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.
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:
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.
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
For OSX users we recommend, after having installed Xcode, to install brew. Brew allows for a seamless installation of additional software via for example
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!
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
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.
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.
$$ \mathbf{A} = \begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \end{bmatrix}\qquad \mathbf{I} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$
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 | \( \sum_k a_{ik} a_{jk}^{ * } = \sum_k a_{ki}^{ * } a_{kj} = \delta_{ij} \) |
For an \( N\times N \) matrix \( \mathbf{A} \) the following properties are all equivalent
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)
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)
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)
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)
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)
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)
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)
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 $$ \hat{\Sigma} = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\ \sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\ \sigma_{zx} & \sigma_{zy} & \sigma_{zz} \end{bmatrix}, $$ where for example $$ \sigma_{xy} =\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})(y_i- \overline{y}). $$ The Numpy function np.cov calculates the covariance elements using the factor \( 1/(n-1) \) instead of \( 1/n \) since it assumes we do not have the exact mean values. The following simple function uses the np.vstack function which takes each vector of dimension \( 1\times n \) and produces a \( 3\times n \) matrix \( \hat{W} \) $$ \hat{W} = \begin{bmatrix} x_0 & y_0 & z_0 \\ x_1 & y_1 & z_1 \\ x_2 & y_2 & z_2 \\ \dots & \dots & \dots \\ x_{n-2} & y_{n-2} & z_{n-2} \\ x_{n-1} & y_{n-1} & z_{n-1} \end{bmatrix}, $$
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)
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()
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.
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!
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 $$ H_{\delta}(a)={\begin{cases}{\frac {1}{2}}{a^{2}}&{\text{for }}|a|\leq \delta ,\\\delta (|a|-{\frac {1}{2}}\delta ),&{\text{otherwise.}}\end{cases}}}. $$ 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))
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:
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()
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))
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()
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.