From 3cb156e3c1be87789922844aebeff9a800d5706a Mon Sep 17 00:00:00 2001 From: mhjensen Date: Thu, 11 Oct 2018 05:06:22 +0200 Subject: [PATCH] Update on log reg --- doc/pub/LogReg/html/._LogReg-bs000.html | 6 +- doc/pub/LogReg/html/._LogReg-bs001.html | 4 +- doc/pub/LogReg/html/._LogReg-bs002.html | 4 +- doc/pub/LogReg/html/._LogReg-bs003.html | 4 +- doc/pub/LogReg/html/._LogReg-bs004.html | 4 +- doc/pub/LogReg/html/._LogReg-bs005.html | 4 +- doc/pub/LogReg/html/._LogReg-bs006.html | 4 +- doc/pub/LogReg/html/._LogReg-bs007.html | 4 +- doc/pub/LogReg/html/._LogReg-bs008.html | 4 +- doc/pub/LogReg/html/._LogReg-bs009.html | 4 +- doc/pub/LogReg/html/._LogReg-bs010.html | 4 +- doc/pub/LogReg/html/._LogReg-bs011.html | 4 +- doc/pub/LogReg/html/._LogReg-bs012.html | 15 +- doc/pub/LogReg/html/._LogReg-bs013.html | 22 +- doc/pub/LogReg/html/._LogReg-bs014.html | 33 ++- doc/pub/LogReg/html/._LogReg-bs015.html | 4 +- doc/pub/LogReg/html/._LogReg-bs016.html | 221 ++++++++++++++++++ doc/pub/LogReg/html/LogReg-bs.html | 6 +- doc/pub/LogReg/html/LogReg-reveal.html | 74 +++++- doc/pub/LogReg/html/LogReg-solarized.html | 64 ++++- doc/pub/LogReg/html/LogReg.html | 64 ++++- doc/pub/LogReg/ipynb/LogReg.ipynb | 170 ++++++++++++-- doc/pub/LogReg/ipynb/ipynb-LogReg-src.tar.gz | Bin 209 -> 210 bytes .../LogReg/pdf/LogReg-beamer-handouts2x3.pdf | Bin 256334 -> 266288 bytes doc/pub/LogReg/pdf/LogReg-beamer.pdf | Bin 246755 -> 255705 bytes doc/pub/LogReg/pdf/LogReg-minted.pdf | Bin 288128 -> 295337 bytes doc/src/LogisticRegression/LogReg.do.txt | 65 +++++- 27 files changed, 710 insertions(+), 78 deletions(-) create mode 100644 doc/pub/LogReg/html/._LogReg-bs016.html diff --git a/doc/pub/LogReg/html/._LogReg-bs000.html b/doc/pub/LogReg/html/._LogReg-bs000.html index b583d2d04..a7a74898b 100644 --- a/doc/pub/LogReg/html/._LogReg-bs000.html +++ b/doc/pub/LogReg/html/._LogReg-bs000.html @@ -53,7 +53,7 @@ Automatically generated HTML file from DocOnce source ('A more compact expression', 2, None, '___sec10'), ('Extending to more predictors', 2, None, '___sec11'), ('Including more classes', 2, None, '___sec12'), - ('Optimizing the cost function', 2, None, '___sec13'), + ('The Softmax function', 2, None, '___sec13'), ('A _scikit-learn_ example', 2, None, '___sec14'), ('A simple classification problem', 2, None, '___sec15')]} end of tocinfo --> @@ -106,7 +106,7 @@ MathJax.Hub.Config({
  • A more compact expression
  • Extending to more predictors
  • Including more classes
  • -
  • Optimizing the cost function
  • +
  • The Softmax function
  • A scikit-learn example
  • A simple classification problem
  • @@ -143,7 +143,7 @@ MathJax.Hub.Config({
    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

    -

    Sep 26, 2018

    +

    Oct 11, 2018


    diff --git a/doc/pub/LogReg/html/._LogReg-bs001.html b/doc/pub/LogReg/html/._LogReg-bs001.html index d07fe970b..101129f7e 100644 --- a/doc/pub/LogReg/html/._LogReg-bs001.html +++ b/doc/pub/LogReg/html/._LogReg-bs001.html @@ -53,7 +53,7 @@ Automatically generated HTML file from DocOnce source ('A more compact expression', 2, None, '___sec10'), ('Extending to more predictors', 2, None, '___sec11'), ('Including more classes', 2, None, '___sec12'), - ('Optimizing the cost function', 2, None, '___sec13'), + ('The Softmax function', 2, None, '___sec13'), ('A _scikit-learn_ example', 2, None, '___sec14'), ('A simple classification problem', 2, None, '___sec15')]} end of tocinfo --> @@ -106,7 +106,7 @@ MathJax.Hub.Config({

