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Commit b1342638 authored by François Févotte's avatar François Févotte
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Ex 04-1 : analyse Challenger en Julia

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# This file is machine-generated - editing it directly is not advised
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module4/Project.toml 0 → 100644
View file @ b1342638
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View file @ b1342638
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<div class="title">
<h1 class="title">Risk Analysis of the Space Shuttle: Pre-Challenger Prediction of Failure</h1>
</div>
<p>In this document we reperform some of the analysis provided in <em>Risk Analysis of the Space Shuttle: Pre-Challenger Prediction of Failure</em> by Siddhartha R. Dalal, Edward B. Fowlkes, Bruce Hoadley published in Journal of the American Statistical Association, Vol. 84, No. 408 &#40;Dec., 1989&#41;, pp. 945-957 and available at <a href="http://www.jstor.org/stable/2290069">http://www.jstor.org/stable/2290069</a>.</p>
<p>On the fourth page of this article, they indicate that the maximum likelihood estimates of the logistic regression using only temperature are: <span class="math">$\hat{\alpha}=5.085$</span> and <span class="math">$\hat{\beta}=-0.1156$</span> and their asymptotic standard errors are <span class="math">$s_{\hat{\alpha}}=3.052$</span> and <span class="math">$s_{\hat{\beta}}=0.047$</span>. The Goodness of fit indicated for this model was <span class="math">$G^2=18.086$</span> with 21 degrees of freedom. Our goal is to reproduce the computation behind these values and the Figure 4 of this article, possibly in a nicer looking way.</p>
<h1>Technical information on the computer on which the analysis is run</h1>
<p>We will be using the <a href="http://julialang.org">Julia</a> language:</p>
<pre class='hljl'>
<span class='hljl-k'>using</span><span class='hljl-t'> </span><span class='hljl-n'>InteractiveUtils</span><span class='hljl-t'>
</span><span class='hljl-nf'>versioninfo</span><span class='hljl-p'>()</span>
</pre>
<pre class="output">
Julia Version 1.4.0
Commit b8e9a9ecc6 &#40;2020-03-21 16:36 UTC&#41;
Platform Info:
OS: Linux &#40;x86_64-pc-linux-gnu&#41;
CPU: Intel&#40;R&#41; Core&#40;TM&#41; i5-6200U CPU @ 2.30GHz
WORD_SIZE: 64
LIBM: libopenlibm
LLVM: libLLVM-8.0.1 &#40;ORCJIT, skylake&#41;
Environment:
JULIA_PROJECT &#61; @.
</pre>
<p>The computations rely on a number of packages in the Julia ecosystem. The direct dependencies are summarized hereafter; the complete environment is described in the <a href="Manifest.toml"><code>Manifest.toml</code></a> file.</p>
<pre class='hljl'>
<span class='hljl-cs'># Setup environment</span><span class='hljl-t'>
</span><span class='hljl-k'>using</span><span class='hljl-t'> </span><span class='hljl-n'>Pkg</span><span class='hljl-t'>
</span><span class='hljl-n'>Pkg</span><span class='hljl-oB'>.</span><span class='hljl-nf'>activate</span><span class='hljl-p'>(</span><span class='hljl-nd'>@__DIR__</span><span class='hljl-p'>)</span><span class='hljl-t'>
</span><span class='hljl-n'>Pkg</span><span class='hljl-oB'>.</span><span class='hljl-nf'>instantiate</span><span class='hljl-p'>()</span><span class='hljl-t'>
</span><span class='hljl-cs'># Load dependencies</span><span class='hljl-t'>
</span><span class='hljl-k'>using</span><span class='hljl-t'> </span><span class='hljl-n'>HTTP</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>CSV</span><span class='hljl-t'>
</span><span class='hljl-k'>using</span><span class='hljl-t'> </span><span class='hljl-n'>Plots</span><span class='hljl-p'>;</span><span class='hljl-t'> </span><span class='hljl-nf'>plotly</span><span class='hljl-p'>()</span><span class='hljl-t'>
</span><span class='hljl-k'>using</span><span class='hljl-t'> </span><span class='hljl-n'>GLM</span><span class='hljl-t'>
</span><span class='hljl-k'>using</span><span class='hljl-t'> </span><span class='hljl-n'>DataFrames</span><span class='hljl-t'>
</span><span class='hljl-k'>using</span><span class='hljl-t'> </span><span class='hljl-n'>Printf</span><span class='hljl-t'>
</span><span class='hljl-nf'>include</span><span class='hljl-p'>(</span><span class='hljl-s'>&quot;utils.jl&quot;</span><span class='hljl-p'>)</span><span class='hljl-t'>
</span><span class='hljl-cs'># Summary</span><span class='hljl-t'>
</span><span class='hljl-n'>Pkg</span><span class='hljl-oB'>.</span><span class='hljl-nf'>status</span><span class='hljl-p'>()</span>
