Morten Hjorth-Jensen
470 lines
30 KiB
HTML
470 lines
30 KiB
HTML
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<a class="navbar-brand" href="week35-bs.html">Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression</a>
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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<ul class="dropdown-menu">
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<!-- navigation toc: --> <li><a href="._week35-bs001.html#plans-for-week-35" style="font-size: 80%;"><b>Plans for week 35</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs001.html#reading-recommendations" style="font-size: 80%;"> Reading recommendations:</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs002.html#for-exercise-sessions-why-linear-regression-aka-ordinary-least-squares-and-family-repeat-from-last-week" style="font-size: 80%;"><b>For exercise sessions: Why Linear Regression (aka Ordinary Least Squares and family), repeat from last week</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs003.html#the-equations-for-ordinary-least-squares" style="font-size: 80%;"><b>The equations for ordinary least squares</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs004.html#the-cost-loss-function" style="font-size: 80%;"><b>The cost/loss function</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs016.html#interpretations-and-optimizing-our-parameters" style="font-size: 80%;"><b>Interpretations and optimizing our parameters</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs016.html#interpretations-and-optimizing-our-parameters" style="font-size: 80%;"><b>Interpretations and optimizing our parameters</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs007.html#some-useful-matrix-and-vector-expressions" style="font-size: 80%;"><b>Some useful matrix and vector expressions</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs008.html#the-jacobian" style="font-size: 80%;"><b>The Jacobian</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs009.html#derivatives-example-1" style="font-size: 80%;"><b>Derivatives, example 1</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs010.html#example-2" style="font-size: 80%;"><b>Example 2</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs011.html#example-3" style="font-size: 80%;"><b>Example 3</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs012.html#example-4" style="font-size: 80%;"><b>Example 4</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs013.html#the-mean-squared-error-and-its-derivative" style="font-size: 80%;"><b>The mean squared error and its derivative</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs014.html#other-useful-relations" style="font-size: 80%;"><b>Other useful relations</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs015.html#meet-the-hessian-matrix" style="font-size: 80%;"><b>Meet the Hessian Matrix</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs016.html#interpretations-and-optimizing-our-parameters" style="font-size: 80%;"><b>Interpretations and optimizing our parameters</b></a></li>
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|
<!-- navigation toc: --> <li><a href="._week35-bs017.html#example-relevant-for-the-exercises" style="font-size: 80%;"><b>Example relevant for the exercises</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs018.html#own-code-for-ordinary-least-squares" style="font-size: 80%;"><b>Own code for Ordinary Least Squares</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs019.html#adding-error-analysis-and-training-set-up" style="font-size: 80%;"><b>Adding error analysis and training set up</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs020.html#splitting-our-data-in-training-and-test-data" style="font-size: 80%;"><b>Splitting our Data in Training and Test data</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs021.html#the-complete-code-with-a-simple-data-set" style="font-size: 80%;"><b>The complete code with a simple data set</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs022.html#making-your-own-test-train-splitting" style="font-size: 80%;"><b>Making your own test-train splitting</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs023.html#reducing-the-number-of-degrees-of-freedom-overarching-view" style="font-size: 80%;"><b>Reducing the number of degrees of freedom, overarching view</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs024.html#preprocessing-our-data" style="font-size: 80%;"><b>Preprocessing our data</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs025.html#functionality-in-scikit-learn" style="font-size: 80%;"><b>Functionality in Scikit-Learn</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs026.html#more-preprocessing" style="font-size: 80%;"><b>More preprocessing</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs027.html#frequently-used-scaling-functions" style="font-size: 80%;"><b>Frequently used scaling functions</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs028.html#example-of-own-standard-scaling" style="font-size: 80%;"><b>Example of own Standard scaling</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs029.html#min-max-scaling" style="font-size: 80%;"><b>Min-Max Scaling</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs030.html#testing-the-means-squared-error-as-function-of-complexity" style="font-size: 80%;"><b>Testing the Means Squared Error as function of Complexity</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs031.html#more-preprocessing-examples-two-dimensional-example-the-franke-function" style="font-size: 80%;"><b>More preprocessing examples, two-dimensional example, the Franke function</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs032.html#to-think-about-first-part" style="font-size: 80%;"><b>To think about, first part</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs033.html#more-thinking" style="font-size: 80%;"><b>More thinking</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs034.html#still-thinking" style="font-size: 80%;"><b>Still thinking</b></a></li>
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|
<!-- navigation toc: --> <li><a href="._week35-bs035.html#what-does-centering-subtracting-the-mean-values-mean-mathematically" style="font-size: 80%;"><b>What does centering (subtracting the mean values) mean mathematically?</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs036.html#further-manipulations" style="font-size: 80%;"><b>Further Manipulations</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs037.html#wrapping-it-up" style="font-size: 80%;"><b>Wrapping it up</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs038.html#linear-regression-code-intercept-handling-first" style="font-size: 80%;"><b>Linear Regression code, Intercept handling first</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs039.html#material-for-lecture-monday-august-26" style="font-size: 80%;"><b>Material for lecture Monday, August 26</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs040.html#mathematical-interpretation-of-ordinary-least-squares" style="font-size: 80%;"><b>Mathematical Interpretation of Ordinary Least Squares</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs041.html#residual-error" style="font-size: 80%;"><b>Residual Error</b></a></li>
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|
