From 48e5ebfd924cff1dd111a1842a789d36b2d15fad Mon Sep 17 00:00:00 2001 From: mhjensen Date: Sun, 20 Oct 2019 23:11:22 +0200 Subject: [PATCH] adding more material, miss theorem and its proof --- doc/pub/DimRed/html/._DimRed-bs000.html | 57 +++-- doc/pub/DimRed/html/._DimRed-bs001.html | 57 +++-- doc/pub/DimRed/html/._DimRed-bs002.html | 57 +++-- doc/pub/DimRed/html/._DimRed-bs003.html | 57 +++-- doc/pub/DimRed/html/._DimRed-bs004.html | 57 +++-- doc/pub/DimRed/html/._DimRed-bs005.html | 57 +++-- doc/pub/DimRed/html/._DimRed-bs006.html | 57 +++-- doc/pub/DimRed/html/._DimRed-bs007.html | 57 +++-- doc/pub/DimRed/html/._DimRed-bs008.html | 57 +++-- doc/pub/DimRed/html/._DimRed-bs009.html | 61 ++--- doc/pub/DimRed/html/DimRed-bs.html | 57 +++-- doc/pub/DimRed/html/DimRed-reveal.html | 193 +++++++++++++--- doc/pub/DimRed/html/DimRed-solarized.html | 222 +++++++++++++++---- doc/pub/DimRed/html/DimRed.html | 222 +++++++++++++++---- doc/pub/DimRed/ipynb/DimRed.ipynb | 210 +++++++++++++++--- doc/pub/DimRed/ipynb/ipynb-DimRed-src.tar.gz | Bin 190 -> 191 bytes doc/pub/DimRed/pdf/DimRed-minted.pdf | Bin 235597 -> 236978 bytes doc/src/DimRed/DimRed.do.txt | 152 +++++++++++-- doc/src/DimRed/covariance.py | 24 ++ doc/src/DimRed/covfrance.py | 43 ++++ doc/src/DimRed/newcov.py | 28 +++ 21 files changed, 1324 insertions(+), 401 deletions(-) create mode 100644 doc/src/DimRed/covariance.py create mode 100644 doc/src/DimRed/covfrance.py create mode 100644 doc/src/DimRed/newcov.py diff --git a/doc/pub/DimRed/html/._DimRed-bs000.html b/doc/pub/DimRed/html/._DimRed-bs000.html index 4d83066c4..74d5b19a6 100644 --- a/doc/pub/DimRed/html/._DimRed-bs000.html +++ b/doc/pub/DimRed/html/._DimRed-bs000.html @@ -76,17 +76,24 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec9'), - ('Classical PCA Theorem', 2, None, '___sec10'), - ('Prof of the PCA Theorem', 2, None, '___sec11'), - ('Getting started with PCA', 2, None, '___sec12'), - ('Principal Component Analysis', 2, None, '___sec13'), - ('PCA and scikit-learn', 2, None, '___sec14'), - ('More on the PCA', 2, None, '___sec15'), - ('Incremental PCA', 2, None, '___sec16'), - ('Randomized PCA', 2, None, '___sec17'), - ('Kernel PCA', 2, None, '___sec18'), - ('LLE', 2, None, '___sec19'), - ('Other techniques', 2, None, '___sec20')]} + ('Covariance Matrix Examples', 2, None, '___sec10'), + ('Correlation Matrix', 2, None, '___sec11'), + ('Correlation Matrix with Pandas', 2, None, '___sec12'), + ('Correlation Matrix with Pandas and the Franke function', + 2, + None, + '___sec13'), + ('Classical PCA Theorem', 2, None, '___sec14'), + ('Prof of the PCA Theorem', 2, None, '___sec15'), + ('Getting started with PCA', 2, None, '___sec16'), + ('Principal Component Analysis', 2, None, '___sec17'), + ('PCA and scikit-learn', 2, None, '___sec18'), + ('More on the PCA', 2, None, '___sec19'), + ('Incremental PCA', 2, None, '___sec20'), + ('Randomized PCA', 2, None, '___sec21'), + ('Kernel PCA', 2, None, '___sec22'), + ('LLE', 2, None, '___sec23'), + ('Other techniques', 2, None, '___sec24')]} end of tocinfo --> @@ -134,17 +141,21 @@ MathJax.Hub.Config({
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    • @@ -203,7 +214,7 @@ MathJax.Hub.Config({
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    • @@ -203,7 +214,7 @@ data.
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    • @@ -202,7 +213,7 @@ ensures that all features are exactly between \( 0 \) and \( 1 \). The
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    • @@ -205,7 +216,7 @@ techniques.
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    • @@ -280,7 +291,7 @@ svm.fit(X_train_scaled, y_train)
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    • @@ -230,7 +241,7 @@ svm.fit(X_train_scaled, y_train)
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    • @@ -209,7 +220,7 @@ logreg.fit(X_train_scaled, y_train)
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    • diff --git a/doc/pub/DimRed/html/._DimRed-bs007.html b/doc/pub/DimRed/html/._DimRed-bs007.html index cd6de3817..e6898fbda 100644 --- a/doc/pub/DimRed/html/._DimRed-bs007.html +++ b/doc/pub/DimRed/html/._DimRed-bs007.html @@ -76,17 +76,24 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec9'), - ('Classical PCA Theorem', 2, None, '___sec10'), - ('Prof of the PCA Theorem', 2, None, '___sec11'), - ('Getting started with PCA', 2, None, '___sec12'), - ('Principal Component Analysis', 2, None, '___sec13'), - ('PCA and scikit-learn', 2, None, '___sec14'), - ('More on the PCA', 2, None, '___sec15'), - ('Incremental PCA', 2, None, '___sec16'), - ('Randomized PCA', 2, None, '___sec17'), - ('Kernel PCA', 2, None, '___sec18'), - ('LLE', 2, None, '___sec19'), - ('Other techniques', 2, None, '___sec20')]} + ('Covariance Matrix Examples', 2, None, '___sec10'), + ('Correlation Matrix', 2, None, '___sec11'), + ('Correlation Matrix with Pandas', 2, None, '___sec12'), + ('Correlation Matrix with Pandas and the Franke function', + 2, + None, + '___sec13'), + ('Classical PCA Theorem', 2, None, '___sec14'), + ('Prof of the PCA Theorem', 2, None, '___sec15'), + ('Getting started with PCA', 2, None, '___sec16'), + ('Principal Component Analysis', 2, None, '___sec17'), + ('PCA and scikit-learn', 2, None, '___sec18'), + ('More on the PCA', 2, None, '___sec19'), + ('Incremental PCA', 2, None, '___sec20'), + ('Randomized PCA', 2, None, '___sec21'), + ('Kernel PCA', 2, None, '___sec22'), + ('LLE', 2, None, '___sec23'), + ('Other techniques', 2, None, '___sec24')]} end of tocinfo --> @@ -134,17 +141,21 @@ MathJax.Hub.Config({
  • Basic ideas of the Principal Component Analysis (PCA)
  • Introducing the Covariance and Correlation functions
  • Correlation Function and Design/Feature Matrix
    • -
  • 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
    • +
  • Covariance Matrix Examples
    • +
  • Correlation Matrix
    • +
  • Correlation Matrix with Pandas
    • +
  • Correlation Matrix with Pandas and the Franke function
    • +
  • 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
    • @@ -264,7 +275,7 @@ applications.
  • 16
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    • diff --git a/doc/pub/DimRed/html/._DimRed-bs008.html b/doc/pub/DimRed/html/._DimRed-bs008.html index 8f476920d..cbc06114e 100644 --- a/doc/pub/DimRed/html/._DimRed-bs008.html +++ b/doc/pub/DimRed/html/._DimRed-bs008.html @@ -76,17 +76,24 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec9'), - ('Classical PCA Theorem', 2, None, '___sec10'), - ('Prof of the PCA Theorem', 2, None, '___sec11'), - ('Getting started with PCA', 2, None, '___sec12'), - ('Principal Component Analysis', 2, None, '___sec13'), - ('PCA and scikit-learn', 2, None, '___sec14'), - ('More on the PCA', 2, None, '___sec15'), - ('Incremental PCA', 2, None, '___sec16'), - ('Randomized PCA', 2, None, '___sec17'), - ('Kernel PCA', 2, None, '___sec18'), - ('LLE', 2, None, '___sec19'), - ('Other techniques', 2, None, '___sec20')]} + ('Covariance Matrix Examples', 2, None, '___sec10'), + ('Correlation Matrix', 2, None, '___sec11'), + ('Correlation Matrix with Pandas', 2, None, '___sec12'), + ('Correlation Matrix with Pandas and the Franke function', + 2, + None, + '___sec13'), + ('Classical PCA Theorem', 2, None, '___sec14'), + ('Prof of the PCA Theorem', 2, None, '___sec15'), + ('Getting started with PCA', 2, None, '___sec16'), + ('Principal Component Analysis', 2, None, '___sec17'), + ('PCA and scikit-learn', 2, None, '___sec18'), + ('More on the PCA', 2, None, '___sec19'), + ('Incremental PCA', 2, None, '___sec20'), + ('Randomized PCA', 2, None, '___sec21'), + ('Kernel PCA', 2, None, '___sec22'), + ('LLE', 2, None, '___sec23'), + ('Other techniques', 2, None, '___sec24')]} end of tocinfo --> @@ -134,17 +141,21 @@ MathJax.Hub.Config({
  • Basic ideas of the Principal Component Analysis (PCA)
  • Introducing the Covariance and Correlation functions
  • Correlation Function and Design/Feature Matrix
    • -
  • 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
    • +
  • Covariance Matrix Examples
    • +
  • Correlation Matrix
    • +
  • Correlation Matrix with Pandas
    • +
  • Correlation Matrix with Pandas and the Franke function
    • +
  • 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
    • @@ -194,7 +205,7 @@ 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 4952e13bb..b8821f107 100644 --- a/doc/pub/DimRed/html/._DimRed-bs009.html +++ b/doc/pub/DimRed/html/._DimRed-bs009.html @@ -76,17 +76,24 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec9'), - ('Classical PCA Theorem', 2, None, '___sec10'), - ('Prof of the PCA Theorem', 2, None, '___sec11'), - ('Getting started with PCA', 2, None, '___sec12'), - ('Principal Component Analysis', 2, None, '___sec13'), - ('PCA and scikit-learn', 2, None, '___sec14'), - ('More on the PCA', 2, None, '___sec15'), - ('Incremental PCA', 2, None, '___sec16'), - ('Randomized PCA', 2, None, '___sec17'), - ('Kernel PCA', 2, None, '___sec18'), - ('LLE', 2, None, '___sec19'), - ('Other techniques', 2, None, '___sec20')]} + ('Covariance Matrix Examples', 2, None, '___sec10'), + ('Correlation Matrix', 2, None, '___sec11'), + ('Correlation Matrix with Pandas', 2, None, '___sec12'), + ('Correlation Matrix with Pandas and the Franke function', + 2, + None, + '___sec13'), + ('Classical PCA Theorem', 2, None, '___sec14'), + ('Prof of the PCA Theorem', 2, None, '___sec15'), + ('Getting started with PCA', 2, None, '___sec16'), + ('Principal Component Analysis', 2, None, '___sec17'), + ('PCA and scikit-learn', 2, None, '___sec18'), + ('More on the PCA', 2, None, '___sec19'), + ('Incremental PCA', 2, None, '___sec20'), + ('Randomized PCA', 2, None, '___sec21'), + ('Kernel PCA', 2, None, '___sec22'), + ('LLE', 2, None, '___sec23'), + ('Other techniques', 2, None, '___sec24')]} end of tocinfo --> @@ -134,17 +141,21 @@ MathJax.Hub.Config({
  • Basic ideas of the Principal Component Analysis (PCA)
  • Introducing the Covariance and Correlation functions
  • Correlation Function and Design/Feature Matrix
    • -
  • 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
    • +
  • Covariance Matrix Examples
    • +
  • Correlation Matrix
    • +
  • Correlation Matrix with Pandas
    • +
  • Correlation Matrix with Pandas and the Franke function
    • +
  • 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
    • @@ -164,7 +175,7 @@ MathJax.Hub.Config({

