diff --git a/doc/pub/DimRed/html/._DimRed-bs019.html b/doc/pub/DimRed/html/._DimRed-bs019.html index 0281b99d7..e0a87c053 100644 --- a/doc/pub/DimRed/html/._DimRed-bs019.html +++ b/doc/pub/DimRed/html/._DimRed-bs019.html @@ -232,6 +232,9 @@ low-dimensional encoding of the data is then given by a set of vectors orthogonal projection of the data onto the columns spanned by the eigenvectors of the covariance(correlations matrix). +

+The proof which follows will be updated by mid January 2020. +

diff --git a/doc/pub/DimRed/html/._DimRed-bs024.html b/doc/pub/DimRed/html/._DimRed-bs024.html index 2be659898..9fd66c361 100644 --- a/doc/pub/DimRed/html/._DimRed-bs024.html +++ b/doc/pub/DimRed/html/._DimRed-bs024.html @@ -206,9 +206,8 @@ 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. diff --git a/doc/pub/DimRed/html/._DimRed-bs029.html b/doc/pub/DimRed/html/._DimRed-bs029.html index 1c29effce..8a4f7a1fd 100644 --- a/doc/pub/DimRed/html/._DimRed-bs029.html +++ b/doc/pub/DimRed/html/._DimRed-bs029.html @@ -213,11 +213,6 @@ algorithm that quickly finds an approximation of the first d principal component complexity is \( O(m \times d^2)+O(d^3) \), instead of \( O(m \times n^2) + O(n^3) \), so it is dramatically faster than the previous algorithms when \( d \) is much smaller than \( n \). -

-

-
- -

diff --git a/doc/pub/DimRed/html/DimRed-reveal.html b/doc/pub/DimRed/html/DimRed-reveal.html index 9c2731ba5..d27980dab 100644 --- a/doc/pub/DimRed/html/DimRed-reveal.html +++ b/doc/pub/DimRed/html/DimRed-reveal.html @@ -1143,6 +1143,9 @@ low-dimensional encoding of the data is then given by a set of vectors \( \boldsymbol{z}_i \) with at most \( l \) vectors, with \( l < < p \), defined by the orthogonal projection of the data onto the columns spanned by the eigenvectors of the covariance(correlations matrix). + +

+The proof which follows will be updated by mid January 2020. @@ -1298,8 +1301,7 @@ This material will be added by mid January 2020.

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. @@ -1477,9 +1479,6 @@ Scikit-Learn offers yet another option to perform PCA, called Randomized PCA. Th algorithm that quickly finds an approximation of the first d principal components. Its computational complexity is \( O(m \times d^2)+O(d^3) \), instead of \( O(m \times n^2) + O(n^3) \), so it is dramatically faster than the previous algorithms when \( d \) is much smaller than \( n \). - - -

diff --git a/doc/pub/DimRed/html/DimRed-solarized.html b/doc/pub/DimRed/html/DimRed-solarized.html index af3fb6759..f34ee2991 100644 --- a/doc/pub/DimRed/html/DimRed-solarized.html +++ b/doc/pub/DimRed/html/DimRed-solarized.html @@ -1115,6 +1115,9 @@ low-dimensional encoding of the data is then given by a set of vectors orthogonal projection of the data onto the columns spanned by the eigenvectors of the covariance(correlations matrix). +

+The proof which follows will be updated by mid January 2020. +











@@ -1245,8 +1248,7 @@ This material will be added by mid January 2020.









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. @@ -1257,7 +1259,7 @@ training set, then extracts the first two principal components. First we center

-

import numpy as np
+
import numpy as np
 import pandas as pd
 from IPython.display import display
 np.random.seed(100)
@@ -1294,7 +1296,7 @@ Selecting this hyperplane ensures that the projection will preserve as much vari
 

 
 
-
W2 = V.T[:, :2]
+
W2 = V.T[:, :2]
 X2D = X_centered.dot(W2)
 

 

@@ -1309,7 +1311,7 @@ that it automatically takes care of centering the data):
 

 
 
-
#thereafter we do a PCA with Scikit-learn
+
#thereafter we do a PCA with Scikit-learn
 from sklearn.decomposition import PCA
 pca = PCA(n_components = 2)
 X2D = pca.fit_transform(X)
@@ -1322,7 +1324,7 @@ principal component is equal to
 

 
 
-
pca.components_.T[:, 0].
+
pca.components_.T[:, 0].
 

