added dot files
This commit is contained in:
Morten Hjorth-Jensen
@@ -131,16 +131,96 @@ desired output of a system. Some of the most common tasks are:
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* Clustering: Data are divided into groups with certain common traits, without knowing the different groups beforehand. It is thus a form of unsupervised learning.
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The methods we cover have three main topics in common, irrespective of
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whether we deal with supervised or unsupervised learning. The first
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ingredient is normally our data set (which can be subdivided into
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training and test data), the second item is a model which is normally a
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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.
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The last ingredient is a so-called _cost_
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function which allows us to present an estimate on how good our model
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is in reproducing the data it is supposed to train.
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At the heart of basically all ML algorithms there are so-called minimization algorithms, often we end up with various variants of _gradient_ methods.
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The methods we cover have three main topics in common, irrespective of
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whether we deal with supervised or unsupervised learning.
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!bpop
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* The first ingredient is normally our data set (which can be subdivided into training, validation and test data). Many find the most difficult part of using Machine Learning to be the set up of your data in a meaningful way.
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* 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.
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* 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.
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!epop
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At the heart of basically all Machine Learning algorithms we will encounter so-called minimization or optimization algorithms. A large family of such methods are so-called _gradient methods_.
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=== A Frequentist approach to data analysis ===
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When you hear phrases like _predictions and estimations_ and
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_correlations and causations_, what do you think of? May be you think
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of the difference between classifying new data points and generating
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new data points.
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Or perhaps you consider that correlations represent some kind of symmetric statements like
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if $A$ is correlated with $B$, then $B$ is correlated with
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$A$. Causation on the other hand is directional, that is if $A$ causes $B$, $B$ does not
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necessarily cause $A$.
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These concepts are in some sense the difference between machine
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learning and statistics. In machine learning and prediction based
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tasks, we are often interested in developing algorithms that are
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capable of learning patterns from given data in an automated fashion,
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and then using these learned patterns to make predictions or
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assessments of newly given data. In many cases, our primary concern
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is the quality of the predictions or assessments, and we are less
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concerned about the underlying patterns that were learned in order
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to make these predictions.
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In machine learning we normally use "a so-called frequentist approach":"https://en.wikipedia.org/wiki/Frequentist_inference",
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where the aim is to make predictions and find correlations. We focus
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less on for example extracting a probability distribution function (PDF). The PDF can be
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used in turn to make estimations and find causations such as given $A$
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what is the likelihood of finding $B$.
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=== What is a good model? ===
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In science and engineering we often end up in situations where we want to infer (or learn) a
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quantitative model $M$ for a given set of sample points $\bm{X} \in [x_1, x_2,\dots x_N]$.
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As we will see repeatedely in these lectures, we could try to fit these data points to a model given by a
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straight line, or if we wish to be more sophisticated to a more complex
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function.
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The reason for inferring such a model is that it
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serves many useful purposes. On the one hand, the model can reveal information
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encoded in the data or underlying mechanisms from which the data were generated. For instance, we could discover important
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corelations that relate interesting physics interpretations.
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In addition, it can simplify the representation of the given data set and help
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us in making predictions about future data samples.
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A first important consideration to keep in mind is that inferring the *correct* model
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for a given data set is an elusive, if not impossible, task. The fundamental difficulty
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is that if we are not specific about what we mean by a *correct* model, there
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could easily be many different models that fit the given data set *equally well*.
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The central question is this: what leads us to say that a model is correct or
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optimal for a given data set? To make the model inference problem well posed, i.e.,
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to guarantee that there is a unique optimal model for the given data, we need to
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impose additional assumptions or restrictions on the class of models considered. To
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this end, we should not be looking for just any model that can describe the data.
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Instead, we should look for a _model_ $M$ that is the best among a restricted class
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of models. In addition, to make the model inference problem computationally
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tractable, we need to specify how restricted the class of models needs to be. A
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common strategy is to start
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with the simplest possible class of models that is just necessary to describe the data
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or solve the problem at hand. More precisely, the model class should be rich enough
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to contain at least one model that can fit the data to a desired accuracy and yet be
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restricted enough that it is relatively simple to find the best model for the given data.
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Thus, the most popular strategy is to start from the
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simplest class of models and increase the complexity of the models only when the
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simpler models become inadequate. For instance, if we work with a regression problem to fit a set of sample points, one
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may first try the simplest class of models, namely linear models, followed obviously by more complex models.
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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.
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@@ -1952,3 +2032,242 @@ print("R2 score for scaled data: {:.2f}".format(clf.score(X_test_scaled,y_test)
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!ec
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===== Exercises =====
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=== Exercise: Setting up various Python environments ===
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The first exercise here is of a mere technical art. We want you to have
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* git as a version control software and to establish a user account on a provider like GitHub. Other providers like GitLab etc are equally fine. You can also use the University of Oslo "GitHub facilities":"https://www.uio.no/tjenester/it/maskin/filer/versjonskontroll/github.html".
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* Install various Python packages
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We will make extensive use of Python as programming language and its
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myriad of available libraries. You will find
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IPython/Jupyter notebooks invaluable in your work. You can run _R_
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codes in the Jupyter/IPython notebooks, with the immediate benefit of
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visualizing your data. You can also use compiled languages like C++,
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Rust, Fortran etc if you prefer. The focus in these lectures will be
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on Python.
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If you have Python installed (we recommend Python3) and you feel
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pretty familiar with installing different packages, we recommend that
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you install the following Python packages via _pip_ as
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o pip install numpy scipy matplotlib ipython scikit-learn sympy pandas pillow
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For _Tensorflow_, we recommend following the instructions in the text of
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"Aurelien Geron, Hands‑On Machine Learning with Scikit‑Learn and TensorFlow, O'Reilly":"http://shop.oreilly.com/product/0636920052289.do"
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We will come back to _tensorflow_ later.
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For Python3, replace _pip_ with _pip3_.
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For OSX users we recommend, after having installed Xcode, to
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install _brew_. Brew allows for a seamless installation of additional
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software via for example
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o brew install python3
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For Linux users, with its variety of distributions like for example the widely popular Ubuntu distribution,
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you can use _pip_ as well and simply install Python as
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o sudo apt-get install python3 (or python for Python2.7)
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If you don't want to perform these operations separately and venture
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into the hassle of exploring how to set up dependencies and paths, we
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recommend two widely used distrubutions which set up all relevant
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dependencies for Python, namely
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* "Anaconda":"https://docs.anaconda.com/",
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which is an open source
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distribution of the Python and R programming languages for large-scale
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data processing, predictive analytics, and scientific computing, that
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aims to simplify package management and deployment. Package versions
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are managed by the package management system _conda_.
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* "Enthought canopy":"https://www.enthought.com/product/canopy/"
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is a Python
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distribution for scientific and analytic computing distribution and
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analysis environment, available for free and under a commercial
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license.
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We recommend using _Anaconda_ if you are not too familiar with setting paths in a terminal environment.
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=== Exercise: making your own data and exploring scikit-learn ===
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We will generate our own dataset for a function $y(x)$ where $x \in [0,1]$ and defined by random numbers computed with the uniform distribution. The function $y$ is a quadratic polynomial in $x$ with added stochastic noise according to the normal distribution $\cal {N}(0,1)$.
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The following simple Python instructions define our $x$ and $y$ values (with 100 data points).
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!bc pycod
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x = np.random.rand(100,1)
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y = 2.0+5*x*x+0.1*np.random.randn(100,1)
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!ec
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o Write your own code (following the examples under the "regression notes":"https://compphysics.github.io/MachineLearning/doc/LectureNotes/_build/html/chapter1.html") for computing the parametrization of the data set fitting a second-order polynomial.
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o Use thereafter _scikit-learn_ (see again the examples in the regression slides) and compare with your own code.
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o Using scikit-learn, compute also the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error defined as
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!bt
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\[ MSE(\hat{y},\hat{\tilde{y}}) = \frac{1}{n}
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\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2,
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\]
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!et
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and the $R^2$ score function.
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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
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!bt
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\[
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R^2(\hat{y}, \tilde{\hat{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2},
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\]
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!et
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where we have defined the mean value of $\hat{y}$ as
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!bt
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\[
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\bar{y} = \frac{1}{n} \sum_{i=0}^{n - 1} y_i.
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\]
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!et
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You can use the functionality included in scikit-learn. If you feel for it, you can use your own program and define functions which compute the above two functions.
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Discuss the meaning of these results. Try also to vary the coefficient in front of the added stochastic noise term and discuss the quality of the fits.
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!bsol
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The code here is an example of where we define our own design matrix and fit parameters $\beta$.
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!bc pycod
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import os
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import numpy as np
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import pandas as pd
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import matplotlib.pyplot as plt
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from sklearn.model_selection import train_test_split
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def save_fig(fig_id):
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plt.savefig(image_path(fig_id) + ".png", format='png')
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def R2(y_data, y_model):
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return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)
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def MSE(y_data,y_model):
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n = np.size(y_model)
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return np.sum((y_data-y_model)**2)/n
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x = np.random.rand(100)
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y = 2.0+5*x*x+0.1*np.random.randn(100)
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# The design matrix now as function of a given polynomial
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X = np.zeros((len(x),3))
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X[:,0] = 1.0
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X[:,1] = x
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X[:,2] = x**2
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# We split the data in test and training data
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X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
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# matrix inversion to find beta
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beta = np.linalg.inv(X_train.T @ X_train) @ X_train.T @ y_train
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print(beta)
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# and then make the prediction
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ytilde = X_train @ beta
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print("Training R2")
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print(R2(y_train,ytilde))
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print("Training MSE")
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print(MSE(y_train,ytilde))
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ypredict = X_test @ beta
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print("Test R2")
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print(R2(y_test,ypredict))
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print("Test MSE")
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print(MSE(y_test,ypredict))
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!ec
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!esol
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=== Exercise: Normalizing our data ===
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A much used approach before starting to train the data is to preprocess our
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data. Normally the data may need a rescaling and/or may be sensitive
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to extreme values. Scaling the data renders our inputs much more
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suitable for the algorithms we want to employ.
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_Scikit-Learn_ has several functions which allow us to rescale the
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data, normally resulting in much better results in terms of various
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accuracy scores. The _StandardScaler_ function in _Scikit-Learn_
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ensures that for each feature/predictor we study the mean value is
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zero and the variance is one (every column in the design/feature
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matrix). This scaling has the drawback that it does not ensure that
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we have a particular maximum or minimum in our data set. Another
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function included in _Scikit-Learn_ is the _MinMaxScaler_ which
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ensures that all features are exactly between $0$ and $1$. The
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The _Normalizer_ scales each data
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point such that the feature vector has a euclidean length of one. In other words, it
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projects a data point on the circle (or sphere in the case of higher dimensions) with a
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radius of 1. This means every data point is scaled by a different number (by the
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inverse of it’s length).
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This normalization is often used when only the direction (or angle) of the data matters,
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not the length of the feature vector.
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The _RobustScaler_ works similarly to the StandardScaler in that it
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ensures statistical properties for each feature that guarantee that
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they are on the same scale. However, the RobustScaler uses the median
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and quartiles, instead of mean and variance. This makes the
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RobustScaler ignore data points that are very different from the rest
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(like measurement errors). These odd data points are also called
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outliers, and might often lead to trouble for other scaling
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techniques.
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It also common to split the data in a _training_ set and a _testing_ set. A typical split is to use $80\%$ of the data for training and the rest
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for testing. This can be done as follows with our design matrix $\bm{X}$ and data $\bm{y}$ (remember to import _scikit-learn_)
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!bc pycod
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# split in training and test data
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X_train, X_test, y_train, y_test = train_test_split(X,y,test_size=0.2)
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!ec
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Then we can use the standard scaler to scale our data as
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!bc pycod
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scaler = StandardScaler()
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scaler.fit(X_train)
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X_train_scaled = scaler.transform(X_train)
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X_test_scaled = scaler.transform(X_test)
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!ec
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|
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In this exercise we want you to to compute the MSE for the training
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data and the test data as function of the complexity of a polynomial,
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that is the degree of a given polynomial. We want you also to compute the $R2$ score as function of the complexity of the model for both training data and test data. You should also run the calculation with and without scaling.
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One of
|
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the aims is to reproduce Figure 2.11 of "Hastie et al":"https://github.com/CompPhysics/MLErasmus/blob/master/doc/Textbooks/elementsstat.pdf".
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|
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|
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|
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Our data is defined by $x\in [-3,3]$ with a total of for example $100$ data points.
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!bc pycod
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np.random.seed()
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n = 100
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maxdegree = 14
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# Make data set.
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x = np.linspace(-3, 3, n).reshape(-1, 1)
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y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)
|
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!ec
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where $y$ is the function we want to fit with a given polynomial.
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!bsubex
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Write a first code which sets up a design matrix $X$ defined by a fifth-order polynomial. Scale your data and split it in training and test data.
|
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!esubex
|
||||
|
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!bsubex
|
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Perform an ordinary least squares and compute the means squared error and the $R2$ factor for the training data and the test data, with and without scaling.
|
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!esubex
|
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|
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!bsubex
|
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Add now a model which allows you to make polynomials up to degree $15$. Perform a standard OLS fitting of the training data and compute the MSE and $R2$ for the training and test data and plot both test and training data MSE and $R2$ as functions of the polynomial degree. Compare what you see with Figure 2.11 of Hastie et al. Comment your results. For which polynomial degree do you find an optimal MSE (smallest value)?
|
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!bsol
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Here you simply need to change the degree of the polynomial in the above code to $n=15$.