    Suppose we have defined two vectors -$\hat{x} and \hat{y} with \( n \) elements each. The covariance matrix $\boldsymbol{C}is defined as +\( \hat{x} \) and \( \hat{y} \) with \( n \) elements each. The covariance matrix \( \boldsymbol{C} \) is defined as $$ \boldsymbol{C}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} cov[\boldsymbol{x},\boldsymbol{x}] & cov[\boldsymbol{x},\boldsymbol{y}] \\ cov[\boldsymbol{y},\boldsymbol{x}] & cov[\boldsymbol{y},\boldsymbol{y}] \\ @@ -196,7 +207,7 @@ introducing instead the correlation matrix defined via the so-called correlation function $$ -corr[\boldsymbol{x},\boldsymbol{y}]=\frac{cov[\boldsymbol{x},\boldsymbol{y}]}{\sqrt{var[\boldsymbol{x}]\var\boldsymbol{y}]}}. +corr[\boldsymbol{x},\boldsymbol{y}]=\frac{cov[\boldsymbol{x},\boldsymbol{y}]}{\sqrt{var[\boldsymbol{x}]\var[\boldsymbol{y}]}}. $$

    @@ -239,7 +250,7 @@ In the above example this is the function we constructed using pandas.

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    • diff --git a/doc/pub/DimRed/html/DimRed-bs.html b/doc/pub/DimRed/html/DimRed-bs.html index 4d83066c4..74d5b19a6 100644 --- a/doc/pub/DimRed/html/DimRed-bs.html +++ b/doc/pub/DimRed/html/DimRed-bs.html @@ -76,17 +76,24 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec9'), - ('Classical PCA Theorem', 2, None, '___sec10'), - ('Prof of the PCA Theorem', 2, None, '___sec11'), - ('Getting started with PCA', 2, None, '___sec12'), - ('Principal Component Analysis', 2, None, '___sec13'), - ('PCA and scikit-learn', 2, None, '___sec14'), - ('More on the PCA', 2, None, '___sec15'), - ('Incremental PCA', 2, None, '___sec16'), - ('Randomized PCA', 2, None, '___sec17'), - ('Kernel PCA', 2, None, '___sec18'), - ('LLE', 2, None, '___sec19'), - ('Other techniques', 2, None, '___sec20')]} + ('Covariance Matrix Examples', 2, None, '___sec10'), + ('Correlation Matrix', 2, None, '___sec11'), + ('Correlation Matrix with Pandas', 2, None, '___sec12'), + ('Correlation Matrix with Pandas and the Franke function', + 2, + None, + '___sec13'), + ('Classical PCA Theorem', 2, None, '___sec14'), + ('Prof of the PCA Theorem', 2, None, '___sec15'), + ('Getting started with PCA', 2, None, '___sec16'), + ('Principal Component Analysis', 2, None, '___sec17'), + ('PCA and scikit-learn', 2, None, '___sec18'), + ('More on the PCA', 2, None, '___sec19'), + ('Incremental PCA', 2, None, '___sec20'), + ('Randomized PCA', 2, None, '___sec21'), + ('Kernel PCA', 2, None, '___sec22'), + ('LLE', 2, None, '___sec23'), + ('Other techniques', 2, None, '___sec24')]} end of tocinfo --> @@ -134,17 +141,21 @@ MathJax.Hub.Config({
  • Basic ideas of the Principal Component Analysis (PCA)
  • Introducing the Covariance and Correlation functions
  • Correlation Function and Design/Feature Matrix
    • -
  • 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
    • +
  • Covariance Matrix Examples
    • +
  • Correlation Matrix
    • +
  • Correlation Matrix with Pandas
    • +
  • Correlation Matrix with Pandas and the Franke function
    • +
  • 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
    • @@ -203,7 +214,7 @@ MathJax.Hub.Config({
  • 9
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    • diff --git a/doc/pub/DimRed/html/DimRed-reveal.html b/doc/pub/DimRed/html/DimRed-reveal.html index 2513c9634..0886daa11 100644 --- a/doc/pub/DimRed/html/DimRed-reveal.html +++ b/doc/pub/DimRed/html/DimRed-reveal.html @@ -526,7 +526,7 @@ We have a data set defined by a design/feature matrix \( \boldsymbol{X} \) (see

