{i}&=\beta_0x_{i0}+\beta_1x_{i1}+\beta_2x_{i2}+\dots+\beta_{n-1}x_{in-1}+\epsilon_i\\ +\dots & \dots \\ +y_{n-1}&=\beta_0x_{n-1,0}+\beta_1x_{n-1,2}+\beta_2x_{n-1,2}+\dots+\beta_{n-1}x_{n-1,n-1}+\epsilon_{n-1}.\\ +\end{align*} +\end{split}\]
+

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

+
+
+

2.9. Generalizing the fitting procedure as a linear algebra problem

+

We redefine in turn the matrix \(\boldsymbol{X}\) as

+
+\[\begin{split} +\boldsymbol{X}= +\begin{bmatrix} +x_{00}& x_{01} &x_{02}& \dots & \dots &x_{0,n-1}\\ +x_{10}& x_{11} &x_{12}& \dots & \dots &x_{1,n-1}\\ +x_{20}& x_{21} &x_{22}& \dots & \dots &x_{2,n-1}\\ +\dots& \dots &\dots& \dots & \dots &\dots\\ +x_{n-1,0}& x_{n-1,1} &x_{n-1,2}& \dots & \dots &x_{n-1,n-1}\\ +\end{bmatrix} +\end{split}\]
+

and without loss of generality we rewrite again our equations as

+
+\[ +\boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta}+\boldsymbol{\epsilon}. +\]
+

The left-hand side of this equation is kwown. Our error vector \(\boldsymbol{\epsilon}\) and the parameter vector \(\boldsymbol{\beta}\) are our unknow quantities. How can we obtain the optimal set of \(\beta_i\) values?

+
+
+

2.10. Optimizing our parameters

+

We have defined the matrix \(\boldsymbol{X}\) via the equations

+
+\[\begin{split} +\begin{align*} +y_0&=\beta_0x_{00}+\beta_1x_{01}+\beta_2x_{02}+\dots+\beta_{n-1}x_{0n-1}+\epsilon_0\\ +y_1&=\beta_0x_{10}+\beta_1x_{11}+\beta_2x_{12}+\dots+\beta_{n-1}x_{1n-1}+\epsilon_1\\ +y_2&=\beta_0x_{20}+\beta_1x_{21}+\beta_2x_{22}+\dots+\beta_{n-1}x_{2n-1}+\epsilon_1\\ +\dots & \dots \\ +y_{i}&=\beta_0x_{i0}+\beta_1x_{i1}+\beta_2x_{i2}+\dots+\beta_{n-1}x_{in-1}+\epsilon_1\\ +\dots & \dots \\ +y_{n-1}&=\beta_0x_{n-1,0}+\beta_1x_{n-1,2}+\beta_2x_{n-1,2}+\dots+\beta_{n-1}x_{n-1,n-1}+\epsilon_{n-1}.\\ +\end{align*} +\end{split}\]
+

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

+
+
+

2.11. Our model for the nuclear binding energies

+

In our introductory notes we looked at the so-called liquid drop model. Let us remind ourselves about what we did by looking at the code.

+

We restate the parts of the code we are most interested in.

+
+
+
%matplotlib inline
+
+# Common imports
+import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+from IPython.display import display
+import os
+
+# Where to save the figures and data files
+PROJECT_ROOT_DIR = "Results"
+FIGURE_ID = "Results/FigureFiles"
+DATA_ID = "DataFiles/"
+
+if not os.path.exists(PROJECT_ROOT_DIR):
+    os.mkdir(PROJECT_ROOT_DIR)
+
+if not os.path.exists(FIGURE_ID):
+    os.makedirs(FIGURE_ID)
+
+if not os.path.exists(DATA_ID):
+    os.makedirs(DATA_ID)
+
+def image_path(fig_id):
+    return os.path.join(FIGURE_ID, fig_id)
+
+def data_path(dat_id):
+    return os.path.join(DATA_ID, dat_id)
+
+def save_fig(fig_id):
+    plt.savefig(image_path(fig_id) + ".png", format='png')
+
+infile = open(data_path("MassEval2016.dat"),'r')
+
+
+# Read the experimental data with Pandas
+Masses = pd.read_fwf(infile, usecols=(2,3,4,6,11),
+              names=('N', 'Z', 'A', 'Element', 'Ebinding'),
+              widths=(1,3,5,5,5,1,3,4,1,13,11,11,9,1,2,11,9,1,3,1,12,11,1),
+              header=39,
+              index_col=False)
+
+# Extrapolated values are indicated by '#' in place of the decimal place, so
+# the Ebinding column won't be numeric. Coerce to float and drop these entries.
+Masses['Ebinding'] = pd.to_numeric(Masses['Ebinding'], errors='coerce')
+Masses = Masses.dropna()
+# Convert from keV to MeV.
+Masses['Ebinding'] /= 1000
+
+# Group the DataFrame by nucleon number, A.
+Masses = Masses.groupby('A')
+# Find the rows of the grouped DataFrame with the maximum binding energy.
+Masses = Masses.apply(lambda t: t[t.Ebinding==t.Ebinding.max()])
+A = Masses['A']
+Z = Masses['Z']
+N = Masses['N']
+Element = Masses['Element']
+Energies = Masses['Ebinding']
+
+# Now we set up the design matrix X
+X = np.zeros((len(A),5))
+X[:,0] = 1
+X[:,1] = A
+X[:,2] = A**(2.0/3.0)
+X[:,3] = A**(-1.0/3.0)
+X[:,4] = A**(-1.0)
+# Then nice printout using pandas
+DesignMatrix = pd.DataFrame(X)
+DesignMatrix.index = A
+DesignMatrix.columns = ['1', 'A', 'A^(2/3)', 'A^(-1/3)', '1/A']
+display(DesignMatrix)
+
+
+
+
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
1AA^(2/3)A^(-1/3)1/A
A
11.01.01.0000001.0000001.000000
21.02.01.5874010.7937010.500000
31.03.02.0800840.6933610.333333
41.04.02.5198420.6299610.250000
51.05.02.9240180.5848040.200000
..................
2641.0264.041.1531060.1558830.003788
2651.0265.041.2569620.1556870.003774
2661.0266.041.3606880.1554910.003759
2691.0269.041.6710890.1549110.003717
2701.0270.041.7743000.1547200.003704
+

267 rows × 5 columns

+
+
+

With \(\boldsymbol{\beta}\in {\mathbb{R}}^{p\times 1}\), it means that we will hereafter write our equations for the approximation as

+
+\[ +\boldsymbol{\tilde{y}}= \boldsymbol{X}\boldsymbol{\beta}, +\]
+

throughout these lectures.

+
+
+

2.12. Optimizing our parameters, more details

+

With the above we use the design matrix to define the approximation \(\boldsymbol{\tilde{y}}\) via the unknown quantity \(\boldsymbol{\beta}\) as

+
+\[ +\boldsymbol{\tilde{y}}= \boldsymbol{X}\boldsymbol{\beta}, +\]
+

and in order to find the optimal parameters \(\beta_i\) instead of solving the above linear algebra problem, we define a function which gives a measure of the spread between the values \(y_i\) (which represent hopefully the exact values) and the parameterized values \(\tilde{y}_i\), namely

+
+\[ +C(\boldsymbol{\beta})=\frac{1}{n}\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2=\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{\tilde{y}}\right)^T\left(\boldsymbol{y}-\boldsymbol{\tilde{y}}\right)\right\}, +\]
+

or using the matrix \(\boldsymbol{X}\) and in a more compact matrix-vector notation as

+
+\[ +C(\boldsymbol{\beta})=\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right\}. +\]
+

This function is one possible way to define the so-called cost function.

+

It is also common to define +the function \(C\) as

+
+\[ +C(\boldsymbol{\beta})=\frac{1}{2n}\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2, +\]
+

since when taking the first derivative with respect to the unknown parameters \(\beta\), the factor of \(2\) cancels out.

+
+
+

2.13. Interpretations and optimizing our parameters

+

The function

+
+\[ +C(\boldsymbol{\beta})=\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right\}, +\]
+

can be linked to the variance of the quantity \(y_i\) if we interpret the latter as the mean value. +When linking (see the discussion below) with the maximum likelihood approach below, we will indeed interpret \(y_i\) as a mean value

+
+\[ +y_{i}=\langle y_i \rangle = \beta_0x_{i,0}+\beta_1x_{i,1}+\beta_2x_{i,2}+\dots+\beta_{n-1}x_{i,n-1}+\epsilon_i, +\]
+

where \(\langle y_i \rangle\) is the mean value. Keep in mind also that +till now we have treated \(y_i\) as the exact value. Normally, the +response (dependent or outcome) variable \(y_i\) the outcome of a +numerical experiment or another type of experiment and is thus only an +approximation to the true value. It is then always accompanied by an +error estimate, often limited to a statistical error estimate given by +the standard deviation discussed earlier. In the discussion here we +will treat \(y_i\) as our exact value for the response variable.

+

In order to find the parameters \(\beta_i\) we will then minimize the spread of \(C(\boldsymbol{\beta})\), that is we are going to solve the problem

+
+\[ +{\displaystyle \min_{\boldsymbol{\beta}\in +{\mathbb{R}}^{p}}}\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right\}. +\]
+

In practical terms it means we will require

+
+\[ +\frac{\partial C(\boldsymbol{\beta})}{\partial \beta_j} = \frac{\partial }{\partial \beta_j}\left[ \frac{1}{n}\sum_{i=0}^{n-1}\left(y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}\right)^2\right]=0, +\]
+

which results in

+
+\[ +\frac{\partial C(\boldsymbol{\beta})}{\partial \beta_j} = -\frac{2}{n}\left[ \sum_{i=0}^{n-1}x_{ij}\left(y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}\right)\right]=0, +\]
+

or in a matrix-vector form as

+
+\[ +\frac{\partial C(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = 0 = \boldsymbol{X}^T\left( \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right). +\]
+
+
+

2.14. Interpretations and optimizing our parameters

+

We can rewrite

+
+\[ +\frac{\partial C(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = 0 = \boldsymbol{X}^T\left( \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right), +\]
+

as

+
+\[ +\boldsymbol{X}^T\boldsymbol{y} = \boldsymbol{X}^T\boldsymbol{X}\boldsymbol{\beta}, +\]
+

and if the matrix \(\boldsymbol{X}^T\boldsymbol{X}\) is invertible we have the solution

+
+\[ +\boldsymbol{\beta} =\left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}. +\]
+

We note also that since our design matrix is defined as \(\boldsymbol{X}\in +{\mathbb{R}}^{n\times p}\), the product \(\boldsymbol{X}^T\boldsymbol{X} \in +{\mathbb{R}}^{p\times p}\). In the above case we have that \(p \ll n\), +in our case \(p=5\) meaning that we end up with inverting a small +\(5\times 5\) matrix. This is a rather common situation, in many cases we end up with low-dimensional +matrices to invert. The methods discussed here and for many other +supervised learning algorithms like classification with logistic +regression or support vector machines, exhibit dimensionalities which +allow for the usage of direct linear algebra methods such as LU decomposition or Singular Value Decomposition (SVD) for finding the inverse of the matrix +\(\boldsymbol{X}^T\boldsymbol{X}\).

+

Small question: Do you think the example we have at hand here (the nuclear binding energies) can lead to problems in inverting the matrix \(\boldsymbol{X}^T\boldsymbol{X}\)? What kind of problems can we expect?

+
+
+

2.15. Some useful matrix and vector expressions

+

The following matrix and vector relation will be useful here and for the rest of the course. Vectors are always written as boldfaced lower case letters and +matrices as upper case boldfaced letters.

+

2 +6

+

< +< +< +! +! +M +A +T +H +_ +B +L +O +C +K

+

2 +7

+

< +< +< +! +! +M +A +T +H +_ +B +L +O +C +K

+

2 +8

+

< +< +< +! +! +M +A +T +H +_ +B +L +O +C +K

+
+\[ +\frac{\partial \log{\vert\boldsymbol{A}\vert}}{\partial \boldsymbol{A}} = (\boldsymbol{A}^{-1})^T. +\]
+
+
+

2.16. Interpretations and optimizing our parameters

+

The residuals \(\boldsymbol{\epsilon}\) are in turn given by

+
+\[ +\boldsymbol{\epsilon} = \boldsymbol{y}-\boldsymbol{\tilde{y}} = \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}, +\]
+

and with

+
+\[ +\boldsymbol{X}^T\left( \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)= 0, +\]
+

we have

+
+\[ +\boldsymbol{X}^T\boldsymbol{\epsilon}=\boldsymbol{X}^T\left( \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)= 0, +\]
+

meaning that the solution for \(\boldsymbol{\beta}\) is the one which minimizes the residuals. Later we will link this with the maximum likelihood approach.

+

Let us now return to our nuclear binding energies and simply code the above equations.

+
+
+

2.17. Own code for Ordinary Least Squares

+

It is rather straightforward to implement the matrix inversion and obtain the parameters \(\boldsymbol{\beta}\). After having defined the matrix \(\boldsymbol{X}\) we simply need to +write

+
+
+
# matrix inversion to find beta
+beta = np.linalg.inv(X.T.dot(X)).dot(X.T).dot(Energies)
+# and then make the prediction
+ytilde = X @ beta
+
+
+
+
+

Alternatively, you can use the least squares functionality in Numpy as

+
+
+
fit = np.linalg.lstsq(X, Energies, rcond =None)[0]
+ytildenp = np.dot(fit,X.T)
+
+
+
+
+

And finally we plot our fit with and compare with data

+
+
+
Masses['Eapprox']  = ytilde
+# Generate a plot comparing the experimental with the fitted values values.
+fig, ax = plt.subplots()
+ax.set_xlabel(r'$A = N + Z$')
+ax.set_ylabel(r'$E_\mathrm{bind}\,/\mathrm{MeV}$')
+ax.plot(Masses['A'], Masses['Ebinding'], alpha=0.7, lw=2,
+            label='Ame2016')
+ax.plot(Masses['A'], Masses['Eapprox'], alpha=0.7, lw=2, c='m',
+            label='Fit')
+ax.legend()
+save_fig("Masses2016OLS")
+plt.show()
+
+
+
+
+_images/chapter3_70_0.png +
+
+
+
+

2.18. Adding error analysis and training set up

+

We can easily test our fit by computing the \(R2\) score that we discussed in connection with the functionality of Scikit-Learn in the introductory slides. +Since we are not using Scikit-Learn here we can define our own \(R2\) function as

+
+
+
def R2(y_data, y_model):
+    return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)
+
+
+
+
+

and we would be using it as

+
+
+
print(R2(Energies,ytilde))
+
+
+
+
+
0.9547578478889096
+
+
+
+
+

We can easily add our MSE score as

+
+
+
def MSE(y_data,y_model):
+    n = np.size(y_model)
+    return np.sum((y_data-y_model)**2)/n
+
+print(MSE(Energies,ytilde))
+
+
+
+
+
0.03787596148305239
+
+
+
+
+

and finally the relative error as

+
+
+
def RelativeError(y_data,y_model):
+    return abs((y_data-y_model)/y_data)
+print(RelativeError(Energies, ytilde))
+
+
+
+
+
A        
+1    0            inf
+2    1       1.123190
+3    2       0.327631
+4    6       0.344172
+5    9       0.044402
+               ...   
+264  3304    0.009911
+265  3310    0.009154
+266  3317    0.007824
+269  3338    0.011347
+270  3344    0.009790
+Name: Ebinding, Length: 267, dtype: float64
+
+
+
+
+
+
+

2.19. The \(\chi^2\) function

+

Normally, the response (dependent or outcome) variable \(y_i\) is the +outcome of a numerical experiment or another type of experiment and is +thus only an approximation to the true value. It is then always +accompanied by an error estimate, often limited to a statistical error +estimate given by the standard deviation discussed earlier. In the +discussion here we will treat \(y_i\) as our exact value for the +response variable.

+

Introducing the standard deviation \(\sigma_i\) for each measurement +\(y_i\), we define now the \(\chi^2\) function (omitting the \(1/n\) term) +as

+
+\[ +\chi^2(\boldsymbol{\beta})=\frac{1}{n}\sum_{i=0}^{n-1}\frac{\left(y_i-\tilde{y}_i\right)^2}{\sigma_i^2}=\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{\tilde{y}}\right)^T\frac{1}{\boldsymbol{\Sigma^2}}\left(\boldsymbol{y}-\boldsymbol{\tilde{y}}\right)\right\}, +\]
+

where the matrix \(\boldsymbol{\Sigma}\) is a diagonal matrix with \(\sigma_i\) as matrix elements.

+
+
+

2.20. The \(\chi^2\) function

+

In order to find the parameters \(\beta_i\) we will then minimize the spread of \(\chi^2(\boldsymbol{\beta})\) by requiring

+
+\[ +\frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \beta_j} = \frac{\partial }{\partial \beta_j}\left[ \frac{1}{n}\sum_{i=0}^{n-1}\left(\frac{y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}}{\sigma_i}\right)^2\right]=0, +\]
+

which results in

+
+\[ +\frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \beta_j} = -\frac{2}{n}\left[ \sum_{i=0}^{n-1}\frac{x_{ij}}{\sigma_i}\left(\frac{y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}}{\sigma_i}\right)\right]=0, +\]
+

or in a matrix-vector form as

+
+\[ +\frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = 0 = \boldsymbol{A}^T\left( \boldsymbol{b}-\boldsymbol{A}\boldsymbol{\beta}\right). +\]
+

where we have defined the matrix \(\boldsymbol{A} =\boldsymbol{X}/\boldsymbol{\Sigma}\) with matrix elements \(a_{ij} = x_{ij}/\sigma_i\) and the vector \(\boldsymbol{b}\) with elements \(b_i = y_i/\sigma_i\).

+
+
+

2.21. The \(\chi^2\) function

+

We can rewrite

+
+\[ +\frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = 0 = \boldsymbol{A}^T\left( \boldsymbol{b}-\boldsymbol{A}\boldsymbol{\beta}\right), +\]
+

as

+
+\[ +\boldsymbol{A}^T\boldsymbol{b} = \boldsymbol{A}^T\boldsymbol{A}\boldsymbol{\beta}, +\]
+

and if the matrix \(\boldsymbol{A}^T\boldsymbol{A}\) is invertible we have the solution

+
+\[ +\boldsymbol{\beta} =\left(\boldsymbol{A}^T\boldsymbol{A}\right)^{-1}\boldsymbol{A}^T\boldsymbol{b}. +\]
+
+
+

2.22. The \(\chi^2\) function

+

If we then introduce the matrix

+
+\[ +\boldsymbol{H} = \left(\boldsymbol{A}^T\boldsymbol{A}\right)^{-1}, +\]
+

we have then the following expression for the parameters \(\beta_j\) (the matrix elements of \(\boldsymbol{H}\) are \(h_{ij}\))

+
+\[ +\beta_j = \sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}\frac{y_i}{\sigma_i}\frac{x_{ik}}{\sigma_i} = \sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}b_ia_{ik} +\]
+

We state without proof the expression for the uncertainty in the parameters \(\beta_j\) as (we leave this as an exercise)

+
+\[ +\sigma^2(\beta_j) = \sum_{i=0}^{n-1}\sigma_i^2\left( \frac{\partial \beta_j}{\partial y_i}\right)^2, +\]
+

resulting in

+
+\[ +\sigma^2(\beta_j) = \left(\sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}a_{ik}\right)\left(\sum_{l=0}^{p-1}h_{jl}\sum_{m=0}^{n-1}a_{ml}\right) = h_{jj}! +\]
+
+
+

2.23. The \(\chi^2\) function

+

The first step here is to approximate the function \(y\) with a first-order polynomial, that is we write

+
+\[ +y=y(x) \rightarrow y(x_i) \approx \beta_0+\beta_1 x_i. +\]
+

By computing the derivatives of \(\chi^2\) with respect to \(\beta_0\) and \(\beta_1\) show that these are given by

+
+\[ +\frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \beta_0} = -2\left[ \frac{1}{n}\sum_{i=0}^{n-1}\left(\frac{y_i-\beta_0-\beta_1x_{i}}{\sigma_i^2}\right)\right]=0, +\]
+

and

+
+\[ +\frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \beta_1} = -\frac{2}{n}\left[ \sum_{i=0}^{n-1}x_i\left(\frac{y_i-\beta_0-\beta_1x_{i}}{\sigma_i^2}\right)\right]=0. +\]
+
+
+

2.24. The \(\chi^2\) function

+

For a linear fit (a first-order polynomial) we don’t need to invert a matrix!!
+Defining

+
+\[ +\gamma = \sum_{i=0}^{n-1}\frac{1}{\sigma_i^2}, +\]
+
+\[ +\gamma_x = \sum_{i=0}^{n-1}\frac{x_{i}}{\sigma_i^2}, +\]
+
+\[ +\gamma_y = \sum_{i=0}^{n-1}\left(\frac{y_i}{\sigma_i^2}\right), +\]
+
+\[ +\gamma_{xx} = \sum_{i=0}^{n-1}\frac{x_ix_{i}}{\sigma_i^2}, +\]
+
+\[ +\gamma_{xy} = \sum_{i=0}^{n-1}\frac{y_ix_{i}}{\sigma_i^2}, +\]
+

we obtain

+
+\[ +\beta_0 = \frac{\gamma_{xx}\gamma_y-\gamma_x\gamma_y}{\gamma\gamma_{xx}-\gamma_x^2}, +\]
+
+\[ +\beta_1 = \frac{\gamma_{xy}\gamma-\gamma_x\gamma_y}{\gamma\gamma_{xx}-\gamma_x^2}. +\]
+

This approach (different linear and non-linear regression) suffers +often from both being underdetermined and overdetermined in the +unknown coefficients \(\beta_i\). A better approach is to use the +Singular Value Decomposition (SVD) method discussed below. Or using +Lasso and Ridge regression. See below.