  • A more compact expression
  • Extending to more predictors
  • Including more classes
  • -
  • Optimizing the cost function
  • +
  • The Softmax function
  • A scikit-learn example
  • A simple classification problem
  • diff --git a/doc/pub/LogReg/html/._LogReg-bs002.html b/doc/pub/LogReg/html/._LogReg-bs002.html index a90ee5fb8..e18efedd6 100644 --- a/doc/pub/LogReg/html/._LogReg-bs002.html +++ b/doc/pub/LogReg/html/._LogReg-bs002.html @@ -53,7 +53,7 @@ Automatically generated HTML file from DocOnce source ('A more compact expression', 2, None, '___sec10'), ('Extending to more predictors', 2, None, '___sec11'), ('Including more classes', 2, None, '___sec12'), - ('Optimizing the cost function', 2, None, '___sec13'), + ('The Softmax function', 2, None, '___sec13'), ('A _scikit-learn_ example', 2, None, '___sec14'), ('A simple classification problem', 2, None, '___sec15')]} end of tocinfo --> @@ -106,7 +106,7 @@ MathJax.Hub.Config({
  • A more compact expression
  • Extending to more predictors
  • Including more classes
  • -
  • Optimizing the cost function
  • +
  • The Softmax function
  • A scikit-learn example
  • A simple classification problem
  • diff --git a/doc/pub/LogReg/html/._LogReg-bs003.html b/doc/pub/LogReg/html/._LogReg-bs003.html index 68633dc0d..09672b236 100644 --- a/doc/pub/LogReg/html/._LogReg-bs003.html +++ b/doc/pub/LogReg/html/._LogReg-bs003.html @@ -53,7 +53,7 @@ Automatically generated HTML file from DocOnce source ('A more compact expression', 2, None, '___sec10'), ('Extending to more predictors', 2, None, '___sec11'), ('Including more classes', 2, None, '___sec12'), - ('Optimizing the cost function', 2, None, '___sec13'), + ('The Softmax function', 2, None, '___sec13'), ('A _scikit-learn_ example', 2, None, '___sec14'), ('A simple classification problem', 2, None, '___sec15')]} end of tocinfo --> @@ -106,7 +106,7 @@ MathJax.Hub.Config({
  • A more compact expression
  • Extending to more predictors
  • Including more classes
  • -
  • Optimizing the cost function
  • +
  • The Softmax function
  • A scikit-learn example
  • A simple classification problem
  • diff --git a/doc/pub/LogReg/html/._LogReg-bs004.html b/doc/pub/LogReg/html/._LogReg-bs004.html index 63a7b3f19..ded12b4f5 100644 --- a/doc/pub/LogReg/html/._LogReg-bs004.html +++ b/doc/pub/LogReg/html/._LogReg-bs004.html @@ -53,7 +53,7 @@ Automatically generated HTML file from DocOnce source ('A more compact expression', 2, None, '___sec10'), ('Extending to more predictors', 2, None, '___sec11'), ('Including more classes', 2, None, '___sec12'), - ('Optimizing the cost function', 2, None, '___sec13'), + ('The Softmax function', 2, None, '___sec13'), ('A _scikit-learn_ example', 2, None, '___sec14'), ('A simple classification problem', 2, None, '___sec15')]} end of tocinfo --> @@ -106,7 +106,7 @@ MathJax.Hub.Config({
  • A more compact expression
  • Extending to more predictors
  • Including more classes
  • -
  • Optimizing the cost function
  • +
  • The Softmax function
  • A scikit-learn example
  • A simple classification problem
  • diff --git a/doc/pub/LogReg/html/._LogReg-bs005.html b/doc/pub/LogReg/html/._LogReg-bs005.html index c5b864af6..1afd49dae 100644 --- a/doc/pub/LogReg/html/._LogReg-bs005.html +++ b/doc/pub/LogReg/html/._LogReg-bs005.html @@ -53,7 +53,7 @@ Automatically generated HTML file from DocOnce source ('A more compact expression', 2, None, '___sec10'), ('Extending to more predictors', 2, None, '___sec11'), ('Including more classes', 2, None, '___sec12'), - ('Optimizing the cost function', 2, None, '___sec13'), + ('The Softmax function', 2, None, '___sec13'), ('A _scikit-learn_ example', 2, None, '___sec14'), ('A simple classification problem', 2, None, '___sec15')]} end of tocinfo --> @@ -106,7 +106,7 @@ MathJax.Hub.Config({
  • A more compact expression
  • Extending to more predictors
  • Including more classes
  • -
  • Optimizing the cost function
  • +
  • The Softmax function
  • A scikit-learn example
  • A simple classification problem
  • diff --git a/doc/pub/LogReg/html/._LogReg-bs006.html b/doc/pub/LogReg/html/._LogReg-bs006.html index b395f7c6d..843fa53ef 100644 --- a/doc/pub/LogReg/html/._LogReg-bs006.html +++ b/doc/pub/LogReg/html/._LogReg-bs006.html @@ -53,7 +53,7 @@ Automatically generated HTML file from DocOnce source ('A more compact expression', 2, None, '___sec10'), ('Extending to more predictors', 2, None, '___sec11'), ('Including more classes', 2, None, '___sec12'), - ('Optimizing the cost function', 2, None, '___sec13'), + ('The Softmax function', 2, None, '___sec13'), ('A _scikit-learn_ example', 2, None, '___sec14'), ('A simple classification problem', 2, None, '___sec15')]} end of tocinfo --> @@ -106,7 +106,7 @@ MathJax.Hub.Config({
  • A more compact expression
  • Extending to more predictors
  • Including more classes
  • -
  • Optimizing the cost function
  • +
  • The Softmax function
  • A scikit-learn example
  • A simple classification problem
  • diff --git a/doc/pub/LogReg/html/._LogReg-bs007.html b/doc/pub/LogReg/html/._LogReg-bs007.html index eab7581ad..55011945a 100644 --- a/doc/pub/LogReg/html/._LogReg-bs007.html +++ b/doc/pub/LogReg/html/._LogReg-bs007.html @@ -53,7 +53,7 @@ Automatically generated HTML file from DocOnce source ('A more compact expression', 2, None, '___sec10'), ('Extending to more predictors', 2, None, '___sec11'), ('Including more classes', 2, None, '___sec12'), - ('Optimizing the cost function', 2, None, '___sec13'), + ('The Softmax function', 2, None, '___sec13'), ('A _scikit-learn_ example', 2, None, '___sec14'), ('A simple classification problem', 2, None, '___sec15')]} end of tocinfo --> @@ -106,7 +106,7 @@ MathJax.Hub.Config({