    Suppose we have defined two vectors -$\hat{x} and \hat{y} with \( n \) elements each. The covariance matrix $\boldsymbol{C}is defined as +\( \hat{x} \) and \( \hat{y} \) with \( n \) elements each. The covariance matrix \( \boldsymbol{C} \) is defined as

     
    $$ \boldsymbol{C}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} cov[\boldsymbol{x},\boldsymbol{x}] & cov[\boldsymbol{x},\boldsymbol{y}] \\ @@ -567,7 +567,7 @@ correlation function

     
    $$ -corr[\boldsymbol{x},\boldsymbol{y}]=\frac{cov[\boldsymbol{x},\boldsymbol{y}]}{\sqrt{var[\boldsymbol{x}]\var\boldsymbol{y}]}}. +corr[\boldsymbol{x},\boldsymbol{y}]=\frac{cov[\boldsymbol{x},\boldsymbol{y}]}{\sqrt{var[\boldsymbol{x}]\var[\boldsymbol{y}]}}. $$

     
    @@ -652,13 +652,22 @@ corr[\boldsymbol{x}_{p-1},\boldsymbol{x}_0] & corr[\boldsymbol{x}_{p-1},\bolds \end{bmatrix}, $$

     
    + + + +

    +

    Covariance Matrix Examples

    -The Numpy function np.cov calculates the covariance elements using the factor \( 1/(n-1) \) instead of \( 1/n \) since it assumes we do not have the exact mean values. -The following simple function uses the np.vstack function which takes each vector of dimension \( 1\times n \) and produces a \( 2\times n \) matrix \( \hat{W} \) +The Numpy function np.cov calculates the covariance elements using +the factor \( 1/(n-1) \) instead of \( 1/n \) since it assumes we do not have +the exact mean values. The following simple function uses the +np.vstack function which takes each vector of dimension \( 1\times n \) +and produces a \( 2\times n \) matrix \( \boldsymbol{W} \) +

     
    $$ -\hat{W} = \begin{bmatrix} x_0 & y_0 \\ +\boldsymbol{W} = \begin{bmatrix} x_0 & y_0 \\ x_1 & y_1 \\ x_2 & y_2\\ \dots & \dots \\ @@ -669,9 +678,9 @@ $$

     

    -which in turn is converted into into the \( 3\times 3 \) covariance matrix -\( \hat{\Sigma} \) via the Numpy function np.cov(). We note that we can also calculate -the mean value of each set of samples \( \hat{x} \) etc using the Numpy +which in turn is converted into into the \( 2\times 2 \) covariance matrix +\( \boldsymbol{C} \) via the Numpy function np.cov(). We note that we can also calculate +the mean value of each set of samples \( \boldsymbol{x} \) etc using the Numpy function np.mean(x). We can also extract the eigenvalues of the covariance matrix through the np.linalg.eig() function. @@ -680,35 +689,165 @@ covariance matrix through the np.linalg.eig() function. 

    # Importing various packages
 import numpy as np
-
 n = 100
 x = np.random.normal(size=n)
 print(np.mean(x))
 y = 4+3*x+np.random.normal(size=n)
 print(np.mean(y))
-z = x**3+np.random.normal(size=n)
-print(np.mean(z))
-W = np.vstack((x, y, z))
-Sigma = np.cov(W)
-print(Sigma)
-Eigvals, Eigvecs = np.linalg.eig(Sigma)
-print(Eigvals)
+W = np.vstack((x, y))
+C = np.cov(W)
+print(C)
    -

    Classical PCA Theorem

    +

    Correlation Matrix

    + +

    +The previous example can be converted into the correlation matrix by +simply scaling the matrix elements with the variances. We should also +subtract the mean values for each column. This leads to the following +code which sets up the correlations matrix for the previous example in +a more brute force way. Here we scale the mean values for each column of the design matrix, calculate the relevant mean values and variances and then finally set up the \( 2\times 2 \) correlation matrix (since we have only two vectors). + +

    + + +

    import numpy as np
+n = 100
+# define two vectors                                                                                           
+x = np.random.random(size=n)
+y = 4+3*x+np.random.normal(size=n)
+#scaling the x and y vectors                                                                                   
+x = x - np.mean(x)
+y = y - np.mean(y)
+variance_x = np.sum(x@x)/n
+variance_y = np.sum(y@y)/n
+print(variance_x)
+print(variance_y)
+cov_xy = np.sum(x@y)/n
+cov_xx = np.sum(x@x)/n
+cov_yy = np.sum(y@y)/n
+C = np.zeros((2,2))
+C[0,0]= cov_xx/variance_x
+C[1,1]= cov_yy/variance_y
+C[0,1]= cov_xy/np.sqrt(variance_y*variance_x)
+C[1,0]= C[0,1]
+print(C)
+
    +

    +We see that the matrix elements along the diagonal are one as they +should be and that the matrix is symmetric. Furthermore, diagonalizing +this matrix we easily see that it is a positive definite matrix. + +

    +The above procedure with numpy can be made more compact if we use pandas.

    -

    Prof of the PCA Theorem

    +

    Correlation Matrix with Pandas

    + +

    +We whow here how we can set up the correlation matrix using pandas, as done in this simple code +

    + + +

    import numpy as np
+import pandas as pd
+n = 10
+x = np.random.normal(size=n)
+x = x - np.mean(x)
+y = 4+3*x+np.random.normal(size=n)
+y = y - np.mean(y)
+X = (np.vstack((x, y))).T
+print(X)
+Xpd = pd.DataFrame(X)
+print(Xpd)
+correlation_matrix = Xpd.corr()
+print(correlation_matrix)
+
    +

    +We expand this model to the Franke function discussed above.

    -

    Getting started with PCA

    +

    Correlation Matrix with Pandas and the Franke function

    + +

    + + +

    # Common imports
+import numpy as np
+import pandas as pd
+
+
+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)    
+
+Xpd = pd.DataFrame(X)
+# subtract the mean values and set up the covariance matrix
+Xpd = Xpd - Xpd.mean()
+covariance_matrix = Xpd.cov()
+print(covariance_matrix)
+
    +

    +We note here that the covariance is zero for the first rows and +columns since all matrix elements in the design matrix were set to one +(we are fitting the function in terms of a polynomial of degree \( n \)). + +

    +This means that the variance for these elements will be zero and will +cause problems when we set up the correlation matrix. We can simply +drop these elements as follows and then construct the correlation +matrix. +

    + + +
    +

    Classical PCA Theorem

    +
    + + +
    +

    Prof of the PCA Theorem

    +
    + + +
    +

    Getting started with PCA

    @@ -724,7 +863,7 @@ X_pca = pca.transform(X_train_scaled)

    -

    Principal Component Analysis

    +

    Principal Component Analysis

    @@ -761,7 +900,7 @@ 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 @@ -792,7 +931,7 @@ 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 @@ -823,7 +962,7 @@ 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 @@ -835,7 +974,7 @@ instances arrive).

    -

    Randomized PCA

    +

    Randomized PCA

    Scikit-Learn offers yet another option to perform PCA, called Randomized PCA. This is a stochastic @@ -849,7 +988,7 @@ previous algorithms when \( d \) is much smaller than \( n \).