</pre>
<pre class="output">
Project Challenger v0.1.0
Status &#96;~/tmp/MOOC-RR/module4/Project.toml&#96;
&#91;336ed68f&#93; CSV v0.6.1
&#91;a93c6f00&#93; DataFrames v0.20.2
&#91;82cc6244&#93; DataInterpolations v2.0.0
&#91;38e38edf&#93; GLM v1.3.9
&#91;cd3eb016&#93; HTTP v0.8.14
&#91;9b87118b&#93; PackageCompiler v1.1.1
&#91;91a5bcdd&#93; Plots v0.29.9
&#91;44d3d7a6&#93; Weave v0.9.4
</pre>
<h1>Loading and inspecting data</h1>
<p>Let&#39;s start by reading data.</p>
<pre class='hljl'>
<span class='hljl-n'>res</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-n'>HTTP</span><span class='hljl-oB'>.</span><span class='hljl-nf'>request</span><span class='hljl-p'>(</span><span class='hljl-sc'>:GET</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-s'>&quot;https://app-learninglab.inria.fr/moocrr/gitlab/moocrr-session3/moocrr-reproducibility-study/raw/master/data/shuttle.csv?inline=false&quot;</span><span class='hljl-p'>)</span><span class='hljl-t'>
</span><span class='hljl-n'>data</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-n'>CSV</span><span class='hljl-oB'>.</span><span class='hljl-nf'>read</span><span class='hljl-p'>(</span><span class='hljl-n'>res</span><span class='hljl-oB'>.</span><span class='hljl-n'>body</span><span class='hljl-p'>)</span>
</pre>
<table class="data-frame"><thead><tr><th></th><th>Date</th><th>Count</th><th>Temperature</th><th>Pressure</th><th>Malfunction</th></tr><tr><th></th><th>String</th><th>Int64</th><th>Int64</th><th>Int64</th><th>Int64</th></tr></thead><tbody><p>23 rows × 5 columns</p><tr><th>1</th><td>4/12/81</td><td>6</td><td>66</td><td>50</td><td>0</td></tr><tr><th>2</th><td>11/12/81</td><td>6</td><td>70</td><td>50</td><td>1</td></tr><tr><th>3</th><td>3/22/82</td><td>6</td><td>69</td><td>50</td><td>0</td></tr><tr><th>4</th><td>11/11/82</td><td>6</td><td>68</td><td>50</td><td>0</td></tr><tr><th>5</th><td>4/04/83</td><td>6</td><td>67</td><td>50</td><td>0</td></tr><tr><th>6</th><td>6/18/82</td><td>6</td><td>72</td><td>50</td><td>0</td></tr><tr><th>7</th><td>8/30/83</td><td>6</td><td>73</td><td>100</td><td>0</td></tr><tr><th>8</th><td>11/28/83</td><td>6</td><td>70</td><td>100</td><td>0</td></tr><tr><th>9</th><td>2/03/84</td><td>6</td><td>57</td><td>200</td><td>1</td></tr><tr><th>10</th><td>4/06/84</td><td>6</td><td>63</td><td>200</td><td>1</td></tr><tr><th>11</th><td>8/30/84</td><td>6</td><td>70</td><td>200</td><td>1</td></tr><tr><th>12</th><td>10/05/84</td><td>6</td><td>78</td><td>200</td><td>0</td></tr><tr><th>13</th><td>11/08/84</td><td>6</td><td>67</td><td>200</td><td>0</td></tr><tr><th>14</th><td>1/24/85</td><td>6</td><td>53</td><td>200</td><td>2</td></tr><tr><th>15</th><td>4/12/85</td><td>6</td><td>67</td><td>200</td><td>0</td></tr><tr><th>16</th><td>4/29/85</td><td>6</td><td>75</td><td>200</td><td>0</td></tr><tr><th>17</th><td>6/17/85</td><td>6</td><td>70</td><td>200</td><td>0</td></tr><tr><th>18</th><td>7/2903/85</td><td>6</td><td>81</td><td>200</td><td>0</td></tr><tr><th>19</th><td>8/27/85</td><td>6</td><td>76</td><td>200</td><td>0</td></tr><tr><th>20</th><td>10/03/85</td><td>6</td><td>79</td><td>200</td><td>0</td></tr><tr><th>21</th><td>10/30/85</td><td>6</td><td>75</td><td>200</td><td>2</td></tr><tr><th>22</th><td>11/26/85</td><td>6</td><td>76</td><td>200</td><td>0</td></tr><tr><th>23</th><td>1/12/86</td><td>6</td><td>58</td><td>200</td><td>1</td></tr></tbody></table>
<p>We know from our previous experience on this data set that filtering data is a really bad idea. We will therefore process it as such.</p>
<pre class='hljl'>
<span class='hljl-n'>data</span><span class='hljl-oB'>.</span><span class='hljl-n'>Frequency</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-n'>data</span><span class='hljl-oB'>.</span><span class='hljl-n'>Malfunction</span><span class='hljl-t'> </span><span class='hljl-oB'>./</span><span class='hljl-t'> </span><span class='hljl-n'>data</span><span class='hljl-oB'>.</span><span class='hljl-n'>Count</span><span class='hljl-t'>
</span><span class='hljl-nf'>plot</span><span class='hljl-p'>(</span><span class='hljl-n'>xlabel</span><span class='hljl-oB'>=</span><span class='hljl-s'>&quot;Temperature [F]&quot;</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>ylabel</span><span class='hljl-oB'>=</span><span class='hljl-s'>&quot;Frequency&quot;</span><span class='hljl-p'>)</span><span class='hljl-t'>
