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Morten Hjorth-Jensen
@@ -7,8 +7,8 @@
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About the course
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@@ -116,7 +122,7 @@ const thebe_selector_output = ".output, .cell_output"
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Review of Statistics with Resampling Techniques and Linear Algebra
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@@ -133,7 +139,7 @@ const thebe_selector_output = ".output, .cell_output"
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<span class="caption-text">
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From Regression to Support Vector Machines
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@@ -170,7 +176,7 @@ const thebe_selector_output = ".output, .cell_output"
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Decision Trees, Ensemble Methods and Boosting
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@@ -187,7 +193,7 @@ const thebe_selector_output = ".output, .cell_output"
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<span class="caption-text">
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Dimensionality Reduction
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@@ -204,7 +210,7 @@ const thebe_selector_output = ".output, .cell_output"
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<span class="caption-text">
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Deep Learning Methods
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@@ -281,7 +287,7 @@ const thebe_selector_output = ".output, .cell_output"
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@@ -299,7 +305,7 @@ const thebe_selector_output = ".output, .cell_output"
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<i class="fas fa-list"></i> Contents
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@@ -384,7 +390,95 @@ const thebe_selector_output = ".output, .cell_output"
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<h1>Ridge and Lasso Regression</h1>
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<li class="toc-h2 nav-item toc-entry">
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<a class="reference internal nav-link" href="#mathematical-interpretation-of-ordinary-least-squares">
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4.1. Mathematical Interpretation of Ordinary Least Squares
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</li>
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<li class="toc-h2 nav-item toc-entry">
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<a class="reference internal nav-link" href="#the-singular-value-decomposition">
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4.2. The singular value decomposition
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</a>
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<li class="toc-h2 nav-item toc-entry">
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<a class="reference internal nav-link" href="#basic-math-of-the-svd">
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4.3. Basic math of the SVD
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</a>
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</li>
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<li class="toc-h2 nav-item toc-entry">
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<a class="reference internal nav-link" href="#codes-for-the-svd">
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4.4. Codes for the SVD
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</a>
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</li>
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<li class="toc-h2 nav-item toc-entry">
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<a class="reference internal nav-link" href="#code-for-svd-and-inversion-of-matrices">
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4.5. Code for SVD and Inversion of Matrices
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</a>
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</li>
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<li class="toc-h2 nav-item toc-entry">
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<a class="reference internal nav-link" href="#mathematics-of-the-svd-and-implications">
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4.6. Mathematics of the SVD and implications
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</a>
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</li>
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<li class="toc-h2 nav-item toc-entry">
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<a class="reference internal nav-link" href="#further-properties-important-for-our-analyses-later">
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4.7. Further properties (important for our analyses later)
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</a>
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</li>
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<li class="toc-h2 nav-item toc-entry">
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<a class="reference internal nav-link" href="#meet-the-covariance-matrix">
