From 7fbad909722060fe142a43563fc0c2fbc8a4600d Mon Sep 17 00:00:00 2001 From: mhjensen Date: Tue, 22 Oct 2019 14:47:55 +0200 Subject: [PATCH] further update, added also --- doc/pub/DimRed/html/._DimRed-bs000.html | 48 ++++---- doc/pub/DimRed/html/._DimRed-bs001.html | 48 ++++---- doc/pub/DimRed/html/._DimRed-bs002.html | 48 ++++---- doc/pub/DimRed/html/._DimRed-bs003.html | 48 ++++---- doc/pub/DimRed/html/._DimRed-bs004.html | 48 ++++---- doc/pub/DimRed/html/._DimRed-bs005.html | 48 ++++---- doc/pub/DimRed/html/._DimRed-bs006.html | 48 ++++---- doc/pub/DimRed/html/._DimRed-bs007.html | 48 ++++---- doc/pub/DimRed/html/._DimRed-bs008.html | 48 ++++---- doc/pub/DimRed/html/._DimRed-bs009.html | 48 ++++---- doc/pub/DimRed/html/._DimRed-bs010.html | 48 ++++---- doc/pub/DimRed/html/._DimRed-bs011.html | 48 ++++---- doc/pub/DimRed/html/._DimRed-bs012.html | 48 ++++---- doc/pub/DimRed/html/._DimRed-bs013.html | 48 ++++---- doc/pub/DimRed/html/._DimRed-bs014.html | 48 ++++---- doc/pub/DimRed/html/._DimRed-bs015.html | 48 ++++---- doc/pub/DimRed/html/._DimRed-bs016.html | 61 ++++++---- doc/pub/DimRed/html/._DimRed-bs017.html | 79 ++++++++---- doc/pub/DimRed/html/._DimRed-bs018.html | 50 ++++---- doc/pub/DimRed/html/._DimRed-bs019.html | 59 ++++----- doc/pub/DimRed/html/._DimRed-bs020.html | 80 +++++------- doc/pub/DimRed/html/._DimRed-bs021.html | 88 ++++++++------ doc/pub/DimRed/html/._DimRed-bs022.html | 82 +++++++------ doc/pub/DimRed/html/._DimRed-bs023.html | 77 +++++++----- doc/pub/DimRed/html/._DimRed-bs024.html | 63 +++++----- doc/pub/DimRed/html/._DimRed-bs025.html | 71 +++++------ doc/pub/DimRed/html/._DimRed-bs026.html | 77 +++++++----- doc/pub/DimRed/html/._DimRed-bs027.html | 121 +++++-------------- doc/pub/DimRed/html/DimRed-bs.html | 48 ++++---- doc/pub/DimRed/html/DimRed-reveal.html | 71 +++++++++-- doc/pub/DimRed/html/DimRed-solarized.html | 96 +++++++++++---- doc/pub/DimRed/html/DimRed.html | 96 +++++++++++---- doc/pub/DimRed/ipynb/DimRed.ipynb | 51 ++++++++ doc/pub/DimRed/ipynb/ipynb-DimRed-src.tar.gz | Bin 191 -> 191 bytes doc/pub/DimRed/pdf/DimRed-minted.pdf | Bin 242104 -> 245191 bytes doc/src/DimRed/DimRed.do.txt | 38 ++++++ doc/src/DimRed/mlpfranke.py | 68 +++++++++++ 37 files changed, 1245 insertions(+), 899 deletions(-) create mode 100644 doc/src/DimRed/mlpfranke.py diff --git a/doc/pub/DimRed/html/._DimRed-bs000.html b/doc/pub/DimRed/html/._DimRed-bs000.html index 701588387..3df86eb8f 100644 --- a/doc/pub/DimRed/html/._DimRed-bs000.html +++ b/doc/pub/DimRed/html/._DimRed-bs000.html @@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source None, '___sec14'), ('Towards the PCA theorem', 2, None, '___sec15'), - ('Classical PCA Theorem', 2, None, '___sec16'), - ('Prof of the PCA Theorem', 2, None, '___sec17'), - ('Getting started with PCA', 2, None, '___sec18'), - ('Principal Component Analysis', 2, None, '___sec19'), - ('PCA and scikit-learn', 2, None, '___sec20'), - ('More on the PCA', 2, None, '___sec21'), - ('Incremental PCA', 2, None, '___sec22'), - ('Randomized PCA', 2, None, '___sec23'), - ('Kernel PCA', 2, None, '___sec24'), - ('LLE', 2, None, '___sec25'), - ('Other techniques', 2, None, '___sec26')]} + ('The Algorithm before the Theorem', 2, None, '___sec16'), + ('Classical PCA Theorem', 2, None, '___sec17'), + ('Prof of the PCA Theorem', 2, None, '___sec18'), + ('Getting started with PCA', 2, None, '___sec19'), + ('Principal Component Analysis', 2, None, '___sec20'), + ('PCA and scikit-learn', 2, None, '___sec21'), + ('More on the PCA', 2, None, '___sec22'), + ('Incremental PCA', 2, None, '___sec23'), + ('Randomized PCA', 2, None, '___sec24'), + ('Kernel PCA', 2, None, '___sec25'), + ('LLE', 2, None, '___sec26'), + ('Other techniques', 2, None, '___sec27')]} end of tocinfo --> @@ -152,17 +153,18 @@ MathJax.Hub.Config({
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    • @@ -221,7 +223,7 @@ MathJax.Hub.Config({
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    • @@ -221,7 +223,7 @@ data.
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    • @@ -220,7 +222,7 @@ ensures that all features are exactly between \( 0 \) and \( 1 \). The
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    • @@ -223,7 +225,7 @@ techniques.
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    • @@ -298,7 +300,7 @@ svm.fit(X_train_scaled, y_train)
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    • @@ -248,7 +250,7 @@ svm.fit(X_train_scaled, y_train)
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  • PCA and scikit-learn
    • +
  • More on the PCA
    • +
  • Incremental PCA
    • +
  • Randomized PCA
    • +
  • Kernel PCA
    • +
  • LLE
    • +
  • Other techniques
    • @@ -282,7 +284,7 @@ applications.
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    • diff --git a/doc/pub/DimRed/html/._DimRed-bs008.html b/doc/pub/DimRed/html/._DimRed-bs008.html index beaf953d2..29b273d42 100644 --- a/doc/pub/DimRed/html/._DimRed-bs008.html +++ b/doc/pub/DimRed/html/._DimRed-bs008.html @@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source None, '___sec14'), ('Towards the PCA theorem', 2, None, '___sec15'), - ('Classical PCA Theorem', 2, None, '___sec16'), - ('Prof of the PCA Theorem', 2, None, '___sec17'), - ('Getting started with PCA', 2, None, '___sec18'), - ('Principal Component Analysis', 2, None, '___sec19'), - ('PCA and scikit-learn', 2, None, '___sec20'), - ('More on the PCA', 2, None, '___sec21'), - ('Incremental PCA', 2, None, '___sec22'), - ('Randomized PCA', 2, None, '___sec23'), - ('Kernel PCA', 2, None, '___sec24'), - ('LLE', 2, None, '___sec25'), - ('Other techniques', 2, None, '___sec26')]} + ('The Algorithm before the Theorem', 2, None, '___sec16'), + ('Classical PCA Theorem', 2, None, '___sec17'), + ('Prof of the PCA Theorem', 2, None, '___sec18'), + ('Getting started with PCA', 2, None, '___sec19'), + ('Principal Component Analysis', 2, None, '___sec20'), + ('PCA and scikit-learn', 2, None, '___sec21'), + ('More on the PCA', 2, None, '___sec22'), + ('Incremental PCA', 2, None, '___sec23'), + ('Randomized PCA', 2, None, '___sec24'), + ('Kernel PCA', 2, None, '___sec25'), + ('LLE', 2, None, '___sec26'), + ('Other techniques', 2, None, '___sec27')]} end of tocinfo --> @@ -152,17 +153,18 @@ MathJax.Hub.Config({
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  • Rewriting the Covariance and/or Correlation Matrix
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  • Getting started with PCA
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  • Principal Component Analysis
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  • PCA and scikit-learn
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  • Incremental PCA
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  • Randomized PCA
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  • Kernel PCA
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    • +
  • Incremental PCA
    • +
  • Randomized PCA
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  • Other techniques
    • @@ -212,7 +214,7 @@ We have a data set defined by a design/feature matrix \( \boldsymbol{X} \) (see
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    • diff --git a/doc/pub/DimRed/html/._DimRed-bs009.html b/doc/pub/DimRed/html/._DimRed-bs009.html index 1c5a84437..0633ff21e 100644 --- a/doc/pub/DimRed/html/._DimRed-bs009.html +++ b/doc/pub/DimRed/html/._DimRed-bs009.html @@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source None, '___sec14'), ('Towards the PCA theorem', 2, None, '___sec15'), - ('Classical PCA Theorem', 2, None, '___sec16'), - ('Prof of the PCA Theorem', 2, None, '___sec17'), - ('Getting started with PCA', 2, None, '___sec18'), - ('Principal Component Analysis', 2, None, '___sec19'), - ('PCA and scikit-learn', 2, None, '___sec20'), - ('More on the PCA', 2, None, '___sec21'), - ('Incremental PCA', 2, None, '___sec22'), - ('Randomized PCA', 2, None, '___sec23'), - ('Kernel PCA', 2, None, '___sec24'), - ('LLE', 2, None, '___sec25'), - ('Other techniques', 2, None, '___sec26')]} + ('The Algorithm before the Theorem', 2, None, '___sec16'), + ('Classical PCA Theorem', 2, None, '___sec17'), + ('Prof of the PCA Theorem', 2, None, '___sec18'), + ('Getting started with PCA', 2, None, '___sec19'), + ('Principal Component Analysis', 2, None, '___sec20'), + ('PCA and scikit-learn', 2, None, '___sec21'), + ('More on the PCA', 2, None, '___sec22'), + ('Incremental PCA', 2, None, '___sec23'), + ('Randomized PCA', 2, None, '___sec24'), + ('Kernel PCA', 2, None, '___sec25'), + ('LLE', 2, None, '___sec26'), + ('Other techniques', 2, None, '___sec27')]} end of tocinfo --> @@ -152,17 +153,18 @@ MathJax.Hub.Config({
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  • Getting started with PCA
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  • PCA and scikit-learn
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  • Incremental PCA
    • +
  • Randomized PCA
    • +
  • Kernel PCA
    • +
  • LLE
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  • Other techniques
    • @@ -261,7 +263,7 @@ In the above example this is the function we constructed using pandas.
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    • diff --git a/doc/pub/DimRed/html/._DimRed-bs010.html b/doc/pub/DimRed/html/._DimRed-bs010.html index e04ad4053..d35499f33 100644 --- a/doc/pub/DimRed/html/._DimRed-bs010.html +++ b/doc/pub/DimRed/html/._DimRed-bs010.html @@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source None, '___sec14'), ('Towards the PCA theorem', 2, None, '___sec15'), - ('Classical PCA Theorem', 2, None, '___sec16'), - ('Prof of the PCA Theorem', 2, None, '___sec17'), - ('Getting started with PCA', 2, None, '___sec18'), - ('Principal Component Analysis', 2, None, '___sec19'), - ('PCA and scikit-learn', 2, None, '___sec20'), - ('More on the PCA', 2, None, '___sec21'), - ('Incremental PCA', 2, None, '___sec22'), - ('Randomized PCA', 2, None, '___sec23'), - ('Kernel PCA', 2, None, '___sec24'), - ('LLE', 2, None, '___sec25'), - ('Other techniques', 2, None, '___sec26')]} + ('The Algorithm before the Theorem', 2, None, '___sec16'), + ('Classical PCA Theorem', 2, None, '___sec17'), + ('Prof of the PCA Theorem', 2, None, '___sec18'), + ('Getting started with PCA', 2, None, '___sec19'), + ('Principal Component Analysis', 2, None, '___sec20'), + ('PCA and scikit-learn', 2, None, '___sec21'), + ('More on the PCA', 2, None, '___sec22'), + ('Incremental PCA', 2, None, '___sec23'), + ('Randomized PCA', 2, None, '___sec24'), + ('Kernel PCA', 2, None, '___sec25'), + ('LLE', 2, None, '___sec26'), + ('Other techniques', 2, None, '___sec27')]} end of tocinfo --> @@ -152,17 +153,18 @@ MathJax.Hub.Config({
  • Correlation Matrix with Pandas and the Franke function
  • Rewriting the Covariance and/or Correlation Matrix
  • Towards the PCA theorem
    • -
  • Classical PCA Theorem
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  • Prof of the PCA Theorem
    • -
  • Getting started with PCA
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  • Principal Component Analysis
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  • PCA and scikit-learn
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  • Incremental PCA
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  • Randomized PCA
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  • Kernel PCA
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    • +
  • Incremental PCA
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  • Randomized PCA
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  • Kernel PCA
    • +
  • LLE
    • +
  • Other techniques
    • @@ -256,7 +258,7 @@ $$
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    • diff --git a/doc/pub/DimRed/html/._DimRed-bs011.html b/doc/pub/DimRed/html/._DimRed-bs011.html index 726ca148a..651f77edc 100644 --- a/doc/pub/DimRed/html/._DimRed-bs011.html +++ b/doc/pub/DimRed/html/._DimRed-bs011.html @@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source None, '___sec14'), ('Towards the PCA theorem', 2, None, '___sec15'), - ('Classical PCA Theorem', 2, None, '___sec16'), - ('Prof of the PCA Theorem', 2, None, '___sec17'), - ('Getting started with PCA', 2, None, '___sec18'), - ('Principal Component Analysis', 2, None, '___sec19'), - ('PCA and scikit-learn', 2, None, '___sec20'), - ('More on the PCA', 2, None, '___sec21'), - ('Incremental PCA', 2, None, '___sec22'), - ('Randomized PCA', 2, None, '___sec23'), - ('Kernel PCA', 2, None, '___sec24'), - ('LLE', 2, None, '___sec25'), - ('Other techniques', 2, None, '___sec26')]} + ('The Algorithm before the Theorem', 2, None, '___sec16'), + ('Classical PCA Theorem', 2, None, '___sec17'), + ('Prof of the PCA Theorem', 2, None, '___sec18'), + ('Getting started with PCA', 2, None, '___sec19'), + ('Principal Component Analysis', 2, None, '___sec20'), + ('PCA and scikit-learn', 2, None, '___sec21'), + ('More on the PCA', 2, None, '___sec22'), + ('Incremental PCA', 2, None, '___sec23'), + ('Randomized PCA', 2, None, '___sec24'), + ('Kernel PCA', 2, None, '___sec25'), + ('LLE', 2, None, '___sec26'), + ('Other techniques', 2, None, '___sec27')]} end of tocinfo --> @@ -152,17 +153,18 @@ MathJax.Hub.Config({
  • Correlation Matrix with Pandas and the Franke function
  • Rewriting the Covariance and/or Correlation Matrix
  • Towards the PCA theorem
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  • Getting started with PCA
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  • Principal Component Analysis
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  • PCA and scikit-learn
    • -
  • More on the PCA
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  • Incremental PCA
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  • Randomized PCA
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  • Kernel PCA
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  • LLE
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    • +
  • Incremental PCA
    • +
  • Randomized PCA
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  • Kernel PCA
    • +
  • LLE
    • +
  • Other techniques
    • @@ -244,7 +246,7 @@ C = np.c
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    • diff --git a/doc/pub/DimRed/html/._DimRed-bs012.html b/doc/pub/DimRed/html/._DimRed-bs012.html index e500f7540..8a75b27c5 100644 --- a/doc/pub/DimRed/html/._DimRed-bs012.html +++ b/doc/pub/DimRed/html/._DimRed-bs012.html @@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source None, '___sec14'), ('Towards the PCA theorem', 2, None, '___sec15'), - ('Classical PCA Theorem', 2, None, '___sec16'), - ('Prof of the PCA Theorem', 2, None, '___sec17'), - ('Getting started with PCA', 2, None, '___sec18'), - ('Principal Component Analysis', 2, None, '___sec19'), - ('PCA and scikit-learn', 2, None, '___sec20'), - ('More on the PCA', 2, None, '___sec21'), - ('Incremental PCA', 2, None, '___sec22'), - ('Randomized PCA', 2, None, '___sec23'), - ('Kernel PCA', 2, None, '___sec24'), - ('LLE', 2, None, '___sec25'), - ('Other techniques', 2, None, '___sec26')]} + ('The Algorithm before the Theorem', 2, None, '___sec16'), + ('Classical PCA Theorem', 2, None, '___sec17'), + ('Prof of the PCA Theorem', 2, None, '___sec18'), + ('Getting started with PCA', 2, None, '___sec19'), + ('Principal Component Analysis', 2, None, '___sec20'), + ('PCA and scikit-learn', 2, None, '___sec21'), + ('More on the PCA', 2, None, '___sec22'), + ('Incremental PCA', 2, None, '___sec23'), + ('Randomized PCA', 2, None, '___sec24'), + ('Kernel PCA', 2, None, '___sec25'), + ('LLE', 2, None, '___sec26'), + ('Other techniques', 2, None, '___sec27')]} end of tocinfo --> @@ -152,17 +153,18 @@ MathJax.Hub.Config({
  • Correlation Matrix with Pandas and the Franke function
  • Rewriting the Covariance and/or Correlation Matrix
  • Towards the PCA theorem
    • -
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    • -
  • Getting started with PCA
    • -
  • Principal Component Analysis
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  • PCA and scikit-learn
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  • More on the PCA
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  • Incremental PCA
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  • Randomized PCA
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  • Kernel PCA
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  • Incremental PCA
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  • Randomized PCA
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  • Kernel PCA
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  • LLE
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  • Other techniques
    • @@ -246,7 +248,7 @@ The above procedure with numpy can be made more compact if we use pand
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    • diff --git a/doc/pub/DimRed/html/._DimRed-bs013.html b/doc/pub/DimRed/html/._DimRed-bs013.html index af63c221c..aadce9ddc 100644 --- a/doc/pub/DimRed/html/._DimRed-bs013.html +++ b/doc/pub/DimRed/html/._DimRed-bs013.html @@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source None, '___sec14'), ('Towards the PCA theorem', 2, None, '___sec15'), - ('Classical PCA Theorem', 2, None, '___sec16'), - ('Prof of the PCA Theorem', 2, None, '___sec17'), - ('Getting started with PCA', 2, None, '___sec18'), - ('Principal Component Analysis', 2, None, '___sec19'), - ('PCA and scikit-learn', 2, None, '___sec20'), - ('More on the PCA', 2, None, '___sec21'), - ('Incremental PCA', 2, None, '___sec22'), - ('Randomized PCA', 2, None, '___sec23'), - ('Kernel PCA', 2, None, '___sec24'), - ('LLE', 2, None, '___sec25'), - ('Other techniques', 2, None, '___sec26')]} + ('The Algorithm before the Theorem', 2, None, '___sec16'), + ('Classical PCA Theorem', 2, None, '___sec17'), + ('Prof of the PCA Theorem', 2, None, '___sec18'), + ('Getting started with PCA', 2, None, '___sec19'), + ('Principal Component Analysis', 2, None, '___sec20'), + ('PCA and scikit-learn', 2, None, '___sec21'), + ('More on the PCA', 2, None, '___sec22'), + ('Incremental PCA', 2, None, '___sec23'), + ('Randomized PCA', 2, None, '___sec24'), + ('Kernel PCA', 2, None, '___sec25'), + ('LLE', 2, None, '___sec26'), + ('Other techniques', 2, None, '___sec27')]} end of tocinfo --> @@ -152,17 +153,18 @@ MathJax.Hub.Config({
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  • Towards the PCA theorem
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  • Getting started with PCA
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  • PCA and scikit-learn
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  • Randomized PCA
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  • LLE
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  • Other techniques
    • @@ -228,7 +230,7 @@ We expand this model to the Franke function discussed above.
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    • diff --git a/doc/pub/DimRed/html/._DimRed-bs014.html b/doc/pub/DimRed/html/._DimRed-bs014.html index 14792183d..847ed1de2 100644 --- a/doc/pub/DimRed/html/._DimRed-bs014.html +++ b/doc/pub/DimRed/html/._DimRed-bs014.html @@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source None, '___sec14'), ('Towards the PCA theorem', 2, None, '___sec15'), - ('Classical PCA Theorem', 2, None, '___sec16'), - ('Prof of the PCA Theorem', 2, None, '___sec17'), - ('Getting started with PCA', 2, None, '___sec18'), - ('Principal Component Analysis', 2, None, '___sec19'), - ('PCA and scikit-learn', 2, None, '___sec20'), - ('More on the PCA', 2, None, '___sec21'), - ('Incremental PCA', 2, None, '___sec22'), - ('Randomized PCA', 2, None, '___sec23'), - ('Kernel PCA', 2, None, '___sec24'), - ('LLE', 2, None, '___sec25'), - ('Other techniques', 2, None, '___sec26')]} + ('The Algorithm before the Theorem', 2, None, '___sec16'), + ('Classical PCA Theorem', 2, None, '___sec17'), + ('Prof of the PCA Theorem', 2, None, '___sec18'), + ('Getting started with PCA', 2, None, '___sec19'), + ('Principal Component Analysis', 2, None, '___sec20'), + ('PCA and scikit-learn', 2, None, '___sec21'), + ('More on the PCA', 2, None, '___sec22'), + ('Incremental PCA', 2, None, '___sec23'), + ('Randomized PCA', 2, None, '___sec24'), + ('Kernel PCA', 2, None, '___sec25'), + ('LLE', 2, None, '___sec26'), + ('Other techniques', 2, None, '___sec27')]} end of tocinfo --> @@ -152,17 +153,18 @@ MathJax.Hub.Config({
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  • Getting started with PCA
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  • Incremental PCA
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    • @@ -264,7 +266,7 @@ matrix.
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    • diff --git a/doc/pub/DimRed/html/._DimRed-bs015.html b/doc/pub/DimRed/html/._DimRed-bs015.html index 39697340b..5a51c64a1 100644 --- a/doc/pub/DimRed/html/._DimRed-bs015.html +++ b/doc/pub/DimRed/html/._DimRed-bs015.html @@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source None, '___sec14'), ('Towards the PCA theorem', 2, None, '___sec15'), - ('Classical PCA Theorem', 2, None, '___sec16'), - ('Prof of the PCA Theorem', 2, None, '___sec17'), - ('Getting started with PCA', 2, None, '___sec18'), - ('Principal Component Analysis', 2, None, '___sec19'), - ('PCA and scikit-learn', 2, None, '___sec20'), - ('More on the PCA', 2, None, '___sec21'), - ('Incremental PCA', 2, None, '___sec22'), - ('Randomized PCA', 2, None, '___sec23'), - ('Kernel PCA', 2, None, '___sec24'), - ('LLE', 2, None, '___sec25'), - ('Other techniques', 2, None, '___sec26')]} + ('The Algorithm before the Theorem', 2, None, '___sec16'), + ('Classical PCA Theorem', 2, None, '___sec17'), + ('Prof of the PCA Theorem', 2, None, '___sec18'), + ('Getting started with PCA', 2, None, '___sec19'), + ('Principal Component Analysis', 2, None, '___sec20'), + ('PCA and scikit-learn', 2, None, '___sec21'), + ('More on the PCA', 2, None, '___sec22'), + ('Incremental PCA', 2, None, '___sec23'), + ('Randomized PCA', 2, None, '___sec24'), + ('Kernel PCA', 2, None, '___sec25'), + ('LLE', 2, None, '___sec26'), + ('Other techniques', 2, None, '___sec27')]} end of tocinfo --> @@ -152,17 +153,18 @@ MathJax.Hub.Config({
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    • @@ -244,7 +246,7 @@ It is easy to generalize this to a matrix \( \boldsymbol{X}\in {\mathbb{R}}^{n\t
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  • PCA and scikit-learn
    • -
  • More on the PCA
    • -
  • Incremental PCA
    • -
  • Randomized PCA
    • -
  • Kernel PCA
    • -
  • LLE
    • -
  • Other techniques
    • +
  • The Algorithm before the Theorem
    • +
  • Classical PCA Theorem
    • +
  • Prof of the PCA Theorem
    • +
  • Getting started with PCA
    • +
  • Principal Component Analysis
    • +
  • PCA and scikit-learn
    • +
  • More on the PCA
    • +
  • Incremental PCA
    • +
  • Randomized PCA
    • +
  • Kernel PCA
    • +
  • LLE
    • +
  • Other techniques
    • @@ -213,6 +215,19 @@ $$ In the derivation of the PCA theorem we will assume that the eigenvalues are ordered in descending order, that is \( \lambda_0 > \lambda_1 > \dots > \lambda_{p-1} \). +

