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mhjensen
2019-09-18 22:11:35 +02:00
parent d0adb4b485
commit 26b3a6a4a6
176 changed files with 2460 additions and 4282 deletions
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doc/pub/LogReg/html/._LogReg-bs000.html
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@@ -55,7 +55,8 @@ Automatically generated HTML file from DocOnce source
('Extending to more predictors', 2, None, '___sec11'),
('Including more classes', 2, None, '___sec12'),
('The Softmax function', 2, None, '___sec13'),
('A simple classification problem', 2, None, '___sec14')]}
('A simple classification problem', 2, None, '___sec14'),
('The Credit Card example', 2, None, '___sec15')]}
end of tocinfo -->
<body>
@@ -108,6 +109,7 @@ MathJax.Hub.Config({
<!-- navigation toc: --> <li><a href="._LogReg-bs013.html#___sec12" style="font-size: 80%;">Including more classes</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs014.html#___sec13" style="font-size: 80%;">The Softmax function</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs015.html#___sec14" style="font-size: 80%;">A simple classification problem</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs016.html#___sec15" style="font-size: 80%;">The Credit Card example</a></li>
</ul>
</li>
@@ -142,7 +144,7 @@ MathJax.Hub.Config({
<center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
<br>
<p>
<center><h4>Sep 16, 2019</h4></center> <!-- date -->
<center><h4>Sep 18, 2019</h4></center> <!-- date -->
<br>
<p>
<p>
@@ -159,7 +161,7 @@ MathJax.Hub.Config({
<li><a href="._LogReg-bs008.html">9</a></li>
<li><a href="._LogReg-bs009.html">10</a></li>
<li><a href="">...</a></li>
<li><a href="._LogReg-bs015.html">16</a></li>
<li><a href="._LogReg-bs016.html">17</a></li>
<li><a href="._LogReg-bs001.html">&raquo;</a></li>
</ul>
<!-- ------------------- end of main content --------------- -->
doc/pub/LogReg/html/._LogReg-bs001.html
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@@ -55,7 +55,8 @@ Automatically generated HTML file from DocOnce source
('Extending to more predictors', 2, None, '___sec11'),
('Including more classes', 2, None, '___sec12'),
('The Softmax function', 2, None, '___sec13'),
('A simple classification problem', 2, None, '___sec14')]}
('A simple classification problem', 2, None, '___sec14'),
('The Credit Card example', 2, None, '___sec15')]}
end of tocinfo -->
<body>
@@ -108,6 +109,7 @@ MathJax.Hub.Config({
<!-- navigation toc: --> <li><a href="._LogReg-bs013.html#___sec12" style="font-size: 80%;">Including more classes</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs014.html#___sec13" style="font-size: 80%;">The Softmax function</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs015.html#___sec14" style="font-size: 80%;">A simple classification problem</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs016.html#___sec15" style="font-size: 80%;">The Credit Card example</a></li>
</ul>
</li>
@@ -176,7 +178,7 @@ failure etc.
<li><a href="._LogReg-bs009.html">10</a></li>
<li><a href="._LogReg-bs010.html">11</a></li>
<li><a href="">...</a></li>
<li><a href="._LogReg-bs015.html">16</a></li>
<li><a href="._LogReg-bs016.html">17</a></li>
<li><a href="._LogReg-bs002.html">&raquo;</a></li>
</ul>
<!-- ------------------- end of main content --------------- -->
doc/pub/LogReg/html/._LogReg-bs002.html
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@@ -55,7 +55,8 @@ Automatically generated HTML file from DocOnce source
('Extending to more predictors', 2, None, '___sec11'),
('Including more classes', 2, None, '___sec12'),
('The Softmax function', 2, None, '___sec13'),
('A simple classification problem', 2, None, '___sec14')]}
('A simple classification problem', 2, None, '___sec14'),
('The Credit Card example', 2, None, '___sec15')]}
end of tocinfo -->
<body>
@@ -108,6 +109,7 @@ MathJax.Hub.Config({
<!-- navigation toc: --> <li><a href="._LogReg-bs013.html#___sec12" style="font-size: 80%;">Including more classes</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs014.html#___sec13" style="font-size: 80%;">The Softmax function</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs015.html#___sec14" style="font-size: 80%;">A simple classification problem</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs016.html#___sec15" style="font-size: 80%;">The Credit Card example</a></li>
</ul>
</li>
@@ -154,7 +156,7 @@ models, as we will see later.
<li><a href="._LogReg-bs010.html">11</a></li>
<li><a href="._LogReg-bs011.html">12</a></li>
<li><a href="">...</a></li>
<li><a href="._LogReg-bs015.html">16</a></li>
<li><a href="._LogReg-bs016.html">17</a></li>
<li><a href="._LogReg-bs003.html">&raquo;</a></li>
</ul>
<!-- ------------------- end of main content --------------- -->
doc/pub/LogReg/html/._LogReg-bs003.html
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@@ -55,7 +55,8 @@ Automatically generated HTML file from DocOnce source
('Extending to more predictors', 2, None, '___sec11'),
('Including more classes', 2, None, '___sec12'),
('The Softmax function', 2, None, '___sec13'),
('A simple classification problem', 2, None, '___sec14')]}
('A simple classification problem', 2, None, '___sec14'),
('The Credit Card example', 2, None, '___sec15')]}
end of tocinfo -->
<body>
@@ -108,6 +109,7 @@ MathJax.Hub.Config({
<!-- navigation toc: --> <li><a href="._LogReg-bs013.html#___sec12" style="font-size: 80%;">Including more classes</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs014.html#___sec13" style="font-size: 80%;">The Softmax function</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs015.html#___sec14" style="font-size: 80%;">A simple classification problem</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs016.html#___sec15" style="font-size: 80%;">The Credit Card example</a></li>
</ul>
</li>
@@ -138,7 +140,11 @@ primary goal is to identify the classes to which new unseen samples
belong.
<p>
Let us specialize to the case of two classes only, with outputs \( y_i=0 \) and \( y_i=1 \). Our outcomes could represent the status of a credit card user who could default or not on her/his credit card debt. That is
Let us specialize to the case of two classes only, with outputs
\( y_i=0 \) and \( y_i=1 \). Our outcomes could represent the status of a
credit card user that could default or not on her/his credit card
debt. That is
$$
y_i = \begin{bmatrix} 0 & \mathrm{no}\\ 1 & \mathrm{yes} \end{bmatrix}.
$$
@@ -162,7 +168,7 @@ $$
<li><a href="._LogReg-bs011.html">12</a></li>
<li><a href="._LogReg-bs012.html">13</a></li>
<li><a href="">...</a></li>
<li><a href="._LogReg-bs015.html">16</a></li>
<li><a href="._LogReg-bs016.html">17</a></li>
<li><a href="._LogReg-bs004.html">&raquo;</a></li>
</ul>
<!-- ------------------- end of main content --------------- -->
doc/pub/LogReg/html/._LogReg-bs004.html
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@@ -55,7 +55,8 @@ Automatically generated HTML file from DocOnce source
('Extending to more predictors', 2, None, '___sec11'),
('Including more classes', 2, None, '___sec12'),
('The Softmax function', 2, None, '___sec13'),
('A simple classification problem', 2, None, '___sec14')]}
('A simple classification problem', 2, None, '___sec14'),
('The Credit Card example', 2, None, '___sec15')]}
end of tocinfo -->
<body>
@@ -108,6 +109,7 @@ MathJax.Hub.Config({
<!-- navigation toc: --> <li><a href="._LogReg-bs013.html#___sec12" style="font-size: 80%;">Including more classes</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs014.html#___sec13" style="font-size: 80%;">The Softmax function</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs015.html#___sec14" style="font-size: 80%;">A simple classification problem</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs016.html#___sec15" style="font-size: 80%;">The Credit Card example</a></li>
</ul>
</li>
@@ -161,7 +163,7 @@ where \( \hat{y} \) is a vector representing the possible outcomes, \( \hat{X} \
<li><a href="._LogReg-bs012.html">13</a></li>
<li><a href="._LogReg-bs013.html">14</a></li>
<li><a href="">...</a></li>
<li><a href="._LogReg-bs015.html">16</a></li>
<li><a href="._LogReg-bs016.html">17</a></li>
<li><a href="._LogReg-bs005.html">&raquo;</a></li>
</ul>
<!-- ------------------- end of main content --------------- -->
doc/pub/LogReg/html/._LogReg-bs005.html
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@@ -55,7 +55,8 @@ Automatically generated HTML file from DocOnce source
('Extending to more predictors', 2, None, '___sec11'),
('Including more classes', 2, None, '___sec12'),
('The Softmax function', 2, None, '___sec13'),
('A simple classification problem', 2, None, '___sec14')]}
('A simple classification problem', 2, None, '___sec14'),
('The Credit Card example', 2, None, '___sec15')]}
end of tocinfo -->
<body>
@@ -108,6 +109,7 @@ MathJax.Hub.Config({
<!-- navigation toc: --> <li><a href="._LogReg-bs013.html#___sec12" style="font-size: 80%;">Including more classes</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs014.html#___sec13" style="font-size: 80%;">The Softmax function</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs015.html#___sec14" style="font-size: 80%;">A simple classification problem</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs016.html#___sec15" style="font-size: 80%;">The Credit Card example</a></li>
</ul>
</li>
@@ -129,10 +131,7 @@ MathJax.Hub.Config({
The main problem with our function is that it
takes values on the entire real axis. In the case of
logistic regression, however, the labels \( y_i \) are discrete
variables.
