tensorflow problems with chapter 12
This commit is contained in:
Morten Hjorth-Jensen
@@ -645,10 +645,10 @@ number <span class="math notranslate nohighlight">\(i\)</span> is left out. Usin
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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>Runtime: 0.135707 sec
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Runtime: 0.135976 sec
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Jackknife Statistics :
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original bias std. error
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100.099 100.089 0.150795
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99.655 99.645 0.148675
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</pre></div>
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@@ -867,7 +867,7 @@ theorem.</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>Bootstrap Statistics :
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original bias std. error
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100.186 15.0063 100.185 0.148455
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100.098 15.248 100.098 0.153966
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</pre></div>
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</div>
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</div>
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@@ -1069,9 +1069,10 @@ Error: 0.32149601703519126
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Bias^2: 0.3123314713548606
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Var: 0.009164545680330616
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0.32149601703519126 >= 0.3123314713548606 + 0.009164545680330616 = 0.3214960170351912
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Polynomial degree:
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</pre></div>
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</div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Polynomial degree: 1
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span> 1
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Error: 0.08426840630693411
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Bias^2: 0.07968918676726028
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Var: 0.004579219539673833
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@@ -1113,14 +1114,14 @@ Error: 0.017355848195591973
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Bias^2: 0.010331721306655588
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Var: 0.007024126888936384
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0.017355848195591973 >= 0.010331721306655588 + 0.007024126888936384 = 0.017355848195591973
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</pre></div>
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</div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Polynomial degree: 9
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Polynomial degree: 9
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Error: 0.026605727637189085
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Bias^2: 0.010018312644140933
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Var: 0.016587414993048166
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0.026605727637189085 >= 0.010018312644140933 + 0.016587414993048166 = 0.0266057276371891
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Polynomial degree: 10
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</pre></div>
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</div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Polynomial degree: 10
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Error: 0.021592704588043153
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Bias^2: 0.010516485576652981
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Var: 0.011076219011390184
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@@ -1135,16 +1136,14 @@ Error: 0.1154777721897675
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Bias^2: 0.01628578269590588
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Var: 0.09919198949386163
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0.1154777721897675 >= 0.01628578269590588 + 0.09919198949386163 = 0.11547777218976751
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</pre></div>
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</div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Polynomial degree: 13
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Polynomial degree: 13
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Error: 0.22842468702166951
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Bias^2: 0.01975416527163567
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Var: 0.20867052175003387
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0.22842468702166951 >= 0.01975416527163567 + 0.20867052175003387 = 0.22842468702166954
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</pre></div>
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</div>
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<img alt="_images/chapter3_62_5.png" src="_images/chapter3_62_5.png" />
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<img alt="_images/chapter3_62_4.png" src="_images/chapter3_62_4.png" />
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</div>
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</div>
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<p>The bias-variance tradeoff summarizes the fundamental tension in
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@@ -1437,12 +1436,12 @@ Mean squared error on test data: 873.95463048
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Degree of polynomial: 23
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Mean squared error on training data: 0.00085890
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Mean squared error on test data: 5535.20053452
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</pre></div>
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</div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 24
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Degree of polynomial: 24
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Mean squared error on training data: 0.00084714
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Mean squared error on test data: 1289.22422186
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Degree of polynomial: 25
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</pre></div>
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</div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 25
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Mean squared error on training data: 0.00079022
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Mean squared error on test data: 136582.88824397
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Degree of polynomial: 26
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@@ -1801,32 +1800,18 @@ When we take the derivative with respect to <span class="math notranslate nohigh
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\[
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\sum_{i=0}^{n-1} \beta_0 = \sum_{i=0}^{n-1}y_i - \sum_{i=0}^{n-1} \sum_{j=1}^{p-1} X_{ij} \beta_j.
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\]</div>
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<p>We assume
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that every column of <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span> is centered, which we can do by subtracting the mean,</p>
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<div class="cell docutils container">
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<div class="cell_input docutils container">
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">X</span> <span class="o">=</span> <span class="n">X</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
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</pre></div>
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</div>
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</div>
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</div>
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<p>This means that we need to rewrite <span class="math notranslate nohighlight">\(X_{ij}\)</span> as <span class="math notranslate nohighlight">\(\tilde{X}_{ij}=X_{ij}-\mu_j\)</span>, where</p>
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<div class="math notranslate nohighlight">
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\[
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\mu_j = \frac{1}{n}\sum_{i=0}^{n-1}X_{ij}.
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\]</div>
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<p>Let us special first to the case where we have only two parameters <span class="math notranslate nohighlight">\(\beta_0\)</span> and <span class="math notranslate nohighlight">\(\beta_1\)</span>.