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!esol
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!esubex
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|
||||
(https://github.com/doconce/doconce/)
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-->
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<html>
|
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<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
|
||||
<meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
|
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<meta name="viewport" content="width=device-width, initial-scale=1.0" />
|
||||
<meta name="description" content="Week 35: Introduction to the course, Logistics and Practicalities">
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|
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<title>Week 35: Introduction to the course, Logistics and Practicalities</title>
|
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|
||||
<!-- Bootstrap style: bootstrap -->
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||||
<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
|
||||
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|
||||
<link href="https://netdna.bootstrapcdn.com/font-awesome/4.0.3/css/font-awesome.css" rel="stylesheet">
|
||||
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|
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|
||||
<style type="text/css">
|
||||
|
||||
/* Add scrollbar to dropdown menus in bootstrap navigation bar */
|
||||
.dropdown-menu {
|
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height: auto;
|
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max-height: 400px;
|
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overflow-x: hidden;
|
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}
|
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|
||||
/* Adds an invisible element before each target to offset for the navigation
|
||||
bar */
|
||||
.anchor::before {
|
||||
content:"";
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display:block;
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height:50px; /* fixed header height for style bootstrap */
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|
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|
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|
||||
|
||||
</head>
|
||||
|
||||
<!-- tocinfo
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||||
{'highest level': 2,
|
||||
'sections': [('Overview of first week', 2, None, 'overview-of-first-week'),
|
||||
('Reading Recommendations', 2, None, 'reading-recommendations'),
|
||||
('Thursday August 26', 2, None, 'thursday-august-26'),
|
||||
('Lectures and ComputerLab', 2, None, 'lectures-and-computerlab'),
|
||||
('Announcement', 2, None, 'announcement'),
|
||||
('Course Format', 2, None, 'course-format'),
|
||||
('Teachers', 2, None, 'teachers'),
|
||||
('Deadlines for projects (tentative)',
|
||||
2,
|
||||
None,
|
||||
'deadlines-for-projects-tentative'),
|
||||
('Recommended textbooks', 2, None, 'recommended-textbooks'),
|
||||
('Prerequisites', 2, None, 'prerequisites'),
|
||||
('Learning outcomes', 2, None, 'learning-outcomes'),
|
||||
('Topics covered in this course: Statistical analysis and '
|
||||
'optimization of data',
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||||
2,
|
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None,
|
||||
'topics-covered-in-this-course-statistical-analysis-and-optimization-of-data'),
|
||||
('Topics covered in this course: Machine Learning',
|
||||
2,
|
||||
None,
|
||||
'topics-covered-in-this-course-machine-learning'),
|
||||
('Extremely useful tools, strongly recommended',
|
||||
2,
|
||||
None,
|
||||
'extremely-useful-tools-strongly-recommended'),
|
||||
('Other courses on Data science and Machine Learning at UiO',
|
||||
2,
|
||||
None,
|
||||
'other-courses-on-data-science-and-machine-learning-at-uio'),
|
||||
('Introduction', 2, None, 'introduction'),
|
||||
('What is Machine Learning?',
|
||||
2,
|
||||
None,
|
||||
'what-is-machine-learning'),
|
||||
('Types of Machine Learning',
|
||||
2,
|
||||
None,
|
||||
'types-of-machine-learning'),
|
||||
('Essential elements of ML', 2, None, 'essential-elements-of-ml'),
|
||||
('An optimization/minimization problem',
|
||||
2,
|
||||
None,
|
||||
'an-optimization-minimization-problem'),
|
||||
('A Frequentist approach to data analysis',
|
||||
2,
|
||||
None,
|
||||
'a-frequentist-approach-to-data-analysis'),
|
||||
('What is a good model?', 2, None, 'what-is-a-good-model'),
|
||||
('What is a good model? Can we define it?',
|
||||
2,
|
||||
None,
|
||||
'what-is-a-good-model-can-we-define-it'),
|
||||
('Software and needed installations',
|
||||
2,
|
||||
None,
|
||||
'software-and-needed-installations'),
|
||||
('Python installers', 2, None, 'python-installers'),
|
||||
('Useful Python libraries', 2, None, 'useful-python-libraries'),
|
||||
('Installing R, C++, cython or Julia',
|
||||
2,
|
||||
None,
|
||||
'installing-r-c-cython-or-julia'),
|
||||
('Installing R, C++, cython, Numba etc',
|
||||
2,
|
||||
None,
|
||||
'installing-r-c-cython-numba-etc'),
|
||||
('Numpy examples and Important Matrix and vector handling '
|
||||
'packages',
|
||||
2,
|
||||
None,
|
||||
'numpy-examples-and-important-matrix-and-vector-handling-packages'),
|
||||
('Basic Matrix Features', 2, None, 'basic-matrix-features'),
|
||||
('Some famous Matrices', 3, None, 'some-famous-matrices'),
|
||||
('More Basic Matrix Features',
|
||||
3,
|
||||
None,
|
||||
'more-basic-matrix-features'),
|
||||
('Numpy and arrays', 2, None, 'numpy-and-arrays'),
|
||||
('Matrices in Python', 2, None, 'matrices-in-python'),
|
||||
('Meet the Pandas', 2, None, 'meet-the-pandas'),
|
||||
('Friday August 27', 2, None, 'friday-august-27'),
|
||||
('Reading Data and fitting', 2, None, 'reading-data-and-fitting'),
|
||||
('Friday August 27', 2, None, 'friday-august-27'),
|
||||
('Simple linear regression model using _scikit-learn_',
|
||||
3,
|
||||
None,
|
||||
'simple-linear-regression-model-using-_scikit-learn_'),
|
||||
('To our real data: nuclear binding energies. Brief reminder on '
|
||||
'masses and binding energies',
|
||||
3,
|
||||
None,
|
||||
'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'),
|
||||
('Organizing our data', 3, None, 'organizing-our-data'),
|
||||
('Seeing the wood for the trees',
|
||||
3,
|
||||
None,
|
||||
'seeing-the-wood-for-the-trees'),
|
||||
('And what about using neural networks?',
|
||||
3,
|
||||
None,
|
||||
'and-what-about-using-neural-networks'),
|
||||
('A first summary', 2, None, 'a-first-summary'),
|
||||
('Exercises for week 36', 2, None, 'exercises-for-week-36'),
|
||||
('Exercise 1: Setting up various Python environments',
|
||||
2,
|
||||
None,
|
||||
'exercise-1-setting-up-various-python-environments'),
|
||||
('Exercise 2: making your own data and exploring scikit-learn',
|
||||
2,
|
||||
None,
|
||||
'exercise-2-making-your-own-data-and-exploring-scikit-learn'),
|
||||
('Exercise 3: Normalizing our data',
|
||||
2,
|
||||
None,
|
||||
'exercise-3-normalizing-our-data')]}
|
||||
end of tocinfo -->
|
||||
|
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|
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<!-- navigation toc: --> <li><a href="._week34-bs001.html#overview-of-first-week" style="font-size: 80%;"><b>Overview of first week</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs009.html#recommended-textbooks" style="font-size: 80%;"><b>Recommended textbooks</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs012.html#topics-covered-in-this-course-statistical-analysis-and-optimization-of-data" style="font-size: 80%;"><b>Topics covered in this course: Statistical analysis and optimization of data</b></a></li>
|
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<!-- navigation toc: --> <li><a href="._week34-bs013.html#topics-covered-in-this-course-machine-learning" style="font-size: 80%;"><b>Topics covered in this course: Machine Learning</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs014.html#extremely-useful-tools-strongly-recommended" style="font-size: 80%;"><b>Extremely useful tools, strongly recommended</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs015.html#other-courses-on-data-science-and-machine-learning-at-uio" style="font-size: 80%;"><b>Other courses on Data science and Machine Learning at UiO</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs016.html#introduction" style="font-size: 80%;"><b>Introduction</b></a></li>
|
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<!-- navigation toc: --> <li><a href="._week34-bs017.html#what-is-machine-learning" style="font-size: 80%;"><b>What is Machine Learning?</b></a></li>
|
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<!-- navigation toc: --> <li><a href="._week34-bs018.html#types-of-machine-learning" style="font-size: 80%;"><b>Types of Machine Learning</b></a></li>
|
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<!-- navigation toc: --> <li><a href="._week34-bs019.html#essential-elements-of-ml" style="font-size: 80%;"><b>Essential elements of ML</b></a></li>
|
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<!-- navigation toc: --> <li><a href="._week34-bs020.html#an-optimization-minimization-problem" style="font-size: 80%;"><b>An optimization/minimization problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs021.html#a-frequentist-approach-to-data-analysis" style="font-size: 80%;"><b>A Frequentist approach to data analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs022.html#what-is-a-good-model" style="font-size: 80%;"><b>What is a good model?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs023.html#what-is-a-good-model-can-we-define-it" style="font-size: 80%;"><b>What is a good model? Can we define it?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs024.html#software-and-needed-installations" style="font-size: 80%;"><b>Software and needed installations</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs025.html#python-installers" style="font-size: 80%;"><b>Python installers</b></a></li>
|
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<!-- navigation toc: --> <li><a href="._week34-bs026.html#useful-python-libraries" style="font-size: 80%;"><b>Useful Python libraries</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs027.html#installing-r-c-cython-or-julia" style="font-size: 80%;"><b>Installing R, C++, cython or Julia</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs028.html#installing-r-c-cython-numba-etc" style="font-size: 80%;"><b>Installing R, C++, cython, Numba etc</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs029.html#numpy-examples-and-important-matrix-and-vector-handling-packages" style="font-size: 80%;"><b>Numpy examples and Important Matrix and vector handling packages</b></a></li>
|
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<!-- navigation toc: --> <li><a href="._week34-bs030.html#basic-matrix-features" style="font-size: 80%;"><b>Basic Matrix Features</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs031.html#some-famous-matrices" style="font-size: 80%;"> Some famous Matrices</a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs032.html#more-basic-matrix-features" style="font-size: 80%;"> More Basic Matrix Features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs033.html#numpy-and-arrays" style="font-size: 80%;"><b>Numpy and arrays</b></a></li>
|
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<!-- navigation toc: --> <li><a href="#matrices-in-python" style="font-size: 80%;"><b>Matrices in Python</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs035.html#meet-the-pandas" style="font-size: 80%;"><b>Meet the Pandas</b></a></li>
|
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<!-- navigation toc: --> <li><a href="._week34-bs038.html#friday-august-27" style="font-size: 80%;"><b>Friday August 27</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs037.html#reading-data-and-fitting" style="font-size: 80%;"><b>Reading Data and fitting</b></a></li>
|
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<!-- navigation toc: --> <li><a href="._week34-bs038.html#friday-august-27" style="font-size: 80%;"><b>Friday August 27</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs039.html#simple-linear-regression-model-using-_scikit-learn_" style="font-size: 80%;"> Simple linear regression model using <b>scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs039.html#to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies" style="font-size: 80%;"> To our real data: nuclear binding energies. Brief reminder on masses and binding energies</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week34-bs039.html#organizing-our-data" style="font-size: 80%;"> Organizing our data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs039.html#seeing-the-wood-for-the-trees" style="font-size: 80%;"> Seeing the wood for the trees</a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs039.html#and-what-about-using-neural-networks" style="font-size: 80%;"> And what about using neural networks?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs039.html#a-first-summary" style="font-size: 80%;"><b>A first summary</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs040.html#exercises-for-week-36" style="font-size: 80%;"><b>Exercises for week 36</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs040.html#exercise-1-setting-up-various-python-environments" style="font-size: 80%;"><b>Exercise 1: Setting up various Python environments</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs040.html#exercise-2-making-your-own-data-and-exploring-scikit-learn" style="font-size: 80%;"><b>Exercise 2: making your own data and exploring scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs040.html#exercise-3-normalizing-our-data" style="font-size: 80%;"><b>Exercise 3: Normalizing our data</b></a></li>
|
||||
|
||||
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|
||||
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<a name="part0034"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="matrices-in-python" class="anchor">Matrices in Python </h2>
|
||||
|
||||
<p>
|
||||
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)
|
||||
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
||||
A <span style="color: #666666">=</span> np<span style="color: #666666">.</span>log(np<span style="color: #666666">.</span>array([ [<span style="color: #666666">4.0</span>, <span style="color: #666666">7.0</span>, <span style="color: #666666">8.0</span>], [<span style="color: #666666">3.0</span>, <span style="color: #666666">10.0</span>, <span style="color: #666666">11.0</span>], [<span style="color: #666666">4.0</span>, <span style="color: #666666">5.0</span>, <span style="color: #666666">7.0</span>] ]))
|
||||
<span style="color: #008000">print</span>(A)
|
||||
</pre></div>
|
||||
<p>
|
||||
If we use the <b>shape</b> 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
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
||||
A <span style="color: #666666">=</span> np<span style="color: #666666">.</span>log(np<span style="color: #666666">.</span>array([ [<span style="color: #666666">4.0</span>, <span style="color: #666666">7.0</span>, <span style="color: #666666">8.0</span>], [<span style="color: #666666">3.0</span>, <span style="color: #666666">10.0</span>, <span style="color: #666666">11.0</span>], [<span style="color: #666666">4.0</span>, <span style="color: #666666">5.0</span>, <span style="color: #666666">7.0</span>] ]))
|
||||
<span style="color: #408080; font-style: italic"># print the first column, row-major order and elements start with 0</span>
|
||||
<span style="color: #008000">print</span>(A[:,<span style="color: #666666">0</span>])
|
||||
</pre></div>
|
||||
<p>
|
||||
We can continue this was by printing out other columns or rows. The example here prints out the second column
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
||||
A <span style="color: #666666">=</span> np<span style="color: #666666">.</span>log(np<span style="color: #666666">.</span>array([ [<span style="color: #666666">4.0</span>, <span style="color: #666666">7.0</span>, <span style="color: #666666">8.0</span>], [<span style="color: #666666">3.0</span>, <span style="color: #666666">10.0</span>, <span style="color: #666666">11.0</span>], [<span style="color: #666666">4.0</span>, <span style="color: #666666">5.0</span>, <span style="color: #666666">7.0</span>] ]))
|
||||
<span style="color: #408080; font-style: italic"># print the first column, row-major order and elements start with 0</span>
|
||||
<span style="color: #008000">print</span>(A[<span style="color: #666666">1</span>,:])
|
||||
</pre></div>
|
||||
<p>
|
||||
Numpy contains many other functionalities that allow us to slice, subdivide etc etc arrays. We strongly recommend that you look up the <a href="http://www.numpy.org/" target="_self">Numpy website for more details</a>. Useful functions when defining a matrix are the <b>np.zeros</b> function which declares a matrix of a given dimension and sets all elements to zero
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
||||
n <span style="color: #666666">=</span> <span style="color: #666666">10</span>
|
||||
<span style="color: #408080; font-style: italic"># define a matrix of dimension 10 x 10 and set all elements to zero</span>
|
||||
A <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros( (n, n) )
|
||||
<span style="color: #008000">print</span>(A)
|
||||
</pre></div>
|
||||
<p>
|
||||
or initializing all elements to
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
||||
n <span style="color: #666666">=</span> <span style="color: #666666">10</span>
|
||||
<span style="color: #408080; font-style: italic"># define a matrix of dimension 10 x 10 and set all elements to one</span>
|
||||
A <span style="color: #666666">=</span> np<span style="color: #666666">.</span>ones( (n, n) )
|
||||
<span style="color: #008000">print</span>(A)
|
||||
</pre></div>
|
||||
<p>
|
||||
or as unitarily distributed random numbers (see the material on random number generators in the statistics part)
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
||||
n <span style="color: #666666">=</span> <span style="color: #666666">10</span>
|
||||
<span style="color: #408080; font-style: italic"># define a matrix of dimension 10 x 10 and set all elements to random numbers with x \in [0, 1]</span>
|
||||
A <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(n, n)
|
||||
<span style="color: #008000">print</span>(A)
|
||||
</pre></div>
|
||||
<p>
|
||||
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 <b>np.cov</b> 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 <b>np.vstack</b> function which takes each vector of dimension \( 1\times n \) and produces a \( 3\times n \) matrix \( \hat{W} \)
|
||||
$$
|
||||
\hat{W} = \begin{bmatrix} x_0 & y_0 & z_0 \\
|
||||
x_1 & y_1 & z_1 \\
|
||||
x_2 & y_2 & z_2 \\
|
||||
\dots & \dots & \dots \\
|
||||
x_{n-2} & y_{n-2} & z_{n-2} \\
|
||||
x_{n-1} & y_{n-1} & z_{n-1}
|
||||
\end{bmatrix},
|
||||
$$
|
||||
|
||||
<p>
|
||||
which in turn is converted into into the \( 3\times 3 \) covariance matrix
|
||||
\( \hat{\Sigma} \) via the Numpy function <b>np.cov()</b>. We note that we can also calculate
|
||||
the mean value of each set of samples \( \hat{x} \) etc using the Numpy
|
||||
function <b>np.mean(x)</b>. We can also extract the eigenvalues of the
|
||||
covariance matrix through the <b>np.linalg.eig()</b> function.