</span><span class='hljl-nf'>plot!</span><span class='hljl-p'>(</span><span class='hljl-n'>data</span><span class='hljl-oB'>.</span><span class='hljl-n'>Temperature</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>data</span><span class='hljl-oB'>.</span><span class='hljl-n'>Frequency</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>seriestype</span><span class='hljl-oB'>=:</span><span class='hljl-n'>scatter</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>label</span><span class='hljl-oB'>=</span><span class='hljl-n'>nothing</span><span class='hljl-p'>)</span><span class='hljl-t'>
</span><span class='hljl-nf'>disp</span><span class='hljl-p'>()</span>
</pre>
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<h1>Logistic regression</h1>
<p>Let&#39;s assume O-rings independently fail with the same probability which solely depends on temperature. A logistic regression should allow us to estimate the influence of temperature.</p>
<pre class='hljl'>
<span class='hljl-n'>model</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>glm</span><span class='hljl-p'>(</span><span class='hljl-nd'>@formula</span><span class='hljl-p'>(</span><span class='hljl-n'>Frequency</span><span class='hljl-t'> </span><span class='hljl-oB'>~</span><span class='hljl-t'> </span><span class='hljl-n'>Temperature</span><span class='hljl-p'>),</span><span class='hljl-t'> </span><span class='hljl-n'>data</span><span class='hljl-p'>,</span><span class='hljl-t'>
</span><span class='hljl-nf'>Binomial</span><span class='hljl-p'>(),</span><span class='hljl-t'> </span><span class='hljl-nf'>LogitLink</span><span class='hljl-p'>())</span><span class='hljl-t'>
</span><span class='hljl-n'>α</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>β</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>coef</span><span class='hljl-p'>(</span><span class='hljl-n'>model</span><span class='hljl-p'>)</span><span class='hljl-t'>
</span><span class='hljl-n'>σα</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>σβ</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>stderror</span><span class='hljl-p'>(</span><span class='hljl-n'>model</span><span class='hljl-p'>)</span><span class='hljl-t'>
</span><span class='hljl-n'>G²</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>deviance</span><span class='hljl-p'>(</span><span class='hljl-n'>model</span><span class='hljl-p'>)</span><span class='hljl-t'>
</span><span class='hljl-n'>nDOF</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>Int</span><span class='hljl-p'>(</span><span class='hljl-nf'>dof_residual</span><span class='hljl-p'>(</span><span class='hljl-n'>model</span><span class='hljl-p'>))</span><span class='hljl-t'>
</span><span class='hljl-n'>model</span>
</pre>
<pre class="output">
StatsModels.TableRegressionModel&#123;GeneralizedLinearModel&#123;GLM.GlmResp&#123;Array&#123;Float64,1&#125;,Binomial&#123;Float64&#125;,LogitLink&#125;,GLM.DensePredChol&#123;Float64,LinearAlgebra.Cholesky&#123;Float64,Array&#123;Float64,2&#125;&#125;&#125;&#125;,Array&#123;Float64,2&#125;&#125;
Frequency ~ 1 &#43; Temperature
Coefficients:
────────────────────────────────────────────────────────────────────────────
Estimate Std. Error z value Pr&#40;&gt;|z|&#41; Lower 95&#37; Upper 95&#37;
────────────────────────────────────────────────────────────────────────────
&#40;Intercept&#41; 5.08498 7.47703 0.68008 0.4965 -9.56973 19.7397
Temperature -0.115601 0.115184 -1.00362 0.3156 -0.341358 0.110156
────────────────────────────────────────────────────────────────────────────
</pre>
<p>The maximum likelyhood estimator of the intercept and of Temperature are thus <span class="math">$\hat{\alpha}=5.085$</span> and <span class="math">$\hat{\beta}=-0.116$</span>.</p>
<p>This corresponds to the values from the article of Dalal et al. The standard errors are <span class="math">$s_{\hat{\alpha}} = 7.477$</span> and <span class="math">$s_{\hat{\beta}} = 0.115$</span>, which is different from the <span class="math">$3.052$</span> and <span class="math">$0.047$</span> reported by Dallal et al. The deviance is <span class="math">$G^2 = 3.014$</span> with 22 degrees of freedom.</p>
<p>I cannot find any value similar to the Goodness of fit &#40;<span class="math">$G^2=18.086$</span>&#41; reported by Dalal et al. However, the number of degrees of freedom is similar to theirs &#40;21&#41;.</p>
<p>There seems to be something wrong. Oh I know, I haven&#39;t indicated that my observations are actually the result of 6 observations for each rocket launch. The correct way to do this would be to weight the data using the <code>Count</code> column. Since I don&#39;t know how to do that with the <a href="https://github.com/JuliaStats/GLM.jl">GLM</a> package I&#39;m using, I will simply duplicate the data:</p>