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Our result for <span class="math notranslate nohighlight">\(\beta_0\)</span> simplifies then to</p>
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<div class="math notranslate nohighlight">
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\[
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n\beta_0 = \sum_{i=0}^{n-1}y_i - \sum_{i=0}^{n-1} X_{i1} \beta_1.
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\]</div>
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<p>Assuming that the matrix elements <span class="math notranslate nohighlight">\(X_{i1}\)</span> are centered, what we have is</p>
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<p>We obtain then</p>
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<div class="math notranslate nohighlight">
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\[
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\beta_0 = \frac{1}{n}\sum_{i=0}^{n-1}y_i - \beta_1\frac{1}{n}\sum_{i=0}^{n-1} \left(X_{i1}-\mu_{1}\right),
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\beta_0 = \frac{1}{n}\sum_{i=0}^{n-1}y_i - \beta_1\frac{1}{n}\sum_{i=0}^{n-1} X_{i1}.
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\]</div>
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<p>where</p>
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<p>If we define</p>
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<div class="math notranslate nohighlight">
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\[
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\mu_1=\frac{1}{n}\sum_{i=0}^{n-1} (X_{i1},
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@@ -1839,21 +1824,14 @@ n\beta_0 = \sum_{i=0}^{n-1}y_i - \sum_{i=0}^{n-1} X_{i1} \beta_1.
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<p>we have</p>
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<div class="math notranslate nohighlight">
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\[
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\beta_0 = \mu_y - \beta_1\frac{1}{n}\sum_{i=0}^{n-1} (X_{i1}-\mu_{1}),
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\beta_0 = \mu_y - \beta_1\mu_{1}.
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\]</div>
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<p>and it is easy to see that the last sum equals zero! This means that we have</p>
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<p>In the general case withmore parameters than <span class="math notranslate nohighlight">\(\beta_0\)</span> and <span class="math notranslate nohighlight">\(\beta_1\)</span>, we have</p>
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<div class="math notranslate nohighlight">
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\[
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\beta_0 = \mu_y,
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\beta_0 = \frac{1}{n}\sum_{i=0}^{n-1}y_i - \frac{1}{n}\sum_{i=0}^{n-1}\sum_{j=1}^{p-1} X_{ij}\beta_j.
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\]</div>
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<p>if the columns of the design matrix are centered. It is straight forward to generalize this results to more values of <span class="math notranslate nohighlight">\(\beta\)</span>.
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We have thus</p>
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<div class="math notranslate nohighlight">
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\[
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\beta_0 = \frac{1}{n}\sum_{i=0}^{n-1} y_i = \overline{\boldsymbol{y}},
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\]</div>
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<p>the average value of <span class="math notranslate nohighlight">\(\boldsymbol{y}\)</span>.</p>
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<p>Replacing <span class="math notranslate nohighlight">\(y_i\)</span> with <span class="math notranslate nohighlight">\(y_i - \beta_0 = y_i - \overline{\boldsymbol{y}}\)</span> and centering also our design matrix results in a cost function (in vector-matrix disguise)</p>
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<p>Replacing <span class="math notranslate nohighlight">\(y_i\)</span> with <span class="math notranslate nohighlight">\(y_i - y_i - \overline{\boldsymbol{y}}\)</span> and centering also our design matrix results in a cost function (in vector-matrix disguise)</p>
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<div class="math notranslate nohighlight">
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\[
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C(\boldsymbol{\beta}) = (\boldsymbol{\tilde{y}} - \tilde{X}\boldsymbol{\beta})^T(\boldsymbol{\tilde{y}} - \tilde{X}\boldsymbol{\beta}).
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@@ -1982,7 +1960,7 @@ MSE with Sklearn intercept
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0.004113634617443135
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</pre></div>
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</div>
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<img alt="_images/chapter3_109_1.png" src="_images/chapter3_109_1.png" />
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<img alt="_images/chapter3_103_1.png" src="_images/chapter3_103_1.png" />
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</div>
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</div>
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<p>The intercept is the value of our output/target variable
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@@ -2177,7 +2155,7 @@ MSE values for Scikit-Learn Ridge implementation
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0.26409315307910036
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</pre></div>
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</div>
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<img alt="_images/chapter3_117_1.png" src="_images/chapter3_117_1.png" />
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<img alt="_images/chapter3_111_1.png" src="_images/chapter3_111_1.png" />
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</div>
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</div>
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<p>The results here agree when we force <strong>Scikit-Learn</strong>’s Ridge function to include the first column in our design matrix.