  • A more compact expression
  • Extending to more predictors
  • Including more classes
  • -
  • Optimizing the cost function
  • +
  • The Softmax function
  • A scikit-learn example
  • A simple classification problem
  • diff --git a/doc/pub/LogReg/html/._LogReg-bs008.html b/doc/pub/LogReg/html/._LogReg-bs008.html index 845f338c3..546bd2af3 100644 --- a/doc/pub/LogReg/html/._LogReg-bs008.html +++ b/doc/pub/LogReg/html/._LogReg-bs008.html @@ -53,7 +53,7 @@ Automatically generated HTML file from DocOnce source ('A more compact expression', 2, None, '___sec10'), ('Extending to more predictors', 2, None, '___sec11'), ('Including more classes', 2, None, '___sec12'), - ('Optimizing the cost function', 2, None, '___sec13'), + ('The Softmax function', 2, None, '___sec13'), ('A _scikit-learn_ example', 2, None, '___sec14'), ('A simple classification problem', 2, None, '___sec15')]} end of tocinfo --> @@ -106,7 +106,7 @@ MathJax.Hub.Config({
  • A more compact expression
  • Extending to more predictors
  • Including more classes
  • -
  • Optimizing the cost function
  • +
  • The Softmax function
  • A scikit-learn example
  • A simple classification problem
  • diff --git a/doc/pub/LogReg/html/._LogReg-bs009.html b/doc/pub/LogReg/html/._LogReg-bs009.html index c7d9b85b1..0f57ffad9 100644 --- a/doc/pub/LogReg/html/._LogReg-bs009.html +++ b/doc/pub/LogReg/html/._LogReg-bs009.html @@ -53,7 +53,7 @@ Automatically generated HTML file from DocOnce source ('A more compact expression', 2, None, '___sec10'), ('Extending to more predictors', 2, None, '___sec11'), ('Including more classes', 2, None, '___sec12'), - ('Optimizing the cost function', 2, None, '___sec13'), + ('The Softmax function', 2, None, '___sec13'), ('A _scikit-learn_ example', 2, None, '___sec14'), ('A simple classification problem', 2, None, '___sec15')]} end of tocinfo --> @@ -106,7 +106,7 @@ MathJax.Hub.Config({
  • A more compact expression
  • Extending to more predictors
  • Including more classes
  • -
  • Optimizing the cost function
  • +
  • The Softmax function
  • A scikit-learn example
  • A simple classification problem
  • diff --git a/doc/pub/LogReg/html/._LogReg-bs010.html b/doc/pub/LogReg/html/._LogReg-bs010.html index d8b894fe4..3b69ab015 100644 --- a/doc/pub/LogReg/html/._LogReg-bs010.html +++ b/doc/pub/LogReg/html/._LogReg-bs010.html @@ -53,7 +53,7 @@ Automatically generated HTML file from DocOnce source ('A more compact expression', 2, None, '___sec10'), ('Extending to more predictors', 2, None, '___sec11'), ('Including more classes', 2, None, '___sec12'), - ('Optimizing the cost function', 2, None, '___sec13'), + ('The Softmax function', 2, None, '___sec13'), ('A _scikit-learn_ example', 2, None, '___sec14'), ('A simple classification problem', 2, None, '___sec15')]} end of tocinfo --> @@ -106,7 +106,7 @@ MathJax.Hub.Config({
  • A more compact expression
  • Extending to more predictors
  • Including more classes
  • -
  • Optimizing the cost function
  • +
  • The Softmax function
  • A scikit-learn example
  • A simple classification problem
  • diff --git a/doc/pub/LogReg/html/._LogReg-bs011.html b/doc/pub/LogReg/html/._LogReg-bs011.html index 26ff89a72..20f433d17 100644 --- a/doc/pub/LogReg/html/._LogReg-bs011.html +++ b/doc/pub/LogReg/html/._LogReg-bs011.html @@ -53,7 +53,7 @@ Automatically generated HTML file from DocOnce source ('A more compact expression', 2, None, '___sec10'), ('Extending to more predictors', 2, None, '___sec11'), ('Including more classes', 2, None, '___sec12'), - ('Optimizing the cost function', 2, None, '___sec13'), + ('The Softmax function', 2, None, '___sec13'), ('A _scikit-learn_ example', 2, None, '___sec14'), ('A simple classification problem', 2, None, '___sec15')]} end of tocinfo --> @@ -106,7 +106,7 @@ MathJax.Hub.Config({
  • A more compact expression
  • Extending to more predictors
  • Including more classes
  • -
  • Optimizing the cost function
  • +
  • The Softmax function
  • A scikit-learn example
  • A simple classification problem
  • diff --git a/doc/pub/LogReg/html/._LogReg-bs012.html b/doc/pub/LogReg/html/._LogReg-bs012.html index 8777c8869..6b86bb12f 100644 --- a/doc/pub/LogReg/html/._LogReg-bs012.html +++ b/doc/pub/LogReg/html/._LogReg-bs012.html @@ -53,7 +53,7 @@ Automatically generated HTML file from DocOnce source ('A more compact expression', 2, None, '___sec10'), ('Extending to more predictors', 2, None, '___sec11'), ('Including more classes', 2, None, '___sec12'), - ('Optimizing the cost function', 2, None, '___sec13'), + ('The Softmax function', 2, None, '___sec13'), ('A _scikit-learn_ example', 2, None, '___sec14'), ('A simple classification problem', 2, None, '___sec15')]} end of tocinfo --> @@ -106,7 +106,7 @@ MathJax.Hub.Config({
  • A more compact expression
  • Extending to more predictors
  • Including more classes
  • -
  • Optimizing the cost function
  • +
  • The Softmax function
  • A scikit-learn example
  • A simple classification problem
  • @@ -126,6 +126,17 @@ MathJax.Hub.Config({