    +The eigenvalues tell us then how much we need to stretch the +corresponding eigenvectors. Dimensions with large eigenvalues have +thus large variations (large variance) and define therefore useful +dimensions. The data points are more spread out in the direction of +these eigenvectors. Smaller eigenvalues mean on the other hand that +the corresponding eigenvectors are shrunk accordingly and the data +points are tightly bunched together and there is not much variation in +these specific directions. Hopefully then we could leave it out +dimensions where the eigenvalues are very small. If \( p \) is very large, +we could then aim at reducing \( p \) to \( l < < p \) and handle only \( l \) +features/predictors. +

    @@ -239,7 +254,7 @@ In the derivation of the PCA theorem we will assume that the eigenvalues are ord

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    • -
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    • diff --git a/doc/pub/DimRed/html/._DimRed-bs017.html b/doc/pub/DimRed/html/._DimRed-bs017.html index 91740f7d8..91074e516 100644 --- a/doc/pub/DimRed/html/._DimRed-bs017.html +++ b/doc/pub/DimRed/html/._DimRed-bs017.html @@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source None, '___sec14'), ('Towards the PCA theorem', 2, None, '___sec15'), - ('Classical PCA Theorem', 2, None, '___sec16'), - ('Prof of the PCA Theorem', 2, None, '___sec17'), - ('Getting started with PCA', 2, None, '___sec18'), - ('Principal Component Analysis', 2, None, '___sec19'), - ('PCA and scikit-learn', 2, None, '___sec20'), - ('More on the PCA', 2, None, '___sec21'), - ('Incremental PCA', 2, None, '___sec22'), - ('Randomized PCA', 2, None, '___sec23'), - ('Kernel PCA', 2, None, '___sec24'), - ('LLE', 2, None, '___sec25'), - ('Other techniques', 2, None, '___sec26')]} + ('The Algorithm before the Theorem', 2, None, '___sec16'), + ('Classical PCA Theorem', 2, None, '___sec17'), + ('Prof of the PCA Theorem', 2, None, '___sec18'), + ('Getting started with PCA', 2, None, '___sec19'), + ('Principal Component Analysis', 2, None, '___sec20'), + ('PCA and scikit-learn', 2, None, '___sec21'), + ('More on the PCA', 2, None, '___sec22'), + ('Incremental PCA', 2, None, '___sec23'), + ('Randomized PCA', 2, None, '___sec24'), + ('Kernel PCA', 2, None, '___sec25'), + ('LLE', 2, None, '___sec26'), + ('Other techniques', 2, None, '___sec27')]} end of tocinfo --> @@ -152,17 +153,18 @@ MathJax.Hub.Config({
  • Correlation Matrix with Pandas and the Franke function
  • Rewriting the Covariance and/or Correlation Matrix
  • Towards the PCA theorem
    • -
  • Classical PCA Theorem
    • -
  • Prof of the PCA Theorem
    • -
  • Getting started with PCA
    • -
  • Principal Component Analysis
    • -
  • PCA and scikit-learn
    • -
  • More on the PCA
    • -
  • Incremental PCA
    • -
  • Randomized PCA
    • -
  • Kernel PCA
    • -
  • LLE
    • -
  • Other techniques
    • +
  • The Algorithm before the Theorem
    • +
  • Classical PCA Theorem
    • +
  • Prof of the PCA Theorem
    • +
  • Getting started with PCA
    • +
  • Principal Component Analysis
    • +
  • PCA and scikit-learn
    • +
  • More on the PCA
    • +
  • Incremental PCA
    • +
  • Randomized PCA
    • +
  • Kernel PCA
    • +
  • LLE
    • +
  • Other techniques
    • @@ -178,7 +180,36 @@ MathJax.Hub.Config({ -

    Classical PCA Theorem

    +

    The Algorithm before theorem

    + +

    +Here's how we would proceed in setting up the algorithm for the PCA, see also discussion below here. + +

    + +$$ +\boldsymbol{X}=\begin{bmatrix} +x_{0,0} & x_{0,1} & x_{0,2}& \dots & \dots x_{0,p-1}\\ +x_{1,0} & x_{1,1} & x_{1,2}& \dots & \dots x_{1,p-1}\\ +x_{2,0} & x_{2,1} & x_{2,2}& \dots & \dots x_{2,p-1}\\ +\dots & \dots & \dots & \dots \dots & \dots \\ +x_{n-2,0} & x_{n-2,1} & x_{n-2,2}& \dots & \dots x_{n-2,p-1}\\ +x_{n-1,0} & x_{n-1,1} & x_{n-1,2}& \dots & \dots x_{n-1,p-1}\\ +\end{bmatrix}, +$$ + + + + +After this we ask ourselves how do we prove the link between the maximum variance and the feature reduction.

