From 6a18aa51b4567204ff48df74b992411e08f72a41 Mon Sep 17 00:00:00 2001 From: mhjensen Date: Thu, 8 Nov 2018 11:33:20 +0100 Subject: [PATCH] Update on sim part --- doc/pub/svm/html/._svm-bs000.html | 26 +- doc/pub/svm/html/._svm-bs001.html | 24 +- doc/pub/svm/html/._svm-bs002.html | 97 ++++++- doc/pub/svm/html/._svm-bs003.html | 24 +- doc/pub/svm/html/._svm-bs004.html | 24 +- doc/pub/svm/html/._svm-bs005.html | 24 +- doc/pub/svm/html/._svm-bs006.html | 24 +- doc/pub/svm/html/._svm-bs007.html | 24 +- doc/pub/svm/html/._svm-bs008.html | 24 +- doc/pub/svm/html/._svm-bs009.html | 24 +- doc/pub/svm/html/._svm-bs010.html | 24 +- doc/pub/svm/html/._svm-bs011.html | 24 +- doc/pub/svm/html/._svm-bs012.html | 24 +- doc/pub/svm/html/._svm-bs013.html | 24 +- doc/pub/svm/html/._svm-bs014.html | 24 +- doc/pub/svm/html/._svm-bs015.html | 24 +- doc/pub/svm/html/._svm-bs016.html | 24 +- doc/pub/svm/html/._svm-bs017.html | 73 ++++- doc/pub/svm/html/._svm-bs018.html | 23 +- doc/pub/svm/html/._svm-bs019.html | 23 +- doc/pub/svm/html/._svm-bs020.html | 23 +- doc/pub/svm/html/._svm-bs021.html | 228 ++++++++++++++-- doc/pub/svm/html/._svm-bs022.html | 54 ++-- doc/pub/svm/html/._svm-bs023.html | 93 ++----- doc/pub/svm/html/._svm-bs024.html | 94 +++++-- doc/pub/svm/html/._svm-bs025.html | 46 +++- doc/pub/svm/html/._svm-bs026.html | 197 ++++++++++++++ doc/pub/svm/html/svm-bs.html | 26 +- doc/pub/svm/html/svm-reveal.html | 334 ++++++++++++++++++++++- doc/pub/svm/html/svm-solarized.html | 342 ++++++++++++++++++++++- doc/pub/svm/html/svm.html | 342 ++++++++++++++++++++++- doc/pub/svm/ipynb/ipynb-svm-src.tar.gz | Bin 208 -> 207 bytes doc/pub/svm/ipynb/svm.ipynb | 363 ++++++++++++++++++++++++- doc/pub/svm/pdf/svm-minted.pdf | Bin 331483 -> 347549 bytes doc/src/SupportVMachines/svm.do.txt | 331 +++++++++++++++++++++- 35 files changed, 2637 insertions(+), 438 deletions(-) create mode 100644 doc/pub/svm/html/._svm-bs026.html diff --git a/doc/pub/svm/html/._svm-bs000.html b/doc/pub/svm/html/._svm-bs000.html index d46b684c2..5a2191ad4 100644 --- a/doc/pub/svm/html/._svm-bs000.html +++ b/doc/pub/svm/html/._svm-bs000.html @@ -63,17 +63,18 @@ Automatically generated HTML file from DocOnce source ('The equations', 2, None, '___sec17'), ('The problem to solve', 2, None, '___sec18'), ("Different kernels and Mercer's theorem", 2, None, '___sec19'), + ('The moons example', 2, None, '___sec20'), ('Mathematical optimization of convex functions', 2, None, - '___sec20'), - ('How do we solve these problems?', 2, None, '___sec21'), - ('A simplex example', 2, None, '___sec22'), - ('Back to the more realistic cases', 2, None, '___sec23'), + '___sec21'), + ('How do we solve these problems?', 2, None, '___sec22'), + ('A simplex example', 2, None, '___sec23'), + ('Back to the more realistic cases', 2, None, '___sec24'), ('Multiclass problems and regression with SVMs', 2, None, - '___sec24')]} + '___sec25')]} end of tocinfo --> @@ -131,11 +132,12 @@ MathJax.Hub.Config({
  • The equations
  • The problem to solve
  • Different kernels and Mercer's theorem
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simplex example
    • -
  • Back to the more realistic cases
    • -
  • Multiclass problems and regression with SVMs
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simplex example
    • +
  • Back to the more realistic cases
    • +
  • Multiclass problems and regression with SVMs
    • @@ -170,7 +172,7 @@ MathJax.Hub.Config({
    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

    -

    Nov 7, 2018

    +

    Nov 8, 2018


    @@ -194,7 +196,7 @@ MathJax.Hub.Config({

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    • -
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  • »
    • diff --git a/doc/pub/svm/html/._svm-bs001.html b/doc/pub/svm/html/._svm-bs001.html index 399026d80..d4b3db13e 100644 --- a/doc/pub/svm/html/._svm-bs001.html +++ b/doc/pub/svm/html/._svm-bs001.html @@ -63,17 +63,18 @@ Automatically generated HTML file from DocOnce source ('The equations', 2, None, '___sec17'), ('The problem to solve', 2, None, '___sec18'), ("Different kernels and Mercer's theorem", 2, None, '___sec19'), + ('The moons example', 2, None, '___sec20'), ('Mathematical optimization of convex functions', 2, None, - '___sec20'), - ('How do we solve these problems?', 2, None, '___sec21'), - ('A simplex example', 2, None, '___sec22'), - ('Back to the more realistic cases', 2, None, '___sec23'), + '___sec21'), + ('How do we solve these problems?', 2, None, '___sec22'), + ('A simplex example', 2, None, '___sec23'), + ('Back to the more realistic cases', 2, None, '___sec24'), ('Multiclass problems and regression with SVMs', 2, None, - '___sec24')]} + '___sec25')]} end of tocinfo --> @@ -131,11 +132,12 @@ MathJax.Hub.Config({
  • The equations
  • The problem to solve
  • Different kernels and Mercer's theorem
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simplex example
    • -
  • Back to the more realistic cases
    • -
  • Multiclass problems and regression with SVMs
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simplex example
    • +
  • Back to the more realistic cases
    • +
  • Multiclass problems and regression with SVMs
    • @@ -199,7 +201,7 @@ unlikely that we can separate classes easily by say straight lines.
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    • diff --git a/doc/pub/svm/html/._svm-bs002.html b/doc/pub/svm/html/._svm-bs002.html index 07b5be673..ea5e950a6 100644 --- a/doc/pub/svm/html/._svm-bs002.html +++ b/doc/pub/svm/html/._svm-bs002.html @@ -63,17 +63,18 @@ Automatically generated HTML file from DocOnce source ('The equations', 2, None, '___sec17'), ('The problem to solve', 2, None, '___sec18'), ("Different kernels and Mercer's theorem", 2, None, '___sec19'), + ('The moons example', 2, None, '___sec20'), ('Mathematical optimization of convex functions', 2, None, - '___sec20'), - ('How do we solve these problems?', 2, None, '___sec21'), - ('A simplex example', 2, None, '___sec22'), - ('Back to the more realistic cases', 2, None, '___sec23'), + '___sec21'), + ('How do we solve these problems?', 2, None, '___sec22'), + ('A simplex example', 2, None, '___sec23'), + ('Back to the more realistic cases', 2, None, '___sec24'), ('Multiclass problems and regression with SVMs', 2, None, - '___sec24')]} + '___sec25')]} end of tocinfo --> @@ -131,11 +132,12 @@ MathJax.Hub.Config({
  • The equations
  • The problem to solve
  • Different kernels and Mercer's theorem
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simplex example
    • -
  • Back to the more realistic cases
    • -
  • Multiclass problems and regression with SVMs
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simplex example
    • +
  • Back to the more realistic cases
    • +
  • Multiclass problems and regression with SVMs
    • @@ -158,14 +160,83 @@ The theory behind support vector machines (SVM hereafter) is based on the mathematical description of so-called hyperplanes. Let us start with a two-dimensional case. This will also allow us to introduce our first SVM examples. These will be tailored to the case of two specific -classes, as displayed in the figure here. +classes, as displayed in the figure here based on the usage of the petal data.

    We assume here that our data set can be well separated into two domains, where a straight line does the job in the separating the two -classes. Here the two classes are represented by either crosses or +classes. Here the two classes are represented by either squares or circles. +

    + +

    from sklearn import datasets
+from sklearn.svm import SVC, LinearSVC
+from sklearn.linear_model import SGDClassifier
+from sklearn.preprocessing import StandardScaler
+import matplotlib
+import matplotlib.pyplot as plt
+plt.rcParams['axes.labelsize'] = 14
+plt.rcParams['xtick.labelsize'] = 12
+plt.rcParams['ytick.labelsize'] = 12
+
+
+iris = datasets.load_iris()
+X = iris["data"][:, (2, 3)]  # petal length, petal width
+y = iris["target"]
+
+setosa_or_versicolor = (y == 0) | (y == 1)
+X = X[setosa_or_versicolor]
+y = y[setosa_or_versicolor]
+
+
+
+C = 5
+alpha = 1 / (C * len(X))
+
+lin_clf = LinearSVC(loss="hinge", C=C, random_state=42)
+svm_clf = SVC(kernel="linear", C=C)
+sgd_clf = SGDClassifier(loss="hinge", learning_rate="constant", eta0=0.001, alpha=alpha,
+                        max_iter=100000, random_state=42)
+
+scaler = StandardScaler()
+X_scaled = scaler.fit_transform(X)
+
+lin_clf.fit(X_scaled, y)
+svm_clf.fit(X_scaled, y)
+sgd_clf.fit(X_scaled, y)
+
+print("LinearSVC:                   ", lin_clf.intercept_, lin_clf.coef_)
+print("SVC:                         ", svm_clf.intercept_, svm_clf.coef_)
+print("SGDClassifier(alpha={:.5f}):".format(sgd_clf.alpha), sgd_clf.intercept_, sgd_clf.coef_)
+
+# Compute the slope and bias of each decision boundary
+w1 = -lin_clf.coef_[0, 0]/lin_clf.coef_[0, 1]
+b1 = -lin_clf.intercept_[0]/lin_clf.coef_[0, 1]
+w2 = -svm_clf.coef_[0, 0]/svm_clf.coef_[0, 1]
+b2 = -svm_clf.intercept_[0]/svm_clf.coef_[0, 1]
+w3 = -sgd_clf.coef_[0, 0]/sgd_clf.coef_[0, 1]
+b3 = -sgd_clf.intercept_[0]/sgd_clf.coef_[0, 1]
+
+# Transform the decision boundary lines back to the original scale
+line1 = scaler.inverse_transform([[-10, -10 * w1 + b1], [10, 10 * w1 + b1]])
+line2 = scaler.inverse_transform([[-10, -10 * w2 + b2], [10, 10 * w2 + b2]])
+line3 = scaler.inverse_transform([[-10, -10 * w3 + b3], [10, 10 * w3 + b3]])
+
+# Plot all three decision boundaries
+plt.figure(figsize=(11, 4))
+plt.plot(line1[:, 0], line1[:, 1], "k:", label="LinearSVC")
+plt.plot(line2[:, 0], line2[:, 1], "b--", linewidth=2, label="SVC")
+plt.plot(line3[:, 0], line3[:, 1], "r-", label="SGDClassifier")
+plt.plot(X[:, 0][y==1], X[:, 1][y==1], "bs") # label="Iris-Versicolor"
+plt.plot(X[:, 0][y==0], X[:, 1][y==0], "yo") # label="Iris-Setosa"
+plt.xlabel("Petal length", fontsize=14)
+plt.ylabel("Petal width", fontsize=14)
+plt.legend(loc="upper center", fontsize=14)
+plt.axis([0, 5.5, 0, 2])
+
+plt.show()
+

    @@ -184,7 +255,7 @@ circles.