|
||||
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #408080; font-style: italic"># Importing various packages</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
||||
|
||||
n <span style="color: #666666">=</span> <span style="color: #666666">100</span>
|
||||
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>normal(size<span style="color: #666666">=</span>n)
|
||||
<span style="color: #008000">print</span>(np<span style="color: #666666">.</span>mean(x))
|
||||
y <span style="color: #666666">=</span> <span style="color: #666666">4+3*</span>x<span style="color: #666666">+</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>normal(size<span style="color: #666666">=</span>n)
|
||||
<span style="color: #008000">print</span>(np<span style="color: #666666">.</span>mean(y))
|
||||
z <span style="color: #666666">=</span> x<span style="color: #666666">**3+</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>normal(size<span style="color: #666666">=</span>n)
|
||||
<span style="color: #008000">print</span>(np<span style="color: #666666">.</span>mean(z))
|
||||
W <span style="color: #666666">=</span> np<span style="color: #666666">.</span>vstack((x, y, z))
|
||||
Sigma <span style="color: #666666">=</span> np<span style="color: #666666">.</span>cov(W)
|
||||
<span style="color: #008000">print</span>(Sigma)
|
||||
Eigvals, Eigvecs <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>eig(Sigma)
|
||||
<span style="color: #008000">print</span>(Eigvals)
|
||||
</pre></div>
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">scipy</span> <span style="color: #008000; font-weight: bold">import</span> sparse
|
||||
eye <span style="color: #666666">=</span> np<span style="color: #666666">.</span>eye(<span style="color: #666666">4</span>)
|
||||
<span style="color: #008000">print</span>(eye)
|
||||
sparse_mtx <span style="color: #666666">=</span> sparse<span style="color: #666666">.</span>csr_matrix(eye)
|
||||
<span style="color: #008000">print</span>(sparse_mtx)
|
||||
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(<span style="color: #666666">-10</span>,<span style="color: #666666">10</span>,<span style="color: #666666">100</span>)
|
||||
y <span style="color: #666666">=</span> np<span style="color: #666666">.</span>sin(x)
|
||||
plt<span style="color: #666666">.</span>plot(x,y,marker<span style="color: #666666">=</span><span style="color: #BA2121">'x'</span>)
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
</pre></div>
|
||||
<p>
|
||||
<p>
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|
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|
||||
|
||||
</head>
|
||||
|
||||
<!-- tocinfo
|
||||
{'highest level': 2,
|
||||
'sections': [('Overview of first week', 2, None, 'overview-of-first-week'),
|
||||
('Reading Recommendations', 2, None, 'reading-recommendations'),
|
||||
('Thursday August 26', 2, None, 'thursday-august-26'),
|
||||
('Lectures and ComputerLab', 2, None, 'lectures-and-computerlab'),
|
||||
('Announcement', 2, None, 'announcement'),
|
||||
('Course Format', 2, None, 'course-format'),
|
||||
('Teachers', 2, None, 'teachers'),
|
||||
('Deadlines for projects (tentative)',
|
||||
2,
|
||||
None,
|
||||
'deadlines-for-projects-tentative'),
|
||||
('Recommended textbooks', 2, None, 'recommended-textbooks'),
|
||||
('Prerequisites', 2, None, 'prerequisites'),
|
||||
('Learning outcomes', 2, None, 'learning-outcomes'),
|
||||
('Topics covered in this course: Statistical analysis and '
|
||||
'optimization of data',
|
||||
2,
|
||||
None,
|
||||
'topics-covered-in-this-course-statistical-analysis-and-optimization-of-data'),
|
||||
('Topics covered in this course: Machine Learning',
|
||||
2,
|
||||
None,
|
||||
'topics-covered-in-this-course-machine-learning'),
|
||||
('Extremely useful tools, strongly recommended',
|
||||
2,
|
||||
None,
|
||||
'extremely-useful-tools-strongly-recommended'),
|
||||
('Other courses on Data science and Machine Learning at UiO',
|
||||
2,
|
||||
None,
|
||||
'other-courses-on-data-science-and-machine-learning-at-uio'),
|
||||
('Introduction', 2, None, 'introduction'),
|
||||
('What is Machine Learning?',
|
||||
2,
|
||||
None,
|
||||
'what-is-machine-learning'),
|
||||
('Types of Machine Learning',
|
||||
2,
|
||||
None,
|
||||
'types-of-machine-learning'),
|
||||
('Essential elements of ML', 2, None, 'essential-elements-of-ml'),
|
||||
('An optimization/minimization problem',
|
||||
2,
|
||||
None,
|
||||
'an-optimization-minimization-problem'),
|
||||
('A Frequentist approach to data analysis',
|
||||
2,
|
||||
None,
|
||||
'a-frequentist-approach-to-data-analysis'),
|
||||
('What is a good model?', 2, None, 'what-is-a-good-model'),
|
||||
('What is a good model? Can we define it?',
|
||||
2,
|
||||
None,
|
||||
'what-is-a-good-model-can-we-define-it'),
|
||||
('Software and needed installations',
|
||||
2,
|
||||
None,
|
||||
'software-and-needed-installations'),
|
||||
('Python installers', 2, None, 'python-installers'),
|
||||
('Useful Python libraries', 2, None, 'useful-python-libraries'),
|
||||
('Installing R, C++, cython or Julia',
|
||||
2,
|
||||
None,
|
||||
'installing-r-c-cython-or-julia'),
|
||||
('Installing R, C++, cython, Numba etc',
|
||||
2,
|
||||
None,
|
||||
'installing-r-c-cython-numba-etc'),
|
||||
('Numpy examples and Important Matrix and vector handling '
|
||||
'packages',
|
||||
2,
|
||||
None,
|
||||
'numpy-examples-and-important-matrix-and-vector-handling-packages'),
|
||||
('Basic Matrix Features', 2, None, 'basic-matrix-features'),
|
||||
('Some famous Matrices', 3, None, 'some-famous-matrices'),
|
||||
('More Basic Matrix Features',
|
||||
3,
|
||||
None,
|
||||
'more-basic-matrix-features'),
|
||||
('Numpy and arrays', 2, None, 'numpy-and-arrays'),
|
||||
('Matrices in Python', 2, None, 'matrices-in-python'),
|
||||
('Meet the Pandas', 2, None, 'meet-the-pandas'),
|
||||
('Friday August 27', 2, None, 'friday-august-27'),
|
||||
('Reading Data and fitting', 2, None, 'reading-data-and-fitting'),
|
||||
('Friday August 27', 2, None, 'friday-august-27'),
|
||||
('Simple linear regression model using _scikit-learn_',
|
||||
3,
|
||||
None,
|
||||
'simple-linear-regression-model-using-_scikit-learn_'),
|
||||
('To our real data: nuclear binding energies. Brief reminder on '
|
||||
'masses and binding energies',
|
||||
3,
|
||||
None,
|
||||
'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'),
|
||||
('Organizing our data', 3, None, 'organizing-our-data'),
|
||||
('Seeing the wood for the trees',
|
||||
3,
|
||||
None,
|
||||
'seeing-the-wood-for-the-trees'),
|
||||
('And what about using neural networks?',
|
||||
3,
|
||||
None,
|
||||
'and-what-about-using-neural-networks'),
|
||||
('A first summary', 2, None, 'a-first-summary'),
|
||||
('Exercises for week 36', 2, None, 'exercises-for-week-36'),
|
||||
('Exercise 1: Setting up various Python environments',
|
||||
2,
|
||||
None,
|
||||
'exercise-1-setting-up-various-python-environments'),
|
||||
('Exercise 2: making your own data and exploring scikit-learn',
|
||||
2,
|
||||
None,
|
||||
'exercise-2-making-your-own-data-and-exploring-scikit-learn'),
|
||||
('Exercise 3: Normalizing our data',
|
||||
2,
|
||||
None,
|
||||
'exercise-3-normalizing-our-data')]}
|
||||
end of tocinfo -->
|
||||
|
||||
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|
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<!-- navigation toc: --> <li><a href="._week34-bs001.html#overview-of-first-week" style="font-size: 80%;"><b>Overview of first week</b></a></li>
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||||
<!-- navigation toc: --> <li><a href="._week34-bs002.html#reading-recommendations" style="font-size: 80%;"><b>Reading Recommendations</b></a></li>
|
||||
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||||
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<!-- navigation toc: --> <li><a href="._week34-bs009.html#recommended-textbooks" style="font-size: 80%;"><b>Recommended textbooks</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs010.html#prerequisites" style="font-size: 80%;"><b>Prerequisites</b></a></li>
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||||
<!-- navigation toc: --> <li><a href="._week34-bs011.html#learning-outcomes" style="font-size: 80%;"><b>Learning outcomes</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs012.html#topics-covered-in-this-course-statistical-analysis-and-optimization-of-data" style="font-size: 80%;"><b>Topics covered in this course: Statistical analysis and optimization of data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs013.html#topics-covered-in-this-course-machine-learning" style="font-size: 80%;"><b>Topics covered in this course: Machine Learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs014.html#extremely-useful-tools-strongly-recommended" style="font-size: 80%;"><b>Extremely useful tools, strongly recommended</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs015.html#other-courses-on-data-science-and-machine-learning-at-uio" style="font-size: 80%;"><b>Other courses on Data science and Machine Learning at UiO</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs016.html#introduction" style="font-size: 80%;"><b>Introduction</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs017.html#what-is-machine-learning" style="font-size: 80%;"><b>What is Machine Learning?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs018.html#types-of-machine-learning" style="font-size: 80%;"><b>Types of Machine Learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs019.html#essential-elements-of-ml" style="font-size: 80%;"><b>Essential elements of ML</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs020.html#an-optimization-minimization-problem" style="font-size: 80%;"><b>An optimization/minimization problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs021.html#a-frequentist-approach-to-data-analysis" style="font-size: 80%;"><b>A Frequentist approach to data analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs022.html#what-is-a-good-model" style="font-size: 80%;"><b>What is a good model?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs023.html#what-is-a-good-model-can-we-define-it" style="font-size: 80%;"><b>What is a good model? Can we define it?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs024.html#software-and-needed-installations" style="font-size: 80%;"><b>Software and needed installations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs025.html#python-installers" style="font-size: 80%;"><b>Python installers</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs026.html#useful-python-libraries" style="font-size: 80%;"><b>Useful Python libraries</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs027.html#installing-r-c-cython-or-julia" style="font-size: 80%;"><b>Installing R, C++, cython or Julia</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs028.html#installing-r-c-cython-numba-etc" style="font-size: 80%;"><b>Installing R, C++, cython, Numba etc</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs029.html#numpy-examples-and-important-matrix-and-vector-handling-packages" style="font-size: 80%;"><b>Numpy examples and Important Matrix and vector handling packages</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs030.html#basic-matrix-features" style="font-size: 80%;"><b>Basic Matrix Features</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs031.html#some-famous-matrices" style="font-size: 80%;"> Some famous Matrices</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs032.html#more-basic-matrix-features" style="font-size: 80%;"> More Basic Matrix Features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs033.html#numpy-and-arrays" style="font-size: 80%;"><b>Numpy and arrays</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs034.html#matrices-in-python" style="font-size: 80%;"><b>Matrices in Python</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="#meet-the-pandas" style="font-size: 80%;"><b>Meet the Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs038.html#friday-august-27" style="font-size: 80%;"><b>Friday August 27</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs037.html#reading-data-and-fitting" style="font-size: 80%;"><b>Reading Data and fitting</b></a></li>
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||||
<!-- navigation toc: --> <li><a href="._week34-bs038.html#friday-august-27" style="font-size: 80%;"><b>Friday August 27</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs039.html#simple-linear-regression-model-using-_scikit-learn_" style="font-size: 80%;"> Simple linear regression model using <b>scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs039.html#to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies" style="font-size: 80%;"> To our real data: nuclear binding energies. Brief reminder on masses and binding energies</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs039.html#organizing-our-data" style="font-size: 80%;"> Organizing our data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs039.html#seeing-the-wood-for-the-trees" style="font-size: 80%;"> Seeing the wood for the trees</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs039.html#and-what-about-using-neural-networks" style="font-size: 80%;"> And what about using neural networks?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs039.html#a-first-summary" style="font-size: 80%;"><b>A first summary</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs040.html#exercises-for-week-36" style="font-size: 80%;"><b>Exercises for week 36</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs040.html#exercise-1-setting-up-various-python-environments" style="font-size: 80%;"><b>Exercise 1: Setting up various Python environments</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs040.html#exercise-2-making-your-own-data-and-exploring-scikit-learn" style="font-size: 80%;"><b>Exercise 2: making your own data and exploring scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs040.html#exercise-3-normalizing-our-data" style="font-size: 80%;"><b>Exercise 3: Normalizing our data</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
</ul>
|
||||
</div>
|
||||
</div>
|
||||
</div> <!-- end of navigation bar -->
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|
||||
<div class="container">
|
||||
|
||||
<p> </p><p> </p><p> </p> <!-- add vertical space -->
|
||||
|
||||
<a name="part0035"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="meet-the-pandas" class="anchor">Meet the Pandas </h2>
|
||||
|
||||
<p>
|
||||
<br /><br /><center><p><img src="fig/pandas.jpg" align="bottom" width=600></p></center><br /><br />
|
||||
|
||||
<p>
|
||||
Another useful Python package is
|
||||
<a href="https://pandas.pydata.org/" target="_self">pandas</a>, which is an open source library
|
||||
providing high-performance, easy-to-use data structures and data
|
||||
analysis tools for Python. <b>pandas</b> stands for panel data, a term borrowed from econometrics and is an efficient library for data analysis with an emphasis on tabular data.
|
||||
<b>pandas</b> has two major classes, the <b>DataFrame</b> class with two-dimensional data objects and tabular data organized in columns and the class <b>Series</b> with a focus on one-dimensional data objects. Both classes allow you to index data easily as we will see in the examples below.
|
||||
<b>pandas</b> allows you also to perform mathematical operations on the data, spanning from simple reshapings of vectors and matrices to statistical operations.
|
||||
|
||||
<p>
|
||||
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 <b>pandas</b>, in particular in connection with classification of data.
|
||||
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">pandas</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">pd</span>
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">IPython.display</span> <span style="color: #008000; font-weight: bold">import</span> display
|
||||
data <span style="color: #666666">=</span> {<span style="color: #BA2121">'First Name'</span>: [<span style="color: #BA2121">"Frodo"</span>, <span style="color: #BA2121">"Bilbo"</span>, <span style="color: #BA2121">"Aragorn II"</span>, <span style="color: #BA2121">"Samwise"</span>],
|
||||
<span style="color: #BA2121">'Last Name'</span>: [<span style="color: #BA2121">"Baggins"</span>, <span style="color: #BA2121">"Baggins"</span>,<span style="color: #BA2121">"Elessar"</span>,<span style="color: #BA2121">"Gamgee"</span>],
|
||||
<span style="color: #BA2121">'Place of birth'</span>: [<span style="color: #BA2121">"Shire"</span>, <span style="color: #BA2121">"Shire"</span>, <span style="color: #BA2121">"Eriador"</span>, <span style="color: #BA2121">"Shire"</span>],
|
||||
<span style="color: #BA2121">'Date of Birth T.A.'</span>: [<span style="color: #666666">2968</span>, <span style="color: #666666">2890</span>, <span style="color: #666666">2931</span>, <span style="color: #666666">2980</span>]
|
||||
}
|
||||
data_pandas <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>DataFrame(data)
|
||||
display(data_pandas)
|
||||
</pre></div>
|
||||
<p>
|
||||
In the above we have imported <b>pandas</b> with the shorthand <b>pd</b>, the latter has become the standard way we import <b>pandas</b>. We make then a list of various variables
|
||||
and reorganize the aboves lists into a <b>DataFrame</b> and then print out a neat table with specific column labels as <em>Name</em>, <em>place of birth</em> and <em>date of birth</em>.
|
||||
Displaying these results, we see that the indices are given by the default numbers from zero to three.
|
||||
<b>pandas</b> is extremely flexible and we can easily change the above indices by defining a new type of indexing as
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span>data_pandas <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>DataFrame(data,index<span style="color: #666666">=</span>[<span style="color: #BA2121">'Frodo'</span>,<span style="color: #BA2121">'Bilbo'</span>,<span style="color: #BA2121">'Aragorn'</span>,<span style="color: #BA2121">'Sam'</span>])
|
||||
display(data_pandas)
|
||||
</pre></div>
|
||||
<p>
|
||||
Thereafter we display the content of the row which begins with the index <b>Aragorn</b>
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span>display(data_pandas<span style="color: #666666">.</span>loc[<span style="color: #BA2121">'Aragorn'</span>])
|
||||
</pre></div>
|
||||
<p>
|
||||
We can easily append data to this, for example
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span>new_hobbit <span style="color: #666666">=</span> {<span style="color: #BA2121">'First Name'</span>: [<span style="color: #BA2121">"Peregrin"</span>],
|
||||
<span style="color: #BA2121">'Last Name'</span>: [<span style="color: #BA2121">"Took"</span>],
|
||||
<span style="color: #BA2121">'Place of birth'</span>: [<span style="color: #BA2121">"Shire"</span>],
|
||||
<span style="color: #BA2121">'Date of Birth T.A.'</span>: [<span style="color: #666666">2990</span>]
|
||||
}
|
||||
data_pandas<span style="color: #666666">=</span>data_pandas<span style="color: #666666">.</span>append(pd<span style="color: #666666">.</span>DataFrame(new_hobbit, index<span style="color: #666666">=</span>[<span style="color: #BA2121">'Pippin'</span>]))
|
||||
display(data_pandas)
|
||||
</pre></div>
|
||||
<p>
|
||||
Here are other examples where we use the <b>DataFrame</b> 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.
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">pandas</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">pd</span>
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">IPython.display</span> <span style="color: #008000; font-weight: bold">import</span> display
|
||||
np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">100</span>)
|
||||
<span style="color: #408080; font-style: italic"># setting up a 10 x 5 matrix</span>
|
||||
rows <span style="color: #666666">=</span> <span style="color: #666666">10</span>
|
||||
cols <span style="color: #666666">=</span> <span style="color: #666666">5</span>
|
||||
a <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(rows,cols)
|
||||
df <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>DataFrame(a)
|
||||
display(df)
|
||||
<span style="color: #008000">print</span>(df<span style="color: #666666">.</span>mean())
|
||||
<span style="color: #008000">print</span>(df<span style="color: #666666">.</span>std())
|
||||
display(df<span style="color: #666666">**2</span>)
|
||||
</pre></div>
|
||||
<p>
|
||||
Thereafter we can select specific columns only and plot final results
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span>df<span style="color: #666666">.</span>columns <span style="color: #666666">=</span> [<span style="color: #BA2121">'First'</span>, <span style="color: #BA2121">'Second'</span>, <span style="color: #BA2121">'Third'</span>, <span style="color: #BA2121">'Fourth'</span>, <span style="color: #BA2121">'Fifth'</span>]
|
||||
df<span style="color: #666666">.</span>index <span style="color: #666666">=</span> np<span style="color: #666666">.</span>arange(<span style="color: #666666">10</span>)
|
||||
|
||||
display(df)
|
||||
<span style="color: #008000">print</span>(df[<span style="color: #BA2121">'Second'</span>]<span style="color: #666666">.</span>mean() )
|
||||
<span style="color: #008000">print</span>(df<span style="color: #666666">.</span>info())
|
||||
<span style="color: #008000">print</span>(df<span style="color: #666666">.</span>describe())
|
||||
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">pylab</span> <span style="color: #008000; font-weight: bold">import</span> plt, mpl
|
||||
plt<span style="color: #666666">.</span>style<span style="color: #666666">.</span>use(<span style="color: #BA2121">'seaborn'</span>)
|
||||
mpl<span style="color: #666666">.</span>rcParams[<span style="color: #BA2121">'font.family'</span>] <span style="color: #666666">=</span> <span style="color: #BA2121">'serif'</span>
|
||||
|
||||
df<span style="color: #666666">.</span>cumsum()<span style="color: #666666">.</span>plot(lw<span style="color: #666666">=2.0</span>, figsize<span style="color: #666666">=</span>(<span style="color: #666666">10</span>,<span style="color: #666666">6</span>))
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
|
||||
|
||||
df<span style="color: #666666">.</span>plot<span style="color: #666666">.</span>bar(figsize<span style="color: #666666">=</span>(<span style="color: #666666">10</span>,<span style="color: #666666">6</span>), rot<span style="color: #666666">=15</span>)
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
</pre></div>
|
||||
<p>
|
||||
We can produce a \( 4\times 4 \) matrix
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span>b <span style="color: #666666">=</span> np<span style="color: #666666">.</span>arange(<span style="color: #666666">16</span>)<span style="color: #666666">.</span>reshape((<span style="color: #666666">4</span>,<span style="color: #666666">4</span>))
|
||||
<span style="color: #008000">print</span>(b)
|
||||
df1 <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>DataFrame(b)
|
||||
<span style="color: #008000">print</span>(df1)
|
||||
</pre></div>
|
||||
<p>
|
||||
and many other operations.
|
||||
|
||||
<p>
|
||||
The <b>Series</b> class is another important class included in
|
||||
<b>pandas</b>. You can view it as a specialization of <b>DataFrame</b> but where
|
||||
we have just a single column of data. It shares many of the same features as _DataFrame. As with <b>DataFrame</b>,
|
||||
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 <a href="http://xarray.pydata.org/en/stable/" target="_self">xarray</a>. <b>xarray</b> has much of the same flexibility as <b>pandas</b>, but allows for the extension to higher dimensions than two. We will see examples later of the usage of both <b>pandas</b> and <b>xarray</b>.