    @@ -206,7 +237,7 @@ MathJax.Hub.Config({

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    • -
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    • diff --git a/doc/pub/DimRed/html/._DimRed-bs018.html b/doc/pub/DimRed/html/._DimRed-bs018.html index ee77ea4d8..0e27b1a8b 100644 --- a/doc/pub/DimRed/html/._DimRed-bs018.html +++ b/doc/pub/DimRed/html/._DimRed-bs018.html @@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source None, '___sec14'), ('Towards the PCA theorem', 2, None, '___sec15'), - ('Classical PCA Theorem', 2, None, '___sec16'), - ('Prof of the PCA Theorem', 2, None, '___sec17'), - ('Getting started with PCA', 2, None, '___sec18'), - ('Principal Component Analysis', 2, None, '___sec19'), - ('PCA and scikit-learn', 2, None, '___sec20'), - ('More on the PCA', 2, None, '___sec21'), - ('Incremental PCA', 2, None, '___sec22'), - ('Randomized PCA', 2, None, '___sec23'), - ('Kernel PCA', 2, None, '___sec24'), - ('LLE', 2, None, '___sec25'), - ('Other techniques', 2, None, '___sec26')]} + ('The Algorithm before the Theorem', 2, None, '___sec16'), + ('Classical PCA Theorem', 2, None, '___sec17'), + ('Prof of the PCA Theorem', 2, None, '___sec18'), + ('Getting started with PCA', 2, None, '___sec19'), + ('Principal Component Analysis', 2, None, '___sec20'), + ('PCA and scikit-learn', 2, None, '___sec21'), + ('More on the PCA', 2, None, '___sec22'), + ('Incremental PCA', 2, None, '___sec23'), + ('Randomized PCA', 2, None, '___sec24'), + ('Kernel PCA', 2, None, '___sec25'), + ('LLE', 2, None, '___sec26'), + ('Other techniques', 2, None, '___sec27')]} end of tocinfo --> @@ -152,17 +153,18 @@ MathJax.Hub.Config({
  • Correlation Matrix with Pandas and the Franke function
  • Rewriting the Covariance and/or Correlation Matrix
  • Towards the PCA theorem
    • -
  • Classical PCA Theorem
    • -
  • Prof of the PCA Theorem
    • -
  • Getting started with PCA
    • -
  • Principal Component Analysis
    • -
  • PCA and scikit-learn
    • -
  • More on the PCA
    • -
  • Incremental PCA
    • -
  • Randomized PCA
    • -
  • Kernel PCA
    • -
  • LLE
    • -
  • Other techniques
    • +
  • The Algorithm before the Theorem
    • +
  • Classical PCA Theorem
    • +
  • Prof of the PCA Theorem
    • +
  • Getting started with PCA
    • +
  • Principal Component Analysis
    • +
  • PCA and scikit-learn
    • +
  • More on the PCA
    • +
  • Incremental PCA
    • +
  • Randomized PCA
    • +
  • Kernel PCA
    • +
  • LLE
    • +
  • Other techniques
    • @@ -178,7 +180,7 @@ MathJax.Hub.Config({ -

    Prof of the PCA Theorem

    +

    Classical PCA Theorem

    @@ -205,6 +207,8 @@ MathJax.Hub.Config({

  • 26
  • 27
  • 28
    • +
  • ...
    • +
  • 29
  • »
    • diff --git a/doc/pub/DimRed/html/._DimRed-bs019.html b/doc/pub/DimRed/html/._DimRed-bs019.html index 30d2d97ef..57a96ad5b 100644 --- a/doc/pub/DimRed/html/._DimRed-bs019.html +++ b/doc/pub/DimRed/html/._DimRed-bs019.html @@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source None, '___sec14'), ('Towards the PCA theorem', 2, None, '___sec15'), - ('Classical PCA Theorem', 2, None, '___sec16'), - ('Prof of the PCA Theorem', 2, None, '___sec17'), - ('Getting started with PCA', 2, None, '___sec18'), - ('Principal Component Analysis', 2, None, '___sec19'), - ('PCA and scikit-learn', 2, None, '___sec20'), - ('More on the PCA', 2, None, '___sec21'), - ('Incremental PCA', 2, None, '___sec22'), - ('Randomized PCA', 2, None, '___sec23'), - ('Kernel PCA', 2, None, '___sec24'), - ('LLE', 2, None, '___sec25'), - ('Other techniques', 2, None, '___sec26')]} + ('The Algorithm before the Theorem', 2, None, '___sec16'), + ('Classical PCA Theorem', 2, None, '___sec17'), + ('Prof of the PCA Theorem', 2, None, '___sec18'), + ('Getting started with PCA', 2, None, '___sec19'), + ('Principal Component Analysis', 2, None, '___sec20'), + ('PCA and scikit-learn', 2, None, '___sec21'), + ('More on the PCA', 2, None, '___sec22'), + ('Incremental PCA', 2, None, '___sec23'), + ('Randomized PCA', 2, None, '___sec24'), + ('Kernel PCA', 2, None, '___sec25'), + ('LLE', 2, None, '___sec26'), + ('Other techniques', 2, None, '___sec27')]} end of tocinfo --> @@ -152,17 +153,18 @@ MathJax.Hub.Config({
  • Correlation Matrix with Pandas and the Franke function
  • Rewriting the Covariance and/or Correlation Matrix
  • Towards the PCA theorem
    • -
  • Classical PCA Theorem
    • -
  • Prof of the PCA Theorem
    • -
  • Getting started with PCA
    • -
  • Principal Component Analysis
    • -
  • PCA and scikit-learn
    • -
  • More on the PCA
    • -
  • Incremental PCA
    • -
  • Randomized PCA
    • -
  • Kernel PCA
    • -
  • LLE
    • -
  • Other techniques
    • +
  • The Algorithm before the Theorem
    • +
  • Classical PCA Theorem
    • +
  • Prof of the PCA Theorem
    • +
  • Getting started with PCA
    • +
  • Principal Component Analysis
    • +
  • PCA and scikit-learn
    • +
  • More on the PCA
    • +
  • Incremental PCA
    • +
  • Randomized PCA
    • +
  • Kernel PCA
    • +
  • LLE
    • +
  • Other techniques
    • @@ -178,18 +180,8 @@ MathJax.Hub.Config({ -

    Getting started with PCA

    +

    Prof of the PCA Theorem

    -

    - - -

    # Now add PCA
-from sklearn.decomposition import PCA
-pca = PCA(n_components = 2)
-pca.fit(X_train_scaled)
-
-X_pca = pca.transform(X_train_scaled)
-

    @@ -214,6 +206,7 @@ X_pca = pca.26

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    • +
  • 29
  • »
    • diff --git a/doc/pub/DimRed/html/._DimRed-bs020.html b/doc/pub/DimRed/html/._DimRed-bs020.html index e5413a86c..5c47f1973 100644 --- a/doc/pub/DimRed/html/._DimRed-bs020.html +++ b/doc/pub/DimRed/html/._DimRed-bs020.html @@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source None, '___sec14'), ('Towards the PCA theorem', 2, None, '___sec15'), - ('Classical PCA Theorem', 2, None, '___sec16'), - ('Prof of the PCA Theorem', 2, None, '___sec17'), - ('Getting started with PCA', 2, None, '___sec18'), - ('Principal Component Analysis', 2, None, '___sec19'), - ('PCA and scikit-learn', 2, None, '___sec20'), - ('More on the PCA', 2, None, '___sec21'), - ('Incremental PCA', 2, None, '___sec22'), - ('Randomized PCA', 2, None, '___sec23'), - ('Kernel PCA', 2, None, '___sec24'), - ('LLE', 2, None, '___sec25'), - ('Other techniques', 2, None, '___sec26')]} + ('The Algorithm before the Theorem', 2, None, '___sec16'), + ('Classical PCA Theorem', 2, None, '___sec17'), + ('Prof of the PCA Theorem', 2, None, '___sec18'), + ('Getting started with PCA', 2, None, '___sec19'), + ('Principal Component Analysis', 2, None, '___sec20'), + ('PCA and scikit-learn', 2, None, '___sec21'), + ('More on the PCA', 2, None, '___sec22'), + ('Incremental PCA', 2, None, '___sec23'), + ('Randomized PCA', 2, None, '___sec24'), + ('Kernel PCA', 2, None, '___sec25'), + ('LLE', 2, None, '___sec26'), + ('Other techniques', 2, None, '___sec27')]} end of tocinfo --> @@ -152,17 +153,18 @@ MathJax.Hub.Config({
  • Correlation Matrix with Pandas and the Franke function
  • Rewriting the Covariance and/or Correlation Matrix
  • Towards the PCA theorem
    • -
  • Classical PCA Theorem
    • -
  • Prof of the PCA Theorem
    • -
  • Getting started with PCA
    • -
  • Principal Component Analysis
    • -
  • PCA and scikit-learn
    • -
  • More on the PCA
    • -
  • Incremental PCA
    • -
  • Randomized PCA
    • -
  • Kernel PCA
    • -
  • LLE
    • -
  • Other techniques
    • +
  • The Algorithm before the Theorem
    • +
  • Classical PCA Theorem
    • +
  • Prof of the PCA Theorem
    • +
  • Getting started with PCA
    • +
  • Principal Component Analysis
    • +
  • PCA and scikit-learn
    • +
  • More on the PCA
    • +
  • Incremental PCA
    • +
  • Randomized PCA
    • +
  • Kernel PCA
    • +
  • LLE
    • +
  • Other techniques
    • @@ -178,38 +180,17 @@ MathJax.Hub.Config({ -

    Principal Component Analysis

    -
    -
    -

    -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. +

    Getting started with PCA

    -

    -The following Python code uses NumPy’s svd() function to obtain all the principal components of the -training set, then extracts the first two principal components

    -

    X_centered = X - X.mean(axis=0)
-U, s, V = np.linalg.svd(X_centered)
-c1 = V.T[:, 0]
-c2 = V.T[:, 1]
-
    -

    -PCA assumes that the dataset is centered around the origin. Scikit-Learn’s PCA classes take care of centering -the data for you. However, if you implement PCA yourself (as in the preceding example), or if you use other libraries, don’t -forget to center the data first. +

    # Now add PCA
+from sklearn.decomposition import PCA
+pca = PCA(n_components = 2)
+pca.fit(X_train_scaled)
 
-
    
-Once you have identified all the principal components, you can reduce the dimensionality of the dataset
-down to \( d \) dimensions by projecting it onto the hyperplane defined by the first \( d \) principal components.
-Selecting this hyperplane ensures that the projection will preserve as much variance as possible. 
-
    
-
-
-
    W2 = V.T[:, :2]
-X2D = X_centered.dot(W2)
+X_pca = pca.transform(X_train_scaled)
 
    
 
    
 
    
@@ -234,6 +215,7 @@ X2D = X_centered26
   
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+  
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    • 
 
 
diff --git a/doc/pub/DimRed/html/._DimRed-bs021.html b/doc/pub/DimRed/html/._DimRed-bs021.html
index 01c56dc53..ede6d1c65 100644
--- a/doc/pub/DimRed/html/._DimRed-bs021.html
+++ b/doc/pub/DimRed/html/._DimRed-bs021.html
@@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source
                None,
                '___sec14'),
               ('Towards the PCA theorem', 2, None, '___sec15'),
-              ('Classical PCA Theorem', 2, None, '___sec16'),
-              ('Prof of the PCA Theorem', 2, None, '___sec17'),
-              ('Getting started with PCA', 2, None, '___sec18'),
-              ('Principal Component Analysis', 2, None, '___sec19'),
-              ('PCA and scikit-learn', 2, None, '___sec20'),
-              ('More on the PCA', 2, None, '___sec21'),
-              ('Incremental PCA', 2, None, '___sec22'),
-              ('Randomized PCA', 2, None, '___sec23'),
-              ('Kernel PCA', 2, None, '___sec24'),
-              ('LLE', 2, None, '___sec25'),
-              ('Other techniques', 2, None, '___sec26')]}
+              ('The Algorithm before the Theorem', 2, None, '___sec16'),
+              ('Classical PCA Theorem', 2, None, '___sec17'),
+              ('Prof of the PCA Theorem', 2, None, '___sec18'),
+              ('Getting started with PCA', 2, None, '___sec19'),
+              ('Principal Component Analysis', 2, None, '___sec20'),
+              ('PCA and scikit-learn', 2, None, '___sec21'),
+              ('More on the PCA', 2, None, '___sec22'),
+              ('Incremental PCA', 2, None, '___sec23'),
+              ('Randomized PCA', 2, None, '___sec24'),
+              ('Kernel PCA', 2, None, '___sec25'),
+              ('LLE', 2, None, '___sec26'),
+              ('Other techniques', 2, None, '___sec27')]}
 end of tocinfo -->
 
