FYS-STK4155/doc/LectureNotes/_build/jupyter_execute/chapter1.ipynb

Linear Regression

Introduction

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

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

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

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

What is Machine Learning?

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

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

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

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

Machine learning is an extremely rich field, in spite of its young age. The increases we have seen during the last three decades in computational capabilities have been followed by developments of methods and techniques for analyzing and handling large date sets, relying heavily on statistics, computer science and mathematics. The field is rather new and developing rapidly. Popular software packages written in Python for machine learning like Scikit-learn, 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.

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, validation and test data). Many find the most difficult part of using Machine Learning to be the set up of your data in a meaningful way.

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

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

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

A Frequentist approach to data analysis

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

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

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

What is a good model?

In science and engineering we often end up in situations where we want to infer (or learn) a quantitative model M for a given set of sample points \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 sets of lectures.

Simple linear regression model using scikit-learn

We start with perhaps our simplest possible example, using Scikit-Learn to perform linear regression analysis on a data set produced by us.

What follows is a simple Python code where we have defined a function y in terms of the variable x. Both are defined as vectors with 100 entries. The numbers in the vector \boldsymbol{x} are given by random numbers generated with a uniform distribution with entries x_i \in [0,1] (more about probability distribution functions later). These values are then used to define a function y(x) (tabulated again as a vector) with a linear dependence on x plus a random noise added via the normal distribution.

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

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

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

For plotting we use the Python package matplotlib which produces publication quality figures. Feel free to explore the extensive gallery of examples. In this example we plot our original values of x and y as well as the prediction ypredict (\tilde{y}), which attempts at fitting our data with a straight line. Note also that Scikit-Learn requires a matrix as input for the input values x and y. In the above code we have solved this by declaring x and y as arrays of dimension n\times 1.

In the code here we have also made a new array for x\in [0,1]. Our prediction is computed for these values, meaning that they were not included in the data set used to train (or fit) the model. This is a recurrring theme in machine learning and data analysis. We would like to train a model on a specific given data set. Thereafter we wish to apply it to data which were not included in the training. Below we will encounter this again in the so-called train-validate-test spliting. We will typically split our data into different sets, oen for training, one for validation and finally, our data from the untouched test vault!

The Python code follows here.

In [1]:

%matplotlib inline

# 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)
# This is our new x-array to which we test our model
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 \boldsymbol{y} -\boldsymbol{\tilde{y}}\vert}{\vert \boldsymbol{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

In [2]:

import numpy as np
import matplotlib.pyplot as plt
from sklearn.linear_model import LinearRegression
# Number of data points
n = 100
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.

In [3]:

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 intercept alpha: 
 [2.08534155]
Coefficient beta : 
 [[4.76326745]]
Mean squared error: 0.26
Variance score: 0.88
Mean squared log error: 0.01
Mean absolute error: 0.42

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(\boldsymbol{y},\boldsymbol{\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 \boldsymbol{y}, disregarding the input features, would get a R^2 score of 0.0.

If \tilde{\boldsymbol{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(\boldsymbol{y}, \tilde{\boldsymbol{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 \boldsymbol{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}(\boldsymbol{y}, \boldsymbol{\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}(\boldsymbol{y}, \boldsymbol{\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

H_{\delta}(\boldsymbol{a})=\left\{\begin{array}{cc}\frac{1}{2} \boldsymbol{a}^{2}& \text{for }|\boldsymbol{a}|\leq \delta\\ \delta (|\boldsymbol{a}|-\frac{1}{2}\delta ),&\text{otherwise}.\end{array}\right.

Here \boldsymbol{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.

In [4]:

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))

0.0050000000000000044

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.

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.

In [5]:

# 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.

In [6]:

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.

In [7]:

"""                                                                                                                         
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.                                                          
"""

Out [7]:

'                                                                                                                         \nThis is taken from the data file of the mass 2016 evaluation.                                                               \nAll 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'

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.