|
||||
|
||||
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|
||||
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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||||
<!-- navigation toc: --> <li><a href="._week34-bs001.html#overview-of-first-week" style="font-size: 80%;"><b>Overview of first week</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs002.html#reading-recommendations" style="font-size: 80%;"><b>Reading Recommendations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs003.html#thursday-august-26" style="font-size: 80%;"><b>Thursday August 26</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs004.html#lectures-and-computerlab" style="font-size: 80%;"><b>Lectures and ComputerLab</b></a></li>
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||||
<!-- navigation toc: --> <li><a href="._week34-bs005.html#announcement" style="font-size: 80%;"><b>Announcement</b></a></li>
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||||
<!-- navigation toc: --> <li><a href="._week34-bs006.html#course-format" style="font-size: 80%;"><b>Course Format</b></a></li>
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||||
<!-- navigation toc: --> <li><a href="._week34-bs007.html#teachers" style="font-size: 80%;"><b>Teachers</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs008.html#deadlines-for-projects-tentative" style="font-size: 80%;"><b>Deadlines for projects (tentative)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs009.html#recommended-textbooks" style="font-size: 80%;"><b>Recommended textbooks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs010.html#prerequisites" style="font-size: 80%;"><b>Prerequisites</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs011.html#learning-outcomes" style="font-size: 80%;"><b>Learning outcomes</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs012.html#topics-covered-in-this-course-statistical-analysis-and-optimization-of-data" style="font-size: 80%;"><b>Topics covered in this course: Statistical analysis and optimization of data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs013.html#topics-covered-in-this-course-machine-learning" style="font-size: 80%;"><b>Topics covered in this course: Machine Learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs014.html#extremely-useful-tools-strongly-recommended" style="font-size: 80%;"><b>Extremely useful tools, strongly recommended</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs015.html#other-courses-on-data-science-and-machine-learning-at-uio" style="font-size: 80%;"><b>Other courses on Data science and Machine Learning at UiO</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs016.html#introduction" style="font-size: 80%;"><b>Introduction</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs017.html#what-is-machine-learning" style="font-size: 80%;"><b>What is Machine Learning?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs018.html#types-of-machine-learning" style="font-size: 80%;"><b>Types of Machine Learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs019.html#essential-elements-of-ml" style="font-size: 80%;"><b>Essential elements of ML</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs020.html#an-optimization-minimization-problem" style="font-size: 80%;"><b>An optimization/minimization problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs021.html#a-frequentist-approach-to-data-analysis" style="font-size: 80%;"><b>A Frequentist approach to data analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs022.html#what-is-a-good-model" style="font-size: 80%;"><b>What is a good model?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs023.html#what-is-a-good-model-can-we-define-it" style="font-size: 80%;"><b>What is a good model? Can we define it?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs024.html#software-and-needed-installations" style="font-size: 80%;"><b>Software and needed installations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs025.html#python-installers" style="font-size: 80%;"><b>Python installers</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs026.html#useful-python-libraries" style="font-size: 80%;"><b>Useful Python libraries</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs027.html#installing-r-c-cython-or-julia" style="font-size: 80%;"><b>Installing R, C++, cython or Julia</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs028.html#installing-r-c-cython-numba-etc" style="font-size: 80%;"><b>Installing R, C++, cython, Numba etc</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs029.html#numpy-examples-and-important-matrix-and-vector-handling-packages" style="font-size: 80%;"><b>Numpy examples and Important Matrix and vector handling packages</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs030.html#basic-matrix-features" style="font-size: 80%;"><b>Basic Matrix Features</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs031.html#some-famous-matrices" style="font-size: 80%;"> Some famous Matrices</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs032.html#more-basic-matrix-features" style="font-size: 80%;"> More Basic Matrix Features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs033.html#numpy-and-arrays" style="font-size: 80%;"><b>Numpy and arrays</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs034.html#matrices-in-python" style="font-size: 80%;"><b>Matrices in Python</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs035.html#meet-the-pandas" style="font-size: 80%;"><b>Meet the Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs038.html#friday-august-27" style="font-size: 80%;"><b>Friday August 27</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs037.html#reading-data-and-fitting" style="font-size: 80%;"><b>Reading Data and fitting</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs038.html#friday-august-27" style="font-size: 80%;"><b>Friday August 27</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs039.html#simple-linear-regression-model-using-_scikit-learn_" style="font-size: 80%;"> Simple linear regression model using <b>scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs039.html#to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies" style="font-size: 80%;"> To our real data: nuclear binding energies. Brief reminder on masses and binding energies</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs039.html#organizing-our-data" style="font-size: 80%;"> Organizing our data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs039.html#seeing-the-wood-for-the-trees" style="font-size: 80%;"> Seeing the wood for the trees</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs039.html#and-what-about-using-neural-networks" style="font-size: 80%;"> And what about using neural networks?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs039.html#a-first-summary" style="font-size: 80%;"><b>A first summary</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs040.html#exercises-for-week-36" style="font-size: 80%;"><b>Exercises for week 36</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs040.html#exercise-1-setting-up-various-python-environments" style="font-size: 80%;"><b>Exercise 1: Setting up various Python environments</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs040.html#exercise-2-making-your-own-data-and-exploring-scikit-learn" style="font-size: 80%;"><b>Exercise 2: making your own data and exploring scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs040.html#exercise-3-normalizing-our-data" style="font-size: 80%;"><b>Exercise 3: Normalizing our data</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
</ul>
|
||||
</div>
|
||||
</div>
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||||
<p> </p><p> </p><p> </p> <!-- add vertical space -->
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||||
|
||||
<a name="part0036"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="friday-august-27" class="anchor">Friday August 27 </h2>
|
||||
|
||||
<p>
|
||||
<a href="https://www.uio.no/studier/emner/matnat/fys/FYS-STK3155/h20/forelesningsvideoer/LectureAug21.mp4?vrtx=view-as-webpage" target="_self">Video of Lecture from fall 2020</a> and <a href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/NotesAugust21.pdf" target="_self">Handwritten notes</a>
|
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|
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|
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<li><a href="._week34-bs028.html">29</a></li>
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@@ -0,0 +1,340 @@
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<a class="navbar-brand" href="week34-bs.html">Week 35: Introduction to the course, Logistics and Practicalities</a>
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
|
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<ul class="dropdown-menu">
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs001.html#overview-of-first-week" style="font-size: 80%;"><b>Overview of first week</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs002.html#reading-recommendations" style="font-size: 80%;"><b>Reading Recommendations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs003.html#thursday-august-26" style="font-size: 80%;"><b>Thursday August 26</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs004.html#lectures-and-computerlab" style="font-size: 80%;"><b>Lectures and ComputerLab</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs005.html#announcement" style="font-size: 80%;"><b>Announcement</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs006.html#course-format" style="font-size: 80%;"><b>Course Format</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs007.html#teachers" style="font-size: 80%;"><b>Teachers</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs008.html#deadlines-for-projects-tentative" style="font-size: 80%;"><b>Deadlines for projects (tentative)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs009.html#recommended-textbooks" style="font-size: 80%;"><b>Recommended textbooks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs010.html#prerequisites" style="font-size: 80%;"><b>Prerequisites</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs011.html#learning-outcomes" style="font-size: 80%;"><b>Learning outcomes</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs012.html#topics-covered-in-this-course-statistical-analysis-and-optimization-of-data" style="font-size: 80%;"><b>Topics covered in this course: Statistical analysis and optimization of data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs013.html#topics-covered-in-this-course-machine-learning" style="font-size: 80%;"><b>Topics covered in this course: Machine Learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs014.html#extremely-useful-tools-strongly-recommended" style="font-size: 80%;"><b>Extremely useful tools, strongly recommended</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs015.html#other-courses-on-data-science-and-machine-learning-at-uio" style="font-size: 80%;"><b>Other courses on Data science and Machine Learning at UiO</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs016.html#introduction" style="font-size: 80%;"><b>Introduction</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs017.html#what-is-machine-learning" style="font-size: 80%;"><b>What is Machine Learning?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs018.html#types-of-machine-learning" style="font-size: 80%;"><b>Types of Machine Learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs019.html#essential-elements-of-ml" style="font-size: 80%;"><b>Essential elements of ML</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs020.html#an-optimization-minimization-problem" style="font-size: 80%;"><b>An optimization/minimization problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs021.html#a-frequentist-approach-to-data-analysis" style="font-size: 80%;"><b>A Frequentist approach to data analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs022.html#what-is-a-good-model" style="font-size: 80%;"><b>What is a good model?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs023.html#what-is-a-good-model-can-we-define-it" style="font-size: 80%;"><b>What is a good model? Can we define it?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs024.html#software-and-needed-installations" style="font-size: 80%;"><b>Software and needed installations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs025.html#python-installers" style="font-size: 80%;"><b>Python installers</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs026.html#useful-python-libraries" style="font-size: 80%;"><b>Useful Python libraries</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs027.html#installing-r-c-cython-or-julia" style="font-size: 80%;"><b>Installing R, C++, cython or Julia</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs028.html#installing-r-c-cython-numba-etc" style="font-size: 80%;"><b>Installing R, C++, cython, Numba etc</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs029.html#numpy-examples-and-important-matrix-and-vector-handling-packages" style="font-size: 80%;"><b>Numpy examples and Important Matrix and vector handling packages</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs030.html#basic-matrix-features" style="font-size: 80%;"><b>Basic Matrix Features</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs031.html#some-famous-matrices" style="font-size: 80%;"> Some famous Matrices</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs032.html#more-basic-matrix-features" style="font-size: 80%;"> More Basic Matrix Features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs033.html#numpy-and-arrays" style="font-size: 80%;"><b>Numpy and arrays</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs034.html#matrices-in-python" style="font-size: 80%;"><b>Matrices in Python</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs035.html#meet-the-pandas" style="font-size: 80%;"><b>Meet the Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs038.html#friday-august-27" style="font-size: 80%;"><b>Friday August 27</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="#reading-data-and-fitting" style="font-size: 80%;"><b>Reading Data and fitting</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs038.html#friday-august-27" style="font-size: 80%;"><b>Friday August 27</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs039.html#simple-linear-regression-model-using-_scikit-learn_" style="font-size: 80%;"> Simple linear regression model using <b>scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs039.html#to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies" style="font-size: 80%;"> To our real data: nuclear binding energies. Brief reminder on masses and binding energies</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs039.html#organizing-our-data" style="font-size: 80%;"> Organizing our data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs039.html#seeing-the-wood-for-the-trees" style="font-size: 80%;"> Seeing the wood for the trees</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs039.html#and-what-about-using-neural-networks" style="font-size: 80%;"> And what about using neural networks?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs039.html#a-first-summary" style="font-size: 80%;"><b>A first summary</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs040.html#exercises-for-week-36" style="font-size: 80%;"><b>Exercises for week 36</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs040.html#exercise-1-setting-up-various-python-environments" style="font-size: 80%;"><b>Exercise 1: Setting up various Python environments</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs040.html#exercise-2-making-your-own-data-and-exploring-scikit-learn" style="font-size: 80%;"><b>Exercise 2: making your own data and exploring scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs040.html#exercise-3-normalizing-our-data" style="font-size: 80%;"><b>Exercise 3: Normalizing our data</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
</ul>
|
||||
</div>
|
||||
</div>
|
||||
</div> <!-- end of navigation bar -->
|
||||
|
||||
<div class="container">
|
||||
|
||||
<p> </p><p> </p><p> </p> <!-- add vertical space -->
|
||||
|
||||
<a name="part0037"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="reading-data-and-fitting" class="anchor">Reading Data and fitting </h2>
|
||||
|
||||
<p>
|
||||
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 <b>design
|
||||
matrix</b> (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).
|
||||
|
||||
<p>
|
||||
In handling various input formats, as discussed above, we will mainly stay with <b>pandas</b>,
|
||||
a Python package which allows us, in a seamless and painless way, to
|
||||
deal with a multitude of formats, from standard <b>csv</b> (comma separated
|
||||
values) files, via <b>excel</b>, <b>html</b> to <b>hdf5</b> formats. With <b>pandas</b>
|
||||
and the <b>DataFrame</b> and <b>Series</b> 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.
|
||||
|
||||
<p>
|
||||
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 merely as an example here of a data set.
|
||||
|
||||
<p>
|
||||
We will show some of the
|
||||
strengths of packages like <b>Scikit-Learn</b> 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.
|
||||
|
||||
<p>
|
||||
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!
|
||||
|
||||
<p>
|
||||
<p>
|
||||
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<li><a href="._week34-bs036.html">«</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week34-bs001.html#overview-of-first-week" style="font-size: 80%;"><b>Overview of first week</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs002.html#reading-recommendations" style="font-size: 80%;"><b>Reading Recommendations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs003.html#thursday-august-26" style="font-size: 80%;"><b>Thursday August 26</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs004.html#lectures-and-computerlab" style="font-size: 80%;"><b>Lectures and ComputerLab</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs005.html#announcement" style="font-size: 80%;"><b>Announcement</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs006.html#course-format" style="font-size: 80%;"><b>Course Format</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs007.html#teachers" style="font-size: 80%;"><b>Teachers</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs008.html#deadlines-for-projects-tentative" style="font-size: 80%;"><b>Deadlines for projects (tentative)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs009.html#recommended-textbooks" style="font-size: 80%;"><b>Recommended textbooks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs010.html#prerequisites" style="font-size: 80%;"><b>Prerequisites</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs011.html#learning-outcomes" style="font-size: 80%;"><b>Learning outcomes</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs012.html#topics-covered-in-this-course-statistical-analysis-and-optimization-of-data" style="font-size: 80%;"><b>Topics covered in this course: Statistical analysis and optimization of data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs013.html#topics-covered-in-this-course-machine-learning" style="font-size: 80%;"><b>Topics covered in this course: Machine Learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs014.html#extremely-useful-tools-strongly-recommended" style="font-size: 80%;"><b>Extremely useful tools, strongly recommended</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs015.html#other-courses-on-data-science-and-machine-learning-at-uio" style="font-size: 80%;"><b>Other courses on Data science and Machine Learning at UiO</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs016.html#introduction" style="font-size: 80%;"><b>Introduction</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs017.html#what-is-machine-learning" style="font-size: 80%;"><b>What is Machine Learning?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs018.html#types-of-machine-learning" style="font-size: 80%;"><b>Types of Machine Learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs019.html#essential-elements-of-ml" style="font-size: 80%;"><b>Essential elements of ML</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs020.html#an-optimization-minimization-problem" style="font-size: 80%;"><b>An optimization/minimization problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs021.html#a-frequentist-approach-to-data-analysis" style="font-size: 80%;"><b>A Frequentist approach to data analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs022.html#what-is-a-good-model" style="font-size: 80%;"><b>What is a good model?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs023.html#what-is-a-good-model-can-we-define-it" style="font-size: 80%;"><b>What is a good model? Can we define it?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs024.html#software-and-needed-installations" style="font-size: 80%;"><b>Software and needed installations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs025.html#python-installers" style="font-size: 80%;"><b>Python installers</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs026.html#useful-python-libraries" style="font-size: 80%;"><b>Useful Python libraries</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs027.html#installing-r-c-cython-or-julia" style="font-size: 80%;"><b>Installing R, C++, cython or Julia</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs028.html#installing-r-c-cython-numba-etc" style="font-size: 80%;"><b>Installing R, C++, cython, Numba etc</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs029.html#numpy-examples-and-important-matrix-and-vector-handling-packages" style="font-size: 80%;"><b>Numpy examples and Important Matrix and vector handling packages</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs030.html#basic-matrix-features" style="font-size: 80%;"><b>Basic Matrix Features</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs031.html#some-famous-matrices" style="font-size: 80%;"> Some famous Matrices</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs032.html#more-basic-matrix-features" style="font-size: 80%;"> More Basic Matrix Features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs033.html#numpy-and-arrays" style="font-size: 80%;"><b>Numpy and arrays</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs034.html#matrices-in-python" style="font-size: 80%;"><b>Matrices in Python</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs035.html#meet-the-pandas" style="font-size: 80%;"><b>Meet the Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="#friday-august-27" style="font-size: 80%;"><b>Friday August 27</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs037.html#reading-data-and-fitting" style="font-size: 80%;"><b>Reading Data and fitting</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="#friday-august-27" style="font-size: 80%;"><b>Friday August 27</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs039.html#simple-linear-regression-model-using-_scikit-learn_" style="font-size: 80%;"> Simple linear regression model using <b>scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs039.html#to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies" style="font-size: 80%;"> To our real data: nuclear binding energies. Brief reminder on masses and binding energies</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs039.html#organizing-our-data" style="font-size: 80%;"> Organizing our data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs039.html#seeing-the-wood-for-the-trees" style="font-size: 80%;"> Seeing the wood for the trees</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs039.html#and-what-about-using-neural-networks" style="font-size: 80%;"> And what about using neural networks?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs039.html#a-first-summary" style="font-size: 80%;"><b>A first summary</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs040.html#exercises-for-week-36" style="font-size: 80%;"><b>Exercises for week 36</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs040.html#exercise-1-setting-up-various-python-environments" style="font-size: 80%;"><b>Exercise 1: Setting up various Python environments</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs040.html#exercise-2-making-your-own-data-and-exploring-scikit-learn" style="font-size: 80%;"><b>Exercise 2: making your own data and exploring scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs040.html#exercise-3-normalizing-our-data" style="font-size: 80%;"><b>Exercise 3: Normalizing our data</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
</ul>
|
||||
</div>
|
||||
</div>
|
||||
</div> <!-- end of navigation bar -->
|
||||
|
||||
<div class="container">
|
||||
|
||||
<p> </p><p> </p><p> </p> <!-- add vertical space -->
|
||||
|
||||
