+
+
+
+
+
+
+
Meet the Pandas
+
+
+
+
+
+Another useful Python package is
+pandas, which is an open source library
+providing high-performance, easy-to-use data structures and data
+analysis tools for Python. pandas stands for panel data, a term borrowed from econometrics and is an efficient library for data analysis with an emphasis on tabular data.
+pandas has two major classes, the DataFrame class with two-dimensional data objects and tabular data organized in columns and the class Series with a focus on one-dimensional data objects. Both classes allow you to index data easily as we will see in the examples below.
+pandas allows you also to perform mathematical operations on the data, spanning from simple reshapings of vectors and matrices to statistical operations.
+
+
+The following simple example shows how we can, in an easy way make tables of our data. Here we define a data set which includes names, place of birth and date of birth, and displays the data in an easy to read way. We will see repeated use of pandas, in particular in connection with classification of data.
+
+
+
+
+
import pandas as pd
+from IPython.display import display
+data = {'First Name': ["Frodo", "Bilbo", "Aragorn II", "Samwise"],
+ 'Last Name': ["Baggins", "Baggins","Elessar","Gamgee"],
+ 'Place of birth': ["Shire", "Shire", "Eriador", "Shire"],
+ 'Date of Birth T.A.': [2968, 2890, 2931, 2980]
+ }
+data_pandas = pd.DataFrame(data)
+display(data_pandas)
+
+
+In the above we have imported pandas with the shorthand pd, the latter has become the standard way we import pandas. We make then a list of various variables
+and reorganize the aboves lists into a DataFrame and then print out a neat table with specific column labels as Name, place of birth and date of birth.
+Displaying these results, we see that the indices are given by the default numbers from zero to three.
+pandas is extremely flexible and we can easily change the above indices by defining a new type of indexing as
+
+
+
+
data_pandas = pd.DataFrame(data,index=['Frodo','Bilbo','Aragorn','Sam'])
+display(data_pandas)
+
+
+Thereafter we display the content of the row which begins with the index Aragorn
+
+
+
+
display(data_pandas.loc['Aragorn'])
+
+
+We can easily append data to this, for example
+
+
+
+
new_hobbit = {'First Name': ["Peregrin"],
+ 'Last Name': ["Took"],
+ 'Place of birth': ["Shire"],
+ 'Date of Birth T.A.': [2990]
+ }
+data_pandas=data_pandas.append(pd.DataFrame(new_hobbit, index=['Pippin']))
+display(data_pandas)
+
+
+Here are other examples where we use the DataFrame functionality to handle arrays, now with more interesting features for us, namely numbers. We set up a matrix
+of dimensionality \( 10\times 5 \) and compute the mean value and standard deviation of each column. Similarly, we can perform mathematial operations like squaring the matrix elements and many other operations.
+
+
+
+
import numpy as np
+import pandas as pd
+from IPython.display import display
+np.random.seed(100)
+# setting up a 10 x 5 matrix
+rows = 10
+cols = 5
+a = np.random.randn(rows,cols)
+df = pd.DataFrame(a)
+display(df)
+print(df.mean())
+print(df.std())
+display(df**2)
+
+
+Thereafter we can select specific columns only and plot final results
+
+
+
+
df.columns = ['First', 'Second', 'Third', 'Fourth', 'Fifth']
+df.index = np.arange(10)
+
+display(df)
+print(df['Second'].mean() )
+print(df.info())
+print(df.describe())
+
+from pylab import plt, mpl
+plt.style.use('seaborn')
+mpl.rcParams['font.family'] = 'serif'
+
+df.cumsum().plot(lw=2.0, figsize=(10,6))
+plt.show()
+
+
+df.plot.bar(figsize=(10,6), rot=15)
+plt.show()
+
+
+We can produce a \( 4\times 4 \) matrix
+
+
+
+
b = np.arange(16).reshape((4,4))
+print(b)
+df1 = pd.DataFrame(b)
+print(df1)
+
+
+and many other operations.
+
+
+The Series class is another important class included in
+pandas. You can view it as a specialization of DataFrame but where
+we have just a single column of data. It shares many of the same features as _DataFrame. As with DataFrame,
+most operations are vectorized, achieving thereby a high performance when dealing with computations of arrays, in particular labeled arrays.
+As we will see below it leads also to a very concice code close to the mathematical operations we may be interested in.