+
+
+

2.25. Fitting an Equation of State for Dense Nuclear Matter

+

Before we continue, let us introduce yet another example. We are going to fit the +nuclear equation of state using results from many-body calculations. +The equation of state we have made available here, as function of +density, has been derived using modern nucleon-nucleon potentials with +the addition of three-body +forces. This +time the file is presented as a standard csv file.

+

The beginning of the Python code here is similar to what you have seen +before, with the same initializations and declarations. We use also +pandas again, rather extensively in order to organize our data.

+

The difference now is that we use Scikit-Learn’s regression tools +instead of our own matrix inversion implementation. Furthermore, we +sneak in Ridge regression (to be discussed below) which includes a +hyperparameter \(\lambda\), also to be explained below.

+
+
+

2.26. The code

+
+
+
# Common imports
+import os
+import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+import matplotlib.pyplot as plt
+import sklearn.linear_model as skl
+from sklearn.metrics import mean_squared_error, r2_score, mean_absolute_error
+
+# Where to save the figures and data files
+PROJECT_ROOT_DIR = "Results"
+FIGURE_ID = "Results/FigureFiles"
+DATA_ID = "DataFiles/"
+
+if not os.path.exists(PROJECT_ROOT_DIR):
+    os.mkdir(PROJECT_ROOT_DIR)
+
+if not os.path.exists(FIGURE_ID):
+    os.makedirs(FIGURE_ID)
+
+if not os.path.exists(DATA_ID):
+    os.makedirs(DATA_ID)
+
+def image_path(fig_id):
+    return os.path.join(FIGURE_ID, fig_id)
+
+def data_path(dat_id):
+    return os.path.join(DATA_ID, dat_id)
+
+def save_fig(fig_id):
+    plt.savefig(image_path(fig_id) + ".png", format='png')
+
+infile = open(data_path("EoS.csv"),'r')
+
+# Read the EoS data as  csv file and organize the data into two arrays with density and energies
+EoS = pd.read_csv(infile, names=('Density', 'Energy'))
+EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce')
+EoS = EoS.dropna()
+Energies = EoS['Energy']
+Density = EoS['Density']
+#  The design matrix now as function of various polytrops
+X = np.zeros((len(Density),4))
+X[:,3] = Density**(4.0/3.0)
+X[:,2] = Density
+X[:,1] = Density**(2.0/3.0)
+X[:,0] = 1
+
+# We use now Scikit-Learn's linear regressor and ridge regressor
+# OLS part
+clf = skl.LinearRegression().fit(X, Energies)
+ytilde = clf.predict(X)
+EoS['Eols']  = ytilde
+# The mean squared error                               
+print("Mean squared error: %.2f" % mean_squared_error(Energies, ytilde))
+# Explained variance score: 1 is perfect prediction                                 
+print('Variance score: %.2f' % r2_score(Energies, ytilde))
+# Mean absolute error                                                           
+print('Mean absolute error: %.2f' % mean_absolute_error(Energies, ytilde))
+print(clf.coef_, clf.intercept_)
+
+# The Ridge regression with a hyperparameter lambda = 0.1
+_lambda = 0.1
+clf_ridge = skl.Ridge(alpha=_lambda).fit(X, Energies)
+yridge = clf_ridge.predict(X)
+EoS['Eridge']  = yridge
+# The mean squared error                               
+print("Mean squared error: %.2f" % mean_squared_error(Energies, yridge))
+# Explained variance score: 1 is perfect prediction                                 
+print('Variance score: %.2f' % r2_score(Energies, yridge))
+# Mean absolute error                                                           
+print('Mean absolute error: %.2f' % mean_absolute_error(Energies, yridge))
+print(clf_ridge.coef_, clf_ridge.intercept_)
+
+fig, ax = plt.subplots()
+ax.set_xlabel(r'$\rho[\mathrm{fm}^{-3}]$')
+ax.set_ylabel(r'Energy per particle')
+ax.plot(EoS['Density'], EoS['Energy'], alpha=0.7, lw=2,
+            label='Theoretical data')
+ax.plot(EoS['Density'], EoS['Eols'], alpha=0.7, lw=2, c='m',
+            label='OLS')
+ax.plot(EoS['Density'], EoS['Eridge'], alpha=0.7, lw=2, c='g',
+            label='Ridge $\lambda = 0.1$')
+ax.legend()
+save_fig("EoSfitting")
+plt.show()
+
+
+
+
+
Mean squared error: 12.36
+Variance score: 1.00
+Mean absolute error: 2.83
+[   0.          618.32047562 -861.13519106 1404.91549644] -11.057088709963637
+Mean squared error: 197.93
+Variance score: 1.00
+Mean absolute error: 11.69
+[  0.          28.18220995 282.79902342 842.30879705] 12.946893955206974
+
+
+_images/chapter3_117_1.png +
+
+

The above simple polynomial in density \(\rho\) gives an excellent fit +to the data.

+

We note also that there is a small deviation between the +standard OLS and the Ridge regression at higher densities. We discuss this in more detail +below.

+
+
+

2.27. Splitting our Data in Training and Test data

+

It is normal in essentially all Machine Learning studies to split the +data in a training set and a test set (sometimes also an additional +validation set). Scikit-Learn has an own function for this. There +is no explicit recipe for how much data should be included as training +data and say test data. An accepted rule of thumb is to use +approximately \(2/3\) to \(4/5\) of the data as training data. We will +postpone a discussion of this splitting to the end of these notes and +our discussion of the so-called bias-variance tradeoff. Here we +limit ourselves to repeat the above equation of state fitting example +but now splitting the data into a training set and a test set.

+
+
+
import os
+import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+from sklearn.model_selection import train_test_split
+# Where to save the figures and data files
+PROJECT_ROOT_DIR = "Results"
+FIGURE_ID = "Results/FigureFiles"
+DATA_ID = "DataFiles/"
+
+if not os.path.exists(PROJECT_ROOT_DIR):
+    os.mkdir(PROJECT_ROOT_DIR)
+
+if not os.path.exists(FIGURE_ID):
+    os.makedirs(FIGURE_ID)
+
+if not os.path.exists(DATA_ID):
+    os.makedirs(DATA_ID)
+
+def image_path(fig_id):
+    return os.path.join(FIGURE_ID, fig_id)
+
+def data_path(dat_id):
+    return os.path.join(DATA_ID, dat_id)
+
+def save_fig(fig_id):
+    plt.savefig(image_path(fig_id) + ".png", format='png')
+
+def R2(y_data, y_model):
+    return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)
+def MSE(y_data,y_model):
+    n = np.size(y_model)
+    return np.sum((y_data-y_model)**2)/n
+
+infile = open(data_path("EoS.csv"),'r')
+
+# Read the EoS data as  csv file and organized into two arrays with density and energies
+EoS = pd.read_csv(infile, names=('Density', 'Energy'))
+EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce')
+EoS = EoS.dropna()
+Energies = EoS['Energy']
+Density = EoS['Density']
+#  The design matrix now as function of various polytrops
+X = np.zeros((len(Density),5))
+X[:,0] = 1
+X[:,1] = Density**(2.0/3.0)
+X[:,2] = Density
+X[:,3] = Density**(4.0/3.0)
+X[:,4] = Density**(5.0/3.0)
+# We split the data in test and training data
+X_train, X_test, y_train, y_test = train_test_split(X, Energies, test_size=0.2)
+# matrix inversion to find beta
+beta = np.linalg.inv(X_train.T.dot(X_train)).dot(X_train.T).dot(y_train)
+# and then make the prediction
+ytilde = X_train @ beta
+print("Training R2")
+print(R2(y_train,ytilde))
+print("Training MSE")
+print(MSE(y_train,ytilde))
+ypredict = X_test @ beta
+print("Test R2")
+print(R2(y_test,ypredict))
+print("Test MSE")
+print(MSE(y_test,ypredict))
+
+
+
+
+
Training R2
+0.9999839096925112
+Training MSE
+6.601101464990597
+Test R2
+0.9999897748408303
+Test MSE
+5.989256390881322
+
+
+
+
+ +
+
+

2.28. The Boston housing data example

+

The Boston housing
+data set was originally a part of UCI Machine Learning Repository +and has been removed now. The data set is now included in Scikit-Learn’s +library. There are 506 samples and 13 feature (predictor) variables +in this data set. The objective is to predict the value of prices of +the house using the features (predictors) listed here.

+

The features/predictors are

+
    +

  1. CRIM: Per capita crime rate by town

    2. +

  2. ZN: Proportion of residential land zoned for lots over 25000 square feet

    3. +

  3. INDUS: Proportion of non-retail business acres per town

    4. +

  4. CHAS: Charles River dummy variable (= 1 if tract bounds river; 0 otherwise)

    5. +

  5. NOX: Nitric oxide concentration (parts per 10 million)

    6. +

  6. RM: Average number of rooms per dwelling

    7. +

  7. AGE: Proportion of owner-occupied units built prior to 1940

    8. +

  8. DIS: Weighted distances to five Boston employment centers

    9. +

  9. RAD: Index of accessibility to radial highways

    10. +

  10. TAX: Full-value property tax rate per USD10000

    11. +

  11. B: \(1000(Bk - 0.63)^2\), where \(Bk\) is the proportion of [people of African American descent] by town

    12. +

  12. LSTAT: Percentage of lower status of the population

    13. +

  13. MEDV: Median value of owner-occupied homes in USD 1000s

    14. +
+
+
+

2.29. Housing data, the code

+

We start by importing the libraries

+
+
+
import numpy as np
+import matplotlib.pyplot as plt 
+
+import pandas as pd  
+import seaborn as sns
+
+
+
+
+

and load the Boston Housing DataSet from Scikit-Learn

+
+
+
from sklearn.datasets import load_boston
+
+boston_dataset = load_boston()
+
+# boston_dataset is a dictionary
+# let's check what it contains
+boston_dataset.keys()
+
+
+
+
+
dict_keys(['data', 'target', 'feature_names', 'DESCR', 'filename'])
+
+
+
+
+

Then we invoke Pandas

+
+
+
boston = pd.DataFrame(boston_dataset.data, columns=boston_dataset.feature_names)
+boston.head()
+boston['MEDV'] = boston_dataset.target
+
+
+
+
+

and preprocess the data

+
+
+
# check for missing values in all the columns
+boston.isnull().sum()
+
+
+
+
+
CRIM       0
+ZN         0
+INDUS      0
+CHAS       0
+NOX        0
+RM         0
+AGE        0
+DIS        0
+RAD        0
+TAX        0
+PTRATIO    0
+B          0
+LSTAT      0
+MEDV       0
+dtype: int64
+
+
+
+
+

We can then visualize the data

+
+
+
# set the size of the figure
+sns.set(rc={'figure.figsize':(11.7,8.27)})
+
+# plot a histogram showing the distribution of the target values
+sns.distplot(boston['MEDV'], bins=30)
+plt.show()
+
+
+
+
+_images/chapter3_129_0.png +
+
+

It is now useful to look at the correlation matrix

+
+
+
# compute the pair wise correlation for all columns  
+correlation_matrix = boston.corr().round(2)
+# use the heatmap function from seaborn to plot the correlation matrix
+# annot = True to print the values inside the square
+sns.heatmap(data=correlation_matrix, annot=True)
+
+
+
+
+
<matplotlib.axes._subplots.AxesSubplot at 0x7ff6a7303b50>
+
+
+_images/chapter3_131_1.png +
+
+

From the above coorelation plot we can see that MEDV is strongly correlated to LSTAT and RM. We see also that RAD and TAX are stronly correlated, but we don’t include this in our features together to avoid multi-colinearity

+
+
+
plt.figure(figsize=(20, 5))
+
+features = ['LSTAT', 'RM']
+target = boston['MEDV']
+
+for i, col in enumerate(features):
+    plt.subplot(1, len(features) , i+1)
+    x = boston[col]
+    y = target
+    plt.scatter(x, y, marker='o')
+    plt.title(col)
+    plt.xlabel(col)
+    plt.ylabel('MEDV')
+
+
+
+
+_images/chapter3_133_0.png +
+
+

Now we start training our model

+
+
+
X = pd.DataFrame(np.c_[boston['LSTAT'], boston['RM']], columns = ['LSTAT','RM'])
+Y = boston['MEDV']
+
+
+
+
+

We split the data into training and test sets

+
+
+
from sklearn.model_selection import train_test_split
+
+# splits the training and test data set in 80% : 20%
+# assign random_state to any value.This ensures consistency.
+X_train, X_test, Y_train, Y_test = train_test_split(X, Y, test_size = 0.2, random_state=5)
+print(X_train.shape)
+print(X_test.shape)
+print(Y_train.shape)
+print(Y_test.shape)
+
+
+
+
+
(404, 2)
+(102, 2)
+(404,)
+(102,)
+
+
+
+
+

Then we use the linear regression functionality from Scikit-Learn

+
+
+
from sklearn.linear_model import LinearRegression
+from sklearn.metrics import mean_squared_error, r2_score
+
+lin_model = LinearRegression()
+lin_model.fit(X_train, Y_train)
+
+# model evaluation for training set
+
+y_train_predict = lin_model.predict(X_train)
+rmse = (np.sqrt(mean_squared_error(Y_train, y_train_predict)))
+r2 = r2_score(Y_train, y_train_predict)
+
+print("The model performance for training set")
+print("--------------------------------------")
+print('RMSE is {}'.format(rmse))
+print('R2 score is {}'.format(r2))
+print("\n")
+
+# model evaluation for testing set
+
+y_test_predict = lin_model.predict(X_test)
+# root mean square error of the model
+rmse = (np.sqrt(mean_squared_error(Y_test, y_test_predict)))
+
+# r-squared score of the model
+r2 = r2_score(Y_test, y_test_predict)
+
+print("The model performance for testing set")
+print("--------------------------------------")
+print('RMSE is {}'.format(rmse))
+print('R2 score is {}'.format(r2))
+
+
+
+
+
The model performance for training set
+--------------------------------------
+RMSE is 5.6371293350711955
+R2 score is 0.6300745149331701
+
+
+The model performance for testing set
+--------------------------------------
+RMSE is 5.137400784702912
+R2 score is 0.6628996975186952
+
+
+
+
+
+
+
# plotting the y_test vs y_pred
+# ideally should have been a straight line
+plt.scatter(Y_test, y_test_predict)
+plt.show()
+
+
+
+
+_images/chapter3_140_0.png +
+
+
+
+

2.30. Reducing the number of degrees of freedom, overarching view

+

Many Machine Learning problems involve thousands or even millions of +features for each training instance. Not only does this make training +extremely slow, it can also make it much harder to find a good +solution, as we will see. This problem is often referred to as the +curse of dimensionality. Fortunately, in real-world problems, it is +often possible to reduce the number of features considerably, turning +an intractable problem into a tractable one.

+

Later we will discuss some of the most popular dimensionality reduction +techniques: the principal component analysis (PCA), Kernel PCA, and +Locally Linear Embedding (LLE).

+

Principal component analysis and its various variants deal with the +problem of fitting a low-dimensional affine +subspace to a set of of +data points in a high-dimensional space. With its family of methods it +is one of the most used tools in data modeling, compression and +visualization.

+
+
+

2.31. Preprocessing our data

+

Before we proceed however, we will discuss how to preprocess our +data. Till now and in connection with our previous examples we have +not met so many cases where we are too sensitive to the scaling of our +data. Normally the data may need a rescaling and/or may be sensitive +to extreme values. Scaling the data renders our inputs much more +suitable for the algorithms we want to employ.

+

Scikit-Learn has several functions which allow us to rescale the +data, normally resulting in much better results in terms of various +accuracy scores. The StandardScaler function in Scikit-Learn +ensures that for each feature/predictor we study the mean value is +zero and the variance is one (every column in the design/feature +matrix). This scaling has the drawback that it does not ensure that +we have a particular maximum or minimum in our data set. Another +function included in Scikit-Learn is the MinMaxScaler which +ensures that all features are exactly between \(0\) and \(1\). The

+
+
+

2.32. More preprocessing

+

The Normalizer scales each data +point such that the feature vector has a euclidean length of one. In other words, it +projects a data point on the circle (or sphere in the case of higher dimensions) with a +radius of 1. This means every data point is scaled by a different number (by the +inverse of it’s length). +This normalization is often used when only the direction (or angle) of the data matters, +not the length of the feature vector.

+

The RobustScaler works similarly to the StandardScaler in that it +ensures statistical properties for each feature that guarantee that +they are on the same scale. However, the RobustScaler uses the median +and quartiles, instead of mean and variance. This makes the +RobustScaler ignore data points that are very different from the rest +(like measurement errors). These odd data points are also called +outliers, and might often lead to trouble for other scaling +techniques.

+
+
+

2.33. Simple preprocessing examples, Franke function and regression

+
+
+
# Common imports
+import os
+import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+import sklearn.linear_model as skl
+from sklearn.metrics import mean_squared_error
+from sklearn.model_selection import  train_test_split
+from sklearn.preprocessing import MinMaxScaler, StandardScaler, Normalizer
+
+# Where to save the figures and data files
+PROJECT_ROOT_DIR = "Results"
+FIGURE_ID = "Results/FigureFiles"
+DATA_ID = "DataFiles/"
+
+if not os.path.exists(PROJECT_ROOT_DIR):
+    os.mkdir(PROJECT_ROOT_DIR)
+
+if not os.path.exists(FIGURE_ID):
+    os.makedirs(FIGURE_ID)
+
+if not os.path.exists(DATA_ID):
+    os.makedirs(DATA_ID)
+
+def image_path(fig_id):
+    return os.path.join(FIGURE_ID, fig_id)
+
+def data_path(dat_id):
+    return os.path.join(DATA_ID, dat_id)
+
+def save_fig(fig_id):
+    plt.savefig(image_path(fig_id) + ".png", format='png')
+
+
+def FrankeFunction(x,y):
+	term1 = 0.75*np.exp(-(0.25*(9*x-2)**2) - 0.25*((9*y-2)**2))
+	term2 = 0.75*np.exp(-((9*x+1)**2)/49.0 - 0.1*(9*y+1))
+	term3 = 0.5*np.exp(-(9*x-7)**2/4.0 - 0.25*((9*y-3)**2))
+	term4 = -0.2*np.exp(-(9*x-4)**2 - (9*y-7)**2)
+	return term1 + term2 + term3 + term4
+
+
+def create_X(x, y, n ):
+	if len(x.shape) > 1:
+		x = np.ravel(x)
+		y = np.ravel(y)
+
+	N = len(x)
+	l = int((n+1)*(n+2)/2)		# Number of elements in beta
+	X = np.ones((N,l))
+
+	for i in range(1,n+1):
+		q = int((i)*(i+1)/2)
+		for k in range(i+1):
+			X[:,q+k] = (x**(i-k))*(y**k)
+
+	return X
+
+
+# Making meshgrid of datapoints and compute Franke's function
+n = 5
+N = 1000
+x = np.sort(np.random.uniform(0, 1, N))
+y = np.sort(np.random.uniform(0, 1, N))
+z = FrankeFunction(x, y)
+X = create_X(x, y, n=n)    
+# split in training and test data
+X_train, X_test, y_train, y_test = train_test_split(X,z,test_size=0.2)
+
+
+clf = skl.LinearRegression().fit(X_train, y_train)
+
+# The mean squared error and R2 score
+print("MSE before scaling: {:.2f}".format(mean_squared_error(clf.predict(X_test), y_test)))
+print("R2 score before scaling {:.2f}".format(clf.score(X_test,y_test)))
+
+scaler = StandardScaler()
+scaler.fit(X_train)
+X_train_scaled = scaler.transform(X_train)
+X_test_scaled = scaler.transform(X_test)
+
+print("Feature min values before scaling:\n {}".format(X_train.min(axis=0)))
+print("Feature max values before scaling:\n {}".format(X_train.max(axis=0)))
+
+print("Feature min values after scaling:\n {}".format(X_train_scaled.min(axis=0)))
+print("Feature max values after scaling:\n {}".format(X_train_scaled.max(axis=0)))
+
+clf = skl.LinearRegression().fit(X_train_scaled, y_train)
+
+
+print("MSE after  scaling: {:.2f}".format(mean_squared_error(clf.predict(X_test_scaled), y_test)))
+print("R2 score for  scaled data: {:.2f}".format(clf.score(X_test_scaled,y_test)))
+
+
+
+
+
MSE before scaling: 0.00
+R2 score before scaling 1.00
+Feature min values before scaling:
+ [1.00000000e+00 2.28444646e-04 1.45426079e-03 5.21869563e-08
+ 3.32218090e-07 2.11487443e-06 1.19218307e-11 7.58934440e-11
+ 4.83131741e-10 3.07557895e-09 2.72347840e-15 1.73374509e-14
+ 1.10368860e-13 7.02599545e-13 4.47269386e-12 6.22164060e-19
+ 3.96064784e-18 2.52131750e-17 1.60505104e-16 1.02176297e-15
+ 6.50446329e-15]
+Feature max values before scaling:
+ [1.         0.99936169 0.99676727 0.99872379 0.99613102 0.99354498
+ 0.9980863  0.99549518 0.9929108  0.99033312 0.99744921 0.99485975
+ 0.99227701 0.98970098 0.98713163 0.99681253 0.99422472 0.99164363
+ 0.98906925 0.98650154 0.9839405 ]
+Feature min values after scaling:
+ [ 0.         -1.73062381 -1.79448669 -1.10522795 -1.11901222 -1.13376089
+ -0.86500438 -0.86965898 -0.874527   -0.87962566 -0.7323435  -0.73424489
+ -0.73620726 -0.73823503 -0.74033294 -0.6467848  -0.64760955 -0.64845217
+ -0.64931411 -0.65019693 -0.65110229]
+Feature max values after scaling:
+ [0.         1.74251315 1.75886038 2.23112789 2.24209788 2.25327217
+ 2.61472545 2.62469494 2.6346596  2.64462605 2.93716229 2.94692844
+ 2.95663429 2.96628194 2.97587366 3.22012801 3.2298595  3.23951215
+ 3.24908673 3.25858404 3.26800498]
+MSE after  scaling: 0.00
+R2 score for  scaled data: 1.00
+
+
+
+
+
+
+

2.34. The singular value decomposition

+

The examples we have looked at so far are cases where we normally can +invert the matrix \(\boldsymbol{X}^T\boldsymbol{X}\). Using a polynomial expansion as we +did both for the masses and the fitting of the equation of state, +leads to row vectors of the design matrix which are essentially +orthogonal due to the polynomial character of our model. Obtaining the inverse of the design matrix is then often done via a so-called LU, QR or Cholesky decomposition.