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@@ -2236,8 +2214,6 @@ Let us see how we can change this code by zero centering.</p>
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<span class="n">OwnRidgeBeta</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">pinv</span><span class="p">(</span><span class="n">X_train_scaled</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">X_train_scaled</span><span class="o">+</span><span class="n">lmb</span><span class="o">*</span><span class="n">I</span><span class="p">)</span> <span class="o">@</span> <span class="n">X_train_scaled</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="p">(</span><span class="n">y_train_scaled</span><span class="p">)</span>
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<span class="n">intercept_</span> <span class="o">=</span> <span class="n">y_scaler</span> <span class="o">-</span> <span class="n">X_train_mean</span><span class="nd">@OwnRidgeBeta</span> <span class="c1">#The intercept can be shifted so the model can predict on uncentered data</span>
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<span class="c1">#Add intercept to prediction</span>
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<span class="n">ypredictOwnRidge</span> <span class="o">=</span> <span class="n">X_test</span> <span class="o">@</span> <span class="n">OwnRidgeBeta</span> <span class="o">+</span> <span class="n">intercept_</span>
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<span class="c1">#Add intercept to prediction</span>
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<span class="n">ypredictOwnRidge</span> <span class="o">=</span> <span class="n">X_test_scaled</span> <span class="o">@</span> <span class="n">OwnRidgeBeta</span> <span class="o">+</span> <span class="n">y_scaler</span>
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<span class="n">RegRidge</span> <span class="o">=</span> <span class="n">linear_model</span><span class="o">.</span><span class="n">Ridge</span><span class="p">(</span><span class="n">lmb</span><span class="p">)</span>
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<span class="n">RegRidge</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span><span class="n">y_train</span><span class="p">)</span>
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@@ -2382,7 +2358,7 @@ MSE values for Scikit-Learn Ridge implementation
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0.002381316302584886
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</pre></div>
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</div>
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<img alt="_images/chapter3_119_1.png" src="_images/chapter3_119_1.png" />
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<img alt="_images/chapter3_113_1.png" src="_images/chapter3_113_1.png" />
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</div>
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</div>
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<p>We see here, when compared to the code which includes explicitely the
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@@ -2596,11 +2572,11 @@ linear system as an equation would reduce this down to
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<div class="cell_output docutils container">
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<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span><ipython-input-21-6f7a6bd7d79f>:7: UserWarning: FixedFormatter should only be used together with FixedLocator
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<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span><ipython-input-20-6f7a6bd7d79f>:7: UserWarning: FixedFormatter should only be used together with FixedLocator
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cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
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</pre></div>
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</div>
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<img alt="_images/chapter3_150_1.png" src="_images/chapter3_150_1.png" />
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<img alt="_images/chapter3_144_1.png" src="_images/chapter3_144_1.png" />
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</div>
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</div>
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<p>It is interesting to note that OLS
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@@ -2740,11 +2716,11 @@ with the form utilized in linear regression, viz.</p>
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</div>
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<div class="cell_output docutils container">
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<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span><ipython-input-26-5dd54edf2138>:7: UserWarning: FixedFormatter should only be used together with FixedLocator
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<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span><ipython-input-25-5dd54edf2138>:7: UserWarning: FixedFormatter should only be used together with FixedLocator
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cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
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</pre></div>
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</div>
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<img alt="_images/chapter3_168_1.png" src="_images/chapter3_168_1.png" />
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<img alt="_images/chapter3_162_1.png" src="_images/chapter3_162_1.png" />
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</div>
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</div>
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<p>The results agree perfectly with our previous discussion where we used our own code.</p>
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@@ -2753,8 +2729,8 @@ regression. In ridge regression we include a <strong>regularizer</strong>. This
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involves a new cost function which leads to a new estimate for the
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weights <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span>. This results in a penalized regression problem. The
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cost function is given by</p>
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<p>6
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0</p>
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<p>5
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8</p>
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<p><
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<