+For multidimensional arrays, we recommend strongly xarray. xarray has much of the same flexibility as pandas, but allows for the extension to higher dimensions than two. We will see examples later of the usage of both pandas and xarray.
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
Simple linear regression model using scikit-learn
+
+
+We start with perhaps our simplest possible example, using Scikit-Learn to perform linear regression analysis on a data set produced by us.
+
+
+What follows is a simple Python code where we have defined a function
+\( y \) in terms of the variable \( x \). Both are defined as vectors with \( 100 \) entries.
+The numbers in the vector \( \hat{x} \) are given
+by random numbers generated with a uniform distribution with entries
+\( x_i \in [0,1] \) (more about probability distribution functions
+later). These values are then used to define a function \( y(x) \)
+(tabulated again as a vector) with a linear dependence on \( x \) plus a
+random noise added via the normal distribution.
+
+
+The Numpy functions are imported used the import numpy as np
+statement and the random number generator for the uniform distribution
+is called using the function np.random.rand(), where we specificy
+that we want \( 100 \) random variables. Using Numpy we define
+automatically an array with the specified number of elements, \( 100 \) in
+our case. With the Numpy function randn() we can compute random
+numbers with the normal distribution (mean value \( \mu \) equal to zero and
+variance \( \sigma^2 \) set to one) and produce the values of \( y \) assuming a linear
+dependence as function of \( x \)
+
+$$
+y = 2x+N(0,1),
+$$
+
+
+where \( N(0,1) \) represents random numbers generated by the normal
+distribution. From Scikit-Learn we import then the
+LinearRegression functionality and make a prediction \( \tilde{y} =
+\alpha + \beta x \) using the function fit(x,y). We call the set of
+data \( (\hat{x},\hat{y}) \) for our training data. The Python package
+scikit-learn has also a functionality which extracts the above
+fitting parameters \( \alpha \) and \( \beta \) (see below). Later we will
+distinguish between training data and test data.
+
+
+For plotting we use the Python package
+matplotlib which produces publication
+quality figures. Feel free to explore the extensive
+gallery of examples. In
+this example we plot our original values of \( x \) and \( y \) as well as the
+prediction ypredict (\( \tilde{y} \)), which attempts at fitting our
+data with a straight line.
+
+
+The Python code follows here.
+
+
+
+
# Importing various packages
+import numpy as np
+import matplotlib.pyplot as plt
+from sklearn.linear_model import LinearRegression
+
+x = np.random.rand(100,1)
+y = 2*x+np.random.randn(100,1)
+linreg = LinearRegression()
+linreg.fit(x,y)
+xnew = np.array([[0],[1]])
+ypredict = linreg.predict(xnew)
+
+plt.plot(xnew, ypredict, "r-")
+plt.plot(x, y ,'ro')
+plt.axis([0,1.0,0, 5.0])
+plt.xlabel(r'$x$')
+plt.ylabel(r'$y$')
+plt.title(r'Simple Linear Regression')
+plt.show()
+
+
+This example serves several aims. It allows us to demonstrate several
+aspects of data analysis and later machine learning algorithms. The
+immediate visualization shows that our linear fit is not
+impressive. It goes through the data points, but there are many
+outliers which are not reproduced by our linear regression. We could
+now play around with this small program and change for example the
+factor in front of \( x \) and the normal distribution. Try to change the
+function \( y \) to
+
+$$
+y = 10x+0.01 \times N(0,1),
+$$
+
+
+where \( x \) is defined as before. Does the fit look better? Indeed, by
+reducing the role of the noise given by the normal distribution we see immediately that
+our linear prediction seemingly reproduces better the training
+set. However, this testing 'by the eye' is obviouly not satisfactory in the
+long run. Here we have only defined the training data and our model, and
+have not discussed a more rigorous approach to the cost function.
+
+
+We need more rigorous criteria in defining whether we have succeeded or
+not in modeling our training data. You will be surprised to see that
+many scientists seldomly venture beyond this 'by the eye' approach. A
+standard approach for the cost function is the so-called \( \chi^2 \)
+function (a variant of the mean-squared error (MSE))
+
+$$ \chi^2 = \frac{1}{n}
+\sum_{i=0}^{n-1}\frac{(y_i-\tilde{y}_i)^2}{\sigma_i^2},
+$$
+
+
+where \( \sigma_i^2 \) is the variance (to be defined later) of the entry
+\( y_i \). We may not know the explicit value of \( \sigma_i^2 \), it serves
+however the aim of scaling the equations and make the cost function
+dimensionless.
