Morten Hjorth-Jensen
669 lines
41 KiB
HTML
669 lines
41 KiB
HTML
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('Reading Recommendations', 2, None, 'reading-recommendations'),
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('More Basic Matrix Features',
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('Meet the Pandas', 2, None, 'meet-the-pandas'),
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('Simple linear regression model using _scikit-learn_',
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('Organizing our data', 3, None, 'organizing-our-data'),
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('Seeing the wood for the trees',
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('And what about using neural networks?',
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('A first summary', 2, None, 'a-first-summary'),
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('Why Linear Regression (aka Ordinary Least Squares and family)',
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2,
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('Regression analysis, overarching aims',
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('Regression analysis, overarching aims II',
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('Rewriting the fitting procedure as a linear algebra problem',
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('Rewriting the fitting procedure as a linear algebra problem, '
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('Optimizing our parameters',
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('Own code for Ordinary Least Squares',
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('Adding error analysis and training set up',
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('The $\\chi^2$ function', 2, None, 'the-chi-2-function'),
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('The $\\chi^2$ function', 2, None, 'the-chi-2-function'),
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('The $\\chi^2$ function', 2, None, 'the-chi-2-function'),
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('The $\\chi^2$ function', 2, None, 'the-chi-2-function'),
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('The $\\chi^2$ function', 2, None, 'the-chi-2-function'),
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('The $\\chi^2$ function', 2, None, 'the-chi-2-function'),
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('Fitting an Equation of State for Dense Nuclear Matter',
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|
|
None,
|
|
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('The code', 2, None, 'the-code'),
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('Splitting our Data in Training and Test data',
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('Exercise 2: making your own data and exploring scikit-learn',
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('Exercise 3: Normalizing our data',
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'exercise-3-normalizing-our-data')]}
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<a class="navbar-brand" href="week34-bs.html">Week 34: 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">
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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#schedule-first-week" style="font-size: 80%;"><b>Schedule first week</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs003.html#reading-recommendations" style="font-size: 80%;"><b>Reading Recommendations</b></a></li>
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<!-- 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#communication-channels" style="font-size: 80%;"><b>Communication channels</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>
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<!-- navigation toc: --> <li><a href="._week34-bs008.html#deadlines-for-projects-tentative" style="font-size: 80%;"><b>Deadlines for projects (tentative)</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-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>
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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" style="font-size: 80%;"><b>Topics covered in this course</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>
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<!-- 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>
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<!-- 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>
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<!-- 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>
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<!-- 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>
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<!-- 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>
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<!-- 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>
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<!-- 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>
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<!-- 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>
|
|
<!-- navigation toc: --> <li><a href="#matrices-in-python" style="font-size: 80%;"><b>Matrices in Python</b></a></li>
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|
<!-- 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-bs036.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#simple-linear-regression-model-using-scikit-learn" style="font-size: 80%;"> Simple linear regression model using <b>scikit-learn</b></a></li>
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|
<!-- navigation toc: --> <li><a href="._week34-bs037.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-bs037.html#organizing-our-data" style="font-size: 80%;"> Organizing our data</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week34-bs037.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-bs037.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-bs037.html#a-first-summary" style="font-size: 80%;"><b>A first summary</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week34-bs038.html#why-linear-regression-aka-ordinary-least-squares-and-family" style="font-size: 80%;"><b>Why Linear Regression (aka Ordinary Least Squares and family)</b></a></li>
|
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<!-- navigation toc: --> <li><a href="._week34-bs039.html#regression-analysis-overarching-aims" style="font-size: 80%;"><b>Regression analysis, overarching aims</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs040.html#regression-analysis-overarching-aims-ii" style="font-size: 80%;"><b>Regression analysis, overarching aims II</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs041.html#examples" style="font-size: 80%;"><b>Examples</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs042.html#general-linear-models" style="font-size: 80%;"><b>General linear models</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs043.html#rewriting-the-fitting-procedure-as-a-linear-algebra-problem" style="font-size: 80%;"><b>Rewriting the fitting procedure as a linear algebra problem</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week34-bs044.html#rewriting-the-fitting-procedure-as-a-linear-algebra-problem-more-details" style="font-size: 80%;"><b>Rewriting the fitting procedure as a linear algebra problem, more details</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs046.html#generalizing-the-fitting-procedure-as-a-linear-algebra-problem" style="font-size: 80%;"><b>Generalizing the fitting procedure as a linear algebra problem</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week34-bs046.html#generalizing-the-fitting-procedure-as-a-linear-algebra-problem" style="font-size: 80%;"><b>Generalizing the fitting procedure as a linear algebra problem</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs047.html#optimizing-our-parameters" style="font-size: 80%;"><b>Optimizing our parameters</b></a></li>
|