    Extending to more predictors

    +

    +Within a binary classification problem, we can easily expand our model to include multiple predictors. Our ratio between likelihoods is then with \( p \) predictors +$$ +\log{ \frac{p(\hat{\beta}\hat{x})}{1-p(\hat{\beta}\hat{x})}} = \beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p. +$$ + +Here we defined \( \hat{x}=[1,x_1,x_2,\dots,x_p] \) and \( \hat{\beta}=[\beta_0, \beta_1, \dots, \beta_p] \) leading to +$$ +p(\hat{\beta}\hat{x})=\frac{ \exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}{1+\exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}. +$$ +

    diff --git a/doc/pub/LogReg/html/._LogReg-bs013.html b/doc/pub/LogReg/html/._LogReg-bs013.html index 49249443a..a70b01ff9 100644 --- a/doc/pub/LogReg/html/._LogReg-bs013.html +++ b/doc/pub/LogReg/html/._LogReg-bs013.html @@ -53,7 +53,7 @@ Automatically generated HTML file from DocOnce source ('A more compact expression', 2, None, '___sec10'), ('Extending to more predictors', 2, None, '___sec11'), ('Including more classes', 2, None, '___sec12'), - ('Optimizing the cost function', 2, None, '___sec13'), + ('The Softmax function', 2, None, '___sec13'), ('A _scikit-learn_ example', 2, None, '___sec14'), ('A simple classification problem', 2, None, '___sec15')]} end of tocinfo --> @@ -106,7 +106,7 @@ MathJax.Hub.Config({

  • A more compact expression
  • Extending to more predictors
  • Including more classes
  • -
  • Optimizing the cost function
  • +
  • The Softmax function
  • A scikit-learn example
  • A simple classification problem
  • @@ -126,6 +126,24 @@ MathJax.Hub.Config({

    Including more classes

    +

    +Till now we have mainly focused on two classes, the so-called binary system. Suppose we wish to extend to \( K \) classes. +Let us for the sake of simplicity assume we have only two predictors. We have then following model +$$ +\log{\frac{p(C=1\vert x)}{p(K\vert x)}} = \beta_{10}+\beta_{11}x_1, +$$ + +$$ +\log{\frac{p(C=2\vert x)}{p(K\vert x)}} = \beta_{20}+\beta_{21}x_1, +$$ + +and so on till the class \( C=K-1 \) class +$$ +\log{\frac{p(C=K-1\vert x)}{p(K\vert x)}} = \beta_{(K-1)0}+\beta_{(K-1)1}x_1, +$$ + +and the model is specified in term of \( K-1 \) so-called log-odds or logit transformations. +

    diff --git a/doc/pub/LogReg/html/._LogReg-bs014.html b/doc/pub/LogReg/html/._LogReg-bs014.html index 83da3d97b..f5f82f6b8 100644 --- a/doc/pub/LogReg/html/._LogReg-bs014.html +++ b/doc/pub/LogReg/html/._LogReg-bs014.html @@ -53,7 +53,7 @@ Automatically generated HTML file from DocOnce source ('A more compact expression', 2, None, '___sec10'), ('Extending to more predictors', 2, None, '___sec11'), ('Including more classes', 2, None, '___sec12'), - ('Optimizing the cost function', 2, None, '___sec13'), + ('The Softmax function', 2, None, '___sec13'), ('A _scikit-learn_ example', 2, None, '___sec14'), ('A simple classification problem', 2, None, '___sec15')]} end of tocinfo --> @@ -106,7 +106,7 @@ MathJax.Hub.Config({

  • A more compact expression
  • Extending to more predictors
  • Including more classes
  • -
  • Optimizing the cost function
  • +
  • The Softmax function
  • A scikit-learn example
  • A simple classification problem
  • @@ -124,10 +124,35 @@ MathJax.Hub.Config({ -

    Optimizing the cost function

    +

    The Softmax function

    -Newton's method and gradient descent methods +In our discussion of neural networks we will encounter the above again in terms of the so-called Softmax function. + +

    +The softmax function is used in various multiclass classification +methods, such as multinomial logistic regression (also known as +softmax regression), multiclass linear discriminant +analysis, naive Bayes classifiers, and artificial neural networks. +Specifically, in multinomial logistic regression and linear +discriminant analysis, the input to the function is the result of \( K \) +distinct linear functions, and the predicted probability for the \( k \)-th +class given a sample vector \( \hat{x} \) and a weighting vector \( \hat{\beta} \) is (with two predictors): + +$$ +p(C=k\vert \mathbf {x} )=\frac{\exp{(\beta_{k0}+\beta_{k1}x_1)}}{1+\sum_{l=1}^{K-1}\exp{(\beta_{l0}+\beta_{l1}x_1)}}. +$$ + +It is easy to extend to more predictors. The final class is +$$ +p(C=K\vert \mathbf {x} )=\frac{1}{1+\sum_{l=1}^{K-1}\exp{(\beta_{l0}+\beta_{l1}x_1)}}, +$$ + +and they sum to one. Our earlier discussions were all specialized to the case with two classes only. It is easy to see from the above that what we derived earlier is compatible with these equations. + +