+
+
+Minimizing the cost function is a central aspect of
+our discussions to come. Finding its minima as function of the model
+parameters (\( \alpha \) and \( \beta \) in our case) will be a recurring
+theme in these series of lectures. Essentially all machine learning
+algorithms we will discuss center around the minimization of the
+chosen cost function. This depends in turn on our specific
+model for describing the data, a typical situation in supervised
+learning. Automatizing the search for the minima of the cost function is a
+central ingredient in all algorithms. Typical methods which are
+employed are various variants of gradient methods. These will be
+discussed in more detail later. Again, you'll be surprised to hear that
+many practitioners minimize the above function ''by the eye', popularly dubbed as
+'chi by the eye'. That is, change a parameter and see (visually and numerically) that
+the \( \chi^2 \) function becomes smaller.
+
+
+There are many ways to define the cost function. A simpler approach is to look at the relative difference between the training data and the predicted data, that is we define
+the relative error (why would we prefer the MSE instead of the relative error?) as
+
+$$
+\epsilon_{\mathrm{relative}}= \frac{\vert \hat{y} -\hat{\tilde{y}}\vert}{\vert \hat{y}\vert}.
+$$
+
+
+The squared cost function results in an arithmetic mean-unbiased
+estimator, and the absolute-value cost function results in a
+median-unbiased estimator (in the one-dimensional case, and a
+geometric median-unbiased estimator for the multi-dimensional
+case). The squared cost function has the disadvantage that it has the tendency
+to be dominated by outliers.
+
+
+We can modify easily the above Python code and plot the relative error instead
+
+
+
+
import numpy as np
+import matplotlib.pyplot as plt
+from sklearn.linear_model import LinearRegression
+
+x = np.random.rand(100,1)
+y = 5*x+0.01*np.random.randn(100,1)
+linreg = LinearRegression()
+linreg.fit(x,y)
+ypredict = linreg.predict(x)
+
+plt.plot(x, np.abs(ypredict-y)/abs(y), "ro")
+plt.axis([0,1.0,0.0, 0.5])
+plt.xlabel(r'$x$')
+plt.ylabel(r'$\epsilon_{\mathrm{relative}}$')
+plt.title(r'Relative error')
+plt.show()
+
+
+Depending on the parameter in front of the normal distribution, we may
+have a small or larger relative error. Try to play around with
+different training data sets and study (graphically) the value of the
+relative error.
+
+
+As mentioned above, Scikit-Learn has an impressive functionality.
+We can for example extract the values of \( \alpha \) and \( \beta \) and
+their error estimates, or the variance and standard deviation and many
+other properties from the statistical data analysis.
+
+
+Here we show an
+example of the functionality of Scikit-Learn.
+
+
+
+
import numpy as np
+import matplotlib.pyplot as plt
+from sklearn.linear_model import LinearRegression
+from sklearn.metrics import mean_squared_error, r2_score, mean_squared_log_error, mean_absolute_error
+
+x = np.random.rand(100,1)
+y = 2.0+ 5*x+0.5*np.random.randn(100,1)
+linreg = LinearRegression()
+linreg.fit(x,y)
+ypredict = linreg.predict(x)
+print('The intercept alpha: \n', linreg.intercept_)
+print('Coefficient beta : \n', linreg.coef_)
+# The mean squared error
+print("Mean squared error: %.2f" % mean_squared_error(y, ypredict))
+# Explained variance score: 1 is perfect prediction
+print('Variance score: %.2f' % r2_score(y, ypredict))
+# Mean squared log error
+print('Mean squared log error: %.2f' % mean_squared_log_error(y, ypredict) )
+# Mean absolute error
+print('Mean absolute error: %.2f' % mean_absolute_error(y, ypredict))
+plt.plot(x, ypredict, "r-")
+plt.plot(x, y ,'ro')
+plt.axis([0.0,1.0,1.5, 7.0])
+plt.xlabel(r'$x$')
+plt.ylabel(r'$y$')
+plt.title(r'Linear Regression fit ')
+plt.show()
+
+
+The function coef gives us the parameter \( \beta \) of our fit while intercept yields
+\( \alpha \). Depending on the constant in front of the normal distribution, we get values near or far from \( \alpha =2 \) and \( \beta =5 \). Try to play around with different parameters in front of the normal distribution. The function meansquarederror gives us the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error or loss defined as
+$$ MSE(\hat{y},\hat{\tilde{y}}) = \frac{1}{n}
+\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2,
+$$
+
+
+The smaller the value, the better the fit. Ideally we would like to
+have an MSE equal zero. The attentive reader has probably recognized
+this function as being similar to the \( \chi^2 \) function defined above.