 
@@ -152,17 +153,18 @@ MathJax.Hub.Config({
       
  • Correlation Matrix with Pandas and the Franke function
    • 
       
  • Rewriting the Covariance and/or Correlation Matrix
    • 
       
  • Towards the PCA theorem
    • 
-      
  • Classical PCA Theorem
    • 
-      
  • Prof of the PCA Theorem
    • 
-      
  • Getting started with PCA
    • 
-      
  • Principal Component Analysis
    • 
-      
  • PCA and scikit-learn
    • 
-      
  • More on the PCA
    • 
-      
  • Incremental PCA
    • 
-      
  • Randomized PCA
    • 
-      
  • Kernel PCA
    • 
-      
  • LLE
    • 
-      
  • Other techniques
    • 
+      
  • The Algorithm before the Theorem
    • 
+      
  • Classical PCA Theorem
    • 
+      
  • Prof of the PCA Theorem
    • 
+      
  • Getting started with PCA
    • 
+      
  • Principal Component Analysis
    • 
+      
  • PCA and scikit-learn
    • 
+      
  • More on the PCA
    • 
+      
  • Incremental PCA
    • 
+      
  • Randomized PCA
    • 
+      
  • Kernel PCA
    • 
+      
  • LLE
    • 
+      
  • Other techniques
    • 
 
         
       
@@ -176,36 +178,41 @@ MathJax.Hub.Config({
 
     
     
     
     
 
 
-
+
 
-
    PCA and scikit-learn 
    
+
    Principal Component Analysis 
    
+
    
+
    
+
     
+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.
 
 
    
-Scikit-Learn’s PCA class implements PCA using SVD decomposition just like we did before. The
-following code applies PCA to reduce the dimensionality of the dataset down to two dimensions (note
-that it automatically takes care of centering the data):
+The following Python code uses NumPy’s svd() function to obtain all the principal components of the
+training set, then extracts the first two principal components
 
    
 
 
-
    from sklearn.decomposition import PCA
-pca = PCA(n_components = 2)
-X2D = pca.fit_transform(X)
+
    X_centered = X - X.mean(axis=0)
+U, s, V = np.linalg.svd(X_centered)
+c1 = V.T[:, 0]
+c2 = V.T[:, 1]
 
    
 
    
-After fitting the PCA transformer to the dataset, you can access the principal components using the
-components variable (note that it contains the PCs as horizontal vectors, so, for example, the first
-principal component is equal to 
+PCA assumes that the dataset is centered around the origin. Scikit-Learn’s PCA classes take care of centering
+the data for you. However, if you implement PCA yourself (as in the preceding example), or if you use other libraries, don’t
+forget to center the data first.
+
+
    
+Once you have identified all the principal components, you can reduce the dimensionality of the dataset
+down to \( d \) dimensions by projecting it onto the hyperplane defined by the first \( d \) principal components.
+Selecting this hyperplane ensures that the projection will preserve as much variance as possible. 
 
    
 
 
-
    pca.components_.T[:, 0]).
+
    W2 = V.T[:, :2]
+X2D = X_centered.dot(W2)
 
    
-
    
-Another very useful piece of information is the explained variance ratio of each principal component,
-available via the \( explained\_variance\_ratio \) variable. It indicates the proportion of the dataset’s
-variance that lies along the axis of each principal component. 
-More material to come here.
-
 
    
 
    
 
@@ -228,6 +235,7 @@ More material to come here.
   
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    • 
   
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    • 
 
 
diff --git a/doc/pub/DimRed/html/._DimRed-bs022.html b/doc/pub/DimRed/html/._DimRed-bs022.html
index afb778037..2b727e37e 100644
--- a/doc/pub/DimRed/html/._DimRed-bs022.html
+++ b/doc/pub/DimRed/html/._DimRed-bs022.html
@@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source
                None,
                '___sec14'),
               ('Towards the PCA theorem', 2, None, '___sec15'),
-              ('Classical PCA Theorem', 2, None, '___sec16'),
-              ('Prof of the PCA Theorem', 2, None, '___sec17'),
-              ('Getting started with PCA', 2, None, '___sec18'),
-              ('Principal Component Analysis', 2, None, '___sec19'),
-              ('PCA and scikit-learn', 2, None, '___sec20'),
-              ('More on the PCA', 2, None, '___sec21'),
-              ('Incremental PCA', 2, None, '___sec22'),
-              ('Randomized PCA', 2, None, '___sec23'),
-              ('Kernel PCA', 2, None, '___sec24'),
-              ('LLE', 2, None, '___sec25'),
-              ('Other techniques', 2, None, '___sec26')]}
+              ('The Algorithm before the Theorem', 2, None, '___sec16'),
+              ('Classical PCA Theorem', 2, None, '___sec17'),
+              ('Prof of the PCA Theorem', 2, None, '___sec18'),
+              ('Getting started with PCA', 2, None, '___sec19'),
+              ('Principal Component Analysis', 2, None, '___sec20'),
+              ('PCA and scikit-learn', 2, None, '___sec21'),
+              ('More on the PCA', 2, None, '___sec22'),
+              ('Incremental PCA', 2, None, '___sec23'),
+              ('Randomized PCA', 2, None, '___sec24'),
+              ('Kernel PCA', 2, None, '___sec25'),
+              ('LLE', 2, None, '___sec26'),
+              ('Other techniques', 2, None, '___sec27')]}
 end of tocinfo -->
 
 
@@ -152,17 +153,18 @@ MathJax.Hub.Config({
       
  • Correlation Matrix with Pandas and the Franke function
    • 
       
  • Rewriting the Covariance and/or Correlation Matrix
    • 
       
  • Towards the PCA theorem
    • 
-      
  • Classical PCA Theorem
    • 
-      
  • Prof of the PCA Theorem
    • 
-      
  • Getting started with PCA
    • 
-      
  • Principal Component Analysis
    • 
-      
  • PCA and scikit-learn
    • 
-      
  • More on the PCA
    • 
-      
  • Incremental PCA
    • 
-      
  • Randomized PCA
    • 
-      
  • Kernel PCA
    • 
-      
  • LLE
    • 
-      
  • Other techniques
    • 
+      
  • The Algorithm before the Theorem
    • 
+      
  • Classical PCA Theorem
    • 
+      
  • Prof of the PCA Theorem
    • 
+      
  • Getting started with PCA
    • 
+      
  • Principal Component Analysis
    • 
+      
  • PCA and scikit-learn
    • 
+      
  • More on the PCA
    • 
+      
  • Incremental PCA
    • 
+      
  • Randomized PCA
    • 
+      
  • Kernel PCA
    • 
+      
  • LLE
    • 
+      
  • Other techniques
    • 
 
         
       
@@ -176,35 +178,36 @@ MathJax.Hub.Config({
 
     
     
     
     
 
 
-
+
 
-
    More on the PCA 
    
+
    PCA and scikit-learn 
    
 
 
    
-Instead of arbitrarily choosing the number of dimensions to reduce down to, it is generally preferable to
-choose the number of dimensions that add up to a sufficiently large portion of the variance (e.g., 95%).
-Unless, of course, you are reducing dimensionality for data visualization — in that case you will
-generally want to reduce the dimensionality down to 2 or 3.
-The following code computes PCA without reducing dimensionality, then computes the minimum number
-of dimensions required to preserve 95% of the training set’s variance:
+Scikit-Learn’s PCA class implements PCA using SVD decomposition just like we did before. The
+following code applies PCA to reduce the dimensionality of the dataset down to two dimensions (note
+that it automatically takes care of centering the data):
 
    
 
 
-
    pca = PCA()
-pca.fit(X)
-cumsum = np.cumsum(pca.explained_variance_ratio_)
-d = np.argmax(cumsum >= 0.95) + 1
+
    from sklearn.decomposition import PCA
+pca = PCA(n_components = 2)
+X2D = pca.fit_transform(X)
 
    
 
    
-You could then set \( n\_components=d \) and run PCA again. However, there is a much better option: instead
-of specifying the number of principal components you want to preserve, you can set \( n\_components \) to be
-a float between 0.0 and 1.0, indicating the ratio of variance you wish to preserve:
+After fitting the PCA transformer to the dataset, you can access the principal components using the
+components variable (note that it contains the PCs as horizontal vectors, so, for example, the first
+principal component is equal to 
 
    
 
 
-
    pca = PCA(n_components=0.95)
-X_reduced = pca.fit_transform(X)
+
    pca.components_.T[:, 0]).
 
    
+
    
+Another very useful piece of information is the explained variance ratio of each principal component,
+available via the \( explained\_variance\_ratio \) variable. It indicates the proportion of the dataset’s
+variance that lies along the axis of each principal component. 
+More material to come here.
+
 
    
 
    
 
@@ -226,6 +229,7 @@ X_reduced = pca
   
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    • 
   
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    • 
   
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    • 
+  
  • 29
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/DimRed/html/._DimRed-bs023.html b/doc/pub/DimRed/html/._DimRed-bs023.html
index dbe918b73..72d6189cd 100644
--- a/doc/pub/DimRed/html/._DimRed-bs023.html
+++ b/doc/pub/DimRed/html/._DimRed-bs023.html
@@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source
                None,
                '___sec14'),
               ('Towards the PCA theorem', 2, None, '___sec15'),
-              ('Classical PCA Theorem', 2, None, '___sec16'),
-              ('Prof of the PCA Theorem', 2, None, '___sec17'),
-              ('Getting started with PCA', 2, None, '___sec18'),
-              ('Principal Component Analysis', 2, None, '___sec19'),
-              ('PCA and scikit-learn', 2, None, '___sec20'),
-              ('More on the PCA', 2, None, '___sec21'),
-              ('Incremental PCA', 2, None, '___sec22'),
-              ('Randomized PCA', 2, None, '___sec23'),
-              ('Kernel PCA', 2, None, '___sec24'),
-              ('LLE', 2, None, '___sec25'),
-              ('Other techniques', 2, None, '___sec26')]}
+              ('The Algorithm before the Theorem', 2, None, '___sec16'),
+              ('Classical PCA Theorem', 2, None, '___sec17'),
+              ('Prof of the PCA Theorem', 2, None, '___sec18'),
+              ('Getting started with PCA', 2, None, '___sec19'),
+              ('Principal Component Analysis', 2, None, '___sec20'),
+              ('PCA and scikit-learn', 2, None, '___sec21'),
+              ('More on the PCA', 2, None, '___sec22'),
+              ('Incremental PCA', 2, None, '___sec23'),
+              ('Randomized PCA', 2, None, '___sec24'),
+              ('Kernel PCA', 2, None, '___sec25'),
+              ('LLE', 2, None, '___sec26'),
+              ('Other techniques', 2, None, '___sec27')]}
 end of tocinfo -->
 
 
@@ -152,17 +153,18 @@ MathJax.Hub.Config({
       
  • Correlation Matrix with Pandas and the Franke function
    • 
       
  • Rewriting the Covariance and/or Correlation Matrix
    • 
       
  • Towards the PCA theorem
    • 
-      
  • Classical PCA Theorem
    • 
-      
  • Prof of the PCA Theorem
    • 
-      
  • Getting started with PCA
    • 
-      
  • Principal Component Analysis
    • 
-      
  • PCA and scikit-learn
    • 
-      
  • More on the PCA
    • 
-      
  • Incremental PCA
    • 
-      
  • Randomized PCA
    • 
-      
  • Kernel PCA
    • 
-      
  • LLE
    • 
-      
  • Other techniques
    • 
+      
  • The Algorithm before the Theorem
    • 
+      
  • Classical PCA Theorem
    • 
+      
  • Prof of the PCA Theorem
    • 
+      
  • Getting started with PCA
    • 
+      
  • Principal Component Analysis
    • 
+      
  • PCA and scikit-learn
    • 
+      
  • More on the PCA
    • 
+      
  • Incremental PCA
    • 
+      
  • Randomized PCA
    • 
+      
  • Kernel PCA
    • 
+      
  • LLE
    • 
+      
  • Other techniques
    • 
 
         
       
@@ -178,15 +180,33 @@ MathJax.Hub.Config({
 
 
 
-
    Incremental PCA 
    
+
    More on the PCA 
    
 
 
    
-One problem with the preceding implementation of PCA is that it requires the whole training set to fit in
-memory in order for the SVD algorithm to run. Fortunately, Incremental PCA (IPCA) algorithms have
-been developed: you can split the training set into mini-batches and feed an IPCA algorithm one minibatch
-at a time. This is useful for large training sets, and also to apply PCA online (i.e., on the fly, as new
-instances arrive).
+Instead of arbitrarily choosing the number of dimensions to reduce down to, it is generally preferable to
+choose the number of dimensions that add up to a sufficiently large portion of the variance (e.g., 95%).
+Unless, of course, you are reducing dimensionality for data visualization — in that case you will
+generally want to reduce the dimensionality down to 2 or 3.
+The following code computes PCA without reducing dimensionality, then computes the minimum number
+of dimensions required to preserve 95% of the training set’s variance:
+
    
 
+
+
    pca = PCA()
+pca.fit(X)
+cumsum = np.cumsum(pca.explained_variance_ratio_)
+d = np.argmax(cumsum >= 0.95) + 1
+
    
+
    
+You could then set \( n\_components=d \) and run PCA again. However, there is a much better option: instead
+of specifying the number of principal components you want to preserve, you can set \( n\_components \) to be
+a float between 0.0 and 1.0, indicating the ratio of variance you wish to preserve:
+
    
+
+
+
    pca = PCA(n_components=0.95)
+X_reduced = pca.fit_transform(X)
+
    
 
    
 
    
 
@@ -207,6 +227,7 @@ instances arrive).
   