+

This may +however not the be case in general and a standard matrix inversion +algorithm based on say LU, QR or Cholesky decomposition may lead to singularities. We will see examples of this below.

+

There is however a way to partially circumvent this problem and also gain some insights about the ordinary least squares approach, and later shrinkage methods like Ridge and Lasso regressions.

+

This is given by the Singular Value Decomposition algorithm, perhaps +the most powerful linear algebra algorithm. Let us look at a +different example where we may have problems with the standard matrix +inversion algorithm. Thereafter we dive into the math of the SVD.

+
+
+

2.35. Linear Regression Problems

+

One of the typical problems we encounter with linear regression, in particular +when the matrix \(\boldsymbol{X}\) (our so-called design matrix) is high-dimensional, +are problems with near singular or singular matrices. The column vectors of \(\boldsymbol{X}\) +may be linearly dependent, normally referred to as super-collinearity.
+This means that the matrix may be rank deficient and it is basically impossible to +to model the data using linear regression. As an example, consider the matrix

+
+\[\begin{split} +\begin{align*} +\mathbf{X} & = \left[ +\begin{array}{rrr} +1 & -1 & 2 +\\ +1 & 0 & 1 +\\ +1 & 2 & -1 +\\ +1 & 1 & 0 +\end{array} \right] +\end{align*} +\end{split}\]
+

The columns of \(\boldsymbol{X}\) are linearly dependent. We see this easily since the +the first column is the row-wise sum of the other two columns. The rank (more correct, +the column rank) of a matrix is the dimension of the space spanned by the +column vectors. Hence, the rank of \(\mathbf{X}\) is equal to the number +of linearly independent columns. In this particular case the matrix has rank 2.

+

Super-collinearity of an \((n \times p)\)-dimensional design matrix \(\mathbf{X}\) implies +that the inverse of the matrix \(\boldsymbol{X}^T\boldsymbol{X}\) (the matrix we need to invert to solve the linear regression equations) is non-invertible. If we have a square matrix that does not have an inverse, we say this matrix singular. The example here demonstrates this

+
+\[\begin{split} +\begin{align*} +\boldsymbol{X} & = \left[ +\begin{array}{rr} +1 & -1 +\\ +1 & -1 +\end{array} \right]. +\end{align*} +\end{split}\]
+

We see easily that \(\mbox{det}(\boldsymbol{X}) = x_{11} x_{22} - x_{12} x_{21} = 1 \times (-1) - 1 \times (-1) = 0\). Hence, \(\mathbf{X}\) is singular and its inverse is undefined. +This is equivalent to saying that the matrix \(\boldsymbol{X}\) has at least an eigenvalue which is zero.

+
+
+

2.36. Fixing the singularity

+

If our design matrix \(\boldsymbol{X}\) which enters the linear regression problem

+ +
+
+\[ +\begin{equation} +\boldsymbol{\beta} = (\boldsymbol{X}^{T} \boldsymbol{X})^{-1} \boldsymbol{X}^{T} \boldsymbol{y}, +\label{_auto1} \tag{1} +\end{equation} +\]
+

has linearly dependent column vectors, we will not be able to compute the inverse +of \(\boldsymbol{X}^T\boldsymbol{X}\) and we cannot find the parameters (estimators) \(\beta_i\). +The estimators are only well-defined if \((\boldsymbol{X}^{T}\boldsymbol{X})^{-1}\) exits. +This is more likely to happen when the matrix \(\boldsymbol{X}\) is high-dimensional. In this case it is likely to encounter a situation where +the regression parameters \(\beta_i\) cannot be estimated.

+

A cheap ad hoc approach is simply to add a small diagonal component to the matrix to invert, that is we change

+
+\[ +\boldsymbol{X}^{T} \boldsymbol{X} \rightarrow \boldsymbol{X}^{T} \boldsymbol{X}+\lambda \boldsymbol{I}, +\]
+

where \(\boldsymbol{I}\) is the identity matrix. When we discuss Ridge regression this is actually what we end up evaluating. The parameter \(\lambda\) is called a hyperparameter. More about this later.

+
+
+

2.37. Basic math of the SVD

+

From standard linear algebra we know that a square matrix \(\boldsymbol{X}\) can be diagonalized if and only it is +a so-called normal matrix, that is if \(\boldsymbol{X}\in {\mathbb{R}}^{n\times n}\) +we have \(\boldsymbol{X}\boldsymbol{X}^T=\boldsymbol{X}^T\boldsymbol{X}\) or if \(\boldsymbol{X}\in {\mathbb{C}}^{n\times n}\) we have \(\boldsymbol{X}\boldsymbol{X}^{\dagger}=\boldsymbol{X}^{\dagger}\boldsymbol{X}\). +The matrix has then a set of eigenpairs

+
+\[ +(\lambda_1,\boldsymbol{u}_1),\dots, (\lambda_n,\boldsymbol{u}_n), +\]
+

and the eigenvalues are given by the diagonal matrix

+
+\[ +\boldsymbol{\Sigma}=\mathrm{Diag}(\lambda_1, \dots,\lambda_n). +\]
+

The matrix \(\boldsymbol{X}\) can be written in terms of an orthogonal/unitary transformation \(\boldsymbol{U}\)

+
+\[ +\boldsymbol{X} = \boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T, +\]
+

with \(\boldsymbol{U}\boldsymbol{U}^T=\boldsymbol{I}\) or \(\boldsymbol{U}\boldsymbol{U}^{\dagger}=\boldsymbol{I}\).

+

Not all square matrices are diagonalizable. A matrix like the one discussed above

+
+\[\begin{split} +\boldsymbol{X} = \begin{bmatrix} +1& -1 \\ +1& -1\\ +\end{bmatrix} +\end{split}\]
+

is not diagonalizable, it is a so-called defective matrix. It is easy to see that the condition +\(\boldsymbol{X}\boldsymbol{X}^T=\boldsymbol{X}^T\boldsymbol{X}\) is not fulfilled.

+
+
+

2.38. The SVD, a Fantastic Algorithm

+

However, and this is the strength of the SVD algorithm, any general +matrix \(\boldsymbol{X}\) can be decomposed in terms of a diagonal matrix and +two orthogonal/unitary matrices. The Singular Value Decompostion +(SVD) theorem +states that a general \(m\times n\) matrix \(\boldsymbol{X}\) can be written in +terms of a diagonal matrix \(\boldsymbol{\Sigma}\) of dimensionality \(m\times n\) +and two orthognal matrices \(\boldsymbol{U}\) and \(\boldsymbol{V}\), where the first has +dimensionality \(m \times m\) and the last dimensionality \(n\times n\). +We have then

+
+\[ +\boldsymbol{X} = \boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T +\]
+

As an example, the above defective matrix can be decomposed as

+
+\[\begin{split} +\boldsymbol{X} = \frac{1}{\sqrt{2}}\begin{bmatrix} 1& 1 \\ 1& -1\\ \end{bmatrix} \begin{bmatrix} 2& 0 \\ 0& 0\\ \end{bmatrix} \frac{1}{\sqrt{2}}\begin{bmatrix} 1& -1 \\ 1& 1\\ \end{bmatrix}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T, +\end{split}\]
+

with eigenvalues \(\sigma_1=2\) and \(\sigma_2=0\). +The SVD exits always!

+

The SVD +decomposition (singular values) gives eigenvalues +\(\sigma_i\geq\sigma_{i+1}\) for all \(i\) and for dimensions larger than \(i=p\), the +eigenvalues (singular values) are zero.

+

In the general case, where our design matrix \(\boldsymbol{X}\) has dimension +\(n\times p\), the matrix is thus decomposed into an \(n\times n\) +orthogonal matrix \(\boldsymbol{U}\), a \(p\times p\) orthogonal matrix \(\boldsymbol{V}\) +and a diagonal matrix \(\boldsymbol{\Sigma}\) with \(r=\mathrm{min}(n,p)\) +singular values \(\sigma_i\geq 0\) on the main diagonal and zeros filling +the rest of the matrix. There are at most \(p\) singular values +assuming that \(n > p\). In our regression examples for the nuclear +masses and the equation of state this is indeed the case, while for +the Ising model we have \(p > n\). These are often cases that lead to +near singular or singular matrices.

+

The columns of \(\boldsymbol{U}\) are called the left singular vectors while the columns of \(\boldsymbol{V}\) are the right singular vectors.

+
+
+

2.39. Economy-size SVD

+

If we assume that \(n > p\), then our matrix \(\boldsymbol{U}\) has dimension \(n +\times n\). The last \(n-p\) columns of \(\boldsymbol{U}\) become however +irrelevant in our calculations since they are multiplied with the +zeros in \(\boldsymbol{\Sigma}\).

+

The economy-size decomposition removes extra rows or columns of zeros +from the diagonal matrix of singular values, \(\boldsymbol{\Sigma}\), along with the columns +in either \(\boldsymbol{U}\) or \(\boldsymbol{V}\) that multiply those zeros in the expression. +Removing these zeros and columns can improve execution time +and reduce storage requirements without compromising the accuracy of +the decomposition.

+

If \(n > p\), we keep only the first \(p\) columns of \(\boldsymbol{U}\) and \(\boldsymbol{\Sigma}\) has dimension \(p\times p\). +If \(p > n\), then only the first \(n\) columns of \(\boldsymbol{V}\) are computed and \(\boldsymbol{\Sigma}\) has dimension \(n\times n\). +The \(n=p\) case is obvious, we retain the full SVD. +In general the economy-size SVD leads to less FLOPS and still conserving the desired accuracy.

+
+
+

2.40. Codes for the SVD

+
+
+
import numpy as np
+# SVD inversion
+def SVDinv(A):
+    ''' Takes as input a numpy matrix A and returns inv(A) based on singular value decomposition (SVD).
+    SVD is numerically more stable than the inversion algorithms provided by
+    numpy and scipy.linalg at the cost of being slower.
+    '''
+    U, s, VT = np.linalg.svd(A)
+#    print('test U')
+#    print( (np.transpose(U) @ U - U @np.transpose(U)))
+#    print('test VT')
+#    print( (np.transpose(VT) @ VT - VT @np.transpose(VT)))
+    print(U)
+    print(s)
+    print(VT)
+
+    D = np.zeros((len(U),len(VT)))
+    for i in range(0,len(VT)):
+        D[i,i]=s[i]
+    UT = np.transpose(U); V = np.transpose(VT); invD = np.linalg.inv(D)
+    return np.matmul(V,np.matmul(invD,UT))
+
+
+X = np.array([ [1.0, -1.0, 2.0], [1.0, 0.0, 1.0], [1.0, 2.0, -1.0], [1.0, 1.0, 0.0] ])
+print(X)
+A = np.transpose(X) @ X
+print(A)
+# Brute force inversion of super-collinear matrix
+#B = np.linalg.inv(A)
+#print(B)
+C = SVDinv(A)
+print(C)
+
+
+
+
+
[[ 1. -1.  2.]
+ [ 1.  0.  1.]
+ [ 1.  2. -1.]
+ [ 1.  1.  0.]]
+[[ 4.  2.  2.]
+ [ 2.  6. -4.]
+ [ 2. -4.  6.]]
+[[-9.57425734e-17  8.16496581e-01 -5.77350269e-01]
+ [-7.07106781e-01  4.08248290e-01  5.77350269e-01]
+ [ 7.07106781e-01  4.08248290e-01  5.77350269e-01]]
+[1.00000000e+01 6.00000000e+00 9.10898112e-32]
+[[ 3.33066907e-17 -7.07106781e-01  7.07106781e-01]
+ [ 8.16496581e-01  4.08248290e-01  4.08248290e-01]
+ [ 5.77350269e-01 -5.77350269e-01 -5.77350269e-01]]
+[[-3.65939208e+30  3.65939208e+30  3.65939208e+30]
+ [ 3.65939208e+30 -3.65939208e+30 -3.65939208e+30]
+ [ 3.65939208e+30 -3.65939208e+30 -3.65939208e+30]]
+
+
+
+
+

The matrix \(\boldsymbol{X}\) has columns that are linearly dependent. The first +column is the row-wise sum of the other two columns. The rank of a +matrix (the column rank) is the dimension of space spanned by the +column vectors. The rank of the matrix is the number of linearly +independent columns, in this case just \(2\). We see this from the +singular values when running the above code. Running the standard +inversion algorithm for matrix inversion with \(\boldsymbol{X}^T\boldsymbol{X}\) results +in the program terminating due to a singular matrix.

+
+
+

2.41. Mathematical Properties

+

There are several interesting mathematical properties which will be +relevant when we are going to discuss the differences between say +ordinary least squares (OLS) and Ridge regression.

+

We have from OLS that the parameters of the linear approximation are given by

+
+\[ +\boldsymbol{\tilde{y}} = \boldsymbol{X}\boldsymbol{\beta} = \boldsymbol{X}\left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}. +\]
+

The matrix to invert can be rewritten in terms of our SVD decomposition as

+
+\[ +\boldsymbol{X}^T\boldsymbol{X} = \boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{U}^T\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T. +\]
+

Using the orthogonality properties of \(\boldsymbol{U}\) we have

+
+\[ +\boldsymbol{X}^T\boldsymbol{X} = \boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{\Sigma}\boldsymbol{V}^T = \boldsymbol{V}\boldsymbol{D}\boldsymbol{V}^T, +\]
+

with \(\boldsymbol{D}\) being a diagonal matrix with values along the diagonal given by the singular values squared.

+

This means that

+
+\[ +(\boldsymbol{X}^T\boldsymbol{X})\boldsymbol{V} = \boldsymbol{V}\boldsymbol{D}, +\]
+

that is the eigenvectors of \((\boldsymbol{X}^T\boldsymbol{X})\) are given by the columns of the right singular matrix of \(\boldsymbol{X}\) and the eigenvalues are the squared singular values. It is easy to show (show this) that

+
+\[ +(\boldsymbol{X}\boldsymbol{X}^T)\boldsymbol{U} = \boldsymbol{U}\boldsymbol{D}, +\]
+

that is, the eigenvectors of \((\boldsymbol{X}\boldsymbol{X})^T\) are the columns of the left singular matrix and the eigenvalues are the same.

+

Going back to our OLS equation we have

+
+\[ +\boldsymbol{X}\boldsymbol{\beta} = \boldsymbol{X}\left(\boldsymbol{V}\boldsymbol{D}\boldsymbol{V}^T \right)^{-1}\boldsymbol{X}^T\boldsymbol{y}=\boldsymbol{U\Sigma V^T}\left(\boldsymbol{V}\boldsymbol{D}\boldsymbol{V}^T \right)^{-1}(\boldsymbol{U\Sigma V^T})^T\boldsymbol{y}=\boldsymbol{U}\boldsymbol{U}^T\boldsymbol{y}. +\]
+

We will come back to this expression when we discuss Ridge regression.

+

$\( \tilde{y}^{OLS}=\boldsymbol{X}\hat{\beta}^{OLS}=\sum_{j=1}^p \boldsymbol{u}_j\boldsymbol{u}_j^T\boldsymbol{y}\)$ and for Ridge we have

+

$\( \tilde{y}^{Ridge}=\boldsymbol{X}\hat{\beta}^{Ridge}=\sum_{j=1}^p \boldsymbol{u}_j\frac{\sigma_j^2}{\sigma_j^2+\lambda}\boldsymbol{u}_j^T\boldsymbol{y}\)$ .

+

It is indeed the economy-sized SVD, note the summation runs up tp $\(p\)\( only and not \)\(n\)$.

+

Here we have that $\(\boldsymbol{X} = \boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T\)\(, with \)\(\Sigma\)\( being an \)\( n\times p\)\( matrix and \)\(\boldsymbol{V}\)\( being a \)\( p\times p\)\( matrix. We also have assumed here that \)\( n > p\)$.

+
+
+

2.42. Ridge and LASSO Regression

+

Let us remind ourselves about the expression for the standard Mean Squared Error (MSE) which we used to define our cost function and the equations for the ordinary least squares (OLS) method, that is +our optimization problem is

+
+\[ +{\displaystyle \min_{\boldsymbol{\beta}\in {\mathbb{R}}^{p}}}\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right\}. +\]
+

or we can state it as

+
+\[ +{\displaystyle \min_{\boldsymbol{\beta}\in +{\mathbb{R}}^{p}}}\frac{1}{n}\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2=\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2, +\]
+

where we have used the definition of a norm-2 vector, that is

+
+\[ +\vert\vert \boldsymbol{x}\vert\vert_2 = \sqrt{\sum_i x_i^2}. +\]
+

By minimizing the above equation with respect to the parameters +\(\boldsymbol{\beta}\) we could then obtain an analytical expression for the +parameters \(\boldsymbol{\beta}\). We can add a regularization parameter \(\lambda\) by +defining a new cost function to be optimized, that is

+
+\[ +{\displaystyle \min_{\boldsymbol{\beta}\in +{\mathbb{R}}^{p}}}\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2+\lambda\vert\vert \boldsymbol{\beta}\vert\vert_2^2 +\]
+

which leads to the Ridge regression minimization problem where we +require that \(\vert\vert \boldsymbol{\beta}\vert\vert_2^2\le t\), where \(t\) is +a finite number larger than zero. By defining

+
+\[ +C(\boldsymbol{X},\boldsymbol{\beta})=\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2+\lambda\vert\vert \boldsymbol{\beta}\vert\vert_1, +\]
+

we have a new optimization equation

+
+\[ +{\displaystyle \min_{\boldsymbol{\beta}\in +{\mathbb{R}}^{p}}}\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2+\lambda\vert\vert \boldsymbol{\beta}\vert\vert_1 +\]
+

which leads to Lasso regression. Lasso stands for least absolute shrinkage and selection operator.

+

Here we have defined the norm-1 as

+
+\[ +\vert\vert \boldsymbol{x}\vert\vert_1 = \sum_i \vert x_i\vert. +\]
+
+
+

2.43. More on Ridge Regression

+

Using the matrix-vector expression for Ridge regression,

+
+\[ +C(\boldsymbol{X},\boldsymbol{\beta})=\frac{1}{n}\left\{(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})^T(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\right\}+\lambda\boldsymbol{\beta}^T\boldsymbol{\beta}, +\]
+

by taking the derivatives with respect to \(\boldsymbol{\beta}\) we obtain then +a slightly modified matrix inversion problem which for finite values +of \(\lambda\) does not suffer from singularity problems. We obtain

+
+\[ +\boldsymbol{\beta}^{\mathrm{Ridge}} = \left(\boldsymbol{X}^T\boldsymbol{X}+\lambda\boldsymbol{I}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}, +\]
+

with \(\boldsymbol{I}\) being a \(p\times p\) identity matrix with the constraint that

+
+\[ +\sum_{i=0}^{p-1} \beta_i^2 \leq t, +\]
+

with \(t\) a finite positive number.

+

We see that Ridge regression is nothing but the standard +OLS with a modified diagonal term added to \(\boldsymbol{X}^T\boldsymbol{X}\). The +consequences, in particular for our discussion of the bias-variance tradeoff +are rather interesting.

