mhjensen
405 lines
22 KiB
HTML
405 lines
22 KiB
HTML
<!--
|
|
Automatically generated HTML file from DocOnce source
|
|
(https://github.com/hplgit/doconce/)
|
|
-->
|
|
<html>
|
|
<head>
|
|
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
|
|
<meta name="generator" content="DocOnce: https://github.com/hplgit/doconce/" />
|
|
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
|
|
<meta name="description" content="Data Analysis and Machine Learning: Preprocessing and Dimensionality Reduction">
|
|
|
|
<title>Data Analysis and Machine Learning: Preprocessing and Dimensionality Reduction</title>
|
|
|
|
<!-- Bootstrap style: bootstrap -->
|
|
<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
|
|
<!-- not necessary
|
|
<link href="https://netdna.bootstrapcdn.com/font-awesome/4.0.3/css/font-awesome.css" rel="stylesheet">
|
|
-->
|
|
|
|
<style type="text/css">
|
|
|
|
/* Add scrollbar to dropdown menus in bootstrap navigation bar */
|
|
.dropdown-menu {
|
|
height: auto;
|
|
max-height: 400px;
|
|
overflow-x: hidden;
|
|
}
|
|
|
|
/* Adds an invisible element before each target to offset for the navigation
|
|
bar */
|
|
.anchor::before {
|
|
content:"";
|
|
display:block;
|
|
height:50px; /* fixed header height for style bootstrap */
|
|
margin:-50px 0 0; /* negative fixed header height */
|
|
}
|
|
</style>
|
|
|
|
|
|
</head>
|
|
|
|
<!-- tocinfo
|
|
{'highest level': 2,
|
|
'sections': [('Reducing the number of degrees of freedom, overarching view',
|
|
2,
|
|
None,
|
|
'___sec0'),
|
|
('Preprocessing our data', 2, None, '___sec1'),
|
|
('More preprocessing', 2, None, '___sec2'),
|
|
('Simple preprocessing examples, Franke function and regression',
|
|
2,
|
|
None,
|
|
'___sec3'),
|
|
('Simple preprocessing examples, breast cancer data and '
|
|
'classification, Support Vector Machines',
|
|
2,
|
|
None,
|
|
'___sec4'),
|
|
('More on Cancer Data, now with Logistic Regression',
|
|
2,
|
|
None,
|
|
'___sec5'),
|
|
('Why should we think of reducing the dimensionality',
|
|
2,
|
|
None,
|
|
'___sec6'),
|
|
('Basic ideas of the Principal Component Analysis (PCA)',
|
|
2,
|
|
None,
|
|
'___sec7'),
|
|
('Introducing the Covariance and Correlation functions',
|
|
2,
|
|
None,
|
|
'___sec8'),
|
|
('Correlation Function and Design/Feature Matrix',
|
|
2,
|
|
None,
|
|
'___sec9'),
|
|
('Covariance Matrix Examples', 2, None, '___sec10'),
|
|
('Correlation Matrix', 2, None, '___sec11'),
|
|
('Correlation Matrix with Pandas', 2, None, '___sec12'),
|
|
('Correlation Matrix with Pandas and the Franke function',
|
|
2,
|
|
None,
|
|
'___sec13'),
|
|
('Rewriting the Covariance and/or Correlation Matrix',
|
|
2,
|
|
None,
|
|
'___sec14'),
|
|
('Towards the PCA theorem', 2, None, '___sec15'),
|
|
('The Algorithm before the Theorem', 2, None, '___sec16'),
|
|
('Writing our own PCA code', 2, None, '___sec17'),
|
|
('Compute the sample mean and center the data',
|
|
3,
|
|
None,
|
|
'___sec18'),
|
|
('Compute the sample covariance', 3, None, '___sec19'),
|
|
('Diagonalize the sample covariance matrix to obtain the '
|
|
'principal components',
|
|
3,
|
|
None,
|
|
'___sec20'),
|
|
('Classical PCA Theorem', 2, None, '___sec21'),
|
|
('Proof of the PCA Theorem', 2, None, '___sec22'),
|
|
('PCA Proof continued', 2, None, '___sec23'),
|
|
('The final step', 2, None, '___sec24'),
|
|
('Geometric Interpretation and link with Singular Value '
|
|
'Decomposition',
|
|
2,
|
|
None,
|
|
'___sec25'),
|
|
('Principal Component Analysis', 2, None, '___sec26'),
|
|
('PCA and scikit-learn', 2, None, '___sec27'),
|
|
('Back to the Cancer Data', 2, None, '___sec28'),
|
|
('More on the PCA', 2, None, '___sec29'),
|
|
('Incremental PCA', 2, None, '___sec30'),
|
|
('Randomized PCA', 2, None, '___sec31'),
|
|
('Kernel PCA', 2, None, '___sec32'),
|
|
('LLE', 2, None, '___sec33'),
|
|
('Other techniques', 2, None, '___sec34')]}
|
|
end of tocinfo -->
|
|
|
|
<body>
|
|
|
|
|
|
|
|
<script type="text/x-mathjax-config">
|
|
MathJax.Hub.Config({
|
|
TeX: {
|
|
equationNumbers: { autoNumber: "none" },
|
|
extensions: ["AMSmath.js", "AMSsymbols.js", "autobold.js", "color.js"]
|
|
}
|
|
});
|
|
</script>
|
|
<script type="text/javascript" async
|
|
src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
|
|
</script>
|
|
|
|
|
|
|
|
|
|
<!-- Bootstrap navigation bar -->
|
|
<div class="navbar navbar-default navbar-fixed-top">
|
|
<div class="navbar-header">
|
|