<a name="part0038"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="friday-august-27" class="anchor">Friday August 27 </h2>
|
||||
|
||||
<p>
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
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<li><a href="._week34-bs037.html">«</a></li>
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<li><a href="._week34-bs000.html">1</a></li>
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<li><a href="">...</a></li>
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<li><a href="._week34-bs030.html">31</a></li>
|
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<li><a href="._week34-bs032.html">33</a></li>
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<li><a href="._week34-bs034.html">35</a></li>
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<li><a href="._week34-bs035.html">36</a></li>
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<li><a href="._week34-bs036.html">37</a></li>
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<li><a href="._week34-bs037.html">38</a></li>
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<li class="active"><a href="._week34-bs038.html">39</a></li>
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<li><a href="._week34-bs039.html">40</a></li>
|
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<li><a href="._week34-bs040.html">41</a></li>
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@@ -0,0 +1,975 @@
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||||
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||||
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|
||||
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|
||||
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||||
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||||
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||||
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|
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|
||||
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|
||||
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|
||||
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|
||||
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|
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||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
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<a class="navbar-brand" href="week34-bs.html">Week 35: Introduction to the course, Logistics and Practicalities</a>
|
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|
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<div class="navbar-collapse collapse navbar-responsive-collapse">
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<ul class="nav navbar-nav navbar-right">
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<li class="dropdown">
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
|
||||
<ul class="dropdown-menu">
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs001.html#overview-of-first-week" style="font-size: 80%;"><b>Overview of first week</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs002.html#reading-recommendations" style="font-size: 80%;"><b>Reading Recommendations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs003.html#thursday-august-26" style="font-size: 80%;"><b>Thursday August 26</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs004.html#lectures-and-computerlab" style="font-size: 80%;"><b>Lectures and ComputerLab</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs005.html#announcement" style="font-size: 80%;"><b>Announcement</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs006.html#course-format" style="font-size: 80%;"><b>Course Format</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs007.html#teachers" style="font-size: 80%;"><b>Teachers</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs008.html#deadlines-for-projects-tentative" style="font-size: 80%;"><b>Deadlines for projects (tentative)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs009.html#recommended-textbooks" style="font-size: 80%;"><b>Recommended textbooks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs010.html#prerequisites" style="font-size: 80%;"><b>Prerequisites</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs011.html#learning-outcomes" style="font-size: 80%;"><b>Learning outcomes</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs012.html#topics-covered-in-this-course-statistical-analysis-and-optimization-of-data" style="font-size: 80%;"><b>Topics covered in this course: Statistical analysis and optimization of data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs013.html#topics-covered-in-this-course-machine-learning" style="font-size: 80%;"><b>Topics covered in this course: Machine Learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs014.html#extremely-useful-tools-strongly-recommended" style="font-size: 80%;"><b>Extremely useful tools, strongly recommended</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs015.html#other-courses-on-data-science-and-machine-learning-at-uio" style="font-size: 80%;"><b>Other courses on Data science and Machine Learning at UiO</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs016.html#introduction" style="font-size: 80%;"><b>Introduction</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs017.html#what-is-machine-learning" style="font-size: 80%;"><b>What is Machine Learning?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs018.html#types-of-machine-learning" style="font-size: 80%;"><b>Types of Machine Learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs019.html#essential-elements-of-ml" style="font-size: 80%;"><b>Essential elements of ML</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs020.html#an-optimization-minimization-problem" style="font-size: 80%;"><b>An optimization/minimization problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs021.html#a-frequentist-approach-to-data-analysis" style="font-size: 80%;"><b>A Frequentist approach to data analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs022.html#what-is-a-good-model" style="font-size: 80%;"><b>What is a good model?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs023.html#what-is-a-good-model-can-we-define-it" style="font-size: 80%;"><b>What is a good model? Can we define it?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs024.html#software-and-needed-installations" style="font-size: 80%;"><b>Software and needed installations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs025.html#python-installers" style="font-size: 80%;"><b>Python installers</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs026.html#useful-python-libraries" style="font-size: 80%;"><b>Useful Python libraries</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs027.html#installing-r-c-cython-or-julia" style="font-size: 80%;"><b>Installing R, C++, cython or Julia</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs028.html#installing-r-c-cython-numba-etc" style="font-size: 80%;"><b>Installing R, C++, cython, Numba etc</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs029.html#numpy-examples-and-important-matrix-and-vector-handling-packages" style="font-size: 80%;"><b>Numpy examples and Important Matrix and vector handling packages</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs030.html#basic-matrix-features" style="font-size: 80%;"><b>Basic Matrix Features</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs031.html#some-famous-matrices" style="font-size: 80%;"> Some famous Matrices</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs032.html#more-basic-matrix-features" style="font-size: 80%;"> More Basic Matrix Features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs033.html#numpy-and-arrays" style="font-size: 80%;"><b>Numpy and arrays</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs034.html#matrices-in-python" style="font-size: 80%;"><b>Matrices in Python</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs035.html#meet-the-pandas" style="font-size: 80%;"><b>Meet the Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs038.html#friday-august-27" style="font-size: 80%;"><b>Friday August 27</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs037.html#reading-data-and-fitting" style="font-size: 80%;"><b>Reading Data and fitting</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs038.html#friday-august-27" style="font-size: 80%;"><b>Friday August 27</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="#simple-linear-regression-model-using-_scikit-learn_" style="font-size: 80%;"> Simple linear regression model using <b>scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="#to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies" style="font-size: 80%;"> To our real data: nuclear binding energies. Brief reminder on masses and binding energies</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#organizing-our-data" style="font-size: 80%;"> Organizing our data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#seeing-the-wood-for-the-trees" style="font-size: 80%;"> Seeing the wood for the trees</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#and-what-about-using-neural-networks" style="font-size: 80%;"> And what about using neural networks?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#a-first-summary" style="font-size: 80%;"><b>A first summary</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs040.html#exercises-for-week-36" style="font-size: 80%;"><b>Exercises for week 36</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs040.html#exercise-1-setting-up-various-python-environments" style="font-size: 80%;"><b>Exercise 1: Setting up various Python environments</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs040.html#exercise-2-making-your-own-data-and-exploring-scikit-learn" style="font-size: 80%;"><b>Exercise 2: making your own data and exploring scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs040.html#exercise-3-normalizing-our-data" style="font-size: 80%;"><b>Exercise 3: Normalizing our data</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
</ul>
|
||||
</div>
|
||||
</div>
|
||||
</div> <!-- end of navigation bar -->
|
||||
|
||||
<div class="container">
|
||||
|
||||
<p> </p><p> </p><p> </p> <!-- add vertical space -->
|
||||
|
||||
<a name="part0039"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h3 id="simple-linear-regression-model-using-_scikit-learn_" class="anchor">Simple linear regression model using <b>scikit-learn</b> </h3>
|
||||
|
||||
<p>
|
||||
We start with perhaps our simplest possible example, using <b>Scikit-Learn</b> to perform linear regression analysis on a data set produced by us.
|
||||
|
||||
<p>
|
||||
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.
|
||||
|
||||
<p>
|
||||
The Numpy functions are imported used the <b>import numpy as np</b>
|
||||
statement and the random number generator for the uniform distribution
|
||||
is called using the function <b>np.random.rand()</b>, 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 <b>randn()</b> 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 \( x \)
|
||||
|
||||
$$
|
||||
y = 2x+N(0,1),
|
||||
$$
|
||||
|
||||
<p>
|
||||
where \( N(0,1) \) represents random numbers generated by the normal
|
||||
distribution. From <b>Scikit-Learn</b> we import then the
|
||||
<b>LinearRegression</b> functionality and make a prediction \( \tilde{y} =
|
||||
\alpha + \beta x \) using the function <b>fit(x,y)</b>. We call the set of
|
||||
data \( (\hat{x},\hat{y}) \) for our training data. The Python package
|
||||
<b>scikit-learn</b> 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.
|
||||
|
||||
<p>
|
||||
For plotting we use the Python package
|
||||
<a href="https://matplotlib.org/" target="_self">matplotlib</a> which produces publication
|
||||
quality figures. Feel free to explore the extensive
|
||||
<a href="https://matplotlib.org/gallery/index.html" target="_self">gallery</a> of examples. In
|
||||
this example we plot our original values of \( x \) and \( y \) as well as the
|
||||
prediction <b>ypredict</b> (\( \tilde{y} \)), which attempts at fitting our
|
||||
data with a straight line.
|
||||
|
||||
<p>
|
||||
The Python code follows here.
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #408080; font-style: italic"># Importing various packages</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> LinearRegression
|
||||
|
||||
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(<span style="color: #666666">100</span>,<span style="color: #666666">1</span>)
|
||||
y <span style="color: #666666">=</span> <span style="color: #666666">2*</span>x<span style="color: #666666">+</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(<span style="color: #666666">100</span>,<span style="color: #666666">1</span>)
|
||||
linreg <span style="color: #666666">=</span> LinearRegression()
|
||||
linreg<span style="color: #666666">.</span>fit(x,y)
|
||||
xnew <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([[<span style="color: #666666">0</span>],[<span style="color: #666666">1</span>]])
|
||||
ypredict <span style="color: #666666">=</span> linreg<span style="color: #666666">.</span>predict(xnew)
|
||||
|
||||
plt<span style="color: #666666">.</span>plot(xnew, ypredict, <span style="color: #BA2121">"r-"</span>)
|
||||
plt<span style="color: #666666">.</span>plot(x, y ,<span style="color: #BA2121">'ro'</span>)
|
||||
plt<span style="color: #666666">.</span>axis([<span style="color: #666666">0</span>,<span style="color: #666666">1.0</span>,<span style="color: #666666">0</span>, <span style="color: #666666">5.0</span>])
|
||||
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">r'$x$'</span>)
|
||||
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">r'$y$'</span>)
|
||||
plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">r'Simple Linear Regression'</span>)
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
</pre></div>
|
||||
<p>
|
||||
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
|
||||
|
||||
$$
|
||||
y = 10x+0.01 \times N(0,1),
|
||||
$$
|
||||
|
||||
<p>
|
||||
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 <b>cost</b> function.
|
||||
|
||||
<p>
|
||||
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 <em>cost</em> function is the so-called \( \chi^2 \)
|
||||
function (a variant of the mean-squared error (MSE))
|
||||
|
||||
$$ \chi^2 = \frac{1}{n}
|
||||
\sum_{i=0}^{n-1}\frac{(y_i-\tilde{y}_i)^2}{\sigma_i^2},
|
||||
$$
|
||||
|
||||
<p>
|
||||
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.
|
||||
|
||||
<p>
|
||||
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 <b>gradient</b> 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.
|
||||
|
||||
<p>
|
||||
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
|
||||
|
||||
$$
|
||||
\epsilon_{\mathrm{relative}}= \frac{\vert \hat{y} -\hat{\tilde{y}}\vert}{\vert \hat{y}\vert}.
|
||||
$$
|
||||
|
||||
<p>
|
||||
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.
|
||||
|
||||
<p>
|
||||
We can modify easily the above Python code and plot the relative error instead
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> LinearRegression
|
||||
|
||||
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(<span style="color: #666666">100</span>,<span style="color: #666666">1</span>)
|
||||
y <span style="color: #666666">=</span> <span style="color: #666666">5*</span>x<span style="color: #666666">+0.01*</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(<span style="color: #666666">100</span>,<span style="color: #666666">1</span>)
|
||||
linreg <span style="color: #666666">=</span> LinearRegression()
|
||||
linreg<span style="color: #666666">.</span>fit(x,y)
|
||||
ypredict <span style="color: #666666">=</span> linreg<span style="color: #666666">.</span>predict(x)
|
||||
|
||||
plt<span style="color: #666666">.</span>plot(x, np<span style="color: #666666">.</span>abs(ypredict<span style="color: #666666">-</span>y)<span style="color: #666666">/</span><span style="color: #008000">abs</span>(y), <span style="color: #BA2121">"ro"</span>)
|
||||
plt<span style="color: #666666">.</span>axis([<span style="color: #666666">0</span>,<span style="color: #666666">1.0</span>,<span style="color: #666666">0.0</span>, <span style="color: #666666">0.5</span>])
|
||||
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">r'$x$'</span>)
|
||||
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">r'$\epsilon_{\mathrm</span><span style="color: #BB6688; font-weight: bold">{relative}</span><span style="color: #BA2121">}$'</span>)
|
||||
plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">r'Relative error'</span>)
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
</pre></div>
|
||||
<p>
|
||||
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.
|
||||
|
||||
<p>
|
||||
As mentioned above, <b>Scikit-Learn</b> 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.
|
||||
|
||||
<p>
|
||||
Here we show an
|
||||
example of the functionality of <b>Scikit-Learn</b>.
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> LinearRegression
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.metrics</span> <span style="color: #008000; font-weight: bold">import</span> mean_squared_error, r2_score, mean_squared_log_error, mean_absolute_error
|
||||
|
||||
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(<span style="color: #666666">100</span>,<span style="color: #666666">1</span>)
|
||||
y <span style="color: #666666">=</span> <span style="color: #666666">2.0+</span> <span style="color: #666666">5*</span>x<span style="color: #666666">+0.5*</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(<span style="color: #666666">100</span>,<span style="color: #666666">1</span>)
|
||||
linreg <span style="color: #666666">=</span> LinearRegression()
|
||||
linreg<span style="color: #666666">.</span>fit(x,y)
|
||||
ypredict <span style="color: #666666">=</span> linreg<span style="color: #666666">.</span>predict(x)
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">'The intercept alpha: </span><span style="color: #BB6622; font-weight: bold">\n</span><span style="color: #BA2121">'</span>, linreg<span style="color: #666666">.</span>intercept_)
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">'Coefficient beta : </span><span style="color: #BB6622; font-weight: bold">\n</span><span style="color: #BA2121">'</span>, linreg<span style="color: #666666">.</span>coef_)
|
||||
<span style="color: #408080; font-style: italic"># The mean squared error </span>
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">"Mean squared error: </span><span style="color: #BB6688; font-weight: bold">%.2f</span><span style="color: #BA2121">"</span> <span style="color: #666666">%</span> mean_squared_error(y, ypredict))
|
||||
<span style="color: #408080; font-style: italic"># Explained variance score: 1 is perfect prediction </span>
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">'Variance score: </span><span style="color: #BB6688; font-weight: bold">%.2f</span><span style="color: #BA2121">'</span> <span style="color: #666666">%</span> r2_score(y, ypredict))
|
||||
<span style="color: #408080; font-style: italic"># Mean squared log error </span>
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">'Mean squared log error: </span><span style="color: #BB6688; font-weight: bold">%.2f</span><span style="color: #BA2121">'</span> <span style="color: #666666">%</span> mean_squared_log_error(y, ypredict) )
|
||||
<span style="color: #408080; font-style: italic"># Mean absolute error </span>
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">'Mean absolute error: </span><span style="color: #BB6688; font-weight: bold">%.2f</span><span style="color: #BA2121">'</span> <span style="color: #666666">%</span> mean_absolute_error(y, ypredict))
|
||||
plt<span style="color: #666666">.</span>plot(x, ypredict, <span style="color: #BA2121">"r-"</span>)
|
||||
plt<span style="color: #666666">.</span>plot(x, y ,<span style="color: #BA2121">'ro'</span>)
|
||||
plt<span style="color: #666666">.</span>axis([<span style="color: #666666">0.0</span>,<span style="color: #666666">1.0</span>,<span style="color: #666666">1.5</span>, <span style="color: #666666">7.0</span>])
|
||||
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">r'$x$'</span>)
|
||||
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">r'$y$'</span>)
|
||||
plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">r'Linear Regression fit '</span>)
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
</pre></div>
|
||||
<p>
|
||||
The function <b>coef</b> gives us the parameter \( \beta \) of our fit while <b>intercept</b> 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 <b>meansquarederror</b> gives us the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error or loss defined as
|
||||
$$ MSE(\hat{y},\hat{\tilde{y}}) = \frac{1}{n}
|
||||
\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2,
|
||||
$$
|
||||
|
||||
<p>
|
||||
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.
|
||||
|
||||
<p>
|
||||
The <b>r2score</b> 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 \( 0.0 \).
|
||||
|
||||
<p>
|
||||
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
|
||||
$$
|
||||
\text{MSLE}(\hat{y}, \hat{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n - 1} (\log_e (1 + y_i) - \log_e (1 + \tilde{y}_i) )^2,
|
||||
$$
|
||||
|
||||
<p>
|
||||
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.
|
||||
|
||||
<p>
|
||||
Finally, another cost function is the Huber cost function used in robust regression.
|
||||
|
||||
<p>
|
||||
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.
|
||||
|
||||
<p>
|
||||
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.
|
||||
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">random</span>
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> Ridge
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> PolynomialFeatures
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.pipeline</span> <span style="color: #008000; font-weight: bold">import</span> make_pipeline
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> LinearRegression
|
||||
|
||||
x<span style="color: #666666">=</span>np<span style="color: #666666">.</span>linspace(<span style="color: #666666">0.02</span>,<span style="color: #666666">0.98</span>,<span style="color: #666666">200</span>)
|
||||
noise <span style="color: #666666">=</span> np<span style="color: #666666">.</span>asarray(random<span style="color: #666666">.</span>sample((<span style="color: #008000">range</span>(<span style="color: #666666">200</span>)),<span style="color: #666666">200</span>))
|
||||
y<span style="color: #666666">=</span>x<span style="color: #666666">**3*</span>noise
|
||||
yn<span style="color: #666666">=</span>x<span style="color: #666666">**3*100</span>
|
||||
poly3 <span style="color: #666666">=</span> PolynomialFeatures(degree<span style="color: #666666">=3</span>)
|
||||
X <span style="color: #666666">=</span> poly3<span style="color: #666666">.</span>fit_transform(x[:,np<span style="color: #666666">.</span>newaxis])
|
||||
clf3 <span style="color: #666666">=</span> LinearRegression()
|
||||
clf3<span style="color: #666666">.</span>fit(X,y)
|
||||
|
||||
Xplot<span style="color: #666666">=</span>poly3<span style="color: #666666">.</span>fit_transform(x[:,np<span style="color: #666666">.</span>newaxis])
|
||||
poly3_plot<span style="color: #666666">=</span>plt<span style="color: #666666">.</span>plot(x, clf3<span style="color: #666666">.</span>predict(Xplot), label<span style="color: #666666">=</span><span style="color: #BA2121">'Cubic Fit'</span>)
|
||||
plt<span style="color: #666666">.</span>plot(x,yn, color<span style="color: #666666">=</span><span style="color: #BA2121">'red'</span>, label<span style="color: #666666">=</span><span style="color: #BA2121">"True Cubic"</span>)
|
||||
plt<span style="color: #666666">.</span>scatter(x, y, label<span style="color: #666666">=</span><span style="color: #BA2121">'Data'</span>, color<span style="color: #666666">=</span><span style="color: #BA2121">'orange'</span>, s<span style="color: #666666">=15</span>)
|
||||
plt<span style="color: #666666">.</span>legend()
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
|
||||
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">error</span>(a):
|
||||
<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> y:
|
||||
err<span style="color: #666666">=</span>(y<span style="color: #666666">-</span>yn)<span style="color: #666666">/</span>yn
|
||||
<span style="color: #008000; font-weight: bold">return</span> <span style="color: #008000">abs</span>(np<span style="color: #666666">.</span>sum(err))<span style="color: #666666">/</span><span style="color: #008000">len</span>(err)
|
||||
|
||||
<span style="color: #008000">print</span> (error(y))
|
||||
</pre></div>
|
||||
|
||||
<h3 id="to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies" class="anchor">To our real data: nuclear binding energies. Brief reminder on masses and binding energies </h3>
|
||||
|
||||
<p>
|
||||
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).
