mhjensen
376 lines
24 KiB
HTML
376 lines
24 KiB
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'sections': [('Overview of first week', 2, None, '___sec0'),
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('Matrices in Python', 2, None, '___sec26'),
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('Meet the Pandas', 2, None, '___sec27'),
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('Friday August 21', 2, None, '___sec28'),
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('Reading Data and fitting', 2, None, '___sec29'),
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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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<!-- navigation toc: --> <li><a href="._week34-bs001.html#___sec0" style="font-size: 80%;"><b>Overview of first week</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs002.html#___sec1" style="font-size: 80%;"><b>Thursday August 20</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs003.html#___sec2" style="font-size: 80%;"><b>Lectures and ComputerLab</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs006.html#___sec5" style="font-size: 80%;"><b>Deadlines for projects (tentative)</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs007.html#___sec6" style="font-size: 80%;"><b>Recommended textbooks</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs008.html#___sec7" style="font-size: 80%;"><b>Prerequisites</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs009.html#___sec8" style="font-size: 80%;"><b>Learning outcomes</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs010.html#___sec9" 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-bs011.html#___sec10" 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-bs012.html#___sec11" style="font-size: 80%;"><b>Extremely useful tools, strongly recommended</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs013.html#___sec12" 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-bs014.html#___sec13" style="font-size: 80%;"><b>Introduction</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs015.html#___sec14" style="font-size: 80%;"><b>What is Machine Learning?</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs016.html#___sec15" style="font-size: 80%;"><b>Types of Machine Learning</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs017.html#___sec16" style="font-size: 80%;"><b>Software and needed installations</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs018.html#___sec17" style="font-size: 80%;"><b>Python installers</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs019.html#___sec18" style="font-size: 80%;"><b>Useful Python libraries</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs020.html#___sec19" style="font-size: 80%;"><b>Installing R, C++, cython or Julia</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs021.html#___sec20" style="font-size: 80%;"><b>Installing R, C++, cython, Numba etc</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs022.html#___sec21" 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-bs023.html#___sec22" style="font-size: 80%;"><b>Basic Matrix Features</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs024.html#___sec23" style="font-size: 80%;"> Some famous Matrices</a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs025.html#___sec24" style="font-size: 80%;"> More Basic Matrix Features</a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs026.html#___sec25" style="font-size: 80%;"><b>Numpy and arrays</b></a></li>
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<!-- navigation toc: --> <li><a href="#___sec26" style="font-size: 80%;"><b>Matrices in Python</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs028.html#___sec27" style="font-size: 80%;"><b>Meet the Pandas</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs029.html#___sec28" style="font-size: 80%;"><b>Friday August 21</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs030.html#___sec29" style="font-size: 80%;"><b>Reading Data and fitting</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs031.html#___sec30" style="font-size: 80%;"><b>Friday August 21</b></a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs032.html#___sec31" 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-bs032.html#___sec32" 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-bs032.html#___sec33" style="font-size: 80%;"> Organizing our data</a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs032.html#___sec34" style="font-size: 80%;"> Seeing the wood for the trees</a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs032.html#___sec35" style="font-size: 80%;"> And what about using neural networks?</a></li>
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<!-- navigation toc: --> <li><a href="._week34-bs032.html#___sec36" style="font-size: 80%;"><b>A first summary</b></a></li>
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<a name="part0027"></a>
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<!-- !split -->
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<h2 id="___sec26" class="anchor">Matrices in Python </h2>
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<p>
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Having defined vectors, we are now ready to try out matrices. We can
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define a \( 3 \times 3 \) real matrix \( \hat{A} \) 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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<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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<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>
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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; font-weight: bold">print</span>(A)
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</pre></div>
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<p>
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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
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<p>
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<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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<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>
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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; font-weight: bold">print</span>(A[:,<span style="color: #666666">0</span>])
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</pre></div>
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<p>
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We can continue this was by printing out other columns or rows. The example here prints out the second column
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<p>
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<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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<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>
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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; font-weight: bold">print</span>(A[<span style="color: #666666">1</span>,:])
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</pre></div>
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<p>
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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
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<p>
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<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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<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>
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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; font-weight: bold">print</span>(A)
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</pre></div>
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<p>
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or initializing all elements to
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<p>
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<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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<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>
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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; font-weight: bold">print</span>(A)
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</pre></div>
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<p>
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or as unitarily distributed random numbers (see the material on random number generators in the statistics part)
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<p>
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<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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<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>
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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; font-weight: bold">print</span>(A)
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</pre></div>
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<p>
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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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\( \hat{x}, \hat{y}, \hat{z} \) with \( n \) elements each. The covariance matrix is defined as
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$$
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\hat{\Sigma} = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\
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\sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\
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\sigma_{zx} & \sigma_{zy} & \sigma_{zz}
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\end{bmatrix},
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$$
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where for example
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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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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 \( \hat{W} \)
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$$
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\hat{W} = \begin{bmatrix} x_0 & y_0 & z_0 \\
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x_1 & y_1 & z_1 \\
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x_2 & y_2 & z_2 \\
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\dots & \dots & \dots \\
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x_{n-2} & y_{n-2} & z_{n-2} \\
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x_{n-1} & y_{n-1} & z_{n-1}
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\end{bmatrix},
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$$
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<p>
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which in turn is converted into into the \( 3\times 3 \) covariance matrix
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\( \hat{\Sigma} \) 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 \( \hat{x} \) 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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<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><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; font-weight: bold">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; font-weight: bold">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; font-weight: bold">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; font-weight: bold">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; font-weight: bold">print</span>(Eigvals)
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</pre></div>
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<p>
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<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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<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>
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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; font-weight: bold">print</span>(eye)
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sparse_mtx <span style="color: #666666">=</span> sparse<span style="color: #666666">.</span>csr_matrix(eye)
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<span style="color: #008000; font-weight: bold">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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</pre></div>
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<p>
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