    -

    Kernel PCA

    +

    Kernel PCA

    @@ -875,7 +1014,7 @@ X_reduced = rbf_pca.fit_transform(X)

    -

    LLE

    +

    LLE

    Locally Linear Embedding (LLE) is another very powerful nonlinear dimensionality reduction @@ -887,7 +1026,7 @@ 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 dbbe0c5ef..c215b4a80 100644 --- a/doc/pub/DimRed/html/DimRed-solarized.html +++ b/doc/pub/DimRed/html/DimRed-solarized.html @@ -96,17 +96,24 @@ div { text-align: justify; text-justify: inter-word; } 2, None, '___sec9'), - ('Classical PCA Theorem', 2, None, '___sec10'), - ('Prof of the PCA Theorem', 2, None, '___sec11'), - ('Getting started with PCA', 2, None, '___sec12'), - ('Principal Component Analysis', 2, None, '___sec13'), - ('PCA and scikit-learn', 2, None, '___sec14'), - ('More on the PCA', 2, None, '___sec15'), - ('Incremental PCA', 2, None, '___sec16'), - ('Randomized PCA', 2, None, '___sec17'), - ('Kernel PCA', 2, None, '___sec18'), - ('LLE', 2, None, '___sec19'), - ('Other techniques', 2, None, '___sec20')]} + ('Covariance Matrix Examples', 2, None, '___sec10'), + ('Correlation Matrix', 2, None, '___sec11'), + ('Correlation Matrix with Pandas', 2, None, '___sec12'), + ('Correlation Matrix with Pandas and the Franke function', + 2, + None, + '___sec13'), + ('Classical PCA Theorem', 2, None, '___sec14'), + ('Prof of the PCA Theorem', 2, None, '___sec15'), + ('Getting started with PCA', 2, None, '___sec16'), + ('Principal Component Analysis', 2, None, '___sec17'), + ('PCA and scikit-learn', 2, None, '___sec18'), + ('More on the PCA', 2, None, '___sec19'), + ('Incremental PCA', 2, None, '___sec20'), + ('Randomized PCA', 2, None, '___sec21'), + ('Kernel PCA', 2, None, '___sec22'), + ('LLE', 2, None, '___sec23'), + ('Other techniques', 2, None, '___sec24')]} end of tocinfo --> @@ -522,7 +529,7 @@ We have a data set defined by a design/feature matrix \( \boldsymbol{X} \) (see

    Suppose we have defined two vectors -$\hat{x} and \hat{y} with \( n \) elements each. The covariance matrix $\boldsymbol{C}is defined as +\( \hat{x} \) and \( \hat{y} \) with \( n \) elements each. The covariance matrix \( \boldsymbol{C} \) is defined as $$ \boldsymbol{C}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} cov[\boldsymbol{x},\boldsymbol{x}] & cov[\boldsymbol{x},\boldsymbol{y}] \\ cov[\boldsymbol{y},\boldsymbol{x}] & cov[\boldsymbol{y},\boldsymbol{y}] \\ @@ -554,7 +561,7 @@ introducing instead the correlation matrix defined via the so-called correlation function $$ -corr[\boldsymbol{x},\boldsymbol{y}]=\frac{cov[\boldsymbol{x},\boldsymbol{y}]}{\sqrt{var[\boldsymbol{x}]\var\boldsymbol{y}]}}. +corr[\boldsymbol{x},\boldsymbol{y}]=\frac{cov[\boldsymbol{x},\boldsymbol{y}]}{\sqrt{var[\boldsymbol{x}]\var[\boldsymbol{y}]}}. $$

    @@ -628,10 +635,19 @@ corr[\boldsymbol{x}_{p-1},\boldsymbol{x}_0] & corr[\boldsymbol{x}_{p-1},\bolds $$

    -The Numpy function np.cov calculates the covariance elements using the factor \( 1/(n-1) \) instead of \( 1/n \) since it assumes we do not have the exact mean values. -The following simple function uses the np.vstack function which takes each vector of dimension \( 1\times n \) and produces a \( 2\times n \) matrix \( \hat{W} \) +









    + +

    Covariance Matrix Examples

    + +

    +The Numpy function np.cov calculates the covariance elements using +the factor \( 1/(n-1) \) instead of \( 1/n \) since it assumes we do not have +the exact mean values. The following simple function uses the +np.vstack function which takes each vector of dimension \( 1\times n \) +and produces a \( 2\times n \) matrix \( \boldsymbol{W} \) + $$ -\hat{W} = \begin{bmatrix} x_0 & y_0 \\ +\boldsymbol{W} = \begin{bmatrix} x_0 & y_0 \\ x_1 & y_1 \\ x_2 & y_2\\ \dots & \dots \\ @@ -641,9 +657,9 @@ $$ $$

    -which in turn is converted into into the \( 3\times 3 \) covariance matrix -\( \hat{\Sigma} \) via the Numpy function np.cov(). We note that we can also calculate -the mean value of each set of samples \( \hat{x} \) etc using the Numpy +which in turn is converted into into the \( 2\times 2 \) covariance matrix +\( \boldsymbol{C} \) via the Numpy function np.cov(). We note that we can also calculate +the mean value of each set of samples \( \boldsymbol{x} \) etc using the Numpy function np.mean(x). We can also extract the eigenvalues of the covariance matrix through the np.linalg.eig() function. @@ -652,34 +668,164 @@ covariance matrix through the np.linalg.eig() function. 

    # Importing various packages
 import numpy as np
-
 n = 100
 x = np.random.normal(size=n)
 print(np.mean(x))
 y = 4+3*x+np.random.normal(size=n)
 print(np.mean(y))
-z = x**3+np.random.normal(size=n)
-print(np.mean(z))
-W = np.vstack((x, y, z))
-Sigma = np.cov(W)
-print(Sigma)
-Eigvals, Eigvecs = np.linalg.eig(Sigma)
-print(Eigvals)
+W = np.vstack((x, y))
+C = np.cov(W)
+print(C)











    -

    Classical PCA Theorem

    +

    Correlation Matrix

    + +

    +The previous example can be converted into the correlation matrix by +simply scaling the matrix elements with the variances. We should also +subtract the mean values for each column. This leads to the following +code which sets up the correlations matrix for the previous example in +a more brute force way. Here we scale the mean values for each column of the design matrix, calculate the relevant mean values and variances and then finally set up the \( 2\times 2 \) correlation matrix (since we have only two vectors). + +

    + + +

    import numpy as np
+n = 100
+# define two vectors                                                                                           
+x = np.random.random(size=n)
+y = 4+3*x+np.random.normal(size=n)
+#scaling the x and y vectors                                                                                   
+x = x - np.mean(x)
+y = y - np.mean(y)
+variance_x = np.sum(x@x)/n
+variance_y = np.sum(y@y)/n
+print(variance_x)
+print(variance_y)
+cov_xy = np.sum(x@y)/n
+cov_xx = np.sum(x@x)/n
+cov_yy = np.sum(y@y)/n
+C = np.zeros((2,2))
+C[0,0]= cov_xx/variance_x
+C[1,1]= cov_yy/variance_y
+C[0,1]= cov_xy/np.sqrt(variance_y*variance_x)
+C[1,0]= C[0,1]
+print(C)
+
    +

    +We see that the matrix elements along the diagonal are one as they +should be and that the matrix is symmetric. Furthermore, diagonalizing +this matrix we easily see that it is a positive definite matrix. + +

    +The above procedure with numpy can be made more compact if we use pandas.











    -

    Prof of the PCA Theorem

    +

    Correlation Matrix with Pandas

    + +

    +We whow here how we can set up the correlation matrix using pandas, as done in this simple code +

    + + +

    import numpy as np
+import pandas as pd
+n = 10
+x = np.random.normal(size=n)
+x = x - np.mean(x)
+y = 4+3*x+np.random.normal(size=n)
+y = y - np.mean(y)
+X = (np.vstack((x, y))).T
+print(X)
+Xpd = pd.DataFrame(X)
+print(Xpd)
+correlation_matrix = Xpd.corr()
+print(correlation_matrix)
+
    +

    +We expand this model to the Franke function discussed above.