<button type="button" class="navbar-toggle" data-toggle="collapse" data-target=".navbar-responsive-collapse">
|
|
<span class="icon-bar"></span>
|
|
<span class="icon-bar"></span>
|
|
<span class="icon-bar"></span>
|
|
</button>
|
|
<a class="navbar-brand" href="DimRed-bs.html">Data Analysis and Machine Learning: Preprocessing and Dimensionality Reduction</a>
|
|
</div>
|
|
|
|
<div class="navbar-collapse collapse navbar-responsive-collapse">
|
|
<ul class="nav navbar-nav navbar-right">
|
|
<li class="dropdown">
|
|
<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
|
|
<ul class="dropdown-menu">
|
|
<!-- navigation toc: --> <li><a href="._DimRed-bs001.html#___sec0" style="font-size: 80%;"><b>Reducing the number of degrees of freedom, overarching view</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._DimRed-bs002.html#___sec1" style="font-size: 80%;"><b>Preprocessing our data</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._DimRed-bs003.html#___sec2" style="font-size: 80%;"><b>More preprocessing</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._DimRed-bs004.html#___sec3" style="font-size: 80%;"><b>Simple preprocessing examples, Franke function and regression</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._DimRed-bs005.html#___sec4" style="font-size: 80%;"><b>Simple preprocessing examples, breast cancer data and classification, Support Vector Machines</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._DimRed-bs006.html#___sec5" style="font-size: 80%;"><b>More on Cancer Data, now with Logistic Regression</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._DimRed-bs007.html#___sec6" style="font-size: 80%;"><b>Why should we think of reducing the dimensionality</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._DimRed-bs008.html#___sec7" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._DimRed-bs009.html#___sec8" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._DimRed-bs010.html#___sec9" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._DimRed-bs011.html#___sec10" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._DimRed-bs012.html#___sec11" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._DimRed-bs013.html#___sec12" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._DimRed-bs014.html#___sec13" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._DimRed-bs015.html#___sec14" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="#___sec17" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="#___sec18" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
|
|
<!-- navigation toc: --> <li><a href="#___sec19" style="font-size: 80%;"> Compute the sample covariance</a></li>
|
|
<!-- navigation toc: --> <li><a href="#___sec20" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
|
|
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec21" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec22" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec23" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec24" style="font-size: 80%;"><b>The final step</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec25" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec26" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec27" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec28" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec29" style="font-size: 80%;"><b>More on the PCA</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec30" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec31" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._DimRed-bs030.html#___sec32" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._DimRed-bs031.html#___sec33" style="font-size: 80%;"><b>LLE</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._DimRed-bs032.html#___sec34" style="font-size: 80%;"><b>Other techniques</b></a></li>
|
|
|
|
</ul>
|
|
</li>
|
|
</ul>
|
|
</div>
|
|
</div>
|
|
</div> <!-- end of navigation bar -->
|
|
|
|
<div class="container">
|
|
|
|
<p> </p><p> </p><p> </p> <!-- add vertical space -->
|
|
|
|
<a name="part0018"></a>
|
|
<!-- !split -->
|
|
|
|
<h2 id="___sec17" class="anchor">Writing our own PCA code </h2>
|
|
|
|
<p>
|
|
We will use a simple example first with two-dimensional data
|
|
drawn from a multivariate normal distribution with the following mean and covariance matrix:
|
|
$$
|
|
\mu = (-1,2) \qquad \Sigma = \begin{bmatrix} 4 & 2 \\
|
|
2 & 2
|
|
\end{bmatrix}
|
|
$$
|
|
|
|
Note that the mean refers to each column of data.