+
+
+The r2score function computes \( R^2 \), the coefficient of
+determination. It provides a measure of how well future samples are
+likely to be predicted by the model. Best possible score is 1.0 and it
+can be negative (because the model can be arbitrarily worse). A
+constant model that always predicts the expected value of \( \hat{y} \),
+disregarding the input features, would get a \( R^2 \) score of \( 0.0 \).
+
+
+If \( \tilde{\hat{y}}_i \) is the predicted value of the \( i-th \) sample and \( y_i \) is the corresponding true value, then the score \( R^2 \) is defined as
+$$
+R^2(\hat{y}, \tilde{\hat{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2},
+$$
+
+where we have defined the mean value of \( \hat{y} \) as
+$$
+\bar{y} = \frac{1}{n} \sum_{i=0}^{n - 1} y_i.
+$$
+
+Another quantity taht we will meet again in our discussions of regression analysis is
+ the mean absolute error (MAE), a risk metric corresponding to the expected value of the absolute error loss or what we call the \( l1 \)-norm loss. In our discussion above we presented the relative error.
+The MAE is defined as follows
+$$
+\text{MAE}(\hat{y}, \hat{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n-1} \left| y_i - \tilde{y}_i \right|.
+$$
+
+We present the
+squared logarithmic (quadratic) error
+$$
+\text{MSLE}(\hat{y}, \hat{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n - 1} (\log_e (1 + y_i) - \log_e (1 + \tilde{y}_i) )^2,
+$$
+
+
+where \( \log_e (x) \) stands for the natural logarithm of \( x \). This error
+estimate is best to use when targets having exponential growth, such
+as population counts, average sales of a commodity over a span of
+years etc.
+
+
+Finally, another cost function is the Huber cost function used in robust regression.
+
+
+The rationale behind this possible cost function is its reduced
+sensitivity to outliers in the data set. In our discussions on
+dimensionality reduction and normalization of data we will meet other
+ways of dealing with outliers.
+
+
+The Huber cost function is defined as
+$$
+H_{\delta}(a)=\left\{\begin{array}\frac{1}{2}a^{2}&{\text{for }}|a|\leq \delta ,\\ \delta (|a|-\frac {1}{2}\delta ),&{\text{otherwise.}\end{array}\right.
+$$
+
+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))
+
+
+
To our real data: nuclear binding energies. Brief reminder on masses and binding energies
+
+
+Let us now dive into nuclear physics and remind ourselves briefly about some basic features about binding
+energies. A basic quantity which can be measured for the ground
+states of nuclei is the atomic mass \( M(N, Z) \) of the neutral atom with
+atomic mass number \( A \) and charge \( Z \). The number of neutrons is \( N \). There are indeed several sophisticated experiments worldwide which allow us to measure this quantity to high precision (parts per million even).
+
+
+Atomic masses are usually tabulated in terms of the mass excess defined by
+$$
+\Delta M(N, Z) = M(N, Z) - uA,
+$$
+
+where \( u \) is the Atomic Mass Unit
+$$
+u = M(^{12}\mathrm{C})/12 = 931.4940954(57) \hspace{0.1cm} \mathrm{MeV}/c^2.
+$$
+
+The nucleon masses are
+$$
+m_p = 1.00727646693(9)u,
+$$
+
+and
+$$
+m_n = 939.56536(8)\hspace{0.1cm} \mathrm{MeV}/c^2 = 1.0086649156(6)u.
+$$
+
+
+In the 2016 mass evaluation of by W.J.Huang, G.Audi, M.Wang, F.G.Kondev, S.Naimi and X.Xu
+there are data on masses and decays of 3437 nuclei.
+
+
+The nuclear binding energy is defined as the energy required to break
+up a given nucleus into its constituent parts of \( N \) neutrons and \( Z \)
+protons. In terms of the atomic masses \( M(N, Z) \) the binding energy is
+defined by
+
+$$
+BE(N, Z) = ZM_H c^2 + Nm_n c^2 - M(N, Z)c^2 ,
+$$
+
+where \( M_H \) is the mass of the hydrogen atom and \( m_n \) is the mass of the neutron.
