Morten Hjorth-Jensen
498 lines
29 KiB
HTML
498 lines
29 KiB
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<!-- navigation toc: --> <li><a href="._week38-bs001.html#plans-for-week-38" style="font-size: 80%;">Plans for week 38</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs002.html#thursday-september-23" style="font-size: 80%;">Thursday September 23</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs003.html#ridge-and-lasso-regression-reminder" style="font-size: 80%;">Ridge and LASSO Regression, reminder</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs004.html#various-steps-in-cross-validation" style="font-size: 80%;">Various steps in cross-validation</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs005.html#how-to-set-up-the-cross-validation-for-ridge-and-or-lasso" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs006.html#cross-validation-in-brief" style="font-size: 80%;">Cross-validation in brief</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs007.html#code-example-for-cross-validation-and-k-fold-cross-validation" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs008.html#to-think-about-first-part" style="font-size: 80%;">To think about, first part</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs009.html#more-thinking" style="font-size: 80%;">More thinking</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs010.html#still-thinking" style="font-size: 80%;">Still thinking</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs011.html#what-does-centering-subtracting-the-mean-values-mean-mathematically" style="font-size: 80%;">What does centering (subtracting the mean values) mean mathematically?</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs012.html#further-manipulations" style="font-size: 80%;">Further Manipulations</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs013.html#wrapping-it-up" style="font-size: 80%;">Wrapping it up</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs014.html#linear-regression-code-intercept-handling-first" style="font-size: 80%;">Linear Regression code, Intercept handling first</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs015.html#code-examples" style="font-size: 80%;">Code Examples</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs016.html#taking-out-the-mean" style="font-size: 80%;">Taking out the mean</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs017.html#more-complicated-example-the-ising-model" style="font-size: 80%;">More complicated Example: The Ising model</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs018.html#reformulating-the-problem-to-suit-regression" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs019.html#linear-regression" style="font-size: 80%;">Linear regression</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs020.html#singular-value-decomposition" style="font-size: 80%;">Singular Value decomposition</a></li>
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<!-- navigation toc: --> <li><a href="#the-one-dimensional-ising-model" style="font-size: 80%;">The one-dimensional Ising model</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs022.html#ridge-regression" style="font-size: 80%;">Ridge regression</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs023.html#lasso-regression" style="font-size: 80%;">LASSO regression</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs024.html#performance-as-function-of-the-regularization-parameter" style="font-size: 80%;">Performance as function of the regularization parameter</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs025.html#finding-the-optimal-value-of-lambda" style="font-size: 80%;">Finding the optimal value of \( \lambda \)</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs026.html#logistic-regression" style="font-size: 80%;">Logistic Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs027.html#classification-problems" style="font-size: 80%;">Classification problems</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs028.html#optimization-and-deep-learning" style="font-size: 80%;">Optimization and Deep learning</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs029.html#basics" style="font-size: 80%;">Basics</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs030.html#linear-classifier" style="font-size: 80%;">Linear classifier</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs031.html#some-selected-properties" style="font-size: 80%;">Some selected properties</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs032.html#simple-example" style="font-size: 80%;">Simple example</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs033.html#plotting-the-mean-value-for-each-group" style="font-size: 80%;">Plotting the mean value for each group</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs034.html#the-logistic-function" style="font-size: 80%;">The logistic function</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs035.html#examples-of-likelihood-functions-used-in-logistic-regression-and-nueral-networks" style="font-size: 80%;">Examples of likelihood functions used in logistic regression and nueral networks</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs036.html#two-parameters" style="font-size: 80%;">Two parameters</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs037.html#maximum-likelihood" style="font-size: 80%;">Maximum likelihood</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs038.html#the-cost-function-rewritten" style="font-size: 80%;">The cost function rewritten</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs039.html#minimizing-the-cross-entropy" style="font-size: 80%;">Minimizing the cross entropy</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs040.html#a-more-compact-expression" style="font-size: 80%;">A more compact expression</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs041.html#extending-to-more-predictors" style="font-size: 80%;">Extending to more predictors</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs042.html#including-more-classes" style="font-size: 80%;">Including more classes</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs043.html#more-classes" style="font-size: 80%;">More classes</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs044.html#friday-september-24" style="font-size: 80%;">Friday September 24</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs045.html#wisconsin-cancer-data" style="font-size: 80%;">Wisconsin Cancer Data</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs046.html#using-the-correlation-matrix" style="font-size: 80%;">Using the correlation matrix</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs047.html#discussing-the-correlation-data" style="font-size: 80%;">Discussing the correlation data</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs048.html#other-measures-in-classification-studies-cancer-data-again" style="font-size: 80%;">Other measures in classification studies: Cancer Data