<!-- navigation toc: --> <li><a href="._week35-bs042.html#simple-case" style="font-size: 80%;"><b>Simple case</b></a></li>
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|
<!-- navigation toc: --> <li><a href="._week35-bs043.html#the-singular-value-decomposition" style="font-size: 80%;"><b>The singular value decomposition</b></a></li>
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|
<!-- navigation toc: --> <li><a href="._week35-bs044.html#linear-regression-problems" style="font-size: 80%;"><b>Linear Regression Problems</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs045.html#fixing-the-singularity" style="font-size: 80%;"><b>Fixing the singularity</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs046.html#basic-math-of-the-svd" style="font-size: 80%;"><b>Basic math of the SVD</b></a></li>
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|
<!-- navigation toc: --> <li><a href="._week35-bs047.html#the-svd-a-fantastic-algorithm" style="font-size: 80%;"><b>The SVD, a Fantastic Algorithm</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week35-bs048.html#economy-size-svd" style="font-size: 80%;"><b>Economy-size SVD</b></a></li>
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<!-- navigation toc: --> <li><a href="#codes-for-the-svd" style="font-size: 80%;"><b>Codes for the SVD</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs050.html#note-about-svd-calculations" style="font-size: 80%;"><b>Note about SVD Calculations</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs051.html#mathematics-of-the-svd-and-implications" style="font-size: 80%;"><b>Mathematics of the SVD and implications</b></a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs066.html#and-finally-boldsymbol-x-boldsymbol-x-t" style="font-size: 80%;"><b>And finally \( \boldsymbol{X}\boldsymbol{X}^T \)</b></a></li>
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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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<a name="part0049"></a>
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<!-- !split -->
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<h2 id="codes-for-the-svd" class="anchor">Codes for the SVD </h2>
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<pre style="line-height: 125%;"><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: #408080; font-style: italic"># SVD inversion</span>
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<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">SVD</span>(A):
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<span style="color: #BA2121; font-style: italic">''' Takes as input a numpy matrix A and returns inv(A) based on singular value decomposition (SVD).</span>
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<span style="color: #BA2121; font-style: italic"> SVD is numerically more stable than the inversion algorithms provided by</span>
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<span style="color: #BA2121; font-style: italic"> numpy and scipy.linalg at the cost of being slower.</span>
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<span style="color: #BA2121; font-style: italic"> '''</span>
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U, S, VT <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>svd(A,full_matrices<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>)
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<span style="color: #008000">print</span>(<span style="color: #BA2121">'test U'</span>)
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<span style="color: #008000">print</span>( (np<span style="color: #666666">.</span>transpose(U) <span style="color: #666666">@</span> U <span style="color: #666666">-</span> U <span style="color: #AA22FF">@np</span><span style="color: #666666">.</span>transpose(U)))
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<span style="color: #008000">print</span>(<span style="color: #BA2121">'test VT'</span>)
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<span style="color: #008000">print</span>( (np<span style="color: #666666">.</span>transpose(VT) <span style="color: #666666">@</span> VT <span style="color: #666666">-</span> VT <span style="color: #AA22FF">@np</span><span style="color: #666666">.</span>transpose(VT)))
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<span style="color: #008000">print</span>(U)
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<span style="color: #008000">print</span>(S)
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<span style="color: #008000">print</span>(VT)
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D <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((<span style="color: #008000">len</span>(U),<span style="color: #008000">len</span>(VT)))
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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>(<span style="color: #666666">0</span>,<span style="color: #008000">len</span>(VT)):
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D[i,i]<span style="color: #666666">=</span>S[i]
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<span style="color: #008000; font-weight: bold">return</span> U <span style="color: #666666">@</span> D <span style="color: #666666">@</span> VT
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X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([ [<span style="color: #666666">1.0</span>,<span style="color: #666666">-1.0</span>], [<span style="color: #666666">1.0</span>,<span style="color: #666666">-1.0</span>]])
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<span style="color: #408080; font-style: italic">#X = np.array([[1, 2], [3, 4], [5, 6]])</span>
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<span style="color: #008000">print</span>(X)
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C <span style="color: #666666">=</span> SVD(X)
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<span style="color: #408080; font-style: italic"># Print the difference between the original matrix and the SVD one</span>
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<span style="color: #008000">print</span>(C<span style="color: #666666">-</span>X)
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</pre>
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<p>The matrix \( \boldsymbol{X} \) has columns that are linearly dependent. The first
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column is the row-wise sum of the other two columns. The rank of a
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matrix (the column rank) is the dimension of space spanned by the
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column vectors. The rank of the matrix is the number of linearly
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independent columns, in this case just \( 2 \). We see this from the
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singular values when running the above code. Running the standard
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inversion algorithm for matrix inversion with \( \boldsymbol{X}^T\boldsymbol{X} \) results
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in the program terminating due to a singular matrix.
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