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    • diff --git a/doc/pub/svm/html/._svm-bs003.html b/doc/pub/svm/html/._svm-bs003.html index bb33133ca..64c4fc579 100644 --- a/doc/pub/svm/html/._svm-bs003.html +++ b/doc/pub/svm/html/._svm-bs003.html @@ -63,17 +63,18 @@ Automatically generated HTML file from DocOnce source ('The equations', 2, None, '___sec17'), ('The problem to solve', 2, None, '___sec18'), ("Different kernels and Mercer's theorem", 2, None, '___sec19'), + ('The moons example', 2, None, '___sec20'), ('Mathematical optimization of convex functions', 2, None, - '___sec20'), - ('How do we solve these problems?', 2, None, '___sec21'), - ('A simplex example', 2, None, '___sec22'), - ('Back to the more realistic cases', 2, None, '___sec23'), + '___sec21'), + ('How do we solve these problems?', 2, None, '___sec22'), + ('A simplex example', 2, None, '___sec23'), + ('Back to the more realistic cases', 2, None, '___sec24'), ('Multiclass problems and regression with SVMs', 2, None, - '___sec24')]} + '___sec25')]} end of tocinfo --> @@ -131,11 +132,12 @@ MathJax.Hub.Config({
  • The equations
  • The problem to solve
  • Different kernels and Mercer's theorem
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simplex example
    • -
  • Back to the more realistic cases
    • -
  • Multiclass problems and regression with SVMs
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simplex example
    • +
  • Back to the more realistic cases
    • +
  • Multiclass problems and regression with SVMs
    • @@ -194,7 +196,7 @@ $$
  • 12
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  • ...
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    • diff --git a/doc/pub/svm/html/._svm-bs004.html b/doc/pub/svm/html/._svm-bs004.html index 749e77be1..939997501 100644 --- a/doc/pub/svm/html/._svm-bs004.html +++ b/doc/pub/svm/html/._svm-bs004.html @@ -63,17 +63,18 @@ Automatically generated HTML file from DocOnce source ('The equations', 2, None, '___sec17'), ('The problem to solve', 2, None, '___sec18'), ("Different kernels and Mercer's theorem", 2, None, '___sec19'), + ('The moons example', 2, None, '___sec20'), ('Mathematical optimization of convex functions', 2, None, - '___sec20'), - ('How do we solve these problems?', 2, None, '___sec21'), - ('A simplex example', 2, None, '___sec22'), - ('Back to the more realistic cases', 2, None, '___sec23'), + '___sec21'), + ('How do we solve these problems?', 2, None, '___sec22'), + ('A simplex example', 2, None, '___sec23'), + ('Back to the more realistic cases', 2, None, '___sec24'), ('Multiclass problems and regression with SVMs', 2, None, - '___sec24')]} + '___sec25')]} end of tocinfo --> @@ -131,11 +132,12 @@ MathJax.Hub.Config({
  • The equations
  • The problem to solve
  • Different kernels and Mercer's theorem
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simplex example
    • -
  • Back to the more realistic cases
    • -
  • Multiclass problems and regression with SVMs
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simplex example
    • +
  • Back to the more realistic cases
    • +
  • Multiclass problems and regression with SVMs
    • @@ -210,7 +212,7 @@ When we try to separate hyperplanes, if it exists, we can use it to construct a
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    • diff --git a/doc/pub/svm/html/._svm-bs005.html b/doc/pub/svm/html/._svm-bs005.html index 58e847731..e25dc7676 100644 --- a/doc/pub/svm/html/._svm-bs005.html +++ b/doc/pub/svm/html/._svm-bs005.html @@ -63,17 +63,18 @@ Automatically generated HTML file from DocOnce source ('The equations', 2, None, '___sec17'), ('The problem to solve', 2, None, '___sec18'), ("Different kernels and Mercer's theorem", 2, None, '___sec19'), + ('The moons example', 2, None, '___sec20'), ('Mathematical optimization of convex functions', 2, None, - '___sec20'), - ('How do we solve these problems?', 2, None, '___sec21'), - ('A simplex example', 2, None, '___sec22'), - ('Back to the more realistic cases', 2, None, '___sec23'), + '___sec21'), + ('How do we solve these problems?', 2, None, '___sec22'), + ('A simplex example', 2, None, '___sec23'), + ('Back to the more realistic cases', 2, None, '___sec24'), ('Multiclass problems and regression with SVMs', 2, None, - '___sec24')]} + '___sec25')]} end of tocinfo --> @@ -131,11 +132,12 @@ MathJax.Hub.Config({
  • The equations
  • The problem to solve
  • Different kernels and Mercer's theorem
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simplex example
    • -
  • Back to the more realistic cases
    • -
  • Multiclass problems and regression with SVMs
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simplex example
    • +
  • Back to the more realistic cases
    • +
  • Multiclass problems and regression with SVMs
    • @@ -196,7 +198,7 @@ for our data sample.
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    • diff --git a/doc/pub/svm/html/._svm-bs006.html b/doc/pub/svm/html/._svm-bs006.html index db52ab566..bd69f3bf9 100644 --- a/doc/pub/svm/html/._svm-bs006.html +++ b/doc/pub/svm/html/._svm-bs006.html @@ -63,17 +63,18 @@ Automatically generated HTML file from DocOnce source ('The equations', 2, None, '___sec17'), ('The problem to solve', 2, None, '___sec18'), ("Different kernels and Mercer's theorem", 2, None, '___sec19'), + ('The moons example', 2, None, '___sec20'), ('Mathematical optimization of convex functions', 2, None, - '___sec20'), - ('How do we solve these problems?', 2, None, '___sec21'), - ('A simplex example', 2, None, '___sec22'), - ('Back to the more realistic cases', 2, None, '___sec23'), + '___sec21'), + ('How do we solve these problems?', 2, None, '___sec22'), + ('A simplex example', 2, None, '___sec23'), + ('Back to the more realistic cases', 2, None, '___sec24'), ('Multiclass problems and regression with SVMs', 2, None, - '___sec24')]} + '___sec25')]} end of tocinfo --> @@ -131,11 +132,12 @@ MathJax.Hub.Config({
  • The equations
  • The problem to solve
  • Different kernels and Mercer's theorem
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simplex example
    • -
  • Back to the more realistic cases
    • -
  • Multiclass problems and regression with SVMs
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simplex example
    • +
  • Back to the more realistic cases
    • +
  • Multiclass problems and regression with SVMs
    • @@ -192,7 +194,7 @@ $$
  • 15
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    • diff --git a/doc/pub/svm/html/._svm-bs007.html b/doc/pub/svm/html/._svm-bs007.html index b748b5532..6c8374c5f 100644 --- a/doc/pub/svm/html/._svm-bs007.html +++ b/doc/pub/svm/html/._svm-bs007.html @@ -63,17 +63,18 @@ Automatically generated HTML file from DocOnce source ('The equations', 2, None, '___sec17'), ('The problem to solve', 2, None, '___sec18'), ("Different kernels and Mercer's theorem", 2, None, '___sec19'), + ('The moons example', 2, None, '___sec20'), ('Mathematical optimization of convex functions', 2, None, - '___sec20'), - ('How do we solve these problems?', 2, None, '___sec21'), - ('A simplex example', 2, None, '___sec22'), - ('Back to the more realistic cases', 2, None, '___sec23'), + '___sec21'), + ('How do we solve these problems?', 2, None, '___sec22'), + ('A simplex example', 2, None, '___sec23'), + ('Back to the more realistic cases', 2, None, '___sec24'), ('Multiclass problems and regression with SVMs', 2, None, - '___sec24')]} + '___sec25')]} end of tocinfo --> @@ -131,11 +132,12 @@ MathJax.Hub.Config({
  • The equations
  • The problem to solve
  • Different kernels and Mercer's theorem
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simplex example
    • -
  • Back to the more realistic cases
    • -
  • Multiclass problems and regression with SVMs
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simplex example
    • +
  • Back to the more realistic cases
    • +
  • Multiclass problems and regression with SVMs
    • @@ -196,7 +198,7 @@ $$
  • 16
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  • ...
    • -
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    • +
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  • »
    • diff --git a/doc/pub/svm/html/._svm-bs008.html b/doc/pub/svm/html/._svm-bs008.html index 48006344a..d78f7d1ea 100644 --- a/doc/pub/svm/html/._svm-bs008.html +++ b/doc/pub/svm/html/._svm-bs008.html @@ -63,17 +63,18 @@ Automatically generated HTML file from DocOnce source ('The equations', 2, None, '___sec17'), ('The problem to solve', 2, None, '___sec18'), ("Different kernels and Mercer's theorem", 2, None, '___sec19'), + ('The moons example', 2, None, '___sec20'), ('Mathematical optimization of convex functions', 2, None, - '___sec20'), - ('How do we solve these problems?', 2, None, '___sec21'), - ('A simplex example', 2, None, '___sec22'), - ('Back to the more realistic cases', 2, None, '___sec23'), + '___sec21'), + ('How do we solve these problems?', 2, None, '___sec22'), + ('A simplex example', 2, None, '___sec23'), + ('Back to the more realistic cases', 2, None, '___sec24'), ('Multiclass problems and regression with SVMs', 2, None, - '___sec24')]} + '___sec25')]} end of tocinfo --> @@ -131,11 +132,12 @@ MathJax.Hub.Config({
  • The equations
  • The problem to solve
  • Different kernels and Mercer's theorem
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simplex example
    • -
  • Back to the more realistic cases
    • -
  • Multiclass problems and regression with SVMs
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simplex example
    • +
  • Back to the more realistic cases
    • +
  • Multiclass problems and regression with SVMs
    • @@ -200,7 +202,7 @@ at all.
  • 17
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    • +
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  • »
    • diff --git a/doc/pub/svm/html/._svm-bs009.html b/doc/pub/svm/html/._svm-bs009.html index 80dfee3db..b1a972e43 100644 --- a/doc/pub/svm/html/._svm-bs009.html +++ b/doc/pub/svm/html/._svm-bs009.html @@ -63,17 +63,18 @@ Automatically generated HTML file from DocOnce source ('The equations', 2, None, '___sec17'), ('The problem to solve', 2, None, '___sec18'), ("Different kernels and Mercer's theorem", 2, None, '___sec19'), + ('The moons example', 2, None, '___sec20'), ('Mathematical optimization of convex functions', 2, None, - '___sec20'), - ('How do we solve these problems?', 2, None, '___sec21'), - ('A simplex example', 2, None, '___sec22'), - ('Back to the more realistic cases', 2, None, '___sec23'), + '___sec21'), + ('How do we solve these problems?', 2, None, '___sec22'), + ('A simplex example', 2, None, '___sec23'), + ('Back to the more realistic cases', 2, None, '___sec24'), ('Multiclass problems and regression with SVMs', 2, None, - '___sec24')]} + '___sec25')]} end of tocinfo --> @@ -131,11 +132,12 @@ MathJax.Hub.Config({
  • The equations
  • The problem to solve
  • Different kernels and Mercer's theorem
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simplex example
    • -
  • Back to the more realistic cases
    • -
  • Multiclass problems and regression with SVMs
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simplex example
    • +
  • Back to the more realistic cases
    • +
  • Multiclass problems and regression with SVMs
    • @@ -212,7 +214,7 @@ We have thus defined our margin as the invers of the norm of \( \boldsymbol{w} \
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    • diff --git a/doc/pub/svm/html/._svm-bs010.html b/doc/pub/svm/html/._svm-bs010.html index e8cac7084..d8d4c2f3f 100644 --- a/doc/pub/svm/html/._svm-bs010.html +++ b/doc/pub/svm/html/._svm-bs010.html @@ -63,17 +63,18 @@ Automatically generated HTML file from DocOnce source ('The equations', 2, None, '___sec17'), ('The problem to solve', 2, None, '___sec18'), ("Different kernels and Mercer's theorem", 2, None, '___sec19'), + ('The moons example', 2, None, '___sec20'), ('Mathematical optimization of convex functions', 2, None, - '___sec20'), - ('How do we solve these problems?', 2, None, '___sec21'), - ('A simplex example', 2, None, '___sec22'), - ('Back to the more realistic cases', 2, None, '___sec23'), + '___sec21'), + ('How do we solve these problems?', 2, None, '___sec22'), + ('A simplex example', 2, None, '___sec23'), + ('Back to the more realistic cases', 2, None, '___sec24'), ('Multiclass problems and regression with SVMs', 2, None, - '___sec24')]} + '___sec25')]} end of tocinfo --> @@ -131,11 +132,12 @@ MathJax.Hub.Config({
  • The equations
  • The problem to solve
  • Different kernels and Mercer's theorem
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simplex example
    • -
  • Back to the more realistic cases
    • -
  • Multiclass problems and regression with SVMs
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simplex example
    • +
  • Back to the more realistic cases
    • +
  • Multiclass problems and regression with SVMs
    • @@ -224,7 +226,7 @@ Then \( dz \) is no longer arbitrary.
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    • -
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  • »
    • diff --git a/doc/pub/svm/html/._svm-bs011.html b/doc/pub/svm/html/._svm-bs011.html index 260e4471a..24b2a6c49 100644 --- a/doc/pub/svm/html/._svm-bs011.html +++ b/doc/pub/svm/html/._svm-bs011.html @@ -63,17 +63,18 @@ Automatically generated HTML file from DocOnce source ('The equations', 2, None, '___sec17'), ('The problem to solve', 2, None, '___sec18'), ("Different kernels and Mercer's theorem", 2, None, '___sec19'), + ('The moons example', 2, None, '___sec20'), ('Mathematical optimization of convex functions', 2, None, - '___sec20'), - ('How do we solve these problems?', 2, None, '___sec21'), - ('A simplex example', 2, None, '___sec22'), - ('Back to the more realistic cases', 2, None, '___sec23'), + '___sec21'), + ('How do we solve these problems?', 2, None, '___sec22'), + ('A simplex example', 2, None, '___sec23'), + ('Back to the more realistic cases', 2, None, '___sec24'), ('Multiclass problems and regression with SVMs', 2, None, - '___sec24')]} + '___sec25')]} end of tocinfo --> @@ -131,11 +132,12 @@ MathJax.Hub.Config({
  • The equations
  • The problem to solve
  • Different kernels and Mercer's theorem
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simplex example
    • -
  • Back to the more realistic cases
    • -
  • Multiclass problems and regression with SVMs
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simplex example
    • +
  • Back to the more realistic cases
    • +
  • Multiclass problems and regression with SVMs
    • @@ -217,7 +219,7 @@ $$
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    • -
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  • »
    • diff --git a/doc/pub/svm/html/._svm-bs012.html b/doc/pub/svm/html/._svm-bs012.html index c73530b2f..47a93895f 100644 --- a/doc/pub/svm/html/._svm-bs012.html +++ b/doc/pub/svm/html/._svm-bs012.html @@ -63,17 +63,18 @@ Automatically generated HTML file from DocOnce source ('The equations', 2, None, '___sec17'), ('The problem to solve', 2, None, '___sec18'), ("Different kernels and Mercer's theorem", 2, None, '___sec19'), + ('The moons example', 2, None, '___sec20'), ('Mathematical optimization of convex functions', 2, None, - '___sec20'), - ('How do we solve these problems?', 2, None, '___sec21'), - ('A simplex example', 2, None, '___sec22'), - ('Back to the more realistic cases', 2, None, '___sec23'), + '___sec21'), + ('How do we solve these problems?', 2, None, '___sec22'), + ('A simplex example', 2, None, '___sec23'), + ('Back to the more realistic cases', 2, None, '___sec24'), ('Multiclass problems and regression with SVMs', 2, None, - '___sec24')]} + '___sec25')]} end of tocinfo --> @@ -131,11 +132,12 @@ MathJax.Hub.Config({
  • The equations
  • The problem to solve
  • Different kernels and Mercer's theorem
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simplex example
    • -
  • Back to the more realistic cases
    • -
  • Multiclass problems and regression with SVMs
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simplex example
    • +
  • Back to the more realistic cases
    • +
  • Multiclass problems and regression with SVMs
    • @@ -215,7 +217,7 @@ When \( \lambda_i > 0 \), the vectors \( \boldsymbol{x}_i \) are called support
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    • diff --git a/doc/pub/svm/html/._svm-bs013.html b/doc/pub/svm/html/._svm-bs013.html index 999868c65..b523ba1b2 100644 --- a/doc/pub/svm/html/._svm-bs013.html +++ b/doc/pub/svm/html/._svm-bs013.html @@ -63,17 +63,18 @@ Automatically generated HTML file from DocOnce source ('The equations', 2, None, '___sec17'), ('The problem to solve', 2, None, '___sec18'), ("Different kernels and Mercer's theorem", 2, None, '___sec19'), + ('The moons example', 2, None, '___sec20'), ('Mathematical optimization of convex functions', 2, None, - '___sec20'), - ('How do we solve these problems?', 2, None, '___sec21'), - ('A simplex example', 2, None, '___sec22'), - ('Back to the more realistic cases', 2, None, '___sec23'), + '___sec21'), + ('How do we solve these problems?', 2, None, '___sec22'), + ('A simplex example', 2, None, '___sec23'), + ('Back to the more realistic cases', 2, None, '___sec24'), ('Multiclass problems and regression with SVMs', 2, None, - '___sec24')]} + '___sec25')]} end of tocinfo --> @@ -131,11 +132,12 @@ MathJax.Hub.Config({
  • The equations
  • The problem to solve
  • Different kernels and Mercer's theorem
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simplex example
    • -
  • Back to the more realistic cases
    • -
  • Multiclass problems and regression with SVMs
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simplex example
    • +
  • Back to the more realistic cases
    • +
  • Multiclass problems and regression with SVMs
    • @@ -198,7 +200,7 @@ subject to \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \). Here we defined the vec
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    • -
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  • »
    • diff --git a/doc/pub/svm/html/._svm-bs014.html b/doc/pub/svm/html/._svm-bs014.html index b56b6ce36..579cadacd 100644 --- a/doc/pub/svm/html/._svm-bs014.html +++ b/doc/pub/svm/html/._svm-bs014.html @@ -63,17 +63,18 @@ Automatically generated HTML file from DocOnce source ('The equations', 2, None, '___sec17'), ('The problem to solve', 2, None, '___sec18'), ("Different kernels and Mercer's theorem", 2, None, '___sec19'), + ('The moons example', 2, None, '___sec20'), ('Mathematical optimization of convex functions', 2, None, - '___sec20'), - ('How do we solve these problems?', 2, None, '___sec21'), - ('A simplex example', 2, None, '___sec22'), - ('Back to the more realistic cases', 2, None, '___sec23'), + '___sec21'), + ('How do we solve these problems?', 2, None, '___sec22'), + ('A simplex example', 2, None, '___sec23'), + ('Back to the more realistic cases', 2, None, '___sec24'), ('Multiclass problems and regression with SVMs', 2, None, - '___sec24')]} + '___sec25')]} end of tocinfo --> @@ -131,11 +132,12 @@ MathJax.Hub.Config({
  • The equations
  • The problem to solve
  • Different kernels and Mercer's theorem
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simplex example
    • -
  • Back to the more realistic cases
    • -
  • Multiclass problems and regression with SVMs
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simplex example
    • +
  • Back to the more realistic cases
    • +
  • Multiclass problems and regression with SVMs
    • @@ -208,7 +210,7 @@ Below we discuss how to find the optimal values of \( \lambda_i \). Before we pr
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    • -
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  • »
    • diff --git a/doc/pub/svm/html/._svm-bs015.html b/doc/pub/svm/html/._svm-bs015.html index 6fc1126c9..5abf7c9cc 100644 --- a/doc/pub/svm/html/._svm-bs015.html +++ b/doc/pub/svm/html/._svm-bs015.html @@ -63,17 +63,18 @@ Automatically generated HTML file from DocOnce source ('The equations', 2, None, '___sec17'), ('The problem to solve', 2, None, '___sec18'), ("Different kernels and Mercer's theorem", 2, None, '___sec19'), + ('The moons example', 2, None, '___sec20'), ('Mathematical optimization of convex functions', 2, None, - '___sec20'), - ('How do we solve these problems?', 2, None, '___sec21'), - ('A simplex example', 2, None, '___sec22'), - ('Back to the more realistic cases', 2, None, '___sec23'), + '___sec21'), + ('How do we solve these problems?', 2, None, '___sec22'), + ('A simplex example', 2, None, '___sec23'), + ('Back to the more realistic cases', 2, None, '___sec24'), ('Multiclass problems and regression with SVMs', 2, None, - '___sec24')]} + '___sec25')]} end of tocinfo --> @@ -131,11 +132,12 @@ MathJax.Hub.Config({
  • The equations
  • The problem to solve
  • Different kernels and Mercer's theorem
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simplex example
    • -
  • Back to the more realistic cases
    • -
  • Multiclass problems and regression with SVMs
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simplex example
    • +
  • Back to the more realistic cases
    • +
  • Multiclass problems and regression with SVMs
    • @@ -209,7 +211,7 @@ misclassifications.
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    • -
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  • »
    • diff --git a/doc/pub/svm/html/._svm-bs016.html b/doc/pub/svm/html/._svm-bs016.html index 7ae6500f9..b4a85d6ec 100644 --- a/doc/pub/svm/html/._svm-bs016.html +++ b/doc/pub/svm/html/._svm-bs016.html @@ -63,17 +63,18 @@ Automatically generated HTML file from DocOnce source ('The equations', 2, None, '___sec17'), ('The problem to solve', 2, None, '___sec18'), ("Different kernels and Mercer's theorem", 2, None, '___sec19'), + ('The moons example', 2, None, '___sec20'), ('Mathematical optimization of convex functions', 2, None, - '___sec20'), - ('How do we solve these problems?', 2, None, '___sec21'), - ('A simplex example', 2, None, '___sec22'), - ('Back to the more realistic cases', 2, None, '___sec23'), + '___sec21'), + ('How do we solve these problems?', 2, None, '___sec22'), + ('A simplex example', 2, None, '___sec23'), + ('Back to the more realistic cases', 2, None, '___sec24'), ('Multiclass problems and regression with SVMs', 2, None, - '___sec24')]} + '___sec25')]} end of tocinfo --> @@ -131,11 +132,12 @@ MathJax.Hub.Config({
  • The equations
  • The problem to solve
  • Different kernels and Mercer's theorem
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simplex example
    • -
  • Back to the more realistic cases
    • -
  • Multiclass problems and regression with SVMs
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simplex example
    • +
  • Back to the more realistic cases
    • +
  • Multiclass problems and regression with SVMs
    • @@ -227,6 +229,8 @@ $$
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    • diff --git a/doc/pub/svm/html/._svm-bs017.html b/doc/pub/svm/html/._svm-bs017.html index b832478ce..bc459cbb1 100644 --- a/doc/pub/svm/html/._svm-bs017.html +++ b/doc/pub/svm/html/._svm-bs017.html @@ -63,17 +63,18 @@ Automatically generated HTML file from DocOnce source ('The equations', 2, None, '___sec17'), ('The problem to solve', 2, None, '___sec18'), ("Different kernels and Mercer's theorem", 2, None, '___sec19'), + ('The moons example', 2, None, '___sec20'), ('Mathematical optimization of convex functions', 2, None, - '___sec20'), - ('How do we solve these problems?', 2, None, '___sec21'), - ('A simplex example', 2, None, '___sec22'), - ('Back to the more realistic cases', 2, None, '___sec23'), + '___sec21'), + ('How do we solve these problems?', 2, None, '___sec22'), + ('A simplex example', 2, None, '___sec23'), + ('Back to the more realistic cases', 2, None, '___sec24'), ('Multiclass problems and regression with SVMs', 2, None, - '___sec24')]} + '___sec25')]} end of tocinfo --> @@ -131,11 +132,12 @@ MathJax.Hub.Config({
  • The equations
  • The problem to solve
  • Different kernels and Mercer's theorem
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simplex example
    • -
  • Back to the more realistic cases
    • -
  • Multiclass problems and regression with SVMs
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simplex example
    • +
  • Back to the more realistic cases
    • +
  • Multiclass problems and regression with SVMs
    • @@ -171,6 +173,56 @@ obtain a separation between the classes which is almost linear. The change of basis, from \( x\rightarrow z=\phi(x) \) leads to the same type of equations to be solved, except that we need to introduce for example a polynomial transformation to a two-dimensional training set. +

    + + +

    import numpy as np
+import os
+
+np.random.seed(42)
+
+# To plot pretty figures
+import matplotlib
+import matplotlib.pyplot as plt
+plt.rcParams['axes.labelsize'] = 14
+plt.rcParams['xtick.labelsize'] = 12
+plt.rcParams['ytick.labelsize'] = 12
+
+
+from sklearn.svm import SVC
+from sklearn import datasets
+
+
+
+X1D = np.linspace(-4, 4, 9).reshape(-1, 1)
+X2D = np.c_[X1D, X1D**2]
+y = np.array([0, 0, 1, 1, 1, 1, 1, 0, 0])
+
+plt.figure(figsize=(11, 4))
+
+plt.subplot(121)
+plt.grid(True, which='both')
+plt.axhline(y=0, color='k')
+plt.plot(X1D[:, 0][y==0], np.zeros(4), "bs")
+plt.plot(X1D[:, 0][y==1], np.zeros(5), "g^")
+plt.gca().get_yaxis().set_ticks([])
+plt.xlabel(r"$x_1$", fontsize=20)
+plt.axis([-4.5, 4.5, -0.2, 0.2])
+
+plt.subplot(122)
+plt.grid(True, which='both')
+plt.axhline(y=0, color='k')
+plt.axvline(x=0, color='k')
+plt.plot(X2D[:, 0][y==0], X2D[:, 1][y==0], "bs")
+plt.plot(X2D[:, 0][y==1], X2D[:, 1][y==1], "g^")
+plt.xlabel(r"$x_1$", fontsize=20)
+plt.ylabel(r"$x_2$", fontsize=20, rotation=0)
+plt.gca().get_yaxis().set_ticks([0, 4, 8, 12, 16])
+plt.plot([-4.5, 4.5], [6.5, 6.5], "r--", linewidth=3)
+plt.axis([-4.5, 4.5, -1, 17])
+plt.subplots_adjust(right=1)
+plt.show()
+

    @@ -195,6 +247,7 @@ we need to introduce for example a polynomial transformation to a two-dimensiona