    +To find the optimal parameters we would typically use a gradient descent method. +Newton's method and gradient descent methods are discussed in the material on optimization methods.

    diff --git a/doc/pub/LogReg/html/._LogReg-bs015.html b/doc/pub/LogReg/html/._LogReg-bs015.html index edca491af..8a7707493 100644 --- a/doc/pub/LogReg/html/._LogReg-bs015.html +++ b/doc/pub/LogReg/html/._LogReg-bs015.html @@ -53,7 +53,7 @@ Automatically generated HTML file from DocOnce source ('A more compact expression', 2, None, '___sec10'), ('Extending to more predictors', 2, None, '___sec11'), ('Including more classes', 2, None, '___sec12'), - ('Optimizing the cost function', 2, None, '___sec13'), + ('The Softmax function', 2, None, '___sec13'), ('A _scikit-learn_ example', 2, None, '___sec14'), ('A simple classification problem', 2, None, '___sec15')]} end of tocinfo --> @@ -106,7 +106,7 @@ MathJax.Hub.Config({

  • A more compact expression
  • Extending to more predictors
  • Including more classes
  • -
  • Optimizing the cost function
  • +
  • The Softmax function
  • A scikit-learn example
  • A simple classification problem
  • diff --git a/doc/pub/LogReg/html/._LogReg-bs016.html b/doc/pub/LogReg/html/._LogReg-bs016.html new file mode 100644 index 000000000..d2b0d497a --- /dev/null +++ b/doc/pub/LogReg/html/._LogReg-bs016.html @@ -0,0 +1,221 @@ + + + + + + + +Data Analysis and Machine Learning: Logistic Regression + + + + + + + + + + + + + + + + + + + + + + + + + + +
    + +

     

     

     

    + + + + +

    A simple classification problem

    +

    + + +

    import numpy as np
    +from sklearn import datasets, linear_model
    +import matplotlib.pyplot as plt
    +
    +
    +def generate_data():
    +    np.random.seed(0)
    +    X, y = datasets.make_moons(200, noise=0.20)
    +    return X, y
    +
    +
    +def visualize(X, y, clf):
    +    # plt.scatter(X[:, 0], X[:, 1], s=40, c=y, cmap=plt.cm.Spectral)
    +    # plt.show()
    +    plot_decision_boundary(lambda x: clf.predict(x), X, y)
    +    plt.title("Logistic Regression")
    +
    +
    +def plot_decision_boundary(pred_func, X, y):
    +    # Set min and max values and give it some padding
    +    x_min, x_max = X[:, 0].min() - .5, X[:, 0].max() + .5
    +    y_min, y_max = X[:, 1].min() - .5, X[:, 1].max() + .5
    +    h = 0.01
    +    # Generate a grid of points with distance h between them
    +    xx, yy = np.meshgrid(np.arange(x_min, x_max, h), np.arange(y_min, y_max, h))
    +    # Predict the function value for the whole gid
    +    Z = pred_func(np.c_[xx.ravel(), yy.ravel()])
    +    Z = Z.reshape(xx.shape)
    +    # Plot the contour and training examples
    +    plt.contourf(xx, yy, Z, cmap=plt.cm.Spectral)
    +    plt.scatter(X[:, 0], X[:, 1], c=y, cmap=plt.cm.Spectral)
    +    plt.show()
    +
    +
    +def classify(X, y):
    +    clf = linear_model.LogisticRegressionCV()
    +    clf.fit(X, y)
    +    return clf
    +
    +
    +def main():
    +    X, y = generate_data()
    +    # visualize(X, y)
    +    clf = classify(X, y)
    +    visualize(X, y, clf)
    +
    +
    +if __name__ == "__main__":
    +    main()
    +
    +

    + +

    + +

    + + +
    + + + + + + + +
    + +
    + + + + + + diff --git a/doc/pub/LogReg/html/LogReg-bs.html b/doc/pub/LogReg/html/LogReg-bs.html index b583d2d04..a7a74898b 100644 --- a/doc/pub/LogReg/html/LogReg-bs.html +++ b/doc/pub/LogReg/html/LogReg-bs.html @@ -53,7 +53,7 @@ Automatically generated HTML file from DocOnce source ('A more compact expression', 2, None, '___sec10'), ('Extending to more predictors', 2, None, '___sec11'), ('Including more classes', 2, None, '___sec12'), - ('Optimizing the cost function', 2, None, '___sec13'), + ('The Softmax function', 2, None, '___sec13'), ('A _scikit-learn_ example', 2, None, '___sec14'), ('A simple classification problem', 2, None, '___sec15')]} end of tocinfo --> @@ -106,7 +106,7 @@ MathJax.Hub.Config({
  • A more compact expression
  • Extending to more predictors
  • Including more classes
  • -
  • Optimizing the cost function
  • +
  • The Softmax function
  • A scikit-learn example
  • A simple classification problem
  • @@ -143,7 +143,7 @@ MathJax.Hub.Config({
    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