again</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs049.html#friday-september-25" style="font-size: 80%;">Friday September 25</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs050.html#optimization-the-central-part-of-any-machine-learning-algortithm" style="font-size: 80%;">Optimization, the central part of any Machine Learning algortithm</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs051.html#revisiting-our-logistic-regression-case" style="font-size: 80%;">Revisiting our Logistic Regression case</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs052.html#the-equations-to-solve" style="font-size: 80%;">The equations to solve</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs053.html#solving-using-newton-raphson-s-method" style="font-size: 80%;">Solving using Newton-Raphson's method</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs054.html#brief-reminder-on-newton-raphson-s-method" style="font-size: 80%;">Brief reminder on Newton-Raphson's method</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs055.html#the-equations" style="font-size: 80%;">The equations</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs056.html#simple-geometric-interpretation" style="font-size: 80%;">Simple geometric interpretation</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs057.html#extending-to-more-than-one-variable" style="font-size: 80%;">Extending to more than one variable</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs058.html#steepest-descent" style="font-size: 80%;">Steepest descent</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs059.html#more-on-steepest-descent" style="font-size: 80%;">More on Steepest descent</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs060.html#the-ideal" style="font-size: 80%;">The ideal</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs061.html#the-sensitiveness-of-the-gradient-descent" style="font-size: 80%;">The sensitiveness of the gradient descent</a></li>
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<div class="container">
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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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<a name="part0021"></a>
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<!-- !split -->
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<h2 id="the-one-dimensional-ising-model" class="anchor">The one-dimensional Ising model </h2>
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<p>
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Let us bring back the Ising model again, but now with an additional
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focus on Ridge and Lasso regression as well. We repeat some of the
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basic parts of the Ising model and the setup of the training and test
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data. The one-dimensional Ising model with nearest neighbor
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interaction, no external field and a constant coupling constant \( J \) is
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given by
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$$
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\begin{align}
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H = -J \sum_{k}^L s_k s_{k + 1},
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\tag{7}
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\end{align}
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$$
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where \( s_i \in \{-1, 1\} \) and \( s_{N + 1} = s_1 \). The number of spins in the system is determined by \( L \). For the one-dimensional system there is no phase transition.
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<p>
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We will look at a system of \( L = 40 \) spins with a coupling constant of \( J = 1 \). To get enough training data we will generate 10000 states with their respective energies.
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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">mpl_toolkits.axes_grid1</span> <span style="color: #008000; font-weight: bold">import</span> make_axes_locatable
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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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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">scipy.linalg</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">scl</span>
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<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: #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>
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">tqdm</span>
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sns<span style="color: #666666">.</span>set(color_codes<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>)
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cmap_args<span style="color: #666666">=</span><span style="color: #008000">dict</span>(vmin<span style="color: #666666">=-1.</span>, vmax<span style="color: #666666">=1.</span>, cmap<span style="color: #666666">=</span><span style="color: #BA2121">'seismic'</span>)
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L <span style="color: #666666">=</span> <span style="color: #666666">40</span>
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n <span style="color: #666666">=</span> <span style="color: #008000">int</span>(<span style="color: #666666">1e4</span>)
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spins <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>choice([<span style="color: #666666">-1</span>, <span style="color: #666666">1</span>], size<span style="color: #666666">=</span>(n, L))
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J <span style="color: #666666">=</span> <span style="color: #666666">1.0</span>
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energies <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(n)
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<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(n):
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energies[i] <span style="color: #666666">=</span> <span style="color: #666666">-</span> J <span style="color: #666666">*</span> np<span style="color: #666666">.</span>dot(spins[i], np<span style="color: #666666">.</span>roll(spins[i], <span style="color: #666666">1</span>))
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</pre></div>
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<p>
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A more general form for the one-dimensional Ising model is
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$$
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\begin{align}
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H = - \sum_j^L \sum_k^L s_j s_k J_{jk}.