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4.8. Meet the Covariance Matrix
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</a>
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</li>
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<li class="toc-h2 nav-item toc-entry">
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<a class="reference internal nav-link" href="#linking-with-the-svd">
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4.9. Linking with the SVD
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</a>
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</li>
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<li class="toc-h2 nav-item toc-entry">
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<a class="reference internal nav-link" href="#id1">
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4.10. Ridge and Lasso Regression
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</a>
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</li>
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<li class="toc-h2 nav-item toc-entry">
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<a class="reference internal nav-link" href="#linking-the-regression-analysis-with-a-statistical-interpretation">
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4.11. Linking the regression analysis with a statistical interpretation
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</a>
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</li>
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<li class="toc-h2 nav-item toc-entry">
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<a class="reference internal nav-link" href="#deriving-ols-from-a-probability-distribution">
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4.12. Deriving OLS from a probability distribution
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</a>
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</li>
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<li class="toc-h2 nav-item toc-entry">
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<a class="reference internal nav-link" href="#bayes-theorem-and-ridge-and-lasso-regression">
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4.13. Bayes’ Theorem and Ridge and Lasso Regression
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</a>
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</li>
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<li class="toc-h2 nav-item toc-entry">
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<a class="reference internal nav-link" href="#linking-bayes-theorem-with-ridge-and-lasso-regression">
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4.14. Linking Bayes’ Theorem with Ridge and Lasso Regression
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</a>
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</li>
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</ul>
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</nav>
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</div>
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</div>
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<div>
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<!-- HTML file automatically generated from DocOnce source (https://github.com/doconce/doconce/)
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@@ -736,13 +830,13 @@ The simple answer is to use the linear algebra function for the pseudoinverse, t
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[2 4 5]
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[3 5 6]]
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test U
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[[ 4.44089210e-16 -4.69484813e-16 -6.67314874e-16]
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[-4.69484813e-16 -4.44089210e-16 -1.54041041e-16]
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[-6.67314874e-16 -1.54041041e-16 1.11022302e-16]]
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[[-2.22044605e-16 -7.49932427e-16 -8.23408962e-16]
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[-7.49932427e-16 0.00000000e+00 4.77954956e-17]
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[-8.23408962e-16 4.77954956e-17 2.22044605e-16]]
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test VT
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[[ 2.22044605e-16 3.78156479e-17 1.85278920e-16]
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[ 3.78156479e-17 0.00000000e+00 -6.33166055e-17]
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[ 1.85278920e-16 -6.33166055e-17 -1.11022302e-16]]
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[[ 3.33066907e-16 -7.32066545e-17 3.32714903e-16]
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[-7.32066545e-17 0.00000000e+00 -1.82997013e-16]
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[ 3.32714903e-16 -1.82997013e-16 -3.33066907e-16]]
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[[0. 0. 0.]
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[0. 0. 0.]
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[0. 0. 0.]]