|
|
We will generate \( n = 1000 \) points \( X = \{ x_1, \ldots, x_N \} \) from
|
|
this distribution, and store them in the \( 1000 \times 2 \) matrix \( \boldsymbol{X} \).
|
|
|
|
<p>
|
|
The following Python code aids in setting up the data and writing out the design matrix.
|
|
Note that the function <b>multivariate</b> returns also the covariance discussed above and that it is defined by dividing by \( n-1 \) instead of \( n \).
|
|
<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">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">import</span> <span style="color: #0000FF; font-weight: bold">pandas</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">pd</span>
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">IPython.display</span> <span style="color: #008000; font-weight: bold">import</span> display
|
|
n <span style="color: #666666">=</span> <span style="color: #666666">100</span>
|
|
mean <span style="color: #666666">=</span> (<span style="color: #666666">-1</span>, <span style="color: #666666">2</span>)
|
|
cov <span style="color: #666666">=</span> [[<span style="color: #666666">4</span>, <span style="color: #666666">2</span>], [<span style="color: #666666">2</span>, <span style="color: #666666">2</span>]]
|
|
X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>multivariate_normal(mean, cov, n)
|
|
<span style="color: #408080; font-style: italic"># Print the X-matrix</span>
|
|
<span style="color: #008000; font-weight: bold">print</span>(X)
|
|
</pre></div>
|
|
<p>
|
|
Try to add to this code your own calculation of the covariance matrix.
|
|
|
|
<p>
|
|
Now we are going to implement the PCA algorithm. We will break it down into various substeps.
|
|
|
|
<h3 id="___sec18" class="anchor">Compute the sample mean and center the data </h3>
|
|
|
|
<p>
|
|
The first step of PCA is to compute the sample mean of the data and use it to center the data. Recall that the sample mean is
|
|
$$
|
|
\mu_n = \frac{1}{n} \sum_{i=1}^n x_i
|
|
$$
|
|
|
|
and the mean-centered data \( \bar{X} = \{ \bar{x}_1, \ldots, \bar{x}_n \} \) takes the form
|
|
$$
|
|
\bar{x}_i = x_i - \mu_n.
|
|
$$
|
|
|
|
When you are done with these steps, print out \( \mu_n \) to verify it is
|
|
close to \( \mu \) and plot your mean centered data to verify it is
|
|
centered at the origin! Compare your code with the functionality from <b>Scikit-Learn</b> discussed above.
|
|
The following code elements perform these operations using <b>pandas</b> or using our own functionality for doing so. The latter, using <b>numpy</b> is rather simply through the <b>mean()</b> function.