 

 Another very useful piece of information is the explained variance ratio of each principal component,
@@ -1338,7 +1340,7 @@ Here we compute performance scores on the training data using logistic regressio
 

 
 
-
import matplotlib.pyplot as plt
+
import matplotlib.pyplot as plt
 import numpy as np
 from sklearn.model_selection import  train_test_split 
 from sklearn.datasets import load_breast_cancer
@@ -1385,7 +1387,7 @@ of dimensions required to preserve 95% of the training set’s variance:
 

 
 
-
pca = PCA()
+
pca = PCA()
 pca.fit(X)
 cumsum = np.cumsum(pca.explained_variance_ratio_)
 d = np.argmax(cumsum >= 0.95) + 1
@@ -1397,7 +1399,7 @@ a float between 0.0 and 1.0, indicating the ratio of variance you wish to preser
 

 
 
-
pca = PCA(n_components=0.95)
+
pca = PCA(n_components=0.95)
 X_reduced = pca.fit_transform(X)
 

 

@@ -1423,10 +1425,6 @@ algorithm that quickly finds an approximation of the first d principal component
 complexity is \( O(m \times d^2)+O(d^3) \), instead of \( O(m \times n^2) + O(n^3) \), so it is dramatically faster than the
 previous algorithms when \( d \) is much smaller than \( n \).
 
-
-

-
-
 

 










 
diff --git a/doc/pub/DimRed/html/DimRed.html b/doc/pub/DimRed/html/DimRed.html
index e9a79a081..e5348ddef 100644
--- a/doc/pub/DimRed/html/DimRed.html
+++ b/doc/pub/DimRed/html/DimRed.html
@@ -1120,6 +1120,9 @@ low-dimensional encoding of the data is then given by a set of vectors
 orthogonal projection of the data onto the columns spanned by the
 eigenvectors of the covariance(correlations matrix).
 
+

+The proof which follows will be updated by mid January 2020.
+
 

 










 
@@ -1250,8 +1253,7 @@ This material will be added by mid January 2020.
 










 
 
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.
@@ -1428,10 +1430,6 @@ algorithm that quickly finds an approximation of the first d principal component
 complexity is \( O(m \times d^2)+O(d^3) \), instead of \( O(m \times n^2) + O(n^3) \), so it is dramatically faster than the
 previous algorithms when \( d \) is much smaller than \( n \).
 
-
-

-
-
 

 










 
diff --git a/doc/pub/DimRed/ipynb/DimRed.ipynb b/doc/pub/DimRed/ipynb/DimRed.ipynb
index ec03b06a5..da58c850c 100644
--- a/doc/pub/DimRed/ipynb/DimRed.ipynb
+++ b/doc/pub/DimRed/ipynb/DimRed.ipynb
@@ -1253,7 +1253,7 @@
     "orthogonal projection of the data onto the columns spanned by the\n",
     "eigenvectors of the covariance(correlations matrix).\n",
     "\n",
-    "\n",
+    "The proof which follows will be updated by mid January 2020.\n",
     "\n",
     "## Proof of the PCA Theorem\n",
     "\n",
@@ -1492,6 +1492,7 @@
     "\n",
     "\n",
     "## Principal Component Analysis\n",
+    "\n",
     "Principal Component Analysis (PCA) is by far the most popular dimensionality reduction algorithm.\n",
     "First it identifies the hyperplane that lies closest to the data, and then it projects the data onto it.\n",
     "\n",
diff --git a/doc/pub/DimRed/ipynb/ipynb-DimRed-src.tar.gz b/doc/pub/DimRed/ipynb/ipynb-DimRed-src.tar.gz
index 363947881..81a89e92b 100644
Binary files a/doc/pub/DimRed/ipynb/ipynb-DimRed-src.tar.gz and b/doc/pub/DimRed/ipynb/ipynb-DimRed-src.tar.gz differ
diff --git a/doc/pub/DimRed/pdf/DimRed-minted.pdf b/doc/pub/DimRed/pdf/DimRed-minted.pdf
index a72e6b2de..5ea117840 100644
Binary files a/doc/pub/DimRed/pdf/DimRed-minted.pdf and b/doc/pub/DimRed/pdf/DimRed-minted.pdf differ
diff --git a/doc/src/DimRed/DimRed.do.txt b/doc/src/DimRed/DimRed.do.txt
index 621e66422..d347bf325 100644
--- a/doc/src/DimRed/DimRed.do.txt
+++ b/doc/src/DimRed/DimRed.do.txt
@@ -872,7 +872,7 @@ $\bm{z}_i$ with at most $l$ vectors, with $l << p$, defined by the
 orthogonal projection of the data onto the columns spanned by the
 eigenvectors of the covariance(correlations matrix).
 
-
+The proof which follows will be updated by mid January 2020.
 
 !split
 ===== Proof of the PCA Theorem  =====
@@ -1001,7 +1001,7 @@ This material will be added by mid January 2020.
 
 !split
 ===== Principal Component Analysis =====
-!bblock
+
 Principal Component Analysis (PCA) is by far the most popular dimensionality reduction algorithm.
 First it identifies the hyperplane that lies closest to the data, and then it projects the data onto it.
 
@@ -1148,7 +1148,6 @@ complexity is $O(m \times d^2)+O(d^3)$, instead of $O(m \times n^2) + O(n^3)$, s
 previous algorithms when $d$ is much smaller than $n$.
 
 
-!eblock
 
 
 !split