  • 26
    • 
   
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    • 
   
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    • 
+  
  • 29
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/DimRed/html/._DimRed-bs024.html b/doc/pub/DimRed/html/._DimRed-bs024.html
index 471ff293c..9c87e99a1 100644
--- a/doc/pub/DimRed/html/._DimRed-bs024.html
+++ b/doc/pub/DimRed/html/._DimRed-bs024.html
@@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source
                None,
                '___sec14'),
               ('Towards the PCA theorem', 2, None, '___sec15'),
-              ('Classical PCA Theorem', 2, None, '___sec16'),
-              ('Prof of the PCA Theorem', 2, None, '___sec17'),
-              ('Getting started with PCA', 2, None, '___sec18'),
-              ('Principal Component Analysis', 2, None, '___sec19'),
-              ('PCA and scikit-learn', 2, None, '___sec20'),
-              ('More on the PCA', 2, None, '___sec21'),
-              ('Incremental PCA', 2, None, '___sec22'),
-              ('Randomized PCA', 2, None, '___sec23'),
-              ('Kernel PCA', 2, None, '___sec24'),
-              ('LLE', 2, None, '___sec25'),
-              ('Other techniques', 2, None, '___sec26')]}
+              ('The Algorithm before the Theorem', 2, None, '___sec16'),
+              ('Classical PCA Theorem', 2, None, '___sec17'),
+              ('Prof of the PCA Theorem', 2, None, '___sec18'),
+              ('Getting started with PCA', 2, None, '___sec19'),
+              ('Principal Component Analysis', 2, None, '___sec20'),
+              ('PCA and scikit-learn', 2, None, '___sec21'),
+              ('More on the PCA', 2, None, '___sec22'),
+              ('Incremental PCA', 2, None, '___sec23'),
+              ('Randomized PCA', 2, None, '___sec24'),
+              ('Kernel PCA', 2, None, '___sec25'),
+              ('LLE', 2, None, '___sec26'),
+              ('Other techniques', 2, None, '___sec27')]}
 end of tocinfo -->
 
 
@@ -152,17 +153,18 @@ MathJax.Hub.Config({
       
  • Correlation Matrix with Pandas and the Franke function
    • 
       
  • Rewriting the Covariance and/or Correlation Matrix
    • 
       
  • Towards the PCA theorem
    • 
-      
  • Classical PCA Theorem
    • 
-      
  • Prof of the PCA Theorem
    • 
-      
  • Getting started with PCA
    • 
-      
  • Principal Component Analysis
    • 
-      
  • PCA and scikit-learn
    • 
-      
  • More on the PCA
    • 
-      
  • Incremental PCA
    • 
-      
  • Randomized PCA
    • 
-      
  • Kernel PCA
    • 
-      
  • LLE
    • 
-      
  • Other techniques
    • 
+      
  • The Algorithm before the Theorem
    • 
+      
  • Classical PCA Theorem
    • 
+      
  • Prof of the PCA Theorem
    • 
+      
  • Getting started with PCA
    • 
+      
  • Principal Component Analysis
    • 
+      
  • PCA and scikit-learn
    • 
+      
  • More on the PCA
    • 
+      
  • Incremental PCA
    • 
+      
  • Randomized PCA
    • 
+      
  • Kernel PCA
    • 
+      
  • LLE
    • 
+      
  • Other techniques
    • 
 
         
       
@@ -178,18 +180,14 @@ MathJax.Hub.Config({
 
 
 
-
    Randomized PCA 
    
+
    Incremental PCA 
    
 
 
    
-Scikit-Learn offers yet another option to perform PCA, called Randomized PCA. This is a stochastic
-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 \( n \).
-
-
    
-
    
-
    
-
+One problem with the preceding implementation of PCA is that it requires the whole training set to fit in
+memory in order for the SVD algorithm to run. Fortunately, Incremental PCA (IPCA) algorithms have
+been developed: you can split the training set into mini-batches and feed an IPCA algorithm one minibatch
+at a time. This is useful for large training sets, and also to apply PCA online (i.e., on the fly, as new
+instances arrive).
 
 
    
 
    
@@ -210,6 +208,7 @@ previous algorithms when \( d \) is much smaller than \( n \).
   
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    • 
   
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    • 
   
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    • 
+  
  • 29
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/DimRed/html/._DimRed-bs025.html b/doc/pub/DimRed/html/._DimRed-bs025.html
index 1f84c5fec..8da2c7966 100644
--- a/doc/pub/DimRed/html/._DimRed-bs025.html
+++ b/doc/pub/DimRed/html/._DimRed-bs025.html
@@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source
                None,
                '___sec14'),
               ('Towards the PCA theorem', 2, None, '___sec15'),
-              ('Classical PCA Theorem', 2, None, '___sec16'),
-              ('Prof of the PCA Theorem', 2, None, '___sec17'),
-              ('Getting started with PCA', 2, None, '___sec18'),
-              ('Principal Component Analysis', 2, None, '___sec19'),
-              ('PCA and scikit-learn', 2, None, '___sec20'),
-              ('More on the PCA', 2, None, '___sec21'),
-              ('Incremental PCA', 2, None, '___sec22'),
-              ('Randomized PCA', 2, None, '___sec23'),
-              ('Kernel PCA', 2, None, '___sec24'),
-              ('LLE', 2, None, '___sec25'),
-              ('Other techniques', 2, None, '___sec26')]}
+              ('The Algorithm before the Theorem', 2, None, '___sec16'),
+              ('Classical PCA Theorem', 2, None, '___sec17'),
+              ('Prof of the PCA Theorem', 2, None, '___sec18'),
+              ('Getting started with PCA', 2, None, '___sec19'),
+              ('Principal Component Analysis', 2, None, '___sec20'),
+              ('PCA and scikit-learn', 2, None, '___sec21'),
+              ('More on the PCA', 2, None, '___sec22'),
+              ('Incremental PCA', 2, None, '___sec23'),
+              ('Randomized PCA', 2, None, '___sec24'),
+              ('Kernel PCA', 2, None, '___sec25'),
+              ('LLE', 2, None, '___sec26'),
+              ('Other techniques', 2, None, '___sec27')]}
 end of tocinfo -->
 
 
@@ -152,17 +153,18 @@ MathJax.Hub.Config({
       
  • Correlation Matrix with Pandas and the Franke function
    • 
       
  • Rewriting the Covariance and/or Correlation Matrix
    • 
       
  • Towards the PCA theorem
    • 
-      
  • Classical PCA Theorem
    • 
-      
  • Prof of the PCA Theorem
    • 
-      
  • Getting started with PCA
    • 
-      
  • Principal Component Analysis
    • 
-      
  • PCA and scikit-learn
    • 
-      
  • More on the PCA
    • 
-      
  • Incremental PCA
    • 
-      
  • Randomized PCA
    • 
-      
  • Kernel PCA
    • 
-      
  • LLE
    • 
-      
  • Other techniques
    • 
+      
  • The Algorithm before the Theorem
    • 
+      
  • Classical PCA Theorem
    • 
+      
  • Prof of the PCA Theorem
    • 
+      
  • Getting started with PCA
    • 
+      
  • Principal Component Analysis
    • 
+      
  • PCA and scikit-learn
    • 
+      
  • More on the PCA
    • 
+      
  • Incremental PCA
    • 
+      
  • Randomized PCA
    • 
+      
  • Kernel PCA
    • 
+      
  • LLE
    • 
+      
  • Other techniques
    • 
 
         
       
@@ -178,28 +180,14 @@ MathJax.Hub.Config({
 
 
 
-
    Kernel PCA 
    
-
    
-
    
-
     
+
    Randomized PCA 
    
 
 
    
-The kernel trick is a mathematical technique that implicitly maps instances into a
-very high-dimensional space (called the feature space), enabling nonlinear classification and regression
-with Support Vector Machines. Recall that a linear decision boundary in the high-dimensional feature
-space corresponds to a complex nonlinear decision boundary in the original space.
-It turns out that the same trick can be applied to PCA, making it possible to perform complex nonlinear
-projections for dimensionality reduction. This is called Kernel PCA (kPCA). It is often good at
-preserving clusters of instances after projection, or sometimes even unrolling datasets that lie close to a
-twisted manifold.
-For example, the following code uses Scikit-Learn’s KernelPCA class to perform kPCA with an
-
    
+Scikit-Learn offers yet another option to perform PCA, called Randomized PCA. This is a stochastic
+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 \( n \).
 
-
-
    from sklearn.decomposition import KernelPCA
-rbf_pca = KernelPCA(n_components = 2, kernel="rbf", gamma=0.04)
-X_reduced = rbf_pca.fit_transform(X)
-
    
 
    
 
    
 
    
@@ -223,6 +211,7 @@ X_reduced = rbf_pca26
   
  • 27
    • 
   
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    • 
+  
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    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/DimRed/html/._DimRed-bs026.html b/doc/pub/DimRed/html/._DimRed-bs026.html
index c3bcffb10..52f5488e9 100644
--- a/doc/pub/DimRed/html/._DimRed-bs026.html
+++ b/doc/pub/DimRed/html/._DimRed-bs026.html
@@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source
                None,
                '___sec14'),
               ('Towards the PCA theorem', 2, None, '___sec15'),
-              ('Classical PCA Theorem', 2, None, '___sec16'),
-              ('Prof of the PCA Theorem', 2, None, '___sec17'),
-              ('Getting started with PCA', 2, None, '___sec18'),
-              ('Principal Component Analysis', 2, None, '___sec19'),
-              ('PCA and scikit-learn', 2, None, '___sec20'),
-              ('More on the PCA', 2, None, '___sec21'),
-              ('Incremental PCA', 2, None, '___sec22'),
-              ('Randomized PCA', 2, None, '___sec23'),
-              ('Kernel PCA', 2, None, '___sec24'),
-              ('LLE', 2, None, '___sec25'),
-              ('Other techniques', 2, None, '___sec26')]}
+              ('The Algorithm before the Theorem', 2, None, '___sec16'),
+              ('Classical PCA Theorem', 2, None, '___sec17'),
+              ('Prof of the PCA Theorem', 2, None, '___sec18'),
+              ('Getting started with PCA', 2, None, '___sec19'),
+              ('Principal Component Analysis', 2, None, '___sec20'),
+              ('PCA and scikit-learn', 2, None, '___sec21'),
+              ('More on the PCA', 2, None, '___sec22'),
+              ('Incremental PCA', 2, None, '___sec23'),
+              ('Randomized PCA', 2, None, '___sec24'),
+              ('Kernel PCA', 2, None, '___sec25'),
+              ('LLE', 2, None, '___sec26'),
+              ('Other techniques', 2, None, '___sec27')]}
 end of tocinfo -->
 
 
@@ -152,17 +153,18 @@ MathJax.Hub.Config({
       
  • Correlation Matrix with Pandas and the Franke function
    • 
       
  • Rewriting the Covariance and/or Correlation Matrix
    • 
       
  • Towards the PCA theorem
    • 
-      
  • Classical PCA Theorem
    • 
-      
  • Prof of the PCA Theorem
    • 
-      
  • Getting started with PCA
    • 
-      
  • Principal Component Analysis
    • 
-      
  • PCA and scikit-learn
    • 
-      
  • More on the PCA
    • 
-      
  • Incremental PCA
    • 
-      
  • Randomized PCA
    • 
-      
  • Kernel PCA
    • 
-      
  • LLE
    • 
-      
  • Other techniques
    • 
+      
  • The Algorithm before the Theorem
    • 
+      
  • Classical PCA Theorem
    • 
+      
  • Prof of the PCA Theorem
    • 
+      
  • Getting started with PCA
    • 
+      
  • Principal Component Analysis
    • 
+      
  • PCA and scikit-learn
    • 
+      
  • More on the PCA
    • 
+      
  • Incremental PCA
    • 
+      
  • Randomized PCA
    • 
+      
  • Kernel PCA
    • 
+      
  • LLE
    • 
+      
  • Other techniques
    • 
 
         
       
@@ -178,14 +180,32 @@ MathJax.Hub.Config({
 
 
 
-
    LLE 
    
+
    Kernel PCA 
    
+
    
+
    
+
     
 
 
    
-Locally Linear Embedding (LLE) is another very powerful nonlinear dimensionality reduction
-(NLDR) technique. It is a Manifold Learning technique that does not rely on projections like the previous
-algorithms. In a nutshell, LLE works by first measuring how each training instance linearly relates to its
-closest neighbors (c.n.), and then looking for a low-dimensional representation of the training set where
-these local relationships are best preserved (more details shortly).
+The kernel trick is a mathematical technique that implicitly maps instances into a
+very high-dimensional space (called the feature space), enabling nonlinear classification and regression
+with Support Vector Machines. Recall that a linear decision boundary in the high-dimensional feature
+space corresponds to a complex nonlinear decision boundary in the original space.
+It turns out that the same trick can be applied to PCA, making it possible to perform complex nonlinear
+projections for dimensionality reduction. This is called Kernel PCA (kPCA). It is often good at
+preserving clusters of instances after projection, or sometimes even unrolling datasets that lie close to a
+twisted manifold.
+For example, the following code uses Scikit-Learn’s KernelPCA class to perform kPCA with an
+
    
+
+
+
    from sklearn.decomposition import KernelPCA
+rbf_pca = KernelPCA(n_components = 2, kernel="rbf", gamma=0.04)
+X_reduced = rbf_pca.fit_transform(X)
+
    
+
    
+
    
+
    
+
 
 
    
 
    
@@ -204,6 +224,7 @@ these local relationships are best preserved (more details shortly).
   