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@@ -1113,10 +1207,10 @@ covariance matrix through the <strong>np.linalg.eig()</strong> function.</p>
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</div>
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</div>
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<div class="cell_output docutils container">
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>-0.02697521163514974
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3.9835443722554817
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[[ 0.98467494 3.11283168]
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[ 3.11283168 10.68965135]]
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.04413933503955871
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4.12330280229368
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[[0.80162359 2.38222896]
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[2.38222896 8.12167821]]
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</pre></div>
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</div>
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</div>
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@@ -1153,10 +1247,10 @@ a more brute force way. Here we scale the mean values for each column of the des
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</div>
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</div>
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<div class="cell_output docutils container">
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.07898165660100093
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1.6984511994530214
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[[1. 0.63862189]
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[0.63862189 1. ]]
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.06786925114666595
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1.9635449873404844
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[[1. 0.65522261]
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[0.65522261 1. ]]
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</pre></div>
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</div>
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</div>
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@@ -1186,30 +1280,30 @@ this matrix we easily see that it is a positive definite matrix.</p>
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</div>
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</div>
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<div class="cell_output docutils container">
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[[ 1.49901244 6.93685631]
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[-1.07535606 -4.3823886 ]
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[-1.23168292 -4.3014751 ]
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[ 1.37683438 3.52093124]
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[ 1.31424359 3.63367582]
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[-1.15269628 -1.86879198]
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[-0.93188452 -4.26291585]
|
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[-0.66295776 -1.21924917]
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[ 0.29883607 -0.44270138]
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[ 0.56565106 2.38605872]]
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[[-0.27091656 -1.29083183]
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[ 0.31980301 0.87495119]
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[-0.10835935 1.61413333]
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[ 0.5188328 2.80380438]
|
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[-0.04996008 -1.95742107]
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||||
[ 1.19432526 2.68719389]
|
||||
[ 0.19710439 1.35590603]
|
||||
[-0.23857423 -2.50104946]
|
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[-0.94054854 -2.09034902]
|
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[-0.62170669 -1.49633743]]
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0 1
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||||
0 1.499012 6.936856
|
||||
1 -1.075356 -4.382389
|
||||
2 -1.231683 -4.301475
|
||||
3 1.376834 3.520931
|
||||
4 1.314244 3.633676
|
||||
5 -1.152696 -1.868792
|
||||
6 -0.931885 -4.262916
|
||||
7 -0.662958 -1.219249
|
||||
8 0.298836 -0.442701
|
||||
9 0.565651 2.386059
|
||||
0 -0.270917 -1.290832
|
||||
1 0.319803 0.874951
|
||||
2 -0.108359 1.614133
|
||||
3 0.518833 2.803804
|
||||
4 -0.049960 -1.957421
|
||||
5 1.194325 2.687194
|
||||
6 0.197104 1.355906
|
||||
7 -0.238574 -2.501049
|
||||
8 -0.940549 -2.090349
|
||||
9 -0.621707 -1.496337
|
||||
0 1
|
||||
0 1.000000 0.946393
|
||||
1 0.946393 1.000000
|
||||
0 1.000000 0.800615
|
||||
1 0.800615 1.000000
|
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</pre></div>
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</div>
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</div>
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@@ -1266,37 +1360,37 @@ this matrix we easily see that it is a positive definite matrix.</p>