<pre class='hljl'>
<span class='hljl-n'>weighted_data</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>DataFrame</span><span class='hljl-p'>(</span><span class='hljl-n'>Temperature</span><span class='hljl-oB'>=</span><span class='hljl-n'>Int</span><span class='hljl-p'>[],</span><span class='hljl-t'> </span><span class='hljl-n'>Frequency</span><span class='hljl-oB'>=</span><span class='hljl-n'>Float64</span><span class='hljl-p'>[])</span><span class='hljl-t'>
</span><span class='hljl-k'>for</span><span class='hljl-t'> </span><span class='hljl-n'>row</span><span class='hljl-t'> </span><span class='hljl-kp'>in</span><span class='hljl-t'> </span><span class='hljl-nf'>eachrow</span><span class='hljl-p'>(</span><span class='hljl-n'>data</span><span class='hljl-p'>)</span><span class='hljl-t'>
</span><span class='hljl-k'>for</span><span class='hljl-t'> </span><span class='hljl-n'>_</span><span class='hljl-t'> </span><span class='hljl-kp'>in</span><span class='hljl-t'> </span><span class='hljl-ni'>1</span><span class='hljl-oB'>:</span><span class='hljl-n'>row</span><span class='hljl-oB'>.</span><span class='hljl-n'>Count</span><span class='hljl-t'>
</span><span class='hljl-nf'>push!</span><span class='hljl-p'>(</span><span class='hljl-n'>weighted_data</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-p'>(</span><span class='hljl-n'>Temperature</span><span class='hljl-oB'>=</span><span class='hljl-n'>row</span><span class='hljl-oB'>.</span><span class='hljl-n'>Temperature</span><span class='hljl-p'>,</span><span class='hljl-t'>
</span><span class='hljl-n'>Frequency</span><span class='hljl-oB'>=</span><span class='hljl-n'>row</span><span class='hljl-oB'>.</span><span class='hljl-n'>Frequency</span><span class='hljl-p'>))</span><span class='hljl-t'>
</span><span class='hljl-k'>end</span><span class='hljl-t'>
</span><span class='hljl-k'>end</span><span class='hljl-t'>
</span><span class='hljl-n'>model</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>glm</span><span class='hljl-p'>(</span><span class='hljl-nd'>@formula</span><span class='hljl-p'>(</span><span class='hljl-n'>Frequency</span><span class='hljl-t'> </span><span class='hljl-oB'>~</span><span class='hljl-t'> </span><span class='hljl-n'>Temperature</span><span class='hljl-p'>),</span><span class='hljl-t'> </span><span class='hljl-n'>weighted_data</span><span class='hljl-p'>,</span><span class='hljl-t'>
</span><span class='hljl-nf'>Binomial</span><span class='hljl-p'>(),</span><span class='hljl-t'> </span><span class='hljl-nf'>LogitLink</span><span class='hljl-p'>())</span><span class='hljl-t'>
</span><span class='hljl-n'>α</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>β</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>coef</span><span class='hljl-p'>(</span><span class='hljl-n'>model</span><span class='hljl-p'>)</span><span class='hljl-t'>
</span><span class='hljl-n'>σα</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>σβ</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>stderror</span><span class='hljl-p'>(</span><span class='hljl-n'>model</span><span class='hljl-p'>)</span><span class='hljl-t'>
</span><span class='hljl-n'>G²</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>deviance</span><span class='hljl-p'>(</span><span class='hljl-n'>model</span><span class='hljl-p'>)</span><span class='hljl-t'>
</span><span class='hljl-n'>nDOF</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>Int</span><span class='hljl-p'>(</span><span class='hljl-nf'>dof_residual</span><span class='hljl-p'>(</span><span class='hljl-n'>model</span><span class='hljl-p'>))</span><span class='hljl-t'>
</span><span class='hljl-n'>model</span>
</pre>
<pre class="output">
StatsModels.TableRegressionModel&#123;GeneralizedLinearModel&#123;GLM.GlmResp&#123;Array&#123;Float64,1&#125;,Binomial&#123;Float64&#125;,LogitLink&#125;,GLM.DensePredChol&#123;Float64,LinearAlgebra.Cholesky&#123;Float64,Array&#123;Float64,2&#125;&#125;&#125;&#125;,Array&#123;Float64,2&#125;&#125;
Frequency ~ 1 &#43; Temperature
Coefficients:
─────────────────────────────────────────────────────────────────────────────
Estimate Std. Error z value Pr&#40;&gt;|z|&#41; Lower 95&#37; Upper 95&#37;
─────────────────────────────────────────────────────────────────────────────
&#40;Intercept&#41; 5.08498 3.05248 1.66585 0.0957 -0.897782 11.0677
Temperature -0.115601 0.0470238 -2.45835 0.0140 -0.207766 -0.0234362