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    • diff --git a/doc/pub/svm/html/._svm-bs018.html b/doc/pub/svm/html/._svm-bs018.html index d79c46dfd..f9dd5ae52 100644 --- a/doc/pub/svm/html/._svm-bs018.html +++ b/doc/pub/svm/html/._svm-bs018.html @@ -63,17 +63,18 @@ Automatically generated HTML file from DocOnce source ('The equations', 2, None, '___sec17'), ('The problem to solve', 2, None, '___sec18'), ("Different kernels and Mercer's theorem", 2, None, '___sec19'), + ('The moons example', 2, None, '___sec20'), ('Mathematical optimization of convex functions', 2, None, - '___sec20'), - ('How do we solve these problems?', 2, None, '___sec21'), - ('A simplex example', 2, None, '___sec22'), - ('Back to the more realistic cases', 2, None, '___sec23'), + '___sec21'), + ('How do we solve these problems?', 2, None, '___sec22'), + ('A simplex example', 2, None, '___sec23'), + ('Back to the more realistic cases', 2, None, '___sec24'), ('Multiclass problems and regression with SVMs', 2, None, - '___sec24')]} + '___sec25')]} end of tocinfo --> @@ -131,11 +132,12 @@ MathJax.Hub.Config({
  • The equations
  • The problem to solve
  • Different kernels and Mercer's theorem
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simplex example
    • -
  • Back to the more realistic cases
    • -
  • Multiclass problems and regression with SVMs
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simplex example
    • +
  • Back to the more realistic cases
    • +
  • Multiclass problems and regression with SVMs
    • @@ -213,6 +215,7 @@ the trouble of performing the transformation
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    • diff --git a/doc/pub/svm/html/._svm-bs019.html b/doc/pub/svm/html/._svm-bs019.html index 2817079ed..5158782c4 100644 --- a/doc/pub/svm/html/._svm-bs019.html +++ b/doc/pub/svm/html/._svm-bs019.html @@ -63,17 +63,18 @@ Automatically generated HTML file from DocOnce source ('The equations', 2, None, '___sec17'), ('The problem to solve', 2, None, '___sec18'), ("Different kernels and Mercer's theorem", 2, None, '___sec19'), + ('The moons example', 2, None, '___sec20'), ('Mathematical optimization of convex functions', 2, None, - '___sec20'), - ('How do we solve these problems?', 2, None, '___sec21'), - ('A simplex example', 2, None, '___sec22'), - ('Back to the more realistic cases', 2, None, '___sec23'), + '___sec21'), + ('How do we solve these problems?', 2, None, '___sec22'), + ('A simplex example', 2, None, '___sec23'), + ('Back to the more realistic cases', 2, None, '___sec24'), ('Multiclass problems and regression with SVMs', 2, None, - '___sec24')]} + '___sec25')]} end of tocinfo --> @@ -131,11 +132,12 @@ MathJax.Hub.Config({
  • The equations
  • The problem to solve
  • Different kernels and Mercer's theorem
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simplex example
    • -
  • Back to the more realistic cases
    • -
  • Multiclass problems and regression with SVMs
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simplex example
    • +
  • Back to the more realistic cases
    • +
  • Multiclass problems and regression with SVMs
    • @@ -206,6 +208,7 @@ Given a kernel \( K \) and the targets \( y_i \) this matrix is easy to set up.
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    • diff --git a/doc/pub/svm/html/._svm-bs020.html b/doc/pub/svm/html/._svm-bs020.html index 711036856..8f90bec88 100644 --- a/doc/pub/svm/html/._svm-bs020.html +++ b/doc/pub/svm/html/._svm-bs020.html @@ -63,17 +63,18 @@ Automatically generated HTML file from DocOnce source ('The equations', 2, None, '___sec17'), ('The problem to solve', 2, None, '___sec18'), ("Different kernels and Mercer's theorem", 2, None, '___sec19'), + ('The moons example', 2, None, '___sec20'), ('Mathematical optimization of convex functions', 2, None, - '___sec20'), - ('How do we solve these problems?', 2, None, '___sec21'), - ('A simplex example', 2, None, '___sec22'), - ('Back to the more realistic cases', 2, None, '___sec23'), + '___sec21'), + ('How do we solve these problems?', 2, None, '___sec22'), + ('A simplex example', 2, None, '___sec23'), + ('Back to the more realistic cases', 2, None, '___sec24'), ('Multiclass problems and regression with SVMs', 2, None, - '___sec24')]} + '___sec25')]} end of tocinfo --> @@ -131,11 +132,12 @@ MathJax.Hub.Config({
  • The equations
  • The problem to solve
  • Different kernels and Mercer's theorem
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simplex example
    • -
  • Back to the more realistic cases
    • -
  • Multiclass problems and regression with SVMs
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simplex example
    • +
  • Back to the more realistic cases
    • +
  • Multiclass problems and regression with SVMs
    • @@ -200,6 +202,7 @@ well in practice.
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  • »
    • diff --git a/doc/pub/svm/html/._svm-bs021.html b/doc/pub/svm/html/._svm-bs021.html index 6bb3920cd..981db5f7d 100644 --- a/doc/pub/svm/html/._svm-bs021.html +++ b/doc/pub/svm/html/._svm-bs021.html @@ -63,17 +63,18 @@ Automatically generated HTML file from DocOnce source ('The equations', 2, None, '___sec17'), ('The problem to solve', 2, None, '___sec18'), ("Different kernels and Mercer's theorem", 2, None, '___sec19'), + ('The moons example', 2, None, '___sec20'), ('Mathematical optimization of convex functions', 2, None, - '___sec20'), - ('How do we solve these problems?', 2, None, '___sec21'), - ('A simplex example', 2, None, '___sec22'), - ('Back to the more realistic cases', 2, None, '___sec23'), + '___sec21'), + ('How do we solve these problems?', 2, None, '___sec22'), + ('A simplex example', 2, None, '___sec23'), + ('Back to the more realistic cases', 2, None, '___sec24'), ('Multiclass problems and regression with SVMs', 2, None, - '___sec24')]} + '___sec25')]} end of tocinfo --> @@ -131,11 +132,12 @@ MathJax.Hub.Config({
  • The equations
  • The problem to solve
  • Different kernels and Mercer's theorem
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simplex example
    • -
  • Back to the more realistic cases
    • -
  • Multiclass problems and regression with SVMs
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simplex example
    • +
  • Back to the more realistic cases
    • +
  • Multiclass problems and regression with SVMs
    • @@ -151,28 +153,199 @@ MathJax.Hub.Config({ -

    Mathematical optimization of convex functions

    - +

    The moons example

    -A mathematical (quadratic) optimization problem, or just optimization problem, has the form -$$ -\begin{align*} - &\mathrm{min}_{\lambda}\hspace{0.2cm} \frac{1}{2}\boldsymbol{\lambda}^T\boldsymbol{P}\boldsymbol{\lambda}+\boldsymbol{q}^T\boldsymbol{\lambda},\\ \nonumber - &\mathrm{subject\hspace{0.1cm}to} \hspace{0.2cm} \boldsymbol{G}\boldsymbol{\lambda} \preceq \boldsymbol{h} \wedge \boldsymbol{A}\boldsymbol{\lambda}=f. -\end{align*} -$$ -subject to some constraints for say a selected set \( i=1,2,\dots, n \). -In our case we are optimizing with respect to the Lagrangian multipliers \( \lambda_i \), and the -vector \( \boldsymbol{\lambda}=[\lambda_1, \lambda_2,\dots, \lambda_n] \) is the optimization variable we are dealing with. + +

    from __future__ import division, print_function, unicode_literals
 
-
    
-In our case we are particularly interested in a class of optimization problems called convex optmization problems. 
-In our discussion on gradient descent methods we discussed at length the definition of a convex function.
+import numpy as np
+np.random.seed(42)
 
-
    
-Convex optimization problems play a central role in applied mathematics and we recommend strongly Boyd and Vandenberghe's text on the topics.
+import matplotlib
+import matplotlib.pyplot as plt
+plt.rcParams['axes.labelsize'] = 14
+plt.rcParams['xtick.labelsize'] = 12
+plt.rcParams['ytick.labelsize'] = 12
 
+
+from sklearn.svm import SVC
+from sklearn import datasets
+
+
+
+from sklearn.pipeline import Pipeline
+from sklearn.preprocessing import StandardScaler
+from sklearn.svm import LinearSVC
+
+
+from sklearn.datasets import make_moons
+X, y = make_moons(n_samples=100, noise=0.15, random_state=42)
+
+def plot_dataset(X, y, axes):
+    plt.plot(X[:, 0][y==0], X[:, 1][y==0], "bs")
+    plt.plot(X[:, 0][y==1], X[:, 1][y==1], "g^")
+    plt.axis(axes)
+    plt.grid(True, which='both')
+    plt.xlabel(r"$x_1$", fontsize=20)
+    plt.ylabel(r"$x_2$", fontsize=20, rotation=0)
+
+plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
+plt.show()
+
+from sklearn.datasets import make_moons
+from sklearn.pipeline import Pipeline
+from sklearn.preprocessing import PolynomialFeatures
+
+polynomial_svm_clf = Pipeline([
+        ("poly_features", PolynomialFeatures(degree=3)),
+        ("scaler", StandardScaler()),
+        ("svm_clf", LinearSVC(C=10, loss="hinge", random_state=42))
+    ])
+
+polynomial_svm_clf.fit(X, y)
+
+def plot_predictions(clf, axes):
+    x0s = np.linspace(axes[0], axes[1], 100)
+    x1s = np.linspace(axes[2], axes[3], 100)
+    x0, x1 = np.meshgrid(x0s, x1s)
+    X = np.c_[x0.ravel(), x1.ravel()]
+    y_pred = clf.predict(X).reshape(x0.shape)
+    y_decision = clf.decision_function(X).reshape(x0.shape)
+    plt.contourf(x0, x1, y_pred, cmap=plt.cm.brg, alpha=0.2)
+    plt.contourf(x0, x1, y_decision, cmap=plt.cm.brg, alpha=0.1)
+
+plot_predictions(polynomial_svm_clf, [-1.5, 2.5, -1, 1.5])
+plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
+
+plt.show()
+
+
+from sklearn.svm import SVC
+
+poly_kernel_svm_clf = Pipeline([
+        ("scaler", StandardScaler()),
+        ("svm_clf", SVC(kernel="poly", degree=3, coef0=1, C=5))
+    ])
+poly_kernel_svm_clf.fit(X, y)
+
+poly100_kernel_svm_clf = Pipeline([
+        ("scaler", StandardScaler()),
+        ("svm_clf", SVC(kernel="poly", degree=10, coef0=100, C=5))
+    ])
+poly100_kernel_svm_clf.fit(X, y)
+
+plt.figure(figsize=(11, 4))
+
+plt.subplot(121)
+plot_predictions(poly_kernel_svm_clf, [-1.5, 2.5, -1, 1.5])
+plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
+plt.title(r"$d=3, r=1, C=5$", fontsize=18)
+
+plt.subplot(122)
+plot_predictions(poly100_kernel_svm_clf, [-1.5, 2.5, -1, 1.5])
+plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
+plt.title(r"$d=10, r=100, C=5$", fontsize=18)
+
+plt.show()
+
+def gaussian_rbf(x, landmark, gamma):
+    return np.exp(-gamma * np.linalg.norm(x - landmark, axis=1)**2)
+
+gamma = 0.3
+
+x1s = np.linspace(-4.5, 4.5, 200).reshape(-1, 1)
+x2s = gaussian_rbf(x1s, -2, gamma)
+x3s = gaussian_rbf(x1s, 1, gamma)
+
+XK = np.c_[gaussian_rbf(X1D, -2, gamma), gaussian_rbf(X1D, 1, gamma)]
+yk = np.array([0, 0, 1, 1, 1, 1, 1, 0, 0])
+
+plt.figure(figsize=(11, 4))
+
+plt.subplot(121)
+plt.grid(True, which='both')
+plt.axhline(y=0, color='k')
+plt.scatter(x=[-2, 1], y=[0, 0], s=150, alpha=0.5, c="red")
+plt.plot(X1D[:, 0][yk==0], np.zeros(4), "bs")
+plt.plot(X1D[:, 0][yk==1], np.zeros(5), "g^")
+plt.plot(x1s, x2s, "g--")
+plt.plot(x1s, x3s, "b:")
+plt.gca().get_yaxis().set_ticks([0, 0.25, 0.5, 0.75, 1])
+plt.xlabel(r"$x_1$", fontsize=20)
+plt.ylabel(r"Similarity", fontsize=14)
+plt.annotate(r'$\mathbf{x}$',
+             xy=(X1D[3, 0], 0),
+             xytext=(-0.5, 0.20),
+             ha="center",
+             arrowprops=dict(facecolor='black', shrink=0.1),
+             fontsize=18,
+            )
+plt.text(-2, 0.9, "$x_2$", ha="center", fontsize=20)
+plt.text(1, 0.9, "$x_3$", ha="center", fontsize=20)
+plt.axis([-4.5, 4.5, -0.1, 1.1])
+
+plt.subplot(122)
+plt.grid(True, which='both')
+plt.axhline(y=0, color='k')
+plt.axvline(x=0, color='k')
+plt.plot(XK[:, 0][yk==0], XK[:, 1][yk==0], "bs")
+plt.plot(XK[:, 0][yk==1], XK[:, 1][yk==1], "g^")
+plt.xlabel(r"$x_2$", fontsize=20)
+plt.ylabel(r"$x_3$  ", fontsize=20, rotation=0)
+plt.annotate(r'$\phi\left(\mathbf{x}\right)$',
+             xy=(XK[3, 0], XK[3, 1]),
+             xytext=(0.65, 0.50),
+             ha="center",
+             arrowprops=dict(facecolor='black', shrink=0.1),
+             fontsize=18,
+            )
+plt.plot([-0.1, 1.1], [0.57, -0.1], "r--", linewidth=3)
+plt.axis([-0.1, 1.1, -0.1, 1.1])
+    
+plt.subplots_adjust(right=1)
+
+plt.show()
+
+
+x1_example = X1D[3, 0]
+for landmark in (-2, 1):
+    k = gaussian_rbf(np.array([[x1_example]]), np.array([[landmark]]), gamma)
+    print("Phi({}, {}) = {}".format(x1_example, landmark, k))
+
+rbf_kernel_svm_clf = Pipeline([
+        ("scaler", StandardScaler()),
+        ("svm_clf", SVC(kernel="rbf", gamma=5, C=0.001))
+    ])
+rbf_kernel_svm_clf.fit(X, y)
+
+
+from sklearn.svm import SVC
+
+gamma1, gamma2 = 0.1, 5
+C1, C2 = 0.001, 1000
+hyperparams = (gamma1, C1), (gamma1, C2), (gamma2, C1), (gamma2, C2)
+
+svm_clfs = []
+for gamma, C in hyperparams:
+    rbf_kernel_svm_clf = Pipeline([
+            ("scaler", StandardScaler()),
+            ("svm_clf", SVC(kernel="rbf", gamma=gamma, C=C))
+        ])
+    rbf_kernel_svm_clf.fit(X, y)
+    svm_clfs.append(rbf_kernel_svm_clf)
+
+plt.figure(figsize=(11, 7))
+
+for i, svm_clf in enumerate(svm_clfs):
+    plt.subplot(221 + i)
+    plot_predictions(svm_clf, [-1.5, 2.5, -1, 1.5])
+    plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
+    gamma, C = hyperparams[i]
+    plt.title(r"$\gamma = {}, C = {}$".format(gamma, C), fontsize=16)
+
+plt.show()
+

    @@ -193,6 +366,7 @@ Convex optimization problems play a central role in applied mathematics and we r

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  • »
    • diff --git a/doc/pub/svm/html/._svm-bs022.html b/doc/pub/svm/html/._svm-bs022.html index 4244211c1..0ae797109 100644 --- a/doc/pub/svm/html/._svm-bs022.html +++ b/doc/pub/svm/html/._svm-bs022.html @@ -63,17 +63,18 @@ Automatically generated HTML file from DocOnce source ('The equations', 2, None, '___sec17'), ('The problem to solve', 2, None, '___sec18'), ("Different kernels and Mercer's theorem", 2, None, '___sec19'), + ('The moons example', 2, None, '___sec20'), ('Mathematical optimization of convex functions', 2, None, - '___sec20'), - ('How do we solve these problems?', 2, None, '___sec21'), - ('A simplex example', 2, None, '___sec22'), - ('Back to the more realistic cases', 2, None, '___sec23'), + '___sec21'), + ('How do we solve these problems?', 2, None, '___sec22'), + ('A simplex example', 2, None, '___sec23'), + ('Back to the more realistic cases', 2, None, '___sec24'), ('Multiclass problems and regression with SVMs', 2, None, - '___sec24')]} + '___sec25')]} end of tocinfo --> @@ -131,11 +132,12 @@ MathJax.Hub.Config({
  • The equations
  • The problem to solve
  • Different kernels and Mercer's theorem
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simplex example
    • -
  • Back to the more realistic cases
    • -
  • Multiclass problems and regression with SVMs
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simplex example
    • +
  • Back to the more realistic cases
    • +
  • Multiclass problems and regression with SVMs
    • @@ -151,28 +153,27 @@ MathJax.Hub.Config({ -

    How do we solve these problems?

    +

    Mathematical optimization of convex functions

    -If we use Python as programming language and wish to venture beyond -scikit-learn, tensorflow and similar software which makes our -lives so much easier, we need to dive into the wonderful world of -quadratic programming. We can, if we wish, solve the minimization -problem using say standard gradient methods or conjugate gradient -methods. However, these methods tend to exhibit a rather slow -converge. So, welcome to the promised land of quadratic programming. +A mathematical (quadratic) optimization problem, or just optimization problem, has the form +$$ +\begin{align*} + &\mathrm{min}_{\lambda}\hspace{0.2cm} \frac{1}{2}\boldsymbol{\lambda}^T\boldsymbol{P}\boldsymbol{\lambda}+\boldsymbol{q}^T\boldsymbol{\lambda},\\ \nonumber + &\mathrm{subject\hspace{0.1cm}to} \hspace{0.2cm} \boldsymbol{G}\boldsymbol{\lambda} \preceq \boldsymbol{h} \wedge \boldsymbol{A}\boldsymbol{\lambda}=f. +\end{align*} +$$ + +subject to some constraints for say a selected set \( i=1,2,\dots, n \). +In our case we are optimizing with respect to the Lagrangian multipliers \( \lambda_i \), and the +vector \( \boldsymbol{\lambda}=[\lambda_1, \lambda_2,\dots, \lambda_n] \) is the optimization variable we are dealing with.

    -The functions we need are contained in the quadratic programming package CVXOPT and we need to import it together with numpy as +In our case we are particularly interested in a class of optimization problems called convex optmization problems. +In our discussion on gradient descent methods we discussed at length the definition of a convex function.

    - - -

    import numpy
-import cvxopt
-
    -

    -This will make our life much easier. You don't need t write your own optimizer. +Convex optimization problems play a central role in applied mathematics and we recommend strongly Boyd and Vandenberghe's text on the topics.

    @@ -193,6 +194,7 @@ This will make our life much easier. You don't need t write your own optimizer.

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  • 26
    • +
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  • »
    • diff --git a/doc/pub/svm/html/._svm-bs023.html b/doc/pub/svm/html/._svm-bs023.html index a8134b546..f2c0886d5 100644 --- a/doc/pub/svm/html/._svm-bs023.html +++ b/doc/pub/svm/html/._svm-bs023.html @@ -63,17 +63,18 @@ Automatically generated HTML file from DocOnce source ('The equations', 2, None, '___sec17'), ('The problem to solve', 2, None, '___sec18'), ("Different kernels and Mercer's theorem", 2, None, '___sec19'), + ('The moons example', 2, None, '___sec20'), ('Mathematical optimization of convex functions', 2, None, - '___sec20'), - ('How do we solve these problems?', 2, None, '___sec21'), - ('A simplex example', 2, None, '___sec22'), - ('Back to the more realistic cases', 2, None, '___sec23'), + '___sec21'), + ('How do we solve these problems?', 2, None, '___sec22'), + ('A simplex example', 2, None, '___sec23'), + ('Back to the more realistic cases', 2, None, '___sec24'), ('Multiclass problems and regression with SVMs', 2, None, - '___sec24')]} + '___sec25')]} end of tocinfo --> @@ -131,11 +132,12 @@ MathJax.Hub.Config({
  • The equations
  • The problem to solve
  • Different kernels and Mercer's theorem
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simplex example
    • -
  • Back to the more realistic cases
    • -
  • Multiclass problems and regression with SVMs
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simplex example
    • +
  • Back to the more realistic cases
    • +
  • Multiclass problems and regression with SVMs
    • @@ -151,71 +153,29 @@ MathJax.Hub.Config({ -

    A simplex example

    +

    How do we solve these problems?