  • 26
    • 
   
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    • 
   
  • 28
    • 
+  
  • 29
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/DimRed/html/._DimRed-bs027.html b/doc/pub/DimRed/html/._DimRed-bs027.html
index 6bb35afea..ca75ce89c 100644
--- a/doc/pub/DimRed/html/._DimRed-bs027.html
+++ b/doc/pub/DimRed/html/._DimRed-bs027.html
@@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source
                None,
                '___sec14'),
               ('Towards the PCA theorem', 2, None, '___sec15'),
-              ('Classical PCA Theorem', 2, None, '___sec16'),
-              ('Prof of the PCA Theorem', 2, None, '___sec17'),
-              ('Getting started with PCA', 2, None, '___sec18'),
-              ('Principal Component Analysis', 2, None, '___sec19'),
-              ('PCA and scikit-learn', 2, None, '___sec20'),
-              ('More on the PCA', 2, None, '___sec21'),
-              ('Incremental PCA', 2, None, '___sec22'),
-              ('Randomized PCA', 2, None, '___sec23'),
-              ('Kernel PCA', 2, None, '___sec24'),
-              ('LLE', 2, None, '___sec25'),
-              ('Other techniques', 2, None, '___sec26')]}
+              ('The Algorithm before the Theorem', 2, None, '___sec16'),
+              ('Classical PCA Theorem', 2, None, '___sec17'),
+              ('Prof of the PCA Theorem', 2, None, '___sec18'),
+              ('Getting started with PCA', 2, None, '___sec19'),
+              ('Principal Component Analysis', 2, None, '___sec20'),
+              ('PCA and scikit-learn', 2, None, '___sec21'),
+              ('More on the PCA', 2, None, '___sec22'),
+              ('Incremental PCA', 2, None, '___sec23'),
+              ('Randomized PCA', 2, None, '___sec24'),
+              ('Kernel PCA', 2, None, '___sec25'),
+              ('LLE', 2, None, '___sec26'),
+              ('Other techniques', 2, None, '___sec27')]}
 end of tocinfo -->
 
 
@@ -152,17 +153,18 @@ MathJax.Hub.Config({
       
  • Correlation Matrix with Pandas and the Franke function
    • 
       
  • Rewriting the Covariance and/or Correlation Matrix
    • 
       
  • Towards the PCA theorem
    • 
-      
  • Classical PCA Theorem
    • 
-      
  • Prof of the PCA Theorem
    • 
-      
  • Getting started with PCA
    • 
-      
  • Principal Component Analysis
    • 
-      
  • PCA and scikit-learn
    • 
-      
  • More on the PCA
    • 
-      
  • Incremental PCA
    • 
-      
  • Randomized PCA
    • 
-      
  • Kernel PCA
    • 
-      
  • LLE
    • 
-      
  • Other techniques
    • 
+      
  • The Algorithm before the Theorem
    • 
+      
  • Classical PCA Theorem
    • 
+      
  • Prof of the PCA Theorem
    • 
+      
  • Getting started with PCA
    • 
+      
  • Principal Component Analysis
    • 
+      
  • PCA and scikit-learn
    • 
+      
  • More on the PCA
    • 
+      
  • Incremental PCA
    • 
+      
  • Randomized PCA
    • 
+      
  • Kernel PCA
    • 
+      
  • LLE
    • 
+      
  • Other techniques
    • 
 
         
       
@@ -178,77 +180,16 @@ MathJax.Hub.Config({
 
 
 
-
    Other techniques 
    
+
    LLE 
    
 
 
    
-There are many other dimensionality reduction techniques, several of which are available in Scikit-Learn.
+Locally Linear Embedding (LLE) is another very powerful nonlinear dimensionality reduction
+(NLDR) technique. It is a Manifold Learning technique that does not rely on projections like the previous
+algorithms. In a nutshell, LLE works by first measuring how each training instance linearly relates to its
+closest neighbors (c.n.), and then looking for a low-dimensional representation of the training set where
+these local relationships are best preserved (more details shortly).
 
 
    
-Here are some of the most popular:
-
-
      
-
    •  Multidimensional Scaling (MDS) reduces dimensionality while trying to preserve the distances between the instances.
      • 
-
    •  Isomap creates a graph by connecting each instance to its nearest neighbors, then reduces dimensionality while trying to preserve the geodesic distances between the instances.
      • 
-
    •  t-Distributed Stochastic Neighbor Embedding (t-SNE) reduces dimensionality while trying to keep similar instances close and dissimilar instances apart. It is mostly used for visualization, in particular to visualize clusters of instances in high-dimensional space (e.g., to visualize the MNIST images in 2D).
      • 
-
    •  Linear Discriminant Analysis (LDA) is actually a classification algorithm, but during training it learns the most discriminative axes between the classes, and these axes can then be used to define a hyperplane onto which to project the data. The benefit is that the projection will keep classes as far apart as possible, so LDA is a good technique to reduce dimensionality before running another classification algorithm such as a Support Vector Machine (SVM) classifier discussed in the SVM lectures.
      • 
-
    
-
-Here are other examples where we use the DataFrame functionality to handle arrays, now with more interesting features for us, namely numbers. We set up a matrix 
-of dimensionality \( 10\times 5 \) and compute the mean value and standard deviation of each column. Similarly, we can perform mathematial operations like squaring the matrix elements and many other operations. 
-
    
-
-
-
    import numpy as np
-import pandas as pd
-from IPython.display import display
-np.random.seed(100)
-# setting up a 10 x 5 matrix
-rows = 10
-cols = 5
-a = np.random.randn(rows,cols)
-df = pd.DataFrame(a)
-display(df)
-print(df.mean())
-print(df.std())
-display(df**2)
-
    
-
    
-Thereafter we can select specific columns only and plot final results
-
    
-
-
-
    df.columns = ['First', 'Second', 'Third', 'Fourth', 'Fifth']
-df.index = np.arange(10)
-
-display(df)
-print(df['Second'].mean() )
-print(df.info())
-print(df.describe())
-
-from pylab import plt, mpl
-plt.style.use('seaborn')
-mpl.rcParams['font.family'] = 'serif'
-
-df.cumsum().plot(lw=2.0, figsize=(10,6))
-plt.show()
-
-
-df.plot.bar(figsize=(10,6), rot=15)
-plt.show()
-
    
-
    
-We can produce a \( 4\times 4 \) matrix
-
    
-
-
-
    b = np.arange(16).reshape((4,4))
-print(b)
-df1 = pd.DataFrame(b)
-print(df1)
-
    
-
    
-and many other operations. 
-
 
    
 
 
      
@@ -264,6 +205,8 @@ and many other operations.
   
    • 26
      • 
   
    • 27
      • 
   
    • 28
      • 
+  
    • 29
      • 
+  
    • »
      • 
 
    
 
 
diff --git a/doc/pub/DimRed/html/DimRed-bs.html b/doc/pub/DimRed/html/DimRed-bs.html
index 701588387..3df86eb8f 100644
--- a/doc/pub/DimRed/html/DimRed-bs.html
+++ b/doc/pub/DimRed/html/DimRed-bs.html
@@ -88,17 +88,18 @@ Automatically generated HTML file from DocOnce source
                None,
                '___sec14'),
               ('Towards the PCA theorem', 2, None, '___sec15'),
-              ('Classical PCA Theorem', 2, None, '___sec16'),
-              ('Prof of the PCA Theorem', 2, None, '___sec17'),
-              ('Getting started with PCA', 2, None, '___sec18'),
-              ('Principal Component Analysis', 2, None, '___sec19'),
-              ('PCA and scikit-learn', 2, None, '___sec20'),
-              ('More on the PCA', 2, None, '___sec21'),
-              ('Incremental PCA', 2, None, '___sec22'),
-              ('Randomized PCA', 2, None, '___sec23'),
-              ('Kernel PCA', 2, None, '___sec24'),
-              ('LLE', 2, None, '___sec25'),
-              ('Other techniques', 2, None, '___sec26')]}
+              ('The Algorithm before the Theorem', 2, None, '___sec16'),
+              ('Classical PCA Theorem', 2, None, '___sec17'),
+              ('Prof of the PCA Theorem', 2, None, '___sec18'),
+              ('Getting started with PCA', 2, None, '___sec19'),
+              ('Principal Component Analysis', 2, None, '___sec20'),
+              ('PCA and scikit-learn', 2, None, '___sec21'),
+              ('More on the PCA', 2, None, '___sec22'),
+              ('Incremental PCA', 2, None, '___sec23'),
+              ('Randomized PCA', 2, None, '___sec24'),
+              ('Kernel PCA', 2, None, '___sec25'),
+              ('LLE', 2, None, '___sec26'),
+              ('Other techniques', 2, None, '___sec27')]}
 end of tocinfo -->
 
 
@@ -152,17 +153,18 @@ MathJax.Hub.Config({
       
  • Correlation Matrix with Pandas and the Franke function
    • 
       
  • Rewriting the Covariance and/or Correlation Matrix
    • 
       
  • Towards the PCA theorem
    • 
-      
  • Classical PCA Theorem
    • 
-      
  • Prof of the PCA Theorem
    • 
-      
  • Getting started with PCA
    • 
-      
  • Principal Component Analysis
    • 
-      
  • PCA and scikit-learn
    • 
-      
  • More on the PCA
    • 
-      
  • Incremental PCA
    • 
-      
  • Randomized PCA
    • 
-      
  • Kernel PCA
    • 
-      
  • LLE
    • 
-      
  • Other techniques
    • 
+      
  • The Algorithm before the Theorem
    • 
+      
  • Classical PCA Theorem
    • 
+      
  • Prof of the PCA Theorem
    • 
+      
  • Getting started with PCA
    • 
+      
  • Principal Component Analysis
    • 
+      
  • PCA and scikit-learn
    • 
+      
  • More on the PCA
    • 
+      
  • Incremental PCA
    • 
+      
  • Randomized PCA
    • 
+      
  • Kernel PCA
    • 
+      
  • LLE
    • 
+      
  • Other techniques
    • 
 
         
       
@@ -221,7 +223,7 @@ MathJax.Hub.Config({
   
  • 9
    • 
   
  • 10
    • 
   
  • ...
    • 
-  
  • 28
    • 
+  
  • 29
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/DimRed/html/DimRed-reveal.html b/doc/pub/DimRed/html/DimRed-reveal.html
index d17f342fa..221d0834e 100644
--- a/doc/pub/DimRed/html/DimRed-reveal.html
+++ b/doc/pub/DimRed/html/DimRed-reveal.html
@@ -936,21 +936,70 @@ $$
 
    
 In the derivation of the PCA theorem we will assume that the eigenvalues are ordered in descending order, that is
 \( \lambda_0 > \lambda_1 > \dots > \lambda_{p-1} \).
+
+
    
+The eigenvalues tell us then how much we need to stretch the
+corresponding eigenvectors. Dimensions with large eigenvalues have
+thus large variations (large variance) and define therefore useful
+dimensions. The data points are more spread out in the direction of
+these eigenvectors.  Smaller eigenvalues mean on the other hand that
+the corresponding eigenvectors are shrunk accordingly and the data
+points are tightly bunched together and there is not much variation in
+these specific directions. Hopefully then we could leave it out
+dimensions where the eigenvalues are very small. If \( p \) is very large,
+we could then aim at reducing \( p \) to \( l < < p \) and handle only \( l \)
+features/predictors.
 
 
 
 
    
-
    Classical PCA Theorem   
    
+
    The Algorithm before theorem 
    
+
+
    
+Here's how we would proceed in setting up the algorithm for the PCA, see also discussion below here. 
+
+
      
+
    •  Set up the datapoints for the design/feature matrix \( \boldsymbol{X} \) with \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \), with the predictors/features \( p \)  referring to the column numbers and the entries \( n \) being the row elements.
      • 
+
    
+
     
    
+$$
+\boldsymbol{X}=\begin{bmatrix}
+x_{0,0} & x_{0,1} & x_{0,2}& \dots & \dots x_{0,p-1}\\
+x_{1,0} & x_{1,1} & x_{1,2}& \dots & \dots x_{1,p-1}\\
+x_{2,0} & x_{2,1} & x_{2,2}& \dots & \dots x_{2,p-1}\\
+\dots & \dots & \dots & \dots \dots & \dots \\
+x_{n-2,0} & x_{n-2,1} & x_{n-2,2}& \dots & \dots x_{n-2,p-1}\\
+x_{n-1,0} & x_{n-1,1} & x_{n-1,2}& \dots & \dots x_{n-1,p-1}\\
+\end{bmatrix},
+$$
+
     
    
+
+
+
      
+
    •  Center the data by subtracting the mean value for each column. This leads to a new matrix \( \boldsymbol{X}\rightarrow \overline{\boldsymbol{X}} \).
      • 
+
    •  Compute then the covariance/correlation matrix $\mathbb{E}[\overline{\boldsymbol{X}}\overline{\boldsymbol{X}}^T].
      • 
+
    •  Find the eigenpairs of \( \boldsymbol{C} \) with eigenvalues \( [\lambda_0,\lambda_1,\dots,\lambda_{p-1}] \) and eigenvectors \( [\boldsymbol{s}_0,\boldsymbol{s}_1,\dots,\boldsymbol{s}_{p-1}] \).
      • 
+
    •  Order the eigenvalue (and the eigenvectors accordingly) in order of decreasing eigenvalues.
      • 
+
    •  Keep only those \( l \) eigenvalues larger than a selected threshold value, discarding thus \( p-l \) features since we expect small variations in the data here.
      • 
+
    
+
    
+
+After this we ask ourselves how do we prove the link between the maximum variance and the feature reduction.
 