─────────────────────────────────────────────────────────────────────────────
</pre>
<p>Good, now I have recovered the asymptotic standard errors <span class="math">$s_{\hat{\alpha}} = 3.052$</span> and <span class="math">$s_{\hat{\beta}} = 0.047$</span>,</p>
<p>The Goodness of fit &#40;Deviance&#41; indicated for this model is <span class="math">$G^2 = 18.086$</span> with 137 degrees of freedom. Now <span class="math">$G^2$</span> is in good accordance to the results of the Dalal <em>et al.</em> article, but the number of degrees of freedom is 6 times larger than i should, due to my tampering of the data to duplicate them instead of weighting them.</p>
<h1>Predicting failure probability</h1>
<p>The temperature when launching the shuttle was 31°F. Let&#39;s try to estimate the failure probability for such temperature using our model:</p>
<pre class='hljl'>
<span class='hljl-n'>prediction</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>DataFrame</span><span class='hljl-p'>(</span><span class='hljl-n'>Temperature</span><span class='hljl-oB'>=</span><span class='hljl-ni'>30</span><span class='hljl-oB'>:</span><span class='hljl-nfB'>0.25</span><span class='hljl-oB'>:</span><span class='hljl-ni'>90</span><span class='hljl-p'>)</span><span class='hljl-t'>
</span><span class='hljl-n'>prediction</span><span class='hljl-oB'>.</span><span class='hljl-n'>Frequency</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>predict</span><span class='hljl-p'>(</span><span class='hljl-n'>model</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>prediction</span><span class='hljl-p'>)</span><span class='hljl-t'>
</span><span class='hljl-nf'>plot</span><span class='hljl-p'>(</span><span class='hljl-n'>xlabel</span><span class='hljl-oB'>=</span><span class='hljl-s'>&quot;Temperature [F]&quot;</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>ylabel</span><span class='hljl-oB'>=</span><span class='hljl-s'>&quot;Frequency&quot;</span><span class='hljl-p'>)</span><span class='hljl-t'>
</span><span class='hljl-nf'>plot!</span><span class='hljl-p'>(</span><span class='hljl-n'>data</span><span class='hljl-oB'>.</span><span class='hljl-n'>Temperature</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>data</span><span class='hljl-oB'>.</span><span class='hljl-n'>Frequency</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>seriestype</span><span class='hljl-oB'>=:</span><span class='hljl-n'>scatter</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>label</span><span class='hljl-oB'>=</span><span class='hljl-s'>&quot;data&quot;</span><span class='hljl-p'>)</span><span class='hljl-t'>
</span><span class='hljl-nf'>plot!</span><span class='hljl-p'>(</span><span class='hljl-n'>prediction</span><span class='hljl-oB'>.</span><span class='hljl-n'>Temperature</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>prediction</span><span class='hljl-oB'>.</span><span class='hljl-n'>Frequency</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>label</span><span class='hljl-oB'>=</span><span class='hljl-s'>&quot;prediction&quot;</span><span class='hljl-p'>)</span><span class='hljl-t'>
</span><span class='hljl-nf'>disp</span><span class='hljl-p'>()</span>
</pre>
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<p>This figure is very similar to the Figure 4 of Dalal <em>et al</em>.</p>
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Published from <a href="challenger.jmd">challenger.jmd</a> using
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---
title : "Risk Analysis of the Space Shuttle: Pre-Challenger Prediction of Failure"
options:
css: skeleton_css.css
template: julia_html.tpl
---
In this document we reperform some of the analysis provided in *Risk Analysis of
the Space Shuttle: Pre-Challenger Prediction of Failure* by Siddhartha R. Dalal,
Edward B. Fowlkes, Bruce Hoadley published in Journal of the American
Statistical Association, Vol. 84, No. 408 (Dec., 1989), pp. 945-957 and
available at [http://www.jstor.org/stable/2290069](http://www.jstor.org/stable/2290069).
On the fourth page of this article, they indicate that the maximum likelihood
estimates of the logistic regression using only temperature are:
$\hat{\alpha}=5.085$ and $\hat{\beta}=-0.1156$ and their asymptotic standard
errors are $s_{\hat{\alpha}}=3.052$ and $s_{\hat{\beta}}=0.047$. The Goodness of
fit indicated for this model was $G^2=18.086$ with 21 degrees of freedom. Our
goal is to reproduce the computation behind these values and the Figure 4 of
this article, possibly in a nicer looking way.