<p>
The table shows an example where we have a followed a group of students during a whole semester, with the aim to see if there are correlations between the number of hours they study and the number of hours they sleep. The output \( y_i \) is labelled as either \( 0 \) (a a grade below average) and \( 1 \), the latter corresponding to a grade above average.
variables. A typical example is the credit card data discussed below here, where we can set the state of defaulting the debt to \( y_i=1 \) and not to \( y_i=0 \) for one the persons in the data set (see the full example below).
<p>
One simple way to get a discrete output is to have sign
@@ -143,13 +142,6 @@ literature. This model is extremely simple. However, in many cases it is more
favorable to use a ``soft" classifier that outputs
the probability of a given category. This leads us to the logistic function.
<p>
The code for plotting the perceptron can be seen here. This si nothing but the standard <a href="https://en.wikipedia.org/wiki/Heaviside_step_function" target="_self">Heaviside step function</a>.
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>
</pre></div>
<p>
<p>
<!-- navigation buttons at the bottom of the page -->
@@ -171,7 +163,7 @@ The code for plotting the perceptron can be seen here. This si nothing but the s
<li><a href="._LogReg-bs013.html">14</a></li>
<li><a href="._LogReg-bs014.html">15</a></li>
<li><a href="">...</a></li>
<li><a href="._LogReg-bs015.html">16</a></li>
<li><a href="._LogReg-bs016.html">17</a></li>
<li><a href="._LogReg-bs006.html">&raquo;</a></li>
</ul>
<!-- ------------------- end of main content --------------- -->
doc/pub/LogReg/html/._LogReg-bs006.html
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@@ -55,7 +55,8 @@ Automatically generated HTML file from DocOnce source
('Extending to more predictors', 2, None, '___sec11'),
('Including more classes', 2, None, '___sec12'),
('The Softmax function', 2, None, '___sec13'),
('A simple classification problem', 2, None, '___sec14')]}
('A simple classification problem', 2, None, '___sec14'),
('The Credit Card example', 2, None, '___sec15')]}
end of tocinfo -->
<body>
@@ -108,6 +109,7 @@ MathJax.Hub.Config({
<!-- navigation toc: --> <li><a href="._LogReg-bs013.html#___sec12" style="font-size: 80%;">Including more classes</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs014.html#___sec13" style="font-size: 80%;">The Softmax function</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs015.html#___sec14" style="font-size: 80%;">A simple classification problem</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs016.html#___sec15" style="font-size: 80%;">The Credit Card example</a></li>
</ul>
</li>
@@ -141,11 +143,81 @@ p(t) = \frac{1}{1+\mathrm \exp{-t}}=\frac{\exp{t}}{1+\mathrm \exp{t}}.
$$
Note that \( 1-p(t)= p(-t) \).
The following code plots the logistic function.
The following code plots the logistic function, the step function and other functions we will encounter from here and on.
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #BA2121; font-style: italic">&quot;&quot;&quot;The sigmoid function (or the logistic curve) is a</span>
<span style="color: #BA2121; font-style: italic">function that takes any real number, z, and outputs a number (0,1).</span>
<span style="color: #BA2121; font-style: italic">It is useful in neural networks for assigning weights on a relative scale.</span>
<span style="color: #BA2121; font-style: italic">The value z is the weighted sum of parameters involved in the learning algorithm.&quot;&quot;&quot;</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</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: #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>
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>)
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)))
sigma <span style="color: #666666">=</span> sigma_fn(z)
fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure()
ax <span style="color: #666666">=</span> fig<span style="color: #666666">.</span>add_subplot(<span style="color: #666666">111</span>)
ax<span style="color: #666666">.</span>plot(z, sigma)
ax<span style="color: #666666">.</span>set_ylim([<span style="color: #666666">-0.1</span>, <span style="color: #666666">1.1</span>])
ax<span style="color: #666666">.</span>set_xlim([<span style="color: #666666">-5</span>,<span style="color: #666666">5</span>])
ax<span style="color: #666666">.</span>grid(<span style="color: #008000">True</span>)
ax<span style="color: #666666">.</span>set_xlabel(<span style="color: #BA2121">&#39;z&#39;</span>)
ax<span style="color: #666666">.</span>set_title(<span style="color: #BA2121">&#39;sigmoid function&#39;</span>)
plt<span style="color: #666666">.</span>show()
<span style="color: #BA2121; font-style: italic">&quot;&quot;&quot;Step Function&quot;&quot;&quot;</span>
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>)
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">&gt;=</span> <span style="color: #666666">0.0</span> <span style="color: #008000; font-weight: bold">else</span> <span style="color: #666666">0.0</span>)
step <span style="color: #666666">=</span> step_fn(z)
fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure()
ax <span style="color: #666666">=</span> fig<span style="color: #666666">.</span>add_subplot(<span style="color: #666666">111</span>)
ax<span style="color: #666666">.</span>plot(z, step)
ax<span style="color: #666666">.</span>set_ylim([<span style="color: #666666">-0.5</span>, <span style="color: #666666">1.5</span>])
ax<span style="color: #666666">.</span>set_xlim([<span style="color: #666666">-5</span>,<span style="color: #666666">5</span>])
ax<span style="color: #666666">.</span>grid(<span style="color: #008000">True</span>)
ax<span style="color: #666666">.</span>set_xlabel(<span style="color: #BA2121">&#39;z&#39;</span>)
ax<span style="color: #666666">.</span>set_title(<span style="color: #BA2121">&#39;step function&#39;</span>)
plt<span style="color: #666666">.</span>show()
<span style="color: #BA2121; font-style: italic">&quot;&quot;&quot;Sine Function&quot;&quot;&quot;</span>
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>)
t <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>sin(z)
fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure()
ax <span style="color: #666666">=</span> fig<span style="color: #666666">.</span>add_subplot(<span style="color: #666666">111</span>)
ax<span style="color: #666666">.</span>plot(z, t)
ax<span style="color: #666666">.</span>set_ylim([<span style="color: #666666">-1.0</span>, <span style="color: #666666">1.0</span>])
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])
ax<span style="color: #666666">.</span>grid(<span style="color: #008000">True</span>)
ax<span style="color: #666666">.</span>set_xlabel(<span style="color: #BA2121">&#39;z&#39;</span>)
ax<span style="color: #666666">.</span>set_title(<span style="color: #BA2121">&#39;sine function&#39;</span>)
plt<span style="color: #666666">.</span>show()
<span style="color: #BA2121; font-style: italic">&quot;&quot;&quot;Plots a graph of the squashing function used by a rectified linear</span>
<span style="color: #BA2121; font-style: italic">unit&quot;&quot;&quot;</span>
z <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>arange(<span style="color: #666666">-2</span>, <span style="color: #666666">2</span>, <span style="color: #666666">.1</span>)
zero <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>zeros(<span style="color: #008000">len</span>(z))
y <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>max([zero, z], axis<span style="color: #666666">=0</span>)
fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure()
ax <span style="color: #666666">=</span> fig<span style="color: #666666">.</span>add_subplot(<span style="color: #666666">111</span>)
ax<span style="color: #666666">.</span>plot(z, y)
ax<span style="color: #666666">.</span>set_ylim([<span style="color: #666666">-2.0</span>, <span style="color: #666666">2.0</span>])
ax<span style="color: #666666">.</span>set_xlim([<span style="color: #666666">-2.0</span>, <span style="color: #666666">2.0</span>])
ax<span style="color: #666666">.</span>grid(<span style="color: #008000">True</span>)
ax<span style="color: #666666">.</span>set_xlabel(<span style="color: #BA2121">&#39;z&#39;</span>)
ax<span style="color: #666666">.</span>set_title(<span style="color: #BA2121">&#39;Rectified linear unit&#39;</span>)
plt<span style="color: #666666">.</span>show()
</pre></div>
<p>
<p>
@@ -168,6 +240,8 @@ The following code plots the logistic function.
