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<!-- navigation toc: --> <li><a href="._week36-bs001.html#___sec0" style="font-size: 80%;">Plans for week 36</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs002.html#___sec1" style="font-size: 80%;">Thursday September 3</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs003.html#___sec2" style="font-size: 80%;">Why resampling methods</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs004.html#___sec3" style="font-size: 80%;">Resampling methods</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs007.html#___sec6" style="font-size: 80%;">Statistical analysis</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs009.html#___sec8" style="font-size: 80%;">Assumptions made</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs012.html#___sec11" style="font-size: 80%;">Resampling methods</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs022.html#___sec21" style="font-size: 80%;">Various steps in cross-validation</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs023.html#___sec22" style="font-size: 80%;">How to set up the cross-validation for Ridge and/or Lasso</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs024.html#___sec23" style="font-size: 80%;">Cross-validation in brief</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs025.html#___sec24" style="font-size: 80%;">Code Example for Cross-validation and \( k \)-fold Cross-validation</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs026.html#___sec25" style="font-size: 80%;">The bias-variance tradeoff</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs029.html#___sec28" style="font-size: 80%;">Summing up</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs030.html#___sec29" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
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<!-- navigation toc: --> <li><a href="._week36-bs032.html#___sec31" style="font-size: 80%;">The same example but now with cross-validation</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs033.html#___sec32" style="font-size: 80%;">Cross-validation with Ridge</a></li>
<!-- navigation toc: --> <li><a href="._week36-bs034.html#___sec33" style="font-size: 80%;">Friday September 4</a></li>
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<h2 id="___sec20" class="anchor">Code example for the Bootstrap method </h2>
<p>
The following code starts with a Gaussian distribution with mean value
\( \mu =100 \) and variance \( \sigma=15 \). We use this to generate the data
used in the bootstrap analysis. The bootstrap analysis returns a data
set after a given number of bootstrap operations (as many as we have
data points). This data set consists of estimated mean values for each
bootstrap operation. The histogram generated by the bootstrap method
shows that the distribution for these mean values is also a Gaussian,
centered around the mean value \( \mu=100 \) but with standard deviation
\( \sigma/\sqrt{n} \), where \( n \) is the number of bootstrap samples (in
this case the same as the number of original data points). The value
of the standard deviation is what we expect from the central limit
theorem.
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">import</span> <span style="color: #666666">*</span>
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">numpy.random</span> <span style="color: #008000; font-weight: bold">import</span> randint, randn
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">time</span> <span style="color: #008000; font-weight: bold">import</span> time
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.mlab</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">mlab</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
<span style="color: #408080; font-style: italic"># Returns mean of bootstrap samples </span>
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">stat</span>(data):
<span style="color: #008000; font-weight: bold">return</span> mean(data)
<span style="color: #408080; font-style: italic"># Bootstrap algorithm</span>
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">bootstrap</span>(data, statistic, R):
t <span style="color: #666666">=</span> zeros(R); n <span style="color: #666666">=</span> <span style="color: #008000">len</span>(data); inds <span style="color: #666666">=</span> arange(n); t0 <span style="color: #666666">=</span> time()
<span style="color: #408080; font-style: italic"># non-parametric bootstrap </span>
<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>(R):
t[i] <span style="color: #666666">=</span> statistic(data[randint(<span style="color: #666666">0</span>,n,n)])
<span style="color: #408080; font-style: italic"># analysis </span>
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&quot;Runtime: </span><span style="color: #BB6688; font-weight: bold">%g</span><span style="color: #BA2121"> sec&quot;</span> <span style="color: #666666">%</span> (time()<span style="color: #666666">-</span>t0)); <span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&quot;Bootstrap Statistics :&quot;</span>)
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&quot;original bias std. error&quot;</span>)
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&quot;</span><span style="color: #BB6688; font-weight: bold">%8g</span><span style="color: #BA2121"> </span><span style="color: #BB6688; font-weight: bold">%8g</span><span style="color: #BA2121"> </span><span style="color: #BB6688; font-weight: bold">%14g</span><span style="color: #BA2121"> </span><span style="color: #BB6688; font-weight: bold">%15g</span><span style="color: #BA2121">&quot;</span> <span style="color: #666666">%</span> (statistic(data), std(data),mean(t),std(t)))
<span style="color: #008000; font-weight: bold">return</span> t
mu, sigma <span style="color: #666666">=</span> <span style="color: #666666">100</span>, <span style="color: #666666">15</span>
datapoints <span style="color: #666666">=</span> <span style="color: #666666">10000</span>
x <span style="color: #666666">=</span> mu <span style="color: #666666">+</span> sigma<span style="color: #666666">*</span>random<span style="color: #666666">.</span>randn(datapoints)
<span style="color: #408080; font-style: italic"># bootstrap returns the data sample </span>
t <span style="color: #666666">=</span> bootstrap(x, stat, datapoints)
<span style="color: #408080; font-style: italic"># the histogram of the bootstrapped data </span>
n, binsboot, patches <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>hist(t, <span style="color: #666666">50</span>, normed<span style="color: #666666">=1</span>, facecolor<span style="color: #666666">=</span><span style="color: #BA2121">&#39;red&#39;</span>, alpha<span style="color: #666666">=0.75</span>)
<span style="color: #408080; font-style: italic"># add a &#39;best fit&#39; line </span>
y <span style="color: #666666">=</span> mlab<span style="color: #666666">.</span>normpdf( binsboot, mean(t), std(t))
lt <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>plot(binsboot, y, <span style="color: #BA2121">&#39;r--&#39;</span>, linewidth<span style="color: #666666">=1</span>)
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">&#39;Smarts&#39;</span>)
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">&#39;Probability&#39;</span>)
plt<span style="color: #666666">.</span>axis([<span style="color: #666666">99.5</span>, <span style="color: #666666">100.6</span>, <span style="color: #666666">0</span>, <span style="color: #666666">3.0</span>])
plt<span style="color: #666666">.</span>grid(<span style="color: #008000">True</span>)
plt<span style="color: #666666">.</span>show()
</pre></div>
<p>
<p>
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