|
||||
|
||||
<p>
|
||||
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
|
||||
$$
|
||||
m_n = 939.56536(8)\hspace{0.1cm} \mathrm{MeV}/c^2 = 1.0086649156(6)u.
|
||||
$$
|
||||
|
||||
<p>
|
||||
In the <a href="http://nuclearmasses.org/resources_folder/Wang_2017_Chinese_Phys_C_41_030003.pdf" target="_self">2016 mass evaluation of by W.J.Huang, G.Audi, M.Wang, F.G.Kondev, S.Naimi and X.Xu</a>
|
||||
there are data on masses and decays of 3437 nuclei.
|
||||
|
||||
<p>
|
||||
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.
|
||||
|
||||
<p>
|
||||
A popular and physically intuitive model which can be used to parametrize
|
||||
the experimental binding energies as function of \( A \), is the so-called
|
||||
<b>liquid drop model</b>. The ansatz is based on the following expression
|
||||
|
||||
$$
|
||||
BE(N,Z) = a_1A-a_2A^{2/3}-a_3\frac{Z^2}{A^{1/3}}-a_4\frac{(N-Z)^2}{A},
|
||||
$$
|
||||
|
||||
<p>
|
||||
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.
|
||||
|
||||
<p>
|
||||
To arrive at the above expression we have assumed that we can make the following assumptions:
|
||||
|
||||
<ul>
|
||||
<li> 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.</li>
|
||||
<li> 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.</li>
|
||||
<li> 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.</li>
|
||||
<li> 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.</li>
|
||||
</ul>
|
||||
|
||||
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.
|
||||
|
||||
<h3 id="organizing-our-data" class="anchor">Organizing our data </h3>
|
||||
|
||||
<p>
|
||||
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.
|
||||
|
||||
<p>
|
||||
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 <b>scikit-learn</b>.
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #408080; font-style: italic"># Common imports</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">pandas</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">pd</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">skl</span>
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.metrics</span> <span style="color: #008000; font-weight: bold">import</span> mean_squared_error, r2_score, mean_absolute_error
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">os</span>
|
||||
|
||||
<span style="color: #408080; font-style: italic"># Where to save the figures and data files</span>
|
||||
PROJECT_ROOT_DIR <span style="color: #666666">=</span> <span style="color: #BA2121">"Results"</span>
|
||||
FIGURE_ID <span style="color: #666666">=</span> <span style="color: #BA2121">"Results/FigureFiles"</span>
|
||||
DATA_ID <span style="color: #666666">=</span> <span style="color: #BA2121">"DataFiles/"</span>
|
||||
|
||||
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(PROJECT_ROOT_DIR):
|
||||
os<span style="color: #666666">.</span>mkdir(PROJECT_ROOT_DIR)
|
||||
|
||||
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(FIGURE_ID):
|
||||
os<span style="color: #666666">.</span>makedirs(FIGURE_ID)
|
||||
|
||||
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(DATA_ID):
|
||||
os<span style="color: #666666">.</span>makedirs(DATA_ID)
|
||||
|
||||
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">image_path</span>(fig_id):
|
||||
<span style="color: #008000; font-weight: bold">return</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>join(FIGURE_ID, fig_id)
|
||||
|
||||
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">data_path</span>(dat_id):
|
||||
<span style="color: #008000; font-weight: bold">return</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>join(DATA_ID, dat_id)
|
||||
|
||||
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">save_fig</span>(fig_id):
|
||||
plt<span style="color: #666666">.</span>savefig(image_path(fig_id) <span style="color: #666666">+</span> <span style="color: #BA2121">".png"</span>, <span style="color: #008000">format</span><span style="color: #666666">=</span><span style="color: #BA2121">'png'</span>)
|
||||
|
||||
infile <span style="color: #666666">=</span> <span style="color: #008000">open</span>(data_path(<span style="color: #BA2121">"MassEval2016.dat"</span>),<span style="color: #BA2121">'r'</span>)
|
||||
</pre></div>
|
||||
<p>
|
||||
Before we proceed, we define also a function for making our plots. You can obviously avoid this and simply set up various <b>matplotlib</b> 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.
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">pylab</span> <span style="color: #008000; font-weight: bold">import</span> plt, mpl
|
||||
plt<span style="color: #666666">.</span>style<span style="color: #666666">.</span>use(<span style="color: #BA2121">'seaborn'</span>)
|
||||
mpl<span style="color: #666666">.</span>rcParams[<span style="color: #BA2121">'font.family'</span>] <span style="color: #666666">=</span> <span style="color: #BA2121">'serif'</span>
|
||||
|
||||
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">MakePlot</span>(x,y, styles, labels, axlabels):
|
||||
plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">10</span>,<span style="color: #666666">6</span>))
|
||||
<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #008000">len</span>(x)):
|
||||
plt<span style="color: #666666">.</span>plot(x[i], y[i], styles[i], label <span style="color: #666666">=</span> labels[i])
|
||||
plt<span style="color: #666666">.</span>xlabel(axlabels[<span style="color: #666666">0</span>])
|
||||
plt<span style="color: #666666">.</span>ylabel(axlabels[<span style="color: #666666">1</span>])
|
||||
plt<span style="color: #666666">.</span>legend(loc<span style="color: #666666">=0</span>)
|
||||
</pre></div>
|
||||
<p>
|
||||
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 <b>pandas</b>. 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!
|
||||
|
||||
<p>
|
||||
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 <b>pandas</b>. The file begins with some basic format information.
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #BA2121; font-style: italic">""" </span>
|
||||
<span style="color: #BA2121; font-style: italic">This is taken from the data file of the mass 2016 evaluation. </span>
|
||||
<span style="color: #BA2121; font-style: italic">All files are 3436 lines long with 124 character per line. </span>
|
||||
<span style="color: #BA2121; font-style: italic"> Headers are 39 lines long. </span>
|
||||
<span style="color: #BA2121; font-style: italic"> col 1 : Fortran character control: 1 = page feed 0 = line feed </span>
|
||||
<span style="color: #BA2121; font-style: italic"> 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 </span>
|
||||
<span style="color: #BA2121; font-style: italic"> These formats are reflected in the pandas widths variable below, see the statement </span>
|
||||
<span style="color: #BA2121; font-style: italic"> widths=(1,3,5,5,5,1,3,4,1,13,11,11,9,1,2,11,9,1,3,1,12,11,1), </span>
|
||||
<span style="color: #BA2121; font-style: italic"> Pandas has also a variable header, with length 39 in this case. </span>
|
||||
<span style="color: #BA2121; font-style: italic">"""</span>
|
||||
</pre></div>
|
||||
<p>
|
||||
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 <b>pandas</b> DataFrame structure.
|
||||
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #408080; font-style: italic"># Read the experimental data with Pandas</span>
|
||||
Masses <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>read_fwf(infile, usecols<span style="color: #666666">=</span>(<span style="color: #666666">2</span>,<span style="color: #666666">3</span>,<span style="color: #666666">4</span>,<span style="color: #666666">6</span>,<span style="color: #666666">11</span>),
|
||||
names<span style="color: #666666">=</span>(<span style="color: #BA2121">'N'</span>, <span style="color: #BA2121">'Z'</span>, <span style="color: #BA2121">'A'</span>, <span style="color: #BA2121">'Element'</span>, <span style="color: #BA2121">'Ebinding'</span>),
|
||||
widths<span style="color: #666666">=</span>(<span style="color: #666666">1</span>,<span style="color: #666666">3</span>,<span style="color: #666666">5</span>,<span style="color: #666666">5</span>,<span style="color: #666666">5</span>,<span style="color: #666666">1</span>,<span style="color: #666666">3</span>,<span style="color: #666666">4</span>,<span style="color: #666666">1</span>,<span style="color: #666666">13</span>,<span style="color: #666666">11</span>,<span style="color: #666666">11</span>,<span style="color: #666666">9</span>,<span style="color: #666666">1</span>,<span style="color: #666666">2</span>,<span style="color: #666666">11</span>,<span style="color: #666666">9</span>,<span style="color: #666666">1</span>,<span style="color: #666666">3</span>,<span style="color: #666666">1</span>,<span style="color: #666666">12</span>,<span style="color: #666666">11</span>,<span style="color: #666666">1</span>),
|
||||
header<span style="color: #666666">=39</span>,
|
||||
index_col<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">False</span>)
|
||||
|
||||
<span style="color: #408080; font-style: italic"># Extrapolated values are indicated by '#' in place of the decimal place, so</span>
|
||||
<span style="color: #408080; font-style: italic"># the Ebinding column won't be numeric. Coerce to float and drop these entries.</span>
|
||||
Masses[<span style="color: #BA2121">'Ebinding'</span>] <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>to_numeric(Masses[<span style="color: #BA2121">'Ebinding'</span>], errors<span style="color: #666666">=</span><span style="color: #BA2121">'coerce'</span>)
|
||||
Masses <span style="color: #666666">=</span> Masses<span style="color: #666666">.</span>dropna()
|
||||
<span style="color: #408080; font-style: italic"># Convert from keV to MeV.</span>
|
||||
Masses[<span style="color: #BA2121">'Ebinding'</span>] <span style="color: #666666">/=</span> <span style="color: #666666">1000</span>
|
||||
|
||||
<span style="color: #408080; font-style: italic"># Group the DataFrame by nucleon number, A.</span>
|
||||
Masses <span style="color: #666666">=</span> Masses<span style="color: #666666">.</span>groupby(<span style="color: #BA2121">'A'</span>)
|
||||
<span style="color: #408080; font-style: italic"># Find the rows of the grouped DataFrame with the maximum binding energy.</span>
|
||||
Masses <span style="color: #666666">=</span> Masses<span style="color: #666666">.</span>apply(<span style="color: #008000; font-weight: bold">lambda</span> t: t[t<span style="color: #666666">.</span>Ebinding<span style="color: #666666">==</span>t<span style="color: #666666">.</span>Ebinding<span style="color: #666666">.</span>max()])
|
||||
</pre></div>
|
||||
<p>
|
||||
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 <b>pandas</b>. 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 <b>numpy</b> and
|
||||
<b>Scikit-Learn</b> afterwards.
|
||||
|
||||
<p>
|
||||
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.
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span>A <span style="color: #666666">=</span> Masses[<span style="color: #BA2121">'A'</span>]
|
||||
Z <span style="color: #666666">=</span> Masses[<span style="color: #BA2121">'Z'</span>]
|
||||
N <span style="color: #666666">=</span> Masses[<span style="color: #BA2121">'N'</span>]
|
||||
Element <span style="color: #666666">=</span> Masses[<span style="color: #BA2121">'Element'</span>]
|
||||
Energies <span style="color: #666666">=</span> Masses[<span style="color: #BA2121">'Ebinding'</span>]
|
||||
<span style="color: #008000">print</span>(Masses)
|
||||
</pre></div>
|
||||
<p>
|
||||
The next step, and we will define this mathematically later, is to set up the so-called <b>design matrix</b>. 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.
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #408080; font-style: italic"># Now we set up the design matrix X</span>
|
||||
X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((<span style="color: #008000">len</span>(A),<span style="color: #666666">5</span>))
|
||||
X[:,<span style="color: #666666">0</span>] <span style="color: #666666">=</span> <span style="color: #666666">1</span>
|
||||
X[:,<span style="color: #666666">1</span>] <span style="color: #666666">=</span> A
|
||||
X[:,<span style="color: #666666">2</span>] <span style="color: #666666">=</span> A<span style="color: #666666">**</span>(<span style="color: #666666">2.0/3.0</span>)
|
||||
X[:,<span style="color: #666666">3</span>] <span style="color: #666666">=</span> A<span style="color: #666666">**</span>(<span style="color: #666666">-1.0/3.0</span>)
|
||||
X[:,<span style="color: #666666">4</span>] <span style="color: #666666">=</span> A<span style="color: #666666">**</span>(<span style="color: #666666">-1.0</span>)
|
||||
</pre></div>
|
||||
<p>
|
||||
With <b>scikitlearn</b> we are now ready to use linear regression and fit our data.
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span>clf <span style="color: #666666">=</span> skl<span style="color: #666666">.</span>LinearRegression()<span style="color: #666666">.</span>fit(X, Energies)
|
||||
fity <span style="color: #666666">=</span> clf<span style="color: #666666">.</span>predict(X)
|
||||
</pre></div>
|
||||
<p>
|
||||
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.
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #408080; font-style: italic"># The mean squared error </span>
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">"Mean squared error: </span><span style="color: #BB6688; font-weight: bold">%.2f</span><span style="color: #BA2121">"</span> <span style="color: #666666">%</span> mean_squared_error(Energies, fity))
|
||||
<span style="color: #408080; font-style: italic"># Explained variance score: 1 is perfect prediction </span>
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">'Variance score: </span><span style="color: #BB6688; font-weight: bold">%.2f</span><span style="color: #BA2121">'</span> <span style="color: #666666">%</span> r2_score(Energies, fity))
|
||||
<span style="color: #408080; font-style: italic"># Mean absolute error </span>
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">'Mean absolute error: </span><span style="color: #BB6688; font-weight: bold">%.2f</span><span style="color: #BA2121">'</span> <span style="color: #666666">%</span> mean_absolute_error(Energies, fity))
|
||||
<span style="color: #008000">print</span>(clf<span style="color: #666666">.</span>coef_, clf<span style="color: #666666">.</span>intercept_)
|
||||
|
||||
Masses[<span style="color: #BA2121">'Eapprox'</span>] <span style="color: #666666">=</span> fity
|
||||
<span style="color: #408080; font-style: italic"># Generate a plot comparing the experimental with the fitted values values.</span>
|
||||
fig, ax <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>subplots()
|
||||
ax<span style="color: #666666">.</span>set_xlabel(<span style="color: #BA2121">r'$A = N + Z$'</span>)
|
||||
ax<span style="color: #666666">.</span>set_ylabel(<span style="color: #BA2121">r'$E_\mathrm</span><span style="color: #BB6688; font-weight: bold">{bind}</span><span style="color: #BA2121">\,/\mathrm</span><span style="color: #BB6688; font-weight: bold">{MeV}</span><span style="color: #BA2121">$'</span>)
|
||||
ax<span style="color: #666666">.</span>plot(Masses[<span style="color: #BA2121">'A'</span>], Masses[<span style="color: #BA2121">'Ebinding'</span>], alpha<span style="color: #666666">=0.7</span>, lw<span style="color: #666666">=2</span>,
|
||||
label<span style="color: #666666">=</span><span style="color: #BA2121">'Ame2016'</span>)
|
||||
ax<span style="color: #666666">.</span>plot(Masses[<span style="color: #BA2121">'A'</span>], Masses[<span style="color: #BA2121">'Eapprox'</span>], alpha<span style="color: #666666">=0.7</span>, lw<span style="color: #666666">=2</span>, c<span style="color: #666666">=</span><span style="color: #BA2121">'m'</span>,
|
||||
label<span style="color: #666666">=</span><span style="color: #BA2121">'Fit'</span>)
|
||||
ax<span style="color: #666666">.</span>legend()
|
||||
save_fig(<span style="color: #BA2121">"Masses2016"</span>)
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
</pre></div>
|
||||
|
||||
<h3 id="seeing-the-wood-for-the-trees" class="anchor">Seeing the wood for the trees </h3>
|
||||
|
||||
<p>
|
||||
As a teaser, let us now see how we can do this with decision trees using <b>scikit-learn</b>. Later we will switch to so-called <b>random forests</b>!
