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('Simple preprocessing examples, Franke function and regression',
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<!-- navigation toc: --> <li><a href="._week35-bs001.html#___sec0" style="font-size: 80%;">Plans for week 35, August 24-28</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs007.html#___sec6" style="font-size: 80%;">General linear models</a></li>
<!-- 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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<!-- navigation toc: --> <li><a href="._week35-bs011.html#___sec10" style="font-size: 80%;">Generalizing the fitting procedure as a linear algebra problem</a></li>
<!-- navigation toc: --> <li><a href="._week35-bs012.html#___sec11" style="font-size: 80%;">Optimizing our parameters</a></li>
<!-- navigation toc: --> <li><a href="._week35-bs013.html#___sec12" style="font-size: 80%;">Our model for the nuclear binding energies</a></li>
<!-- navigation toc: --> <li><a href="._week35-bs014.html#___sec13" style="font-size: 80%;">Optimizing our parameters, more details</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs019.html#___sec18" style="font-size: 80%;">Own code for Ordinary Least Squares</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs021.html#___sec20" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
<!-- navigation toc: --> <li><a href="._week35-bs022.html#___sec21" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
<!-- navigation toc: --> <li><a href="._week35-bs023.html#___sec22" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
<!-- navigation toc: --> <li><a href="._week35-bs024.html#___sec23" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs029.html#___sec28" style="font-size: 80%;">Splitting our Data in Training and Test data</a></li>
<!-- navigation toc: --> <li><a href="._week35-bs030.html#___sec29" style="font-size: 80%;">The Boston housing data example</a></li>
<!-- navigation toc: --> <li><a href="._week35-bs031.html#___sec30" style="font-size: 80%;">Housing data, the code</a></li>
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<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">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">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.metrics</span> <span style="color: #008000; font-weight: bold">import</span> mean_squared_error, r2_score, mean_absolute_error
<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;EoS.csv&quot;</span>),<span style="color: #BA2121">&#39;r&#39;</span>)
<span style="color: #408080; font-style: italic"># Read the EoS data as csv file and organize the data into two arrays with density and energies</span>
EoS <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>read_csv(infile, names<span style="color: #666666">=</span>(<span style="color: #BA2121">&#39;Density&#39;</span>, <span style="color: #BA2121">&#39;Energy&#39;</span>))
EoS[<span style="color: #BA2121">&#39;Energy&#39;</span>] <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>to_numeric(EoS[<span style="color: #BA2121">&#39;Energy&#39;</span>], errors<span style="color: #666666">=</span><span style="color: #BA2121">&#39;coerce&#39;</span>)
EoS <span style="color: #666666">=</span> EoS<span style="color: #666666">.</span>dropna()
Energies <span style="color: #666666">=</span> EoS[<span style="color: #BA2121">&#39;Energy&#39;</span>]
Density <span style="color: #666666">=</span> EoS[<span style="color: #BA2121">&#39;Density&#39;</span>]
<span style="color: #408080; font-style: italic"># The design matrix now as function of various polytrops</span>
X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((<span style="color: #008000">len</span>(Density),<span style="color: #666666">4</span>))
X[:,<span style="color: #666666">3</span>] <span style="color: #666666">=</span> Density<span style="color: #666666">**</span>(<span style="color: #666666">4.0/3.0</span>)
X[:,<span style="color: #666666">2</span>] <span style="color: #666666">=</span> Density
X[:,<span style="color: #666666">1</span>] <span style="color: #666666">=</span> Density<span style="color: #666666">**</span>(<span style="color: #666666">2.0/3.0</span>)
X[:,<span style="color: #666666">0</span>] <span style="color: #666666">=</span> <span style="color: #666666">1</span>
<span style="color: #408080; font-style: italic"># We use now Scikit-Learn&#39;s linear regressor and ridge regressor</span>
<span style="color: #408080; font-style: italic"># OLS part</span>
clf <span style="color: #666666">=</span> skl<span style="color: #666666">.</span>LinearRegression()<span style="color: #666666">.</span>fit(X, Energies)
ytilde <span style="color: #666666">=</span> clf<span style="color: #666666">.</span>predict(X)
EoS[<span style="color: #BA2121">&#39;Eols&#39;</span>] <span style="color: #666666">=</span> ytilde
<span style="color: #408080; font-style: italic"># The mean squared error </span>
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&quot;Mean squared error: </span><span style="color: #BB6688; font-weight: bold">%.2f</span><span style="color: #BA2121">&quot;</span> <span style="color: #666666">%</span> mean_squared_error(Energies, ytilde))