+In terms of the mass excess the binding energy is given by
+$$
+BE(N, Z) = Z\Delta_H c^2 + N\Delta_n c^2 -\Delta(N, Z)c^2 ,
+$$
+
+where \( \Delta_H c^2 = 7.2890 \) MeV and \( \Delta_n c^2 = 8.0713 \) MeV.
+
+
+A popular and physically intuitive model which can be used to parametrize
+the experimental binding energies as function of \( A \), is the so-called
+liquid drop model. The ansatz is based on the following expression
+
+$$
+BE(N,Z) = a_1A-a_2A^{2/3}-a_3\frac{Z^2}{A^{1/3}}-a_4\frac{(N-Z)^2}{A},
+$$
+
+
+where \( A \) stands for the number of nucleons and the $a_i$s are parameters which are determined by a fit
+to the experimental data.
+
+
+To arrive at the above expression we have assumed that we can make the following assumptions:
+
+
+ - There is a volume term \( a_1A \) proportional with the number of nucleons (the energy is also an extensive quantity). When an assembly of nucleons of the same size is packed together into the smallest volume, each interior nucleon has a certain number of other nucleons in contact with it. This contribution is proportional to the volume.
+ - There is a surface energy term \( a_2A^{2/3} \). The assumption here is that a nucleon at the surface of a nucleus interacts with fewer other nucleons than one in the interior of the nucleus and hence its binding energy is less. This surface energy term takes that into account and is therefore negative and is proportional to the surface area.
+ - There is a Coulomb energy term \( a_3\frac{Z^2}{A^{1/3}} \). The electric repulsion between each pair of protons in a nucleus yields less binding.
+ - There is an asymmetry term \( a_4\frac{(N-Z)^2}{A} \). This term is associated with the Pauli exclusion principle and reflects the fact that the proton-neutron interaction is more attractive on the average than the neutron-neutron and proton-proton interactions.
+
+
+We could also add a so-called pairing term, which is a correction term that
+arises from the tendency of proton pairs and neutron pairs to
+occur. An even number of particles is more stable than an odd number.
+
+
Organizing our data
+
+
+Let us start with reading and organizing our data.
+We start with the compilation of masses and binding energies from 2016.
+After having downloaded this file to our own computer, we are now ready to read the file and start structuring our data.
+
+
+We start with preparing folders for storing our calculations and the data file over masses and binding energies. We import also various modules that we will find useful in order to present various Machine Learning methods. Here we focus mainly on the functionality of scikit-learn.
+
+
+
+
# Common imports
+import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+import sklearn.linear_model as skl
+from sklearn.model_selection import train_test_split
+from sklearn.metrics import mean_squared_error, r2_score, mean_absolute_error
+import os
+
+# Where to save the figures and data files
+PROJECT_ROOT_DIR = "Results"
+FIGURE_ID = "Results/FigureFiles"
+DATA_ID = "DataFiles/"
+
+if not os.path.exists(PROJECT_ROOT_DIR):
+ os.mkdir(PROJECT_ROOT_DIR)
+
+if not os.path.exists(FIGURE_ID):
+ os.makedirs(FIGURE_ID)
+
+if not os.path.exists(DATA_ID):
+ os.makedirs(DATA_ID)
+
+def image_path(fig_id):
+ return os.path.join(FIGURE_ID, fig_id)
+
+def data_path(dat_id):
+ return os.path.join(DATA_ID, dat_id)
+
+def save_fig(fig_id):
+ plt.savefig(image_path(fig_id) + ".png", format='png')
+
+infile = open(data_path("MassEval2016.dat"),'r')
+
+
+Before we proceed, we define also a function for making our plots. You can obviously avoid this and simply set up various matplotlib commands every time you need them. You may however find it convenient to collect all such commands in one function and simply call this function.
+
+
+
+
from pylab import plt, mpl
+plt.style.use('seaborn')
+mpl.rcParams['font.family'] = 'serif'
+
+def MakePlot(x,y, styles, labels, axlabels):
+ plt.figure(figsize=(10,6))
+ for i in range(len(x)):
+ plt.plot(x[i], y[i], styles[i], label = labels[i])
+ plt.xlabel(axlabels[0])
+ plt.ylabel(axlabels[1])
+ plt.legend(loc=0)
+
+
+Our next step is to read the data on experimental binding energies and
+reorganize them as functions of the mass number \( A \), the number of
+protons \( Z \) and neutrons \( N \) using pandas. Before we do this it is
+always useful (unless you have a binary file or other types of compressed
+data) to actually open the file and simply take a look at it!