<li><a href="._LogReg-bs013.html">14</a></li>
<li><a href="._LogReg-bs014.html">15</a></li>
<li><a href="._LogReg-bs015.html">16</a></li>
<li><a href="">...</a></li>
<li><a href="._LogReg-bs016.html">17</a></li>
<li><a href="._LogReg-bs007.html">&raquo;</a></li>
</ul>
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doc/pub/LogReg/html/._LogReg-bs007.html
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@@ -55,7 +55,8 @@ Automatically generated HTML file from DocOnce source
('Extending to more predictors', 2, None, '___sec11'),
('Including more classes', 2, None, '___sec12'),
('The Softmax function', 2, None, '___sec13'),
('A simple classification problem', 2, None, '___sec14')]}
('A simple classification problem', 2, None, '___sec14'),
('The Credit Card example', 2, None, '___sec15')]}
end of tocinfo -->
<body>
@@ -108,6 +109,7 @@ MathJax.Hub.Config({
<!-- navigation toc: --> <li><a href="._LogReg-bs013.html#___sec12" style="font-size: 80%;">Including more classes</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs014.html#___sec13" style="font-size: 80%;">The Softmax function</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs015.html#___sec14" style="font-size: 80%;">A simple classification problem</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs016.html#___sec15" style="font-size: 80%;">The Credit Card example</a></li>
</ul>
</li>
@@ -163,6 +165,7 @@ $$
<li><a href="._LogReg-bs013.html">14</a></li>
<li><a href="._LogReg-bs014.html">15</a></li>
<li><a href="._LogReg-bs015.html">16</a></li>
<li><a href="._LogReg-bs016.html">17</a></li>
<li><a href="._LogReg-bs008.html">&raquo;</a></li>
</ul>
<!-- ------------------- end of main content --------------- -->
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('Extending to more predictors', 2, None, '___sec11'),
('Including more classes', 2, None, '___sec12'),
('The Softmax function', 2, None, '___sec13'),
('A simple classification problem', 2, None, '___sec14')]}
('A simple classification problem', 2, None, '___sec14'),
('The Credit Card example', 2, None, '___sec15')]}
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<body>
@@ -108,6 +109,7 @@ MathJax.Hub.Config({
<!-- navigation toc: --> <li><a href="._LogReg-bs013.html#___sec12" style="font-size: 80%;">Including more classes</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs014.html#___sec13" style="font-size: 80%;">The Softmax function</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs015.html#___sec14" style="font-size: 80%;">A simple classification problem</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs016.html#___sec15" style="font-size: 80%;">The Credit Card example</a></li>
</ul>
</li>
@@ -164,6 +166,7 @@ $$
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('Extending to more predictors', 2, None, '___sec11'),
('Including more classes', 2, None, '___sec12'),
('The Softmax function', 2, None, '___sec13'),
('A simple classification problem', 2, None, '___sec14')]}
('A simple classification problem', 2, None, '___sec14'),
('The Credit Card example', 2, None, '___sec15')]}
end of tocinfo -->
<body>
@@ -108,6 +109,7 @@ MathJax.Hub.Config({
<!-- navigation toc: --> <li><a href="._LogReg-bs013.html#___sec12" style="font-size: 80%;">Including more classes</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs014.html#___sec13" style="font-size: 80%;">The Softmax function</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs015.html#___sec14" style="font-size: 80%;">A simple classification problem</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs016.html#___sec15" style="font-size: 80%;">The Credit Card example</a></li>
</ul>
</li>
@@ -162,6 +164,7 @@ in practice we often supplement the cross-entropy with additional regularization
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('Extending to more predictors', 2, None, '___sec11'),
('Including more classes', 2, None, '___sec12'),
('The Softmax function', 2, None, '___sec13'),
('A simple classification problem', 2, None, '___sec14')]}
('A simple classification problem', 2, None, '___sec14'),
('The Credit Card example', 2, None, '___sec15')]}
end of tocinfo -->
<body>
@@ -108,6 +109,7 @@ MathJax.Hub.Config({
<!-- navigation toc: --> <li><a href="._LogReg-bs013.html#___sec12" style="font-size: 80%;">Including more classes</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs014.html#___sec13" style="font-size: 80%;">The Softmax function</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs015.html#___sec14" style="font-size: 80%;">A simple classification problem</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs016.html#___sec15" style="font-size: 80%;">The Credit Card example</a></li>
</ul>
</li>
@@ -163,6 +165,7 @@ $$
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<li><a href="._LogReg-bs011.html">&raquo;</a></li>
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('Extending to more predictors', 2, None, '___sec11'),
('Including more classes', 2, None, '___sec12'),
('The Softmax function', 2, None, '___sec13'),
('A simple classification problem', 2, None, '___sec14')]}
('A simple classification problem', 2, None, '___sec14'),
('The Credit Card example', 2, None, '___sec15')]}
end of tocinfo -->
<body>
@@ -108,6 +109,7 @@ MathJax.Hub.Config({
<!-- navigation toc: --> <li><a href="._LogReg-bs013.html#___sec12" style="font-size: 80%;">Including more classes</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs014.html#___sec13" style="font-size: 80%;">The Softmax function</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs015.html#___sec14" style="font-size: 80%;">A simple classification problem</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs016.html#___sec15" style="font-size: 80%;">The Credit Card example</a></li>
</ul>
</li>
@@ -163,6 +165,7 @@ $$
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('Extending to more predictors', 2, None, '___sec11'),
('Including more classes', 2, None, '___sec12'),
('The Softmax function', 2, None, '___sec13'),
('A simple classification problem', 2, None, '___sec14')]}
('A simple classification problem', 2, None, '___sec14'),
('The Credit Card example', 2, None, '___sec15')]}
end of tocinfo -->
<body>
@@ -108,6 +109,7 @@ MathJax.Hub.Config({
<!-- navigation toc: --> <li><a href="._LogReg-bs013.html#___sec12" style="font-size: 80%;">Including more classes</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs014.html#___sec13" style="font-size: 80%;">The Softmax function</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs015.html#___sec14" style="font-size: 80%;">A simple classification problem</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs016.html#___sec15" style="font-size: 80%;">The Credit Card example</a></li>
</ul>
</li>
@@ -155,6 +157,7 @@ $$
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('Extending to more predictors', 2, None, '___sec11'),
('Including more classes', 2, None, '___sec12'),
('The Softmax function', 2, None, '___sec13'),
('A simple classification problem', 2, None, '___sec14')]}
('A simple classification problem', 2, None, '___sec14'),
('The Credit Card example', 2, None, '___sec15')]}
end of tocinfo -->
<body>
@@ -108,6 +109,7 @@ MathJax.Hub.Config({
<!-- navigation toc: --> <li><a href="#___sec12" style="font-size: 80%;">Including more classes</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs014.html#___sec13" style="font-size: 80%;">The Softmax function</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs015.html#___sec14" style="font-size: 80%;">A simple classification problem</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs016.html#___sec15" style="font-size: 80%;">The Credit Card example</a></li>
</ul>
</li>
@@ -166,6 +168,7 @@ and the model is specified in term of \( K-1 \) so-called log-odds or
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('Extending to more predictors', 2, None, '___sec11'),
('Including more classes', 2, None, '___sec12'),
('The Softmax function', 2, None, '___sec13'),
('A simple classification problem', 2, None, '___sec14')]}
('A simple classification problem', 2, None, '___sec14'),
('The Credit Card example', 2, None, '___sec15')]}
end of tocinfo -->
<body>
@@ -108,6 +109,7 @@ MathJax.Hub.Config({
<!-- navigation toc: --> <li><a href="._LogReg-bs013.html#___sec12" style="font-size: 80%;">Including more classes</a></li>
<!-- navigation toc: --> <li><a href="#___sec13" style="font-size: 80%;">The Softmax function</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs015.html#___sec14" style="font-size: 80%;">A simple classification problem</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs016.html#___sec15" style="font-size: 80%;">The Credit Card example</a></li>
</ul>
</li>
@@ -177,6 +179,7 @@ methods</a>.