    
 
 
 
    
-
    Prof of the PCA Theorem  
    
+
    Classical PCA Theorem   
    
 
    
 
 
 
    
-
    Getting started with PCA 
    
+
    Prof of the PCA Theorem  
    
+
    
+
+
+
    
+
    Getting started with PCA 
    
 
 
    
 
@@ -966,7 +1015,7 @@ X_pca = pca.transform(X_train_scaled)
 
 
 
    
-
    Principal Component Analysis 
    
+
    Principal Component Analysis 
    
 
    
 
 
    
@@ -1003,7 +1052,7 @@ X2D = X_centered.dot(W2)
 
 
 
    
-
    PCA and scikit-learn 
    
+
    PCA and scikit-learn 
    
 
 
    
 Scikit-Learn’s PCA class implements PCA using SVD decomposition just like we did before. The
@@ -1034,7 +1083,7 @@ More material to come here.
 
 
 
    
-
    More on the PCA 
    
+
    More on the PCA 
    
 
 
    
 Instead of arbitrarily choosing the number of dimensions to reduce down to, it is generally preferable to
@@ -1065,7 +1114,7 @@ X_reduced = pca.fit_transform(X)
 
 
 
    
-
    Incremental PCA 
    
+
    Incremental PCA 
    
 
 
    
 One problem with the preceding implementation of PCA is that it requires the whole training set to fit in
@@ -1077,7 +1126,7 @@ instances arrive).
 
 
 
    
-
    Randomized PCA 
    
+
    Randomized PCA 
    
 
 
    
 Scikit-Learn offers yet another option to perform PCA, called Randomized PCA. This is a stochastic
@@ -1091,7 +1140,7 @@ previous algorithms when \( d \) is much smaller than \( n \).
 
 
 
    
-
    Kernel PCA 
    
+
    Kernel PCA 
    
 
    
 
 
    
@@ -1117,7 +1166,7 @@ X_reduced = rbf_pca.fit_transform(X)
 
 
 
    
-
    LLE 
    
+
    LLE 
    
 
 
    
 Locally Linear Embedding (LLE) is another very powerful nonlinear dimensionality reduction
@@ -1129,7 +1178,7 @@ these local relationships are best preserved (more details shortly).
 
 
 
    
-
    Other techniques 
    
+
    Other techniques 
    
 
 
    
 There are many other dimensionality reduction techniques, several of which are available in Scikit-Learn.
diff --git a/doc/pub/DimRed/html/DimRed-solarized.html b/doc/pub/DimRed/html/DimRed-solarized.html
index 959e73c3b..006dc5c14 100644
--- a/doc/pub/DimRed/html/DimRed-solarized.html
+++ b/doc/pub/DimRed/html/DimRed-solarized.html
@@ -108,17 +108,18 @@ div { text-align: justify; text-justify: inter-word; }
                None,
                '___sec14'),
               ('Towards the PCA theorem', 2, None, '___sec15'),
-              ('Classical PCA Theorem', 2, None, '___sec16'),
-              ('Prof of the PCA Theorem', 2, None, '___sec17'),
-              ('Getting started with PCA', 2, None, '___sec18'),
-              ('Principal Component Analysis', 2, None, '___sec19'),
-              ('PCA and scikit-learn', 2, None, '___sec20'),
-              ('More on the PCA', 2, None, '___sec21'),
-              ('Incremental PCA', 2, None, '___sec22'),
-              ('Randomized PCA', 2, None, '___sec23'),
-              ('Kernel PCA', 2, None, '___sec24'),
-              ('LLE', 2, None, '___sec25'),
-              ('Other techniques', 2, None, '___sec26')]}
+              ('The Algorithm before the Theorem', 2, None, '___sec16'),
+              ('Classical PCA Theorem', 2, None, '___sec17'),
+              ('Prof of the PCA Theorem', 2, None, '___sec18'),
+              ('Getting started with PCA', 2, None, '___sec19'),
+              ('Principal Component Analysis', 2, None, '___sec20'),
+              ('PCA and scikit-learn', 2, None, '___sec21'),
+              ('More on the PCA', 2, None, '___sec22'),
+              ('Incremental PCA', 2, None, '___sec23'),
+              ('Randomized PCA', 2, None, '___sec24'),
+              ('Kernel PCA', 2, None, '___sec25'),
+              ('LLE', 2, None, '___sec26'),
+              ('Other techniques', 2, None, '___sec27')]}
 end of tocinfo -->
 
 
@@ -903,19 +904,66 @@ In the derivation of the PCA theorem we will assume that the eigenvalues are ord
 \( \lambda_0 > \lambda_1 > \dots > \lambda_{p-1} \).
 
 
    
-









    
-
-
    Classical PCA Theorem   
    
+The eigenvalues tell us then how much we need to stretch the
+corresponding eigenvectors. Dimensions with large eigenvalues have
+thus large variations (large variance) and define therefore useful
+dimensions. The data points are more spread out in the direction of
+these eigenvectors.  Smaller eigenvalues mean on the other hand that
+the corresponding eigenvectors are shrunk accordingly and the data
+points are tightly bunched together and there is not much variation in
+these specific directions. Hopefully then we could leave it out
+dimensions where the eigenvalues are very small. If \( p \) is very large,
+we could then aim at reducing \( p \) to \( l < < p \) and handle only \( l \)
+features/predictors.
 
 
    
 









    
 
-
    Prof of the PCA Theorem  
    
+
    The Algorithm before theorem 
    
+
+
    
+Here's how we would proceed in setting up the algorithm for the PCA, see also discussion below here. 
+
+
      
+
    •  Set up the datapoints for the design/feature matrix \( \boldsymbol{X} \) with \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \), with the predictors/features \( p \)  referring to the column numbers and the entries \( n \) being the row elements.
      • 
+
    
+
+$$
+\boldsymbol{X}=\begin{bmatrix}
+x_{0,0} & x_{0,1} & x_{0,2}& \dots & \dots x_{0,p-1}\\
+x_{1,0} & x_{1,1} & x_{1,2}& \dots & \dots x_{1,p-1}\\
+x_{2,0} & x_{2,1} & x_{2,2}& \dots & \dots x_{2,p-1}\\
+\dots & \dots & \dots & \dots \dots & \dots \\
+x_{n-2,0} & x_{n-2,1} & x_{n-2,2}& \dots & \dots x_{n-2,p-1}\\
+x_{n-1,0} & x_{n-1,1} & x_{n-1,2}& \dots & \dots x_{n-1,p-1}\\
+\end{bmatrix},
+$$
+
+
+
      
+
    •  Center the data by subtracting the mean value for each column. This leads to a new matrix \( \boldsymbol{X}\rightarrow \overline{\boldsymbol{X}} \).
      • 
+
    •  Compute then the covariance/correlation matrix $\mathbb{E}[\overline{\boldsymbol{X}}\overline{\boldsymbol{X}}^T].
      • 
+
    •  Find the eigenpairs of \( \boldsymbol{C} \) with eigenvalues \( [\lambda_0,\lambda_1,\dots,\lambda_{p-1}] \) and eigenvectors \( [\boldsymbol{s}_0,\boldsymbol{s}_1,\dots,\boldsymbol{s}_{p-1}] \).
      • 
+
    •  Order the eigenvalue (and the eigenvectors accordingly) in order of decreasing eigenvalues.
      • 
+
    •  Keep only those \( l \) eigenvalues larger than a selected threshold value, discarding thus \( p-l \) features since we expect small variations in the data here.
      • 
+
    
+
+After this we ask ourselves how do we prove the link between the maximum variance and the feature reduction.
 
 
    
 









    
 
-
    Getting started with PCA 
    
+
    Classical PCA Theorem   
    
+
+
    
+









    
+
+
    Prof of the PCA Theorem  
    
+
+
    
+









    
+
+
    Getting started with PCA 
    
 
 
    
 
@@ -930,7 +978,7 @@ X_pca = pca.transform(X_train_scaled)
 
    
 









    
 
-
    Principal Component Analysis 
    
+
    Principal Component Analysis 
    
 
    
 
 
    
@@ -966,7 +1014,7 @@ X2D = X_centered.dot(W2)
 
    
 
 
-
    PCA and scikit-learn 
    
+
    PCA and scikit-learn 
    
 
 
    
 Scikit-Learn’s PCA class implements PCA using SVD decomposition just like we did before. The
@@ -997,7 +1045,7 @@ More material to come here.
 
    
 









    
 
-
    More on the PCA 
    
+
    More on the PCA 
    
 
 
    
 Instead of arbitrarily choosing the number of dimensions to reduce down to, it is generally preferable to
@@ -1027,7 +1075,7 @@ X_reduced = pca.fit_transform(X)
 
    
 









    
 
-
    Incremental PCA 
    
+
    Incremental PCA 
    
 
 
    
 One problem with the preceding implementation of PCA is that it requires the whole training set to fit in
@@ -1039,7 +1087,7 @@ instances arrive).
 
    
 









    
 
-
    Randomized PCA 
    
+
    Randomized PCA 
    
 
 
    
 Scikit-Learn offers yet another option to perform PCA, called Randomized PCA. This is a stochastic
@@ -1054,7 +1102,7 @@ previous algorithms when \( d \) is much smaller than \( n \).
 
    
 









    
 
-
    Kernel PCA 
    
+
    Kernel PCA 
    
 
    
 
 
    
@@ -1083,7 +1131,7 @@ X_reduced = rbf_pca.fit_transform(X)
 
    
 









    
 
-
    LLE 
    
+
    LLE 
    
 
 
    
 Locally Linear Embedding (LLE) is another very powerful nonlinear dimensionality reduction
@@ -1095,7 +1143,7 @@ these local relationships are best preserved (more details shortly).
 
    
 









    
 
-
    Other techniques 
    
+
    Other techniques 
    
 
 
    
 There are many other dimensionality reduction techniques, several of which are available in Scikit-Learn.
diff --git a/doc/pub/DimRed/html/DimRed.html b/doc/pub/DimRed/html/DimRed.html
index f30d112ee..f6e2efdb3 100644
--- a/doc/pub/DimRed/html/DimRed.html
+++ b/doc/pub/DimRed/html/DimRed.html
@@ -113,17 +113,18 @@ div { text-align: justify; text-justify: inter-word; }
                None,
                '___sec14'),
               ('Towards the PCA theorem', 2, None, '___sec15'),
-              ('Classical PCA Theorem', 2, None, '___sec16'),
-              ('Prof of the PCA Theorem', 2, None, '___sec17'),
-              ('Getting started with PCA', 2, None, '___sec18'),
-              ('Principal Component Analysis', 2, None, '___sec19'),
-              ('PCA and scikit-learn', 2, None, '___sec20'),
-              ('More on the PCA', 2, None, '___sec21'),
-              ('Incremental PCA', 2, None, '___sec22'),
-              ('Randomized PCA', 2, None, '___sec23'),
-              ('Kernel PCA', 2, None, '___sec24'),
-              ('LLE', 2, None, '___sec25'),
-              ('Other techniques', 2, None, '___sec26')]}
+              ('The Algorithm before the Theorem', 2, None, '___sec16'),
+              ('Classical PCA Theorem', 2, None, '___sec17'),
+              ('Prof of the PCA Theorem', 2, None, '___sec18'),
+              ('Getting started with PCA', 2, None, '___sec19'),
+              ('Principal Component Analysis', 2, None, '___sec20'),
+              ('PCA and scikit-learn', 2, None, '___sec21'),
+              ('More on the PCA', 2, None, '___sec22'),
+              ('Incremental PCA', 2, None, '___sec23'),
+              ('Randomized PCA', 2, None, '___sec24'),
+              ('Kernel PCA', 2, None, '___sec25'),
+              ('LLE', 2, None, '___sec26'),
+              ('Other techniques', 2, None, '___sec27')]}
 end of tocinfo -->
 
 
@@ -908,19 +909,66 @@ In the derivation of the PCA theorem we will assume that the eigenvalues are ord
 \( \lambda_0 > \lambda_1 > \dots > \lambda_{p-1} \).
 
 
    
-









    
-
-
    Classical PCA Theorem   
    
+The eigenvalues tell us then how much we need to stretch the
+corresponding eigenvectors. Dimensions with large eigenvalues have
+thus large variations (large variance) and define therefore useful
+dimensions. The data points are more spread out in the direction of
+these eigenvectors.  Smaller eigenvalues mean on the other hand that
+the corresponding eigenvectors are shrunk accordingly and the data
+points are tightly bunched together and there is not much variation in
+these specific directions. Hopefully then we could leave it out
+dimensions where the eigenvalues are very small. If \( p \) is very large,
+we could then aim at reducing \( p \) to \( l < < p \) and handle only \( l \)
+features/predictors.
 
