mhjensen
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414 lines
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'sections': [('Plans for week 35, August 24-28', 2, None, '___sec0'),
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('Interpretations and optimizing our parameters',
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('The $\\chi^2$ function', 2, None, '___sec21'),
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('The $\\chi^2$ function', 2, None, '___sec22'),
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('The $\\chi^2$ function', 2, None, '___sec23'),
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('The $\\chi^2$ function', 2, None, '___sec24'),
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('The $\\chi^2$ function', 2, None, '___sec25'),
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('Fitting an Equation of State for Dense Nuclear Matter',
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2,
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None,
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'___sec26'),
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('The code', 2, None, '___sec27'),
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('Splitting our Data in Training and Test data',
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2,
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('Reducing the number of degrees of freedom, overarching view',
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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-bs002.html#___sec1" style="font-size: 80%;">Thursday August 27</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs003.html#___sec2" style="font-size: 80%;">Why Linear Regression (aka Ordinary Least Squares and family)</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs004.html#___sec3" style="font-size: 80%;">Regression analysis, overarching aims</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs005.html#___sec4" style="font-size: 80%;">Regression analysis, overarching aims II</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs006.html#___sec5" style="font-size: 80%;">Examples</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs007.html#___sec6" style="font-size: 80%;">General linear models</a></li>
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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>
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<!-- 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-bs010.html#___sec9" style="font-size: 80%;">Generalizing the fitting procedure as a linear algebra problem</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>
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<!-- navigation toc: --> <li><a href="._week35-bs012.html#___sec11" style="font-size: 80%;">Optimizing our parameters</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs013.html#___sec12" style="font-size: 80%;">Our model for the nuclear binding energies</a></li>
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<!-- 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-bs015.html#___sec14" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs016.html#___sec15" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs017.html#___sec16" style="font-size: 80%;">Some useful matrix and vector expressions</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs018.html#___sec17" style="font-size: 80%;">Interpretations and optimizing our parameters</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-bs020.html#___sec19" style="font-size: 80%;">Adding error analysis and training set up</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>
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<!-- navigation toc: --> <li><a href="._week35-bs022.html#___sec21" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs023.html#___sec22" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- 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-bs025.html#___sec24" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs026.html#___sec25" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs027.html#___sec26" style="font-size: 80%;">Fitting an Equation of State for Dense Nuclear Matter</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs028.html#___sec27" style="font-size: 80%;">The code</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>
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<!-- navigation toc: --> <li><a href="._week35-bs030.html#___sec29" style="font-size: 80%;">The Boston housing data example</a></li>
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<!-- navigation toc: --> <li><a href="#___sec30" style="font-size: 80%;">Housing data, the code</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs032.html#___sec31" style="font-size: 80%;">Reducing the number of degrees of freedom, overarching view</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs033.html#___sec32" style="font-size: 80%;">Preprocessing our data</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs034.html#___sec33" style="font-size: 80%;">More preprocessing</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs035.html#___sec34" style="font-size: 80%;">Simple preprocessing examples, Franke function and regression</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs036.html#___sec35" style="font-size: 80%;">Friday August 28</a></li>
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<h2 id="___sec30" class="anchor">Housing data, the code </h2>
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We start by importing the libraries
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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">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>
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<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>
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</pre></div>
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<p>
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and load the Boston Housing DataSet from <b>Scikit-Learn</b>
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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">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.datasets</span> <span style="color: #008000; font-weight: bold">import</span> load_boston
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boston_dataset <span style="color: #666666">=</span> load_boston()
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<span style="color: #408080; font-style: italic"># boston_dataset is a dictionary</span>
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<span style="color: #408080; font-style: italic"># let's check what it contains</span>
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boston_dataset<span style="color: #666666">.</span>keys()
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</pre></div>
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<p>
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Then we invoke Pandas
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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>boston <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>DataFrame(boston_dataset<span style="color: #666666">.</span>data, columns<span style="color: #666666">=</span>boston_dataset<span style="color: #666666">.</span>feature_names)
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boston<span style="color: #666666">.</span>head()
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boston[<span style="color: #BA2121">'MEDV'</span>] <span style="color: #666666">=</span> boston_dataset<span style="color: #666666">.</span>target
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</pre></div>
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<p>
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and preprocess the data
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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"># check for missing values in all the columns</span>
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boston<span style="color: #666666">.</span>isnull()<span style="color: #666666">.</span>sum()
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</pre></div>
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<p>
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We can then visualize the data
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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"># set the size of the figure</span>
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sns<span style="color: #666666">.</span>set(rc<span style="color: #666666">=</span>{<span style="color: #BA2121">'figure.figsize'</span>:(<span style="color: #666666">11.7</span>,<span style="color: #666666">8.27</span>)})
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<span style="color: #408080; font-style: italic"># plot a histogram showing the distribution of the target values</span>
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sns<span style="color: #666666">.</span>distplot(boston[<span style="color: #BA2121">'MEDV'</span>], bins<span style="color: #666666">=30</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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It is now useful to look at the correlation matrix
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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"># compute the pair wise correlation for all columns </span>
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correlation_matrix <span style="color: #666666">=</span> boston<span style="color: #666666">.</span>corr()<span style="color: #666666">.</span>round(<span style="color: #666666">2</span>)
