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mhjensen
2019-12-29 21:59:45 +01:00
parent c2f533862f
commit 204804be23
10 changed files with 28 additions and 36 deletions
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doc/pub/DimRed/html/._DimRed-bs019.html
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@@ -232,6 +232,9 @@ low-dimensional encoding of the data is then given by a set of vectors
orthogonal projection of the data onto the columns spanned by the
eigenvectors of the covariance(correlations matrix).
<p>
The proof which follows will be updated by mid January 2020.
<p>
<p>
<!-- navigation buttons at the bottom of the page -->
doc/pub/DimRed/html/._DimRed-bs024.html
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@@ -206,9 +206,8 @@ MathJax.Hub.Config({
<!-- !split -->
<h2 id="___sec26" class="anchor">Principal Component Analysis </h2>
<div class="panel panel-default">
<div class="panel-body">
<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
<p>
Principal Component Analysis (PCA) is by far the most popular dimensionality reduction algorithm.
First it identifies the hyperplane that lies closest to the data, and then it projects the data onto it.
doc/pub/DimRed/html/._DimRed-bs029.html
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@@ -213,11 +213,6 @@ algorithm that quickly finds an approximation of the first d principal component
complexity is \( O(m \times d^2)+O(d^3) \), instead of \( O(m \times n^2) + O(n^3) \), so it is dramatically faster than the
previous algorithms when \( d \) is much smaller than \( n \).
<p>
</div>
</div>
<p>
<p>
<!-- navigation buttons at the bottom of the page -->
doc/pub/DimRed/html/DimRed-reveal.html
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@@ -1143,6 +1143,9 @@ low-dimensional encoding of the data is then given by a set of vectors
\( \boldsymbol{z}_i \) with at most \( l \) vectors, with \( l < < p \), defined by the
orthogonal projection of the data onto the columns spanned by the
eigenvectors of the covariance(correlations matrix).
<p>
The proof which follows will be updated by mid January 2020.
</section>
@@ -1298,8 +1301,7 @@ This material will be added by mid January 2020.
<section>
<h2 id="___sec26">Principal Component Analysis </h2>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<p>
Principal Component Analysis (PCA) is by far the most popular dimensionality reduction algorithm.
First it identifies the hyperplane that lies closest to the data, and then it projects the data onto it.
@@ -1477,9 +1479,6 @@ Scikit-Learn offers yet another option to perform PCA, called Randomized PCA. Th
algorithm that quickly finds an approximation of the first d principal components. Its computational
complexity is \( O(m \times d^2)+O(d^3) \), instead of \( O(m \times n^2) + O(n^3) \), so it is dramatically faster than the
previous algorithms when \( d \) is much smaller than \( n \).
</div>
</section>
doc/pub/DimRed/html/DimRed-solarized.html
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@@ -1115,6 +1115,9 @@ low-dimensional encoding of the data is then given by a set of vectors
orthogonal projection of the data onto the columns spanned by the
eigenvectors of the covariance(correlations matrix).
<p>
The proof which follows will be updated by mid January 2020.
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
@@ -1245,8 +1248,7 @@ This material will be added by mid January 2020.
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec26">Principal Component Analysis </h2>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<p>
Principal Component Analysis (PCA) is by far the most popular dimensionality reduction algorithm.
First it identifies the hyperplane that lies closest to the data, and then it projects the data onto it.
@@ -1257,7 +1259,7 @@ training set, then extracts the first two principal components. First we center
<p>
<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
<div class="highlight" style="background: #eee8d5"><pre style="line-height: 125%"><span></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">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
<div class="highlight" style="background: #eeeedd"><pre style="line-height: 125%"><span></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">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">pandas</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">pd</span>
<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">IPython.display</span> <span style="color: #8B008B; font-weight: bold">import</span> display
np.random.seed(<span style="color: #B452CD">100</span>)
@@ -1294,7 +1296,7 @@ Selecting this hyperplane ensures that the projection will preserve as much vari
<p>
<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
<div class="highlight" style="background: #eee8d5"><pre style="line-height: 125%"><span></span>W2 = V.T[:, :<span style="color: #B452CD">2</span>]
<div class="highlight" style="background: #eeeedd"><pre style="line-height: 125%"><span></span>W2 = V.T[:, :<span style="color: #B452CD">2</span>]
X2D = X_centered.dot(W2)
</pre></div>
<p>
@@ -1309,7 +1311,7 @@ that it automatically takes care of centering the data):
<p>
<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
<div class="highlight" style="background: #eee8d5"><pre style="line-height: 125%"><span></span><span style="color: #228B22">#thereafter we do a PCA with Scikit-learn</span>
<div class="highlight" style="background: #eeeedd"><pre style="line-height: 125%"><span></span><span style="color: #228B22">#thereafter we do a PCA with Scikit-learn</span>
<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.decomposition</span> <span style="color: #8B008B; font-weight: bold">import</span> PCA
pca = PCA(n_components = <span style="color: #B452CD">2</span>)
X2D = pca.fit_transform(X)
@@ -1322,7 +1324,7 @@ principal component is equal to
<p>
<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
<div class="highlight" style="background: #eee8d5"><pre style="line-height: 125%"><span></span>pca.components_.T[:, <span style="color: #B452CD">0</span>].