In [8]:

# 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.

In [9]:

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

            N    Z    A Element  Ebinding
A                                        
1   0       0    1    1       H  0.000000
2   1       1    1    2       H  1.112283
3   2       2    1    3       H  2.827265
4   6       2    2    4      He  7.073915
5   9       3    2    5      He  5.512132
...       ...  ...  ...     ...       ...
264 3304  156  108  264      Hs  7.298375
265 3310  157  108  265      Hs  7.296247
266 3317  158  108  266      Hs  7.298273
269 3338  159  110  269      Ds  7.250154
270 3344  160  110  270      Ds  7.253775

[267 rows x 5 columns]

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 n\times p, 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.

In [10]:

# 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)

Note well that we have made life simple here. We perform a fit in terms of the number of nucleons only. A more sophisticated fit can be done by including an explicit dependence on the number of protons and neutrons in the asymmetry and Coulomb terms. We leave this as an exercise to you the reader.

With Scikit-Learn we are now ready to use linear regression and fit our data.

In [11]:

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.

In [12]:

# 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))

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()

Mean squared error: 0.04
Variance score: 0.95
Mean absolute error: 0.05

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!

In [13]:

#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))

            N    Z    A Element  Ebinding   Eapprox
A                                                  
1   0       0    1    1       H  0.000000  0.000000
2   1       1    1    2       H  1.112283  1.112283
3   2       2    1    3       H  2.827265  2.827265
4   6       2    2    4      He  7.073915  7.073915
5   9       3    2    5      He  5.512132  5.512132
...       ...  ...  ...     ...       ...       ...
264 3304  156  108  264      Hs  7.298375  7.298375
265 3310  157  108  265      Hs  7.296247  7.297260
266 3317  158  108  266      Hs  7.298273  7.297260
269 3338  159  110  269      Ds  7.250154  7.250154
270 3344  160  110  270      Ds  7.253775  7.253775

[267 rows x 6 columns]
0.009883615646716186

With a deeper and deeper tree level, we can almost reproduce every single data point by increasing the max depth of the tree. We can actually decide to make a decision tree which fits every single point. As we will see later, this has the benefit that we can really train a model which traverses every single data point. However, the price we pay is that we will easily overfit. That is, if we apply our model to unseen data, we will most likely fail miserably in our attempt at making predictions. As an exercise, try to make the tree level larger by adjusting the maximum depth variable. When printing out the predicition, you will note that the binding energy of every nucleus is accurately reproduced.

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.

In [14]:

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()

/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
  warnings.warn(
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
  warnings.warn(
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
  warnings.warn(

/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
  warnings.warn(
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
  warnings.warn(
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
  warnings.warn(

/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
  warnings.warn(
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
  warnings.warn(
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
  warnings.warn(

/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
  warnings.warn(
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
  warnings.warn(
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
  warnings.warn(

/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
  warnings.warn(
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
  warnings.warn(
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
  warnings.warn(

/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
  warnings.warn(
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
  warnings.warn(
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
  warnings.warn(

/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
  warnings.warn(
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
  warnings.warn(
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
  warnings.warn(
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
  warnings.warn(

/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
  warnings.warn(
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
  warnings.warn(
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
  warnings.warn(

/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
  warnings.warn(
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
  warnings.warn(
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
  warnings.warn(

/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
  warnings.warn(

/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
  warnings.warn(
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
  warnings.warn(

/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
  warnings.warn(
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
  warnings.warn(

Linear Regression, basic elements

Video of Lecture.

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.

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.

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.

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.

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.

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

\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*}

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

\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}

we can rewrite our equations as

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

The above design matrix is called a Vandermonde matrix.

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{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*}

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

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

\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}

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?

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

\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*}

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.

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.

In [15]:

# 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)

	1	A	A^(2/3)	A^(-1/3)	1/A
A
1	1.0	1.0	1.000000	1.000000	1.000000
2	1.0	2.0	1.587401	0.793701	0.500000
3	1.0	3.0	2.080084	0.693361	0.333333
4	1.0	4.0	2.519842	0.629961	0.250000
5	1.0	5.0	2.924018	0.584804	0.200000
...	...	...	...	...	...
264	1.0	264.0	41.153106	0.155883	0.003788
265	1.0	265.0	41.256962	0.155687	0.003774
266	1.0	266.0	41.360688	0.155491	0.003759
269	1.0	269.0	41.671089	0.154911	0.003717
270	1.0	270.0	41.774300	0.154720	0.003704

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.

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.

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\},

Warning:

Output truncated. This notebook contains too many cells to display efficiently.