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<li><a href="._LogReg-bs015.html">&raquo;</a></li>
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('Extending to more predictors', 2, None, '___sec11'),
('Including more classes', 2, None, '___sec12'),
('The Softmax function', 2, None, '___sec13'),
('A simple classification problem', 2, None, '___sec14')]}
('A simple classification problem', 2, None, '___sec14'),
('The Credit Card example', 2, None, '___sec15')]}
end of tocinfo -->
<body>
@@ -108,6 +109,7 @@ MathJax.Hub.Config({
<!-- navigation toc: --> <li><a href="._LogReg-bs013.html#___sec12" style="font-size: 80%;">Including more classes</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs014.html#___sec13" style="font-size: 80%;">The Softmax function</a></li>
<!-- navigation toc: --> <li><a href="#___sec14" style="font-size: 80%;">A simple classification problem</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs016.html#___sec15" style="font-size: 80%;">The Credit Card example</a></li>
</ul>
</li>
@@ -178,7 +180,6 @@ MathJax.Hub.Config({
main()
</pre></div>
<p>
<p>
<!-- navigation buttons at the bottom of the page -->
<ul class="pagination">
@@ -194,6 +195,8 @@ MathJax.Hub.Config({
<li><a href="._LogReg-bs013.html">14</a></li>
<li><a href="._LogReg-bs014.html">15</a></li>
<li class="active"><a href="._LogReg-bs015.html">16</a></li>
<li><a href="._LogReg-bs016.html">17</a></li>
<li><a href="._LogReg-bs016.html">&raquo;</a></li>
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<head>
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<meta name="description" content="Data Analysis and Machine Learning: Logistic Regression">
<title>Data Analysis and Machine Learning: Logistic Regression</title>
@@ -54,18 +55,8 @@ Automatically generated HTML file from DocOnce source
('Extending to more predictors', 2, None, '___sec11'),
('Including more classes', 2, None, '___sec12'),
('The Softmax function', 2, None, '___sec13'),
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('A simple classification problem', 2, None, '___sec15'),
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None,
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('Reading in the data', 2, None, '___sec17'),
('Logistic regression', 2, None, '___sec18'),
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('Accuracy of a classification model', 2, None, '___sec20'),
('Analyzing the results', 2, None, '___sec21')]}
('A simple classification problem', 2, None, '___sec14'),
('The Credit Card example', 2, None, '___sec15')]}
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<body>
@@ -117,14 +108,8 @@ MathJax.Hub.Config({
<!-- navigation toc: --> <li><a href="._LogReg-bs012.html#___sec11" style="font-size: 80%;">Extending to more predictors</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs013.html#___sec12" style="font-size: 80%;">Including more classes</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs014.html#___sec13" style="font-size: 80%;">The Softmax function</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs015.html#___sec14" style="font-size: 80%;">A <b>scikit-learn</b> example</a></li>
<!-- navigation toc: --> <li><a href="#___sec15" style="font-size: 80%;">A simple classification problem</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs017.html#___sec16" style="font-size: 80%;">The two-dimensional Ising model, Predicting phase transition of the two-dimensional Ising model</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs018.html#___sec17" style="font-size: 80%;">Reading in the data</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs019.html#___sec18" style="font-size: 80%;">Logistic regression</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs020.html#___sec19" style="font-size: 80%;">Exploring the logistic regression</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs021.html#___sec20" style="font-size: 80%;">Accuracy of a classification model</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs022.html#___sec21" style="font-size: 80%;">Analyzing the results</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs015.html#___sec14" style="font-size: 80%;">A simple classification problem</a></li>
<!-- navigation toc: --> <li><a href="#___sec15" style="font-size: 80%;">The Credit Card example</a></li>
</ul>
</li>
@@ -140,61 +125,8 @@ MathJax.Hub.Config({
<a name="part0016"></a>
<!-- !split -->
<h2 id="___sec15" class="anchor">A simple classification problem </h2>
<p>
<h2 id="___sec15" class="anchor">The Credit Card example </h2>
<!-- 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">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn</span> <span style="color: #008000; font-weight: bold">import</span> datasets, linear_model
<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: #008000; font-weight: bold">def</span> <span style="color: #0000FF">generate_data</span>():
np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">0</span>)
X, y <span style="color: #666666">=</span> datasets<span style="color: #666666">.</span>make_moons(<span style="color: #666666">200</span>, noise<span style="color: #666666">=0.20</span>)
<span style="color: #008000; font-weight: bold">return</span> X, y
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">visualize</span>(X, y, clf):
<span style="color: #408080; font-style: italic"># plt.scatter(X[:, 0], X[:, 1], s=40, c=y, cmap=plt.cm.Spectral)</span>
<span style="color: #408080; font-style: italic"># plt.show()</span>
plot_decision_boundary(<span style="color: #008000; font-weight: bold">lambda</span> x: clf<span style="color: #666666">.</span>predict(x), X, y)
plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">&quot;Logistic Regression&quot;</span>)
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">plot_decision_boundary</span>(pred_func, X, y):
<span style="color: #408080; font-style: italic"># Set min and max values and give it some padding</span>
x_min, x_max <span style="color: #666666">=</span> X[:, <span style="color: #666666">0</span>]<span style="color: #666666">.</span>min() <span style="color: #666666">-</span> <span style="color: #666666">.5</span>, X[:, <span style="color: #666666">0</span>]<span style="color: #666666">.</span>max() <span style="color: #666666">+</span> <span style="color: #666666">.5</span>
y_min, y_max <span style="color: #666666">=</span> X[:, <span style="color: #666666">1</span>]<span style="color: #666666">.</span>min() <span style="color: #666666">-</span> <span style="color: #666666">.5</span>, X[:, <span style="color: #666666">1</span>]<span style="color: #666666">.</span>max() <span style="color: #666666">+</span> <span style="color: #666666">.5</span>
h <span style="color: #666666">=</span> <span style="color: #666666">0.01</span>
<span style="color: #408080; font-style: italic"># Generate a grid of points with distance h between them</span>
xx, yy <span style="color: #666666">=</span> np<span style="color: #666666">.</span>meshgrid(np<span style="color: #666666">.</span>arange(x_min, x_max, h), np<span style="color: #666666">.</span>arange(y_min, y_max, h))
<span style="color: #408080; font-style: italic"># Predict the function value for the whole gid</span>
Z <span style="color: #666666">=</span> pred_func(np<span style="color: #666666">.</span>c_[xx<span style="color: #666666">.</span>ravel(), yy<span style="color: #666666">.</span>ravel()])
Z <span style="color: #666666">=</span> Z<span style="color: #666666">.</span>reshape(xx<span style="color: #666666">.</span>shape)
<span style="color: #408080; font-style: italic"># Plot the contour and training examples</span>
plt<span style="color: #666666">.</span>contourf(xx, yy, Z, cmap<span style="color: #666666">=</span>plt<span style="color: #666666">.</span>cm<span style="color: #666666">.</span>Spectral)
plt<span style="color: #666666">.</span>scatter(X[:, <span style="color: #666666">0</span>], X[:, <span style="color: #666666">1</span>], c<span style="color: #666666">=</span>y, cmap<span style="color: #666666">=</span>plt<span style="color: #666666">.</span>cm<span style="color: #666666">.</span>Spectral)
plt<span style="color: #666666">.</span>show()
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">classify</span>(X, y):
clf <span style="color: #666666">=</span> linear_model<span style="color: #666666">.</span>LogisticRegressionCV()
clf<span style="color: #666666">.</span>fit(X, y)
<span style="color: #008000; font-weight: bold">return</span> clf
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">main</span>():
X, y <span style="color: #666666">=</span> generate_data()
<span style="color: #408080; font-style: italic"># visualize(X, y)</span>
clf <span style="color: #666666">=</span> classify(X, y)
visualize(X, y, clf)
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #19177C">__name__</span> <span style="color: #666666">==</span> <span style="color: #BA2121">&quot;__main__&quot;</span>:
main()
</pre></div>
<p>
<p>
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@@ -210,13 +142,6 @@ MathJax.Hub.Config({
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('Extending to more predictors', 2, None, '___sec11'),
('Including more classes', 2, None, '___sec12'),
('The Softmax function', 2, None, '___sec13'),
('A simple classification problem', 2, None, '___sec14')]}
('A simple classification problem', 2, None, '___sec14'),
('The Credit Card example', 2, None, '___sec15')]}
end of tocinfo -->
<body>
@@ -108,6 +109,7 @@ MathJax.Hub.Config({
<!-- navigation toc: --> <li><a href="._LogReg-bs013.html#___sec12" style="font-size: 80%;">Including more classes</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs014.html#___sec13" style="font-size: 80%;">The Softmax function</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs015.html#___sec14" style="font-size: 80%;">A simple classification problem</a></li>
<!-- navigation toc: --> <li><a href="._LogReg-bs016.html#___sec15" style="font-size: 80%;">The Credit Card example</a></li>
</ul>
</li>
@@ -142,7 +144,7 @@ MathJax.Hub.Config({
<center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
<br>
<p>
<center><h4>Sep 16, 2019</h4></center> <!-- date -->
<center><h4>Sep 18, 2019</h4></center> <!-- date -->
<br>
<p>
<p>
@@ -159,7 +161,7 @@ MathJax.Hub.Config({
<li><a href="._LogReg-bs008.html">9</a></li>
<li><a href="._LogReg-bs009.html">10</a></li>
<li><a href="">...</a></li>
<li><a href="._LogReg-bs015.html">16</a></li>
<li><a href="._LogReg-bs016.html">17</a></li>
<li><a href="._LogReg-bs001.html">&raquo;</a></li>
</ul>
<!-- ------------------- end of main content --------------- -->
doc/pub/LogReg/html/LogReg-reveal.html
+84 -16
View File
@@ -148,7 +148,7 @@ MathJax.Hub.Config({
<center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
<br>
<p>&nbsp;<br>
<center><h4>Sep 16, 2019</h4></center> <!-- date -->
<center><h4>Sep 18, 2019</h4></center> <!-- date -->
<br>
<p>
@@ -226,7 +226,11 @@ primary goal is to identify the classes to which new unseen samples
belong.