+

Furthermore, if we use the result above in terms of the SVD decomposition (our analysis was done for the OLS method), we had

+
+\[ +(\boldsymbol{X}\boldsymbol{X}^T)\boldsymbol{U} = \boldsymbol{U}\boldsymbol{D}. +\]
+

We can analyse the OLS solutions in terms of the eigenvectors (the columns) of the right singular value matrix \(\boldsymbol{U}\) as

+
+\[ +\boldsymbol{X}\boldsymbol{\beta} = \boldsymbol{X}\left(\boldsymbol{V}\boldsymbol{D}\boldsymbol{V}^T \right)^{-1}\boldsymbol{X}^T\boldsymbol{y}=\boldsymbol{U\Sigma V^T}\left(\boldsymbol{V}\boldsymbol{D}\boldsymbol{V}^T \right)^{-1}(\boldsymbol{U\Sigma V^T})^T\boldsymbol{y}=\boldsymbol{U}\boldsymbol{U}^T\boldsymbol{y} +\]
+

For Ridge regression this becomes

+
+\[ +\boldsymbol{X}\boldsymbol{\beta}^{\mathrm{Ridge}} = \boldsymbol{U\Sigma V^T}\left(\boldsymbol{V}\boldsymbol{D}\boldsymbol{V}^T+\lambda\boldsymbol{I} \right)^{-1}(\boldsymbol{U\Sigma V^T})^T\boldsymbol{y}=\sum_{j=0}^{p-1}\boldsymbol{u}_j\boldsymbol{u}_j^T\frac{\sigma_j^2}{\sigma_j^2+\lambda}\boldsymbol{y}, +\]
+

with the vectors \(\boldsymbol{u}_j\) being the columns of \(\boldsymbol{U}\).

+
+
+

2.44. Interpreting the Ridge results

+

Since \(\lambda \geq 0\), it means that compared to OLS, we have

+
+\[ +\frac{\sigma_j^2}{\sigma_j^2+\lambda} \leq 1. +\]
+

Ridge regression finds the coordinates of \(\boldsymbol{y}\) with respect to the +orthonormal basis \(\boldsymbol{U}\), it then shrinks the coordinates by +\(\frac{\sigma_j^2}{\sigma_j^2+\lambda}\). Recall that the SVD has +eigenvalues ordered in a descending way, that is \(\sigma_i \geq +\sigma_{i+1}\).

+

For small eigenvalues \(\sigma_i\) it means that their contributions become less important, a fact which can be used to reduce the number of degrees of freedom. +Actually, calculating the variance of \(\boldsymbol{X}\boldsymbol{v}_j\) shows that this quantity is equal to \(\sigma_j^2/n\). +With a parameter \(\lambda\) we can thus shrink the role of specific parameters.

+
+
+

2.45. More interpretations

+

For the sake of simplicity, let us assume that the design matrix is orthonormal, that is

+
+\[ +\boldsymbol{X}^T\boldsymbol{X}=(\boldsymbol{X}^T\boldsymbol{X})^{-1} =\boldsymbol{I}. +\]
+

In this case the standard OLS results in

+
+\[ +\boldsymbol{\beta}^{\mathrm{OLS}} = \boldsymbol{X}^T\boldsymbol{y}=\sum_{i=0}^{p-1}\boldsymbol{u}_j\boldsymbol{u}_j^T\boldsymbol{y}, +\]
+

and

+
+\[ +\boldsymbol{\beta}^{\mathrm{Ridge}} = \left(\boldsymbol{I}+\lambda\boldsymbol{I}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}=\left(1+\lambda\right)^{-1}\boldsymbol{\beta}^{\mathrm{OLS}}, +\]
+

that is the Ridge estimator scales the OLS estimator by the inverse of a factor \(1+\lambda\), and +the Ridge estimator converges to zero when the hyperparameter goes to +infinity.

+

We will come back to more interpreations after we have gone through some of the statistical analysis part.

+

For more discussions of Ridge and Lasso regression, Wessel van Wieringen’s article is highly recommended. +Similarly, Mehta et al’s article is also recommended.

+ +
+
+

2.46. A better understanding of regularization

+

The parameter \(\lambda\) that we have introduced in the Ridge (and +Lasso as well) regression is often called a regularization parameter +or shrinkage parameter. It is common to call it a hyperparameter. What does it mean mathemtically?

+

Here we will first look at how to analyze the difference between the +standard OLS equations and the Ridge expressions in terms of a linear +algebra analysis using the SVD algorithm. Thereafter, we will link +(see the material on the bias-variance tradeoff below) these +observation to the statisical analysis of the results. In particular +we consider how the variance of the parameters \(\boldsymbol{\beta}\) is +affected by changing the parameter \(\lambda\).

+
+
+

2.47. Decomposing the OLS and Ridge expressions

+

We have our design matrix +\(\boldsymbol{X}\in {\mathbb{R}}^{n\times p}\). With the SVD we decompose it as

+
+\[ +\boldsymbol{X} = \boldsymbol{U\Sigma V^T}, +\]
+

with \(\boldsymbol{U}\in {\mathbb{R}}^{n\times n}\), \(\boldsymbol{\Sigma}\in {\mathbb{R}}^{n\times p}\) +and \(\boldsymbol{V}\in {\mathbb{R}}^{p\times p}\).

+

The matrices \(\boldsymbol{U}\) and \(\boldsymbol{V}\) are unitary/orthonormal matrices, that is in case the matrices are real we have \(\boldsymbol{U}^T\boldsymbol{U}=\boldsymbol{U}\boldsymbol{U}^T=\boldsymbol{I}\) and \(\boldsymbol{V}^T\boldsymbol{V}=\boldsymbol{V}\boldsymbol{V}^T=\boldsymbol{I}\).

+
+
+

2.48. Introducing the Covariance and Correlation functions

+

Before we discuss the link between for example Ridge regression and the singular value decomposition, we need to remind ourselves about +the definition of the covariance and the correlation function. These are quantities

+

Suppose we have defined two vectors +\(\hat{x}\) and \(\hat{y}\) with \(n\) elements each. The covariance matrix \(\boldsymbol{C}\) is defined as

+
+\[\begin{split} +\boldsymbol{C}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} \mathrm{cov}[\boldsymbol{x},\boldsymbol{x}] & \mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] \\ + \mathrm{cov}[\boldsymbol{y},\boldsymbol{x}] & \mathrm{cov}[\boldsymbol{y},\boldsymbol{y}] \\ + \end{bmatrix}, +\end{split}\]
+

where for example

+
+\[ +\mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] =\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})(y_i- \overline{y}). +\]
+

With this definition and recalling that the variance is defined as

+
+\[ +\mathrm{var}[\boldsymbol{x}]=\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})^2, +\]
+

we can rewrite the covariance matrix as

+
+\[\begin{split} +\boldsymbol{C}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} \mathrm{var}[\boldsymbol{x}] & \mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] \\ + \mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] & \mathrm{var}[\boldsymbol{y}] \\ + \end{bmatrix}. +\end{split}\]
+

The covariance takes values between zero and infinity and may thus +lead to problems with loss of numerical precision for particularly +large values. It is common to scale the covariance matrix by +introducing instead the correlation matrix defined via the so-called +correlation function

+
+\[ +\mathrm{corr}[\boldsymbol{x},\boldsymbol{y}]=\frac{\mathrm{cov}[\boldsymbol{x},\boldsymbol{y}]}{\sqrt{\mathrm{var}[\boldsymbol{x}] \mathrm{var}[\boldsymbol{y}]}}. +\]
+

The correlation function is then given by values \(\mathrm{corr}[\boldsymbol{x},\boldsymbol{y}] +\in [-1,1]\). This avoids eventual problems with too large values. We +can then define the correlation matrix for the two vectors \(\boldsymbol{x}\) +and \(\boldsymbol{y}\) as

+
+\[\begin{split} +\boldsymbol{K}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} 1 & \mathrm{corr}[\boldsymbol{x},\boldsymbol{y}] \\ + \mathrm{corr}[\boldsymbol{y},\boldsymbol{x}] & 1 \\ + \end{bmatrix}, +\end{split}\]
+

In the above example this is the function we constructed using pandas.

+
+
+

2.49. Correlation Function and Design/Feature Matrix

+

In our derivation of the various regression algorithms like Ordinary Least Squares or Ridge regression +we defined the design/feature matrix \(\boldsymbol{X}\) as

+
+\[\begin{split} +\boldsymbol{X}=\begin{bmatrix} +x_{0,0} & x_{0,1} & x_{0,2}& \dots & \dots x_{0,p-1}\\ +x_{1,0} & x_{1,1} & x_{1,2}& \dots & \dots x_{1,p-1}\\ +x_{2,0} & x_{2,1} & x_{2,2}& \dots & \dots x_{2,p-1}\\ +\dots & \dots & \dots & \dots \dots & \dots \\ +x_{n-2,0} & x_{n-2,1} & x_{n-2,2}& \dots & \dots x_{n-2,p-1}\\ +x_{n-1,0} & x_{n-1,1} & x_{n-1,2}& \dots & \dots x_{n-1,p-1}\\ +\end{bmatrix}, +\end{split}\]
+

with \(\boldsymbol{X}\in {\mathbb{R}}^{n\times p}\), with the predictors/features \(p\) refering to the column numbers and the +entries \(n\) being the row elements. +We can rewrite the design/feature matrix in terms of its column vectors as

+
+\[ +\boldsymbol{X}=\begin{bmatrix} \boldsymbol{x}_0 & \boldsymbol{x}_1 & \boldsymbol{x}_2 & \dots & \dots & \boldsymbol{x}_{p-1}\end{bmatrix}, +\]
+

with a given vector

+
+\[ +\boldsymbol{x}_i^T = \begin{bmatrix}x_{0,i} & x_{1,i} & x_{2,i}& \dots & \dots x_{n-1,i}\end{bmatrix}. +\]
+

With these definitions, we can now rewrite our \(2\times 2\) +correaltion/covariance matrix in terms of a moe general design/feature +matrix \(\boldsymbol{X}\in {\mathbb{R}}^{n\times p}\). This leads to a \(p\times p\) +covariance matrix for the vectors \(\boldsymbol{x}_i\) with \(i=0,1,\dots,p-1\)

+
+\[\begin{split} +\boldsymbol{C}[\boldsymbol{x}] = \begin{bmatrix} +\mathrm{var}[\boldsymbol{x}_0] & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_1] & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_2] & \dots & \dots & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_{p-1}]\\ +\mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_0] & \mathrm{var}[\boldsymbol{x}_1] & \mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_2] & \dots & \dots & \mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_{p-1}]\\ +\mathrm{cov}[\boldsymbol{x}_2,\boldsymbol{x}_0] & \mathrm{cov}[\boldsymbol{x}_2,\boldsymbol{x}_1] & \mathrm{var}[\boldsymbol{x}_2] & \dots & \dots & \mathrm{cov}[\boldsymbol{x}_2,\boldsymbol{x}_{p-1}]\\ +\dots & \dots & \dots & \dots & \dots & \dots \\ +\dots & \dots & \dots & \dots & \dots & \dots \\ +\mathrm{cov}[\boldsymbol{x}_{p-1},\boldsymbol{x}_0] & \mathrm{cov}[\boldsymbol{x}_{p-1},\boldsymbol{x}_1] & \mathrm{cov}[\boldsymbol{x}_{p-1},\boldsymbol{x}_{2}] & \dots & \dots & \mathrm{var}[\boldsymbol{x}_{p-1}]\\ +\end{bmatrix}, +\end{split}\]
+

and the correlation matrix

+
+\[\begin{split} +\boldsymbol{K}[\boldsymbol{x}] = \begin{bmatrix} +1 & \mathrm{corr}[\boldsymbol{x}_0,\boldsymbol{x}_1] & \mathrm{corr}[\boldsymbol{x}_0,\boldsymbol{x}_2] & \dots & \dots & \mathrm{corr}[\boldsymbol{x}_0,\boldsymbol{x}_{p-1}]\\ +\mathrm{corr}[\boldsymbol{x}_1,\boldsymbol{x}_0] & 1 & \mathrm{corr}[\boldsymbol{x}_1,\boldsymbol{x}_2] & \dots & \dots & \mathrm{corr}[\boldsymbol{x}_1,\boldsymbol{x}_{p-1}]\\ +\mathrm{corr}[\boldsymbol{x}_2,\boldsymbol{x}_0] & \mathrm{corr}[\boldsymbol{x}_2,\boldsymbol{x}_1] & 1 & \dots & \dots & \mathrm{corr}[\boldsymbol{x}_2,\boldsymbol{x}_{p-1}]\\ +\dots & \dots & \dots & \dots & \dots & \dots \\ +\dots & \dots & \dots & \dots & \dots & \dots \\ +\mathrm{corr}[\boldsymbol{x}_{p-1},\boldsymbol{x}_0] & \mathrm{corr}[\boldsymbol{x}_{p-1},\boldsymbol{x}_1] & \mathrm{corr}[\boldsymbol{x}_{p-1},\boldsymbol{x}_{2}] & \dots & \dots & 1\\ +\end{bmatrix}, +\end{split}\]
+
+
+

2.50. Covariance Matrix Examples

+

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 \(2\times n\) matrix \(\boldsymbol{W}\)

+
+\[\begin{split} +\boldsymbol{W} = \begin{bmatrix} x_0 & y_0 \\ + x_1 & y_1 \\ + x_2 & y_2\\ + \dots & \dots \\ + x_{n-2} & y_{n-2}\\ + x_{n-1} & y_{n-1} & + \end{bmatrix}, +\end{split}\]
+

which in turn is converted into into the \(2\times 2\) covariance matrix +\(\boldsymbol{C}\) via the Numpy function np.cov(). We note that we can also calculate +the mean value of each set of samples \(\boldsymbol{x}\) etc using the Numpy +function np.mean(x). We can also extract the eigenvalues of the +covariance matrix through the np.linalg.eig() function.

+
+
+
# 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))
+W = np.vstack((x, y))
+C = np.cov(W)
+print(C)
+
+
+
+
+
-0.2130387083960083
+3.3985018954897055
+[[0.92782183 2.79556832]
+ [2.79556832 9.43064124]]
+
+
+
+
+
+
+

2.51. Correlation Matrix

+

The previous example can be converted into the correlation matrix by +simply scaling the matrix elements with the variances. We should also +subtract the mean values for each column. This leads to the following +code which sets up the correlations matrix for the previous example in +a more brute force way. Here we scale the mean values for each column of the design matrix, calculate the relevant mean values and variances and then finally set up the \(2\times 2\) correlation matrix (since we have only two vectors).

+
+
+
import numpy as np
+n = 100
+# define two vectors                                                                                           
+x = np.random.random(size=n)
+y = 4+3*x+np.random.normal(size=n)
+#scaling the x and y vectors                                                                                   
+x = x - np.mean(x)
+y = y - np.mean(y)
+variance_x = np.sum(x@x)/n
+variance_y = np.sum(y@y)/n
+print(variance_x)
+print(variance_y)
+cov_xy = np.sum(x@y)/n
+cov_xx = np.sum(x@x)/n
+cov_yy = np.sum(y@y)/n
+C = np.zeros((2,2))
+C[0,0]= cov_xx/variance_x
+C[1,1]= cov_yy/variance_y
+C[0,1]= cov_xy/np.sqrt(variance_y*variance_x)
+C[1,0]= C[0,1]
+print(C)
+
+
+
+
+
0.07244432041213644
+1.7626405470635576
+[[1.         0.68117768]
+ [0.68117768 1.        ]]
+
+
+
+
+

We see that the matrix elements along the diagonal are one as they +should be and that the matrix is symmetric. Furthermore, diagonalizing +this matrix we easily see that it is a positive definite matrix.

+

The above procedure with numpy can be made more compact if we use pandas.

+
+
+

2.52. Correlation Matrix with Pandas

+

We whow here how we can set up the correlation matrix using pandas, as done in this simple code

+
+
+
import numpy as np
+import pandas as pd
+n = 10
+x = np.random.normal(size=n)
+x = x - np.mean(x)
+y = 4+3*x+np.random.normal(size=n)
+y = y - np.mean(y)
+X = (np.vstack((x, y))).T
+print(X)
+Xpd = pd.DataFrame(X)
+print(Xpd)
+correlation_matrix = Xpd.corr()
+print(correlation_matrix)
+
+
+
+
+
[[-1.3459346  -4.25337129]
+ [ 0.36376194  0.11355734]
+ [-0.73140048 -2.43802224]
+ [-1.01008165 -2.99654052]
+ [ 1.82227838  6.21343095]
+ [ 0.69549376  2.18497475]
+ [ 0.32668536  1.1544353 ]
+ [-0.41758614 -0.87152289]
+ [-0.60054519 -0.66392823]
+ [ 0.89732863  1.55698683]]
+          0         1
+0 -1.345935 -4.253371
+1  0.363762  0.113557
+2 -0.731400 -2.438022
+3 -1.010082 -2.996541
+4  1.822278  6.213431
+5  0.695494  2.184975
+6  0.326685  1.154435
+7 -0.417586 -0.871523
+8 -0.600545 -0.663928
+9  0.897329  1.556987
+          0         1
+0  1.000000  0.972722
+1  0.972722  1.000000
+
+
+
+
+

We expand this model to the Franke function discussed above.

+
+
+

2.53. Correlation Matrix with Pandas and the Franke function

+
+
+
# Common imports
+import numpy as np
+import pandas as pd
+
+
+def FrankeFunction(x,y):
+	term1 = 0.75*np.exp(-(0.25*(9*x-2)**2) - 0.25*((9*y-2)**2))
+	term2 = 0.75*np.exp(-((9*x+1)**2)/49.0 - 0.1*(9*y+1))
+	term3 = 0.5*np.exp(-(9*x-7)**2/4.0 - 0.25*((9*y-3)**2))
+	term4 = -0.2*np.exp(-(9*x-4)**2 - (9*y-7)**2)
+	return term1 + term2 + term3 + term4
+
+
+def create_X(x, y, n ):
+	if len(x.shape) > 1:
+		x = np.ravel(x)
+		y = np.ravel(y)
+
+	N = len(x)
+	l = int((n+1)*(n+2)/2)		# Number of elements in beta
+	X = np.ones((N,l))
+
+	for i in range(1,n+1):
+		q = int((i)*(i+1)/2)
+		for k in range(i+1):
+			X[:,q+k] = (x**(i-k))*(y**k)
+
+	return X
+
+
+# Making meshgrid of datapoints and compute Franke's function
+n = 4
+N = 100
+x = np.sort(np.random.uniform(0, 1, N))
+y = np.sort(np.random.uniform(0, 1, N))
+z = FrankeFunction(x, y)
+X = create_X(x, y, n=n)    
+
+Xpd = pd.DataFrame(X)
+# subtract the mean values and set up the covariance matrix
+Xpd = Xpd - Xpd.mean()
+covariance_matrix = Xpd.cov()
+print(covariance_matrix)
+
+
+
+
+
     0         1         2         3         4         5         6         7   \
+0   0.0  0.000000  0.000000  0.000000  0.000000  0.000000  0.000000  0.000000   
+1   0.0  0.085922  0.084669  0.082116  0.079841  0.077867  0.071472  0.069589   
+2   0.0  0.084669  0.085414  0.083563  0.082564  0.081677  0.074521  0.073493   
+3   0.0  0.082116  0.083563  0.083634  0.082948  0.082334  0.075811  0.074915   
+4   0.0  0.079841  0.082564  0.082948  0.083198  0.083394  0.076380  0.076155   
+5   0.0  0.077867  0.081677  0.082334  0.083394  0.084292  0.076860  0.077220   
+6   0.0  0.071472  0.074521  0.075811  0.076380  0.076860  0.070720  0.070720   
+7   0.0  0.069589  0.073493  0.074915  0.076155  0.077220  0.070720  0.071220   
+8   0.0  0.068015  0.072657  0.074197  0.076020  0.077596  0.070782  0.071721   
+9   0.0  0.066715  0.071996  0.073645  0.075978  0.078005  0.070913  0.072238   
+10  0.0  0.061510  0.065384  0.067169  0.068543  0.069733  0.064041  0.064670   
+11  0.0  0.060162  0.064633  0.066471  0.068331  0.069949  0.063981  0.064981   
+12  0.0  0.059062  0.064053  0.065946  0.068232  0.070227  0.064014  0.065343   
+13  0.0  0.058180  0.063630  0.065577  0.068240  0.070571  0.064138  0.065758   
+14  0.0  0.057490  0.063348  0.065350  0.068349  0.070980  0.064347  0.066230   
+
+          8         9         10        11        12        13        14  
+0   0.000000  0.000000  0.000000  0.000000  0.000000  0.000000  0.000000  
+1   0.068015  0.066715  0.061510  0.060162  0.059062  0.058180  0.057490  
+2   0.072657  0.071996  0.065384  0.064633  0.064053  0.063630  0.063348  
+3   0.074197  0.073645  0.067169  0.066471  0.065946  0.065577  0.065350  
+4   0.076020  0.075978  0.068543  0.068331  0.068232  0.068240  0.068349  
+5   0.077596  0.078005  0.069733  0.069949  0.070227  0.070571  0.070980  
+6   0.070782  0.070913  0.064041  0.063981  0.064014  0.064138  0.064347  
+7   0.071721  0.072238  0.064670  0.064981  0.065343  0.065758  0.066230  
+8   0.072607  0.073467  0.065283  0.065922  0.066576  0.067253  0.067960  
+9   0.073467  0.074632  0.065897  0.066829  0.067744  0.068656  0.069576  
+10  0.065283  0.065897  0.059007  0.059415  0.059860  0.060345  0.060875  
+11  0.065922  0.066829  0.059415  0.060106  0.060801  0.061512  0.062245  
+12  0.066576  0.067744  0.059860  0.060801  0.061722  0.062635  0.063553  
+13  0.067253  0.068656  0.060345  0.061512  0.062635  0.063733  0.064819  
+14  0.067960  0.069576  0.060875  0.062245  0.063553  0.064819  0.066059  
+
+
+
+
+

We note here that the covariance is zero for the first rows and +columns since all matrix elements in the design matrix were set to one +(we are fitting the function in terms of a polynomial of degree \(n\)).