# Technical information on the computer on which the analysis is run
We will be using the [Julia](http://julialang.org) language:
```julia
using InteractiveUtils
versioninfo()
```
The computations rely on a number of packages in the Julia ecosystem. The direct
dependencies are summarized hereafter; the complete environment is described in
the [`Manifest.toml`](Manifest.toml) file.
```julia
# Setup environment
using Pkg
Pkg.activate(@__DIR__)
Pkg.instantiate()
# Load dependencies
using HTTP, CSV
using Plots; plotly()
using GLM
using DataFrames
using Printf
include("utils.jl")
# Summary
Pkg.status()
```
# Loading and inspecting data
Let's start by reading data.
```julia
res = HTTP.request(:GET, "https://app-learninglab.inria.fr/moocrr/gitlab/moocrr-session3/moocrr-reproducibility-study/raw/master/data/shuttle.csv?inline=false")
data = CSV.read(res.body)
```
We know from our previous experience on this data set that filtering data is a
really bad idea. We will therefore process it as such.
```julia; results="raw"
data.Frequency = data.Malfunction ./ data.Count
plot(xlabel="Temperature [F]", ylabel="Frequency")
plot!(data.Temperature, data.Frequency, seriestype=:scatter, label=nothing)
disp()
```
# Logistic regression
Let's assume O-rings independently fail with the same probability which solely
depends on temperature. A logistic regression should allow us to estimate the
influence of temperature.
```julia; wrap=false; hold=true
model = glm(@formula(Frequency ~ Temperature), data,
Binomial(), LogitLink())
α, β = coef(model)
σα, σβ = stderror(model)
G² = deviance(model)
nDOF = Int(dof_residual(model))
model
```
The maximum likelyhood estimator of the intercept and of Temperature are thus
$\hat{\alpha}=`j @printf "%.3f" α`$ and
$\hat{\beta}=`j @printf "%.3f" β`$.
This corresponds to the values from the article of Dalal et al. The standard
errors are
$s_{\hat{\alpha}} = `j @printf "%.3f" σα`$ and
$s_{\hat{\beta}} = `j @printf "%.3f" σβ`$,
which is different from the $3.052$ and $0.047$ reported by Dallal et al. The
deviance is
$G^2 = `j @printf "%.3f" G²`$ with `j nDOF` degrees of freedom.
I cannot find any value similar to the Goodness of fit ($G^2=18.086$) reported
by Dalal et al. However, the number of degrees of freedom is similar to theirs
(21).
There seems to be something wrong. Oh I know, I haven't indicated that my
observations are actually the result of 6 observations for each rocket
launch. The correct way to do this would be to weight the data using the `Count`
column. Since I don't know how to do that with the
[GLM](https://github.com/JuliaStats/GLM.jl) package I'm using, I will simply
duplicate the data:
```julia; wrap=false; hold=true
weighted_data = DataFrame(Temperature=Int[], Frequency=Float64[])
for row in eachrow(data)
for _ in 1:row.Count
push!(weighted_data, (Temperature=row.Temperature,
Frequency=row.Frequency))
end
end
model = glm(@formula(Frequency ~ Temperature), weighted_data,
Binomial(), LogitLink())
α, β = coef(model)
σα, σβ = stderror(model)
G² = deviance(model)
nDOF = Int(dof_residual(model))
model
```
Good, now I have recovered the asymptotic standard errors
$s_{\hat{\alpha}} = `j @printf "%.3f" σα`$ and
$s_{\hat{\beta}} = `j @printf "%.3f" σβ`$,
The Goodness of fit (Deviance) indicated for this model is
$G^2 = `j @printf "%.3f" G²`$ with `j nDOF` degrees of freedom. Now $G^2$ is in
good accordance to the results of the Dalal *et al.* article, but the number of
degrees of freedom is 6 times larger than i should, due to my tampering of the
data to duplicate them instead of weighting them.
# Predicting failure probability
The temperature when launching the shuttle was 31°F. Let's try to estimate the
failure probability for such temperature using our model:
```julia; results="raw"
prediction = DataFrame(Temperature=30:0.25:90)
prediction.Frequency = predict(model, prediction)
plot(xlabel="Temperature [F]", ylabel="Frequency")
plot!(data.Temperature, data.Frequency, seriestype=:scatter, label="data")
plot!(prediction.Temperature, prediction.Frequency, label="prediction")
disp()
```
This figure is very similar to the Figure 4 of Dalal *et al*.