    -We remind ourselves about the general problem we want to solve -$$ -\begin{align*} - &\mathrm{min}_{x}\hspace{0.2cm} \frac{1}{2}\boldsymbol{x}^T\boldsymbol{P}\boldsymbol{x}+\boldsymbol{q}^T\boldsymbol{x},\\ \nonumber - &\mathrm{subject\hspace{0.1cm} to} \hspace{0.2cm} \boldsymbol{G}\boldsymbol{x} \preceq \boldsymbol{h} \wedge \boldsymbol{A}\boldsymbol{x}=f. -\end{align*} -$$ +If we use Python as programming language and wish to venture beyond +scikit-learn, tensorflow and similar software which makes our +lives so much easier, we need to dive into the wonderful world of +quadratic programming. We can, if we wish, solve the minimization +problem using say standard gradient methods or conjugate gradient +methods. However, these methods tend to exhibit a rather slow +converge. So, welcome to the promised land of quadratic programming.

    -Let us show how to perform the optmization using a simple case. Assume we want to optimize the following problem -$$ -\begin{align*} - &\mathrm{min}_{x}\hspace{0.2cm} \frac{1}{2}x^2+5x+3y \\ \nonumber - &\mathrm{subject to} \\ \nonumber - &x, y \geq 0 \\ \nonumber - &x+3y \geq 15 \\ \nonumber - &2x+5y \leq 100 \\ \nonumber - &3x+4y \leq 80. \\ \nonumber -\end{align*} -$$ +The functions we need are contained in the quadratic programming package CVXOPT and we need to import it together with numpy as -The minimization problem can be rewritten in terms of vectors and matrices as (with \( x \) and \( y \) being the unknowns) -$$ -\frac{1}{2}\begin{bmatrix} x\\ y \end{bmatrix}^T \begin{bmatrix} 1 & 0\\ 0 & 0 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix}3\\ 4 \end{bmatrix}^T \begin{bmatrix}x \\ y \end{bmatrix}. -$$ - -Similarly, we can now set up the inequalities (we need to change \( \geq \) to \( \leq \) by multiplying with \( -1 \) on bot sides) as the following matrix-vector equation -$$ -\begin{bmatrix} -1 & 0 \\ 0 & -1 \\ -1 & -3 \\ 2 & 5 \\ 3 & 4\end{bmatrix}\begin{bmatrix} x \\ y\end{bmatrix} \preceq \begin{bmatrix}0 \\ 0\\ -15 \\ 100 \\ 80\end{bmatrix}. -$$ - -We have collapsed all the inequalities into a single matrix \( \boldsymbol{G} \). We see also that our matrix -$$ -\boldsymbol{P} =\begin{bmatrix} 1 & 0\\ 0 & 0 \end{bmatrix} -$$ - -is clearly positive semi-definite (all eigenvalues larger or equal zero). -Finally, the vector \( \boldsymbol{h} \) is defined as -$$ -\boldsymbol{h} = \begin{bmatrix}0 \\ 0\\ -15 \\ 100 \\ 80\end{bmatrix}. -$$ - -

    -Since we don't have any equalities the matrix \( \boldsymbol{A} \) is set to zero -The following code solves the equations for us

    -

    # Import the necessary packages
-import numpy
-from cvxopt import matrix
-from cvxopt import solvers
-P = matrix(numpy.diag([1,0]), tc=’d’)
-q = matrix(numpy.array([3,4]), tc=’d’)
-G = matrix(numpy.array([[-1,0],[0,-1],[-1,-3],[2,5],[3,4]]), tc=’d’)
-h = matrix(numpy.array([0,0,-15,100,80]), tc=’d’)
-# Construct the QP, invoke solver
-sol = solvers.qp(P,q,G,h)
-# Extract optimal value and solution
-sol[’x’] 
-sol[’primal objective’]
+
    import numpy
+import cvxopt
 
    
+
    
+This will make our life much easier. You don't need t write your own optimizer.
+
 
    
 
    
 
@@ -234,6 +194,7 @@ sol[’primal objective’]
   
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+  
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    • 
 
 
diff --git a/doc/pub/svm/html/._svm-bs024.html b/doc/pub/svm/html/._svm-bs024.html
index aa7fb5b20..87521650f 100644
--- a/doc/pub/svm/html/._svm-bs024.html
+++ b/doc/pub/svm/html/._svm-bs024.html
@@ -63,17 +63,18 @@ Automatically generated HTML file from DocOnce source
               ('The equations', 2, None, '___sec17'),
               ('The problem to solve', 2, None, '___sec18'),
               ("Different kernels and Mercer's theorem", 2, None, '___sec19'),
+              ('The moons example', 2, None, '___sec20'),
               ('Mathematical optimization of convex functions',
                2,
                None,
-               '___sec20'),
-              ('How do we solve these problems?', 2, None, '___sec21'),
-              ('A simplex example', 2, None, '___sec22'),
-              ('Back to the more realistic cases', 2, None, '___sec23'),
+               '___sec21'),
+              ('How do we solve these problems?', 2, None, '___sec22'),
+              ('A simplex example', 2, None, '___sec23'),
+              ('Back to the more realistic cases', 2, None, '___sec24'),
               ('Multiclass problems and regression with SVMs',
                2,
                None,
-               '___sec24')]}
+               '___sec25')]}
 end of tocinfo -->
 
 
@@ -131,11 +132,12 @@ MathJax.Hub.Config({
       
  • The equations
    • 
       
  • The problem to solve
    • 
       
  • Different kernels and Mercer's theorem
    • 
-      
  • Mathematical optimization of convex functions
    • 
-      
  • How do we solve these problems?
    • 
-      
  • A simplex example
    • 
-      
  • Back to the more realistic cases
    • 
-      
  • Multiclass problems and regression with SVMs
    • 
+      
  • The moons example
    • 
+      
  • Mathematical optimization of convex functions
    • 
+      
  • How do we solve these problems?
    • 
+      
  • A simplex example
    • 
+      
  • Back to the more realistic cases
    • 
+      
  • Multiclass problems and regression with SVMs
    • 
 
         
       
@@ -151,26 +153,71 @@ MathJax.Hub.Config({
 
 
 
-
    Back to the more realistic cases 
    
+
    A simplex example 
    
 
 
    
-We are now ready to return to our setup of the optmization problem for a more realistic case. Introducint the slack parameter \( C \) we have
+We remind ourselves about the general problem we want to solve
 $$
-\frac{1}{2} \boldsymbol{\lambda}^T\begin{bmatrix} y_1y_1K(\boldsymbol{x}_1,\boldsymbol{x}_1) & y_1y_2K(\boldsymbol{x}_1,\boldsymbol{x}_2) & \dots & \dots & y_1y_nK(\boldsymbol{x}_1,\boldsymbol{x}_n) \\
-y_2y_1K(\boldsymbol{x}_2,\boldsymbol{x}_1) & y_2y_2K(\boldsymbol{x}_2,\boldsymbol{x}_2) & \dots & \dots & y_1y_nK(\boldsymbol{x}_2,\boldsymbol{x}_n) \\
-\dots & \dots & \dots & \dots & \dots \\
-\dots & \dots & \dots & \dots & \dots \\
-y_ny_1K(\boldsymbol{x}_n,\boldsymbol{x}_1) & y_ny_2K(\boldsymbol{x}_n\boldsymbol{x}_2) & \dots & \dots & y_ny_nK(\boldsymbol{x}_n,\boldsymbol{x}_n) \\
-\end{bmatrix}\boldsymbol{\lambda}-\mathbb{I}\boldsymbol{\lambda}, 
+\begin{align*}
+    &\mathrm{min}_{x}\hspace{0.2cm} \frac{1}{2}\boldsymbol{x}^T\boldsymbol{P}\boldsymbol{x}+\boldsymbol{q}^T\boldsymbol{x},\\ \nonumber
+    &\mathrm{subject\hspace{0.1cm} to} \hspace{0.2cm} \boldsymbol{G}\boldsymbol{x} \preceq \boldsymbol{h} \wedge  \boldsymbol{A}\boldsymbol{x}=f.
+\end{align*}
 $$
 
-subject to \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \). Here we defined the vectors \( \boldsymbol{\lambda} =[\lambda_1,\lambda_2,\dots,\lambda_n] \) and 
-\( \boldsymbol{y}=[y_1,y_2,\dots,y_n] \). 
-With  the slack constants this leads to the additional constraint \( 0\leq \lambda_i \leq C \).
-
 
    
-code will be added
+Let us show how to perform the optmization using a simple case. Assume we want to optimize the following problem
+$$
+\begin{align*}
+    &\mathrm{min}_{x}\hspace{0.2cm} \frac{1}{2}x^2+5x+3y \\ \nonumber
+    &\mathrm{subject to} \\ \nonumber
+    &x, y \geq 0 \\ \nonumber
+    &x+3y  \geq 15 \\ \nonumber
+    &2x+5y  \leq  100 \\ \nonumber
+    &3x+4y  \leq  80.  \\ \nonumber
+\end{align*}
+$$
 
+The minimization problem can be rewritten in terms of vectors and matrices as (with \( x \) and \( y \) being the unknowns)
+$$
+\frac{1}{2}\begin{bmatrix} x\\ y \end{bmatrix}^T   \begin{bmatrix} 1 & 0\\ 0 & 0  \end{bmatrix}  \begin{bmatrix} x \\ y \end{bmatrix}  + \begin{bmatrix}3\\ 4  \end{bmatrix}^T \begin{bmatrix}x \\ y  \end{bmatrix}.  
+$$
+
+Similarly, we can now set up the inequalities (we need to change \( \geq \) to \( \leq \) by multiplying with \( -1 \) on bot sides) as the following matrix-vector equation
+$$
+\begin{bmatrix} -1 & 0 \\ 0 & -1 \\ -1 & -3 \\ 2 & 5 \\ 3 & 4\end{bmatrix}\begin{bmatrix} x \\ y\end{bmatrix} \preceq \begin{bmatrix}0 \\ 0\\ -15 \\ 100 \\ 80\end{bmatrix}.
+$$
+
+We have collapsed all the inequalities into a single matrix \( \boldsymbol{G} \). We see also that our matrix 
+$$
+\boldsymbol{P} =\begin{bmatrix} 1 & 0\\ 0 & 0  \end{bmatrix}
+$$
+
+is clearly positive semi-definite (all eigenvalues larger or equal zero). 
+Finally, the vector \( \boldsymbol{h} \) is defined as 
+$$
+\boldsymbol{h} = \begin{bmatrix}0 \\ 0\\ -15 \\ 100 \\ 80\end{bmatrix}.
+$$
+
+
    
+Since we don't have any equalities the matrix \( \boldsymbol{A} \) is set to zero
+The following code solves the equations for us
+
    
+
+
+
    # Import the necessary packages
+import numpy
+from cvxopt import matrix
+from cvxopt import solvers
+P = matrix(numpy.diag([1,0]), tc=’d’)
+q = matrix(numpy.array([3,4]), tc=’d’)
+G = matrix(numpy.array([[-1,0],[0,-1],[-1,-3],[2,5],[3,4]]), tc=’d’)
+h = matrix(numpy.array([0,0,-15,100,80]), tc=’d’)
+# Construct the QP, invoke solver
+sol = solvers.qp(P,q,G,h)
+# Extract optimal value and solution
+sol[’x’] 
+sol[’primal objective’]
+
    
 
    
 
    
 
@@ -188,6 +235,7 @@ With  the slack constants this leads to the additional constraint \( 0\leq \lamb
   
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    • 
   
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+  
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    • 
 
 
diff --git a/doc/pub/svm/html/._svm-bs025.html b/doc/pub/svm/html/._svm-bs025.html
index 6a1bdc7ab..265f3e28d 100644
--- a/doc/pub/svm/html/._svm-bs025.html
+++ b/doc/pub/svm/html/._svm-bs025.html
@@ -63,17 +63,18 @@ Automatically generated HTML file from DocOnce source
               ('The equations', 2, None, '___sec17'),
               ('The problem to solve', 2, None, '___sec18'),
               ("Different kernels and Mercer's theorem", 2, None, '___sec19'),
+              ('The moons example', 2, None, '___sec20'),
               ('Mathematical optimization of convex functions',
                2,
                None,
-               '___sec20'),
-              ('How do we solve these problems?', 2, None, '___sec21'),
-              ('A simplex example', 2, None, '___sec22'),
-              ('Back to the more realistic cases', 2, None, '___sec23'),
+               '___sec21'),
+              ('How do we solve these problems?', 2, None, '___sec22'),
+              ('A simplex example', 2, None, '___sec23'),
+              ('Back to the more realistic cases', 2, None, '___sec24'),
               ('Multiclass problems and regression with SVMs',
                2,
                None,
-               '___sec24')]}
+               '___sec25')]}
 end of tocinfo -->
 
 
@@ -131,11 +132,12 @@ MathJax.Hub.Config({
       
  • The equations
    • 
       
  • The problem to solve
    • 
       
  • Different kernels and Mercer's theorem
    • 
-      
  • Mathematical optimization of convex functions
    • 
-      
  • How do we solve these problems?
    • 
-      
  • A simplex example
    • 
-      
  • Back to the more realistic cases
    • 
-      
  • Multiclass problems and regression with SVMs
    • 
+      
  • The moons example
    • 
+      
  • Mathematical optimization of convex functions
    • 
+      
  • How do we solve these problems?
    • 
+      
  • A simplex example
    • 
+      
  • Back to the more realistic cases
    • 
+      
  • Multiclass problems and regression with SVMs
    • 
 
         
       
@@ -151,9 +153,27 @@ MathJax.Hub.Config({
 
 
 
-
    Multiclass problems and regression with SVMs 
    
-This material will be added later.
+
    Back to the more realistic cases 
    
 
+
    
+We are now ready to return to our setup of the optmization problem for a more realistic case. Introducint the slack parameter \( C \) we have
+$$
+\frac{1}{2} \boldsymbol{\lambda}^T\begin{bmatrix} y_1y_1K(\boldsymbol{x}_1,\boldsymbol{x}_1) & y_1y_2K(\boldsymbol{x}_1,\boldsymbol{x}_2) & \dots & \dots & y_1y_nK(\boldsymbol{x}_1,\boldsymbol{x}_n) \\
+y_2y_1K(\boldsymbol{x}_2,\boldsymbol{x}_1) & y_2y_2K(\boldsymbol{x}_2,\boldsymbol{x}_2) & \dots & \dots & y_1y_nK(\boldsymbol{x}_2,\boldsymbol{x}_n) \\
+\dots & \dots & \dots & \dots & \dots \\
+\dots & \dots & \dots & \dots & \dots \\
+y_ny_1K(\boldsymbol{x}_n,\boldsymbol{x}_1) & y_ny_2K(\boldsymbol{x}_n\boldsymbol{x}_2) & \dots & \dots & y_ny_nK(\boldsymbol{x}_n,\boldsymbol{x}_n) \\
+\end{bmatrix}\boldsymbol{\lambda}-\mathbb{I}\boldsymbol{\lambda}, 
+$$
+
+subject to \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \). Here we defined the vectors \( \boldsymbol{\lambda} =[\lambda_1,\lambda_2,\dots,\lambda_n] \) and 
+\( \boldsymbol{y}=[y_1,y_2,\dots,y_n] \). 
+With  the slack constants this leads to the additional constraint \( 0\leq \lambda_i \leq C \).
+
+
    
+code will be added
+
+
    
 
    
 
 
      
@@ -169,6 +189,8 @@ This material will be added later.
   
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+  
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+  
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      • 
 
    
 
 
diff --git a/doc/pub/svm/html/._svm-bs026.html b/doc/pub/svm/html/._svm-bs026.html
new file mode 100644
index 000000000..256c542f6
--- /dev/null
+++ b/doc/pub/svm/html/._svm-bs026.html
@@ -0,0 +1,197 @@
+
+
+
+
+
+
+
+Data Analysis and Machine Learning: Support Vector Machines
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+    
+
+
    
+  
    
+    
+    Data Analysis and Machine Learning: Support Vector Machines
+  
    
+
+  
    
+    
+  
    
+
    
+
    + +
    + +

     

     

     

    + + + + +

    Multiclass problems and regression with SVMs

    +This material will be added later. + +

    + +

    + + +
    + + + + + + + +
    + +
    + + + + + + diff --git a/doc/pub/svm/html/svm-bs.html b/doc/pub/svm/html/svm-bs.html index d46b684c2..5a2191ad4 100644 --- a/doc/pub/svm/html/svm-bs.html +++ b/doc/pub/svm/html/svm-bs.html @@ -63,17 +63,18 @@ Automatically generated HTML file from DocOnce source ('The equations', 2, None, '___sec17'), ('The problem to solve', 2, None, '___sec18'), ("Different kernels and Mercer's theorem", 2, None, '___sec19'), + ('The moons example', 2, None, '___sec20'), ('Mathematical optimization of convex functions', 2, None, - '___sec20'), - ('How do we solve these problems?', 2, None, '___sec21'), - ('A simplex example', 2, None, '___sec22'), - ('Back to the more realistic cases', 2, None, '___sec23'), + '___sec21'), + ('How do we solve these problems?', 2, None, '___sec22'), + ('A simplex example', 2, None, '___sec23'), + ('Back to the more realistic cases', 2, None, '___sec24'), ('Multiclass problems and regression with SVMs', 2, None, - '___sec24')]} + '___sec25')]} end of tocinfo --> @@ -131,11 +132,12 @@ MathJax.Hub.Config({
  • The equations
  • The problem to solve
  • Different kernels and Mercer's theorem
    • -
  • Mathematical optimization of convex functions
    • -
  • How do we solve these problems?
    • -
  • A simplex example
    • -
  • Back to the more realistic cases
    • -
  • Multiclass problems and regression with SVMs
    • +
  • The moons example
    • +
  • Mathematical optimization of convex functions
    • +
  • How do we solve these problems?
    • +
  • A simplex example
    • +
  • Back to the more realistic cases
    • +
  • Multiclass problems and regression with SVMs
    • @@ -170,7 +172,7 @@ MathJax.Hub.Config({
    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

    -

    Nov 7, 2018

    +

    Nov 8, 2018


    @@ -194,7 +196,7 @@ MathJax.Hub.Config({

  • 9
  • 10
  • ...
    • -
  • 26
    • +
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  • »
    • diff --git a/doc/pub/svm/html/svm-reveal.html b/doc/pub/svm/html/svm-reveal.html index e68b85bc0..26583d00f 100644 --- a/doc/pub/svm/html/svm-reveal.html +++ b/doc/pub/svm/html/svm-reveal.html @@ -148,7 +148,7 @@ MathJax.Hub.Config({
    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

     
    -

    Nov 7, 2018

    +

    Nov 8, 2018


    @@ -200,13 +200,83 @@ The theory behind support vector machines (SVM hereafter) is based on the mathematical description of so-called hyperplanes. Let us start with a two-dimensional case. This will also allow us to introduce our first SVM examples. These will be tailored to the case of two specific -classes, as displayed in the figure here. +classes, as displayed in the figure here based on the usage of the petal data.