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>df <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>DataFrame(X)
|
|
<span style="color: #408080; font-style: italic"># Pandas does the centering for us</span>
|
|
df <span style="color: #666666">=</span> df <span style="color: #666666">-</span>df<span style="color: #666666">.</span>mean()
|
|
display(df)
|
|
|
|
<span style="color: #408080; font-style: italic"># we center it ourselves</span>
|
|
X_centered <span style="color: #666666">=</span> X <span style="color: #666666">-</span> X<span style="color: #666666">.</span>mean(axis<span style="color: #666666">=0</span>)
|
|
<span style="color: #408080; font-style: italic"># test that we get the same as Pandas</span>
|
|
<span style="color: #008000; font-weight: bold">print</span>(X_centered<span style="color: #666666">-</span>df)
|
|
</pre></div>
|
|
<p>
|
|
Alternatively, you could also have used the functions we discussed earlier for scaling the data set.
|
|
That is, we could have used the <b>StandardScaler</b> function in <b>Scikit-Learn</b>, a function which
|
|
ensures that for each feature/predictor we study the mean value is
|
|
zero and the variance is one (every column in the design/feature
|
|
matrix). You would then not get the same results, since we divide by the variance. For diagonal covariance matrix elements will then be one, while the non-diagonal ones will be divided by \( 2\sqrt{2} \) for our specific case.
|
|
|
|
<h3 id="___sec19" class="anchor">Compute the sample covariance </h3>
|
|
|
|
<p>
|
|
Now we are going to use the mean centered data to compute the sample covariance of the data. Recall it is given by:
|
|
$$
|
|
\begin{equation*}
|
|
\Sigma_n = \frac{1}{n-1} \sum_{i=1}^n \bar{x}_i^T \bar{x}_i = \frac{1}{n-1} \sum_{i=1}^n (x_i - \mu_n)^T (x_i - \mu_n)
|
|
\end{equation*}
|
|
$$
|
|
|
|
where the data points \( x_i \in \mathbb{R}^p \) (here in this example \( p = 2 \)) are column vectors and \( x^T \) is the transpose of \( x \).
|
|
We can write our own code or simply use either the functionaly of <b>numpy</b> or that of <b>pandas</b>, as follows
|
|
<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">print</span>(np<span style="color: #666666">.</span>cov(X<span style="color: #666666">.</span>T))
|
|
<span style="color: #008000; font-weight: bold">print</span>(df<span style="color: #666666">.</span>cov())
|
|
<span style="color: #008000; font-weight: bold">print</span>(np<span style="color: #666666">.</span>cov(X_centered<span style="color: #666666">.</span>T))
|
|
</pre></div>
|
|
<p>
|
|
Depending on the number of points \( n \), we will get results that are close to the covariance values defined above.
|
|
|
|
<h3 id="___sec20" class="anchor">Diagonalize the sample covariance matrix to obtain the principal components </h3>
|
|
|
|
<p>
|
|
Now we are ready to solve for the principal components! To do so we
|
|
diagonalize the sample covariance matrix \( \Sigma_n \). We can use the
|
|
function <b>np.linalg.eig</b> to do so. It will return the eigenvalues and
|
|
eigenvectors of \( \Sigma_n \). Once we have these we can perform the
|
|
following tasks:
|
|
|
|
<ul>
|
|
<li> We compute the percentage of the total variance captured by the first principal component</li>
|
|
<li> We plot the mean centered data and lines along the first and second principal components</li>
|
|
<li> Then we project the mean centered data onto the first and second principal components, and plot the projected data.</li>
|
|
<li> Finally, we approximate the data as</li>
|
|
</ul>
|
|
|
|
$$
|
|
\begin{equation*}
|
|
x_i \approx \tilde{x}_i = \mu_n + \langle x_i, v_0 \rangle v_0
|
|
\end{equation*}
|
|
$$
|
|
|
|
where \( v_0 \) is the first principal component.
|
|
|
|
<p>
|
|
Collecting all these steps we can write our own PCA function and
|
|
compare this with the functionality included in <b>Scikit-Learn</b>.
|
|
|
|
<p>
|
|
The code here outlines some of the elements we could include in the analysis. Feel free to extend upon this.