+
+
+In particular, the program that outputs the final nuclear masses is written in Fortran with a specific format. It means that we need to figure out the format and which columns contain the data we are interested in. Pandas comes with a function that reads formatted output. After having admired the file, we are now ready to start massaging it with pandas. The file begins with some basic format information.
+
+
+
+
"""
+This is taken from the data file of the mass 2016 evaluation.
+All files are 3436 lines long with 124 character per line.
+ Headers are 39 lines long.
+ col 1 : Fortran character control: 1 = page feed 0 = line feed
+ format : a1,i3,i5,i5,i5,1x,a3,a4,1x,f13.5,f11.5,f11.3,f9.3,1x,a2,f11.3,f9.3,1x,i3,1x,f12.5,f11.5
+ These formats are reflected in the pandas widths variable below, see the statement
+ widths=(1,3,5,5,5,1,3,4,1,13,11,11,9,1,2,11,9,1,3,1,12,11,1),
+ Pandas has also a variable header, with length 39 in this case.
+"""
+
+
+The data we are interested in are in columns 2, 3, 4 and 11, giving us
+the number of neutrons, protons, mass numbers and binding energies,
+respectively. We add also for the sake of completeness the element name. The data are in fixed-width formatted lines and we will
+covert them into the pandas DataFrame structure.
+
+
+
+
+
# Read the experimental data with Pandas
+Masses = pd.read_fwf(infile, usecols=(2,3,4,6,11),
+ names=('N', 'Z', 'A', 'Element', 'Ebinding'),
+ widths=(1,3,5,5,5,1,3,4,1,13,11,11,9,1,2,11,9,1,3,1,12,11,1),
+ header=39,
+ index_col=False)
+
+# Extrapolated values are indicated by '#' in place of the decimal place, so
+# the Ebinding column won't be numeric. Coerce to float and drop these entries.
+Masses['Ebinding'] = pd.to_numeric(Masses['Ebinding'], errors='coerce')
+Masses = Masses.dropna()
+# Convert from keV to MeV.
+Masses['Ebinding'] /= 1000
+
+# Group the DataFrame by nucleon number, A.
+Masses = Masses.groupby('A')
+# Find the rows of the grouped DataFrame with the maximum binding energy.
+Masses = Masses.apply(lambda t: t[t.Ebinding==t.Ebinding.max()])
+
+
+We have now read in the data, grouped them according to the variables we are interested in.
+We see how easy it is to reorganize the data using pandas. If we
+were to do these operations in C/C++ or Fortran, we would have had to
+write various functions/subroutines which perform the above
+reorganizations for us. Having reorganized the data, we can now start
+to make some simple fits using both the functionalities in numpy and
+Scikit-Learn afterwards.
+
+
+Now we define five variables which contain
+the number of nucleons \( A \), the number of protons \( Z \) and the number of neutrons \( N \), the element name and finally the energies themselves.
+
+
+
+
A = Masses['A']
+Z = Masses['Z']
+N = Masses['N']
+Element = Masses['Element']
+Energies = Masses['Ebinding']
+print(Masses)
+
+
+The next step, and we will define this mathematically later, is to set up the so-called design matrix. We will throughout call this matrix \( \boldsymbol{X} \).
+It has dimensionality \( p\times n \), where \( n \) is the number of data points and \( p \) are the so-called predictors. In our case here they are given by the number of polynomials in \( A \) we wish to include in the fit.
+
+
+
+
# Now we set up the design matrix X
+X = np.zeros((len(A),5))
+X[:,0] = 1
+X[:,1] = A
+X[:,2] = A**(2.0/3.0)
+X[:,3] = A**(-1.0/3.0)
+X[:,4] = A**(-1.0)
+
+
+With scikitlearn we are now ready to use linear regression and fit our data.
+
+
+
+
clf = skl.LinearRegression().fit(X, Energies)
+fity = clf.predict(X)
+
+
+Pretty simple!
+Now we can print measures of how our fit is doing, the coefficients from the fits and plot the final fit together with our data.