<p>
Let us specialize to the case of two classes only, with outputs \( y_i=0 \) and \( y_i=1 \). Our outcomes could represent the status of a credit card user who could default or not on her/his credit card debt. That is
Let us specialize to the case of two classes only, with outputs
\( y_i=0 \) and \( y_i=1 \). Our outcomes could represent the status of a
credit card user that could default or not on her/his credit card
debt. That is
<p>&nbsp;<br>
$$
y_i = \begin{bmatrix} 0 & \mathrm{no}\\ 1 & \mathrm{yes} \end{bmatrix}.
@@ -265,10 +269,7 @@ where \( \hat{y} \) is a vector representing the possible outcomes, \( \hat{X} \
The main problem with our function is that it
takes values on the entire real axis. In the case of
logistic regression, however, the labels \( y_i \) are discrete
variables.
<p>
The table shows an example where we have a followed a group of students during a whole semester, with the aim to see if there are correlations between the number of hours they study and the number of hours they sleep. The output \( y_i \) is labelled as either \( 0 \) (a a grade below average) and \( 1 \), the latter corresponding to a grade above average.
variables. A typical example is the credit card data discussed below here, where we can set the state of defaulting the debt to \( y_i=1 \) and not to \( y_i=0 \) for one the persons in the data set (see the full example below).
<p>
One simple way to get a discrete output is to have sign
@@ -278,14 +279,6 @@ We will encounter this model in our first demonstration of neural networks. Hist
literature. This model is extremely simple. However, in many cases it is more
favorable to use a ``soft" classifier that outputs
the probability of a given category. This leads us to the logistic function.
<p>
The code for plotting the perceptron can be seen here. This si nothing but the standard <a href="https://en.wikipedia.org/wiki/Heaviside_step_function" target="_blank">Heaviside step function</a>.
<p>
<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
<div class="highlight" style="background: #eeeedd"><pre style="font-size: 80%; line-height: 125%"><span></span>
</pre></div>
</section>
@@ -310,11 +303,81 @@ $$
<p>&nbsp;<br>
Note that \( 1-p(t)= p(-t) \).
The following code plots the logistic function.
The following code plots the logistic function, the step function and other functions we will encounter from here and on.
<p>
<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
<div class="highlight" style="background: #eeeedd"><pre style="font-size: 80%; line-height: 125%"><span></span>
<div class="highlight" style="background: #eeeedd"><pre style="font-size: 80%; line-height: 125%"><span></span><span style="color: #CD5555">&quot;&quot;&quot;The sigmoid function (or the logistic curve) is a</span>
<span style="color: #CD5555">function that takes any real number, z, and outputs a number (0,1).</span>
<span style="color: #CD5555">It is useful in neural networks for assigning weights on a relative scale.</span>
<span style="color: #CD5555">The value z is the weighted sum of parameters involved in the learning algorithm.&quot;&quot;&quot;</span>
<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">numpy</span>
<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">matplotlib.pyplot</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">plt</span>
<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">math</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">mt</span>
z = numpy.arange(-<span style="color: #B452CD">5</span>, <span style="color: #B452CD">5</span>, .<span style="color: #B452CD">1</span>)
sigma_fn = numpy.vectorize(<span style="color: #8B008B; font-weight: bold">lambda</span> z: <span style="color: #B452CD">1</span>/(<span style="color: #B452CD">1</span>+numpy.exp(-z)))
sigma = sigma_fn(z)
fig = plt.figure()
ax = fig.add_subplot(<span style="color: #B452CD">111</span>)
ax.plot(z, sigma)
ax.set_ylim([-<span style="color: #B452CD">0.1</span>, <span style="color: #B452CD">1.1</span>])
ax.set_xlim([-<span style="color: #B452CD">5</span>,<span style="color: #B452CD">5</span>])
ax.grid(<span style="color: #658b00">True</span>)
ax.set_xlabel(<span style="color: #CD5555">&#39;z&#39;</span>)
ax.set_title(<span style="color: #CD5555">&#39;sigmoid function&#39;</span>)
plt.show()
<span style="color: #CD5555">&quot;&quot;&quot;Step Function&quot;&quot;&quot;</span>
z = numpy.arange(-<span style="color: #B452CD">5</span>, <span style="color: #B452CD">5</span>, .<span style="color: #B452CD">02</span>)
step_fn = numpy.vectorize(<span style="color: #8B008B; font-weight: bold">lambda</span> z: <span style="color: #B452CD">1.0</span> <span style="color: #8B008B; font-weight: bold">if</span> z &gt;= <span style="color: #B452CD">0.0</span> <span style="color: #8B008B; font-weight: bold">else</span> <span style="color: #B452CD">0.0</span>)
step = step_fn(z)
fig = plt.figure()
ax = fig.add_subplot(<span style="color: #B452CD">111</span>)
ax.plot(z, step)
ax.set_ylim([-<span style="color: #B452CD">0.5</span>, <span style="color: #B452CD">1.5</span>])
ax.set_xlim([-<span style="color: #B452CD">5</span>,<span style="color: #B452CD">5</span>])
ax.grid(<span style="color: #658b00">True</span>)
ax.set_xlabel(<span style="color: #CD5555">&#39;z&#39;</span>)
ax.set_title(<span style="color: #CD5555">&#39;step function&#39;</span>)
plt.show()
<span style="color: #CD5555">&quot;&quot;&quot;Sine Function&quot;&quot;&quot;</span>
z = numpy.arange(-<span style="color: #B452CD">2</span>*mt.pi, <span style="color: #B452CD">2</span>*mt.pi, <span style="color: #B452CD">0.1</span>)
t = numpy.sin(z)
fig = plt.figure()
ax = fig.add_subplot(<span style="color: #B452CD">111</span>)
ax.plot(z, t)
ax.set_ylim([-<span style="color: #B452CD">1.0</span>, <span style="color: #B452CD">1.0</span>])
ax.set_xlim([-<span style="color: #B452CD">2</span>*mt.pi,<span style="color: #B452CD">2</span>*mt.pi])
ax.grid(<span style="color: #658b00">True</span>)
ax.set_xlabel(<span style="color: #CD5555">&#39;z&#39;</span>)
ax.set_title(<span style="color: #CD5555">&#39;sine function&#39;</span>)
plt.show()
<span style="color: #CD5555">&quot;&quot;&quot;Plots a graph of the squashing function used by a rectified linear</span>
<span style="color: #CD5555">unit&quot;&quot;&quot;</span>
z = numpy.arange(-<span style="color: #B452CD">2</span>, <span style="color: #B452CD">2</span>, .<span style="color: #B452CD">1</span>)
zero = numpy.zeros(<span style="color: #658b00">len</span>(z))
y = numpy.max([zero, z], axis=<span style="color: #B452CD">0</span>)
fig = plt.figure()
ax = fig.add_subplot(<span style="color: #B452CD">111</span>)
ax.plot(z, y)
ax.set_ylim([-<span style="color: #B452CD">2.0</span>, <span style="color: #B452CD">2.0</span>])
ax.set_xlim([-<span style="color: #B452CD">2.0</span>, <span style="color: #B452CD">2.0</span>])
ax.grid(<span style="color: #658b00">True</span>)
ax.set_xlabel(<span style="color: #CD5555">&#39;z&#39;</span>)
ax.set_title(<span style="color: #CD5555">&#39;Rectified linear unit&#39;</span>)
plt.show()
</pre></div>
</section>
@@ -606,6 +669,11 @@ methods</a>.