+

This means that the variance for these elements will be zero and will +cause problems when we set up the correlation matrix. We can simply +drop these elements and construct a correlation +matrix without these elements.

+
+
+

2.54. Rewriting the Covariance and/or Correlation Matrix

+

We can rewrite the covariance matrix in a more compact form in terms of the design/feature matrix \(\boldsymbol{X}\) as

+
+\[ +\boldsymbol{C}[\boldsymbol{x}] = \frac{1}{n}\boldsymbol{X}^T\boldsymbol{X}= \mathbb{E}[\boldsymbol{X}^T\boldsymbol{X}]. +\]
+

To see this let us simply look at a design matrix \(\boldsymbol{X}\in {\mathbb{R}}^{2\times 2}\)

+
+\[\begin{split} +\boldsymbol{X}=\begin{bmatrix} +x_{00} & x_{01}\\ +x_{10} & x_{11}\\ +\end{bmatrix}=\begin{bmatrix} +\boldsymbol{x}_{0} & \boldsymbol{x}_{1}\\ +\end{bmatrix}. +\end{split}\]
+

If we then compute the expectation value

+
+\[\begin{split} +\mathbb{E}[\boldsymbol{X}^T\boldsymbol{X}] = \frac{1}{n}\boldsymbol{X}^T\boldsymbol{X}=\begin{bmatrix} +x_{00}^2+x_{01}^2 & x_{00}x_{10}+x_{01}x_{11}\\ +x_{10}x_{00}+x_{11}x_{01} & x_{10}^2+x_{11}^2\\ +\end{bmatrix}, +\end{split}\]
+

which is just

+
+\[\begin{split} +\boldsymbol{C}[\boldsymbol{x}_0,\boldsymbol{x}_1] = \boldsymbol{C}[\boldsymbol{x}]=\begin{bmatrix} \mathrm{var}[\boldsymbol{x}_0] & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_1] \\ + \mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_0] & \mathrm{var}[\boldsymbol{x}_1] \\ + \end{bmatrix}, +\end{split}\]
+

where we wrote $\(\boldsymbol{C}[\boldsymbol{x}_0,\boldsymbol{x}_1] = \boldsymbol{C}[\boldsymbol{x}]\)\( to indicate that this the covariance of the vectors \)\boldsymbol{x}\( of the design/feature matrix \)\boldsymbol{X}$.

+

It is easy to generalize this to a matrix \(\boldsymbol{X}\in {\mathbb{R}}^{n\times p}\).

+
+
+

2.55. Linking with SVD

+

See lecture september 11. More text to be added here soon.

+
+
+

2.56. Where are we going?

+

Before we proceed, we need to rethink what we have been doing. In our +eager to fit the data, we have omitted several important elements in +our regression analysis. In what follows we will

+
    +

  1. look at statistical properties, including a discussion of mean values, variance and the so-called bias-variance tradeoff

    2. +

  2. introduce resampling techniques like cross-validation, bootstrapping and jackknife and more

    3. +
+

This will allow us to link the standard linear algebra methods we have discussed above to a statistical interpretation of the methods.

+
+
+

2.57. Resampling methods

+

Resampling methods are an indispensable tool in modern +statistics. They involve repeatedly drawing samples from a training +set and refitting a model of interest on each sample in order to +obtain additional information about the fitted model. For example, in +order to estimate the variability of a linear regression fit, we can +repeatedly draw different samples from the training data, fit a linear +regression to each new sample, and then examine the extent to which +the resulting fits differ. Such an approach may allow us to obtain +information that would not be available from fitting the model only +once using the original training sample.

+

Two resampling methods are often used in Machine Learning analyses,

+
    +

  1. The bootstrap method

    2. +

  2. and Cross-Validation

    3. +
+

In addition there are several other methods such as the Jackknife and the Blocking methods. We will discuss in particular +cross-validation and the bootstrap method.

+
+
+

2.58. Resampling approaches can be computationally expensive

+

Resampling approaches can be computationally expensive, because they +involve fitting the same statistical method multiple times using +different subsets of the training data. However, due to recent +advances in computing power, the computational requirements of +resampling methods generally are not prohibitive. In this chapter, we +discuss two of the most commonly used resampling methods, +cross-validation and the bootstrap. Both methods are important tools +in the practical application of many statistical learning +procedures. For example, cross-validation can be used to estimate the +test error associated with a given statistical learning method in +order to evaluate its performance, or to select the appropriate level +of flexibility. The process of evaluating a model’s performance is +known as model assessment, whereas the process of selecting the proper +level of flexibility for a model is known as model selection. The +bootstrap is widely used.

+
+
+

2.59. Why resampling methods ?

+

Statistical analysis.

+
    +

  • Our simulations can be treated as computer experiments. This is particularly the case for Monte Carlo methods

    • +

  • 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.

    • +
+
+
+

2.60. Statistical analysis

+
    +

  • As in other experiments, many numerical 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.

    • +
+ +
+
+

2.61. Linking the regression analysis with a statistical interpretation

+

The +advantage of doing linear regression is that we actually end up with +analytical expressions for several statistical quantities.
+Standard least squares and Ridge regression allow us to +derive quantities like the variance and other expectation values in a +rather straightforward way.

+

It is assumed that \(\varepsilon_i +\sim \mathcal{N}(0, \sigma^2)\) and the \(\varepsilon_{i}\) are +independent, i.e.:

+
+\[\begin{split} +\begin{align*} +\mbox{Cov}(\varepsilon_{i_1}, +\varepsilon_{i_2}) & = \left\{ \begin{array}{lcc} \sigma^2 & \mbox{if} +& i_1 = i_2, \\ 0 & \mbox{if} & i_1 \not= i_2. \end{array} \right. +\end{align*} +\end{split}\]
+

The randomness of \(\varepsilon_i\) implies that +\(\mathbf{y}_i\) is also a random variable. In particular, +\(\mathbf{y}_i\) is normally distributed, because \(\varepsilon_i \sim +\mathcal{N}(0, \sigma^2)\) and \(\mathbf{X}_{i,\ast} \, \boldsymbol{\beta}\) is a +non-random scalar. To specify the parameters of the distribution of +\(\mathbf{y}_i\) we need to calculate its first two moments.

+

Recall that \(\boldsymbol{X}\) is a matrix of dimensionality \(n\times p\). The +notation above \(\mathbf{X}_{i,\ast}\) means that we are looking at the +row number \(i\) and perform a sum over all values \(p\).

+
+
+

2.62. Assumptions made

+

The assumption we have made here can be summarized as (and this is going to be useful when we discuss the bias-variance trade off) +that there exists a function \(f(\boldsymbol{x})\) and a normal distributed error \(\boldsymbol{\varepsilon}\sim \mathcal{N}(0, \sigma^2)\) +which describe our data

+
+\[ +\boldsymbol{y} = f(\boldsymbol{x})+\boldsymbol{\varepsilon} +\]
+

We approximate this function with our model from the solution of the linear regression equations, that is our +function \(f\) is approximated by \(\boldsymbol{\tilde{y}}\) where we want to minimize \((\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\), our MSE, with

+
+\[ +\boldsymbol{\tilde{y}} = \boldsymbol{X}\boldsymbol{\beta}. +\]
+
+
+

2.63. Expectation value and variance

+

We can calculate the expectation value of \(\boldsymbol{y}\) for a given element \(i\)

+
+\[ +\begin{align*} +\mathbb{E}(y_i) & = +\mathbb{E}(\mathbf{X}_{i, \ast} \, \boldsymbol{\beta}) + \mathbb{E}(\varepsilon_i) +\, \, \, = \, \, \, \mathbf{X}_{i, \ast} \, \beta, +\end{align*} +\]
+

while +its variance is

+
+\[\begin{split} +\begin{align*} \mbox{Var}(y_i) & = \mathbb{E} \{ [y_i +- \mathbb{E}(y_i)]^2 \} \, \, \, = \, \, \, \mathbb{E} ( y_i^2 ) - +[\mathbb{E}(y_i)]^2 \\ & = \mathbb{E} [ ( \mathbf{X}_{i, \ast} \, +\beta + \varepsilon_i )^2] - ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 \\ & += \mathbb{E} [ ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 + 2 \varepsilon_i +\mathbf{X}_{i, \ast} \, \boldsymbol{\beta} + \varepsilon_i^2 ] - ( \mathbf{X}_{i, +\ast} \, \beta)^2 \\ & = ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 + 2 +\mathbb{E}(\varepsilon_i) \mathbf{X}_{i, \ast} \, \boldsymbol{\beta} + +\mathbb{E}(\varepsilon_i^2 ) - ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 +\\ & = \mathbb{E}(\varepsilon_i^2 ) \, \, \, = \, \, \, +\mbox{Var}(\varepsilon_i) \, \, \, = \, \, \, \sigma^2. +\end{align*} +\end{split}\]
+

Hence, \(y_i \sim \mathcal{N}( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta}, \sigma^2)\), that is \(\boldsymbol{y}\) follows a normal distribution with +mean value \(\boldsymbol{X}\boldsymbol{\beta}\) and variance \(\sigma^2\) (not be confused with the singular values of the SVD).

+
+
+

2.64. Expectation value and variance for \(\boldsymbol{\beta}\)

+

With the OLS expressions for the parameters \(\boldsymbol{\beta}\) we can evaluate the expectation value

+
+\[ +\mathbb{E}(\boldsymbol{\beta}) = \mathbb{E}[ (\mathbf{X}^{\top} \mathbf{X})^{-1}\mathbf{X}^{T} \mathbf{Y}]=(\mathbf{X}^{T} \mathbf{X})^{-1}\mathbf{X}^{T} \mathbb{E}[ \mathbf{Y}]=(\mathbf{X}^{T} \mathbf{X})^{-1} \mathbf{X}^{T}\mathbf{X}\boldsymbol{\beta}=\boldsymbol{\beta}. +\]
+

This means that the estimator of the regression parameters is unbiased.

+

We can also calculate the variance

+

The variance of \(\boldsymbol{\beta}\) is

+
+\[\begin{split} +\begin{eqnarray*} +\mbox{Var}(\boldsymbol{\beta}) & = & \mathbb{E} \{ [\boldsymbol{\beta} - \mathbb{E}(\boldsymbol{\beta})] [\boldsymbol{\beta} - \mathbb{E}(\boldsymbol{\beta})]^{T} \} +\\ +& = & \mathbb{E} \{ [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} - \boldsymbol{\beta}] \, [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} - \boldsymbol{\beta}]^{T} \} +\\ +% & = & \mathbb{E} \{ [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y}] \, [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y}]^{T} \} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} +% \\ +% & = & \mathbb{E} \{ (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} \, \mathbf{Y}^{T} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} \} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} +% \\ +& = & (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \, \mathbb{E} \{ \mathbf{Y} \, \mathbf{Y}^{T} \} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} +\\ +& = & (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \, \{ \mathbf{X} \, \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} \, \mathbf{X}^{T} + \sigma^2 \} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} +% \\ +% & = & (\mathbf{X}^T \mathbf{X})^{-1} \, \mathbf{X}^T \, \mathbf{X} \, \boldsymbol{\beta} \, \boldsymbol{\beta}^T \, \mathbf{X}^T \, \mathbf{X} \, (\mathbf{X}^T % \mathbf{X})^{-1} +% \\ +% & & + \, \, \sigma^2 \, (\mathbf{X}^T \mathbf{X})^{-1} \, \mathbf{X}^T \, \mathbf{X} \, (\mathbf{X}^T \mathbf{X})^{-1} - \boldsymbol{\beta} \boldsymbol{\beta}^T +\\ +& = & \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} + \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} +\, \, \, = \, \, \, \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1}, +\end{eqnarray*} +\end{split}\]
+

where we have used that \(\mathbb{E} (\mathbf{Y} \mathbf{Y}^{T}) = +\mathbf{X} \, \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} \, \mathbf{X}^{T} + +\sigma^2 \, \mathbf{I}_{nn}\). From \(\mbox{Var}(\boldsymbol{\beta}) = \sigma^2 +\, (\mathbf{X}^{T} \mathbf{X})^{-1}\), one obtains an estimate of the +variance of the estimate of the \(j\)-th regression coefficient: +\(\boldsymbol{\sigma}^2 (\boldsymbol{\beta}_j ) = \boldsymbol{\sigma}^2 \sqrt{ +[(\mathbf{X}^{T} \mathbf{X})^{-1}]_{jj} }\). This may be used to +construct a confidence interval for the estimates.

+

In a similar way, we can obtain analytical expressions for say the +expectation values of the parameters \(\boldsymbol{\beta}\) and their variance +when we employ Ridge regression, allowing us again to define a confidence interval.

+

It is rather straightforward to show that

+
+\[ +\mathbb{E} \big[ \boldsymbol{\beta}^{\mathrm{Ridge}} \big]=(\mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I}_{pp})^{-1} (\mathbf{X}^{\top} \mathbf{X})\boldsymbol{\beta}^{\mathrm{OLS}}. +\]
+

We see clearly that +\(\mathbb{E} \big[ \boldsymbol{\beta}^{\mathrm{Ridge}} \big] \not= \boldsymbol{\beta}^{\mathrm{OLS}}\) for any \(\lambda > 0\). We say then that the ridge estimator is biased.

+

We can also compute the variance as

+
+\[ +\mbox{Var}[\boldsymbol{\beta}^{\mathrm{Ridge}}]=\sigma^2[ \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1} \mathbf{X}^{T} \mathbf{X} \{ [ \mathbf{X}^{\top} \mathbf{X} + \lambda \mathbf{I} ]^{-1}\}^{T}, +\]
+

and it is easy to see that if the parameter \(\lambda\) goes to infinity then the variance of Ridge parameters \(\boldsymbol{\beta}\) goes to zero.

+

With this, we can compute the difference

+
+\[ +\mbox{Var}[\boldsymbol{\beta}^{\mathrm{OLS}}]-\mbox{Var}(\boldsymbol{\beta}^{\mathrm{Ridge}})=\sigma^2 [ \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}[ 2\lambda\mathbf{I} + \lambda^2 (\mathbf{X}^{T} \mathbf{X})^{-1} ] \{ [ \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}\}^{T}. +\]
+

The difference is non-negative definite since each component of the +matrix product is non-negative definite. +This means the variance we obtain with the standard OLS will always for \(\lambda > 0\) be larger than the variance of \(\boldsymbol{\beta}\) obtained with the Ridge estimator. This has interesting consequences when we discuss the so-called bias-variance trade-off below.

+
+
+

2.65. Resampling methods

+

With all these analytical equations for both the OLS and Ridge +regression, we will now outline how to assess a given model. This will +lead us to a discussion of the so-called bias-variance tradeoff (see +below) and so-called resampling methods.

+

One of the quantities we have discussed as a way to measure errors is +the mean-squared error (MSE), mainly used for fitting of continuous +functions. Another choice is the absolute error.

+

In the discussions below we will focus on the MSE and in particular since we will split the data into test and training data, +we discuss the

+
    +

  1. prediction error or simply the test error \(\mathrm{Err_{Test}}\), where we have a fixed training set and the test error is the MSE arising from the data reserved for testing. We discuss also the

    2. +

  2. training error \(\mathrm{Err_{Train}}\), which is the average loss over the training data.

    3. +
+

As our model becomes more and more complex, more of the training data tends to used. The training may thence adapt to more complicated structures in the data. This may lead to a decrease in the bias (see below for code example) and a slight increase of the variance for the test error. +For a certain level of complexity the test error will reach minimum, before starting to increase again. The +training error reaches a saturation.

+
+
+

2.66. Resampling methods: Jackknife and Bootstrap

+

Two famous +resampling methods are the independent bootstrap and the jackknife.

+

The jackknife is a special case of the independent bootstrap. Still, the jackknife was made +popular prior to the independent bootstrap. And as the popularity of +the independent bootstrap soared, new variants, such as the dependent bootstrap.

+

The Jackknife and independent bootstrap work for +independent, identically distributed random variables. +If these conditions are not +satisfied, the methods will fail. Yet, it should be said that if the data are +independent, identically distributed, and we only want to estimate the +variance of \(\overline{X}\) (which often is the case), then there is no +need for bootstrapping.

+
+
+

2.67. Resampling methods: Jackknife

+

The Jackknife works by making many replicas of the estimator \(\widehat{\theta}\). +The jackknife is a resampling method where we systematically leave out one observation from the vector of observed values \(\boldsymbol{x} = (x_1,x_2,\cdots,X_n)\). +Let \(\boldsymbol{x}_i\) denote the vector

+
+\[ +\boldsymbol{x}_i = (x_1,x_2,\cdots,x_{i-1},x_{i+1},\cdots,x_n), +\]
+

which equals the vector \(\boldsymbol{x}\) with the exception that observation +number \(i\) is left out. Using this notation, define +\(\widehat{\theta}_i\) to be the estimator +\(\widehat{\theta}\) computed using \(\vec{X}_i\).

+
+
+

2.68. Jackknife code example

+
+
+
from numpy import *
+from numpy.random import randint, randn
+from time import time
+
+def jackknife(data, stat):
+    n = len(data);t = zeros(n); inds = arange(n); t0 = time()
+    ## 'jackknifing' by leaving out an observation for each i                                                                                                                      
+    for i in range(n):
+        t[i] = stat(delete(data,i) )
+
+    # analysis                                                                                                                                                                     
+    print("Runtime: %g sec" % (time()-t0)); print("Jackknife Statistics :")
+    print("original           bias      std. error")
+    print("%8g %14g %15g" % (stat(data),(n-1)*mean(t)/n, (n*var(t))**.5))
+
+    return t
+
+
+# Returns mean of data samples                                                                                                                                                     
+def stat(data):
+    return mean(data)
+
+
+mu, sigma = 100, 15
+datapoints = 10000
+x = mu + sigma*random.randn(datapoints)
+# jackknife returns the data sample                                                                                                                                                
+t = jackknife(x, stat)
+
+
+
+
+
Runtime: 0.425398 sec
+Jackknife Statistics :
+original           bias      std. error
+ 99.9357        99.9258        0.149923
+
+
+
+
+
+
+

2.69. Resampling methods: Bootstrap

+

Bootstrapping is a nonparametric approach to statistical inference +that substitutes computation for more traditional distributional +assumptions and asymptotic results. Bootstrapping offers a number of +advantages:

+
    +

  1. The bootstrap is quite general, although there are some cases in which it fails.

    2. +

  2. Because it does not require distributional assumptions (such as normally distributed errors), the bootstrap can provide more accurate inferences when the data are not well behaved or when the sample size is small.

    3. +

  3. It is possible to apply the bootstrap to statistics with sampling distributions that are difficult to derive, even asymptotically.

    4. +

  4. It is relatively simple to apply the bootstrap to complex data-collection plans (such as stratified and clustered samples).

    5. +
+
+
+

2.70. Resampling methods: Bootstrap background

+

Since \(\widehat{\theta} = \widehat{\theta}(\boldsymbol{X})\) is a function of random variables, +\(\widehat{\theta}\) itself must be a random variable. Thus it has +a pdf, call this function \(p(\boldsymbol{t})\). The aim of the bootstrap is to +estimate \(p(\boldsymbol{t})\) by the relative frequency of +\(\widehat{\theta}\). You can think of this as using a histogram +in the place of \(p(\boldsymbol{t})\). If the relative frequency closely +resembles \(p(\vec{t})\), then using numerics, it is straight forward to +estimate all the interesting parameters of \(p(\boldsymbol{t})\) using point +estimators.

+
+
+

2.71. Resampling methods: More Bootstrap background

+

In the case that \(\widehat{\theta}\) has +more than one component, and the components are independent, we use the +same estimator on each component separately. If the probability +density function of \(X_i\), \(p(x)\), had been known, then it would have +been straight forward to do this by:

+
    +

  1. Drawing lots of numbers from \(p(x)\), suppose we call one such set of numbers \((X_1^*, X_2^*, \cdots, X_n^*)\).

    2. +

  2. Then using these numbers, we could compute a replica of \(\widehat{\theta}\) called \(\widehat{\theta}^*\).

    3. +
+

By repeated use of (1) and (2), many +estimates of \(\widehat{\theta}\) could have been obtained. The +idea is to use the relative frequency of \(\widehat{\theta}^*\) +(think of a histogram) as an estimate of \(p(\boldsymbol{t})\).