<span style="color: #408080; font-style: italic"># Explained variance score: 1 is perfect prediction </span>
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&#39;Variance score: </span><span style="color: #BB6688; font-weight: bold">%.2f</span><span style="color: #BA2121">&#39;</span> <span style="color: #666666">%</span> r2_score(Energies, ytilde))
<span style="color: #408080; font-style: italic"># Mean absolute error </span>
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&#39;Mean absolute error: </span><span style="color: #BB6688; font-weight: bold">%.2f</span><span style="color: #BA2121">&#39;</span> <span style="color: #666666">%</span> mean_absolute_error(Energies, ytilde))
<span style="color: #008000; font-weight: bold">print</span>(clf<span style="color: #666666">.</span>coef_, clf<span style="color: #666666">.</span>intercept_)
<span style="color: #408080; font-style: italic"># The Ridge regression with a hyperparameter lambda = 0.1</span>
_lambda <span style="color: #666666">=</span> <span style="color: #666666">0.1</span>
clf_ridge <span style="color: #666666">=</span> skl<span style="color: #666666">.</span>Ridge(alpha<span style="color: #666666">=</span>_lambda)<span style="color: #666666">.</span>fit(X, Energies)
yridge <span style="color: #666666">=</span> clf_ridge<span style="color: #666666">.</span>predict(X)
EoS[<span style="color: #BA2121">&#39;Eridge&#39;</span>] <span style="color: #666666">=</span> yridge
<span style="color: #408080; font-style: italic"># The mean squared error </span>
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&quot;Mean squared error: </span><span style="color: #BB6688; font-weight: bold">%.2f</span><span style="color: #BA2121">&quot;</span> <span style="color: #666666">%</span> mean_squared_error(Energies, yridge))
<span style="color: #408080; font-style: italic"># Explained variance score: 1 is perfect prediction </span>
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&#39;Variance score: </span><span style="color: #BB6688; font-weight: bold">%.2f</span><span style="color: #BA2121">&#39;</span> <span style="color: #666666">%</span> r2_score(Energies, yridge))
<span style="color: #408080; font-style: italic"># Mean absolute error </span>
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&#39;Mean absolute error: </span><span style="color: #BB6688; font-weight: bold">%.2f</span><span style="color: #BA2121">&#39;</span> <span style="color: #666666">%</span> mean_absolute_error(Energies, yridge))
<span style="color: #008000; font-weight: bold">print</span>(clf_ridge<span style="color: #666666">.</span>coef_, clf_ridge<span style="color: #666666">.</span>intercept_)
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&#39;$\rho[\mathrm{fm}^{-3}]$&#39;</span>)
ax<span style="color: #666666">.</span>set_ylabel(<span style="color: #BA2121">r&#39;Energy per particle&#39;</span>)
ax<span style="color: #666666">.</span>plot(EoS[<span style="color: #BA2121">&#39;Density&#39;</span>], EoS[<span style="color: #BA2121">&#39;Energy&#39;</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">&#39;Theoretical data&#39;</span>)
ax<span style="color: #666666">.</span>plot(EoS[<span style="color: #BA2121">&#39;Density&#39;</span>], EoS[<span style="color: #BA2121">&#39;Eols&#39;</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">&#39;m&#39;</span>,
label<span style="color: #666666">=</span><span style="color: #BA2121">&#39;OLS&#39;</span>)
ax<span style="color: #666666">.</span>plot(EoS[<span style="color: #BA2121">&#39;Density&#39;</span>], EoS[<span style="color: #BA2121">&#39;Eridge&#39;</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">&#39;g&#39;</span>,
label<span style="color: #666666">=</span><span style="color: #BA2121">&#39;Ridge $\lambda = 0.1$&#39;</span>)
ax<span style="color: #666666">.</span>legend()
save_fig(<span style="color: #BA2121">&quot;EoSfitting&quot;</span>)
plt<span style="color: #666666">.</span>show()
</pre></div>
<p>
The above simple polynomial in density \( \rho \) gives an excellent fit
to the data.
<p>
We note also that there is a small deviation between the
standard OLS and the Ridge regression at higher densities. We discuss this in more detail
below.
<p>
<p>
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<li><a href="._week35-bs025.html">26</a></li>
<li><a href="._week35-bs026.html">27</a></li>
<li><a href="._week35-bs027.html">28</a></li>
<li class="active"><a href="._week35-bs028.html">29</a></li>
<li><a href="._week35-bs029.html">30</a></li>
<li><a href="._week35-bs030.html">31</a></li>
<li><a href="._week35-bs031.html">32</a></li>
<li><a href="._week35-bs032.html">33</a></li>
<li><a href="._week35-bs033.html">34</a></li>
<li><a href="._week35-bs034.html">35</a></li>
<li><a href="._week35-bs035.html">36</a></li>
<li><a href="._week35-bs036.html">37</a></li>
<li><a href="._week35-bs029.html">&raquo;</a></li>
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