</section>
<section>
<h2 id="___sec15">The Credit Card example </h2>
</section>
</div> <!-- class="slides" -->
</div> <!-- class="reveal" -->
doc/pub/LogReg/html/LogReg-solarized.html
+84 -16
View File
@@ -49,7 +49,8 @@ div { text-align: justify; text-justify: inter-word; }
('Extending to more predictors', 2, None, '___sec11'),
('Including more classes', 2, None, '___sec12'),
('The Softmax function', 2, None, '___sec13'),
('A simple classification problem', 2, None, '___sec14')]}
('A simple classification problem', 2, None, '___sec14'),
('The Credit Card example', 2, None, '___sec15')]}
end of tocinfo -->
<body>
@@ -91,7 +92,7 @@ MathJax.Hub.Config({
<center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
<br>
<p>
<center><h4>Sep 16, 2019</h4></center> <!-- date -->
<center><h4>Sep 18, 2019</h4></center> <!-- date -->
<br>
<p>
<!-- !split -->
@@ -163,7 +164,11 @@ primary goal is to identify the classes to which new unseen samples
belong.
<p>
Let us specialize to the case of two classes only, with outputs \( y_i=0 \) and \( y_i=1 \). Our outcomes could represent the status of a credit card user who could default or not on her/his credit card debt. That is
Let us specialize to the case of two classes only, with outputs
\( y_i=0 \) and \( y_i=1 \). Our outcomes could represent the status of a
credit card user that could default or not on her/his credit card
debt. That is
$$
y_i = \begin{bmatrix} 0 & \mathrm{no}\\ 1 & \mathrm{yes} \end{bmatrix}.
$$
@@ -198,10 +203,7 @@ where \( \hat{y} \) is a vector representing the possible outcomes, \( \hat{X} \
The main problem with our function is that it
takes values on the entire real axis. In the case of
logistic regression, however, the labels \( y_i \) are discrete
variables.
<p>
The table shows an example where we have a followed a group of students during a whole semester, with the aim to see if there are correlations between the number of hours they study and the number of hours they sleep. The output \( y_i \) is labelled as either \( 0 \) (a a grade below average) and \( 1 \), the latter corresponding to a grade above average.
variables. A typical example is the credit card data discussed below here, where we can set the state of defaulting the debt to \( y_i=1 \) and not to \( y_i=0 \) for one the persons in the data set (see the full example below).
<p>
One simple way to get a discrete output is to have sign
@@ -212,13 +214,6 @@ literature. This model is extremely simple. However, in many cases it is more
favorable to use a ``soft" classifier that outputs
the probability of a given category. This leads us to the logistic function.
<p>
The code for plotting the perceptron can be seen here. This si nothing but the standard <a href="https://en.wikipedia.org/wiki/Heaviside_step_function" target="_blank">Heaviside step function</a>.
<p>
<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
<div class="highlight" style="background: #eeeedd"><pre style="line-height: 125%"><span></span>
</pre></div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
@@ -240,11 +235,81 @@ p(t) = \frac{1}{1+\mathrm \exp{-t}}=\frac{\exp{t}}{1+\mathrm \exp{t}}.
$$
Note that \( 1-p(t)= p(-t) \).
The following code plots the logistic function.
The following code plots the logistic function, the step function and other functions we will encounter from here and on.
<p>
<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
<div class="highlight" style="background: #eeeedd"><pre style="line-height: 125%"><span></span>
<div class="highlight" style="background: #eeeedd"><pre style="line-height: 125%"><span></span><span style="color: #CD5555">&quot;&quot;&quot;The sigmoid function (or the logistic curve) is a</span>
<span style="color: #CD5555">function that takes any real number, z, and outputs a number (0,1).</span>
<span style="color: #CD5555">It is useful in neural networks for assigning weights on a relative scale.</span>
<span style="color: #CD5555">The value z is the weighted sum of parameters involved in the learning algorithm.&quot;&quot;&quot;</span>
<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">numpy</span>
<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">matplotlib.pyplot</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">plt</span>
<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">math</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">mt</span>
z = numpy.arange(-<span style="color: #B452CD">5</span>, <span style="color: #B452CD">5</span>, .<span style="color: #B452CD">1</span>)
sigma_fn = numpy.vectorize(<span style="color: #8B008B; font-weight: bold">lambda</span> z: <span style="color: #B452CD">1</span>/(<span style="color: #B452CD">1</span>+numpy.exp(-z)))
sigma = sigma_fn(z)
fig = plt.figure()
ax = fig.add_subplot(<span style="color: #B452CD">111</span>)
ax.plot(z, sigma)
ax.set_ylim([-<span style="color: #B452CD">0.1</span>, <span style="color: #B452CD">1.1</span>])
ax.set_xlim([-<span style="color: #B452CD">5</span>,<span style="color: #B452CD">5</span>])
ax.grid(<span style="color: #658b00">True</span>)
ax.set_xlabel(<span style="color: #CD5555">&#39;z&#39;</span>)
ax.set_title(<span style="color: #CD5555">&#39;sigmoid function&#39;</span>)
plt.show()
<span style="color: #CD5555">&quot;&quot;&quot;Step Function&quot;&quot;&quot;</span>
z = numpy.arange(-<span style="color: #B452CD">5</span>, <span style="color: #B452CD">5</span>, .<span style="color: #B452CD">02</span>)
step_fn = numpy.vectorize(<span style="color: #8B008B; font-weight: bold">lambda</span> z: <span style="color: #B452CD">1.0</span> <span style="color: #8B008B; font-weight: bold">if</span> z &gt;= <span style="color: #B452CD">0.0</span> <span style="color: #8B008B; font-weight: bold">else</span> <span style="color: #B452CD">0.0</span>)
step = step_fn(z)
fig = plt.figure()
ax = fig.add_subplot(<span style="color: #B452CD">111</span>)
ax.plot(z, step)
ax.set_ylim([-<span style="color: #B452CD">0.5</span>, <span style="color: #B452CD">1.5</span>])
ax.set_xlim([-<span style="color: #B452CD">5</span>,<span style="color: #B452CD">5</span>])
ax.grid(<span style="color: #658b00">True</span>)
ax.set_xlabel(<span style="color: #CD5555">&#39;z&#39;</span>)
ax.set_title(<span style="color: #CD5555">&#39;step function&#39;</span>)
plt.show()
<span style="color: #CD5555">&quot;&quot;&quot;Sine Function&quot;&quot;&quot;</span>
z = numpy.arange(-<span style="color: #B452CD">2</span>*mt.pi, <span style="color: #B452CD">2</span>*mt.pi, <span style="color: #B452CD">0.1</span>)
t = numpy.sin(z)
fig = plt.figure()
ax = fig.add_subplot(<span style="color: #B452CD">111</span>)
ax.plot(z, t)
ax.set_ylim([-<span style="color: #B452CD">1.0</span>, <span style="color: #B452CD">1.0</span>])
ax.set_xlim([-<span style="color: #B452CD">2</span>*mt.pi,<span style="color: #B452CD">2</span>*mt.pi])
ax.grid(<span style="color: #658b00">True</span>)
ax.set_xlabel(<span style="color: #CD5555">&#39;z&#39;</span>)
ax.set_title(<span style="color: #CD5555">&#39;sine function&#39;</span>)
plt.show()
<span style="color: #CD5555">&quot;&quot;&quot;Plots a graph of the squashing function used by a rectified linear</span>
<span style="color: #CD5555">unit&quot;&quot;&quot;</span>
z = numpy.arange(-<span style="color: #B452CD">2</span>, <span style="color: #B452CD">2</span>, .<span style="color: #B452CD">1</span>)
zero = numpy.zeros(<span style="color: #658b00">len</span>(z))
y = numpy.max([zero, z], axis=<span style="color: #B452CD">0</span>)
fig = plt.figure()
ax = fig.add_subplot(<span style="color: #B452CD">111</span>)
ax.plot(z, y)
ax.set_ylim([-<span style="color: #B452CD">2.0</span>, <span style="color: #B452CD">2.0</span>])
ax.set_xlim([-<span style="color: #B452CD">2.0</span>, <span style="color: #B452CD">2.0</span>])
ax.grid(<span style="color: #658b00">True</span>)
ax.set_xlabel(<span style="color: #CD5555">&#39;z&#39;</span>)
ax.set_title(<span style="color: #CD5555">&#39;Rectified linear unit&#39;</span>)
plt.show()
</pre></div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
@@ -499,6 +564,9 @@ methods</a>.
main()
</pre></div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec15">The Credit Card example </h2>
<!-- ------------------- end of main content --------------- -->
doc/pub/LogReg/html/LogReg.html
+84 -16
View File
@@ -54,7 +54,8 @@ div { text-align: justify; text-justify: inter-word; }
('Extending to more predictors', 2, None, '___sec11'),
('Including more classes', 2, None, '___sec12'),
('The Softmax function', 2, None, '___sec13'),
('A simple classification problem', 2, None, '___sec14')]}
('A simple classification problem', 2, None, '___sec14'),
('The Credit Card example', 2, None, '___sec15')]}
end of tocinfo -->
<body>
@@ -96,7 +97,7 @@ MathJax.Hub.Config({
<center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
<br>
<p>
<center><h4>Sep 16, 2019</h4></center> <!-- date -->
<center><h4>Sep 18, 2019</h4></center> <!-- date -->
<br>
<p>
<!-- !split -->
@@ -168,7 +169,11 @@ primary goal is to identify the classes to which new unseen samples
belong.