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<span style="color: #408080; font-style: italic"># use the heatmap function from seaborn to plot the correlation matrix</span>
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<span style="color: #408080; font-style: italic"># annot = True to print the values inside the square</span>
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sns<span style="color: #666666">.</span>heatmap(data<span style="color: #666666">=</span>correlation_matrix, annot<span style="color: #666666">=</span><span style="color: #008000">True</span>)
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</pre></div>
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<p>
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From the above coorelation plot we can see that <b>MEDV</b> is strongly correlated to <b>LSTAT</b> and <b>RM</b>. We see also that <b>RAD</b> and <b>TAX</b> are stronly correlated, but we don't include this in our features together to avoid multi-colinearity
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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>plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">20</span>, <span style="color: #666666">5</span>))
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features <span style="color: #666666">=</span> [<span style="color: #BA2121">'LSTAT'</span>, <span style="color: #BA2121">'RM'</span>]
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target <span style="color: #666666">=</span> boston[<span style="color: #BA2121">'MEDV'</span>]
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<span style="color: #008000; font-weight: bold">for</span> i, col <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">enumerate</span>(features):
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plt<span style="color: #666666">.</span>subplot(<span style="color: #666666">1</span>, <span style="color: #008000">len</span>(features) , i<span style="color: #666666">+1</span>)
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x <span style="color: #666666">=</span> boston[col]
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y <span style="color: #666666">=</span> target
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plt<span style="color: #666666">.</span>scatter(x, y, marker<span style="color: #666666">=</span><span style="color: #BA2121">'o'</span>)
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plt<span style="color: #666666">.</span>title(col)
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plt<span style="color: #666666">.</span>xlabel(col)
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plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">'MEDV'</span>)
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</pre></div>
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<p>
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Now we start training our model
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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>X <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>DataFrame(np<span style="color: #666666">.</span>c_[boston[<span style="color: #BA2121">'LSTAT'</span>], boston[<span style="color: #BA2121">'RM'</span>]], columns <span style="color: #666666">=</span> [<span style="color: #BA2121">'LSTAT'</span>,<span style="color: #BA2121">'RM'</span>])
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Y <span style="color: #666666">=</span> boston[<span style="color: #BA2121">'MEDV'</span>]
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</pre></div>
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<p>
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We split the data into training and test sets
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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">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
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<span style="color: #408080; font-style: italic"># splits the training and test data set in 80% : 20%</span>
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<span style="color: #408080; font-style: italic"># assign random_state to any value.This ensures consistency.</span>
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X_train, X_test, Y_train, Y_test <span style="color: #666666">=</span> train_test_split(X, Y, test_size <span style="color: #666666">=</span> <span style="color: #666666">0.2</span>, random_state<span style="color: #666666">=5</span>)
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<span style="color: #008000; font-weight: bold">print</span>(X_train<span style="color: #666666">.</span>shape)
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<span style="color: #008000; font-weight: bold">print</span>(X_test<span style="color: #666666">.</span>shape)
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<span style="color: #008000; font-weight: bold">print</span>(Y_train<span style="color: #666666">.</span>shape)
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<span style="color: #008000; font-weight: bold">print</span>(Y_test<span style="color: #666666">.</span>shape)
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</pre></div>
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<p>
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Then we use the linear regression functionality from <b>Scikit-Learn</b>
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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">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> LinearRegression
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<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
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lin_model <span style="color: #666666">=</span> LinearRegression()
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lin_model<span style="color: #666666">.</span>fit(X_train, Y_train)
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<span style="color: #408080; font-style: italic"># model evaluation for training set</span>
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y_train_predict <span style="color: #666666">=</span> lin_model<span style="color: #666666">.</span>predict(X_train)
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rmse <span style="color: #666666">=</span> (np<span style="color: #666666">.</span>sqrt(mean_squared_error(Y_train, y_train_predict)))
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r2 <span style="color: #666666">=</span> r2_score(Y_train, y_train_predict)
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<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">"The model performance for training set"</span>)
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<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">"--------------------------------------"</span>)
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<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">'RMSE is {}'</span><span style="color: #666666">.</span>format(rmse))
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<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">'R2 score is {}'</span><span style="color: #666666">.</span>format(r2))
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<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">"</span><span style="color: #BB6622; font-weight: bold">\n</span><span style="color: #BA2121">"</span>)
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<span style="color: #408080; font-style: italic"># model evaluation for testing set</span>
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y_test_predict <span style="color: #666666">=</span> lin_model<span style="color: #666666">.</span>predict(X_test)
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<span style="color: #408080; font-style: italic"># root mean square error of the model</span>
|
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rmse <span style="color: #666666">=</span> (np<span style="color: #666666">.</span>sqrt(mean_squared_error(Y_test, y_test_predict)))
|
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<span style="color: #408080; font-style: italic"># r-squared score of the model</span>
|
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r2 <span style="color: #666666">=</span> r2_score(Y_test, y_test_predict)
|
|
|
|
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">"The model performance for testing set"</span>)
|
|
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">"--------------------------------------"</span>)
|
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<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">'RMSE is {}'</span><span style="color: #666666">.</span>format(rmse))
|
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<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">'R2 score is {}'</span><span style="color: #666666">.</span>format(r2))
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</pre></div>
|
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<p>
|
|
|
|
<!-- 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"># plotting the y_test vs y_pred</span>
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<span style="color: #408080; font-style: italic"># ideally should have been a straight line</span>
|
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plt<span style="color: #666666">.</span>scatter(Y_test, y_test_predict)
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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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<p>
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