    We assume here that our data set can be well separated into two domains, where a straight line does the job in the separating the two -classes. Here the two classes are represented by either crosses or +classes. Here the two classes are represented by either squares or circles. +

    + + +

    from sklearn import datasets
+from sklearn.svm import SVC, LinearSVC
+from sklearn.linear_model import SGDClassifier
+from sklearn.preprocessing import StandardScaler
+import matplotlib
+import matplotlib.pyplot as plt
+plt.rcParams['axes.labelsize'] = 14
+plt.rcParams['xtick.labelsize'] = 12
+plt.rcParams['ytick.labelsize'] = 12
+
+
+iris = datasets.load_iris()
+X = iris["data"][:, (2, 3)]  # petal length, petal width
+y = iris["target"]
+
+setosa_or_versicolor = (y == 0) | (y == 1)
+X = X[setosa_or_versicolor]
+y = y[setosa_or_versicolor]
+
+
+
+C = 5
+alpha = 1 / (C * len(X))
+
+lin_clf = LinearSVC(loss="hinge", C=C, random_state=42)
+svm_clf = SVC(kernel="linear", C=C)
+sgd_clf = SGDClassifier(loss="hinge", learning_rate="constant", eta0=0.001, alpha=alpha,
+                        max_iter=100000, random_state=42)
+
+scaler = StandardScaler()
+X_scaled = scaler.fit_transform(X)
+
+lin_clf.fit(X_scaled, y)
+svm_clf.fit(X_scaled, y)
+sgd_clf.fit(X_scaled, y)
+
+print("LinearSVC:                   ", lin_clf.intercept_, lin_clf.coef_)
+print("SVC:                         ", svm_clf.intercept_, svm_clf.coef_)
+print("SGDClassifier(alpha={:.5f}):".format(sgd_clf.alpha), sgd_clf.intercept_, sgd_clf.coef_)
+
+# Compute the slope and bias of each decision boundary
+w1 = -lin_clf.coef_[0, 0]/lin_clf.coef_[0, 1]
+b1 = -lin_clf.intercept_[0]/lin_clf.coef_[0, 1]
+w2 = -svm_clf.coef_[0, 0]/svm_clf.coef_[0, 1]
+b2 = -svm_clf.intercept_[0]/svm_clf.coef_[0, 1]
+w3 = -sgd_clf.coef_[0, 0]/sgd_clf.coef_[0, 1]
+b3 = -sgd_clf.intercept_[0]/sgd_clf.coef_[0, 1]
+
+# Transform the decision boundary lines back to the original scale
+line1 = scaler.inverse_transform([[-10, -10 * w1 + b1], [10, 10 * w1 + b1]])
+line2 = scaler.inverse_transform([[-10, -10 * w2 + b2], [10, 10 * w2 + b2]])
+line3 = scaler.inverse_transform([[-10, -10 * w3 + b3], [10, 10 * w3 + b3]])
+
+# Plot all three decision boundaries
+plt.figure(figsize=(11, 4))
+plt.plot(line1[:, 0], line1[:, 1], "k:", label="LinearSVC")
+plt.plot(line2[:, 0], line2[:, 1], "b--", linewidth=2, label="SVC")
+plt.plot(line3[:, 0], line3[:, 1], "r-", label="SGDClassifier")
+plt.plot(X[:, 0][y==1], X[:, 1][y==1], "bs") # label="Iris-Versicolor"
+plt.plot(X[:, 0][y==0], X[:, 1][y==0], "yo") # label="Iris-Setosa"
+plt.xlabel("Petal length", fontsize=14)
+plt.ylabel("Petal width", fontsize=14)
+plt.legend(loc="upper center", fontsize=14)
+plt.axis([0, 5.5, 0, 2])
+
+plt.show()
+
    @@ -828,6 +898,57 @@ obtain a separation between the classes which is almost linear.

    The change of basis, from \( x\rightarrow z=\phi(x) \) leads to the same type of equations to be solved, except that we need to introduce for example a polynomial transformation to a two-dimensional training set. + +

    + + +

    import numpy as np
+import os
+
+np.random.seed(42)
+
+# To plot pretty figures
+import matplotlib
+import matplotlib.pyplot as plt
+plt.rcParams['axes.labelsize'] = 14
+plt.rcParams['xtick.labelsize'] = 12
+plt.rcParams['ytick.labelsize'] = 12
+
+
+from sklearn.svm import SVC
+from sklearn import datasets
+
+
+
+X1D = np.linspace(-4, 4, 9).reshape(-1, 1)
+X2D = np.c_[X1D, X1D**2]
+y = np.array([0, 0, 1, 1, 1, 1, 1, 0, 0])
+
+plt.figure(figsize=(11, 4))
+
+plt.subplot(121)
+plt.grid(True, which='both')
+plt.axhline(y=0, color='k')
+plt.plot(X1D[:, 0][y==0], np.zeros(4), "bs")
+plt.plot(X1D[:, 0][y==1], np.zeros(5), "g^")
+plt.gca().get_yaxis().set_ticks([])
+plt.xlabel(r"$x_1$", fontsize=20)
+plt.axis([-4.5, 4.5, -0.2, 0.2])
+
+plt.subplot(122)
+plt.grid(True, which='both')
+plt.axhline(y=0, color='k')
+plt.axvline(x=0, color='k')
+plt.plot(X2D[:, 0][y==0], X2D[:, 1][y==0], "bs")
+plt.plot(X2D[:, 0][y==1], X2D[:, 1][y==1], "g^")
+plt.xlabel(r"$x_1$", fontsize=20)
+plt.ylabel(r"$x_2$", fontsize=20, rotation=0)
+plt.gca().get_yaxis().set_ticks([0, 4, 8, 12, 16])
+plt.plot([-4.5, 4.5], [6.5, 6.5], "r--", linewidth=3)
+plt.axis([-4.5, 4.5, -1, 17])
+plt.subplots_adjust(right=1)
+plt.show()
+
    @@ -960,7 +1081,204 @@ well in practice.
    -

    Mathematical optimization of convex functions

    +

    The moons example

    +

    + + +

    from __future__ import division, print_function, unicode_literals
+
+import numpy as np
+np.random.seed(42)
+
+import matplotlib
+import matplotlib.pyplot as plt
+plt.rcParams['axes.labelsize'] = 14
+plt.rcParams['xtick.labelsize'] = 12
+plt.rcParams['ytick.labelsize'] = 12
+
+
+from sklearn.svm import SVC
+from sklearn import datasets
+
+
+
+from sklearn.pipeline import Pipeline
+from sklearn.preprocessing import StandardScaler
+from sklearn.svm import LinearSVC
+
+
+from sklearn.datasets import make_moons
+X, y = make_moons(n_samples=100, noise=0.15, random_state=42)
+
+def plot_dataset(X, y, axes):
+    plt.plot(X[:, 0][y==0], X[:, 1][y==0], "bs")
+    plt.plot(X[:, 0][y==1], X[:, 1][y==1], "g^")
+    plt.axis(axes)
+    plt.grid(True, which='both')
+    plt.xlabel(r"$x_1$", fontsize=20)
+    plt.ylabel(r"$x_2$", fontsize=20, rotation=0)
+
+plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
+plt.show()
+
+from sklearn.datasets import make_moons
+from sklearn.pipeline import Pipeline
+from sklearn.preprocessing import PolynomialFeatures
+
+polynomial_svm_clf = Pipeline([
+        ("poly_features", PolynomialFeatures(degree=3)),
+        ("scaler", StandardScaler()),
+        ("svm_clf", LinearSVC(C=10, loss="hinge", random_state=42))
+    ])
+
+polynomial_svm_clf.fit(X, y)
+
+def plot_predictions(clf, axes):
+    x0s = np.linspace(axes[0], axes[1], 100)
+    x1s = np.linspace(axes[2], axes[3], 100)
+    x0, x1 = np.meshgrid(x0s, x1s)
+    X = np.c_[x0.ravel(), x1.ravel()]
+    y_pred = clf.predict(X).reshape(x0.shape)
+    y_decision = clf.decision_function(X).reshape(x0.shape)
+    plt.contourf(x0, x1, y_pred, cmap=plt.cm.brg, alpha=0.2)
+    plt.contourf(x0, x1, y_decision, cmap=plt.cm.brg, alpha=0.1)
+
+plot_predictions(polynomial_svm_clf, [-1.5, 2.5, -1, 1.5])
+plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
+
+plt.show()
+
+
+from sklearn.svm import SVC
+
+poly_kernel_svm_clf = Pipeline([
+        ("scaler", StandardScaler()),
+        ("svm_clf", SVC(kernel="poly", degree=3, coef0=1, C=5))
+    ])
+poly_kernel_svm_clf.fit(X, y)
+
+poly100_kernel_svm_clf = Pipeline([
+        ("scaler", StandardScaler()),
+        ("svm_clf", SVC(kernel="poly", degree=10, coef0=100, C=5))
+    ])
+poly100_kernel_svm_clf.fit(X, y)
+
+plt.figure(figsize=(11, 4))
+
+plt.subplot(121)
+plot_predictions(poly_kernel_svm_clf, [-1.5, 2.5, -1, 1.5])
+plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
+plt.title(r"$d=3, r=1, C=5$", fontsize=18)
+
+plt.subplot(122)
+plot_predictions(poly100_kernel_svm_clf, [-1.5, 2.5, -1, 1.5])
+plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
+plt.title(r"$d=10, r=100, C=5$", fontsize=18)
+
+plt.show()
+
+def gaussian_rbf(x, landmark, gamma):
+    return np.exp(-gamma * np.linalg.norm(x - landmark, axis=1)**2)
+
+gamma = 0.3
+
+x1s = np.linspace(-4.5, 4.5, 200).reshape(-1, 1)
+x2s = gaussian_rbf(x1s, -2, gamma)
+x3s = gaussian_rbf(x1s, 1, gamma)
+
+XK = np.c_[gaussian_rbf(X1D, -2, gamma), gaussian_rbf(X1D, 1, gamma)]
+yk = np.array([0, 0, 1, 1, 1, 1, 1, 0, 0])
+
+plt.figure(figsize=(11, 4))
+
+plt.subplot(121)
+plt.grid(True, which='both')
+plt.axhline(y=0, color='k')
+plt.scatter(x=[-2, 1], y=[0, 0], s=150, alpha=0.5, c="red")
+plt.plot(X1D[:, 0][yk==0], np.zeros(4), "bs")
+plt.plot(X1D[:, 0][yk==1], np.zeros(5), "g^")
+plt.plot(x1s, x2s, "g--")
+plt.plot(x1s, x3s, "b:")
+plt.gca().get_yaxis().set_ticks([0, 0.25, 0.5, 0.75, 1])
+plt.xlabel(r"$x_1$", fontsize=20)
+plt.ylabel(r"Similarity", fontsize=14)
+plt.annotate(r'$\mathbf{x}$',
+             xy=(X1D[3, 0], 0),
+             xytext=(-0.5, 0.20),
+             ha="center",
+             arrowprops=dict(facecolor='black', shrink=0.1),
+             fontsize=18,
+            )
+plt.text(-2, 0.9, "$x_2$", ha="center", fontsize=20)
+plt.text(1, 0.9, "$x_3$", ha="center", fontsize=20)
+plt.axis([-4.5, 4.5, -0.1, 1.1])
+
+plt.subplot(122)
+plt.grid(True, which='both')
+plt.axhline(y=0, color='k')
+plt.axvline(x=0, color='k')
+plt.plot(XK[:, 0][yk==0], XK[:, 1][yk==0], "bs")
+plt.plot(XK[:, 0][yk==1], XK[:, 1][yk==1], "g^")
+plt.xlabel(r"$x_2$", fontsize=20)
+plt.ylabel(r"$x_3$  ", fontsize=20, rotation=0)
+plt.annotate(r'$\phi\left(\mathbf{x}\right)$',
+             xy=(XK[3, 0], XK[3, 1]),
+             xytext=(0.65, 0.50),
+             ha="center",
+             arrowprops=dict(facecolor='black', shrink=0.1),
+             fontsize=18,
+            )
+plt.plot([-0.1, 1.1], [0.57, -0.1], "r--", linewidth=3)
+plt.axis([-0.1, 1.1, -0.1, 1.1])
+    
+plt.subplots_adjust(right=1)
+
+plt.show()
+
+
+x1_example = X1D[3, 0]
+for landmark in (-2, 1):
+    k = gaussian_rbf(np.array([[x1_example]]), np.array([[landmark]]), gamma)
+    print("Phi({}, {}) = {}".format(x1_example, landmark, k))
+
+rbf_kernel_svm_clf = Pipeline([
+        ("scaler", StandardScaler()),
+        ("svm_clf", SVC(kernel="rbf", gamma=5, C=0.001))
+    ])
+rbf_kernel_svm_clf.fit(X, y)
+
+
+from sklearn.svm import SVC
+
+gamma1, gamma2 = 0.1, 5
+C1, C2 = 0.001, 1000
+hyperparams = (gamma1, C1), (gamma1, C2), (gamma2, C1), (gamma2, C2)
+
+svm_clfs = []
+for gamma, C in hyperparams:
+    rbf_kernel_svm_clf = Pipeline([
+            ("scaler", StandardScaler()),
+            ("svm_clf", SVC(kernel="rbf", gamma=gamma, C=C))
+        ])
+    rbf_kernel_svm_clf.fit(X, y)
+    svm_clfs.append(rbf_kernel_svm_clf)
+
+plt.figure(figsize=(11, 7))
+
+for i, svm_clf in enumerate(svm_clfs):
+    plt.subplot(221 + i)
+    plot_predictions(svm_clf, [-1.5, 2.5, -1, 1.5])
+    plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
+    gamma, C = hyperparams[i]
+    plt.title(r"$\gamma = {}, C = {}$".format(gamma, C), fontsize=16)
+
+plt.show()
+
    +
    + + +
    +

    Mathematical optimization of convex functions

    A mathematical (quadratic) optimization problem, or just optimization problem, has the form @@ -987,7 +1305,7 @@ Convex optimization problems play a central role in applied mathematics and we r

    -

    How do we solve these problems?

    +

    How do we solve these problems?

    If we use Python as programming language and wish to venture beyond @@ -1013,7 +1331,7 @@ This will make our life much easier. You don't need t write your own optimizer.

    -

    A simplex example

    +

    A simplex example

    We remind ourselves about the general problem we want to solve @@ -1094,7 +1412,7 @@ sol[’primal obj

    -

    Back to the more realistic cases

    +

    Back to the more realistic cases

    We are now ready to return to our setup of the optmization problem for a more realistic case. Introducint the slack parameter \( C \) we have @@ -1119,7 +1437,7 @@ With the slack constants this leads to the additional constraint \( 0\leq \lamb

    -

    Multiclass problems and regression with SVMs

    +

    Multiclass problems and regression with SVMs

    This material will be added later.
    diff --git a/doc/pub/svm/html/svm-solarized.html b/doc/pub/svm/html/svm-solarized.html index 582cd6470..3a6d1342e 100644 --- a/doc/pub/svm/html/svm-solarized.html +++ b/doc/pub/svm/html/svm-solarized.html @@ -57,17 +57,18 @@ div { text-align: justify; text-justify: inter-word; } ('The equations', 2, None, '___sec17'), ('The problem to solve', 2, None, '___sec18'), ("Different kernels and Mercer's theorem", 2, None, '___sec19'), + ('The moons example', 2, None, '___sec20'), ('Mathematical optimization of convex functions', 2, None, - '___sec20'), - ('How do we solve these problems?', 2, None, '___sec21'), - ('A simplex example', 2, None, '___sec22'), - ('Back to the more realistic cases', 2, None, '___sec23'), + '___sec21'), + ('How do we solve these problems?', 2, None, '___sec22'), + ('A simplex example', 2, None, '___sec23'), + ('Back to the more realistic cases', 2, None, '___sec24'), ('Multiclass problems and regression with SVMs', 2, None, - '___sec24')]} + '___sec25')]} end of tocinfo --> @@ -109,7 +110,7 @@ MathJax.Hub.Config({
    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

    -

    Nov 7, 2018

    +

    Nov 8, 2018












    @@ -155,14 +156,83 @@ The theory behind support vector machines (SVM hereafter) is based on the mathematical description of so-called hyperplanes. Let us start with a two-dimensional case. This will also allow us to introduce our first SVM examples. These will be tailored to the case of two specific -classes, as displayed in the figure here. +classes, as displayed in the figure here based on the usage of the petal data.

    We assume here that our data set can be well separated into two domains, where a straight line does the job in the separating the two -classes. Here the two classes are represented by either crosses or +classes. Here the two classes are represented by either squares or circles. +

    + +

    from sklearn import datasets
+from sklearn.svm import SVC, LinearSVC
+from sklearn.linear_model import SGDClassifier
+from sklearn.preprocessing import StandardScaler
+import matplotlib
+import matplotlib.pyplot as plt
+plt.rcParams['axes.labelsize'] = 14
+plt.rcParams['xtick.labelsize'] = 12
+plt.rcParams['ytick.labelsize'] = 12
+
+
+iris = datasets.load_iris()
+X = iris["data"][:, (2, 3)]  # petal length, petal width
+y = iris["target"]
+
+setosa_or_versicolor = (y == 0) | (y == 1)
+X = X[setosa_or_versicolor]
+y = y[setosa_or_versicolor]
+
+
+
+C = 5
+alpha = 1 / (C * len(X))
+
+lin_clf = LinearSVC(loss="hinge", C=C, random_state=42)
+svm_clf = SVC(kernel="linear", C=C)
+sgd_clf = SGDClassifier(loss="hinge", learning_rate="constant", eta0=0.001, alpha=alpha,
+                        max_iter=100000, random_state=42)
+
+scaler = StandardScaler()
+X_scaled = scaler.fit_transform(X)
+
+lin_clf.fit(X_scaled, y)
+svm_clf.fit(X_scaled, y)
+sgd_clf.fit(X_scaled, y)
+
+print("LinearSVC:                   ", lin_clf.intercept_, lin_clf.coef_)
+print("SVC:                         ", svm_clf.intercept_, svm_clf.coef_)
+print("SGDClassifier(alpha={:.5f}):".format(sgd_clf.alpha), sgd_clf.intercept_, sgd_clf.coef_)
+
+# Compute the slope and bias of each decision boundary
+w1 = -lin_clf.coef_[0, 0]/lin_clf.coef_[0, 1]
+b1 = -lin_clf.intercept_[0]/lin_clf.coef_[0, 1]
+w2 = -svm_clf.coef_[0, 0]/svm_clf.coef_[0, 1]
+b2 = -svm_clf.intercept_[0]/svm_clf.coef_[0, 1]
+w3 = -sgd_clf.coef_[0, 0]/sgd_clf.coef_[0, 1]
+b3 = -sgd_clf.intercept_[0]/sgd_clf.coef_[0, 1]
+
+# Transform the decision boundary lines back to the original scale
+line1 = scaler.inverse_transform([[-10, -10 * w1 + b1], [10, 10 * w1 + b1]])
+line2 = scaler.inverse_transform([[-10, -10 * w2 + b2], [10, 10 * w2 + b2]])
+line3 = scaler.inverse_transform([[-10, -10 * w3 + b3], [10, 10 * w3 + b3]])
+
+# Plot all three decision boundaries
+plt.figure(figsize=(11, 4))
+plt.plot(line1[:, 0], line1[:, 1], "k:", label="LinearSVC")
+plt.plot(line2[:, 0], line2[:, 1], "b--", linewidth=2, label="SVC")
+plt.plot(line3[:, 0], line3[:, 1], "r-", label="SGDClassifier")
+plt.plot(X[:, 0][y==1], X[:, 1][y==1], "bs") # label="Iris-Versicolor"
+plt.plot(X[:, 0][y==0], X[:, 1][y==0], "yo") # label="Iris-Setosa"
+plt.xlabel("Petal length", fontsize=14)
+plt.ylabel("Petal width", fontsize=14)
+plt.legend(loc="upper center", fontsize=14)
+plt.axis([0, 5.5, 0, 2])
+
+plt.show()
+











    @@ -677,6 +747,56 @@ obtain a separation between the classes which is almost linear. The change of basis, from \( x\rightarrow z=\phi(x) \) leads to the same type of equations to be solved, except that we need to introduce for example a polynomial transformation to a two-dimensional training set. +

    + + +

    import numpy as np
+import os
+
+np.random.seed(42)
+
+# To plot pretty figures
+import matplotlib
+import matplotlib.pyplot as plt
+plt.rcParams['axes.labelsize'] = 14
+plt.rcParams['xtick.labelsize'] = 12
+plt.rcParams['ytick.labelsize'] = 12
+
+
+from sklearn.svm import SVC
+from sklearn import datasets
+
+
+
+X1D = np.linspace(-4, 4, 9).reshape(-1, 1)
+X2D = np.c_[X1D, X1D**2]
+y = np.array([0, 0, 1, 1, 1, 1, 1, 0, 0])
+
+plt.figure(figsize=(11, 4))
+
+plt.subplot(121)
+plt.grid(True, which='both')
+plt.axhline(y=0, color='k')
+plt.plot(X1D[:, 0][y==0], np.zeros(4), "bs")
+plt.plot(X1D[:, 0][y==1], np.zeros(5), "g^")
+plt.gca().get_yaxis().set_ticks([])
+plt.xlabel(r"$x_1$", fontsize=20)
+plt.axis([-4.5, 4.5, -0.2, 0.2])
+
+plt.subplot(122)
+plt.grid(True, which='both')
+plt.axhline(y=0, color='k')
+plt.axvline(x=0, color='k')
+plt.plot(X2D[:, 0][y==0], X2D[:, 1][y==0], "bs")
+plt.plot(X2D[:, 0][y==1], X2D[:, 1][y==1], "g^")
+plt.xlabel(r"$x_1$", fontsize=20)
+plt.ylabel(r"$x_2$", fontsize=20, rotation=0)
+plt.gca().get_yaxis().set_ticks([0, 4, 8, 12, 16])
+plt.plot([-4.5, 4.5], [6.5, 6.5], "r--", linewidth=3)
+plt.axis([-4.5, 4.5, -1, 17])
+plt.subplots_adjust(right=1)
+plt.show()
+











    @@ -789,7 +909,203 @@ well in practice.