    -

    Getting started with PCA

    +

    Correlation Matrix with Pandas and the Franke function

    + +

    + + +

    # Common imports
+import numpy as np
+import pandas as pd
+
+
+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)    
+
+Xpd = pd.DataFrame(X)
+# subtract the mean values and set up the covariance matrix
+Xpd = Xpd - Xpd.mean()
+covariance_matrix = Xpd.cov()
+print(covariance_matrix)
+
    +

    +We note here that the covariance is zero for the first rows and +columns since all matrix elements in the design matrix were set to one +(we are fitting the function in terms of a polynomial of degree \( n \)). + +

    +This means that the variance for these elements will be zero and will +cause problems when we set up the correlation matrix. We can simply +drop these elements as follows and then construct the correlation +matrix. + +

    +









    + +

    Classical PCA Theorem

    + +

    +









    + +

    Prof of the PCA Theorem

    + +

    +









    + +

    Getting started with PCA

    @@ -694,7 +840,7 @@ X_pca = pca.transform(X_train_scaled)











    -

    Principal Component Analysis

    +

    Principal Component Analysis

    @@ -730,7 +876,7 @@ 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 @@ -761,7 +907,7 @@ 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 @@ -791,7 +937,7 @@ 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 @@ -803,7 +949,7 @@ instances arrive).











    -

    Randomized PCA

    +

    Randomized PCA

    Scikit-Learn offers yet another option to perform PCA, called Randomized PCA. This is a stochastic @@ -818,7 +964,7 @@ previous algorithms when \( d \) is much smaller than \( n \).











    -

    Kernel PCA

    +

    Kernel PCA

    @@ -847,7 +993,7 @@ X_reduced = rbf_pca.fit_transform(X)











    -

    LLE

    +

    LLE

    Locally Linear Embedding (LLE) is another very powerful nonlinear dimensionality reduction @@ -859,7 +1005,7 @@ 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 8478d5b7f..d3244342a 100644 --- a/doc/pub/DimRed/html/DimRed.html +++ b/doc/pub/DimRed/html/DimRed.html @@ -101,17 +101,24 @@ div { text-align: justify; text-justify: inter-word; } 2, None, '___sec9'), - ('Classical PCA Theorem', 2, None, '___sec10'), - ('Prof of the PCA Theorem', 2, None, '___sec11'), - ('Getting started with PCA', 2, None, '___sec12'), - ('Principal Component Analysis', 2, None, '___sec13'), - ('PCA and scikit-learn', 2, None, '___sec14'), - ('More on the PCA', 2, None, '___sec15'), - ('Incremental PCA', 2, None, '___sec16'), - ('Randomized PCA', 2, None, '___sec17'), - ('Kernel PCA', 2, None, '___sec18'), - ('LLE', 2, None, '___sec19'), - ('Other techniques', 2, None, '___sec20')]} + ('Covariance Matrix Examples', 2, None, '___sec10'), + ('Correlation Matrix', 2, None, '___sec11'), + ('Correlation Matrix with Pandas', 2, None, '___sec12'), + ('Correlation Matrix with Pandas and the Franke function', + 2, + None, + '___sec13'), + ('Classical PCA Theorem', 2, None, '___sec14'), + ('Prof of the PCA Theorem', 2, None, '___sec15'), + ('Getting started with PCA', 2, None, '___sec16'), + ('Principal Component Analysis', 2, None, '___sec17'), + ('PCA and scikit-learn', 2, None, '___sec18'), + ('More on the PCA', 2, None, '___sec19'), + ('Incremental PCA', 2, None, '___sec20'), + ('Randomized PCA', 2, None, '___sec21'), + ('Kernel PCA', 2, None, '___sec22'), + ('LLE', 2, None, '___sec23'), + ('Other techniques', 2, None, '___sec24')]} end of tocinfo --> @@ -527,7 +534,7 @@ We have a data set defined by a design/feature matrix \( \boldsymbol{X} \) (see

    Suppose we have defined two vectors -$\hat{x} and \hat{y} with \( n \) elements each. The covariance matrix $\boldsymbol{C}is defined as +\( \hat{x} \) and \( \hat{y} \) with \( n \) elements each. The covariance matrix \( \boldsymbol{C} \) is defined as $$ \boldsymbol{C}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} cov[\boldsymbol{x},\boldsymbol{x}] & cov[\boldsymbol{x},\boldsymbol{y}] \\ cov[\boldsymbol{y},\boldsymbol{x}] & cov[\boldsymbol{y},\boldsymbol{y}] \\ @@ -559,7 +566,7 @@ introducing instead the correlation matrix defined via the so-called correlation function $$ -corr[\boldsymbol{x},\boldsymbol{y}]=\frac{cov[\boldsymbol{x},\boldsymbol{y}]}{\sqrt{var[\boldsymbol{x}]\var\boldsymbol{y}]}}. +corr[\boldsymbol{x},\boldsymbol{y}]=\frac{cov[\boldsymbol{x},\boldsymbol{y}]}{\sqrt{var[\boldsymbol{x}]\var[\boldsymbol{y}]}}. $$

    @@ -633,10 +640,19 @@ corr[\boldsymbol{x}_{p-1},\boldsymbol{x}_0] & corr[\boldsymbol{x}_{p-1},\bolds $$

    -The Numpy function np.cov calculates the covariance elements using the factor \( 1/(n-1) \) instead of \( 1/n \) since it assumes we do not have the exact mean values. -The following simple function uses the np.vstack function which takes each vector of dimension \( 1\times n \) and produces a \( 2\times n \) matrix \( \hat{W} \) +









    + +

    Covariance Matrix Examples

    + +

    +The Numpy function np.cov calculates the covariance elements using +the factor \( 1/(n-1) \) instead of \( 1/n \) since it assumes we do not have +the exact mean values. The following simple function uses the +np.vstack function which takes each vector of dimension \( 1\times n \) +and produces a \( 2\times n \) matrix \( \boldsymbol{W} \) + $$ -\hat{W} = \begin{bmatrix} x_0 & y_0 \\ +\boldsymbol{W} = \begin{bmatrix} x_0 & y_0 \\ x_1 & y_1 \\ x_2 & y_2\\ \dots & \dots \\ @@ -646,9 +662,9 @@ $$ $$

    -which in turn is converted into into the \( 3\times 3 \) covariance matrix -\( \hat{\Sigma} \) via the Numpy function np.cov(). We note that we can also calculate -the mean value of each set of samples \( \hat{x} \) etc using the Numpy +which in turn is converted into into the \( 2\times 2 \) covariance matrix +\( \boldsymbol{C} \) via the Numpy function np.cov(). We note that we can also calculate +the mean value of each set of samples \( \boldsymbol{x} \) etc using the Numpy function np.mean(x). We can also extract the eigenvalues of the covariance matrix through the np.linalg.eig() function. @@ -657,34 +673,164 @@ covariance matrix through the np.linalg.eig() function. 