 
    
 









    
 
-
    Prof of the PCA Theorem  
    
+
    The Algorithm before theorem 
    
+
+
    
+Here's how we would proceed in setting up the algorithm for the PCA, see also discussion below here. 
+
+
      
+
    •  Set up the datapoints for the design/feature matrix \( \boldsymbol{X} \) with \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \), with the predictors/features \( p \)  referring to the column numbers and the entries \( n \) being the row elements.
      • 
+
    
+
+$$
+\boldsymbol{X}=\begin{bmatrix}
+x_{0,0} & x_{0,1} & x_{0,2}& \dots & \dots x_{0,p-1}\\
+x_{1,0} & x_{1,1} & x_{1,2}& \dots & \dots x_{1,p-1}\\
+x_{2,0} & x_{2,1} & x_{2,2}& \dots & \dots x_{2,p-1}\\
+\dots & \dots & \dots & \dots \dots & \dots \\
+x_{n-2,0} & x_{n-2,1} & x_{n-2,2}& \dots & \dots x_{n-2,p-1}\\
+x_{n-1,0} & x_{n-1,1} & x_{n-1,2}& \dots & \dots x_{n-1,p-1}\\
+\end{bmatrix},
+$$
+
+
+
      
+
    •  Center the data by subtracting the mean value for each column. This leads to a new matrix \( \boldsymbol{X}\rightarrow \overline{\boldsymbol{X}} \).
      • 
+
    •  Compute then the covariance/correlation matrix $\mathbb{E}[\overline{\boldsymbol{X}}\overline{\boldsymbol{X}}^T].
      • 
+
    •  Find the eigenpairs of \( \boldsymbol{C} \) with eigenvalues \( [\lambda_0,\lambda_1,\dots,\lambda_{p-1}] \) and eigenvectors \( [\boldsymbol{s}_0,\boldsymbol{s}_1,\dots,\boldsymbol{s}_{p-1}] \).
      • 
+
    •  Order the eigenvalue (and the eigenvectors accordingly) in order of decreasing eigenvalues.
      • 
+
    •  Keep only those \( l \) eigenvalues larger than a selected threshold value, discarding thus \( p-l \) features since we expect small variations in the data here.
      • 
+
    
+
+After this we ask ourselves how do we prove the link between the maximum variance and the feature reduction.
 
 
    
 









    
 
-
    Getting started with PCA 
    
+
    Classical PCA Theorem   
    
+
+
    
+









    
+
+
    Prof of the PCA Theorem  
    
+
+
    
+









    
+
+
    Getting started with PCA 
    
 
 
    
 
@@ -935,7 +983,7 @@ X_pca = pca.
 









    
 
-
    Principal Component Analysis 
    
+
    Principal Component Analysis 
    
 
    
 
 
    
@@ -971,7 +1019,7 @@ X2D = X_centered
 
 
-
    PCA and scikit-learn 
    
+
    PCA and scikit-learn 
    
 
 
    
 Scikit-Learn’s PCA class implements PCA using SVD decomposition just like we did before. The
@@ -1002,7 +1050,7 @@ More material to come here.
 
    
 









    
 
-
    More on the PCA 
    
+
    More on the PCA 
    
 
 
    
 Instead of arbitrarily choosing the number of dimensions to reduce down to, it is generally preferable to
@@ -1032,7 +1080,7 @@ X_reduced = pca
 
    
 









    
 
-
    Incremental PCA 
    
+
    Incremental PCA 
    
 
 
    
 One problem with the preceding implementation of PCA is that it requires the whole training set to fit in
@@ -1044,7 +1092,7 @@ instances arrive).
 
    
 









    
 
-
    Randomized PCA 
    
+
    Randomized PCA 
    
 
 
    
 Scikit-Learn offers yet another option to perform PCA, called Randomized PCA. This is a stochastic
@@ -1059,7 +1107,7 @@ previous algorithms when \( d \) is much smaller than \( n \).
 
    
 









    
 
-
    Kernel PCA 
    
+
    Kernel PCA 
    
 
    
 
 
    
@@ -1088,7 +1136,7 @@ X_reduced = rbf_pcaLLE 
+
    LLE 
    
 
 
    
 Locally Linear Embedding (LLE) is another very powerful nonlinear dimensionality reduction
@@ -1100,7 +1148,7 @@ these local relationships are best preserved (more details shortly).
 
    
 









    
 
-
    Other techniques 
    
+
    Other techniques 
    
 
 
    
 There are many other dimensionality reduction techniques, several of which are available in Scikit-Learn.
diff --git a/doc/pub/DimRed/ipynb/DimRed.ipynb b/doc/pub/DimRed/ipynb/DimRed.ipynb
index f2c182c39..538fd7b6f 100644
--- a/doc/pub/DimRed/ipynb/DimRed.ipynb
+++ b/doc/pub/DimRed/ipynb/DimRed.ipynb
@@ -1000,6 +1000,57 @@
     "In the derivation of the PCA theorem we will assume that the eigenvalues are ordered in descending order, that is\n",
     "$\\lambda_0 > \\lambda_1 > \\dots > \\lambda_{p-1}$. \n",
     "\n",
+    "\n",
+    "The eigenvalues tell us then how much we need to stretch the\n",
+    "corresponding eigenvectors. Dimensions with large eigenvalues have\n",
+    "thus large variations (large variance) and define therefore useful\n",
+    "dimensions. The data points are more spread out in the direction of\n",
+    "these eigenvectors.  Smaller eigenvalues mean on the other hand that\n",
+    "the corresponding eigenvectors are shrunk accordingly and the data\n",
+    "points are tightly bunched together and there is not much variation in\n",
+    "these specific directions. Hopefully then we could leave it out\n",
+    "dimensions where the eigenvalues are very small. If $p$ is very large,\n",
+    "we could then aim at reducing $p$ to $l << p$ and handle only $l$\n",
+    "features/predictors.\n",
+    "\n",
+    "## The Algorithm before theorem\n",
+    "\n",
+    "Here's how we would proceed in setting up the algorithm for the PCA, see also discussion below here. \n",
+    "* Set up the datapoints for the design/feature matrix $\\boldsymbol{X}$ with $\\boldsymbol{X}\\in {\\mathbb{R}}^{n\\times p}$, with the predictors/features $p$  referring to the column numbers and the entries $n$ being the row elements."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$$\n",
+    "\\boldsymbol{X}=\\begin{bmatrix}\n",
+    "x_{0,0} & x_{0,1} & x_{0,2}& \\dots & \\dots x_{0,p-1}\\\\\n",
+    "x_{1,0} & x_{1,1} & x_{1,2}& \\dots & \\dots x_{1,p-1}\\\\\n",
+    "x_{2,0} & x_{2,1} & x_{2,2}& \\dots & \\dots x_{2,p-1}\\\\\n",
+    "\\dots & \\dots & \\dots & \\dots \\dots & \\dots \\\\\n",
+    "x_{n-2,0} & x_{n-2,1} & x_{n-2,2}& \\dots & \\dots x_{n-2,p-1}\\\\\n",
+    "x_{n-1,0} & x_{n-1,1} & x_{n-1,2}& \\dots & \\dots x_{n-1,p-1}\\\\\n",
+    "\\end{bmatrix},\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "* Center the data by subtracting the mean value for each column. This leads to a new matrix $\\boldsymbol{X}\\rightarrow \\overline{\\boldsymbol{X}}$.\n",
+    "\n",
+    "* Compute then the covariance/correlation matrix $\\mathbb{E}[\\overline{\\boldsymbol{X}}\\overline{\\boldsymbol{X}}^T].\n",
+    "\n",
+    "* Find the eigenpairs of $\\boldsymbol{C}$ with eigenvalues $[\\lambda_0,\\lambda_1,\\dots,\\lambda_{p-1}]$ and eigenvectors $[\\boldsymbol{s}_0,\\boldsymbol{s}_1,\\dots,\\boldsymbol{s}_{p-1}]$.\n",
+    "\n",
+    "* Order the eigenvalue (and the eigenvectors accordingly) in order of decreasing eigenvalues.\n",
+    "\n",
+    "* Keep only those $l$ eigenvalues larger than a selected threshold value, discarding thus $p-l$ features since we expect small variations in the data here.\n",
+    "\n",
+    "After this we ask ourselves how do we prove the link between the maximum variance and the feature reduction.\n",
+    "\n",
     "## Classical PCA Theorem\n",
     "\n",
     "\n",
diff --git a/doc/pub/DimRed/ipynb/ipynb-DimRed-src.tar.gz b/doc/pub/DimRed/ipynb/ipynb-DimRed-src.tar.gz
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diff --git a/doc/src/DimRed/DimRed.do.txt b/doc/src/DimRed/DimRed.do.txt
index 3db103be5..a7124c982 100644
--- a/doc/src/DimRed/DimRed.do.txt
+++ b/doc/src/DimRed/DimRed.do.txt
@@ -703,6 +703,44 @@ and since $\bm{C}[\bm{y}]$ is diagonal we have for a given eigenvalue $i$ of the
 In the derivation of the PCA theorem we will assume that the eigenvalues are ordered in descending order, that is
 $\lambda_0 > \lambda_1 > \dots > \lambda_{p-1}$. 
 
+
+The eigenvalues tell us then how much we need to stretch the
+corresponding eigenvectors. Dimensions with large eigenvalues have
+thus large variations (large variance) and define therefore useful
+dimensions. The data points are more spread out in the direction of
+these eigenvectors.  Smaller eigenvalues mean on the other hand that
+the corresponding eigenvectors are shrunk accordingly and the data
+points are tightly bunched together and there is not much variation in
+these specific directions. Hopefully then we could leave it out
+dimensions where the eigenvalues are very small. If $p$ is very large,
+we could then aim at reducing $p$ to $l << p$ and handle only $l$
+features/predictors.
+
+!split
+===== The Algorithm before the Theorem =====
+
+Here's how we would proceed in setting up the algorithm for the PCA, see also discussion below here. 
+* Set up the datapoints for the design/feature matrix $\bm{X}$ with $\bm{X}\in {\mathbb{R}}^{n\times p}$, with the predictors/features $p$  referring to the column numbers and the entries $n$ being the row elements.
+!bt
+\[
+\bm{X}=\begin{bmatrix}
+x_{0,0} & x_{0,1} & x_{0,2}& \dots & \dots x_{0,p-1}\\
+x_{1,0} & x_{1,1} & x_{1,2}& \dots & \dots x_{1,p-1}\\
+x_{2,0} & x_{2,1} & x_{2,2}& \dots & \dots x_{2,p-1}\\
+\dots & \dots & \dots & \dots \dots & \dots \\
+x_{n-2,0} & x_{n-2,1} & x_{n-2,2}& \dots & \dots x_{n-2,p-1}\\
+x_{n-1,0} & x_{n-1,1} & x_{n-1,2}& \dots & \dots x_{n-1,p-1}\\
+\end{bmatrix},
+\]
+!et
+* Center the data by subtracting the mean value for each column. This leads to a new matrix $\bm{X}\rightarrow \overline{\bm{X}}$.
+* Compute then the covariance/correlation matrix $\mathbb{E}[\overline{\bm{X}}\overline{\bm{X}}^T].
+* Find the eigenpairs of $\bm{C}$ with eigenvalues $[\lambda_0,\lambda_1,\dots,\lambda_{p-1}]$ and eigenvectors $[\bm{s}_0,\bm{s}_1,\dots,\bm{s}_{p-1}]$.
+* Order the eigenvalue (and the eigenvectors accordingly) in order of decreasing eigenvalues.
+* Keep only those $l$ eigenvalues larger than a selected threshold value, discarding thus $p-l$ features since we expect small variations in the data here.
+
+After this we ask ourselves how do we prove the link between the maximum variance and the feature reduction.
+
 !split
 ===== Classical PCA Theorem   =====
 
diff --git a/doc/src/DimRed/mlpfranke.py b/doc/src/DimRed/mlpfranke.py
new file mode 100644
index 000000000..639e605ef
--- /dev/null
+++ b/doc/src/DimRed/mlpfranke.py
@@ -0,0 +1,68 @@
+# Common imports
+import numpy as np
+from sklearn.neural_network import MLPRegressor
+from sklearn.metrics import accuracy_score
+import seaborn as sns
+import matplotlib.pyplot as plt
+
+def FrankeFunction(x,y):
+	term1 = 0.75*np.exp(-(0.25*(9*x-2)**2) - 0.25*((9*y-2)**2))
+	term2 = 0.75*np.exp(-((9*x+1)**2)/49.0 - 0.1*(9*y+1))
+	term3 = 0.5*np.exp(-(9*x-7)**2/4.0 - 0.25*((9*y-3)**2))
+	term4 = -0.2*np.exp(-(9*x-4)**2 - (9*y-7)**2)
+	return term1 + term2 + term3 + term4
+
+
+def create_X(x, y, n ):
+	if len(x.shape) > 1:
+		x = np.ravel(x)
+		y = np.ravel(y)
+
+	N = len(x)
+	l = int((n+1)*(n+2)/2)		# Number of elements in beta
+	X = np.ones((N,l))
+
+	for i in range(1,n+1):
+		q = int((i)*(i+1)/2)
+		for k in range(i+1):
+			X[:,q+k] = (x**(i-k))*(y**k)
+
+	return X
+
+
+# Making meshgrid of datapoints and compute Franke's function
+n = 4
+N = 100
+x = np.sort(np.random.uniform(0, 1, N))
+y = np.sort(np.random.uniform(0, 1, N))
+z = FrankeFunction(x, y)
+X = create_X(x, y, n=n)    
+
+# only training data, no advanced splitting
+X_train = X
+Y_train = z
+# only one simple layer with 100 neurons
+n_hidden_neurons = 100
+epochs = 100
+# store models for later use
+eta_vals = np.logspace(-5, 1, 7)
+lmbd_vals = np.logspace(-5, 1, 7)
+# store the models for later use
+DNN_scikit = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
+train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
+sns.set()
+for i, eta in enumerate(eta_vals):
+    for j, lmbd in enumerate(lmbd_vals):
+        dnn = MLPRegressor(hidden_layer_sizes=(n_hidden_neurons), activation='logistic',
+                            alpha=lmbd, learning_rate_init=eta, max_iter=epochs)
+        dnn.fit(X_train, Y_train)
+        DNN_scikit[i][j] = dnn
+        train_accuracy[i][j] = dnn.score(X_train, Y_train)
+
+fig, ax = plt.subplots(figsize = (10, 10))
+sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis")
+ax.set_title("Training Accuracy")
+ax.set_ylabel("$\eta$")
+ax.set_xlabel("$\lambda$")
+plt.show()
+