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<!-- navigation toc: --> <li><a href="._week34-bs048.html#our-model-for-the-nuclear-binding-energies" style="font-size: 80%;"><b>Our model for the nuclear binding energies</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs049.html#optimizing-our-parameters-more-details" style="font-size: 80%;"><b>Optimizing our parameters, more details</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs053.html#interpretations-and-optimizing-our-parameters" style="font-size: 80%;"><b>Interpretations and optimizing our parameters</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs053.html#interpretations-and-optimizing-our-parameters" style="font-size: 80%;"><b>Interpretations and optimizing our parameters</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs052.html#some-useful-matrix-and-vector-expressions" style="font-size: 80%;"><b>Some useful matrix and vector expressions</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs053.html#interpretations-and-optimizing-our-parameters" style="font-size: 80%;"><b>Interpretations and optimizing our parameters</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs054.html#own-code-for-ordinary-least-squares" style="font-size: 80%;"><b>Own code for Ordinary Least Squares</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs055.html#adding-error-analysis-and-training-set-up" style="font-size: 80%;"><b>Adding error analysis and training set up</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs061.html#the-chi-2-function" style="font-size: 80%;"><b>The \( \chi^2 \) function</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs061.html#the-chi-2-function" style="font-size: 80%;"><b>The \( \chi^2 \) function</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs061.html#the-chi-2-function" style="font-size: 80%;"><b>The \( \chi^2 \) function</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs061.html#the-chi-2-function" style="font-size: 80%;"><b>The \( \chi^2 \) function</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs061.html#the-chi-2-function" style="font-size: 80%;"><b>The \( \chi^2 \) function</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs061.html#the-chi-2-function" style="font-size: 80%;"><b>The \( \chi^2 \) function</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs062.html#fitting-an-equation-of-state-for-dense-nuclear-matter" style="font-size: 80%;"><b>Fitting an Equation of State for Dense Nuclear Matter</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs063.html#the-code" style="font-size: 80%;"><b>The code</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs064.html#splitting-our-data-in-training-and-test-data" style="font-size: 80%;"><b>Splitting our Data in Training and Test data</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs065.html#exercises" style="font-size: 80%;"><b>Exercises</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs065.html#exercise-1-setting-up-various-python-environments" style="font-size: 80%;"><b>Exercise 1: Setting up various Python environments</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs065.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>
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<!-- navigation toc: --> <li><a href="._week34-bs065.html#exercise-3-normalizing-our-data" style="font-size: 80%;"><b>Exercise 3: Normalizing our data</b></a></li>
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</ul>
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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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<a name="part0034"></a>
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<h2 id="matrices-in-python" class="anchor">Matrices in Python </h2>
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<p>Having defined vectors, we are now ready to try out matrices. We can
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define a \( 3 \times 3 \) real matrix \( \boldsymbol{A} \) as (recall that we user
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lowercase letters for vectors and uppercase letters for matrices)
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</p>
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<pre style="line-height: 125%;"><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>
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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>] ]))
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<span style="color: #008000">print</span>(A)
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</pre>
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<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>
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<pre style="line-height: 125%;"><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>
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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>] ]))
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<span style="color: #408080; font-style: italic"># print the first column, row-major order and elements start with 0</span>
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<span style="color: #008000">print</span>(A[:,<span style="color: #666666">0</span>])
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</pre>
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<p>We can continue this was by printing out other columns or rows. The example here prints out the second column</p>
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<pre style="line-height: 125%;"><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>
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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>] ]))
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<span style="color: #408080; font-style: italic"># print the first column, row-major order and elements start with 0</span>
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<span style="color: #008000">print</span>(A[<span style="color: #666666">1</span>,:])
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</pre>
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<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>
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<pre style="line-height: 125%;"><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>
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n <span style="color: #666666">=</span> <span style="color: #666666">10</span>
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<span style="color: #408080; font-style: italic"># define a matrix of dimension 10 x 10 and set all elements to zero</span>
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A <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros( (n, n) )
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<span style="color: #008000">print</span>(A)
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</pre>
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<p>or initializing all elements to </p>
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<pre style="line-height: 125%;"><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>
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n <span style="color: #666666">=</span> <span style="color: #666666">10</span>
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<span style="color: #408080; font-style: italic"># define a matrix of dimension 10 x 10 and set all elements to one</span>
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A <span style="color: #666666">=</span> np<span style="color: #666666">.</span>ones( (n, n) )
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<span style="color: #008000">print</span>(A)
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</pre>
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<p>or as unitarily distributed random numbers (see the material on random number generators in the statistics part)</p>
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<pre style="line-height: 125%;"><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>
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n <span style="color: #666666">=</span> <span style="color: #666666">10</span>
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<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>
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A <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(n, n)
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<span style="color: #008000">print</span>(A)
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</pre>
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<p>As we will see throughout these lectures, there are several extremely useful functionalities in Numpy.