    -

    Mathematical optimization of convex functions

    +

    The moons example

    +

    + + +

    from __future__ import division, print_function, unicode_literals
+
+import numpy as np
+np.random.seed(42)
+
+import matplotlib
+import matplotlib.pyplot as plt
+plt.rcParams['axes.labelsize'] = 14
+plt.rcParams['xtick.labelsize'] = 12
+plt.rcParams['ytick.labelsize'] = 12
+
+
+from sklearn.svm import SVC
+from sklearn import datasets
+
+
+
+from sklearn.pipeline import Pipeline
+from sklearn.preprocessing import StandardScaler
+from sklearn.svm import LinearSVC
+
+
+from sklearn.datasets import make_moons
+X, y = make_moons(n_samples=100, noise=0.15, random_state=42)
+
+def plot_dataset(X, y, axes):
+    plt.plot(X[:, 0][y==0], X[:, 1][y==0], "bs")
+    plt.plot(X[:, 0][y==1], X[:, 1][y==1], "g^")
+    plt.axis(axes)
+    plt.grid(True, which='both')
+    plt.xlabel(r"$x_1$", fontsize=20)
+    plt.ylabel(r"$x_2$", fontsize=20, rotation=0)
+
+plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
+plt.show()
+
+from sklearn.datasets import make_moons
+from sklearn.pipeline import Pipeline
+from sklearn.preprocessing import PolynomialFeatures
+
+polynomial_svm_clf = Pipeline([
+        ("poly_features", PolynomialFeatures(degree=3)),
+        ("scaler", StandardScaler()),
+        ("svm_clf", LinearSVC(C=10, loss="hinge", random_state=42))
+    ])
+
+polynomial_svm_clf.fit(X, y)
+
+def plot_predictions(clf, axes):
+    x0s = np.linspace(axes[0], axes[1], 100)
+    x1s = np.linspace(axes[2], axes[3], 100)
+    x0, x1 = np.meshgrid(x0s, x1s)
+    X = np.c_[x0.ravel(), x1.ravel()]
+    y_pred = clf.predict(X).reshape(x0.shape)
+    y_decision = clf.decision_function(X).reshape(x0.shape)
+    plt.contourf(x0, x1, y_pred, cmap=plt.cm.brg, alpha=0.2)
+    plt.contourf(x0, x1, y_decision, cmap=plt.cm.brg, alpha=0.1)
+
+plot_predictions(polynomial_svm_clf, [-1.5, 2.5, -1, 1.5])
+plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
+
+plt.show()
+
+
+from sklearn.svm import SVC
+
+poly_kernel_svm_clf = Pipeline([
+        ("scaler", StandardScaler()),
+        ("svm_clf", SVC(kernel="poly", degree=3, coef0=1, C=5))
+    ])
+poly_kernel_svm_clf.fit(X, y)
+
+poly100_kernel_svm_clf = Pipeline([
+        ("scaler", StandardScaler()),
+        ("svm_clf", SVC(kernel="poly", degree=10, coef0=100, C=5))
+    ])
+poly100_kernel_svm_clf.fit(X, y)
+
+plt.figure(figsize=(11, 4))
+
+plt.subplot(121)
+plot_predictions(poly_kernel_svm_clf, [-1.5, 2.5, -1, 1.5])
+plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
+plt.title(r"$d=3, r=1, C=5$", fontsize=18)
+
+plt.subplot(122)
+plot_predictions(poly100_kernel_svm_clf, [-1.5, 2.5, -1, 1.5])
+plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
+plt.title(r"$d=10, r=100, C=5$", fontsize=18)
+
+plt.show()
+
+def gaussian_rbf(x, landmark, gamma):
+    return np.exp(-gamma * np.linalg.norm(x - landmark, axis=1)**2)
+
+gamma = 0.3
+
+x1s = np.linspace(-4.5, 4.5, 200).reshape(-1, 1)
+x2s = gaussian_rbf(x1s, -2, gamma)
+x3s = gaussian_rbf(x1s, 1, gamma)
+
+XK = np.c_[gaussian_rbf(X1D, -2, gamma), gaussian_rbf(X1D, 1, gamma)]
+yk = np.array([0, 0, 1, 1, 1, 1, 1, 0, 0])
+
+plt.figure(figsize=(11, 4))
+
+plt.subplot(121)
+plt.grid(True, which='both')
+plt.axhline(y=0, color='k')
+plt.scatter(x=[-2, 1], y=[0, 0], s=150, alpha=0.5, c="red")
+plt.plot(X1D[:, 0][yk==0], np.zeros(4), "bs")
+plt.plot(X1D[:, 0][yk==1], np.zeros(5), "g^")
+plt.plot(x1s, x2s, "g--")
+plt.plot(x1s, x3s, "b:")
+plt.gca().get_yaxis().set_ticks([0, 0.25, 0.5, 0.75, 1])
+plt.xlabel(r"$x_1$", fontsize=20)
+plt.ylabel(r"Similarity", fontsize=14)
+plt.annotate(r'$\mathbf{x}$',
+             xy=(X1D[3, 0], 0),
+             xytext=(-0.5, 0.20),
+             ha="center",
+             arrowprops=dict(facecolor='black', shrink=0.1),
+             fontsize=18,
+            )
+plt.text(-2, 0.9, "$x_2$", ha="center", fontsize=20)
+plt.text(1, 0.9, "$x_3$", ha="center", fontsize=20)
+plt.axis([-4.5, 4.5, -0.1, 1.1])
+
+plt.subplot(122)
+plt.grid(True, which='both')
+plt.axhline(y=0, color='k')
+plt.axvline(x=0, color='k')
+plt.plot(XK[:, 0][yk==0], XK[:, 1][yk==0], "bs")
+plt.plot(XK[:, 0][yk==1], XK[:, 1][yk==1], "g^")
+plt.xlabel(r"$x_2$", fontsize=20)
+plt.ylabel(r"$x_3$  ", fontsize=20, rotation=0)
+plt.annotate(r'$\phi\left(\mathbf{x}\right)$',
+             xy=(XK[3, 0], XK[3, 1]),
+             xytext=(0.65, 0.50),
+             ha="center",
+             arrowprops=dict(facecolor='black', shrink=0.1),
+             fontsize=18,
+            )
+plt.plot([-0.1, 1.1], [0.57, -0.1], "r--", linewidth=3)
+plt.axis([-0.1, 1.1, -0.1, 1.1])
+    
+plt.subplots_adjust(right=1)
+
+plt.show()
+
+
+x1_example = X1D[3, 0]
+for landmark in (-2, 1):
+    k = gaussian_rbf(np.array([[x1_example]]), np.array([[landmark]]), gamma)
+    print("Phi({}, {}) = {}".format(x1_example, landmark, k))
+
+rbf_kernel_svm_clf = Pipeline([
+        ("scaler", StandardScaler()),
+        ("svm_clf", SVC(kernel="rbf", gamma=5, C=0.001))
+    ])
+rbf_kernel_svm_clf.fit(X, y)
+
+
+from sklearn.svm import SVC
+
+gamma1, gamma2 = 0.1, 5
+C1, C2 = 0.001, 1000
+hyperparams = (gamma1, C1), (gamma1, C2), (gamma2, C1), (gamma2, C2)
+
+svm_clfs = []
+for gamma, C in hyperparams:
+    rbf_kernel_svm_clf = Pipeline([
+            ("scaler", StandardScaler()),
+            ("svm_clf", SVC(kernel="rbf", gamma=gamma, C=C))
+        ])
+    rbf_kernel_svm_clf.fit(X, y)
+    svm_clfs.append(rbf_kernel_svm_clf)
+
+plt.figure(figsize=(11, 7))
+
+for i, svm_clf in enumerate(svm_clfs):
+    plt.subplot(221 + i)
+    plot_predictions(svm_clf, [-1.5, 2.5, -1, 1.5])
+    plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
+    gamma, C = hyperparams[i]
+    plt.title(r"$\gamma = {}, C = {}$".format(gamma, C), fontsize=16)
+
+plt.show()
+
    +

    +









    + +

    Mathematical optimization of convex functions

    A mathematical (quadratic) optimization problem, or just optimization problem, has the form @@ -814,7 +1130,7 @@ Convex optimization problems play a central role in applied mathematics and we r











    -

    How do we solve these problems?

    +

    How do we solve these problems?

    If we use Python as programming language and wish to venture beyond @@ -840,7 +1156,7 @@ This will make our life much easier. You don't need t write your own optimizer.











    -

    A simplex example

    +

    A simplex example

    We remind ourselves about the general problem we want to solve @@ -908,7 +1224,7 @@ sol[’primal obj











    -

    Back to the more realistic cases

    +

    Back to the more realistic cases

    We are now ready to return to our setup of the optmization problem for a more realistic case. Introducint the slack parameter \( C \) we have @@ -931,7 +1247,7 @@ With the slack constants this leads to the additional constraint \( 0\leq \lamb











    -

    Multiclass problems and regression with SVMs

    +

    Multiclass problems and regression with SVMs

    This material will be added later. diff --git a/doc/pub/svm/html/svm.html b/doc/pub/svm/html/svm.html index 7b4651af8..a41c5358e 100644 --- a/doc/pub/svm/html/svm.html +++ b/doc/pub/svm/html/svm.html @@ -62,17 +62,18 @@ div { text-align: justify; text-justify: inter-word; } ('The equations', 2, None, '___sec17'), ('The problem to solve', 2, None, '___sec18'), ("Different kernels and Mercer's theorem", 2, None, '___sec19'), + ('The moons example', 2, None, '___sec20'), ('Mathematical optimization of convex functions', 2, None, - '___sec20'), - ('How do we solve these problems?', 2, None, '___sec21'), - ('A simplex example', 2, None, '___sec22'), - ('Back to the more realistic cases', 2, None, '___sec23'), + '___sec21'), + ('How do we solve these problems?', 2, None, '___sec22'), + ('A simplex example', 2, None, '___sec23'), + ('Back to the more realistic cases', 2, None, '___sec24'), ('Multiclass problems and regression with SVMs', 2, None, - '___sec24')]} + '___sec25')]} end of tocinfo --> @@ -114,7 +115,7 @@ MathJax.Hub.Config({
    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

    -

    Nov 7, 2018

    +

    Nov 8, 2018












    @@ -160,14 +161,83 @@ The theory behind support vector machines (SVM hereafter) is based on the mathematical description of so-called hyperplanes. Let us start with a two-dimensional case. This will also allow us to introduce our first SVM examples. These will be tailored to the case of two specific -classes, as displayed in the figure here. +classes, as displayed in the figure here based on the usage of the petal data.

    We assume here that our data set can be well separated into two domains, where a straight line does the job in the separating the two -classes. Here the two classes are represented by either crosses or +classes. Here the two classes are represented by either squares or circles. +

    + +

    from sklearn import datasets
+from sklearn.svm import SVC, LinearSVC
+from sklearn.linear_model import SGDClassifier
+from sklearn.preprocessing import StandardScaler
+import matplotlib
+import matplotlib.pyplot as plt
+plt.rcParams['axes.labelsize'] = 14
+plt.rcParams['xtick.labelsize'] = 12
+plt.rcParams['ytick.labelsize'] = 12
+
+
+iris = datasets.load_iris()
+X = iris["data"][:, (2, 3)]  # petal length, petal width
+y = iris["target"]
+
+setosa_or_versicolor = (y == 0) | (y == 1)
+X = X[setosa_or_versicolor]
+y = y[setosa_or_versicolor]
+
+
+
+C = 5
+alpha = 1 / (C * len(X))
+
+lin_clf = LinearSVC(loss="hinge", C=C, random_state=42)
+svm_clf = SVC(kernel="linear", C=C)
+sgd_clf = SGDClassifier(loss="hinge", learning_rate="constant", eta0=0.001, alpha=alpha,
+                        max_iter=100000, random_state=42)
+
+scaler = StandardScaler()
+X_scaled = scaler.fit_transform(X)
+
+lin_clf.fit(X_scaled, y)
+svm_clf.fit(X_scaled, y)
+sgd_clf.fit(X_scaled, y)
+
+print("LinearSVC:                   ", lin_clf.intercept_, lin_clf.coef_)
+print("SVC:                         ", svm_clf.intercept_, svm_clf.coef_)
+print("SGDClassifier(alpha={:.5f}):".format(sgd_clf.alpha), sgd_clf.intercept_, sgd_clf.coef_)
+
+# Compute the slope and bias of each decision boundary
+w1 = -lin_clf.coef_[0, 0]/lin_clf.coef_[0, 1]
+b1 = -lin_clf.intercept_[0]/lin_clf.coef_[0, 1]
+w2 = -svm_clf.coef_[0, 0]/svm_clf.coef_[0, 1]
+b2 = -svm_clf.intercept_[0]/svm_clf.coef_[0, 1]
+w3 = -sgd_clf.coef_[0, 0]/sgd_clf.coef_[0, 1]
+b3 = -sgd_clf.intercept_[0]/sgd_clf.coef_[0, 1]
+
+# Transform the decision boundary lines back to the original scale
+line1 = scaler.inverse_transform([[-10, -10 * w1 + b1], [10, 10 * w1 + b1]])
+line2 = scaler.inverse_transform([[-10, -10 * w2 + b2], [10, 10 * w2 + b2]])
+line3 = scaler.inverse_transform([[-10, -10 * w3 + b3], [10, 10 * w3 + b3]])
+
+# Plot all three decision boundaries
+plt.figure(figsize=(11, 4))
+plt.plot(line1[:, 0], line1[:, 1], "k:", label="LinearSVC")
+plt.plot(line2[:, 0], line2[:, 1], "b--", linewidth=2, label="SVC")
+plt.plot(line3[:, 0], line3[:, 1], "r-", label="SGDClassifier")
+plt.plot(X[:, 0][y==1], X[:, 1][y==1], "bs") # label="Iris-Versicolor"
+plt.plot(X[:, 0][y==0], X[:, 1][y==0], "yo") # label="Iris-Setosa"
+plt.xlabel("Petal length", fontsize=14)
+plt.ylabel("Petal width", fontsize=14)
+plt.legend(loc="upper center", fontsize=14)
+plt.axis([0, 5.5, 0, 2])
+
+plt.show()
+











    @@ -682,6 +752,56 @@ obtain a separation between the classes which is almost linear. The change of basis, from \( x\rightarrow z=\phi(x) \) leads to the same type of equations to be solved, except that we need to introduce for example a polynomial transformation to a two-dimensional training set. +

    + + +

    import numpy as np
+import os
+
+np.random.seed(42)
+
+# To plot pretty figures
+import matplotlib
+import matplotlib.pyplot as plt
+plt.rcParams['axes.labelsize'] = 14
+plt.rcParams['xtick.labelsize'] = 12
+plt.rcParams['ytick.labelsize'] = 12
+
+
+from sklearn.svm import SVC
+from sklearn import datasets
+
+
+
+X1D = np.linspace(-4, 4, 9).reshape(-1, 1)
+X2D = np.c_[X1D, X1D**2]
+y = np.array([0, 0, 1, 1, 1, 1, 1, 0, 0])
+
+plt.figure(figsize=(11, 4))
+
+plt.subplot(121)
+plt.grid(True, which='both')
+plt.axhline(y=0, color='k')
+plt.plot(X1D[:, 0][y==0], np.zeros(4), "bs")
+plt.plot(X1D[:, 0][y==1], np.zeros(5), "g^")
+plt.gca().get_yaxis().set_ticks([])
+plt.xlabel(r"$x_1$", fontsize=20)
+plt.axis([-4.5, 4.5, -0.2, 0.2])
+
+plt.subplot(122)
+plt.grid(True, which='both')
+plt.axhline(y=0, color='k')
+plt.axvline(x=0, color='k')
+plt.plot(X2D[:, 0][y==0], X2D[:, 1][y==0], "bs")
+plt.plot(X2D[:, 0][y==1], X2D[:, 1][y==1], "g^")
+plt.xlabel(r"$x_1$", fontsize=20)
+plt.ylabel(r"$x_2$", fontsize=20, rotation=0)
+plt.gca().get_yaxis().set_ticks([0, 4, 8, 12, 16])
+plt.plot([-4.5, 4.5], [6.5, 6.5], "r--", linewidth=3)
+plt.axis([-4.5, 4.5, -1, 17])
+plt.subplots_adjust(right=1)
+plt.show()
+











    @@ -794,7 +914,203 @@ well in practice.