<p>
Let us specialize to the case of two classes only, with outputs \( y_i=0 \) and \( y_i=1 \). Our outcomes could represent the status of a credit card user who could default or not on her/his credit card debt. That is
Let us specialize to the case of two classes only, with outputs
\( y_i=0 \) and \( y_i=1 \). Our outcomes could represent the status of a
credit card user that could default or not on her/his credit card
debt. That is
$$
y_i = \begin{bmatrix} 0 & \mathrm{no}\\ 1 & \mathrm{yes} \end{bmatrix}.
$$
@@ -203,10 +208,7 @@ where \( \hat{y} \) is a vector representing the possible outcomes, \( \hat{X} \
The main problem with our function is that it
takes values on the entire real axis. In the case of
logistic regression, however, the labels \( y_i \) are discrete
variables.
<p>
The table shows an example where we have a followed a group of students during a whole semester, with the aim to see if there are correlations between the number of hours they study and the number of hours they sleep. The output \( y_i \) is labelled as either \( 0 \) (a a grade below average) and \( 1 \), the latter corresponding to a grade above average.
variables. A typical example is the credit card data discussed below here, where we can set the state of defaulting the debt to \( y_i=1 \) and not to \( y_i=0 \) for one the persons in the data set (see the full example below).
<p>
One simple way to get a discrete output is to have sign
@@ -217,13 +219,6 @@ literature. This model is extremely simple. However, in many cases it is more
favorable to use a ``soft" classifier that outputs
the probability of a given category. This leads us to the logistic function.
<p>
The code for plotting the perceptron can be seen here. This si nothing but the standard <a href="https://en.wikipedia.org/wiki/Heaviside_step_function" target="_blank">Heaviside step function</a>.
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>
</pre></div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
@@ -245,11 +240,81 @@ p(t) = \frac{1}{1+\mathrm \exp{-t}}=\frac{\exp{t}}{1+\mathrm \exp{t}}.
$$
Note that \( 1-p(t)= p(-t) \).
The following code plots the logistic function.
The following code plots the logistic function, the step function and other functions we will encounter from here and on.
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #BA2121; font-style: italic">&quot;&quot;&quot;The sigmoid function (or the logistic curve) is a</span>
<span style="color: #BA2121; font-style: italic">function that takes any real number, z, and outputs a number (0,1).</span>
<span style="color: #BA2121; font-style: italic">It is useful in neural networks for assigning weights on a relative scale.</span>
<span style="color: #BA2121; font-style: italic">The value z is the weighted sum of parameters involved in the learning algorithm.&quot;&quot;&quot;</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</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: #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>
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>)
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)))
sigma <span style="color: #666666">=</span> sigma_fn(z)
fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure()
ax <span style="color: #666666">=</span> fig<span style="color: #666666">.</span>add_subplot(<span style="color: #666666">111</span>)
ax<span style="color: #666666">.</span>plot(z, sigma)
ax<span style="color: #666666">.</span>set_ylim([<span style="color: #666666">-0.1</span>, <span style="color: #666666">1.1</span>])
ax<span style="color: #666666">.</span>set_xlim([<span style="color: #666666">-5</span>,<span style="color: #666666">5</span>])
ax<span style="color: #666666">.</span>grid(<span style="color: #008000">True</span>)
ax<span style="color: #666666">.</span>set_xlabel(<span style="color: #BA2121">&#39;z&#39;</span>)
ax<span style="color: #666666">.</span>set_title(<span style="color: #BA2121">&#39;sigmoid function&#39;</span>)
plt<span style="color: #666666">.</span>show()
<span style="color: #BA2121; font-style: italic">&quot;&quot;&quot;Step Function&quot;&quot;&quot;</span>
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>)
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">&gt;=</span> <span style="color: #666666">0.0</span> <span style="color: #008000; font-weight: bold">else</span> <span style="color: #666666">0.0</span>)
step <span style="color: #666666">=</span> step_fn(z)
fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure()
ax <span style="color: #666666">=</span> fig<span style="color: #666666">.</span>add_subplot(<span style="color: #666666">111</span>)
ax<span style="color: #666666">.</span>plot(z, step)
ax<span style="color: #666666">.</span>set_ylim([<span style="color: #666666">-0.5</span>, <span style="color: #666666">1.5</span>])
ax<span style="color: #666666">.</span>set_xlim([<span style="color: #666666">-5</span>,<span style="color: #666666">5</span>])
ax<span style="color: #666666">.</span>grid(<span style="color: #008000">True</span>)
ax<span style="color: #666666">.</span>set_xlabel(<span style="color: #BA2121">&#39;z&#39;</span>)
ax<span style="color: #666666">.</span>set_title(<span style="color: #BA2121">&#39;step function&#39;</span>)
plt<span style="color: #666666">.</span>show()
<span style="color: #BA2121; font-style: italic">&quot;&quot;&quot;Sine Function&quot;&quot;&quot;</span>
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>)
t <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>sin(z)
fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure()
ax <span style="color: #666666">=</span> fig<span style="color: #666666">.</span>add_subplot(<span style="color: #666666">111</span>)
ax<span style="color: #666666">.</span>plot(z, t)
ax<span style="color: #666666">.</span>set_ylim([<span style="color: #666666">-1.0</span>, <span style="color: #666666">1.0</span>])
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])
ax<span style="color: #666666">.</span>grid(<span style="color: #008000">True</span>)
ax<span style="color: #666666">.</span>set_xlabel(<span style="color: #BA2121">&#39;z&#39;</span>)
ax<span style="color: #666666">.</span>set_title(<span style="color: #BA2121">&#39;sine function&#39;</span>)
plt<span style="color: #666666">.</span>show()
<span style="color: #BA2121; font-style: italic">&quot;&quot;&quot;Plots a graph of the squashing function used by a rectified linear</span>
<span style="color: #BA2121; font-style: italic">unit&quot;&quot;&quot;</span>
z <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>arange(<span style="color: #666666">-2</span>, <span style="color: #666666">2</span>, <span style="color: #666666">.1</span>)
zero <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>zeros(<span style="color: #008000">len</span>(z))
y <span style="color: #666666">=</span> numpy<span style="color: #666666">.</span>max([zero, z], axis<span style="color: #666666">=0</span>)
fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure()
ax <span style="color: #666666">=</span> fig<span style="color: #666666">.</span>add_subplot(<span style="color: #666666">111</span>)
ax<span style="color: #666666">.</span>plot(z, y)
ax<span style="color: #666666">.</span>set_ylim([<span style="color: #666666">-2.0</span>, <span style="color: #666666">2.0</span>])
ax<span style="color: #666666">.</span>set_xlim([<span style="color: #666666">-2.0</span>, <span style="color: #666666">2.0</span>])
ax<span style="color: #666666">.</span>grid(<span style="color: #008000">True</span>)
ax<span style="color: #666666">.</span>set_xlabel(<span style="color: #BA2121">&#39;z&#39;</span>)
ax<span style="color: #666666">.</span>set_title(<span style="color: #BA2121">&#39;Rectified linear unit&#39;</span>)
plt<span style="color: #666666">.</span>show()
</pre></div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
@@ -504,6 +569,9 @@ methods</a>.
main()
</pre></div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec15">The Credit Card example </h2>
<!-- ------------------- end of main content --------------- -->
doc/pub/LogReg/ipynb/LogReg.ipynb
+98 -17
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@@ -10,7 +10,7 @@
"<!-- Author: --> \n",
"**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University\n",
"\n",
"Date: **Sep 16, 2019**\n",
"Date: **Sep 18, 2019**\n",
"\n",
"Copyright 1999-2019, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n",
"\n",
@@ -74,7 +74,10 @@
"primary goal is to identify the classes to which new unseen samples\n",
"belong.\n",
"\n",
"Let us specialize to the case of two classes only, with outputs $y_i=0$ and $y_i=1$. Our outcomes could represent the status of a credit card user who could default or not on her/his credit card debt. That is"
"Let us specialize to the case of two classes only, with outputs\n",
"$y_i=0$ and $y_i=1$. Our outcomes could represent the status of a\n",
"credit card user that could default or not on her/his credit card\n",
"debt. That is"
]
},
{
@@ -125,10 +128,7 @@
"The main problem with our function is that it \n",
"takes values on the entire real axis. In the case of\n",
"logistic regression, however, the labels $y_i$ are discrete\n",
"variables. \n",
"\n",
"The table shows an example where we have a followed a group of students during a whole semester, with the aim to see if there are correlations between the number of hours they study and the number of hours they sleep. The output $y_i$ is labelled as either $0$ (a a grade below average) and $1$, the latter corresponding to a grade above average. \n",
"\n",
"variables. A typical example is the credit card data discussed below here, where we can set the state of defaulting the debt to $y_i=1$ and not to $y_i=0$ for one the persons in the data set (see the full example below).\n",
"\n",
"One simple way to get a discrete output is to have sign\n",
"functions that map the output of a linear regressor to values $\\{0,1\\}$,\n",
@@ -138,13 +138,7 @@
"favorable to use a ``soft\" classifier that outputs\n",
"the probability of a given category. This leads us to the logistic function.\n",
"\n",
"The code for plotting the perceptron can be seen here. This si nothing but the standard [Heaviside step function](https://en.wikipedia.org/wiki/Heaviside_step_function)."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"## The logistic function\n",
"\n",
"The perceptron is an example of a ``hard classification\" model. We\n",
@@ -173,7 +167,89 @@
"metadata": {},
"source": [
"Note that $1-p(t)= p(-t)$.\n",
"The following code plots the logistic function."