<div class="highlight" style="background: #eeeedd"><pre style="line-height: 125%"><span></span>pca.components_.T[:, <span style="color: #B452CD">0</span>].
</pre></div>
<p>
Another very useful piece of information is the explained variance ratio of each principal component,
@@ -1338,7 +1340,7 @@ Here we compute performance scores on the training data using logistic regressio
<p>
<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
<div class="highlight" style="background: #eee8d5"><pre style="line-height: 125%"><span></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>
<div class="highlight" style="background: #eeeedd"><pre style="line-height: 125%"><span></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">numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.model_selection</span> <span style="color: #8B008B; font-weight: bold">import</span> train_test_split
<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.datasets</span> <span style="color: #8B008B; font-weight: bold">import</span> load_breast_cancer
@@ -1385,7 +1387,7 @@ of dimensions required to preserve 95% of the training set&#8217;s variance:
<p>
<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
<div class="highlight" style="background: #eee8d5"><pre style="line-height: 125%"><span></span>pca = PCA()
<div class="highlight" style="background: #eeeedd"><pre style="line-height: 125%"><span></span>pca = PCA()
pca.fit(X)
cumsum = np.cumsum(pca.explained_variance_ratio_)
d = np.argmax(cumsum &gt;= <span style="color: #B452CD">0.95</span>) + <span style="color: #B452CD">1</span>
@@ -1397,7 +1399,7 @@ a float between 0.0 and 1.0, indicating the ratio of variance you wish to preser
<p>
<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
<div class="highlight" style="background: #eee8d5"><pre style="line-height: 125%"><span></span>pca = PCA(n_components=<span style="color: #B452CD">0.95</span>)
<div class="highlight" style="background: #eeeedd"><pre style="line-height: 125%"><span></span>pca = PCA(n_components=<span style="color: #B452CD">0.95</span>)
X_reduced = pca.fit_transform(X)
</pre></div>
<p>
@@ -1423,10 +1425,6 @@ algorithm that quickly finds an approximation of the first d principal component
complexity is \( O(m \times d^2)+O(d^3) \), instead of \( O(m \times n^2) + O(n^3) \), so it is dramatically faster than the
previous algorithms when \( d \) is much smaller than \( n \).
</div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
doc/pub/DimRed/html/DimRed.html
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@@ -1120,6 +1120,9 @@ low-dimensional encoding of the data is then given by a set of vectors
orthogonal projection of the data onto the columns spanned by the
eigenvectors of the covariance(correlations matrix).
<p>
The proof which follows will be updated by mid January 2020.
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
@@ -1250,8 +1253,7 @@ This material will be added by mid January 2020.
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="___sec26">Principal Component Analysis </h2>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<p>
Principal Component Analysis (PCA) is by far the most popular dimensionality reduction algorithm.
First it identifies the hyperplane that lies closest to the data, and then it projects the data onto it.
@@ -1428,10 +1430,6 @@ algorithm that quickly finds an approximation of the first d principal component
complexity is \( O(m \times d^2)+O(d^3) \), instead of \( O(m \times n^2) + O(n^3) \), so it is dramatically faster than the
previous algorithms when \( d \) is much smaller than \( n \).
</div>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
doc/pub/DimRed/ipynb/DimRed.ipynb
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@@ -1253,7 +1253,7 @@
"orthogonal projection of the data onto the columns spanned by the\n",
"eigenvectors of the covariance(correlations matrix).\n",
"\n",
"\n",
"The proof which follows will be updated by mid January 2020.\n",
"\n",
"## Proof of the PCA Theorem\n",
"\n",
@@ -1492,6 +1492,7 @@
"\n",
"\n",
"## Principal Component Analysis\n",
"\n",
"Principal Component Analysis (PCA) is by far the most popular dimensionality reduction algorithm.\n",
"First it identifies the hyperplane that lies closest to the data, and then it projects the data onto it.\n",
"\n",
doc/pub/DimRed/ipynb/ipynb-DimRed-src.tar.gz
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doc/pub/DimRed/pdf/DimRed-minted.pdf
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doc/src/DimRed/DimRed.do.txt
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@@ -872,7 +872,7 @@ $\bm{z}_i$ with at most $l$ vectors, with $l << p$, defined by the
orthogonal projection of the data onto the columns spanned by the
eigenvectors of the covariance(correlations matrix).
The proof which follows will be updated by mid January 2020.
!split
===== Proof of the PCA Theorem =====
@@ -1001,7 +1001,7 @@ This material will be added by mid January 2020.
!split
===== Principal Component Analysis =====
!bblock
Principal Component Analysis (PCA) is by far the most popular dimensionality reduction algorithm.
First it identifies the hyperplane that lies closest to the data, and then it projects the data onto it.
@@ -1148,7 +1148,6 @@ complexity is $O(m \times d^2)+O(d^3)$, instead of $O(m \times n^2) + O(n^3)$, s
previous algorithms when $d$ is much smaller than $n$.
!eblock
!split