<!-- Local Variables: -->
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.eight.columns { width: 65.3333333333%; }
.nine.columns { width: 74.0%; }
.ten.columns { width: 82.6666666667%; }
.eleven.columns { width: 91.3333333333%; }
.twelve.columns { width: 100%; margin-left: 0; }
.one-third.column { width: 30.6666666667%; }
.two-thirds.column { width: 65.3333333333%; }
.one-half.column { width: 48%; }
/* Offsets */
.offset-by-one.column,
.offset-by-one.columns { margin-left: 8.66666666667%; }
.offset-by-two.column,
.offset-by-two.columns { margin-left: 17.3333333333%; }
.offset-by-three.column,
.offset-by-three.columns { margin-left: 26%; }
.offset-by-four.column,
.offset-by-four.columns { margin-left: 34.6666666667%; }
.offset-by-five.column,
.offset-by-five.columns { margin-left: 43.3333333333%; }
.offset-by-six.column,
.offset-by-six.columns { margin-left: 52%; }
.offset-by-seven.column,
.offset-by-seven.columns { margin-left: 60.6666666667%; }
.offset-by-eight.column,
.offset-by-eight.columns { margin-left: 69.3333333333%; }
.offset-by-nine.column,
.offset-by-nine.columns { margin-left: 78.0%; }
.offset-by-ten.column,
.offset-by-ten.columns { margin-left: 86.6666666667%; }
.offset-by-eleven.column,
.offset-by-eleven.columns { margin-left: 95.3333333333%; }
.offset-by-one-third.column,
.offset-by-one-third.columns { margin-left: 34.6666666667%; }
.offset-by-two-thirds.column,
.offset-by-two-thirds.columns { margin-left: 69.3333333333%; }
.offset-by-one-half.column,
.offset-by-one-half.columns { margin-left: 52%; }
}
html {
font-size: 62.5%; }
body {
font-size: 1.5em; /* currently ems cause chrome bug misinterpreting rems on body element */
line-height: 1.6;
font-weight: 400;
font-family: "Raleway", "HelveticaNeue", "Helvetica Neue", Helvetica, Arial, sans-serif;
color: #222; }
h1, h2, h3, h4, h5, h6 {
margin-top: 2rem;
margin-bottom: 0;
font-weight: 300; }
h1 { font-size: 3.6rem; line-height: 1.2; letter-spacing: -.1rem;}
h2 { font-size: 3.0rem; line-height: 1.25; letter-spacing: -.1rem; }
h3 { font-size: 2.8rem; line-height: 1.3; letter-spacing: -.1rem; }
h4 { font-size: 2.6rem; line-height: 1.35; letter-spacing: -.08rem; }
h5 { font-size: 2.4rem; line-height: 1.5; letter-spacing: -.05rem; }
h6 { font-size: 1.5rem; line-height: 1.6; letter-spacing: 0; }
p {
margin-top: 0; }
a {
color: #1EAEDB; }
a:hover {
color: #0FA0CE; }
.button,
button,
input[type="submit"],
input[type="reset"],
input[type="button"] {
display: inline-block;
height: 38px;
padding: 0 30px;
color: #555;
text-align: center;
font-size: 11px;
font-weight: 600;
line-height: 38px;
letter-spacing: .1rem;
text-transform: uppercase;
text-decoration: none;
white-space: nowrap;
background-color: transparent;
border-radius: 4px;
border: 1px solid #bbb;
cursor: pointer;
box-sizing: border-box; }
.button:hover,
button:hover,
input[type="submit"]:hover,
input[type="reset"]:hover,
input[type="button"]:hover,
.button:focus,
button:focus,
input[type="submit"]:focus,
input[type="reset"]:focus,
input[type="button"]:focus {
color: #333;
border-color: #888;
outline: 0; }
.button.button-primary,
button.button-primary,
input[type="submit"].button-primary,
input[type="reset"].button-primary,
input[type="button"].button-primary {
color: #FFF;
background-color: #33C3F0;
border-color: #33C3F0; }
.button.button-primary:hover,
button.button-primary:hover,
input[type="submit"].button-primary:hover,
input[type="reset"].button-primary:hover,
input[type="button"].button-primary:hover,
.button.button-primary:focus,
button.button-primary:focus,
input[type="submit"].button-primary:focus,
input[type="reset"].button-primary:focus,
input[type="button"].button-primary:focus {
color: #FFF;
background-color: #1EAEDB;
border-color: #1EAEDB; }
input[type="email"],
input[type="number"],
input[type="search"],
input[type="text"],
input[type="tel"],
input[type="url"],
input[type="password"],
textarea,
select {
height: 38px;
padding: 6px 10px; /* The 6px vertically centers text on FF, ignored by Webkit */
background-color: #fff;
border: 1px solid #D1D1D1;
border-radius: 4px;
box-shadow: none;
box-sizing: border-box; }
/* Removes awkward default styles on some inputs for iOS */
input[type="email"],
input[type="number"],
input[type="search"],