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As an example, consider the discussion of the covariance matrix. Suppose we have defined three vectors
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\( \boldsymbol{x}, \boldsymbol{y}, \boldsymbol{z} \) with \( n \) elements each. The covariance matrix is defined as
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</p>
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$$
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\boldsymbol{\Sigma} = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\
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\sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\
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\sigma_{zx} & \sigma_{zy} & \sigma_{zz}
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\end{bmatrix},
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$$
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<p>where for example</p>
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$$
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\sigma_{xy} =\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})(y_i- \overline{y}).
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$$
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<p>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.
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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 \( \boldsymbol{W} \)
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</p>
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$$
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\boldsymbol{W} = \begin{bmatrix} x_0 & x_1 & x_2 & \dots & x_{n-2} & x_{n-1} \\
|
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y_0 & y_1 & y_2 & \dots & y_{n-2} & y_{n-1} \\
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z_0 & z_1 & z_2 & \dots & z_{n-2} & z_{n-1} \\
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\end{bmatrix},
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$$
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<p>which in turn is converted into into the \( 3\times 3 \) covariance matrix
|
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\( \boldsymbol{\Sigma} \) via the Numpy function <b>np.cov()</b>. We note that we can also calculate
|
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the mean value of each set of samples \( \boldsymbol{x} \) etc using the Numpy
|
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function <b>np.mean(x)</b>. We can also extract the eigenvalues of the
|
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covariance matrix through the <b>np.linalg.eig()</b> function.
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</p>
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<pre style="line-height: 125%;"><span style="color: #408080; font-style: italic"># Importing various packages</span>
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<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>
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n <span style="color: #666666">=</span> <span style="color: #666666">100</span>
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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)
|
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<span style="color: #008000">print</span>(np<span style="color: #666666">.</span>mean(x))
|
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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)
|
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<span style="color: #008000">print</span>(np<span style="color: #666666">.</span>mean(y))
|
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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)
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<span style="color: #008000">print</span>(np<span style="color: #666666">.</span>mean(z))
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W <span style="color: #666666">=</span> np<span style="color: #666666">.</span>vstack((x, y, z))
|
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Sigma <span style="color: #666666">=</span> np<span style="color: #666666">.</span>cov(W)
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<span style="color: #008000">print</span>(Sigma)
|
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Eigvals, Eigvecs <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>eig(Sigma)
|
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<span style="color: #008000">print</span>(Eigvals)
|
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<pre style="line-height: 125%;"><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>
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<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>
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<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
|
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eye <span style="color: #666666">=</span> np<span style="color: #666666">.</span>eye(<span style="color: #666666">4</span>)
|
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<span style="color: #008000">print</span>(eye)
|
|
sparse_mtx <span style="color: #666666">=</span> sparse<span style="color: #666666">.</span>csr_matrix(eye)
|
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<span style="color: #008000">print</span>(sparse_mtx)
|
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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>)
|
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y <span style="color: #666666">=</span> np<span style="color: #666666">.</span>sin(x)
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plt<span style="color: #666666">.</span>plot(x,y,marker<span style="color: #666666">=</span><span style="color: #BA2121">'x'</span>)
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plt<span style="color: #666666">.</span>show()
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