    -

    Sep 26, 2018

    +

    Oct 11, 2018


    diff --git a/doc/pub/LogReg/html/LogReg-reveal.html b/doc/pub/LogReg/html/LogReg-reveal.html index 2615d7966..9534ff30a 100644 --- a/doc/pub/LogReg/html/LogReg-reveal.html +++ b/doc/pub/LogReg/html/LogReg-reveal.html @@ -148,7 +148,7 @@ MathJax.Hub.Config({

    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

     
    -

    Sep 26, 2018

    +

    Oct 11, 2018


    @@ -448,19 +448,87 @@ $$

    Extending to more predictors

    + +

    +Within a binary classification problem, we can easily expand our model to include multiple predictors. Our ratio between likelihoods is then with \( p \) predictors +

     
    +$$ +\log{ \frac{p(\hat{\beta}\hat{x})}{1-p(\hat{\beta}\hat{x})}} = \beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p. +$$ +

     
    + +Here we defined \( \hat{x}=[1,x_1,x_2,\dots,x_p] \) and \( \hat{\beta}=[\beta_0, \beta_1, \dots, \beta_p] \) leading to +

     
    +$$ +p(\hat{\beta}\hat{x})=\frac{ \exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}{1+\exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}. +$$ +

     

    Including more classes

    + +

    +Till now we have mainly focused on two classes, the so-called binary system. Suppose we wish to extend to \( K \) classes. +Let us for the sake of simplicity assume we have only two predictors. We have then following model +

     
    +$$ +\log{\frac{p(C=1\vert x)}{p(K\vert x)}} = \beta_{10}+\beta_{11}x_1, +$$ +

     
    + +

     
    +$$ +\log{\frac{p(C=2\vert x)}{p(K\vert x)}} = \beta_{20}+\beta_{21}x_1, +$$ +

     
    + +and so on till the class \( C=K-1 \) class +

     
    +$$ +\log{\frac{p(C=K-1\vert x)}{p(K\vert x)}} = \beta_{(K-1)0}+\beta_{(K-1)1}x_1, +$$ +

     
    + +and the model is specified in term of \( K-1 \) so-called log-odds or logit transformations.

    -

    Optimizing the cost function

    +

    The Softmax function

    -Newton's method and gradient descent methods +In our discussion of neural networks we will encounter the above again in terms of the so-called Softmax function. + +

    +The softmax function is used in various multiclass classification +methods, such as multinomial logistic regression (also known as +softmax regression), multiclass linear discriminant +analysis, naive Bayes classifiers, and artificial neural networks. +Specifically, in multinomial logistic regression and linear +discriminant analysis, the input to the function is the result of \( K \) +distinct linear functions, and the predicted probability for the \( k \)-th +class given a sample vector \( \hat{x} \) and a weighting vector \( \hat{\beta} \) is (with two predictors): + +

     
    +$$ +p(C=k\vert \mathbf {x} )=\frac{\exp{(\beta_{k0}+\beta_{k1}x_1)}}{1+\sum_{l=1}^{K-1}\exp{(\beta_{l0}+\beta_{l1}x_1)}}. +$$ +

     
    + +It is easy to extend to more predictors. The final class is +

     
    +$$ +p(C=K\vert \mathbf {x} )=\frac{1}{1+\sum_{l=1}^{K-1}\exp{(\beta_{l0}+\beta_{l1}x_1)}}, +$$ +

     
    + +and they sum to one. Our earlier discussions were all specialized to the case with two classes only. It is easy to see from the above that what we derived earlier is compatible with these equations. + +

    +To find the optimal parameters we would typically use a gradient descent method. +Newton's method and gradient descent methods are discussed in the material on optimization methods.

    diff --git a/doc/pub/LogReg/html/LogReg-solarized.html b/doc/pub/LogReg/html/LogReg-solarized.html index 6211d4f48..3571525a5 100644 --- a/doc/pub/LogReg/html/LogReg-solarized.html +++ b/doc/pub/LogReg/html/LogReg-solarized.html @@ -47,7 +47,7 @@ div { text-align: justify; text-justify: inter-word; } ('A more compact expression', 2, None, '___sec10'), ('Extending to more predictors', 2, None, '___sec11'), ('Including more classes', 2, None, '___sec12'), - ('Optimizing the cost function', 2, None, '___sec13'), + ('The Softmax function', 2, None, '___sec13'), ('A _scikit-learn_ example', 2, None, '___sec14'), ('A simple classification problem', 2, None, '___sec15')]} end of tocinfo --> @@ -91,7 +91,7 @@ MathJax.Hub.Config({
    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

    -

    Sep 26, 2018

    +

    Oct 11, 2018


    @@ -358,18 +358,72 @@ $$

    Extending to more predictors

    +

    +Within a binary classification problem, we can easily expand our model to include multiple predictors. Our ratio between likelihoods is then with \( p \) predictors +$$ +\log{ \frac{p(\hat{\beta}\hat{x})}{1-p(\hat{\beta}\hat{x})}} = \beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p. +$$ + +Here we defined \( \hat{x}=[1,x_1,x_2,\dots,x_p] \) and \( \hat{\beta}=[\beta_0, \beta_1, \dots, \beta_p] \) leading to +$$ +p(\hat{\beta}\hat{x})=\frac{ \exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}{1+\exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}. +$$ +











    Including more classes

    -









    +Till now we have mainly focused on two classes, the so-called binary system. Suppose we wish to extend to \( K \) classes. +Let us for the sake of simplicity assume we have only two predictors. We have then following model +$$ +\log{\frac{p(C=1\vert x)}{p(K\vert x)}} = \beta_{10}+\beta_{11}x_1, +$$ -

    Optimizing the cost function

    +$$ +\log{\frac{p(C=2\vert x)}{p(K\vert x)}} = \beta_{20}+\beta_{21}x_1, +$$ + +and so on till the class \( C=K-1 \) class +$$ +\log{\frac{p(C=K-1\vert x)}{p(K\vert x)}} = \beta_{(K-1)0}+\beta_{(K-1)1}x_1, +$$ + +and the model is specified in term of \( K-1 \) so-called log-odds or logit transformations.

