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<!-- navigation toc: --> <li><a href="._week35-bs008.html#___sec7" style="font-size: 80%;">Rewriting the fitting procedure as a linear algebra problem</a></li>
<!-- navigation toc: --> <li><a href="._week35-bs009.html#___sec8" style="font-size: 80%;">Rewriting the fitting procedure as a linear algebra problem, more details</a></li>
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<h2 id="___sec12" class="anchor">Our model for the nuclear binding energies </h2>
<p>
In our <a href="https://compphysics.github.io/MachineLearning/doc/pub/How2ReadData/html/How2ReadData.html" target="_self">introductory notes</a> we looked at the so-called <a href="https://en.wikipedia.org/wiki/Semi-empirical_mass_formula" target="_self">liquid drop model</a>. Let us remind ourselves about what we did by looking at the code.
<p>
We restate the parts of the code we are most interested in.
<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">from</span> <span style="color: #0000FF; font-weight: bold">IPython.display</span> <span style="color: #008000; font-weight: bold">import</span> display
<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">&quot;Results&quot;</span>
FIGURE_ID <span style="color: #666666">=</span> <span style="color: #BA2121">&quot;Results/FigureFiles&quot;</span>
DATA_ID <span style="color: #666666">=</span> <span style="color: #BA2121">&quot;DataFiles/&quot;</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">&quot;.png&quot;</span>, format<span style="color: #666666">=</span><span style="color: #BA2121">&#39;png&#39;</span>)
infile <span style="color: #666666">=</span> <span style="color: #008000">open</span>(data_path(<span style="color: #BA2121">&quot;MassEval2016.dat&quot;</span>),<span style="color: #BA2121">&#39;r&#39;</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">&#39;N&#39;</span>, <span style="color: #BA2121">&#39;Z&#39;</span>, <span style="color: #BA2121">&#39;A&#39;</span>, <span style="color: #BA2121">&#39;Element&#39;</span>, <span style="color: #BA2121">&#39;Ebinding&#39;</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">False</span>)
<span style="color: #408080; font-style: italic"># Extrapolated values are indicated by &#39;#&#39; in place of the decimal place, so</span>
<span style="color: #408080; font-style: italic"># the Ebinding column won&#39;t be numeric. Coerce to float and drop these entries.</span>
Masses[<span style="color: #BA2121">&#39;Ebinding&#39;</span>] <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>to_numeric(Masses[<span style="color: #BA2121">&#39;Ebinding&#39;</span>], errors<span style="color: #666666">=</span><span style="color: #BA2121">&#39;coerce&#39;</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">&#39;Ebinding&#39;</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">&#39;A&#39;</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()])
A <span style="color: #666666">=</span> Masses[<span style="color: #BA2121">&#39;A&#39;</span>]
Z <span style="color: #666666">=</span> Masses[<span style="color: #BA2121">&#39;Z&#39;</span>]
N <span style="color: #666666">=</span> Masses[<span style="color: #BA2121">&#39;N&#39;</span>]
Element <span style="color: #666666">=</span> Masses[<span style="color: #BA2121">&#39;Element&#39;</span>]
Energies <span style="color: #666666">=</span> Masses[<span style="color: #BA2121">&#39;Ebinding&#39;</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>)
<span style="color: #408080; font-style: italic"># Then nice printout using pandas</span>
DesignMatrix <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>DataFrame(X)
DesignMatrix<span style="color: #666666">.</span>index <span style="color: #666666">=</span> A
DesignMatrix<span style="color: #666666">.</span>columns <span style="color: #666666">=</span> [<span style="color: #BA2121">&#39;1&#39;</span>, <span style="color: #BA2121">&#39;A&#39;</span>, <span style="color: #BA2121">&#39;A^(2/3)&#39;</span>, <span style="color: #BA2121">&#39;A^(-1/3)&#39;</span>, <span style="color: #BA2121">&#39;1/A&#39;</span>]
display(DesignMatrix)
</pre></div>
<p>
With \( \boldsymbol{\beta}\in {\mathbb{R}}^{p\times 1} \), it means that we will hereafter write our equations for the approximation as
$$
\boldsymbol{\tilde{y}}= \boldsymbol{X}\boldsymbol{\beta},
$$
throughout these lectures.
<p>
<p>
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