    # Importing various packages
 import numpy as np
-
 n = 100
 x = np.random.normal(size=n)
 print(np.mean(x))
 y = 4+3*x+np.random.normal(size=n)
 print(np.mean(y))
-z = x**3+np.random.normal(size=n)
-print(np.mean(z))
-W = np.vstack((x, y, z))
-Sigma = np.cov(W)
-print(Sigma)
-Eigvals, Eigvecs = np.linalg.eig(Sigma)
-print(Eigvals)
+W = np.vstack((x, y))
+C = np.cov(W)
+print(C)











    -

    Classical PCA Theorem

    +

    Correlation Matrix

    + +

    +The previous example can be converted into the correlation matrix by +simply scaling the matrix elements with the variances. We should also +subtract the mean values for each column. This leads to the following +code which sets up the correlations matrix for the previous example in +a more brute force way. Here we scale the mean values for each column of the design matrix, calculate the relevant mean values and variances and then finally set up the \( 2\times 2 \) correlation matrix (since we have only two vectors). + +

    + + +

    import numpy as np
+n = 100
+# define two vectors                                                                                           
+x = np.random.random(size=n)
+y = 4+3*x+np.random.normal(size=n)
+#scaling the x and y vectors                                                                                   
+x = x - np.mean(x)
+y = y - np.mean(y)
+variance_x = np.sum(x@x)/n
+variance_y = np.sum(y@y)/n
+print(variance_x)
+print(variance_y)
+cov_xy = np.sum(x@y)/n
+cov_xx = np.sum(x@x)/n
+cov_yy = np.sum(y@y)/n
+C = np.zeros((2,2))
+C[0,0]= cov_xx/variance_x
+C[1,1]= cov_yy/variance_y
+C[0,1]= cov_xy/np.sqrt(variance_y*variance_x)
+C[1,0]= C[0,1]
+print(C)
+
    +

    +We see that the matrix elements along the diagonal are one as they +should be and that the matrix is symmetric. Furthermore, diagonalizing +this matrix we easily see that it is a positive definite matrix. + +

    +The above procedure with numpy can be made more compact if we use pandas.











    -

    Prof of the PCA Theorem

    +

    Correlation Matrix with Pandas

    + +

    +We whow here how we can set up the correlation matrix using pandas, as done in this simple code +

    + + +

    import numpy as np
+import pandas as pd
+n = 10
+x = np.random.normal(size=n)
+x = x - np.mean(x)
+y = 4+3*x+np.random.normal(size=n)
+y = y - np.mean(y)
+X = (np.vstack((x, y))).T
+print(X)
+Xpd = pd.DataFrame(X)
+print(Xpd)
+correlation_matrix = Xpd.corr()
+print(correlation_matrix)
+
    +

    +We expand this model to the Franke function discussed above.











    -

    Getting started with PCA

    +

    Correlation Matrix with Pandas and the Franke function

    + +

    + + +

    # Common imports
+import numpy as np
+import pandas as pd
+
+
+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)    
+
+Xpd = pd.DataFrame(X)
+# subtract the mean values and set up the covariance matrix
+Xpd = Xpd - Xpd.mean()
+covariance_matrix = Xpd.cov()
+print(covariance_matrix)
+
    +

    +We note here that the covariance is zero for the first rows and +columns since all matrix elements in the design matrix were set to one +(we are fitting the function in terms of a polynomial of degree \( n \)). + +

    +This means that the variance for these elements will be zero and will +cause problems when we set up the correlation matrix. We can simply +drop these elements as follows and then construct the correlation +matrix. + +

    +









    + +

    Classical PCA Theorem

    + +

    +









    + +

    Prof of the PCA Theorem

    + +

    +









    + +

    Getting started with PCA

    @@ -699,7 +845,7 @@ X_pca = pca.









    -

    Principal Component Analysis

    +

    Principal Component Analysis

    @@ -735,7 +881,7 @@ 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 @@ -766,7 +912,7 @@ 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 @@ -796,7 +942,7 @@ 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 @@ -808,7 +954,7 @@ instances arrive).











    -

    Randomized PCA

    +

    Randomized PCA

    Scikit-Learn offers yet another option to perform PCA, called Randomized PCA. This is a stochastic @@ -823,7 +969,7 @@ previous algorithms when \( d \) is much smaller than \( n \).