|
||||
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #408080; font-style: italic">#Decision Tree Regression</span>
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.tree</span> <span style="color: #008000; font-weight: bold">import</span> DecisionTreeRegressor
|
||||
regr_1<span style="color: #666666">=</span>DecisionTreeRegressor(max_depth<span style="color: #666666">=5</span>)
|
||||
regr_2<span style="color: #666666">=</span>DecisionTreeRegressor(max_depth<span style="color: #666666">=7</span>)
|
||||
regr_3<span style="color: #666666">=</span>DecisionTreeRegressor(max_depth<span style="color: #666666">=9</span>)
|
||||
regr_1<span style="color: #666666">.</span>fit(X, Energies)
|
||||
regr_2<span style="color: #666666">.</span>fit(X, Energies)
|
||||
regr_3<span style="color: #666666">.</span>fit(X, Energies)
|
||||
|
||||
|
||||
y_1 <span style="color: #666666">=</span> regr_1<span style="color: #666666">.</span>predict(X)
|
||||
y_2 <span style="color: #666666">=</span> regr_2<span style="color: #666666">.</span>predict(X)
|
||||
y_3<span style="color: #666666">=</span>regr_3<span style="color: #666666">.</span>predict(X)
|
||||
Masses[<span style="color: #BA2121">'Eapprox'</span>] <span style="color: #666666">=</span> y_3
|
||||
<span style="color: #408080; font-style: italic"># Plot the results</span>
|
||||
plt<span style="color: #666666">.</span>figure()
|
||||
plt<span style="color: #666666">.</span>plot(A, Energies, color<span style="color: #666666">=</span><span style="color: #BA2121">"blue"</span>, label<span style="color: #666666">=</span><span style="color: #BA2121">"Data"</span>, linewidth<span style="color: #666666">=2</span>)
|
||||
plt<span style="color: #666666">.</span>plot(A, y_1, color<span style="color: #666666">=</span><span style="color: #BA2121">"red"</span>, label<span style="color: #666666">=</span><span style="color: #BA2121">"max_depth=5"</span>, linewidth<span style="color: #666666">=2</span>)
|
||||
plt<span style="color: #666666">.</span>plot(A, y_2, color<span style="color: #666666">=</span><span style="color: #BA2121">"green"</span>, label<span style="color: #666666">=</span><span style="color: #BA2121">"max_depth=7"</span>, linewidth<span style="color: #666666">=2</span>)
|
||||
plt<span style="color: #666666">.</span>plot(A, y_3, color<span style="color: #666666">=</span><span style="color: #BA2121">"m"</span>, label<span style="color: #666666">=</span><span style="color: #BA2121">"max_depth=9"</span>, linewidth<span style="color: #666666">=2</span>)
|
||||
|
||||
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">"$A$"</span>)
|
||||
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">"$E$[MeV]"</span>)
|
||||
plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">"Decision Tree Regression"</span>)
|
||||
plt<span style="color: #666666">.</span>legend()
|
||||
save_fig(<span style="color: #BA2121">"Masses2016Trees"</span>)
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
<span style="color: #008000">print</span>(Masses)
|
||||
<span style="color: #008000">print</span>(np<span style="color: #666666">.</span>mean( (Energies<span style="color: #666666">-</span>y_1)<span style="color: #666666">**2</span>))
|
||||
</pre></div>
|
||||
|
||||
<h3 id="and-what-about-using-neural-networks" class="anchor">And what about using neural networks? </h3>
|
||||
|
||||
The <b>seaborn</b> package allows us to visualize data in an efficient way. Note that we use <b>scikit-learn</b>'s multi-layer perceptron (or feed forward neural network)
|
||||
functionality.
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.neural_network</span> <span style="color: #008000; font-weight: bold">import</span> MLPRegressor
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.metrics</span> <span style="color: #008000; font-weight: bold">import</span> accuracy_score
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">seaborn</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">sns</span>
|
||||
|
||||
X_train <span style="color: #666666">=</span> X
|
||||
Y_train <span style="color: #666666">=</span> Energies
|
||||
n_hidden_neurons <span style="color: #666666">=</span> <span style="color: #666666">100</span>
|
||||
epochs <span style="color: #666666">=</span> <span style="color: #666666">100</span>
|
||||
<span style="color: #408080; font-style: italic"># store models for later use</span>
|
||||
eta_vals <span style="color: #666666">=</span> np<span style="color: #666666">.</span>logspace(<span style="color: #666666">-5</span>, <span style="color: #666666">1</span>, <span style="color: #666666">7</span>)
|
||||
lmbd_vals <span style="color: #666666">=</span> np<span style="color: #666666">.</span>logspace(<span style="color: #666666">-5</span>, <span style="color: #666666">1</span>, <span style="color: #666666">7</span>)
|
||||
<span style="color: #408080; font-style: italic"># store the models for later use</span>
|
||||
DNN_scikit <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((<span style="color: #008000">len</span>(eta_vals), <span style="color: #008000">len</span>(lmbd_vals)), dtype<span style="color: #666666">=</span><span style="color: #008000">object</span>)
|
||||
train_accuracy <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((<span style="color: #008000">len</span>(eta_vals), <span style="color: #008000">len</span>(lmbd_vals)))
|
||||
sns<span style="color: #666666">.</span>set()
|
||||
<span style="color: #008000; font-weight: bold">for</span> i, eta <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">enumerate</span>(eta_vals):
|
||||
<span style="color: #008000; font-weight: bold">for</span> j, lmbd <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">enumerate</span>(lmbd_vals):
|
||||
dnn <span style="color: #666666">=</span> MLPRegressor(hidden_layer_sizes<span style="color: #666666">=</span>(n_hidden_neurons), activation<span style="color: #666666">=</span><span style="color: #BA2121">'logistic'</span>,
|
||||
alpha<span style="color: #666666">=</span>lmbd, learning_rate_init<span style="color: #666666">=</span>eta, max_iter<span style="color: #666666">=</span>epochs)
|
||||
dnn<span style="color: #666666">.</span>fit(X_train, Y_train)
|
||||
DNN_scikit[i][j] <span style="color: #666666">=</span> dnn
|
||||
train_accuracy[i][j] <span style="color: #666666">=</span> dnn<span style="color: #666666">.</span>score(X_train, Y_train)
|
||||
|
||||
fig, ax <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>subplots(figsize <span style="color: #666666">=</span> (<span style="color: #666666">10</span>, <span style="color: #666666">10</span>))
|
||||
sns<span style="color: #666666">.</span>heatmap(train_accuracy, annot<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>, ax<span style="color: #666666">=</span>ax, cmap<span style="color: #666666">=</span><span style="color: #BA2121">"viridis"</span>)
|
||||
ax<span style="color: #666666">.</span>set_title(<span style="color: #BA2121">"Training Accuracy"</span>)
|
||||
ax<span style="color: #666666">.</span>set_ylabel(<span style="color: #BA2121">"$\eta$"</span>)
|
||||
ax<span style="color: #666666">.</span>set_xlabel(<span style="color: #BA2121">"$\lambda$"</span>)
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
</pre></div>
|
||||
|
||||
<h2 id="a-first-summary" class="anchor">A first summary </h2>
|
||||
|
||||
<p>
|
||||
The aim behind these introductory words was to present to you various
|
||||
Python libraries and their functionalities, in particular libraries like
|
||||
<b>numpy</b>, <b>pandas</b>, <b>xarray</b> and <b>matplotlib</b> and other that make our life much easier
|
||||
in handling various data sets and visualizing data.
|
||||
|
||||
<p>
|
||||
Furthermore,
|
||||
<b>Scikit-Learn</b> allows us with few lines of code to implement popular
|
||||
Machine Learning algorithms for supervised learning. Later we will meet <b>Tensorflow</b>, 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.
|
||||
|
||||
<p>
|
||||
<p>
|
||||
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|
||||
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||||
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||||
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<style type="text/css">
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|
||||
/* Add scrollbar to dropdown menus in bootstrap navigation bar */
|
||||
.dropdown-menu {
|
||||
height: auto;
|
||||
max-height: 400px;
|
||||
overflow-x: hidden;
|
||||
}
|
||||
|
||||
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|
||||
bar */
|
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.anchor::before {
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content:"";
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display:block;
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height:50px; /* fixed header height for style bootstrap */
|
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|
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|
||||
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|
||||
|
||||
|
||||
</head>
|
||||
|
||||
<!-- tocinfo
|
||||
{'highest level': 2,
|
||||
'sections': [('Overview of first week', 2, None, 'overview-of-first-week'),
|
||||
('Reading Recommendations', 2, None, 'reading-recommendations'),
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|
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|
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2,
|
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None,
|
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|
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|
||||
2,
|
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None,
|
||||
'topics-covered-in-this-course-machine-learning'),
|
||||
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|
||||
2,
|
||||
None,
|
||||
'extremely-useful-tools-strongly-recommended'),
|
||||
('Other courses on Data science and Machine Learning at UiO',
|
||||
2,
|
||||
None,
|
||||
'other-courses-on-data-science-and-machine-learning-at-uio'),
|
||||
('Introduction', 2, None, 'introduction'),
|
||||
('What is Machine Learning?',
|
||||
2,
|
||||
None,
|
||||
'what-is-machine-learning'),
|
||||
('Types of Machine Learning',
|
||||
2,
|
||||
None,
|
||||
'types-of-machine-learning'),
|
||||
('Essential elements of ML', 2, None, 'essential-elements-of-ml'),
|
||||
('An optimization/minimization problem',
|
||||
2,
|
||||
None,
|
||||
'an-optimization-minimization-problem'),
|
||||
('A Frequentist approach to data analysis',
|
||||
2,
|
||||
None,
|
||||
'a-frequentist-approach-to-data-analysis'),
|
||||
('What is a good model?', 2, None, 'what-is-a-good-model'),
|
||||
('What is a good model? Can we define it?',
|
||||
2,
|
||||
None,
|
||||
'what-is-a-good-model-can-we-define-it'),
|
||||
('Software and needed installations',
|
||||
2,
|
||||
None,
|
||||
'software-and-needed-installations'),
|
||||
('Python installers', 2, None, 'python-installers'),
|
||||
('Useful Python libraries', 2, None, 'useful-python-libraries'),
|
||||
('Installing R, C++, cython or Julia',
|
||||
2,
|
||||
None,
|
||||
'installing-r-c-cython-or-julia'),
|
||||
('Installing R, C++, cython, Numba etc',
|
||||
2,
|
||||
None,
|
||||
'installing-r-c-cython-numba-etc'),
|
||||
('Numpy examples and Important Matrix and vector handling '
|
||||
'packages',
|
||||
2,
|
||||
None,
|
||||
'numpy-examples-and-important-matrix-and-vector-handling-packages'),
|
||||
('Basic Matrix Features', 2, None, 'basic-matrix-features'),
|
||||
('Some famous Matrices', 3, None, 'some-famous-matrices'),
|
||||
('More Basic Matrix Features',
|
||||
3,
|
||||
None,
|
||||
'more-basic-matrix-features'),
|
||||
('Numpy and arrays', 2, None, 'numpy-and-arrays'),
|
||||
('Matrices in Python', 2, None, 'matrices-in-python'),
|
||||
('Meet the Pandas', 2, None, 'meet-the-pandas'),
|
||||
('Friday August 27', 2, None, 'friday-august-27'),
|
||||
('Reading Data and fitting', 2, None, 'reading-data-and-fitting'),
|
||||
('Friday August 27', 2, None, 'friday-august-27'),
|
||||
('Simple linear regression model using _scikit-learn_',
|
||||
3,
|
||||
None,
|
||||
'simple-linear-regression-model-using-_scikit-learn_'),
|
||||
('To our real data: nuclear binding energies. Brief reminder on '
|
||||
'masses and binding energies',
|
||||
3,
|
||||
None,
|
||||
'to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies'),
|
||||
('Organizing our data', 3, None, 'organizing-our-data'),
|
||||
('Seeing the wood for the trees',
|
||||
3,
|
||||
None,
|
||||
'seeing-the-wood-for-the-trees'),
|
||||
('And what about using neural networks?',
|
||||
3,
|
||||
None,
|
||||
'and-what-about-using-neural-networks'),
|
||||
('A first summary', 2, None, 'a-first-summary'),
|
||||
('Exercises for week 36', 2, None, 'exercises-for-week-36'),
|
||||
('Exercise 1: Setting up various Python environments',
|
||||
2,
|
||||
None,
|
||||
'exercise-1-setting-up-various-python-environments'),
|
||||
('Exercise 2: making your own data and exploring scikit-learn',
|
||||
2,
|
||||
None,
|
||||
'exercise-2-making-your-own-data-and-exploring-scikit-learn'),
|
||||
('Exercise 3: Normalizing our data',
|
||||
2,
|
||||
None,
|
||||
'exercise-3-normalizing-our-data')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
|
||||
|
||||
|
||||
<script type="text/x-mathjax-config">
|
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|
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|
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|
||||
}
|
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});
|
||||
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|
||||
<script type="text/javascript" async
|
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<!-- Bootstrap navigation bar -->
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<div class="navbar navbar-default navbar-fixed-top">
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<div class="navbar-header">
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<button type="button" class="navbar-toggle" data-toggle="collapse" data-target=".navbar-responsive-collapse">
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<span class="icon-bar"></span>
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<a class="navbar-brand" href="week34-bs.html">Week 35: Introduction to the course, Logistics and Practicalities</a>
|
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</div>
|
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|
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<div class="navbar-collapse collapse navbar-responsive-collapse">
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<ul class="nav navbar-nav navbar-right">
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<li class="dropdown">
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
|
||||
<ul class="dropdown-menu">
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs001.html#overview-of-first-week" style="font-size: 80%;"><b>Overview of first week</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs002.html#reading-recommendations" style="font-size: 80%;"><b>Reading Recommendations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs003.html#thursday-august-26" style="font-size: 80%;"><b>Thursday August 26</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs004.html#lectures-and-computerlab" style="font-size: 80%;"><b>Lectures and ComputerLab</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs005.html#announcement" style="font-size: 80%;"><b>Announcement</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs006.html#course-format" style="font-size: 80%;"><b>Course Format</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs007.html#teachers" style="font-size: 80%;"><b>Teachers</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs008.html#deadlines-for-projects-tentative" style="font-size: 80%;"><b>Deadlines for projects (tentative)</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs009.html#recommended-textbooks" style="font-size: 80%;"><b>Recommended textbooks</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs010.html#prerequisites" style="font-size: 80%;"><b>Prerequisites</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs011.html#learning-outcomes" style="font-size: 80%;"><b>Learning outcomes</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs012.html#topics-covered-in-this-course-statistical-analysis-and-optimization-of-data" style="font-size: 80%;"><b>Topics covered in this course: Statistical analysis and optimization of data</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs013.html#topics-covered-in-this-course-machine-learning" style="font-size: 80%;"><b>Topics covered in this course: Machine Learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs014.html#extremely-useful-tools-strongly-recommended" style="font-size: 80%;"><b>Extremely useful tools, strongly recommended</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs015.html#other-courses-on-data-science-and-machine-learning-at-uio" style="font-size: 80%;"><b>Other courses on Data science and Machine Learning at UiO</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs016.html#introduction" style="font-size: 80%;"><b>Introduction</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs017.html#what-is-machine-learning" style="font-size: 80%;"><b>What is Machine Learning?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs018.html#types-of-machine-learning" style="font-size: 80%;"><b>Types of Machine Learning</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs019.html#essential-elements-of-ml" style="font-size: 80%;"><b>Essential elements of ML</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs020.html#an-optimization-minimization-problem" style="font-size: 80%;"><b>An optimization/minimization problem</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs021.html#a-frequentist-approach-to-data-analysis" style="font-size: 80%;"><b>A Frequentist approach to data analysis</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs022.html#what-is-a-good-model" style="font-size: 80%;"><b>What is a good model?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs023.html#what-is-a-good-model-can-we-define-it" style="font-size: 80%;"><b>What is a good model? Can we define it?</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs024.html#software-and-needed-installations" style="font-size: 80%;"><b>Software and needed installations</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs025.html#python-installers" style="font-size: 80%;"><b>Python installers</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs026.html#useful-python-libraries" style="font-size: 80%;"><b>Useful Python libraries</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs027.html#installing-r-c-cython-or-julia" style="font-size: 80%;"><b>Installing R, C++, cython or Julia</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs028.html#installing-r-c-cython-numba-etc" style="font-size: 80%;"><b>Installing R, C++, cython, Numba etc</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs029.html#numpy-examples-and-important-matrix-and-vector-handling-packages" style="font-size: 80%;"><b>Numpy examples and Important Matrix and vector handling packages</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs030.html#basic-matrix-features" style="font-size: 80%;"><b>Basic Matrix Features</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs031.html#some-famous-matrices" style="font-size: 80%;"> Some famous Matrices</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs032.html#more-basic-matrix-features" style="font-size: 80%;"> More Basic Matrix Features</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs033.html#numpy-and-arrays" style="font-size: 80%;"><b>Numpy and arrays</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs034.html#matrices-in-python" style="font-size: 80%;"><b>Matrices in Python</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs035.html#meet-the-pandas" style="font-size: 80%;"><b>Meet the Pandas</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs038.html#friday-august-27" style="font-size: 80%;"><b>Friday August 27</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs037.html#reading-data-and-fitting" style="font-size: 80%;"><b>Reading Data and fitting</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs038.html#friday-august-27" style="font-size: 80%;"><b>Friday August 27</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs039.html#simple-linear-regression-model-using-_scikit-learn_" style="font-size: 80%;"> Simple linear regression model using <b>scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs039.html#to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies" style="font-size: 80%;"> To our real data: nuclear binding energies. Brief reminder on masses and binding energies</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs039.html#organizing-our-data" style="font-size: 80%;"> Organizing our data</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs039.html#seeing-the-wood-for-the-trees" style="font-size: 80%;"> Seeing the wood for the trees</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs039.html#and-what-about-using-neural-networks" style="font-size: 80%;"> And what about using neural networks?</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._week34-bs039.html#a-first-summary" style="font-size: 80%;"><b>A first summary</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="#exercises-for-week-36" style="font-size: 80%;"><b>Exercises for week 36</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="#exercise-1-setting-up-various-python-environments" style="font-size: 80%;"><b>Exercise 1: Setting up various Python environments</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="#exercise-2-making-your-own-data-and-exploring-scikit-learn" style="font-size: 80%;"><b>Exercise 2: making your own data and exploring scikit-learn</b></a></li>
|
||||
<!-- navigation toc: --> <li><a href="#exercise-3-normalizing-our-data" style="font-size: 80%;"><b>Exercise 3: Normalizing our data</b></a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
</ul>
|
||||
</div>
|
||||
</div>
|
||||
</div> <!-- end of navigation bar -->
|
||||
|
||||
<div class="container">
|
||||
|
||||
<p> </p><p> </p><p> </p> <!-- add vertical space -->
|
||||
|
||||
<a name="part0040"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="exercises-for-week-36" class="anchor">Exercises for week 36 </h2>
|
||||
Here are three possible exercises for week 36 and the lab sessions of Wednesday September 1..