input[type="text"],
input[type="tel"],
input[type="url"],
input[type="password"],
textarea {
-webkit-appearance: none;
-moz-appearance: none;
appearance: none; }
textarea {
min-height: 65px;
padding-top: 6px;
padding-bottom: 6px; }
input[type="email"]:focus,
input[type="number"]:focus,
input[type="search"]:focus,
input[type="text"]:focus,
input[type="tel"]:focus,
input[type="url"]:focus,
input[type="password"]:focus,
textarea:focus,
select:focus {
border: 1px solid #33C3F0;
outline: 0; }
label,
legend {
display: block;
margin-bottom: .5rem;
font-weight: 600; }
fieldset {
padding: 0;
border-width: 0; }
input[type="checkbox"],
input[type="radio"] {
display: inline; }
label > .label-body {
display: inline-block;
margin-left: .5rem;
font-weight: normal; }
ul {
list-style: circle; }
ol {
list-style: decimal; }
ul ul,
ul ol,
ol ol,
ol ul {
margin: 1.5rem 0 1.5rem 3rem;
font-size: 90%; }
li > p {margin : 0;}
th,
td {
padding: 5px 15px;
text-align: left;
border-bottom: 1px solid #E1E1E1; }
th:first-child,
td:first-child {
padding-left: 0; }
th:last-child,
td:last-child {
padding-right: 0; }
button,
.button {
margin-bottom: 1rem; }
input,
textarea,
select,
fieldset {
margin-bottom: 1.5rem; }
pre,
blockquote,
dl,
figure,
table,
p,
ul,
ol,
form {
margin-bottom: 1.0rem; }
.u-full-width {
width: 100%;
box-sizing: border-box; }
.u-max-full-width {
max-width: 100%;
box-sizing: border-box; }
.u-pull-right {
float: right; }
.u-pull-left {
float: left; }
hr {
margin-top: 3rem;
margin-bottom: 3.5rem;
border-width: 0;
border-top: 1px solid #E1E1E1; }
.container:after,
.row:after,
.u-cf {
content: "";
display: table;
clear: both; }
pre {
display: block;
padding: 9.5px;
margin: 0 0 10px;
font-size: 13px;
line-height: 1.42857143;
word-break: break-all;
word-wrap: break-word;
border: 1px solid #ccc;
border-radius: 4px;
}
pre.hljl {
margin: 0 0 10px;
display: block;
background: #f5f5f5;
border-radius: 4px;
padding : 5px;
}
pre.output {
background: #ffffff;
}
pre.code {
background: #ffffff;
}
pre.julia-error {
color : red
}
code,
kbd,
pre,
samp {
font-family: Menlo, Monaco, Consolas, "Courier New", monospace;
font-size: 0.9em;
}
@media (min-width: 400px) {}
@media (min-width: 550px) {}
@media (min-width: 750px) {}
@media (min-width: 1000px) {}
@media (min-width: 1200px) {}
h1.title {margin-top : 20px}
img {max-width : 100%}
div.title {text-align: center;}
body {counter-reset: h1counter;}
h1 {counter-reset: h2counter;}
h2 {counter-reset: h3counter;}
h1.title:before {
content: none;
counter-increment: none;
}
h1:before {
content: counter(h1counter) "\0000a0-\0000a0";
counter-increment: h1counter;
}
h2:before {
content: counter(h1counter) "." counter(h2counter) "\0000a0-\0000a0";
counter-increment: h2counter;
}
h3:before {
content: counter(h1counter) "." counter(h2counter) "." counter(h3counter) "\0000a0-\0000a0";
counter-increment: h3counter;
}
module4/utils.jl 0 → 100644
View file @ b1342638
# Affichage des DataFrames dans la sortie HTML
using DataFrames
using Printf
function info(df::DataFrame)
try
WEAVE_ARGS
catch
display(df)
return
end
pretty(x) = x
pretty(x::Float64) = @sprintf "%.6f" x
pretty(::Type{Union{T,Missing}}) where {T} = "$T?"
function printrow(i)
print("<tr>")
print("<th>$i</th>")
for j in 1:ncol(df)
print("<td>$(pretty(df[i,j]))</td>")
end
print("</tr>")
end
function printrows()
i = 1
while i<=nrow(df) && i<=3
printrow(i)
i += 1
end
i>nrow(df) && return
if i < nrow(df)-2
print("<tr><td>...</td></tr>")
end
i = max(i, nrow(df)-2)
while i<=nrow(df)
printrow(i)
i += 1
end
end
println("$(nrow(df)) rows × $(ncol(df)) columns")
print("<table>")
print("<thead>")
print("<tr><th></th>")
for col in names(df)
print("<th>$col</th>")
end
print("</tr>")
print("<tr><th></th>")
for col in names(df)
print("<th>$(pretty(eltype(df[!,col])))</th>")
end
print("</tr>")
print("</thead>")
print("<tbody>")
printrows()
print("</tbody>")
print("</table>\n")
end
# Meilleure intégration entre Plotly et Weave
function disp()
try
WEAVE_ARGS
catch
display(plot!())
return
end
buf = IOBuffer()
show(buf, MIME("text/html"), plot!())
seekstart(buf)
str = String(read(buf))
m = match(r"<body>(.*)</body>"s, str)
return if m === nothing
str
else
m.captures[1]
end |> print
end
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