    -

    Mathematical optimization of convex functions

    +

    The moons example

    +

    + + +

    from __future__ import division, print_function, unicode_literals
+
+import numpy as np
+np.random.seed(42)
+
+import matplotlib
+import matplotlib.pyplot as plt
+plt.rcParams['axes.labelsize'] = 14
+plt.rcParams['xtick.labelsize'] = 12
+plt.rcParams['ytick.labelsize'] = 12
+
+
+from sklearn.svm import SVC
+from sklearn import datasets
+
+
+
+from sklearn.pipeline import Pipeline
+from sklearn.preprocessing import StandardScaler
+from sklearn.svm import LinearSVC
+
+
+from sklearn.datasets import make_moons
+X, y = make_moons(n_samples=100, noise=0.15, random_state=42)
+
+def plot_dataset(X, y, axes):
+    plt.plot(X[:, 0][y==0], X[:, 1][y==0], "bs")
+    plt.plot(X[:, 0][y==1], X[:, 1][y==1], "g^")
+    plt.axis(axes)
+    plt.grid(True, which='both')
+    plt.xlabel(r"$x_1$", fontsize=20)
+    plt.ylabel(r"$x_2$", fontsize=20, rotation=0)
+
+plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
+plt.show()
+
+from sklearn.datasets import make_moons
+from sklearn.pipeline import Pipeline
+from sklearn.preprocessing import PolynomialFeatures
+
+polynomial_svm_clf = Pipeline([
+        ("poly_features", PolynomialFeatures(degree=3)),
+        ("scaler", StandardScaler()),
+        ("svm_clf", LinearSVC(C=10, loss="hinge", random_state=42))
+    ])
+
+polynomial_svm_clf.fit(X, y)
+
+def plot_predictions(clf, axes):
+    x0s = np.linspace(axes[0], axes[1], 100)
+    x1s = np.linspace(axes[2], axes[3], 100)
+    x0, x1 = np.meshgrid(x0s, x1s)
+    X = np.c_[x0.ravel(), x1.ravel()]
+    y_pred = clf.predict(X).reshape(x0.shape)
+    y_decision = clf.decision_function(X).reshape(x0.shape)
+    plt.contourf(x0, x1, y_pred, cmap=plt.cm.brg, alpha=0.2)
+    plt.contourf(x0, x1, y_decision, cmap=plt.cm.brg, alpha=0.1)
+
+plot_predictions(polynomial_svm_clf, [-1.5, 2.5, -1, 1.5])
+plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
+
+plt.show()
+
+
+from sklearn.svm import SVC
+
+poly_kernel_svm_clf = Pipeline([
+        ("scaler", StandardScaler()),
+        ("svm_clf", SVC(kernel="poly", degree=3, coef0=1, C=5))
+    ])
+poly_kernel_svm_clf.fit(X, y)
+
+poly100_kernel_svm_clf = Pipeline([
+        ("scaler", StandardScaler()),
+        ("svm_clf", SVC(kernel="poly", degree=10, coef0=100, C=5))
+    ])
+poly100_kernel_svm_clf.fit(X, y)
+
+plt.figure(figsize=(11, 4))
+
+plt.subplot(121)
+plot_predictions(poly_kernel_svm_clf, [-1.5, 2.5, -1, 1.5])
+plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
+plt.title(r"$d=3, r=1, C=5$", fontsize=18)
+
+plt.subplot(122)
+plot_predictions(poly100_kernel_svm_clf, [-1.5, 2.5, -1, 1.5])
+plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
+plt.title(r"$d=10, r=100, C=5$", fontsize=18)
+
+plt.show()
+
+def gaussian_rbf(x, landmark, gamma):
+    return np.exp(-gamma * np.linalg.norm(x - landmark, axis=1)**2)
+
+gamma = 0.3
+
+x1s = np.linspace(-4.5, 4.5, 200).reshape(-1, 1)
+x2s = gaussian_rbf(x1s, -2, gamma)
+x3s = gaussian_rbf(x1s, 1, gamma)
+
+XK = np.c_[gaussian_rbf(X1D, -2, gamma), gaussian_rbf(X1D, 1, gamma)]
+yk = np.array([0, 0, 1, 1, 1, 1, 1, 0, 0])
+
+plt.figure(figsize=(11, 4))
+
+plt.subplot(121)
+plt.grid(True, which='both')
+plt.axhline(y=0, color='k')
+plt.scatter(x=[-2, 1], y=[0, 0], s=150, alpha=0.5, c="red")
+plt.plot(X1D[:, 0][yk==0], np.zeros(4), "bs")
+plt.plot(X1D[:, 0][yk==1], np.zeros(5), "g^")
+plt.plot(x1s, x2s, "g--")
+plt.plot(x1s, x3s, "b:")
+plt.gca().get_yaxis().set_ticks([0, 0.25, 0.5, 0.75, 1])
+plt.xlabel(r"$x_1$", fontsize=20)
+plt.ylabel(r"Similarity", fontsize=14)
+plt.annotate(r'$\mathbf{x}$',
+             xy=(X1D[3, 0], 0),
+             xytext=(-0.5, 0.20),
+             ha="center",
+             arrowprops=dict(facecolor='black', shrink=0.1),
+             fontsize=18,
+            )
+plt.text(-2, 0.9, "$x_2$", ha="center", fontsize=20)
+plt.text(1, 0.9, "$x_3$", ha="center", fontsize=20)
+plt.axis([-4.5, 4.5, -0.1, 1.1])
+
+plt.subplot(122)
+plt.grid(True, which='both')
+plt.axhline(y=0, color='k')
+plt.axvline(x=0, color='k')
+plt.plot(XK[:, 0][yk==0], XK[:, 1][yk==0], "bs")
+plt.plot(XK[:, 0][yk==1], XK[:, 1][yk==1], "g^")
+plt.xlabel(r"$x_2$", fontsize=20)
+plt.ylabel(r"$x_3$  ", fontsize=20, rotation=0)
+plt.annotate(r'$\phi\left(\mathbf{x}\right)$',
+             xy=(XK[3, 0], XK[3, 1]),
+             xytext=(0.65, 0.50),
+             ha="center",
+             arrowprops=dict(facecolor='black', shrink=0.1),
+             fontsize=18,
+            )
+plt.plot([-0.1, 1.1], [0.57, -0.1], "r--", linewidth=3)
+plt.axis([-0.1, 1.1, -0.1, 1.1])
+    
+plt.subplots_adjust(right=1)
+
+plt.show()
+
+
+x1_example = X1D[3, 0]
+for landmark in (-2, 1):
+    k = gaussian_rbf(np.array([[x1_example]]), np.array([[landmark]]), gamma)
+    print("Phi({}, {}) = {}".format(x1_example, landmark, k))
+
+rbf_kernel_svm_clf = Pipeline([
+        ("scaler", StandardScaler()),
+        ("svm_clf", SVC(kernel="rbf", gamma=5, C=0.001))
+    ])
+rbf_kernel_svm_clf.fit(X, y)
+
+
+from sklearn.svm import SVC
+
+gamma1, gamma2 = 0.1, 5
+C1, C2 = 0.001, 1000
+hyperparams = (gamma1, C1), (gamma1, C2), (gamma2, C1), (gamma2, C2)
+
+svm_clfs = []
+for gamma, C in hyperparams:
+    rbf_kernel_svm_clf = Pipeline([
+            ("scaler", StandardScaler()),
+            ("svm_clf", SVC(kernel="rbf", gamma=gamma, C=C))
+        ])
+    rbf_kernel_svm_clf.fit(X, y)
+    svm_clfs.append(rbf_kernel_svm_clf)
+
+plt.figure(figsize=(11, 7))
+
+for i, svm_clf in enumerate(svm_clfs):
+    plt.subplot(221 + i)
+    plot_predictions(svm_clf, [-1.5, 2.5, -1, 1.5])
+    plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
+    gamma, C = hyperparams[i]
+    plt.title(r"$\gamma = {}, C = {}$".format(gamma, C), fontsize=16)
+
+plt.show()
+
    +

    +









    + +

    Mathematical optimization of convex functions

    A mathematical (quadratic) optimization problem, or just optimization problem, has the form @@ -819,7 +1135,7 @@ Convex optimization problems play a central role in applied mathematics and we r











    -

    How do we solve these problems?

    +

    How do we solve these problems?

    If we use Python as programming language and wish to venture beyond @@ -845,7 +1161,7 @@ This will make our life much easier. You don't need t write your own optimizer.











    -

    A simplex example

    +

    A simplex example

    We remind ourselves about the general problem we want to solve @@ -913,7 +1229,7 @@ sol[’primal objective’]











    -

    Back to the more realistic cases

    +

    Back to the more realistic cases

    We are now ready to return to our setup of the optmization problem for a more realistic case. Introducint the slack parameter \( C \) we have @@ -936,7 +1252,7 @@ With the slack constants this leads to the additional constraint \( 0\leq \lamb