|
|
<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: #408080; font-style: italic">#Now we do an SVD</span>
|
|
U, s, V <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>svd(X_centered)
|
|
c1 <span style="color: #666666">=</span> V<span style="color: #666666">.</span>T[:, <span style="color: #666666">0</span>]
|
|
c2 <span style="color: #666666">=</span> V<span style="color: #666666">.</span>T[:, <span style="color: #666666">1</span>]
|
|
W2 <span style="color: #666666">=</span> V<span style="color: #666666">.</span>T[:, :<span style="color: #666666">2</span>]
|
|
X2D <span style="color: #666666">=</span> X_centered<span style="color: #666666">.</span>dot(W2)
|
|
<span style="color: #008000; font-weight: bold">print</span>(X2D)
|
|
<span style="color: #408080; font-style: italic">#thereafter we do a PCA with Scikit-learn</span>
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.decomposition</span> <span style="color: #008000; font-weight: bold">import</span> PCA
|
|
pca <span style="color: #666666">=</span> PCA(n_components <span style="color: #666666">=</span> <span style="color: #666666">2</span>)
|
|
X2Dsl <span style="color: #666666">=</span> pca<span style="color: #666666">.</span>fit_transform(X)
|
|
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">"Check that we get the same"</span>)
|
|
<span style="color: #008000; font-weight: bold">print</span>(X2D<span style="color: #666666">-</span>X2Dsl)
|
|
|
|
<span style="color: #008000; font-weight: bold">print</span>(pca<span style="color: #666666">.</span>components_<span style="color: #666666">.</span>T[:, <span style="color: #666666">0</span>])
|
|
</pre></div>
|
|
<p>
|
|
<p>
|
|
<!-- navigation buttons at the bottom of the page -->
|
|
<ul class="pagination">
|
|
<li><a href="._DimRed-bs017.html">«</a></li>
|
|
<li><a href="._DimRed-bs000.html">1</a></li>
|
|
<li><a href="">...</a></li>
|
|
<li><a href="._DimRed-bs010.html">11</a></li>
|
|
<li><a href="._DimRed-bs011.html">12</a></li>
|
|
<li><a href="._DimRed-bs012.html">13</a></li>
|
|
<li><a href="._DimRed-bs013.html">14</a></li>
|
|
<li><a href="._DimRed-bs014.html">15</a></li>
|
|
<li><a href="._DimRed-bs015.html">16</a></li>
|
|
<li><a href="._DimRed-bs016.html">17</a></li>
|
|
<li><a href="._DimRed-bs017.html">18</a></li>
|
|
<li class="active"><a href="._DimRed-bs018.html">19</a></li>
|
|
<li><a href="._DimRed-bs019.html">20</a></li>
|
|
<li><a href="._DimRed-bs020.html">21</a></li>
|
|
<li><a href="._DimRed-bs021.html">22</a></li>
|
|
<li><a href="._DimRed-bs022.html">23</a></li>
|
|
<li><a href="._DimRed-bs023.html">24</a></li>
|
|
<li><a href="._DimRed-bs024.html">25</a></li>
|
|
<li><a href="._DimRed-bs025.html">26</a></li>
|
|
<li><a href="._DimRed-bs026.html">27</a></li>
|
|
<li><a href="._DimRed-bs027.html">28</a></li>
|
|
<li><a href="">...</a></li>
|
|
<li><a href="._DimRed-bs032.html">33</a></li>
|
|
<li><a href="._DimRed-bs019.html">»</a></li>
|
|
</ul>
|
|
<!-- ------------------- end of main content --------------- -->
|
|
|
|
</div> <!-- end container -->
|
|
<!-- include javascript, jQuery *first* -->
|
|
<script src="https://ajax.googleapis.com/ajax/libs/jquery/1.10.2/jquery.min.js"></script>
|
|
<script src="https://netdna.bootstrapcdn.com/bootstrap/3.0.0/js/bootstrap.min.js"></script>
|
|
|
|
<!-- Bootstrap footer
|
|
<footer>
|
|
<a href="http://..."><img width="250" align=right src="http://..."></a>
|
|
</footer>
|
|
-->
|
|
|
|
|
|
<center style="font-size:80%">
|
|
<!-- copyright only on the titlepage -->
|
|
</center>
|
|
|
|
|
|
</body>
|
|
</html>
|
|
|
|
|