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<
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@@ -2788,11 +2764,11 @@ K</p>
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</div>
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<div class="cell_output docutils container">
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<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span><ipython-input-27-fe5b9d300cc0>:10: UserWarning: FixedFormatter should only be used together with FixedLocator
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<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span><ipython-input-26-fe5b9d300cc0>:10: UserWarning: FixedFormatter should only be used together with FixedLocator
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cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
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</pre></div>
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</div>
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<img alt="_images/chapter3_171_1.png" src="_images/chapter3_171_1.png" />
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<img alt="_images/chapter3_165_1.png" src="_images/chapter3_165_1.png" />
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</div>
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</div>
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<p>In the <strong>Least Absolute Shrinkage and Selection Operator</strong> (LASSO)-method we get a third cost function.</p>
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@@ -2823,11 +2799,11 @@ K</p>
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</div>
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<div class="cell_output docutils container">
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<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span><ipython-input-28-25845e8df859>:9: UserWarning: FixedFormatter should only be used together with FixedLocator
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<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span><ipython-input-27-25845e8df859>:9: UserWarning: FixedFormatter should only be used together with FixedLocator
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cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
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</pre></div>
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</div>
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</div>
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<p>It is quite striking how LASSO breaks the symmetry of the coupling
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@@ -2885,40 +2861,40 @@ constant as opposed to ridge and OLS. We get a sparse solution with
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<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span> 60%|██████ | 6/10 [00:01<00:00, 4.61it/s]
|
||||
</pre></div>
|
||||
</div>
|
||||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span> 70%|███████ | 7/10 [00:01<00:00, 5.95it/s]
|
||||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span> 70%|███████ | 7/10 [00:01<00:00, 5.17it/s]
|
||||
</pre></div>
|
||||
</div>
|
||||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span> 80%|████████ | 8/10 [00:01<00:00, 6.55it/s]
|
||||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span> 80%|████████ | 8/10 [00:01<00:00, 5.73it/s]
|
||||
</pre></div>
|
||||
</div>
|
||||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span> 90%|█████████ | 9/10 [00:01<00:00, 6.99it/s]
|
||||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span> 90%|█████████ | 9/10 [00:01<00:00, 6.46it/s]
|
||||
</pre></div>
|
||||
</div>
|
||||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>100%|██████████| 10/10 [00:01<00:00, 7.35it/s]
|
||||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>100%|██████████| 10/10 [00:01<00:00, 6.87it/s]
|
||||
</pre></div>
|
||||
</div>
|
||||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>100%|██████████| 10/10 [00:01<00:00, 5.87it/s]
|
||||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>100%|██████████| 10/10 [00:01<00:00, 5.22it/s]
|
||||
</pre></div>
|
||||
</div>
|
||||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>
|
||||
</pre></div>
|
||||
</div>
|
||||
<img alt="_images/chapter3_177_13.png" src="_images/chapter3_177_13.png" />
|
||||
<img alt="_images/chapter3_171_13.png" src="_images/chapter3_171_13.png" />
|
||||
</div>
|
||||
</div>
|
||||
<p>We see that LASSO reaches a good solution for low
|
||||
@@ -2967,7 +2943,7 @@ testing set that is close to the accuracy of the training set.</p>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<img alt="_images/chapter3_179_0.png" src="_images/chapter3_179_0.png" />
|
||||
<img alt="_images/chapter3_173_0.png" src="_images/chapter3_173_0.png" />
|
||||
</div>
|
||||
</div>
|
||||
<p>From the above figure we can see that LASSO with <span class="math notranslate nohighlight">\(\lambda = 10^{-2}\)</span>
|
||||
@@ -3059,7 +3035,7 @@ which polynomial fits the data best.</p>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<img alt="_images/chapter3_183_0.png" src="_images/chapter3_183_0.png" />
|
||||
<img alt="_images/chapter3_177_0.png" src="_images/chapter3_177_0.png" />
|
||||
</div>
|
||||
</div>
|
||||
<div class="section" id="exercise-ordinary-least-square-ols-on-the-franke-function">
|
||||
@@ -3216,7 +3192,7 @@ Python program using</p>
|
||||
<div class="cell_output docutils container">
|
||||
<div class="output traceback highlight-ipythontb notranslate"><div class="highlight"><pre><span></span><span class="gt">---------------------------------------------------------------------------</span>
|
||||
<span class="ne">NameError</span><span class="g g-Whitespace"> </span>Traceback (most recent call last)
|
||||
<span class="o"><</span><span class="n">ipython</span><span class="o">-</span><span class="nb">input</span><span class="o">-</span><span class="mi">32</span><span class="o">-</span><span class="n">d985fb40c43d</span><span class="o">></span> <span class="ow">in</span> <span class="o"><</span><span class="n">module</span><span class="o">></span>
|
||||
<span class="o"><</span><span class="n">ipython</span><span class="o">-</span><span class="nb">input</span><span class="o">-</span><span class="mi">31</span><span class="o">-</span><span class="n">d985fb40c43d</span><span class="o">></span> <span class="ow">in</span> <span class="o"><</span><span class="n">module</span><span class="o">></span>
|
||||
<span class="ne">----> </span><span class="mi">1</span> <span class="n">scipy</span><span class="o">.</span><span class="n">misc</span><span class="o">.</span><span class="n">imread</span>
|
||||
|
||||
<span class="ne">NameError</span>: name 'scipy' is not defined
|
||||
|
||||
Reference in New Issue
Block a user