mhjensen
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HTML
248 lines
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'sections': [('Logistic Regression', 2, None, '___sec0'),
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('A simple classification problem', 2, None, '___sec15'),
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<!-- navigation toc: --> <li><a href="._LogReg-bs001.html#___sec0" style="font-size: 80%;">Logistic Regression</a></li>
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<!-- navigation toc: --> <li><a href="._LogReg-bs006.html#___sec5" style="font-size: 80%;">The logistic function</a></li>
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<!-- navigation toc: --> <li><a href="#___sec6" style="font-size: 80%;">Examples of likelihood functions used in logistic regression and nueral networks</a></li>
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<h2 id="___sec6" class="anchor">Examples of likelihood functions used in logistic regression and nueral networks </h2>
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<p>
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The following code plots the logistic function, the step function and other functions we will encounter from here and on.
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<p>
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<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #BA2121; font-style: italic">"""The sigmoid function (or the logistic curve) is a</span>
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<span style="color: #BA2121; font-style: italic">function that takes any real number, z, and outputs a number (0,1).</span>
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<span style="color: #BA2121; font-style: italic">It is useful in neural networks for assigning weights on a relative scale.</span>
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<span style="color: #BA2121; font-style: italic">The value z is the weighted sum of parameters involved in the learning algorithm."""</span>
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span>
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<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>
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">math</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">mt</span>
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z <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>arange(<span style="color: #666666">-5</span>, <span style="color: #666666">5</span>, <span style="color: #666666">.1</span>)
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sigma_fn <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>vectorize(<span style="color: #008000; font-weight: bold">lambda</span> z: <span style="color: #666666">1/</span>(<span style="color: #666666">1+</span>numpy<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>z)))
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sigma <span style="color: #666666">=</span> sigma_fn(z)
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fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure()
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ax <span style="color: #666666">=</span> fig<span style="color: #666666">.</span>add_subplot(<span style="color: #666666">111</span>)
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ax<span style="color: #666666">.</span>plot(z, sigma)
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ax<span style="color: #666666">.</span>set_ylim([<span style="color: #666666">-0.1</span>, <span style="color: #666666">1.1</span>])
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ax<span style="color: #666666">.</span>set_xlim([<span style="color: #666666">-5</span>,<span style="color: #666666">5</span>])
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ax<span style="color: #666666">.</span>grid(<span style="color: #008000">True</span>)
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ax<span style="color: #666666">.</span>set_xlabel(<span style="color: #BA2121">'z'</span>)
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ax<span style="color: #666666">.</span>set_title(<span style="color: #BA2121">'sigmoid function'</span>)
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plt<span style="color: #666666">.</span>show()
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<span style="color: #BA2121; font-style: italic">"""Step Function"""</span>
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z <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>arange(<span style="color: #666666">-5</span>, <span style="color: #666666">5</span>, <span style="color: #666666">.02</span>)
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step_fn <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>vectorize(<span style="color: #008000; font-weight: bold">lambda</span> z: <span style="color: #666666">1.0</span> <span style="color: #008000; font-weight: bold">if</span> z <span style="color: #666666">>=</span> <span style="color: #666666">0.0</span> <span style="color: #008000; font-weight: bold">else</span> <span style="color: #666666">0.0</span>)
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step <span style="color: #666666">=</span> step_fn(z)
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fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure()
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ax <span style="color: #666666">=</span> fig<span style="color: #666666">.</span>add_subplot(<span style="color: #666666">111</span>)
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ax<span style="color: #666666">.</span>plot(z, step)
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ax<span style="color: #666666">.</span>set_ylim([<span style="color: #666666">-0.5</span>, <span style="color: #666666">1.5</span>])
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ax<span style="color: #666666">.</span>set_xlim([<span style="color: #666666">-5</span>,<span style="color: #666666">5</span>])
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ax<span style="color: #666666">.</span>grid(<span style="color: #008000">True</span>)
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ax<span style="color: #666666">.</span>set_xlabel(<span style="color: #BA2121">'z'</span>)
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ax<span style="color: #666666">.</span>set_title(<span style="color: #BA2121">'step function'</span>)
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plt<span style="color: #666666">.</span>show()
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<span style="color: #BA2121; font-style: italic">"""tanh Function"""</span>
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z <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>arange(<span style="color: #666666">-2*</span>mt<span style="color: #666666">.</span>pi, <span style="color: #666666">2*</span>mt<span style="color: #666666">.</span>pi, <span style="color: #666666">0.1</span>)
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t <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>tanh(z)
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fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure()
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ax <span style="color: #666666">=</span> fig<span style="color: #666666">.</span>add_subplot(<span style="color: #666666">111</span>)
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ax<span style="color: #666666">.</span>plot(z, t)
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ax<span style="color: #666666">.</span>set_ylim([<span style="color: #666666">-1.0</span>, <span style="color: #666666">1.0</span>])
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ax<span style="color: #666666">.</span>set_xlim([<span style="color: #666666">-2*</span>mt<span style="color: #666666">.</span>pi,<span style="color: #666666">2*</span>mt<span style="color: #666666">.</span>pi])
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ax<span style="color: #666666">.</span>grid(<span style="color: #008000">True</span>)
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ax<span style="color: #666666">.</span>set_xlabel(<span style="color: #BA2121">'z'</span>)
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ax<span style="color: #666666">.</span>set_title(<span style="color: #BA2121">'tanh function'</span>)
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plt<span style="color: #666666">.</span>show()
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</pre></div>
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