    -

    Kernel PCA

    +

    Kernel PCA

    @@ -852,7 +998,7 @@ X_reduced = rbf_pcaLLE +

    LLE

    Locally Linear Embedding (LLE) is another very powerful nonlinear dimensionality reduction @@ -864,7 +1010,7 @@ 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 802dd81f3..0e28b6e62 100644 --- a/doc/pub/DimRed/ipynb/DimRed.ipynb +++ b/doc/pub/DimRed/ipynb/DimRed.ipynb @@ -412,7 +412,7 @@ "## Introducing the Covariance and Correlation functions\n", "\n", "Suppose we have defined two vectors\n", - "$\\hat{x} and \\hat{y} with $n$ elements each. The covariance matrix $\\boldsymbol{C}is defined as" + "$\\hat{x}$ and $\\hat{y}$ with $n$ elements each. The covariance matrix $\\boldsymbol{C}$ is defined as" ] }, { @@ -492,7 +492,7 @@ "metadata": {}, "source": [ "$$\n", - "corr[\\boldsymbol{x},\\boldsymbol{y}]=\\frac{cov[\\boldsymbol{x},\\boldsymbol{y}]}{\\sqrt{var[\\boldsymbol{x}]\\var\\boldsymbol{y}]}}.\n", + "corr[\\boldsymbol{x},\\boldsymbol{y}]=\\frac{cov[\\boldsymbol{x},\\boldsymbol{y}]}{\\sqrt{var[\\boldsymbol{x}]\\var[\\boldsymbol{y}]}}.\n", "$$" ] }, @@ -628,8 +628,14 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "The Numpy function **np.cov** calculates the covariance elements using the factor $1/(n-1)$ instead of $1/n$ since it assumes we do not have the exact mean values. \n", - "The following simple function uses the **np.vstack** function which takes each vector of dimension $1\\times n$ and produces a $2\\times n$ matrix $\\hat{W}$" + "## Covariance Matrix Examples\n", + "\n", + "\n", + "The Numpy function **np.cov** calculates the covariance elements using\n", + "the factor $1/(n-1)$ instead of $1/n$ since it assumes we do not have\n", + "the exact mean values. The following simple function uses the\n", + "**np.vstack** function which takes each vector of dimension $1\\times n$\n", + "and produces a $2\\times n$ matrix $\\boldsymbol{W}$" ] }, { @@ -637,7 +643,7 @@ "metadata": {}, "source": [ "$$\n", - "\\hat{W} = \\begin{bmatrix} x_0 & y_0 \\\\\n", + "\\boldsymbol{W} = \\begin{bmatrix} x_0 & y_0 \\\\\n", " x_1 & y_1 \\\\\n", " x_2 & y_2\\\\\n", " \\dots & \\dots \\\\\n", @@ -651,9 +657,9 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "which in turn is converted into into the $3\\times 3$ covariance matrix\n", - "$\\hat{\\Sigma}$ via the Numpy function **np.cov()**. We note that we can also calculate\n", - "the mean value of each set of samples $\\hat{x}$ etc using the Numpy\n", + "which in turn is converted into into the $2\\times 2$ covariance matrix\n", + "$\\boldsymbol{C}$ via the Numpy function **np.cov()**. We note that we can also calculate\n", + "the mean value of each set of samples $\\boldsymbol{x}$ etc using the Numpy\n", "function **np.mean(x)**. We can also extract the eigenvalues of the\n", "covariance matrix through the **np.linalg.eig()** function." ] @@ -668,25 +674,175 @@ "source": [ "# Importing various packages\n", "import numpy as np\n", - "\n", "n = 100\n", "x = np.random.normal(size=n)\n", "print(np.mean(x))\n", "y = 4+3*x+np.random.normal(size=n)\n", "print(np.mean(y))\n", - "z = x**3+np.random.normal(size=n)\n", - "print(np.mean(z))\n", - "W = np.vstack((x, y, z))\n", - "Sigma = np.cov(W)\n", - "print(Sigma)\n", - "Eigvals, Eigvecs = np.linalg.eig(Sigma)\n", - "print(Eigvals)" + "W = np.vstack((x, y))\n", + "C = np.cov(W)\n", + "print(C)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ + "## Correlation Matrix\n", + "\n", + "The previous example can be converted into the correlation matrix by\n", + "simply scaling the matrix elements with the variances. We should also\n", + "subtract the mean values for each column. This leads to the following\n", + "code which sets up the correlations matrix for the previous example in\n", + "a more brute force way. Here we scale the mean values for each column of the design matrix, calculate the relevant mean values and variances and then finally set up the $2\\times 2$ correlation matrix (since we have only two vectors)." + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "import numpy as np\n", + "n = 100\n", + "# define two vectors \n", + "x = np.random.random(size=n)\n", + "y = 4+3*x+np.random.normal(size=n)\n", + "#scaling the x and y vectors \n", + "x = x - np.mean(x)\n", + "y = y - np.mean(y)\n", + "variance_x = np.sum(x@x)/n\n", + "variance_y = np.sum(y@y)/n\n", + "print(variance_x)\n", + "print(variance_y)\n", + "cov_xy = np.sum(x@y)/n\n", + "cov_xx = np.sum(x@x)/n\n", + "cov_yy = np.sum(y@y)/n\n", + "C = np.zeros((2,2))\n", + "C[0,0]= cov_xx/variance_x\n", + "C[1,1]= cov_yy/variance_y\n", + "C[0,1]= cov_xy/np.sqrt(variance_y*variance_x)\n", + "C[1,0]= C[0,1]\n", + "print(C)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We see that the matrix elements along the diagonal are one as they\n", + "should be and that the matrix is symmetric. Furthermore, diagonalizing\n", + "this matrix we easily see that it is a positive definite matrix.\n", + "\n", + "The above procedure with **numpy** can be made more compact if we use **pandas**.\n", + "\n", + "## Correlation Matrix with Pandas\n", + "\n", + "We whow here how we can set up the correlation matrix using **pandas**, as done in this simple code" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "import numpy as np\n", + "import pandas as pd\n", + "n = 10\n", + "x = np.random.normal(size=n)\n", + "x = x - np.mean(x)\n", + "y = 4+3*x+np.random.normal(size=n)\n", + "y = y - np.mean(y)\n", + "X = (np.vstack((x, y))).T\n", + "print(X)\n", + "Xpd = pd.DataFrame(X)\n", + "print(Xpd)\n", + "correlation_matrix = Xpd.corr()\n", + "print(correlation_matrix)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We expand this model to the Franke function discussed above.\n", + "\n", + "## Correlation Matrix with Pandas and the Franke function" + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "# Common imports\n", + "import numpy as np\n", + "import pandas as pd\n", + "\n", + "\n", + "def FrankeFunction(x,y):\n", + "\tterm1 = 0.75*np.exp(-(0.25*(9*x-2)**2) - 0.25*((9*y-2)**2))\n", + "\tterm2 = 0.75*np.exp(-((9*x+1)**2)/49.0 - 0.1*(9*y+1))\n", + "\tterm3 = 0.5*np.exp(-(9*x-7)**2/4.0 - 0.25*((9*y-3)**2))\n", + "\tterm4 = -0.2*np.exp(-(9*x-4)**2 - (9*y-7)**2)\n", + "\treturn term1 + term2 + term3 + term4\n", + "\n", + "\n", + "def create_X(x, y, n ):\n", + "\tif len(x.shape) > 1:\n", + "\t\tx = np.ravel(x)\n", + "\t\ty = np.ravel(y)\n", + "\n", + "\tN = len(x)\n", + "\tl = int((n+1)*(n+2)/2)\t\t# Number of elements in beta\n", + "\tX = np.ones((N,l))\n", + "\n", + "\tfor i in range(1,n+1):\n", + "\t\tq = int((i)*(i+1)/2)\n", + "\t\tfor k in range(i+1):\n", + "\t\t\tX[:,q+k] = (x**(i-k))*(y**k)\n", + "\n", + "\treturn X\n", + "\n", + "\n", + "# Making meshgrid of datapoints and compute Franke's function\n", + "n = 4\n", + "N = 100\n", + "x = np.sort(np.random.uniform(0, 1, N))\n", + "y = np.sort(np.random.uniform(0, 1, N))\n", + "z = FrankeFunction(x, y)\n", + "X = create_X(x, y, n=n) \n", + "\n", + "Xpd = pd.DataFrame(X)\n", + "# subtract the mean values and set up the covariance matrix\n", + "Xpd = Xpd - Xpd.mean()\n", + "covariance_matrix = Xpd.cov()\n", + "print(covariance_matrix)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We note here that the covariance is zero for the first rows and\n", + "columns since all matrix elements in the design matrix were set to one\n", + "(we are fitting the function in terms of a polynomial of degree $n$).