    -Newton's method and gradient descent methods +









    + +

    The Softmax function

    + +

    +In our discussion of neural networks we will encounter the above again in terms of the so-called Softmax function. + +

    +The softmax function is used in various multiclass classification +methods, such as multinomial logistic regression (also known as +softmax regression), multiclass linear discriminant +analysis, naive Bayes classifiers, and artificial neural networks. +Specifically, in multinomial logistic regression and linear +discriminant analysis, the input to the function is the result of \( K \) +distinct linear functions, and the predicted probability for the \( k \)-th +class given a sample vector \( \hat{x} \) and a weighting vector \( \hat{\beta} \) is (with two predictors): + +$$ +p(C=k\vert \mathbf {x} )=\frac{\exp{(\beta_{k0}+\beta_{k1}x_1)}}{1+\sum_{l=1}^{K-1}\exp{(\beta_{l0}+\beta_{l1}x_1)}}. +$$ + +It is easy to extend to more predictors. The final class is +$$ +p(C=K\vert \mathbf {x} )=\frac{1}{1+\sum_{l=1}^{K-1}\exp{(\beta_{l0}+\beta_{l1}x_1)}}, +$$ + +and they sum to one. Our earlier discussions were all specialized to the case with two classes only. It is easy to see from the above that what we derived earlier is compatible with these equations. + +

    +To find the optimal parameters we would typically use a gradient descent method. +Newton's method and gradient descent methods are discussed in the material on optimization methods.











    diff --git a/doc/pub/LogReg/html/LogReg.html b/doc/pub/LogReg/html/LogReg.html index 02994c5c2..3a86df613 100644 --- a/doc/pub/LogReg/html/LogReg.html +++ b/doc/pub/LogReg/html/LogReg.html @@ -52,7 +52,7 @@ div { text-align: justify; text-justify: inter-word; } ('A more compact expression', 2, None, '___sec10'), ('Extending to more predictors', 2, None, '___sec11'), ('Including more classes', 2, None, '___sec12'), - ('Optimizing the cost function', 2, None, '___sec13'), + ('The Softmax function', 2, None, '___sec13'), ('A _scikit-learn_ example', 2, None, '___sec14'), ('A simple classification problem', 2, None, '___sec15')]} end of tocinfo --> @@ -96,7 +96,7 @@ MathJax.Hub.Config({

    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

    -

    Sep 26, 2018

    +

    Oct 11, 2018


    @@ -363,18 +363,72 @@ $$

    Extending to more predictors

    +

    +Within a binary classification problem, we can easily expand our model to include multiple predictors. Our ratio between likelihoods is then with \( p \) predictors +$$ +\log{ \frac{p(\hat{\beta}\hat{x})}{1-p(\hat{\beta}\hat{x})}} = \beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p. +$$ + +Here we defined \( \hat{x}=[1,x_1,x_2,\dots,x_p] \) and \( \hat{\beta}=[\beta_0, \beta_1, \dots, \beta_p] \) leading to +$$ +p(\hat{\beta}\hat{x})=\frac{ \exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}{1+\exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}. +$$ +











    Including more classes

    -









    +Till now we have mainly focused on two classes, the so-called binary system. Suppose we wish to extend to \( K \) classes. +Let us for the sake of simplicity assume we have only two predictors. We have then following model +$$ +\log{\frac{p(C=1\vert x)}{p(K\vert x)}} = \beta_{10}+\beta_{11}x_1, +$$ -

    Optimizing the cost function

    +$$ +\log{\frac{p(C=2\vert x)}{p(K\vert x)}} = \beta_{20}+\beta_{21}x_1, +$$ + +and so on till the class \( C=K-1 \) class +$$ +\log{\frac{p(C=K-1\vert x)}{p(K\vert x)}} = \beta_{(K-1)0}+\beta_{(K-1)1}x_1, +$$ + +and the model is specified in term of \( K-1 \) so-called log-odds or logit transformations.

    -Newton's method and gradient descent methods +









    + +

    The Softmax function

    + +

    +In our discussion of neural networks we will encounter the above again in terms of the so-called Softmax function. + +

    +The softmax function is used in various multiclass classification +methods, such as multinomial logistic regression (also known as +softmax regression), multiclass linear discriminant +analysis, naive Bayes classifiers, and artificial neural networks. +Specifically, in multinomial logistic regression and linear +discriminant analysis, the input to the function is the result of \( K \) +distinct linear functions, and the predicted probability for the \( k \)-th +class given a sample vector \( \hat{x} \) and a weighting vector \( \hat{\beta} \) is (with two predictors): + +$$ +p(C=k\vert \mathbf {x} )=\frac{\exp{(\beta_{k0}+\beta_{k1}x_1)}}{1+\sum_{l=1}^{K-1}\exp{(\beta_{l0}+\beta_{l1}x_1)}}. +$$ + +It is easy to extend to more predictors. The final class is +$$ +p(C=K\vert \mathbf {x} )=\frac{1}{1+\sum_{l=1}^{K-1}\exp{(\beta_{l0}+\beta_{l1}x_1)}}, +$$ + +and they sum to one. Our earlier discussions were all specialized to the case with two classes only. It is easy to see from the above that what we derived earlier is compatible with these equations. + +