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<div class="cell_output docutils container">
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span> 0 1 2 3 4 5 6 7 \
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0 0.0 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
|
||||
1 0.0 0.096516 0.086951 0.097634 0.093624 0.089369 0.089758 0.086888
|
||||
2 0.0 0.086951 0.079242 0.086994 0.083832 0.080479 0.079719 0.077419
|
||||
3 0.0 0.097634 0.086994 0.105161 0.100303 0.095184 0.100358 0.096893
|
||||
4 0.0 0.093624 0.083832 0.100303 0.095935 0.091315 0.095522 0.092409
|
||||
5 0.0 0.089369 0.080479 0.095184 0.091315 0.087208 0.090451 0.087689
|
||||
6 0.0 0.089758 0.079719 0.100358 0.095522 0.090451 0.098135 0.094657
|
||||
7 0.0 0.086888 0.077419 0.096893 0.092409 0.087689 0.094657 0.091440
|
||||
8 0.0 0.084096 0.075188 0.093497 0.089355 0.084979 0.091236 0.088271
|
||||
9 0.0 0.081343 0.072994 0.090128 0.086322 0.082286 0.087833 0.085113
|
||||
10 0.0 0.081707 0.072554 0.093593 0.089041 0.084278 0.093078 0.089775
|
||||
11 0.0 0.079268 0.070562 0.090694 0.086420 0.081932 0.090174 0.087081
|
||||
12 0.0 0.076942 0.068664 0.087910 0.083899 0.079676 0.087372 0.084477
|
||||
13 0.0 0.074715 0.066850 0.085224 0.081468 0.077499 0.084657 0.081953
|
||||
14 0.0 0.072575 0.065110 0.082620 0.079110 0.075389 0.082014 0.079493
|
||||
1 0.0 0.084006 0.079882 0.084682 0.084092 0.083417 0.076315 0.076097
|
||||
2 0.0 0.079882 0.077644 0.078542 0.078962 0.079424 0.069534 0.069977
|
||||
3 0.0 0.084682 0.078542 0.090649 0.088758 0.086665 0.085105 0.084008
|
||||
4 0.0 0.084092 0.078962 0.088758 0.087573 0.086250 0.082424 0.081832
|
||||
5 0.0 0.083417 0.079424 0.086665 0.086250 0.085776 0.079486 0.079438
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||||
6 0.0 0.076315 0.069534 0.085105 0.082424 0.079486 0.082288 0.080588
|
||||
7 0.0 0.076097 0.069977 0.084008 0.081832 0.079438 0.080588 0.079264
|
||||
8 0.0 0.076022 0.070618 0.082990 0.081357 0.079553 0.078908 0.077986
|
||||
9 0.0 0.076079 0.071460 0.082027 0.080984 0.079823 0.077219 0.076729
|
||||
10 0.0 0.068075 0.061188 0.078143 0.075043 0.071666 0.077200 0.075149
|
||||
11 0.0 0.067712 0.061308 0.077144 0.074420 0.071445 0.075770 0.074006
|
||||
12 0.0 0.067499 0.061604 0.076264 0.073938 0.071388 0.074418 0.072955
|
||||
13 0.0 0.067443 0.062089 0.075498 0.073597 0.071505 0.073134 0.071991
|
||||
14 0.0 0.067547 0.062777 0.074845 0.073400 0.071804 0.071908 0.071106
|
||||
|
||||
8 9 10 11 12 13 14
|
||||
0 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
|
||||
1 0.084096 0.081343 0.081707 0.079268 0.076942 0.074715 0.072575
|
||||
2 0.075188 0.072994 0.072554 0.070562 0.068664 0.066850 0.065110
|
||||
3 0.093497 0.090128 0.093593 0.090694 0.087910 0.085224 0.082620
|
||||
4 0.089355 0.086322 0.089041 0.086420 0.083899 0.081468 0.079110
|
||||
5 0.084979 0.082286 0.084278 0.081932 0.079676 0.077499 0.075389
|
||||
6 0.091236 0.087833 0.093078 0.090174 0.087372 0.084657 0.082014
|
||||
7 0.088271 0.085113 0.089775 0.087081 0.084477 0.081953 0.079493
|
||||
8 0.085345 0.082426 0.086518 0.084024 0.081613 0.079273 0.076990
|
||||
9 0.082426 0.079741 0.083272 0.080973 0.078749 0.076588 0.074480
|
||||
10 0.086518 0.083272 0.089374 0.086610 0.083935 0.081334 0.078795
|
||||
11 0.084024 0.080973 0.086610 0.084017 0.081503 0.079056 0.076665
|
||||
12 0.081613 0.078749 0.083935 0.081503 0.079142 0.076843 0.074592
|
||||
13 0.079273 0.076588 0.081334 0.079056 0.076843 0.074684 0.072569
|
||||
14 0.076990 0.074480 0.078795 0.076665 0.074592 0.072569 0.070585
|
||||
1 0.076022 0.076079 0.068075 0.067712 0.067499 0.067443 0.067547
|
||||
2 0.070618 0.071460 0.061188 0.061308 0.061604 0.062089 0.062777
|
||||
3 0.082990 0.082027 0.078143 0.077144 0.076264 0.075498 0.074845
|
||||
4 0.081357 0.080984 0.075043 0.074420 0.073938 0.073597 0.073400
|
||||
5 0.079553 0.079823 0.071666 0.071445 0.071388 0.071505 0.071804
|
||||
6 0.078908 0.077219 0.077200 0.075770 0.074418 0.073134 0.071908
|
||||
7 0.077986 0.076729 0.075149 0.074006 0.072955 0.071991 0.071106
|
||||
8 0.077140 0.076349 0.073080 0.072240 0.071510 0.070887 0.070370
|
||||
9 0.076349 0.076066 0.070961 0.070443 0.070056 0.069801 0.069681
|
||||
10 0.073080 0.070961 0.073601 0.071922 0.070288 0.068689 0.067110
|
||||
11 0.072240 0.070443 0.071922 0.070466 0.069065 0.067709 0.066388
|
||||
12 0.071510 0.070056 0.070288 0.069065 0.067907 0.066808 0.065761
|
||||
13 0.070887 0.069801 0.068689 0.067709 0.066808 0.065983 0.065228
|
||||
14 0.070370 0.069681 0.067110 0.066388 0.065761 0.065228 0.064787
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
@@ -2584,7 +2678,7 @@ order to another one.</p>
|
||||
<div class="cell_output docutils container">
|
||||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[ 1.0169643 0.27924636 -1.4087793 1.03308408 0. ]
|
||||
Test MSE OLS
|
||||
0.958228616652075
|
||||
0.9582286166520774
|
||||
</pre></div>
|
||||
</div>
|
||||
<img alt="_images/chapter2_322_1.png" src="_images/chapter2_322_1.png" />
|
||||
@@ -2993,54 +3087,42 @@ decreasing <span class="math notranslate nohighlight">\(\lambda\)</span> and shr
|
||||
|
||||
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