+
+
+

2.72. Resampling methods: Bootstrap approach

+

But +unless there is enough information available about the process that +generated \(X_1,X_2,\cdots,X_n\), \(p(x)\) is in general +unknown. Therefore, Efron in 1979 asked the +question: What if we replace \(p(x)\) by the relative frequency +of the observation \(X_i\); if we draw observations in accordance with +the relative frequency of the observations, will we obtain the same +result in some asymptotic sense? The answer is yes.

+

Instead of generating the histogram for the relative +frequency of the observation \(X_i\), just draw the values +\((X_1^*,X_2^*,\cdots,X_n^*)\) with replacement from the vector +\(\boldsymbol{X}\).

+
+
+

2.73. Resampling methods: Bootstrap steps

+

The independent bootstrap works like this:

+
    +

  1. Draw with replacement \(n\) numbers for the observed variables \(\boldsymbol{x} = (x_1,x_2,\cdots,x_n)\).

    2. +

  2. Define a vector \(\boldsymbol{x}^*\) containing the values which were drawn from \(\boldsymbol{x}\).

    3. +

  3. Using the vector \(\boldsymbol{x}^*\) compute \(\widehat{\theta}^*\) by evaluating \(\widehat \theta\) under the observations \(\boldsymbol{x}^*\).

    4. +

  4. Repeat this process \(k\) times.

    5. +
+

When you are done, you can draw a histogram of the relative frequency +of \(\widehat \theta^*\). This is your estimate of the probability +distribution \(p(t)\). Using this probability distribution you can +estimate any statistics thereof. In principle you never draw the +histogram of the relative frequency of \(\widehat{\theta}^*\). Instead +you use the estimators corresponding to the statistic of interest. For +example, if you are interested in estimating the variance of \(\widehat +\theta\), apply the etsimator \(\widehat \sigma^2\) to the values +\(\widehat \theta ^*\).

+
+
+

2.74. Code example for the Bootstrap method

+

The following code starts with a Gaussian distribution with mean value +\(\mu =100\) and variance \(\sigma=15\). We use this to generate the data +used in the bootstrap analysis. The bootstrap analysis returns a data +set after a given number of bootstrap operations (as many as we have +data points). This data set consists of estimated mean values for each +bootstrap operation. The histogram generated by the bootstrap method +shows that the distribution for these mean values is also a Gaussian, +centered around the mean value \(\mu=100\) but with standard deviation +\(\sigma/\sqrt{n}\), where \(n\) is the number of bootstrap samples (in +this case the same as the number of original data points). The value +of the standard deviation is what we expect from the central limit +theorem.

+
+
+
from numpy import *
+from numpy.random import randint, randn
+from time import time
+import matplotlib.mlab as mlab
+import matplotlib.pyplot as plt
+
+# Returns mean of bootstrap samples                                                                                                                                                
+def stat(data):
+    return mean(data)
+
+# Bootstrap algorithm
+def bootstrap(data, statistic, R):
+    t = zeros(R); n = len(data); inds = arange(n); t0 = time()
+    # non-parametric bootstrap         
+    for i in range(R):
+        t[i] = statistic(data[randint(0,n,n)])
+
+    # analysis    
+    print("Runtime: %g sec" % (time()-t0)); print("Bootstrap Statistics :")
+    print("original           bias      std. error")
+    print("%8g %8g %14g %15g" % (statistic(data), std(data),mean(t),std(t)))
+    return t
+
+
+mu, sigma = 100, 15
+datapoints = 10000
+x = mu + sigma*random.randn(datapoints)
+# bootstrap returns the data sample                                    
+t = bootstrap(x, stat, datapoints)
+# the histogram of the bootstrapped  data                                                                                                    
+n, binsboot, patches = plt.hist(t, 50, normed=1, facecolor='red', alpha=0.75)
+
+# add a 'best fit' line  
+y = mlab.normpdf( binsboot, mean(t), std(t))
+lt = plt.plot(binsboot, y, 'r--', linewidth=1)
+plt.xlabel('Smarts')
+plt.ylabel('Probability')
+plt.axis([99.5, 100.6, 0, 3.0])
+plt.grid(True)
+
+plt.show()
+
+
+
+
+
Runtime: 2.2375 sec
+Bootstrap Statistics :
+original           bias      std. error
+  100.14  15.1574        100.137        0.152441
+
+
+
---------------------------------------------------------------------------
+AttributeError                            Traceback (most recent call last)
+<ipython-input-29-53990135e988> in <module>
+     29 t = bootstrap(x, stat, datapoints)
+     30 # the histogram of the bootstrapped  data
+---> 31 n, binsboot, patches = plt.hist(t, 50, normed=1, facecolor='red', alpha=0.75)
+     32 
+     33 # add a 'best fit' line
+
+~/opt/anaconda3/lib/python3.8/site-packages/matplotlib/pyplot.py in hist(x, bins, range, density, weights, cumulative, bottom, histtype, align, orientation, rwidth, log, color, label, stacked, data, **kwargs)
+   2603         orientation='vertical', rwidth=None, log=False, color=None,
+   2604         label=None, stacked=False, *, data=None, **kwargs):
+-> 2605     return gca().hist(
+   2606         x, bins=bins, range=range, density=density, weights=weights,
+   2607         cumulative=cumulative, bottom=bottom, histtype=histtype,
+
+~/opt/anaconda3/lib/python3.8/site-packages/matplotlib/__init__.py in inner(ax, data, *args, **kwargs)
+   1563     def inner(ax, *args, data=None, **kwargs):
+   1564         if data is None:
+-> 1565             return func(ax, *map(sanitize_sequence, args), **kwargs)
+   1566 
+   1567         bound = new_sig.bind(ax, *args, **kwargs)
+
+~/opt/anaconda3/lib/python3.8/site-packages/matplotlib/axes/_axes.py in hist(self, x, bins, range, density, weights, cumulative, bottom, histtype, align, orientation, rwidth, log, color, label, stacked, **kwargs)
+   6817             if patch:
+   6818                 p = patch[0]
+-> 6819                 p.update(kwargs)
+   6820                 if lbl is not None:
+   6821                     p.set_label(lbl)
+
+~/opt/anaconda3/lib/python3.8/site-packages/matplotlib/artist.py in update(self, props)
+   1004 
+   1005         with cbook._setattr_cm(self, eventson=False):
+-> 1006             ret = [_update_property(self, k, v) for k, v in props.items()]
+   1007 
+   1008         if len(ret):
+
+~/opt/anaconda3/lib/python3.8/site-packages/matplotlib/artist.py in <listcomp>(.0)
+   1004 
+   1005         with cbook._setattr_cm(self, eventson=False):
+-> 1006             ret = [_update_property(self, k, v) for k, v in props.items()]
+   1007 
+   1008         if len(ret):
+
+~/opt/anaconda3/lib/python3.8/site-packages/matplotlib/artist.py in _update_property(self, k, v)
+    999                 func = getattr(self, 'set_' + k, None)
+   1000                 if not callable(func):
+-> 1001                     raise AttributeError('{!r} object has no property {!r}'
+   1002                                          .format(type(self).__name__, k))
+   1003                 return func(v)
+
+AttributeError: 'Rectangle' object has no property 'normed'
+
+
+_images/chapter3_278_2.png +
+
+ +
+
+

2.75. Various steps in cross-validation

+

When the repetitive splitting of the data set is done randomly, +samples may accidently end up in a fast majority of the splits in +either training or test set. Such samples may have an unbalanced +influence on either model building or prediction evaluation. To avoid +this \(k\)-fold cross-validation structures the data splitting. The +samples are divided into \(k\) more or less equally sized exhaustive and +mutually exclusive subsets. In turn (at each split) one of these +subsets plays the role of the test set while the union of the +remaining subsets constitutes the training set. Such a splitting +warrants a balanced representation of each sample in both training and +test set over the splits. Still the division into the \(k\) subsets +involves a degree of randomness. This may be fully excluded when +choosing \(k=n\). This particular case is referred to as leave-one-out +cross-validation (LOOCV).

+ +
+
+

2.76. How to set up the cross-validation for Ridge and/or Lasso

+
    +

  • Define a range of interest for the penalty parameter.

    • +

  • Divide the data set into training and test set comprising samples \(\{1, \ldots, n\} \setminus i\) and \(\{ i \}\), respectively.

    • +

  • Fit the linear regression model by means of ridge estimation for each \(\lambda\) in the grid using the training set, and the corresponding estimate of the error variance \(\boldsymbol{\sigma}_{-i}^2(\lambda)\), as

    • +
+
+\[ +\begin{align*} +\boldsymbol{\beta}_{-i}(\lambda) & = ( \boldsymbol{X}_{-i, \ast}^{T} +\boldsymbol{X}_{-i, \ast} + \lambda \boldsymbol{I}_{pp})^{-1} +\boldsymbol{X}_{-i, \ast}^{T} \boldsymbol{y}_{-i} +\end{align*} +\]
+
    +

  • Evaluate the prediction performance of these models on the test set by \(\log\{L[y_i, \boldsymbol{X}_{i, \ast}; \boldsymbol{\beta}_{-i}(\lambda), \boldsymbol{\sigma}_{-i}^2(\lambda)]\}\). Or, by the prediction error \(|y_i - \boldsymbol{X}_{i, \ast} \boldsymbol{\beta}_{-i}(\lambda)|\), the relative error, the error squared or the R2 score function.

    • +

  • Repeat the first three steps such that each sample plays the role of the test set once.

    • +

  • Average the prediction performances of the test sets at each grid point of the penalty bias/parameter. It is an estimate of the prediction performance of the model corresponding to this value of the penalty parameter on novel data. It is defined as

    • +
+
+\[ +\begin{align*} +\frac{1}{n} \sum_{i = 1}^n \log\{L[y_i, \mathbf{X}_{i, \ast}; \boldsymbol{\beta}_{-i}(\lambda), \boldsymbol{\sigma}_{-i}^2(\lambda)]\}. +\end{align*} +\]
+
+
+

2.77. Cross-validation in brief

+

For the various values of \(k\)

+
    +

  1. shuffle the dataset randomly.

    2. +

  2. Split the dataset into \(k\) groups.

    3. +

  3. For each unique group:

    4. +
+

a. Decide which group to use as set for test data

+

b. Take the remaining groups as a training data set

+

c. Fit a model on the training set and evaluate it on the test set

+

d. Retain the evaluation score and discard the model

+
    +

  1. Summarize the model using the sample of model evaluation scores

    2. +
+
+
+

2.78. Code Example for Cross-validation and \(k\)-fold Cross-validation

+

The code here uses Ridge regression with cross-validation (CV) resampling and \(k\)-fold CV in order to fit a specific polynomial.

+
+
+
import numpy as np
+import matplotlib.pyplot as plt
+from sklearn.model_selection import KFold
+from sklearn.linear_model import Ridge
+from sklearn.model_selection import cross_val_score
+from sklearn.preprocessing import PolynomialFeatures
+
+# A seed just to ensure that the random numbers are the same for every run.
+# Useful for eventual debugging.
+np.random.seed(3155)
+
+# Generate the data.
+nsamples = 100
+x = np.random.randn(nsamples)
+y = 3*x**2 + np.random.randn(nsamples)
+
+## Cross-validation on Ridge regression using KFold only
+
+# Decide degree on polynomial to fit
+poly = PolynomialFeatures(degree = 6)
+
+# Decide which values of lambda to use
+nlambdas = 500
+lambdas = np.logspace(-3, 5, nlambdas)
+
+# Initialize a KFold instance
+k = 5
+kfold = KFold(n_splits = k)
+
+# Perform the cross-validation to estimate MSE
+scores_KFold = np.zeros((nlambdas, k))
+
+i = 0
+for lmb in lambdas:
+    ridge = Ridge(alpha = lmb)
+    j = 0
+    for train_inds, test_inds in kfold.split(x):
+        xtrain = x[train_inds]
+        ytrain = y[train_inds]
+
+        xtest = x[test_inds]
+        ytest = y[test_inds]
+
+        Xtrain = poly.fit_transform(xtrain[:, np.newaxis])
+        ridge.fit(Xtrain, ytrain[:, np.newaxis])
+
+        Xtest = poly.fit_transform(xtest[:, np.newaxis])
+        ypred = ridge.predict(Xtest)
+
+        scores_KFold[i,j] = np.sum((ypred - ytest[:, np.newaxis])**2)/np.size(ypred)
+
+        j += 1
+    i += 1
+
+
+estimated_mse_KFold = np.mean(scores_KFold, axis = 1)
+
+## Cross-validation using cross_val_score from sklearn along with KFold
+
+# kfold is an instance initialized above as:
+# kfold = KFold(n_splits = k)
+
+estimated_mse_sklearn = np.zeros(nlambdas)
+i = 0
+for lmb in lambdas:
+    ridge = Ridge(alpha = lmb)
+
+    X = poly.fit_transform(x[:, np.newaxis])
+    estimated_mse_folds = cross_val_score(ridge, X, y[:, np.newaxis], scoring='neg_mean_squared_error', cv=kfold)
+
+    # cross_val_score return an array containing the estimated negative mse for every fold.
+    # we have to the the mean of every array in order to get an estimate of the mse of the model
+    estimated_mse_sklearn[i] = np.mean(-estimated_mse_folds)
+
+    i += 1
+
+## Plot and compare the slightly different ways to perform cross-validation
+
+plt.figure()
+
+plt.plot(np.log10(lambdas), estimated_mse_sklearn, label = 'cross_val_score')
+plt.plot(np.log10(lambdas), estimated_mse_KFold, 'r--', label = 'KFold')
+
+plt.xlabel('log10(lambda)')
+plt.ylabel('mse')
+
+plt.legend()
+
+plt.show()
+
+
+
+
+
+
+

2.79. The bias-variance tradeoff

+

We will discuss the bias-variance tradeoff in the context of +continuous predictions such as regression. However, many of the +intuitions and ideas discussed here also carry over to classification +tasks. Consider a dataset \(\mathcal{L}\) consisting of the data +\(\mathbf{X}_\mathcal{L}=\{(y_j, \boldsymbol{x}_j), j=0\ldots n-1\}\).

+

Let us assume that the true data is generated from a noisy model

+
+\[ +\boldsymbol{y}=f(\boldsymbol{x}) + \boldsymbol{\epsilon} +\]
+

where \(\epsilon\) is normally distributed with mean zero and standard deviation \(\sigma^2\).

+

In our derivation of the ordinary least squares method we defined then +an approximation to the function \(f\) in terms of the parameters +\(\boldsymbol{\beta}\) and the design matrix \(\boldsymbol{X}\) which embody our model, +that is \(\boldsymbol{\tilde{y}}=\boldsymbol{X}\boldsymbol{\beta}\).

+

Thereafter we found the parameters \(\boldsymbol{\beta}\) by optimizing the means squared error via the so-called cost function

+
+\[ +C(\boldsymbol{X},\boldsymbol{\beta}) =\frac{1}{n}\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2=\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]. +\]
+

We can rewrite this as

+
+\[ +\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\frac{1}{n}\sum_i(f_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2+\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2+\sigma^2. +\]
+

The three terms represent the square of the bias of the learning +method, which can be thought of as the error caused by the simplifying +assumptions built into the method. The second term represents the +variance of the chosen model and finally the last terms is variance of +the error \(\boldsymbol{\epsilon}\).

+

To derive this equation, we need to recall that the variance of \(\boldsymbol{y}\) and \(\boldsymbol{\epsilon}\) are both equal to \(\sigma^2\). The mean value of \(\boldsymbol{\epsilon}\) is by definition equal to zero. Furthermore, the function \(f\) is not a stochastics variable, idem for \(\boldsymbol{\tilde{y}}\). +We use a more compact notation in terms of the expectation value

+
+\[ +\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\mathbb{E}\left[(\boldsymbol{f}+\boldsymbol{\epsilon}-\boldsymbol{\tilde{y}})^2\right], +\]
+

and adding and subtracting \(\mathbb{E}\left[\boldsymbol{\tilde{y}}\right]\) we get

+
+\[ +\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\mathbb{E}\left[(\boldsymbol{f}+\boldsymbol{\epsilon}-\boldsymbol{\tilde{y}}+\mathbb{E}\left[\boldsymbol{\tilde{y}}\right]-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2\right], +\]
+

which, using the abovementioned expectation values can be rewritten as

+
+\[ +\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\mathbb{E}\left[(\boldsymbol{y}-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2\right]+\mathrm{Var}\left[\boldsymbol{\tilde{y}}\right]+\sigma^2, +\]
+

that is the rewriting in terms of the so-called bias, the variance of the model \(\boldsymbol{\tilde{y}}\) and the variance of \(\boldsymbol{\epsilon}\).

+
+
+

2.80. Example code for Bias-Variance tradeoff

+
+
+
import matplotlib.pyplot as plt
+import numpy as np
+from sklearn.linear_model import LinearRegression, Ridge, Lasso
+from sklearn.preprocessing import PolynomialFeatures
+from sklearn.model_selection import train_test_split
+from sklearn.pipeline import make_pipeline
+from sklearn.utils import resample
+
+np.random.seed(2018)
+
+n = 500
+n_boostraps = 100
+degree = 18  # A quite high value, just to show.
+noise = 0.1
+
+# Make data set.
+x = np.linspace(-1, 3, n).reshape(-1, 1)
+y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2) + np.random.normal(0, 0.1, x.shape)
+
+# Hold out some test data that is never used in training.
+x_train, x_test, y_train, y_test = train_test_split(x, y, test_size=0.2)
+
+# Combine x transformation and model into one operation.
+# Not neccesary, but convenient.
+model = make_pipeline(PolynomialFeatures(degree=degree), LinearRegression(fit_intercept=False))
+
+# The following (m x n_bootstraps) matrix holds the column vectors y_pred
+# for each bootstrap iteration.
+y_pred = np.empty((y_test.shape[0], n_boostraps))
+for i in range(n_boostraps):
+    x_, y_ = resample(x_train, y_train)
+
+    # Evaluate the new model on the same test data each time.
+    y_pred[:, i] = model.fit(x_, y_).predict(x_test).ravel()
+
+# Note: Expectations and variances taken w.r.t. different training
+# data sets, hence the axis=1. Subsequent means are taken across the test data
+# set in order to obtain a total value, but before this we have error/bias/variance
+# calculated per data point in the test set.
+# Note 2: The use of keepdims=True is important in the calculation of bias as this 
+# maintains the column vector form. Dropping this yields very unexpected results.
+error = np.mean( np.mean((y_test - y_pred)**2, axis=1, keepdims=True) )
+bias = np.mean( (y_test - np.mean(y_pred, axis=1, keepdims=True))**2 )
+variance = np.mean( np.var(y_pred, axis=1, keepdims=True) )
+print('Error:', error)
+print('Bias^2:', bias)
+print('Var:', variance)
+print('{} >= {} + {} = {}'.format(error, bias, variance, bias+variance))
+
+plt.plot(x[::5, :], y[::5, :], label='f(x)')
+plt.scatter(x_test, y_test, label='Data points')
+plt.scatter(x_test, np.mean(y_pred, axis=1), label='Pred')
+plt.legend()
+plt.show()
+
+
+
+
+
+
+

2.81. Understanding what happens

+
+
+
import matplotlib.pyplot as plt
+import numpy as np
+from sklearn.linear_model import LinearRegression, Ridge, Lasso
+from sklearn.preprocessing import PolynomialFeatures
+from sklearn.model_selection import train_test_split
+from sklearn.pipeline import make_pipeline
+from sklearn.utils import resample
+
+np.random.seed(2018)
+
+n = 40
+n_boostraps = 100
+maxdegree = 14
+
+
+# Make data set.
+x = np.linspace(-3, 3, n).reshape(-1, 1)
+y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)
+error = np.zeros(maxdegree)
+bias = np.zeros(maxdegree)
+variance = np.zeros(maxdegree)
+polydegree = np.zeros(maxdegree)
+x_train, x_test, y_train, y_test = train_test_split(x, y, test_size=0.2)
+
+for degree in range(maxdegree):
+    model = make_pipeline(PolynomialFeatures(degree=degree), LinearRegression(fit_intercept=False))
+    y_pred = np.empty((y_test.shape[0], n_boostraps))
+    for i in range(n_boostraps):
+        x_, y_ = resample(x_train, y_train)
+        y_pred[:, i] = model.fit(x_, y_).predict(x_test).ravel()
+
+    polydegree[degree] = degree
+    error[degree] = np.mean( np.mean((y_test - y_pred)**2, axis=1, keepdims=True) )
+    bias[degree] = np.mean( (y_test - np.mean(y_pred, axis=1, keepdims=True))**2 )
+    variance[degree] = np.mean( np.var(y_pred, axis=1, keepdims=True) )
+    print('Polynomial degree:', degree)
+    print('Error:', error[degree])
+    print('Bias^2:', bias[degree])
+    print('Var:', variance[degree])
+    print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))
+
+plt.plot(polydegree, error, label='Error')
+plt.plot(polydegree, bias, label='bias')
+plt.plot(polydegree, variance, label='Variance')
+plt.legend()
+plt.show()
+
+
+
+
+ +
+
+

2.82. Summing up

+

The bias-variance tradeoff summarizes the fundamental tension in +machine learning, particularly supervised learning, between the +complexity of a model and the amount of training data needed to train +it. Since data is often limited, in practice it is often useful to +use a less-complex model with higher bias, that is a model whose asymptotic +performance is worse than another model because it is easier to +train and less sensitive to sampling noise arising from having a +finite-sized training dataset (smaller variance).