    +To find the optimal parameters we would typically use a gradient descent method. +Newton's method and gradient descent methods are discussed in the material on optimization methods.











    diff --git a/doc/pub/LogReg/ipynb/LogReg.ipynb b/doc/pub/LogReg/ipynb/LogReg.ipynb index baaa9264e..cf3532b4d 100644 --- a/doc/pub/LogReg/ipynb/LogReg.ipynb +++ b/doc/pub/LogReg/ipynb/LogReg.ipynb @@ -10,7 +10,7 @@ " \n", "**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University\n", "\n", - "Date: **Sep 26, 2018**\n", + "Date: **Oct 11, 2018**\n", "\n", "Copyright 1999-2018, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n", "\n", @@ -374,11 +374,147 @@ "source": [ "## Extending to more predictors\n", "\n", + "Within a binary classification problem, we can easily expand our model to include multiple predictors. Our ratio between likelihoods is then with $p$ predictors" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\log{ \\frac{p(\\hat{\\beta}\\hat{x})}{1-p(\\hat{\\beta}\\hat{x})}} = \\beta_0+\\beta_1x_1+\\beta_2x_2+\\dots+\\beta_px_p.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Here we defined $\\hat{x}=[1,x_1,x_2,\\dots,x_p]$ and $\\hat{\\beta}=[\\beta_0, \\beta_1, \\dots, \\beta_p]$ leading to" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "p(\\hat{\\beta}\\hat{x})=\\frac{ \\exp{(\\beta_0+\\beta_1x_1+\\beta_2x_2+\\dots+\\beta_px_p)}}{1+\\exp{(\\beta_0+\\beta_1x_1+\\beta_2x_2+\\dots+\\beta_px_p)}}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ "## Including more classes\n", "\n", - "## Optimizing the cost function\n", + "Till now we have mainly focused on two classes, the so-called binary system. Suppose we wish to extend to $K$ classes.\n", + "Let us for the sake of simplicity assume we have only two predictors. We have then following model" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "1\n", + "5\n", + " \n", + "<\n", + "<\n", + "<\n", + "!\n", + "!\n", + "M\n", + "A\n", + "T\n", + "H\n", + "_\n", + "B\n", + "L\n", + "O\n", + "C\n", + "K" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\log{\\frac{p(C=2\\vert x)}{p(K\\vert x)}} = \\beta_{20}+\\beta_{21}x_1,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and so on till the class $C=K-1$ class" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\log{\\frac{p(C=K-1\\vert x)}{p(K\\vert x)}} = \\beta_{(K-1)0}+\\beta_{(K-1)1}x_1,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and the model is specified in term of $K-1$ so-called log-odds or **logit** transformations.\n", "\n", - "Newton's method and gradient descent methods\n", + "\n", + "## The Softmax function\n", + "\n", + "In our discussion of neural networks we will encounter the above again in terms of the so-called **Softmax** function.\n", + "\n", + "The softmax function is used in various multiclass classification\n", + "methods, such as multinomial logistic regression (also known as\n", + "softmax regression), multiclass linear discriminant\n", + "analysis, naive Bayes classifiers, and artificial neural networks.\n", + "Specifically, in multinomial logistic regression and linear\n", + "discriminant analysis, the input to the function is the result of $K$\n", + "distinct linear functions, and the predicted probability for the $k$-th\n", + "class given a sample vector $\\hat{x}$ and a weighting vector $\\hat{\\beta}$ is (with two predictors):" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "p(C=k\\vert \\mathbf {x} )=\\frac{\\exp{(\\beta_{k0}+\\beta_{k1}x_1)}}{1+\\sum_{l=1}^{K-1}\\exp{(\\beta_{l0}+\\beta_{l1}x_1)}}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "It is easy to extend to more predictors. The final class is" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "p(C=K\\vert \\mathbf {x} )=\\frac{1}{1+\\sum_{l=1}^{K-1}\\exp{(\\beta_{l0}+\\beta_{l1}x_1)}},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and they sum to one. Our earlier discussions were all specialized to the case with two classes only. It is easy to see from the above that what we derived earlier is compatible with these equations. \n", + "\n", + "To find the optimal parameters we would typically use a gradient descent method.\n", + "Newton's method and gradient descent methods are discussed in the material on [optimization methods](https://compphysics.github.io/MachineLearning/doc/pub/Splines/html/Splines-bs.html). \n", "\n", "\n", "\n", @@ -388,7 +524,9 @@ { "cell_type": "code", "execution_count": 1, - "metadata": {}, + "metadata": { + "collapsed": false + }, "outputs": [], "source": [ "%matplotlib inline\n", @@ -423,7 +561,9 @@ { "cell_type": "code", "execution_count": 2, - "metadata": {}, + "metadata": { + "collapsed": false + }, "outputs": [], "source": [ "import numpy as np\n", @@ -478,25 +618,7 @@ ] } ], - "metadata": { - "kernelspec": { - "display_name": "Python 3", - "language": "python", - "name": "python3" - }, - "language_info": { - "codemirror_mode": { - "name": "ipython", - "version": 3 - }, - 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