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\tag{8}
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\end{align}
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$$
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<p>
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Here we allow for interactions beyond the nearest neighbors and a more
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adaptive coupling matrix. This latter expression can be formulated as
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a matrix-product on the form
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$$
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\begin{align}
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H = X J,
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\tag{9}
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\end{align}
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$$
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<p>
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where \( X_{jk} = s_j s_k \) and \( J \) is the matrix consisting of the
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elements \( -J_{jk} \). This form of writing the energy fits perfectly
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with the form utilized in linear regression, viz.
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$$
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\begin{align}
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\boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta} + \boldsymbol{\epsilon}.
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\tag{10}
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\end{align}
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$$
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We organize the data as we did above
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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> np<span style="color: #666666">.</span>zeros((n, L <span style="color: #666666">**</span> <span style="color: #666666">2</span>))
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<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(n):
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X[i] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>outer(spins[i], spins[i])<span style="color: #666666">.</span>ravel()
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y <span style="color: #666666">=</span> energies
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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">=0.96</span>)
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X_train_own <span style="color: #666666">=</span> np<span style="color: #666666">.</span>concatenate(
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(np<span style="color: #666666">.</span>ones(<span style="color: #008000">len</span>(X_train))[:, np<span style="color: #666666">.</span>newaxis], X_train),
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axis<span style="color: #666666">=1</span>
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)
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X_test_own <span style="color: #666666">=</span> np<span style="color: #666666">.</span>concatenate(
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(np<span style="color: #666666">.</span>ones(<span style="color: #008000">len</span>(X_test))[:, np<span style="color: #666666">.</span>newaxis], X_test),
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axis<span style="color: #666666">=1</span>
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)
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</pre></div>
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<p>
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We will do all fitting with <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>clf <span style="color: #666666">=</span> skl<span style="color: #666666">.</span>LinearRegression()<span style="color: #666666">.</span>fit(X_train, y_train)
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</pre></div>
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<p>
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When extracting the \( J \)-matrix we make sure to remove the intercept
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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>J_sk <span style="color: #666666">=</span> clf<span style="color: #666666">.</span>coef_<span style="color: #666666">.</span>reshape(L, L)
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</pre></div>
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<p>
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And then we plot the results
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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>fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">20</span>, <span style="color: #666666">14</span>))
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im <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>imshow(J_sk, <span style="color: #666666">**</span>cmap_args)
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plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">"LinearRegression from Scikit-learn"</span>, fontsize<span style="color: #666666">=18</span>)
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plt<span style="color: #666666">.</span>xticks(fontsize<span style="color: #666666">=18</span>)
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plt<span style="color: #666666">.</span>yticks(fontsize<span style="color: #666666">=18</span>)
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cb <span style="color: #666666">=</span> fig<span style="color: #666666">.</span>colorbar(im)
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cb<span style="color: #666666">.</span>ax<span style="color: #666666">.</span>set_yticklabels(cb<span style="color: #666666">.</span>ax<span style="color: #666666">.</span>get_yticklabels(), fontsize<span style="color: #666666">=18</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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The results perfectly with our previous discussion where we used our own code.
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<p>
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<p>
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