+

The above equations tell us that in +order to minimize the expected test error, we need to select a +statistical learning method that simultaneously achieves low variance +and low bias. Note that variance is inherently a nonnegative quantity, +and squared bias is also nonnegative. Hence, we see that the expected +test MSE can never lie below \(Var(\epsilon)\), the irreducible error.

+

What do we mean by the variance and bias of a statistical learning +method? The variance refers to the amount by which our model would change if we +estimated it using a different training data set. Since the training +data are used to fit the statistical learning method, different +training data sets will result in a different estimate. But ideally the +estimate for our model should not vary too much between training +sets. However, if a method has high variance then small changes in +the training data can result in large changes in the model. In general, more +flexible statistical methods have higher variance.

+

You may also find this recent article of interest.

+
+
+

2.83. Another Example from Scikit-Learn’s Repository

+
+
+
"""
+============================
+Underfitting vs. Overfitting
+============================
+
+This example demonstrates the problems of underfitting and overfitting and
+how we can use linear regression with polynomial features to approximate
+nonlinear functions. The plot shows the function that we want to approximate,
+which is a part of the cosine function. In addition, the samples from the
+real function and the approximations of different models are displayed. The
+models have polynomial features of different degrees. We can see that a
+linear function (polynomial with degree 1) is not sufficient to fit the
+training samples. This is called **underfitting**. A polynomial of degree 4
+approximates the true function almost perfectly. However, for higher degrees
+the model will **overfit** the training data, i.e. it learns the noise of the
+training data.
+We evaluate quantitatively **overfitting** / **underfitting** by using
+cross-validation. We calculate the mean squared error (MSE) on the validation
+set, the higher, the less likely the model generalizes correctly from the
+training data.
+"""
+
+print(__doc__)
+
+import numpy as np
+import matplotlib.pyplot as plt
+from sklearn.pipeline import Pipeline
+from sklearn.preprocessing import PolynomialFeatures
+from sklearn.linear_model import LinearRegression
+from sklearn.model_selection import cross_val_score
+
+
+def true_fun(X):
+    return np.cos(1.5 * np.pi * X)
+
+np.random.seed(0)
+
+n_samples = 30
+degrees = [1, 4, 15]
+
+X = np.sort(np.random.rand(n_samples))
+y = true_fun(X) + np.random.randn(n_samples) * 0.1
+
+plt.figure(figsize=(14, 5))
+for i in range(len(degrees)):
+    ax = plt.subplot(1, len(degrees), i + 1)
+    plt.setp(ax, xticks=(), yticks=())
+
+    polynomial_features = PolynomialFeatures(degree=degrees[i],
+                                             include_bias=False)
+    linear_regression = LinearRegression()
+    pipeline = Pipeline([("polynomial_features", polynomial_features),
+                         ("linear_regression", linear_regression)])
+    pipeline.fit(X[:, np.newaxis], y)
+
+    # Evaluate the models using crossvalidation
+    scores = cross_val_score(pipeline, X[:, np.newaxis], y,
+                             scoring="neg_mean_squared_error", cv=10)
+
+    X_test = np.linspace(0, 1, 100)
+    plt.plot(X_test, pipeline.predict(X_test[:, np.newaxis]), label="Model")
+    plt.plot(X_test, true_fun(X_test), label="True function")
+    plt.scatter(X, y, edgecolor='b', s=20, label="Samples")
+    plt.xlabel("x")
+    plt.ylabel("y")
+    plt.xlim((0, 1))
+    plt.ylim((-2, 2))
+    plt.legend(loc="best")
+    plt.title("Degree {}\nMSE = {:.2e}(+/- {:.2e})".format(
+        degrees[i], -scores.mean(), scores.std()))
+plt.show()
+
+
+
+
+
+
+

2.84. More examples on bootstrap and cross-validation and errors

+
+
+
# Common imports
+import os
+import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+from sklearn.linear_model import LinearRegression, Ridge, Lasso
+from sklearn.model_selection import train_test_split
+from sklearn.utils import resample
+from sklearn.metrics import mean_squared_error
+# Where to save the figures and data files
+PROJECT_ROOT_DIR = "Results"
+FIGURE_ID = "Results/FigureFiles"
+DATA_ID = "DataFiles/"
+
+if not os.path.exists(PROJECT_ROOT_DIR):
+    os.mkdir(PROJECT_ROOT_DIR)
+
+if not os.path.exists(FIGURE_ID):
+    os.makedirs(FIGURE_ID)
+
+if not os.path.exists(DATA_ID):
+    os.makedirs(DATA_ID)
+
+def image_path(fig_id):
+    return os.path.join(FIGURE_ID, fig_id)
+
+def data_path(dat_id):
+    return os.path.join(DATA_ID, dat_id)
+
+def save_fig(fig_id):
+    plt.savefig(image_path(fig_id) + ".png", format='png')
+
+infile = open(data_path("EoS.csv"),'r')
+
+# Read the EoS data as  csv file and organize the data into two arrays with density and energies
+EoS = pd.read_csv(infile, names=('Density', 'Energy'))
+EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce')
+EoS = EoS.dropna()
+Energies = EoS['Energy']
+Density = EoS['Density']
+#  The design matrix now as function of various polytrops
+
+Maxpolydegree = 30
+X = np.zeros((len(Density),Maxpolydegree))
+X[:,0] = 1.0
+testerror = np.zeros(Maxpolydegree)
+trainingerror = np.zeros(Maxpolydegree)
+polynomial = np.zeros(Maxpolydegree)
+
+trials = 100
+for polydegree in range(1, Maxpolydegree):
+    polynomial[polydegree] = polydegree
+    for degree in range(polydegree):
+        X[:,degree] = Density**(degree/3.0)
+
+# loop over trials in order to estimate the expectation value of the MSE
+    testerror[polydegree] = 0.0
+    trainingerror[polydegree] = 0.0
+    for samples in range(trials):
+        x_train, x_test, y_train, y_test = train_test_split(X, Energies, test_size=0.2)
+        model = LinearRegression(fit_intercept=True).fit(x_train, y_train)
+        ypred = model.predict(x_train)
+        ytilde = model.predict(x_test)
+        testerror[polydegree] += mean_squared_error(y_test, ytilde)
+        trainingerror[polydegree] += mean_squared_error(y_train, ypred) 
+
+    testerror[polydegree] /= trials
+    trainingerror[polydegree] /= trials
+    print("Degree of polynomial: %3d"% polynomial[polydegree])
+    print("Mean squared error on training data: %.8f" % trainingerror[polydegree])
+    print("Mean squared error on test data: %.8f" % testerror[polydegree])
+
+plt.plot(polynomial, np.log10(trainingerror), label='Training Error')
+plt.plot(polynomial, np.log10(testerror), label='Test Error')
+plt.xlabel('Polynomial degree')
+plt.ylabel('log10[MSE]')
+plt.legend()
+plt.show()
+
+
+
+
+ +
+
+

2.85. The same example but now with cross-validation

+
+
+
# Common imports
+import os
+import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+from sklearn.linear_model import LinearRegression, Ridge, Lasso
+from sklearn.metrics import mean_squared_error
+from sklearn.model_selection import KFold
+from sklearn.model_selection import cross_val_score
+
+
+# Where to save the figures and data files
+PROJECT_ROOT_DIR = "Results"
+FIGURE_ID = "Results/FigureFiles"
+DATA_ID = "DataFiles/"
+
+if not os.path.exists(PROJECT_ROOT_DIR):
+    os.mkdir(PROJECT_ROOT_DIR)
+
+if not os.path.exists(FIGURE_ID):
+    os.makedirs(FIGURE_ID)
+
+if not os.path.exists(DATA_ID):
+    os.makedirs(DATA_ID)
+
+def image_path(fig_id):
+    return os.path.join(FIGURE_ID, fig_id)
+
+def data_path(dat_id):
+    return os.path.join(DATA_ID, dat_id)
+
+def save_fig(fig_id):
+    plt.savefig(image_path(fig_id) + ".png", format='png')
+
+infile = open(data_path("EoS.csv"),'r')
+
+# Read the EoS data as  csv file and organize the data into two arrays with density and energies
+EoS = pd.read_csv(infile, names=('Density', 'Energy'))
+EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce')
+EoS = EoS.dropna()
+Energies = EoS['Energy']
+Density = EoS['Density']
+#  The design matrix now as function of various polytrops
+
+Maxpolydegree = 30
+X = np.zeros((len(Density),Maxpolydegree))
+X[:,0] = 1.0
+estimated_mse_sklearn = np.zeros(Maxpolydegree)
+polynomial = np.zeros(Maxpolydegree)
+k =5
+kfold = KFold(n_splits = k)
+
+for polydegree in range(1, Maxpolydegree):
+    polynomial[polydegree] = polydegree
+    for degree in range(polydegree):
+        X[:,degree] = Density**(degree/3.0)
+        OLS = LinearRegression()
+# loop over trials in order to estimate the expectation value of the MSE
+    estimated_mse_folds = cross_val_score(OLS, X, Energies, scoring='neg_mean_squared_error', cv=kfold)
+#[:, np.newaxis]
+    estimated_mse_sklearn[polydegree] = np.mean(-estimated_mse_folds)
+
+plt.plot(polynomial, np.log10(estimated_mse_sklearn), label='Test Error')
+plt.xlabel('Polynomial degree')
+plt.ylabel('log10[MSE]')
+plt.legend()
+plt.show()
+
+
+
+
+
+
+

2.86. Cross-validation with Ridge

+
+
+
import numpy as np
+import matplotlib.pyplot as plt
+from sklearn.model_selection import KFold
+from sklearn.linear_model import Ridge
+from sklearn.model_selection import cross_val_score
+from sklearn.preprocessing import PolynomialFeatures
+
+# A seed just to ensure that the random numbers are the same for every run.
+np.random.seed(3155)
+# Generate the data.
+n = 100
+x = np.linspace(-3, 3, n).reshape(-1, 1)
+y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)
+# Decide degree on polynomial to fit
+poly = PolynomialFeatures(degree = 10)
+
+# Decide which values of lambda to use
+nlambdas = 500
+lambdas = np.logspace(-3, 5, nlambdas)
+# Initialize a KFold instance
+k = 5
+kfold = KFold(n_splits = k)
+estimated_mse_sklearn = np.zeros(nlambdas)
+i = 0
+for lmb in lambdas:
+    ridge = Ridge(alpha = lmb)
+    estimated_mse_folds = cross_val_score(ridge, x, y, scoring='neg_mean_squared_error', cv=kfold)
+    estimated_mse_sklearn[i] = np.mean(-estimated_mse_folds)
+    i += 1
+plt.figure()
+plt.plot(np.log10(lambdas), estimated_mse_sklearn, label = 'cross_val_score')
+plt.xlabel('log10(lambda)')
+plt.ylabel('MSE')
+plt.legend()
+plt.show()
+
+
+
+
+
+
+

2.87. The Ising model

+

The one-dimensional Ising model with nearest neighbor interaction, no +external field and a constant coupling constant \(J\) is given by

+ +
+
+\[ +\begin{equation} + H = -J \sum_{k}^L s_k s_{k + 1}, +\label{_auto2} \tag{2} +\end{equation} +\]
+

where \(s_i \in \{-1, 1\}\) and \(s_{N + 1} = s_1\). The number of spins +in the system is determined by \(L\). For the one-dimensional system +there is no phase transition.

+

We will look at a system of \(L = 40\) spins with a coupling constant of +\(J = 1\). To get enough training data we will generate 10000 states +with their respective energies.

+
+
+
import numpy as np
+import matplotlib.pyplot as plt
+from mpl_toolkits.axes_grid1 import make_axes_locatable
+import seaborn as sns
+import scipy.linalg as scl
+from sklearn.model_selection import train_test_split
+import tqdm
+sns.set(color_codes=True)
+cmap_args=dict(vmin=-1., vmax=1., cmap='seismic')
+
+L = 40
+n = int(1e4)
+
+spins = np.random.choice([-1, 1], size=(n, L))
+J = 1.0
+
+energies = np.zeros(n)
+
+for i in range(n):
+    energies[i] = - J * np.dot(spins[i], np.roll(spins[i], 1))
+
+
+
+
+

Here we use ordinary least squares +regression to predict the energy for the nearest neighbor +one-dimensional Ising model on a ring, i.e., the endpoints wrap +around. We will use linear regression to fit a value for +the coupling constant to achieve this.

+
+
+

2.88. Reformulating the problem to suit regression

+

A more general form for the one-dimensional Ising model is

+ +
+
+\[ +\begin{equation} + H = - \sum_j^L \sum_k^L s_j s_k J_{jk}. +\label{_auto3} \tag{3} +\end{equation} +\]
+

Here we allow for interactions beyond the nearest neighbors and a state dependent +coupling constant. This latter expression can be formulated as +a matrix-product

+ +
+
+\[ +\begin{equation} + \boldsymbol{H} = \boldsymbol{X} J, +\label{_auto4} \tag{4} +\end{equation} +\]
+

where \(X_{jk} = s_j s_k\) and \(J\) is a matrix which consists of the +elements \(-J_{jk}\). This form of writing the energy fits perfectly +with the form utilized in linear regression, that is

+ +
+
+\[ +\begin{equation} + \boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta} + \boldsymbol{\epsilon}, +\label{_auto5} \tag{5} +\end{equation} +\]
+

We split the data in training and test data as discussed in the previous example

+
+
+
X = np.zeros((n, L ** 2))
+for i in range(n):
+    X[i] = np.outer(spins[i], spins[i]).ravel()
+y = energies
+X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
+
+
+
+
+
+
+

2.89. Linear regression

+

In the ordinary least squares method we choose the cost function

+ +
+
+\[ +\begin{equation} + C(\boldsymbol{X}, \boldsymbol{\beta})= \frac{1}{n}\left\{(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})^T(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})\right\}. +\label{_auto6} \tag{6} +\end{equation} +\]
+

We then find the extremal point of \(C\) by taking the derivative with respect to \(\boldsymbol{\beta}\) as discussed above. +This yields the expression for \(\boldsymbol{\beta}\) to be

+
+\[ +\boldsymbol{\beta} = \frac{\boldsymbol{X}^T \boldsymbol{y}}{\boldsymbol{X}^T \boldsymbol{X}}, +\]
+

which immediately imposes some requirements on \(\boldsymbol{X}\) as there must exist +an inverse of \(\boldsymbol{X}^T \boldsymbol{X}\). If the expression we are modeling contains an +intercept, i.e., a constant term, we must make sure that the +first column of \(\boldsymbol{X}\) consists of \(1\). We do this here

+
+
+
X_train_own = np.concatenate(
+    (np.ones(len(X_train))[:, np.newaxis], X_train),
+    axis=1
+)
+X_test_own = np.concatenate(
+    (np.ones(len(X_test))[:, np.newaxis], X_test),
+    axis=1
+)
+
+
+
+
+
+
+
def ols_inv(x: np.ndarray, y: np.ndarray) -> np.ndarray:
+    return scl.inv(x.T @ x) @ (x.T @ y)
+beta = ols_inv(X_train_own, y_train)
+
+
+
+
+
+
+

2.90. Singular Value decomposition

+

Doing the inversion directly turns out to be a bad idea since the matrix +\(\boldsymbol{X}^T\boldsymbol{X}\) is singular. An alternative approach is to use the singular +value decomposition. Using the definition of the Moore-Penrose +pseudoinverse we can write the equation for \(\boldsymbol{\beta}\) as

+
+\[ +\boldsymbol{\beta} = \boldsymbol{X}^{+}\boldsymbol{y}, +\]
+

where the pseudoinverse of \(\boldsymbol{X}\) is given by

+
+\[ +\boldsymbol{X}^{+} = \frac{\boldsymbol{X}^T}{\boldsymbol{X}^T\boldsymbol{X}}. +\]
+

Using singular value decomposition we can decompose the matrix \(\boldsymbol{X} = \boldsymbol{U}\boldsymbol{\Sigma} \boldsymbol{V}^T\), +where \(\boldsymbol{U}\) and \(\boldsymbol{V}\) are orthogonal(unitary) matrices and \(\boldsymbol{\Sigma}\) contains the singular values (more details below). +where \(X^{+} = V\Sigma^{+} U^T\). This reduces the equation for +\(\omega\) to

+ +
+
+\[ +\begin{equation} + \boldsymbol{\beta} = \boldsymbol{V}\boldsymbol{\Sigma}^{+} \boldsymbol{U}^T \boldsymbol{y}. +\label{_auto7} \tag{7} +\end{equation} +\]
+

Note that solving this equation by actually doing the pseudoinverse +(which is what we will do) is not a good idea as this operation scales +as \(\mathcal{O}(n^3)\), where \(n\) is the number of elements in a +general matrix. Instead, doing \(QR\)-factorization and solving the +linear system as an equation would reduce this down to +\(\mathcal{O}(n^2)\) operations.

+
+
+
def ols_svd(x: np.ndarray, y: np.ndarray) -> np.ndarray:
+    u, s, v = scl.svd(x)
+    return v.T @ scl.pinv(scl.diagsvd(s, u.shape[0], v.shape[0])) @ u.T @ y
+
+
+
+
+
+
+
beta = ols_svd(X_train_own,y_train)
+
+
+
+
+

When extracting the \(J\)-matrix we need to make sure that we remove the intercept, as is done here

+
+
+
J = beta[1:].reshape(L, L)
+
+
+
+
+

A way of looking at the coefficients in \(J\) is to plot the matrices as images.

+
+
+
fig = plt.figure(figsize=(20, 14))
+im = plt.imshow(J, **cmap_args)
+plt.title("OLS", fontsize=18)
+plt.xticks(fontsize=18)
+plt.yticks(fontsize=18)
+cb = fig.colorbar(im)
+cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
+plt.show()
+
+
+
+
+

It is interesting to note that OLS +considers both \(J_{j, j + 1} = -0.5\) and \(J_{j, j - 1} = -0.5\) as +valid matrix elements for \(J\). +In our discussion below on hyperparameters and Ridge and Lasso regression we will see that +this problem can be removed, partly and only with Lasso regression.

+

In this case our matrix inversion was actually possible. The obvious question now is what is the mathematics behind the SVD?

+
+
+

2.91. The one-dimensional Ising model

+

Let us bring back the Ising model again, but now with an additional +focus on Ridge and Lasso regression as well. We repeat some of the +basic parts of the Ising model and the setup of the training and test +data. The one-dimensional Ising model with nearest neighbor +interaction, no external field and a constant coupling constant \(J\) is +given by

+ +
+
+\[ +\begin{equation} + H = -J \sum_{k}^L s_k s_{k + 1}, +\label{_auto8} \tag{8} +\end{equation} +\]
+

where \(s_i \in \{-1, 1\}\) and \(s_{N + 1} = s_1\). The number of spins in the system is determined by \(L\). For the one-dimensional system there is no phase transition.

+

We will look at a system of \(L = 40\) spins with a coupling constant of \(J = 1\). To get enough training data we will generate 10000 states with their respective energies.

+
+
+
import numpy as np
+import matplotlib.pyplot as plt
+from mpl_toolkits.axes_grid1 import make_axes_locatable
+import seaborn as sns
+import scipy.linalg as scl
+from sklearn.model_selection import train_test_split
+import sklearn.linear_model as skl
+import tqdm
+sns.set(color_codes=True)
+cmap_args=dict(vmin=-1., vmax=1., cmap='seismic')
+
+L = 40
+n = int(1e4)
+
+spins = np.random.choice([-1, 1], size=(n, L))
+J = 1.0
+
+energies = np.zeros(n)
+
+for i in range(n):
+    energies[i] = - J * np.dot(spins[i], np.roll(spins[i], 1))
+
+
+
+
+

A more general form for the one-dimensional Ising model is

+ +
+
+\[ +\begin{equation} + H = - \sum_j^L \sum_k^L s_j s_k J_{jk}. +\label{_auto9} \tag{9} +\end{equation} +\]
+

Here we allow for interactions beyond the nearest neighbors and a more +adaptive coupling matrix. This latter expression can be formulated as +a matrix-product on the form

+ +
+
+\[ +\begin{equation} + H = X J, +\label{_auto10} \tag{10} +\end{equation} +\]
+

where \(X_{jk} = s_j s_k\) and \(J\) is the matrix consisting of the +elements \(-J_{jk}\). This form of writing the energy fits perfectly +with the form utilized in linear regression, viz.

+ +
+
+\[ +\begin{equation} + \boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta} + \boldsymbol{\epsilon}. +\label{_auto11} \tag{11} +\end{equation} +\]
+

We organize the data as we did above

+
+
+
X = np.zeros((n, L ** 2))
+for i in range(n):
+    X[i] = np.outer(spins[i], spins[i]).ravel()
+y = energies
+X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.96)
+
+X_train_own = np.concatenate(
+    (np.ones(len(X_train))[:, np.newaxis], X_train),
+    axis=1
+)
+
+X_test_own = np.concatenate(
+    (np.ones(len(X_test))[:, np.newaxis], X_test),
+    axis=1
+)
+
+
+
+
+

We will do all fitting with Scikit-Learn,

+
+
+
clf = skl.LinearRegression().fit(X_train, y_train)
+
+
+
+
+

When extracting the \(J\)-matrix we make sure to remove the intercept

+
+
+
J_sk = clf.coef_.reshape(L, L)
+
+
+
+
+

And then we plot the results

+
+
+
fig = plt.figure(figsize=(20, 14))
+im = plt.imshow(J_sk, **cmap_args)
+plt.title("LinearRegression from Scikit-learn", fontsize=18)
+plt.xticks(fontsize=18)
+plt.yticks(fontsize=18)
+cb = fig.colorbar(im)
+cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
+plt.show()
+
+
+
+
+

The results perfectly with our previous discussion where we used our own code.