"The following code plots the logistic function, the step function and other functions we will encounter from here and on."
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"%matplotlib inline\n",
"\n",
"\"\"\"The sigmoid function (or the logistic curve) is a\n",
"function that takes any real number, z, and outputs a number (0,1).\n",
"It is useful in neural networks for assigning weights on a relative scale.\n",
"The value z is the weighted sum of parameters involved in the learning algorithm.\"\"\"\n",
"\n",
"import numpy\n",
"import matplotlib.pyplot as plt\n",
"import math as mt\n",
"\n",
"z = numpy.arange(-5, 5, .1)\n",
"sigma_fn = numpy.vectorize(lambda z: 1/(1+numpy.exp(-z)))\n",
"sigma = sigma_fn(z)\n",
"\n",
"fig = plt.figure()\n",
"ax = fig.add_subplot(111)\n",
"ax.plot(z, sigma)\n",
"ax.set_ylim([-0.1, 1.1])\n",
"ax.set_xlim([-5,5])\n",
"ax.grid(True)\n",
"ax.set_xlabel('z')\n",
"ax.set_title('sigmoid function')\n",
"\n",
"plt.show()\n",
"\n",
"\"\"\"Step Function\"\"\"\n",
"z = numpy.arange(-5, 5, .02)\n",
"step_fn = numpy.vectorize(lambda z: 1.0 if z >= 0.0 else 0.0)\n",
"step = step_fn(z)\n",
"\n",
"fig = plt.figure()\n",
"ax = fig.add_subplot(111)\n",
"ax.plot(z, step)\n",
"ax.set_ylim([-0.5, 1.5])\n",
"ax.set_xlim([-5,5])\n",
"ax.grid(True)\n",
"ax.set_xlabel('z')\n",
"ax.set_title('step function')\n",
"\n",
"plt.show()\n",
"\n",
"\"\"\"Sine Function\"\"\"\n",
"z = numpy.arange(-2*mt.pi, 2*mt.pi, 0.1)\n",
"t = numpy.sin(z)\n",
"\n",
"fig = plt.figure()\n",
"ax = fig.add_subplot(111)\n",
"ax.plot(z, t)\n",
"ax.set_ylim([-1.0, 1.0])\n",
"ax.set_xlim([-2*mt.pi,2*mt.pi])\n",
"ax.grid(True)\n",
"ax.set_xlabel('z')\n",
"ax.set_title('sine function')\n",
"\n",
"plt.show()\n",
"\"\"\"Plots a graph of the squashing function used by a rectified linear\n",
"unit\"\"\"\n",
"z = numpy.arange(-2, 2, .1)\n",
"zero = numpy.zeros(len(z))\n",
"y = numpy.max([zero, z], axis=0)\n",
"\n",
"fig = plt.figure()\n",
"ax = fig.add_subplot(111)\n",
"ax.plot(z, y)\n",
"ax.set_ylim([-2.0, 2.0])\n",
"ax.set_xlim([-2.0, 2.0])\n",
"ax.grid(True)\n",
"ax.set_xlabel('z')\n",
"ax.set_title('Rectified linear unit')\n",
"\n",
"plt.show()"
]
},
{
@@ -537,14 +613,12 @@
},
{
"cell_type": "code",
"execution_count": 1,
"execution_count": 2,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"%matplotlib inline\n",
"\n",
"import numpy as np\n",
"from sklearn import datasets, linear_model\n",
"import matplotlib.pyplot as plt\n",
@@ -595,6 +669,13 @@
"if __name__ == \"__main__\":\n",
" main()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## The Credit Card example"
]
}
],
"metadata": {},
doc/pub/LogReg/ipynb/ipynb-LogReg-src.tar.gz
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@@ -374,7 +375,7 @@ MathJax.Hub.Config({
<center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
<br>
<p>
<center><h4>Oct 19, 2018</h4></center> <!-- date -->
<center><h4>Sep 18, 2019</h4></center> <!-- date -->
<br>
<p>
@@ -416,7 +417,7 @@ MathJax.Hub.Config({
<center style="font-size:80%">
<!-- copyright --> &copy; 1999-2018, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license
<!-- copyright --> &copy; 1999-2019, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license
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@@ -371,7 +372,7 @@ $$
and using the Hadamard product of two vectors we can write this as
$$
\hat{\delta}^L = f'(\hat{z}^L)\circ\frac{\partial {\cal C}}{\partial (\hat{a}L)}.
\hat{\delta}^L = f'(\hat{z}^L)\circ\frac{\partial {\cal C}}{\partial (\hat{a}^L)}.
$$
<p>
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@@ -366,17 +367,16 @@ being realizations of this object with different hyperparameters. An implementat
<!-- 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">class</span> <span style="color: #0000FF; font-weight: bold">NeuralNetwork</span>:
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">__init__</span>(
<span style="color: #008000">self</span>,
X_data,
Y_data,
n_hidden_neurons<span style="color: #666666">=50</span>,
n_categories<span style="color: #666666">=10</span>,
epochs<span style="color: #666666">=10</span>,
batch_size<span style="color: #666666">=100</span>,
eta<span style="color: #666666">=0.1</span>,
lmbd<span style="color: #666666">=0.0</span>,
<span style="color: #008000">self</span>,
X_data,
Y_data,
n_hidden_neurons<span style="color: #666666">=50</span>,
n_categories<span style="color: #666666">=10</span>,
epochs<span style="color: #666666">=10</span>,
batch_size<span style="color: #666666">=100</span>,
eta<span style="color: #666666">=0.1</span>,
lmbd<span style="color: #666666">=0.0</span>):
):
<span style="color: #008000">self</span><span style="color: #666666">.</span>X_data_full <span style="color: #666666">=</span> X_data
<span style="color: #008000">self</span><span style="color: #666666">.</span>Y_data_full <span style="color: #666666">=</span> Y_data
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@@ -369,19 +370,18 @@ MathJax.Hub.Config({
<span style="color: #008000; font-weight: bold">class</span> <span style="color: #0000FF; font-weight: bold">NeuralNetworkTensorflow</span>:
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">__init__</span>(
<span style="color: #008000">self</span>,
X_train,
Y_train,
X_test,
Y_test,
n_neurons_layer1<span style="color: #666666">=100</span>,
n_neurons_layer2<span style="color: #666666">=50</span>,
n_categories<span style="color: #666666">=2</span>,
epochs<span style="color: #666666">=10</span>,
batch_size<span style="color: #666666">=100</span>,
eta<span style="color: #666666">=0.1</span>,
lmbd<span style="color: #666666">=0.0</span>,
):
<span style="color: #008000">self</span>,
X_train,
Y_train,
X_test,
Y_test,
n_neurons_layer1<span style="color: #666666">=100</span>,
n_neurons_layer2<span style="color: #666666">=50</span>,
n_categories<span style="color: #666666">=2</span>,
epochs<span style="color: #666666">=10</span>,
batch_size<span style="color: #666666">=100</span>,
eta<span style="color: #666666">=0.1</span>,
lmbd<span style="color: #666666">=0.0</span>):
<span style="color: #408080; font-style: italic"># keep track of number of steps</span>
<span style="color: #008000">self</span><span style="color: #666666">.</span>global_step <span style="color: #666666">=</span> tf<span style="color: #666666">.</span>Variable(<span style="color: #666666">0</span>, dtype<span style="color: #666666">=</span>tf<span style="color: #666666">.</span>int32, trainable<span style="color: #666666">=</span><span style="color: #008000">False</span>, name<span style="color: #666666">=</span><span style="color: #BA2121">&#39;global_step&#39;</span>)
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