|
||||
|
||||
<p>
|
||||
<!-- --- begin exercise --- -->
|
||||
|
||||
<h2 id="exercise-1-setting-up-various-python-environments" class="anchor">Exercise 1: Setting up various Python environments </h2>
|
||||
|
||||
<p>
|
||||
The first exercise here is of a mere technical art. We want you to have
|
||||
|
||||
<ul>
|
||||
<li> git as a version control software and to establish a user account on a provider like GitHub. Other providers like GitLab etc are equally fine. You can also use the University of Oslo <a href="https://www.uio.no/tjenester/it/maskin/filer/versjonskontroll/github.html" target="_self">GitHub facilities</a>.</li>
|
||||
<li> Install various Python packages</li>
|
||||
</ul>
|
||||
|
||||
We will make extensive use of Python as programming language and its
|
||||
myriad of available libraries. You will find
|
||||
IPython/Jupyter notebooks invaluable in your work. You can run <b>R</b>
|
||||
codes in the Jupyter/IPython notebooks, with the immediate benefit of
|
||||
visualizing your data. You can also use compiled languages like C++,
|
||||
Rust, Fortran etc if you prefer. The focus in these lectures will be
|
||||
on Python.
|
||||
|
||||
<p>
|
||||
If you have Python installed (we recommend Python3) and you feel
|
||||
pretty familiar with installing different packages, we recommend that
|
||||
you install the following Python packages via <b>pip</b> as
|
||||
|
||||
<ol>
|
||||
<li> pip install numpy scipy matplotlib ipython scikit-learn sympy pandas pillow</li>
|
||||
</ol>
|
||||
|
||||
For <b>Tensorflow</b>, we recommend following the instructions in the text of
|
||||
<a href="http://shop.oreilly.com/product/0636920052289.do" target="_self">Aurelien Geron, Hands‑On Machine Learning with Scikit‑Learn and TensorFlow, O'Reilly</a>
|
||||
|
||||
<p>
|
||||
We will come back to <b>tensorflow</b> later.
|
||||
|
||||
<p>
|
||||
For Python3, replace <b>pip</b> with <b>pip3</b>.
|
||||
|
||||
<p>
|
||||
For OSX users we recommend, after having installed Xcode, to
|
||||
install <b>brew</b>. Brew allows for a seamless installation of additional
|
||||
software via for example
|
||||
|
||||
<ol>
|
||||
<li> brew install python3</li>
|
||||
</ol>
|
||||
|
||||
For Linux users, with its variety of distributions like for example the widely popular Ubuntu distribution,
|
||||
you can use <b>pip</b> as well and simply install Python as
|
||||
|
||||
<ol>
|
||||
<li> sudo apt-get install python3 (or python for Python2.7)</li>
|
||||
</ol>
|
||||
|
||||
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
|
||||
|
||||
<ul>
|
||||
<li> <a href="https://docs.anaconda.com/" target="_self">Anaconda</a>,</li>
|
||||
</ul>
|
||||
|
||||
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 <b>conda</b>.
|
||||
|
||||
<ul>
|
||||
<li> <a href="https://www.enthought.com/product/canopy/" target="_self">Enthought canopy</a></li>
|
||||
</ul>
|
||||
|
||||
is a Python
|
||||
distribution for scientific and analytic computing distribution and
|
||||
analysis environment, available for free and under a commercial
|
||||
license.
|
||||
|
||||
<p>
|
||||
We recommend using <b>Anaconda</b> if you are not too familiar with setting paths in a terminal environment.
|
||||
|
||||
<p>
|
||||
<!-- --- end exercise --- -->
|
||||
|
||||
<p>
|
||||
<!-- --- begin exercise --- -->
|
||||
|
||||
<h2 id="exercise-2-making-your-own-data-and-exploring-scikit-learn" class="anchor">Exercise 2: making your own data and exploring scikit-learn </h2>
|
||||
|
||||
<p>
|
||||
We will generate our own dataset for a function \( y(x) \) where \( x \in [0,1] \) and defined by random numbers computed with the uniform distribution. The function \( y \) is a quadratic polynomial in \( x \) with added stochastic noise according to the normal distribution \( \cal {N}(0,1) \).
|
||||
The following simple Python instructions define our \( x \) and \( y \) values (with 100 data points).
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span>x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(<span style="color: #666666">100</span>,<span style="color: #666666">1</span>)
|
||||
y <span style="color: #666666">=</span> <span style="color: #666666">2.0+5*</span>x<span style="color: #666666">*</span>x<span style="color: #666666">+0.1*</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(<span style="color: #666666">100</span>,<span style="color: #666666">1</span>)
|
||||
</pre></div>
|
||||
<ol>
|
||||
<li> Write your own code (following the examples under the <a href="https://compphysics.github.io/MachineLearning/doc/LectureNotes/_build/html/chapter1.html" target="_self">regression notes</a>) for computing the parametrization of the data set fitting a second-order polynomial.</li>
|
||||
<li> Use thereafter <b>scikit-learn</b> (see again the examples in the regression slides) and compare with your own code.</li>
|
||||
<li> Using scikit-learn, compute also the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error defined as</li>
|
||||
</ol>
|
||||
|
||||
$$ MSE(\hat{y},\hat{\tilde{y}}) = \frac{1}{n}
|
||||
\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2,
|
||||
$$
|
||||
|
||||
and the \( R^2 \) score function.
|
||||
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
|
||||
$$
|
||||
\bar{y} = \frac{1}{n} \sum_{i=0}^{n - 1} y_i.
|
||||
$$
|
||||
|
||||
You can use the functionality included in scikit-learn. If you feel for it, you can use your own program and define functions which compute the above two functions.
|
||||
Discuss the meaning of these results. Try also to vary the coefficient in front of the added stochastic noise term and discuss the quality of the fits.
|
||||
|
||||
<p>
|
||||
<!-- --- begin solution of exercise --- -->
|
||||
|
||||
<p>
|
||||
<a class="glyphicon glyphicon-hand-right showdetails" data-toggle="collapse"
|
||||
data-target="#exer_2_1" style="font-size: 80%;"></a>
|
||||
<a href="#exer_2_1" data-toggle="collapse">
|
||||
<b>Solution.</b>
|
||||
</a>
|
||||
<div class="collapse-group">
|
||||
<p><div class="collapse" id="exer_2_1">
|
||||
|
||||
<p>
|
||||
The code here is an example of where we define our own design matrix and fit parameters \( \beta \).
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">os</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">pandas</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">pd</span>
|
||||
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
|
||||
|
||||
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">save_fig</span>(fig_id):
|
||||
plt<span style="color: #666666">.</span>savefig(image_path(fig_id) <span style="color: #666666">+</span> <span style="color: #BA2121">".png"</span>, <span style="color: #008000">format</span><span style="color: #666666">=</span><span style="color: #BA2121">'png'</span>)
|
||||
|
||||
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">R2</span>(y_data, y_model):
|
||||
<span style="color: #008000; font-weight: bold">return</span> <span style="color: #666666">1</span> <span style="color: #666666">-</span> np<span style="color: #666666">.</span>sum((y_data <span style="color: #666666">-</span> y_model) <span style="color: #666666">**</span> <span style="color: #666666">2</span>) <span style="color: #666666">/</span> np<span style="color: #666666">.</span>sum((y_data <span style="color: #666666">-</span> np<span style="color: #666666">.</span>mean(y_data)) <span style="color: #666666">**</span> <span style="color: #666666">2</span>)
|
||||
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">MSE</span>(y_data,y_model):
|
||||
n <span style="color: #666666">=</span> np<span style="color: #666666">.</span>size(y_model)
|
||||
<span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>sum((y_data<span style="color: #666666">-</span>y_model)<span style="color: #666666">**2</span>)<span style="color: #666666">/</span>n
|
||||
|
||||
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(<span style="color: #666666">100</span>)
|
||||
y <span style="color: #666666">=</span> <span style="color: #666666">2.0+5*</span>x<span style="color: #666666">*</span>x<span style="color: #666666">+0.1*</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(<span style="color: #666666">100</span>)
|
||||
|
||||
|
||||
<span style="color: #408080; font-style: italic"># The design matrix now as function of a given polynomial</span>
|
||||
X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((<span style="color: #008000">len</span>(x),<span style="color: #666666">3</span>))
|
||||
X[:,<span style="color: #666666">0</span>] <span style="color: #666666">=</span> <span style="color: #666666">1.0</span>
|
||||
X[:,<span style="color: #666666">1</span>] <span style="color: #666666">=</span> x
|
||||
X[:,<span style="color: #666666">2</span>] <span style="color: #666666">=</span> x<span style="color: #666666">**2</span>
|
||||
<span style="color: #408080; font-style: italic"># We split the data in test and training data</span>
|
||||
X_train, X_test, y_train, y_test <span style="color: #666666">=</span> train_test_split(X, y, test_size<span style="color: #666666">=0.2</span>)
|
||||
<span style="color: #408080; font-style: italic"># matrix inversion to find beta</span>
|
||||
beta <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>inv(X_train<span style="color: #666666">.</span>T <span style="color: #666666">@</span> X_train) <span style="color: #666666">@</span> X_train<span style="color: #666666">.</span>T <span style="color: #666666">@</span> y_train
|
||||
<span style="color: #008000">print</span>(beta)
|
||||
<span style="color: #408080; font-style: italic"># and then make the prediction</span>
|
||||
ytilde <span style="color: #666666">=</span> X_train <span style="color: #666666">@</span> beta
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">"Training R2"</span>)
|
||||
<span style="color: #008000">print</span>(R2(y_train,ytilde))
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">"Training MSE"</span>)
|
||||
<span style="color: #008000">print</span>(MSE(y_train,ytilde))
|
||||
ypredict <span style="color: #666666">=</span> X_test <span style="color: #666666">@</span> beta
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">"Test R2"</span>)
|
||||
<span style="color: #008000">print</span>(R2(y_test,ypredict))
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">"Test MSE"</span>)
|
||||
<span style="color: #008000">print</span>(MSE(y_test,ypredict))
|
||||
</pre></div>
|
||||
<p>
|
||||
</div></p>
|
||||
</div>
|
||||
</p>
|
||||
|
||||
<p>
|
||||
<!-- --- end solution of exercise --- -->
|
||||
|
||||
<p>
|
||||
<!-- --- end exercise --- -->
|
||||
|
||||
<p>
|
||||
<!-- --- begin exercise --- -->
|
||||
|
||||
<h2 id="exercise-3-normalizing-our-data" class="anchor">Exercise 3: Normalizing our data </h2>
|
||||
|
||||
<p>
|
||||
A much used approach before starting to train the data is to preprocess our
|
||||
data. Normally the data may need a rescaling and/or may be sensitive
|
||||
to extreme values. Scaling the data renders our inputs much more
|
||||
suitable for the algorithms we want to employ.
|
||||
|
||||
<p>
|
||||
<b>Scikit-Learn</b> has several functions which allow us to rescale the
|
||||
data, normally resulting in much better results in terms of various
|
||||
accuracy scores. The <b>StandardScaler</b> function in <b>Scikit-Learn</b>
|
||||
ensures that for each feature/predictor we study the mean value is
|
||||
zero and the variance is one (every column in the design/feature
|
||||
matrix). This scaling has the drawback that it does not ensure that
|
||||
we have a particular maximum or minimum in our data set. Another
|
||||
function included in <b>Scikit-Learn</b> is the <b>MinMaxScaler</b> which
|
||||
ensures that all features are exactly between \( 0 \) and \( 1 \). The
|
||||
|
||||
<p>
|
||||
The <b>Normalizer</b> scales each data
|
||||
point such that the feature vector has a euclidean length of one. In other words, it
|
||||
projects a data point on the circle (or sphere in the case of higher dimensions) with a
|
||||
radius of 1. This means every data point is scaled by a different number (by the
|
||||
inverse of it’s length).
|
||||
This normalization is often used when only the direction (or angle) of the data matters,
|
||||
not the length of the feature vector.
|
||||
|
||||
<p>
|
||||
The <b>RobustScaler</b> works similarly to the StandardScaler in that it
|
||||
ensures statistical properties for each feature that guarantee that
|
||||
they are on the same scale. However, the RobustScaler uses the median
|
||||
and quartiles, instead of mean and variance. This makes the
|
||||
RobustScaler ignore data points that are very different from the rest
|
||||
(like measurement errors). These odd data points are also called
|
||||
outliers, and might often lead to trouble for other scaling
|
||||
techniques.
|
||||
|
||||
<p>
|
||||
It also common to split the data in a <b>training</b> set and a <b>testing</b> set. A typical split is to use \( 80\% \) of the data for training and the rest
|
||||
for testing. This can be done as follows with our design matrix \( \boldsymbol{X} \) and data \( \boldsymbol{y} \) (remember to import <b>scikit-learn</b>)
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span><span style="color: #408080; font-style: italic"># split in training and test data</span>
|
||||
X_train, X_test, y_train, y_test <span style="color: #666666">=</span> train_test_split(X,y,test_size<span style="color: #666666">=0.2</span>)
|
||||
</pre></div>
|
||||
<p>
|
||||
Then we can use the standard scaler to scale our data as
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span>scaler <span style="color: #666666">=</span> StandardScaler()
|
||||
scaler<span style="color: #666666">.</span>fit(X_train)
|
||||
X_train_scaled <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>transform(X_train)
|
||||
X_test_scaled <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>transform(X_test)
|
||||
</pre></div>
|
||||
<p>
|
||||
In this exercise we want you to to compute the MSE for the training
|
||||
data and the test data as function of the complexity of a polynomial,
|
||||
that is the degree of a given polynomial. We want you also to compute the \( R2 \) score as function of the complexity of the model for both training data and test data. You should also run the calculation with and without scaling.
|
||||
|
||||
<p>
|
||||
One of
|
||||
the aims is to reproduce Figure 2.11 of <a href="https://github.com/CompPhysics/MLErasmus/blob/master/doc/Textbooks/elementsstat.pdf" target="_self">Hastie et al</a>.
|
||||
|
||||
<p>
|
||||
Our data is defined by \( x\in [-3,3] \) with a total of for example \( 100 \) data points.
|
||||
<p>
|
||||
|
||||
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
||||
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%;"><span></span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed()
|
||||
n <span style="color: #666666">=</span> <span style="color: #666666">100</span>
|
||||
maxdegree <span style="color: #666666">=</span> <span style="color: #666666">14</span>
|
||||
<span style="color: #408080; font-style: italic"># Make data set.</span>
|
||||
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(<span style="color: #666666">-3</span>, <span style="color: #666666">3</span>, n)<span style="color: #666666">.</span>reshape(<span style="color: #666666">-1</span>, <span style="color: #666666">1</span>)
|
||||
y <span style="color: #666666">=</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>x<span style="color: #666666">**2</span>) <span style="color: #666666">+</span> <span style="color: #666666">1.5</span> <span style="color: #666666">*</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>(x<span style="color: #666666">-2</span>)<span style="color: #666666">**2</span>)<span style="color: #666666">+</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>normal(<span style="color: #666666">0</span>, <span style="color: #666666">0.1</span>, x<span style="color: #666666">.</span>shape)
|
||||
</pre></div>
|
||||
<p>
|
||||
where \( y \) is the function we want to fit with a given polynomial.
|
||||
|
||||
<p>
|
||||
<b>a)</b>
|
||||
Write a first code which sets up a design matrix \( X \) defined by a fifth-order polynomial. Scale your data and split it in training and test data.
|
||||
|
||||
<p>
|
||||
<b>b)</b>
|
||||
Perform an ordinary least squares and compute the means squared error and the \( R2 \) factor for the training data and the test data, with and without scaling.
|
||||
|
||||
<p>
|
||||
<b>c)</b>
|
||||
Add now a model which allows you to make polynomials up to degree \( 15 \). Perform a standard OLS fitting of the training data and compute the MSE and \( R2 \) for the training and test data and plot both test and training data MSE and \( R2 \) as functions of the polynomial degree. Compare what you see with Figure 2.11 of Hastie et al. Comment your results. For which polynomial degree do you find an optimal MSE (smallest value)?
|
||||
|
||||
<p>
|
||||
<!-- --- begin solution of exercise --- -->
|
||||
|
||||
<p>
|
||||
<a class="glyphicon glyphicon-hand-right showdetails" data-toggle="collapse"
|
||||
data-target="#exer_3_1" style="font-size: 80%;"></a>
|
||||
<a href="#exer_3_1" data-toggle="collapse">
|
||||
<b>Solution.</b>
|
||||
</a>
|
||||
<div class="collapse-group">
|
||||
<p><div class="collapse" id="exer_3_1">
|
||||
|
||||
<p>
|
||||
Here you simply need to change the degree of the polynomial in the above code to \( n=15 \).
|
||||
</div></p>
|
||||
</div>
|
||||
</p>
|
||||
|
||||
<p>
|
||||
<!-- --- end solution of exercise --- -->
|
||||
|
||||
<p>
|
||||
<!-- --- end exercise --- -->
|
||||
|
||||
<p>
|
||||
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||||
|
||||
Reference in New Issue
Block a user