+
+
+

2.92. Ridge regression

+

Having explored the ordinary least squares we move on to ridge +regression. In ridge regression we include a regularizer. This +involves a new cost function which leads to a new estimate for the +weights \(\boldsymbol{\beta}\). This results in a penalized regression problem. The +cost function is given by

+

1 +3 +6

+

< +< +< +! +! +M +A +T +H +_ +B +L +O +C +K

+
+
+
_lambda = 0.1
+clf_ridge = skl.Ridge(alpha=_lambda).fit(X_train, y_train)
+J_ridge_sk = clf_ridge.coef_.reshape(L, L)
+fig = plt.figure(figsize=(20, 14))
+im = plt.imshow(J_ridge_sk, **cmap_args)
+plt.title("Ridge from Scikit-learn", fontsize=18)
+plt.xticks(fontsize=18)
+plt.yticks(fontsize=18)
+cb = fig.colorbar(im)
+cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
+
+plt.show()
+
+
+
+
+
+
+

2.93. LASSO regression

+

In the Least Absolute Shrinkage and Selection Operator (LASSO)-method we get a third cost function.

+ +
+
+\[ +\begin{equation} + C(\boldsymbol{X}, \boldsymbol{\beta}; \lambda) = (\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})^T(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y}) + \lambda \sqrt{\boldsymbol{\beta}^T\boldsymbol{\beta}}. +\label{_auto13} \tag{13} +\end{equation} +\]
+

Finding the extremal point of this cost function is not so straight-forward as in least squares and ridge. We will therefore rely solely on the function Lasso from Scikit-Learn.

+
+
+
clf_lasso = skl.Lasso(alpha=_lambda).fit(X_train, y_train)
+J_lasso_sk = clf_lasso.coef_.reshape(L, L)
+fig = plt.figure(figsize=(20, 14))
+im = plt.imshow(J_lasso_sk, **cmap_args)
+plt.title("Lasso from Scikit-learn", fontsize=18)
+plt.xticks(fontsize=18)
+plt.yticks(fontsize=18)
+cb = fig.colorbar(im)
+cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
+
+plt.show()
+
+
+
+
+

It is quite striking how LASSO breaks the symmetry of the coupling +constant as opposed to ridge and OLS. We get a sparse solution with +\(J_{j, j + 1} = -1\).

+
+
+

2.94. Performance as function of the regularization parameter

+

We see how the different models perform for a different set of values for \(\lambda\).

+
+
+
lambdas = np.logspace(-4, 5, 10)
+
+train_errors = {
+    "ols_sk": np.zeros(lambdas.size),
+    "ridge_sk": np.zeros(lambdas.size),
+    "lasso_sk": np.zeros(lambdas.size)
+}
+
+test_errors = {
+    "ols_sk": np.zeros(lambdas.size),
+    "ridge_sk": np.zeros(lambdas.size),
+    "lasso_sk": np.zeros(lambdas.size)
+}
+
+plot_counter = 1
+
+fig = plt.figure(figsize=(32, 54))
+
+for i, _lambda in enumerate(tqdm.tqdm(lambdas)):
+    for key, method in zip(
+        ["ols_sk", "ridge_sk", "lasso_sk"],
+        [skl.LinearRegression(), skl.Ridge(alpha=_lambda), skl.Lasso(alpha=_lambda)]
+    ):
+        method = method.fit(X_train, y_train)
+
+        train_errors[key][i] = method.score(X_train, y_train)
+        test_errors[key][i] = method.score(X_test, y_test)
+
+        omega = method.coef_.reshape(L, L)
+
+        plt.subplot(10, 5, plot_counter)
+        plt.imshow(omega, **cmap_args)
+        plt.title(r"%s, $\lambda = %.4f$" % (key, _lambda))
+        plot_counter += 1
+
+plt.show()
+
+
+
+
+

We see that LASSO reaches a good solution for low +values of \(\lambda\), but will “wither” when we increase \(\lambda\) too +much. Ridge is more stable over a larger range of values for +\(\lambda\), but eventually also fades away.

+
+
+

2.95. Finding the optimal value of \(\lambda\)

+

To determine which value of \(\lambda\) is best we plot the accuracy of +the models when predicting the training and the testing set. We expect +the accuracy of the training set to be quite good, but if the accuracy +of the testing set is much lower this tells us that we might be +subject to an overfit model. The ideal scenario is an accuracy on the +testing set that is close to the accuracy of the training set.

+
+
+
fig = plt.figure(figsize=(20, 14))
+
+colors = {
+    "ols_sk": "r",
+    "ridge_sk": "y",
+    "lasso_sk": "c"
+}
+
+for key in train_errors:
+    plt.semilogx(
+        lambdas,
+        train_errors[key],
+        colors[key],
+        label="Train {0}".format(key),
+        linewidth=4.0
+    )
+
+for key in test_errors:
+    plt.semilogx(
+        lambdas,
+        test_errors[key],
+        colors[key] + "--",
+        label="Test {0}".format(key),
+        linewidth=4.0
+    )
+plt.legend(loc="best", fontsize=18)
+plt.xlabel(r"$\lambda$", fontsize=18)
+plt.ylabel(r"$R^2$", fontsize=18)
+plt.tick_params(labelsize=18)
+plt.show()
+
+
+
+
+

From the above figure we can see that LASSO with \(\lambda = 10^{-2}\) +achieves a very good accuracy on the test set. This by far surpasses the +other models for all values of \(\lambda\).

+
+
+ + + + +
+ +
+
+ + +
+ + 1. Getting started, our first data and Machine Learning encounters + 3. Logistic Regression + +
+
+
+

+ + By Morten Hjorth-Jensen
+ + © Copyright 2020.
+

+
+
+
+ + +
+
+ + + + + \ No newline at end of file diff --git a/doc/src/LectureNotes/_build/html/chapter4.html b/doc/src/LectureNotes/_build/html/chapter4.html new file mode 100644 index 000000000..cf49e7ddd --- /dev/null +++ b/doc/src/LectureNotes/_build/html/chapter4.html @@ -0,0 +1,598 @@ + + + + + + + + + 3. Logistic Regression — Applied Machine Learning and Data Analysis + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + + + + + + + +
+ +
+
+
+
+ + + +
+ + + +
+ + + + + + +
+ +
+ + +
+ +
+ + + +
+
+ + + + + + + +
+ + +
+
+ Contents +
+ +
+
+
+
+ +
+ +
+

3. Logistic Regression

+
+

3.1. Introduction

+

In linear regression our main interest was centered on learning the +coefficients of a functional fit (say a polynomial) in order to be +able to predict the response of a continuous variable on some unseen +data. The fit to the continuous variable \(y_i\) is based on some +independent variables \(\hat{x}_i\). Linear regression resulted in +analytical expressions for standard ordinary Least Squares or Ridge +regression (in terms of matrices to invert) for several quantities, +ranging from the variance and thereby the confidence intervals of the +parameters \(\hat{\beta}\) to the mean squared error. If we can invert +the product of the design matrices, linear regression gives then a +simple recipe for fitting our data.

+

Classification problems, however, are concerned with outcomes taking +the form of discrete variables (i.e. categories). We may for example, +on the basis of DNA sequencing for a number of patients, like to find +out which mutations are important for a certain disease; or based on +scans of various patients’ brains, figure out if there is a tumor or +not; or given a specific physical system, we’d like to identify its +state, say whether it is an ordered or disordered system (typical +situation in solid state physics); or classify the status of a +patient, whether she/he has a stroke or not and many other similar +situations.

+

The most common situation we encounter when we apply logistic +regression is that of two possible outcomes, normally denoted as a +binary outcome, true or false, positive or negative, success or +failure etc.

+

Logistic regression will also serve as our stepping stone towards +neural network algorithms and supervised deep learning. For logistic +learning, the minimization of the cost function leads to a non-linear +equation in the parameters \(\hat{\beta}\). The optimization of the +problem calls therefore for minimization algorithms. This forms the +bottle neck of all machine learning algorithms, namely how to find +reliable minima of a multi-variable function. This leads us to the +family of gradient descent methods. The latter are the working horses +of basically all modern machine learning algorithms.

+

We note also that many of the topics discussed here on logistic +regression are also commonly used in modern supervised Deep Learning +models, as we will see later.

+
+
+

3.2. Basics

+

We consider the case where the dependent variables, also called the +responses or the outcomes, \(y_i\) are discrete and only take values +from \(k=0,\dots,K-1\) (i.e. \(K\) classes).

+

The goal is to predict the +output classes from the design matrix \(\hat{X}\in\mathbb{R}^{n\times p}\) +made of \(n\) samples, each of which carries \(p\) features or predictors. The +primary goal is to identify the classes to which new unseen samples +belong.

+

Let us specialize to the case of two classes only, with outputs +\(y_i=0\) and \(y_i=1\). Our outcomes could represent the status of a +credit card user that could default or not on her/his credit card +debt. That is

+
+\[\begin{split} +y_i = \begin{bmatrix} 0 & \mathrm{no}\\ 1 & \mathrm{yes} \end{bmatrix}. +\end{split}\]
+

Before moving to the logistic model, let us try to use our linear +regression model to classify these two outcomes. We could for example +fit a linear model to the default case if \(y_i > 0.5\) and the no +default case \(y_i \leq 0.5\).

+

We would then have our +weighted linear combination, namely

+ +
+
+\[ +\begin{equation} +\hat{y} = \hat{X}^T\hat{\beta} + \hat{\epsilon}, +\label{_auto1} \tag{1} +\end{equation} +\]
+

where \(\hat{y}\) is a vector representing the possible outcomes, \(\hat{X}\) is our +\(n\times p\) design matrix and \(\hat{\beta}\) represents our estimators/predictors.

+

The main problem with our function is that it takes values on the +entire real axis. In the case of logistic regression, however, the +labels \(y_i\) are discrete variables. A typical example is the credit +card data discussed below here, where we can set the state of +defaulting the debt to \(y_i=1\) and not to \(y_i=0\) for one the persons +in the data set (see the full example below).

+

One simple way to get a discrete output is to have sign +functions that map the output of a linear regressor to values \(\{0,1\}\), +\(f(s_i)=sign(s_i)=1\) if \(s_i\ge 0\) and 0 if otherwise. +We will encounter this model in our first demonstration of neural networks. Historically it is called the “perceptron” model in the machine learning +literature. This model is extremely simple. However, in many cases it is more +favorable to use a ``soft” classifier that outputs +the probability of a given category. This leads us to the logistic function.

+
+
+

3.3. The logistic function

+

The perceptron is an example of a ``hard classification” model. We +will encounter this model when we discuss neural networks as +well. Each datapoint is deterministically assigned to a category (i.e +\(y_i=0\) or \(y_i=1\)). In many cases, it is favorable to have a “soft” +classifier that outputs the probability of a given category rather +than a single value. For example, given \(x_i\), the classifier +outputs the probability of being in a category \(k\). Logistic regression +is the most common example of a so-called soft classifier. In logistic +regression, the probability that a data point \(x_i\) +belongs to a category \(y_i=\{0,1\}\) is given by the so-called logit function (or Sigmoid) which is meant to represent the likelihood for a given event,

+
+\[ +p(t) = \frac{1}{1+\mathrm \exp{-t}}=\frac{\exp{t}}{1+\mathrm \exp{t}}. +\]
+

Note that \(1-p(t)= p(-t)\).

+

The following code plots the logistic function, the step function and other functions we will encounter from here and on.

+
+
+
%matplotlib inline
+
+"""The sigmoid function (or the logistic curve) is a
+function that takes any real number, z, and outputs a number (0,1).
+It is useful in neural networks for assigning weights on a relative scale.
+The value z is the weighted sum of parameters involved in the learning algorithm."""
+
+import numpy
+import matplotlib.pyplot as plt
+import math as mt
+
+z = numpy.arange(-5, 5, .1)
+sigma_fn = numpy.vectorize(lambda z: 1/(1+numpy.exp(-z)))
+sigma = sigma_fn(z)
+
+fig = plt.figure()
+ax = fig.add_subplot(111)
+ax.plot(z, sigma)
+ax.set_ylim([-0.1, 1.1])
+ax.set_xlim([-5,5])
+ax.grid(True)
+ax.set_xlabel('z')
+ax.set_title('sigmoid function')
+
+plt.show()
+
+"""Step Function"""
+z = numpy.arange(-5, 5, .02)
+step_fn = numpy.vectorize(lambda z: 1.0 if z >= 0.0 else 0.0)
+step = step_fn(z)
+
+fig = plt.figure()
+ax = fig.add_subplot(111)
+ax.plot(z, step)
+ax.set_ylim([-0.5, 1.5])
+ax.set_xlim([-5,5])
+ax.grid(True)
+ax.set_xlabel('z')
+ax.set_title('step function')
+
+plt.show()
+
+"""tanh Function"""
+z = numpy.arange(-2*mt.pi, 2*mt.pi, 0.1)
+t = numpy.tanh(z)
+
+fig = plt.figure()
+ax = fig.add_subplot(111)
+ax.plot(z, t)
+ax.set_ylim([-1.0, 1.0])
+ax.set_xlim([-2*mt.pi,2*mt.pi])
+ax.grid(True)
+ax.set_xlabel('z')
+ax.set_title('tanh function')
+
+plt.show()
+
+
+
+
+_images/chapter4_7_0.png +_images/chapter4_7_1.png +_images/chapter4_7_2.png +
+
+
+
+

3.4. Two parameters

+

We assume now that we have two classes with \(y_i\) either \(0\) or \(1\). Furthermore we assume also that we have only two parameters \(\beta\) in our fitting of the Sigmoid function, that is we define probabilities

+
+\[\begin{split} +\begin{align*} +p(y_i=1|x_i,\hat{\beta}) &= \frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}},\nonumber\\ +p(y_i=0|x_i,\hat{\beta}) &= 1 - p(y_i=1|x_i,\hat{\beta}), +\end{align*} +\end{split}\]
+

where \(\hat{\beta}\) are the weights we wish to extract from data, in our case \(\beta_0\) and \(\beta_1\).

+

Note that we used

+
+\[ +p(y_i=0\vert x_i, \hat{\beta}) = 1-p(y_i=1\vert x_i, \hat{\beta}). +\]
+
+
+

3.5. Maximum likelihood

+

In order to define the total likelihood for all possible outcomes from a
+dataset \(\mathcal{D}=\{(y_i,x_i)\}\), with the binary labels +\(y_i\in\{0,1\}\) and where the data points are drawn independently, we use the so-called Maximum Likelihood Estimation (MLE) principle. +We aim thus at maximizing +the probability of seeing the observed data. We can then approximate the +likelihood in terms of the product of the individual probabilities of a specific outcome \(y_i\), that is

+
+\[\begin{split} +\begin{align*} +P(\mathcal{D}|\hat{\beta})& = \prod_{i=1}^n \left[p(y_i=1|x_i,\hat{\beta})\right]^{y_i}\left[1-p(y_i=1|x_i,\hat{\beta}))\right]^{1-y_i}\nonumber \\ +\end{align*} +\end{split}\]
+

from which we obtain the log-likelihood and our cost/loss function

+
+\[ +\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n \left( y_i\log{p(y_i=1|x_i,\hat{\beta})} + (1-y_i)\log\left[1-p(y_i=1|x_i,\hat{\beta}))\right]\right). +\]
+

Reordering the logarithms, we can rewrite the cost/loss function as

+
+\[ +\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right). +\]
+

The maximum likelihood estimator is defined as the set of parameters that maximize the log-likelihood where we maximize with respect to \(\beta\). +Since the cost (error) function is just the negative log-likelihood, for logistic regression we have that

+
+\[ +\mathcal{C}(\hat{\beta})=-\sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right). +\]
+

This equation is known in statistics as the cross entropy. Finally, we note that just as in linear regression, +in practice we often supplement the cross-entropy with additional regularization terms, usually \(L_1\) and \(L_2\) regularization as we did for Ridge and Lasso regression.

+

The cross entropy is a convex function of the weights \(\hat{\beta}\) and, +therefore, any local minimizer is a global minimizer.

+

Minimizing this +cost function with respect to the two parameters \(\beta_0\) and \(\beta_1\) we obtain

+
+\[ +\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \beta_0} = -\sum_{i=1}^n \left(y_i -\frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}}\right), +\]
+

and

+
+\[ +\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \beta_1} = -\sum_{i=1}^n \left(y_ix_i -x_i\frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}}\right). +\]
+

Let us now define a vector \(\hat{y}\) with \(n\) elements \(y_i\), an +\(n\times p\) matrix \(\hat{X}\) which contains the \(x_i\) values and a +vector \(\hat{p}\) of fitted probabilities \(p(y_i\vert x_i,\hat{\beta})\). We can rewrite in a more compact form the first +derivative of cost function as

+
+\[ +\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}} = -\hat{X}^T\left(\hat{y}-\hat{p}\right). +\]
+

If we in addition define a diagonal matrix \(\hat{W}\) with elements +\(p(y_i\vert x_i,\hat{\beta})(1-p(y_i\vert x_i,\hat{\beta})\), we can obtain a compact expression of the second derivative as

+
+\[ +\frac{\partial^2 \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}\partial \hat{\beta}^T} = \hat{X}^T\hat{W}\hat{X}. +\]
+

Within a binary classification problem, we can easily expand our model to include multiple predictors. Our ratio between likelihoods is then with \(p\) predictors

+
+\[ +\log{ \frac{p(\hat{\beta}\hat{x})}{1-p(\hat{\beta}\hat{x})}} = \beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p. +\]
+

Here we defined \(\hat{x}=[1,x_1,x_2,\dots,x_p]\) and \(\hat{\beta}=[\beta_0, \beta_1, \dots, \beta_p]\) leading to

+
+\[ +p(\hat{\beta}\hat{x})=\frac{ \exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}{1+\exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}. +\]
+
+
+

3.6. Including more classes

+

Till now we have mainly focused on two classes, the so-called binary +system. Suppose we wish to extend to \(K\) classes. Let us for the sake +of simplicity assume we have only two predictors. We have then +following model

+

1 +5

+

< +< +< +! +! +M +A +T +H +_ +B +L +O +C +K

+
+\[ +\log{\frac{p(C=2\vert x)}{p(K\vert x)}} = \beta_{20}+\beta_{21}x_1, +\]
+

and so on till the class \(C=K-1\) class

+
+\[ +\log{\frac{p(C=K-1\vert x)}{p(K\vert x)}} = \beta_{(K-1)0}+\beta_{(K-1)1}x_1, +\]
+

and the model is specified in term of \(K-1\) so-called log-odds or +logit transformations.

+

In our discussion of neural networks we will encounter the above again +in terms of a slightly modified function, the so-called Softmax function.

+

The softmax function is used in various multiclass classification +methods, such as multinomial logistic regression (also known as +softmax regression), multiclass linear discriminant analysis, naive +Bayes classifiers, and artificial neural networks. Specifically, in +multinomial logistic regression and linear discriminant analysis, the +input to the function is the result of \(K\) distinct linear functions, +and the predicted probability for the \(k\)-th class given a sample +vector \(\hat{x}\) and a weighting vector \(\hat{\beta}\) is (with two +predictors):

+
+\[ +p(C=k\vert \mathbf {x} )=\frac{\exp{(\beta_{k0}+\beta_{k1}x_1)}}{1+\sum_{l=1}^{K-1}\exp{(\beta_{l0}+\beta_{l1}x_1)}}. +\]
+

It is easy to extend to more predictors. The final class is

+
+\[ +p(C=K\vert \mathbf {x} )=\frac{1}{1+\sum_{l=1}^{K-1}\exp{(\beta_{l0}+\beta_{l1}x_1)}}, +\]
+

and they sum to one. Our earlier discussions were all specialized to +the case with two classes only. It is easy to see from the above that +what we derived earlier is compatible with these equations.

+

To find the optimal parameters we would typically use a gradient +descent method. Newton’s method and gradient descent methods are +discussed in the material on optimization +methods.

+
+
+ + + + +
+ +
+
+ + +
+ + 2. Linear Regression and more Advanced Regression Analysis + +
+
+
+

+ + By Morten Hjorth-Jensen
+ + © Copyright 2020.
+

+
+
+
+ + +
+
+ + + + + \ No newline at end of file diff --git a/doc/src/LectureNotes/_build/html/content.html b/doc/src/LectureNotes/_build/html/content.html deleted file mode 100644 index 6339c23b0..000000000 --- a/doc/src/LectureNotes/_build/html/content.html +++ /dev/null @@ -1,236 +0,0 @@ - - - - - - - - - Content in Jupyter Book — My sample book - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
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