    -

    Multiclass problems and regression with SVMs

    +

    Multiclass problems and regression with SVMs

    This material will be added later. diff --git a/doc/pub/svm/ipynb/ipynb-svm-src.tar.gz b/doc/pub/svm/ipynb/ipynb-svm-src.tar.gz index dc344b8f0e308d3b66b96dac6d07b7419fbbd521..3aff4b71ff45dc112f2f48088f2c9e075d7b57db 100644 GIT binary patch literal 207 zcmb2|=3qD`@Fbdn`R)0GSxklk#|qEe9kn@VR9I>&{%T>n)5LuaDaq!!zBxime}7xY z_3g?%mnHx2KTU2+oDr!fCpTX_^rzn9)qz*jtfnTPn|)KWv}m1w&&-aMuXhG6-fFhU zIqCW|Y>gr}+l5SrUmv8zAh`Ci8vP{JXmF;*R|1&o=)3*KT_~{K>TXBgOfP*K7}dwO9E0 \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: **Nov 7, 2018**\n", + "Date: **Nov 8, 2018**\n", "\n", "Copyright 1999-2018, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n", "\n", @@ -49,13 +49,96 @@ "the mathematical description of so-called hyperplanes. Let us start\n", "with a two-dimensional case. This will also allow us to introduce our\n", "first SVM examples. These will be tailored to the case of two specific\n", - "classes, as displayed in the figure here.\n", + "classes, as displayed in the figure here based on the usage of the petal data.\n", "\n", "We assume here that our data set can be well separated into two\n", "domains, where a straight line does the job in the separating the two\n", - "classes. Here the two classes are represented by either crosses or\n", - "circles.\n", + "classes. Here the two classes are represented by either squares or\n", + "circles." + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "%matplotlib inline\n", "\n", + "from sklearn import datasets\n", + "from sklearn.svm import SVC, LinearSVC\n", + "from sklearn.linear_model import SGDClassifier\n", + "from sklearn.preprocessing import StandardScaler\n", + "import matplotlib\n", + "import matplotlib.pyplot as plt\n", + "plt.rcParams['axes.labelsize'] = 14\n", + "plt.rcParams['xtick.labelsize'] = 12\n", + "plt.rcParams['ytick.labelsize'] = 12\n", + "\n", + "\n", + "iris = datasets.load_iris()\n", + "X = iris[\"data\"][:, (2, 3)] # petal length, petal width\n", + "y = iris[\"target\"]\n", + "\n", + "setosa_or_versicolor = (y == 0) | (y == 1)\n", + "X = X[setosa_or_versicolor]\n", + "y = y[setosa_or_versicolor]\n", + "\n", + "\n", + "\n", + "C = 5\n", + "alpha = 1 / (C * len(X))\n", + "\n", + "lin_clf = LinearSVC(loss=\"hinge\", C=C, random_state=42)\n", + "svm_clf = SVC(kernel=\"linear\", C=C)\n", + "sgd_clf = SGDClassifier(loss=\"hinge\", learning_rate=\"constant\", eta0=0.001, alpha=alpha,\n", + " max_iter=100000, random_state=42)\n", + "\n", + "scaler = StandardScaler()\n", + "X_scaled = scaler.fit_transform(X)\n", + "\n", + "lin_clf.fit(X_scaled, y)\n", + "svm_clf.fit(X_scaled, y)\n", + "sgd_clf.fit(X_scaled, y)\n", + "\n", + "print(\"LinearSVC: \", lin_clf.intercept_, lin_clf.coef_)\n", + "print(\"SVC: \", svm_clf.intercept_, svm_clf.coef_)\n", + "print(\"SGDClassifier(alpha={:.5f}):\".format(sgd_clf.alpha), sgd_clf.intercept_, sgd_clf.coef_)\n", + "\n", + "# Compute the slope and bias of each decision boundary\n", + "w1 = -lin_clf.coef_[0, 0]/lin_clf.coef_[0, 1]\n", + "b1 = -lin_clf.intercept_[0]/lin_clf.coef_[0, 1]\n", + "w2 = -svm_clf.coef_[0, 0]/svm_clf.coef_[0, 1]\n", + "b2 = -svm_clf.intercept_[0]/svm_clf.coef_[0, 1]\n", + "w3 = -sgd_clf.coef_[0, 0]/sgd_clf.coef_[0, 1]\n", + "b3 = -sgd_clf.intercept_[0]/sgd_clf.coef_[0, 1]\n", + "\n", + "# Transform the decision boundary lines back to the original scale\n", + "line1 = scaler.inverse_transform([[-10, -10 * w1 + b1], [10, 10 * w1 + b1]])\n", + "line2 = scaler.inverse_transform([[-10, -10 * w2 + b2], [10, 10 * w2 + b2]])\n", + "line3 = scaler.inverse_transform([[-10, -10 * w3 + b3], [10, 10 * w3 + b3]])\n", + "\n", + "# Plot all three decision boundaries\n", + "plt.figure(figsize=(11, 4))\n", + "plt.plot(line1[:, 0], line1[:, 1], \"k:\", label=\"LinearSVC\")\n", + "plt.plot(line2[:, 0], line2[:, 1], \"b--\", linewidth=2, label=\"SVC\")\n", + "plt.plot(line3[:, 0], line3[:, 1], \"r-\", label=\"SGDClassifier\")\n", + "plt.plot(X[:, 0][y==1], X[:, 1][y==1], \"bs\") # label=\"Iris-Versicolor\"\n", + "plt.plot(X[:, 0][y==0], X[:, 1][y==0], \"yo\") # label=\"Iris-Setosa\"\n", + "plt.xlabel(\"Petal length\", fontsize=14)\n", + "plt.ylabel(\"Petal width\", fontsize=14)\n", + "plt.legend(loc=\"upper center\", fontsize=14)\n", + "plt.axis([0, 5.5, 0, 2])\n", + "\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ "## What is a hyperplane?\n", "\n", "The aim of the SVM algorithm is to find a hyperplane in an $p$-dimensional space, where $p$ is the number of features that distinctly classifies the data points. \n", @@ -1079,8 +1162,69 @@ "obtain a separation between the classes which is almost linear. \n", "\n", "The change of basis, from $x\\rightarrow z=\\phi(x)$ leads to the same type of equations to be solved, except that\n", - "we need to introduce for example a polynomial transformation to a two-dimensional training set.\n", + "we need to introduce for example a polynomial transformation to a two-dimensional training set." + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "import numpy as np\n", + "import os\n", "\n", + "np.random.seed(42)\n", + "\n", + "# To plot pretty figures\n", + "import matplotlib\n", + "import matplotlib.pyplot as plt\n", + "plt.rcParams['axes.labelsize'] = 14\n", + "plt.rcParams['xtick.labelsize'] = 12\n", + "plt.rcParams['ytick.labelsize'] = 12\n", + "\n", + "\n", + "from sklearn.svm import SVC\n", + "from sklearn import datasets\n", + "\n", + "\n", + "\n", + "X1D = np.linspace(-4, 4, 9).reshape(-1, 1)\n", + "X2D = np.c_[X1D, X1D**2]\n", + "y = np.array([0, 0, 1, 1, 1, 1, 1, 0, 0])\n", + "\n", + "plt.figure(figsize=(11, 4))\n", + "\n", + "plt.subplot(121)\n", + "plt.grid(True, which='both')\n", + "plt.axhline(y=0, color='k')\n", + "plt.plot(X1D[:, 0][y==0], np.zeros(4), \"bs\")\n", + "plt.plot(X1D[:, 0][y==1], np.zeros(5), \"g^\")\n", + "plt.gca().get_yaxis().set_ticks([])\n", + "plt.xlabel(r\"$x_1$\", fontsize=20)\n", + "plt.axis([-4.5, 4.5, -0.2, 0.2])\n", + "\n", + "plt.subplot(122)\n", + "plt.grid(True, which='both')\n", + "plt.axhline(y=0, color='k')\n", + "plt.axvline(x=0, color='k')\n", + "plt.plot(X2D[:, 0][y==0], X2D[:, 1][y==0], \"bs\")\n", + "plt.plot(X2D[:, 0][y==1], X2D[:, 1][y==1], \"g^\")\n", + "plt.xlabel(r\"$x_1$\", fontsize=20)\n", + "plt.ylabel(r\"$x_2$\", fontsize=20, rotation=0)\n", + "plt.gca().get_yaxis().set_ticks([0, 4, 8, 12, 16])\n", + "plt.plot([-4.5, 4.5], [6.5, 6.5], \"r--\", linewidth=3)\n", + "plt.axis([-4.5, 4.5, -1, 17])\n", + "plt.subplots_adjust(right=1)\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ "## The equations\n", "\n", "Suppose we define a polynomial transformation of degree two only (we continue to live in a plane with $x_i$ and $y_i$ as variables)" @@ -1277,6 +1421,211 @@ "well in practice.\n", "\n", "\n", + "## The moons example" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "from __future__ import division, print_function, unicode_literals\n", + "\n", + "import numpy as np\n", + "np.random.seed(42)\n", + "\n", + "import matplotlib\n", + "import matplotlib.pyplot as plt\n", + "plt.rcParams['axes.labelsize'] = 14\n", + "plt.rcParams['xtick.labelsize'] = 12\n", + "plt.rcParams['ytick.labelsize'] = 12\n", + "\n", + "\n", + "from sklearn.svm import SVC\n", + "from sklearn import datasets\n", + "\n", + "\n", + "\n", + "from sklearn.pipeline import Pipeline\n", + "from sklearn.preprocessing import StandardScaler\n", + "from sklearn.svm import LinearSVC\n", + "\n", + "\n", + "from sklearn.datasets import make_moons\n", + "X, y = make_moons(n_samples=100, noise=0.15, random_state=42)\n", + "\n", + "def plot_dataset(X, y, axes):\n", + " plt.plot(X[:, 0][y==0], X[:, 1][y==0], \"bs\")\n", + " plt.plot(X[:, 0][y==1], X[:, 1][y==1], \"g^\")\n", + " plt.axis(axes)\n", + " plt.grid(True, which='both')\n", + " plt.xlabel(r\"$x_1$\", fontsize=20)\n", + " plt.ylabel(r\"$x_2$\", fontsize=20, rotation=0)\n", + "\n", + "plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])\n", + "plt.show()\n", + "\n", + "from sklearn.datasets import make_moons\n", + "from sklearn.pipeline import Pipeline\n", + "from sklearn.preprocessing import PolynomialFeatures\n", + "\n", + "polynomial_svm_clf = Pipeline([\n", + " (\"poly_features\", PolynomialFeatures(degree=3)),\n", + " (\"scaler\", StandardScaler()),\n", + " (\"svm_clf\", LinearSVC(C=10, loss=\"hinge\", random_state=42))\n", + " ])\n", + "\n", + "polynomial_svm_clf.fit(X, y)\n", + "\n", + "def plot_predictions(clf, axes):\n", + " x0s = np.linspace(axes[0], axes[1], 100)\n", + " x1s = np.linspace(axes[2], axes[3], 100)\n", + " x0, x1 = np.meshgrid(x0s, x1s)\n", + " X = np.c_[x0.ravel(), x1.ravel()]\n", + " y_pred = clf.predict(X).reshape(x0.shape)\n", + " y_decision = clf.decision_function(X).reshape(x0.shape)\n", + " plt.contourf(x0, x1, y_pred, cmap=plt.cm.brg, alpha=0.2)\n", + " plt.contourf(x0, x1, y_decision, cmap=plt.cm.brg, alpha=0.1)\n", + "\n", + "plot_predictions(polynomial_svm_clf, [-1.5, 2.5, -1, 1.5])\n", + "plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])\n", + "\n", + "plt.show()\n", + "\n", + "\n", + "from sklearn.svm import SVC\n", + "\n", + "poly_kernel_svm_clf = Pipeline([\n", + " (\"scaler\", StandardScaler()),\n", + " (\"svm_clf\", SVC(kernel=\"poly\", degree=3, coef0=1, C=5))\n", + " ])\n", + "poly_kernel_svm_clf.fit(X, y)\n", + "\n", + "poly100_kernel_svm_clf = Pipeline([\n", + " (\"scaler\", StandardScaler()),\n", + " (\"svm_clf\", SVC(kernel=\"poly\", degree=10, coef0=100, C=5))\n", + " ])\n", + "poly100_kernel_svm_clf.fit(X, y)\n", + "\n", + "plt.figure(figsize=(11, 4))\n", + "\n", + "plt.subplot(121)\n", + "plot_predictions(poly_kernel_svm_clf, [-1.5, 2.5, -1, 1.5])\n", + "plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])\n", + "plt.title(r\"$d=3, r=1, C=5$\", fontsize=18)\n", + "\n", + "plt.subplot(122)\n", + "plot_predictions(poly100_kernel_svm_clf, [-1.5, 2.5, -1, 1.5])\n", + "plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])\n", + "plt.title(r\"$d=10, r=100, C=5$\", fontsize=18)\n", + "\n", + "plt.show()\n", + "\n", + "def gaussian_rbf(x, landmark, gamma):\n", + " return np.exp(-gamma * np.linalg.norm(x - landmark, axis=1)**2)\n", + "\n", + "gamma = 0.3\n", + "\n", + "x1s = np.linspace(-4.5, 4.5, 200).reshape(-1, 1)\n", + "x2s = gaussian_rbf(x1s, -2, gamma)\n", + "x3s = gaussian_rbf(x1s, 1, gamma)\n", + "\n", + "XK = np.c_[gaussian_rbf(X1D, -2, gamma), gaussian_rbf(X1D, 1, gamma)]\n", + "yk = np.array([0, 0, 1, 1, 1, 1, 1, 0, 0])\n", + "\n", + "plt.figure(figsize=(11, 4))\n", + "\n", + "plt.subplot(121)\n", + "plt.grid(True, which='both')\n", + "plt.axhline(y=0, color='k')\n", + "plt.scatter(x=[-2, 1], y=[0, 0], s=150, alpha=0.5, c=\"red\")\n", + "plt.plot(X1D[:, 0][yk==0], np.zeros(4), \"bs\")\n", + "plt.plot(X1D[:, 0][yk==1], np.zeros(5), \"g^\")\n", + "plt.plot(x1s, x2s, \"g--\")\n", + "plt.plot(x1s, x3s, \"b:\")\n", + "plt.gca().get_yaxis().set_ticks([0, 0.25, 0.5, 0.75, 1])\n", + "plt.xlabel(r\"$x_1$\", fontsize=20)\n", + "plt.ylabel(r\"Similarity\", fontsize=14)\n", + "plt.annotate(r'$\\mathbf{x}$',\n", + " xy=(X1D[3, 0], 0),\n", + " xytext=(-0.5, 0.20),\n", + " ha=\"center\",\n", + " arrowprops=dict(facecolor='black', shrink=0.1),\n", + " fontsize=18,\n", + " )\n", + "plt.text(-2, 0.9, \"$x_2$\", ha=\"center\", fontsize=20)\n", + "plt.text(1, 0.9, \"$x_3$\", ha=\"center\", fontsize=20)\n", + "plt.axis([-4.5, 4.5, -0.1, 1.1])\n", + "\n", + "plt.subplot(122)\n", + "plt.grid(True, which='both')\n", + "plt.axhline(y=0, color='k')\n", + "plt.axvline(x=0, color='k')\n", + "plt.plot(XK[:, 0][yk==0], XK[:, 1][yk==0], \"bs\")\n", + "plt.plot(XK[:, 0][yk==1], XK[:, 1][yk==1], \"g^\")\n", + "plt.xlabel(r\"$x_2$\", fontsize=20)\n", + "plt.ylabel(r\"$x_3$ \", fontsize=20, rotation=0)\n", + "plt.annotate(r'$\\phi\\left(\\mathbf{x}\\right)$',\n", + " xy=(XK[3, 0], XK[3, 1]),\n", + " xytext=(0.65, 0.50),\n", + " ha=\"center\",\n", + " arrowprops=dict(facecolor='black', shrink=0.1),\n", + " fontsize=18,\n", + " )\n", + "plt.plot([-0.1, 1.1], [0.57, -0.1], \"r--\", linewidth=3)\n", + "plt.axis([-0.1, 1.1, -0.1, 1.1])\n", + " \n", + "plt.subplots_adjust(right=1)\n", + "\n", + "plt.show()\n", + "\n", + "\n", + "x1_example = X1D[3, 0]\n", + "for landmark in (-2, 1):\n", + " k = gaussian_rbf(np.array([[x1_example]]), np.array([[landmark]]), gamma)\n", + " print(\"Phi({}, {}) = {}\".format(x1_example, landmark, k))\n", + "\n", + "rbf_kernel_svm_clf = Pipeline([\n", + " (\"scaler\", StandardScaler()),\n", + " (\"svm_clf\", SVC(kernel=\"rbf\", gamma=5, C=0.001))\n", + " ])\n", + "rbf_kernel_svm_clf.fit(X, y)\n", + "\n", + "\n", + "from sklearn.svm import SVC\n", + "\n", + "gamma1, gamma2 = 0.1, 5\n", + "C1, C2 = 0.001, 1000\n", + "hyperparams = (gamma1, C1), (gamma1, C2), (gamma2, C1), (gamma2, C2)\n", + "\n", + "svm_clfs = []\n", + "for gamma, C in hyperparams:\n", + " rbf_kernel_svm_clf = Pipeline([\n", + " (\"scaler\", StandardScaler()),\n", + " (\"svm_clf\", SVC(kernel=\"rbf\", gamma=gamma, C=C))\n", + " ])\n", + " rbf_kernel_svm_clf.fit(X, y)\n", + " svm_clfs.append(rbf_kernel_svm_clf)\n", + "\n", + "plt.figure(figsize=(11, 7))\n", + "\n", + "for i, svm_clf in enumerate(svm_clfs):\n", + " plt.subplot(221 + i)\n", + " plot_predictions(svm_clf, [-1.5, 2.5, -1, 1.5])\n", + " plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])\n", + " gamma, C = hyperparams[i]\n", + " plt.title(r\"$\\gamma = {}, C = {}$\".format(gamma, C), fontsize=16)\n", + "\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ "## Mathematical optimization of convex functions\n", "\n", "A mathematical (quadratic) optimization problem, or just optimization problem, has the form" @@ -1324,7 +1673,7 @@ }, { "cell_type": "code", - "execution_count": 1, + "execution_count": 4, "metadata": { "collapsed": false }, @@ -1456,7 +1805,7 @@ }, { "cell_type": "code", - "execution_count": 2, + "execution_count": 5, "metadata": { "collapsed": false }, diff --git a/doc/pub/svm/pdf/svm-minted.pdf b/doc/pub/svm/pdf/svm-minted.pdf index 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Let us start with a two-dimensional case. This will also allow us to introduce our first SVM examples. These will be tailored to the case of two specific -classes, as displayed in the figure here. +classes, as displayed in the figure here based on the usage of the petal data. We assume here that our data set can be well separated into two domains, where a straight line does the job in the separating the two -classes. Here the two classes are represented by either crosses or +classes. Here the two classes are represented by either squares or circles. +!bc pycod +from sklearn import datasets +from sklearn.svm import SVC, LinearSVC +from sklearn.linear_model import SGDClassifier +from sklearn.preprocessing import StandardScaler +import matplotlib +import matplotlib.pyplot as plt +plt.rcParams['axes.labelsize'] = 14 +plt.rcParams['xtick.labelsize'] = 12 +plt.rcParams['ytick.labelsize'] = 12 + + +iris = datasets.load_iris() +X = iris["data"][:, (2, 3)] # petal length, petal width +y = iris["target"] + +setosa_or_versicolor = (y == 0) | (y == 1) +X = X[setosa_or_versicolor] +y = y[setosa_or_versicolor] + + + +C = 5 +alpha = 1 / (C * len(X)) + +lin_clf = LinearSVC(loss="hinge", C=C, random_state=42) +svm_clf = SVC(kernel="linear", C=C) +sgd_clf = SGDClassifier(loss="hinge", learning_rate="constant", eta0=0.001, alpha=alpha, + max_iter=100000, random_state=42) + +scaler = StandardScaler() +X_scaled = scaler.fit_transform(X) + +lin_clf.fit(X_scaled, y) +svm_clf.fit(X_scaled, y) +sgd_clf.fit(X_scaled, y) + +print("LinearSVC: ", lin_clf.intercept_, lin_clf.coef_) +print("SVC: ", svm_clf.intercept_, svm_clf.coef_) +print("SGDClassifier(alpha={:.5f}):".format(sgd_clf.alpha), sgd_clf.intercept_, sgd_clf.coef_) + +# Compute the slope and bias of each decision boundary +w1 = -lin_clf.coef_[0, 0]/lin_clf.coef_[0, 1] +b1 = -lin_clf.intercept_[0]/lin_clf.coef_[0, 1] +w2 = -svm_clf.coef_[0, 0]/svm_clf.coef_[0, 1] +b2 = -svm_clf.intercept_[0]/svm_clf.coef_[0, 1] +w3 = -sgd_clf.coef_[0, 0]/sgd_clf.coef_[0, 1] +b3 = -sgd_clf.intercept_[0]/sgd_clf.coef_[0, 1] + +# Transform the decision boundary lines back to the original scale +line1 = scaler.inverse_transform([[-10, -10 * w1 + b1], [10, 10 * w1 + b1]]) +line2 = scaler.inverse_transform([[-10, -10 * w2 + b2], [10, 10 * w2 + b2]]) +line3 = scaler.inverse_transform([[-10, -10 * w3 + b3], [10, 10 * w3 + b3]]) + +# Plot all three decision boundaries +plt.figure(figsize=(11, 4)) +plt.plot(line1[:, 0], line1[:, 1], "k:", label="LinearSVC") +plt.plot(line2[:, 0], line2[:, 1], "b--", linewidth=2, label="SVC") +plt.plot(line3[:, 0], line3[:, 1], "r-", label="SGDClassifier") +plt.plot(X[:, 0][y==1], X[:, 1][y==1], "bs") # label="Iris-Versicolor" +plt.plot(X[:, 0][y==0], X[:, 1][y==0], "yo") # label="Iris-Setosa" +plt.xlabel("Petal length", fontsize=14) +plt.ylabel("Petal width", fontsize=14) +plt.legend(loc="upper center", fontsize=14) +plt.axis([0, 5.5, 0, 2]) + +plt.show() + + + + + + + + +!ec + + + !split ===== What is a hyperplane? ===== @@ -553,6 +632,58 @@ obtain a separation between the classes which is almost linear. The change of basis, from $x\rightarrow z=\phi(x)$ leads to the same type of equations to be solved, except that we need to introduce for example a polynomial transformation to a two-dimensional training set. +!bc pycod +import numpy as np +import os + +np.random.seed(42) + +# To plot pretty figures +import matplotlib +import matplotlib.pyplot as plt +plt.rcParams['axes.labelsize'] = 14 +plt.rcParams['xtick.labelsize'] = 12 +plt.rcParams['ytick.labelsize'] = 12 + + +from sklearn.svm import SVC +from sklearn import datasets + + + +X1D = np.linspace(-4, 4, 9).reshape(-1, 1) +X2D = np.c_[X1D, X1D**2] +y = np.array([0, 0, 1, 1, 1, 1, 1, 0, 0]) + +plt.figure(figsize=(11, 4)) + +plt.subplot(121) +plt.grid(True, which='both') +plt.axhline(y=0, color='k') +plt.plot(X1D[:, 0][y==0], np.zeros(4), "bs") +plt.plot(X1D[:, 0][y==1], np.zeros(5), "g^") +plt.gca().get_yaxis().set_ticks([]) +plt.xlabel(r"$x_1$", fontsize=20) +plt.axis([-4.5, 4.5, -0.2, 0.2]) + +plt.subplot(122) +plt.grid(True, which='both') +plt.axhline(y=0, color='k') +plt.axvline(x=0, color='k') +plt.plot(X2D[:, 0][y==0], X2D[:, 1][y==0], "bs") +plt.plot(X2D[:, 0][y==1], X2D[:, 1][y==1], "g^") +plt.xlabel(r"$x_1$", fontsize=20) +plt.ylabel(r"$x_2$", fontsize=20, rotation=0) +plt.gca().get_yaxis().set_ticks([0, 4, 8, 12, 16]) +plt.plot([-4.5, 4.5], [6.5, 6.5], "r--", linewidth=3) +plt.axis([-4.5, 4.5, -1, 17]) +plt.subplots_adjust(right=1) +plt.show() + +!ec + + + !split ===== The equations ===== @@ -658,6 +789,202 @@ Note that some frequently used kernels (such as the Sigmoid kernel) don’t resp well in practice. +!split +===== The moons example ===== +!bc pycod +from __future__ import division, print_function, unicode_literals + +import numpy as np +np.random.seed(42) + +import matplotlib +import matplotlib.pyplot as plt +plt.rcParams['axes.labelsize'] = 14 +plt.rcParams['xtick.labelsize'] = 12 +plt.rcParams['ytick.labelsize'] = 12 + + +from sklearn.svm import SVC +from sklearn import datasets + + + +from sklearn.pipeline import Pipeline +from sklearn.preprocessing import StandardScaler +from sklearn.svm import LinearSVC + + +from sklearn.datasets import make_moons +X, y = make_moons(n_samples=100, noise=0.15, random_state=42) + +def plot_dataset(X, y, axes): + plt.plot(X[:, 0][y==0], X[:, 1][y==0], "bs") + plt.plot(X[:, 0][y==1], X[:, 1][y==1], "g^") + plt.axis(axes) + plt.grid(True, which='both') + plt.xlabel(r"$x_1$", fontsize=20) + plt.ylabel(r"$x_2$", fontsize=20, rotation=0) + +plot_dataset(X, y, [-1.5, 2.5, -1, 1.5]) +plt.show() + +from sklearn.datasets import make_moons +from sklearn.pipeline import Pipeline +from sklearn.preprocessing import PolynomialFeatures + +polynomial_svm_clf = Pipeline([ + ("poly_features", PolynomialFeatures(degree=3)), + ("scaler", StandardScaler()), + ("svm_clf", LinearSVC(C=10, loss="hinge", random_state=42)) + ]) + +polynomial_svm_clf.fit(X, y) + +def plot_predictions(clf, axes): + x0s = np.linspace(axes[0], axes[1], 100) + x1s = np.linspace(axes[2], axes[3], 100) + x0, x1 = np.meshgrid(x0s, x1s) + X = np.c_[x0.ravel(), x1.ravel()] + y_pred = clf.predict(X).reshape(x0.shape) + y_decision = clf.decision_function(X).reshape(x0.shape) + plt.contourf(x0, x1, y_pred, cmap=plt.cm.brg, alpha=0.2) + plt.contourf(x0, x1, y_decision, cmap=plt.cm.brg, alpha=0.1) + +plot_predictions(polynomial_svm_clf, [-1.5, 2.5, -1, 1.5]) +plot_dataset(X, y, [-1.5, 2.5, -1, 1.5]) + +plt.show() + + +from sklearn.svm import SVC + +poly_kernel_svm_clf = Pipeline([ + ("scaler", StandardScaler()), + ("svm_clf", SVC(kernel="poly", degree=3, coef0=1, C=5)) + ]) +poly_kernel_svm_clf.fit(X, y) + +poly100_kernel_svm_clf = Pipeline([ + ("scaler", StandardScaler()), + ("svm_clf", SVC(kernel="poly", degree=10, coef0=100, C=5)) + ]) +poly100_kernel_svm_clf.fit(X, y) + +plt.figure(figsize=(11, 4)) + +plt.subplot(121) +plot_predictions(poly_kernel_svm_clf, [-1.5, 2.5, -1, 1.5]) +plot_dataset(X, y, [-1.5, 2.5, -1, 1.5]) +plt.title(r"$d=3, r=1, C=5$", fontsize=18) + +plt.subplot(122) +plot_predictions(poly100_kernel_svm_clf, [-1.5, 2.5, -1, 1.5]) +plot_dataset(X, y, [-1.5, 2.5, -1, 1.5]) +plt.title(r"$d=10, r=100, C=5$", fontsize=18) + +plt.show() + +def gaussian_rbf(x, landmark, gamma): + return np.exp(-gamma * np.linalg.norm(x - landmark, axis=1)**2) + +gamma = 0.3 + +x1s = np.linspace(-4.5, 4.5, 200).reshape(-1, 1) +x2s = gaussian_rbf(x1s, -2, gamma) +x3s = gaussian_rbf(x1s, 1, gamma) + +XK = np.c_[gaussian_rbf(X1D, -2, gamma), gaussian_rbf(X1D, 1, gamma)] +yk = np.array([0, 0, 1, 1, 1, 1, 1, 0, 0]) + +plt.figure(figsize=(11, 4)) + +plt.subplot(121) +plt.grid(True, which='both') +plt.axhline(y=0, color='k') +plt.scatter(x=[-2, 1], y=[0, 0], s=150, alpha=0.5, c="red") +plt.plot(X1D[:, 0][yk==0], np.zeros(4), "bs") +plt.plot(X1D[:, 0][yk==1], np.zeros(5), "g^") +plt.plot(x1s, x2s, "g--") +plt.plot(x1s, x3s, "b:") +plt.gca().get_yaxis().set_ticks([0, 0.25, 0.5, 0.75, 1]) +plt.xlabel(r"$x_1$", fontsize=20) +plt.ylabel(r"Similarity", fontsize=14) +plt.annotate(r'$\mathbf{x}$', + xy=(X1D[3, 0], 0), + xytext=(-0.5, 0.20), + ha="center", + arrowprops=dict(facecolor='black', shrink=0.1), + fontsize=18, + ) +plt.text(-2, 0.9, "$x_2$", ha="center", fontsize=20) +plt.text(1, 0.9, "$x_3$", ha="center", fontsize=20) +plt.axis([-4.5, 4.5, -0.1, 1.1]) + +plt.subplot(122) +plt.grid(True, which='both') +plt.axhline(y=0, color='k') +plt.axvline(x=0, color='k') +plt.plot(XK[:, 0][yk==0], XK[:, 1][yk==0], "bs") +plt.plot(XK[:, 0][yk==1], XK[:, 1][yk==1], "g^") +plt.xlabel(r"$x_2$", fontsize=20) +plt.ylabel(r"$x_3$ ", fontsize=20, rotation=0) +plt.annotate(r'$\phi\left(\mathbf{x}\right)$', + xy=(XK[3, 0], XK[3, 1]), + xytext=(0.65, 0.50), + ha="center", + arrowprops=dict(facecolor='black', shrink=0.1), + fontsize=18, + ) +plt.plot([-0.1, 1.1], [0.57, -0.1], "r--", linewidth=3) +plt.axis([-0.1, 1.1, -0.1, 1.1]) + +plt.subplots_adjust(right=1) + +plt.show() + + +x1_example = X1D[3, 0] +for landmark in (-2, 1): + k = gaussian_rbf(np.array([[x1_example]]), np.array([[landmark]]), gamma) + print("Phi({}, {}) = {}".format(x1_example, landmark, k)) + +rbf_kernel_svm_clf = Pipeline([ + ("scaler", StandardScaler()), + ("svm_clf", SVC(kernel="rbf", gamma=5, C=0.001)) + ]) +rbf_kernel_svm_clf.fit(X, y) + + +from sklearn.svm import SVC + +gamma1, gamma2 = 0.1, 5 +C1, C2 = 0.001, 1000 +hyperparams = (gamma1, C1), (gamma1, C2), (gamma2, C1), (gamma2, C2) + +svm_clfs = [] +for gamma, C in hyperparams: + rbf_kernel_svm_clf = Pipeline([ + ("scaler", StandardScaler()), + ("svm_clf", SVC(kernel="rbf", gamma=gamma, C=C)) + ]) + rbf_kernel_svm_clf.fit(X, y) + svm_clfs.append(rbf_kernel_svm_clf) + +plt.figure(figsize=(11, 7)) + +for i, svm_clf in enumerate(svm_clfs): + plt.subplot(221 + i) + plot_predictions(svm_clf, [-1.5, 2.5, -1, 1.5]) + plot_dataset(X, y, [-1.5, 2.5, -1, 1.5]) + gamma, C = hyperparams[i] + plt.title(r"$\gamma = {}, C = {}$".format(gamma, C), fontsize=16) + +plt.show() + +!ec + + + !split ===== Mathematical optimization of convex functions =====