+
+
+
+
# The mean squared error
+print("Mean squared error: %.2f" % mean_squared_error(Energies, fity))
+# Explained variance score: 1 is perfect prediction
+print('Variance score: %.2f' % r2_score(Energies, fity))
+# Mean absolute error
+print('Mean absolute error: %.2f' % mean_absolute_error(Energies, fity))
+print(clf.coef_, clf.intercept_)
+
+Masses['Eapprox'] = fity
+# Generate a plot comparing the experimental with the fitted values values.
+fig, ax = plt.subplots()
+ax.set_xlabel(r'$A = N + Z$')
+ax.set_ylabel(r'$E_\mathrm{bind}\,/\mathrm{MeV}$')
+ax.plot(Masses['A'], Masses['Ebinding'], alpha=0.7, lw=2,
+ label='Ame2016')
+ax.plot(Masses['A'], Masses['Eapprox'], alpha=0.7, lw=2, c='m',
+ label='Fit')
+ax.legend()
+save_fig("Masses2016")
+plt.show()
+
+
+
Seeing the wood for the trees
+
+
+As a teaser, let us now see how we can do this with decision trees using scikit-learn. Later we will switch to so-called random forests!
+
+
+
+
+
#Decision Tree Regression
+from sklearn.tree import DecisionTreeRegressor
+regr_1=DecisionTreeRegressor(max_depth=5)
+regr_2=DecisionTreeRegressor(max_depth=7)
+regr_3=DecisionTreeRegressor(max_depth=9)
+regr_1.fit(X, Energies)
+regr_2.fit(X, Energies)
+regr_3.fit(X, Energies)
+
+
+y_1 = regr_1.predict(X)
+y_2 = regr_2.predict(X)
+y_3=regr_3.predict(X)
+Masses['Eapprox'] = y_3
+# Plot the results
+plt.figure()
+plt.plot(A, Energies, color="blue", label="Data", linewidth=2)
+plt.plot(A, y_1, color="red", label="max_depth=5", linewidth=2)
+plt.plot(A, y_2, color="green", label="max_depth=7", linewidth=2)
+plt.plot(A, y_3, color="m", label="max_depth=9", linewidth=2)
+
+plt.xlabel("$A$")
+plt.ylabel("$E$[MeV]")
+plt.title("Decision Tree Regression")
+plt.legend()
+save_fig("Masses2016Trees")
+plt.show()
+print(Masses)
+print(np.mean( (Energies-y_1)**2))
+
+
+
And what about using neural networks?
+
+The
seaborn package allows us to visualize data in an efficient way. Note that we use
scikit-learn's multi-layer perceptron (or feed forward neural network)
+functionality.
+
+
+
+
from sklearn.neural_network import MLPRegressor
+from sklearn.metrics import accuracy_score
+import seaborn as sns
+
+X_train = X
+Y_train = Energies
+n_hidden_neurons = 100
+epochs = 100
+# store models for later use
+eta_vals = np.logspace(-5, 1, 7)
+lmbd_vals = np.logspace(-5, 1, 7)
+# store the models for later use
+DNN_scikit = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
+train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
+sns.set()
+for i, eta in enumerate(eta_vals):
+ for j, lmbd in enumerate(lmbd_vals):
+ dnn = MLPRegressor(hidden_layer_sizes=(n_hidden_neurons), activation='logistic',
+ alpha=lmbd, learning_rate_init=eta, max_iter=epochs)
+ dnn.fit(X_train, Y_train)
+ DNN_scikit[i][j] = dnn
+ train_accuracy[i][j] = dnn.score(X_train, Y_train)
+
+fig, ax = plt.subplots(figsize = (10, 10))
+sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis")
+ax.set_title("Training Accuracy")
+ax.set_ylabel("$\eta$")
+ax.set_xlabel("$\lambda$")
+plt.show()
+
+
+
A first summary
+
+
+The aim behind these introductory words was to present to you various
+Python libraries and their functionalities, in particular libraries like
+numpy, pandas, xarray and matplotlib and other that make our life much easier
+in handling various data sets and visualizing data.
+
+
+Furthermore,
+Scikit-Learn allows us with few lines of code to implement popular
+Machine Learning algorithms for supervised learning. Later we will meet Tensorflow, a powerful library for deep learning.
+Now it is time to dive more into the details of various methods. We will start with linear regression and try to take a deeper look at what it entails.
+
+
+
+
+
+
+
+
+