\n", + "\n", + "This means that the variance for these elements will be zero and will\n", + "cause problems when we set up the correlation matrix. We can simply\n", + "drop these elements as follows and then construct the correlation\n", + "matrix. \n", + "\n", + "\n", + "\n", "## Classical PCA Theorem\n", "\n", "\n", @@ -702,7 +858,7 @@ }, { "cell_type": "code", - "execution_count": 8, + "execution_count": 11, "metadata": { "collapsed": false }, @@ -730,7 +886,7 @@ }, { "cell_type": "code", - "execution_count": 9, + "execution_count": 12, "metadata": { "collapsed": false }, @@ -757,7 +913,7 @@ }, { "cell_type": "code", - "execution_count": 10, + "execution_count": 13, "metadata": { "collapsed": false }, @@ -781,7 +937,7 @@ }, { "cell_type": "code", - "execution_count": 11, + "execution_count": 14, "metadata": { "collapsed": false }, @@ -803,7 +959,7 @@ }, { "cell_type": "code", - "execution_count": 12, + "execution_count": 15, "metadata": { "collapsed": false }, @@ -833,7 +989,7 @@ }, { "cell_type": "code", - "execution_count": 13, + "execution_count": 16, "metadata": { "collapsed": false }, @@ -856,7 +1012,7 @@ }, { "cell_type": "code", - "execution_count": 14, + "execution_count": 17, "metadata": { "collapsed": false }, @@ -903,7 +1059,7 @@ }, { "cell_type": "code", - "execution_count": 15, + "execution_count": 18, "metadata": { "collapsed": false }, @@ -948,7 +1104,7 @@ }, { "cell_type": "code", - "execution_count": 16, + "execution_count": 19, "metadata": { "collapsed": false }, @@ -978,7 +1134,7 @@ }, { "cell_type": "code", - "execution_count": 17, + "execution_count": 20, "metadata": { "collapsed": false }, @@ -1013,7 +1169,7 @@ }, { "cell_type": "code", - "execution_count": 18, + "execution_count": 21, "metadata": { "collapsed": false }, diff --git a/doc/pub/DimRed/ipynb/ipynb-DimRed-src.tar.gz b/doc/pub/DimRed/ipynb/ipynb-DimRed-src.tar.gz index 990230aad97bf79bbaaa66d1999de27f7d7f6341..359365f9f6c231d9d10f5ea769acba5b5d717deb 100644 GIT binary patch literal 191 zcmV;w06_mAiwFQ<&8%Gj1MSaC3c@fD2H>uHia9|^Oj6fEyXZoY;ssKY+Ne!xl7hXx 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$\bm{X}$ (see below for it ===== Introducing the Covariance and Correlation functions ===== Suppose we have defined two vectors -$\hat{x} and \hat{y} with $n$ elements each. The covariance matrix $\bm{C}is defined as +$\hat{x}$ and $\hat{y}$ with $n$ elements each. The covariance matrix $\bm{C}$ is defined as !bt \[ \bm{C}[\bm{x},\bm{y}] = \begin{bmatrix} cov[\bm{x},\bm{x}] & cov[\bm{x},\bm{y}] \\ @@ -378,7 +378,7 @@ correlation function !bt \[ -corr[\bm{x},\bm{y}]=\frac{cov[\bm{x},\bm{y}]}{\sqrt{var[\bm{x}]\var\bm{y}]}}. +corr[\bm{x},\bm{y}]=\frac{cov[\bm{x},\bm{y}]}{\sqrt{var[\bm{x}]\var[\bm{y}]}}. \] !et @@ -456,14 +456,20 @@ corr[\bm{x}_{p-1},\bm{x}_0] & corr[\bm{x}_{p-1},\bm{x}_1] & corr[\bm{x}_{p-1}, !et +!split +===== Covariance Matrix Examples ===== +The Numpy function _np.cov_ calculates the covariance elements using +the factor $1/(n-1)$ instead of $1/n$ since it assumes we do not have +the exact mean values. The following simple function uses the +_np.vstack_ function which takes each vector of dimension $1\times n$ +and produces a $2\times n$ matrix $\bm{W}$ + -The Numpy function _np.cov_ calculates the covariance elements using the factor $1/(n-1)$ instead of $1/n$ since it assumes we do not have the exact mean values. -The following simple function uses the _np.vstack_ function which takes each vector of dimension $1\times n$ and produces a $2\times n$ matrix $\hat{W}$ !bt \[ -\hat{W} = \begin{bmatrix} x_0 & y_0 \\ +\bm{W} = \begin{bmatrix} x_0 & y_0 \\ x_1 & y_1 \\ x_2 & y_2\\ \dots & \dots \\ @@ -473,30 +479,144 @@ The following simple function uses the _np.vstack_ function which takes each vec \] !et -which in turn is converted into into the $3\times 3$ covariance matrix -$\hat{\Sigma}$ via the Numpy function _np.cov()_. We note that we can also calculate -the mean value of each set of samples $\hat{x}$ etc using the Numpy +which in turn is converted into into the $2\times 2$ covariance matrix +$\bm{C}$ via the Numpy function _np.cov()_. We note that we can also calculate +the mean value of each set of samples $\bm{x}$ etc using the Numpy function _np.mean(x)_. We can also extract the eigenvalues of the covariance matrix through the _np.linalg.eig()_ function. !bc pycod # Importing various packages import numpy as np - n = 100 x = np.random.normal(size=n) print(np.mean(x)) y = 4+3*x+np.random.normal(size=n) print(np.mean(y)) -z = x**3+np.random.normal(size=n) -print(np.mean(z)) -W = np.vstack((x, y, z)) -Sigma = np.cov(W) -print(Sigma) -Eigvals, Eigvecs = np.linalg.eig(Sigma) -print(Eigvals) +W = np.vstack((x, y)) +C = np.cov(W) +print(C) !ec +!split +===== Correlation Matrix ===== + +The previous example can be converted into the correlation matrix by +simply scaling the matrix elements with the variances. We should also +subtract the mean values for each column. This leads to the following +code which sets up the correlations matrix for the previous example in +a more brute force way. Here we scale the mean values for each column of the design matrix, calculate the relevant mean values and variances and then finally set up the $2\times 2$ correlation matrix (since we have only two vectors). + +!bc pycod +import numpy as np +n = 100 +# define two vectors +x = np.random.random(size=n) +y = 4+3*x+np.random.normal(size=n) +#scaling the x and y vectors +x = x - np.mean(x) +y = y - np.mean(y) +variance_x = np.sum(x@x)/n +variance_y = np.sum(y@y)/n +print(variance_x) +print(variance_y) +cov_xy = np.sum(x@y)/n +cov_xx = np.sum(x@x)/n +cov_yy = np.sum(y@y)/n +C = np.zeros((2,2)) +C[0,0]= cov_xx/variance_x +C[1,1]= cov_yy/variance_y +C[0,1]= cov_xy/np.sqrt(variance_y*variance_x) +C[1,0]= C[0,1] +print(C) +!ec + +We see that the matrix elements along the diagonal are one as they +should be and that the matrix is symmetric. Furthermore, diagonalizing +this matrix we easily see that it is a positive definite matrix. + +The above procedure with _numpy_ can be made more compact if we use _pandas_. + +!split +===== Correlation Matrix with Pandas ===== + +We whow here how we can set up the correlation matrix using _pandas_, as done in this simple code +!bc pycod +import numpy as np +import pandas as pd +n = 10 +x = np.random.normal(size=n) +x = x - np.mean(x) +y = 4+3*x+np.random.normal(size=n) +y = y - np.mean(y) +X = (np.vstack((x, y))).T +print(X) +Xpd = pd.DataFrame(X) +print(Xpd) +correlation_matrix = Xpd.corr() +print(correlation_matrix) +!ec + + +We expand this model to the Franke function discussed above. + +!split +===== Correlation Matrix with Pandas and the Franke function ===== + +!bc pycod +# Common imports +import numpy as np +import pandas as pd + + +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) + +Xpd = pd.DataFrame(X) +# subtract the mean values and set up the covariance matrix +Xpd = Xpd - Xpd.mean() +covariance_matrix = Xpd.cov() +print(covariance_matrix) +!ec + +We note here that the covariance is zero for the first rows and +columns since all matrix elements in the design matrix were set to one +(we are fitting the function in terms of a polynomial of degree $n$). + +This means that the variance for these elements will be zero and will +cause problems when we set up the correlation matrix. We can simply +drop these elements as follows and then construct the correlation +matrix. diff --git a/doc/src/DimRed/covariance.py b/doc/src/DimRed/covariance.py new file mode 100644 index 000000000..5f47aeac5 --- /dev/null +++ b/doc/src/DimRed/covariance.py @@ -0,0 +1,24 @@ +# Importing various packages +import numpy as np +n = 100 +# define two vectors +x = np.random.random(size=n) +y = 4+3*x+np.random.normal(size=n) +#scaling the x and y vectors +x = x - np.mean(x) +y = y - np.mean(y) +variance_x = np.sum(x@x)/n +variance_y = np.sum(y@y)/n +print(variance_x) +print(variance_y) +cov_xy = np.sum(x@y)/n +cov_xx = np.sum(x@x)/n +cov_yy = np.sum(y@y)/n +C = np.zeros((2,2)) +C[0,0]= cov_xx/variance_x +C[1,1]= cov_yy/variance_y +C[0,1]= cov_xy/np.sqrt(variance_y*variance_x) +C[1,0]= C[0,1] +print(C) +Eigvals, Eigvecs = np.linalg.eig(C) +print(Eigvals) diff --git a/doc/src/DimRed/covfrance.py b/doc/src/DimRed/covfrance.py new file mode 100644 index 000000000..ab57bf2bd --- /dev/null +++ b/doc/src/DimRed/covfrance.py @@ -0,0 +1,43 @@ +# Common imports +import numpy as np +import pandas as pd + + +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 = 1000 +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) + +Xpd = pd.DataFrame(X) +Xpd = Xpd - Xpd.mean() +correlation_matrix = Xpd.cov() +print(correlation_matrix) + diff --git a/doc/src/DimRed/newcov.py b/doc/src/DimRed/newcov.py new file mode 100644 index 000000000..8e6fab13e --- /dev/null +++ b/doc/src/DimRed/newcov.py @@ -0,0 +1,28 @@ +# Importing various packages +import numpy as np +n = 10 +x = np.random.normal(size=n) +x = x - np.mean(x) +y = 4+3*x+np.random.normal(size=n) +y = y - np.mean(y) +X = (np.vstack((x, y))).T +print(X) +import pandas as pd +Xpd = pd.DataFrame(X) +print(Xpd) +correlation_matrix = Xpd.corr() +print(correlation_matrix) + + + +variance_x = np.sum(x@x)/n +variance_y = np.sum(y@y)/n +cov_xy = np.sum(x@y)/n +cov_xx = np.sum(x@x)/n +cov_yy = np.sum(y@y)/n +C = np.zeros((2,2)) +C[0,0]= cov_xx/variance_x +C[1,1]= cov_yy/variance_y +C[0,1]= cov_xy/np.sqrt(variance_y*variance_x) +C[1,0]= C[0,1] +print(C)