From 49df21377c4da0ed9d33826ace70e9bfe3634dc7 Mon Sep 17 00:00:00 2001 From: mhjensen Date: Thu, 1 Nov 2018 06:40:07 +0100 Subject: [PATCH] update of sim slides, more to come --- doc/Programs/DimRed/covariance.py | 4 +- doc/pub/svm/html/svm-bs.html | 164 ++++++++++++++++----- doc/pub/svm/html/svm-reveal.html | 153 ++++++++++++++----- doc/pub/svm/html/svm-solarized.html | 160 +++++++++++++++----- doc/pub/svm/html/svm.html | 160 +++++++++++++++----- doc/pub/svm/ipynb/ipynb-svm-src.tar.gz | Bin 207 -> 207 bytes doc/pub/svm/ipynb/svm.ipynb | 164 +++++++++++++++++---- doc/pub/svm/pdf/svm-beamer-handouts2x3.pdf | Bin 112703 -> 139987 bytes doc/pub/svm/pdf/svm-beamer.pdf | Bin 109561 -> 135245 bytes doc/pub/svm/pdf/svm-minted.pdf | Bin 229897 -> 247764 bytes doc/src/DecisionTrees/DecisionTrees.do.txt | 10 +- doc/src/SupportVMachines/svm.do.txt | 130 +++++++++++----- 12 files changed, 734 insertions(+), 211 deletions(-) diff --git a/doc/Programs/DimRed/covariance.py b/doc/Programs/DimRed/covariance.py index b610073d3..601b19c96 100644 --- a/doc/Programs/DimRed/covariance.py +++ b/doc/Programs/DimRed/covariance.py @@ -13,8 +13,8 @@ def covariance(x, y, n): n = 10 x = np.random.normal(size=n) -y = x*np.random.normal(size=n) -z = x*x+y*np.random.normal(size=n) +y = np.random.normal(size=n) +z = x*x*x+y*y +0.5*np.random.normal(size=n) covxx = covariance(x,x,n) covxy = covariance(x,y,n) covxz = covariance(x,z,n) diff --git a/doc/pub/svm/html/svm-bs.html b/doc/pub/svm/html/svm-bs.html index 143c4afdb..bb4c7ceaf 100644 --- a/doc/pub/svm/html/svm-bs.html +++ b/doc/pub/svm/html/svm-bs.html @@ -40,10 +40,11 @@ Automatically generated HTML file from DocOnce source @@ -66,6 +67,10 @@ end of tocinfo --> Contents @@ -99,7 +104,7 @@ end of tocinfo -->
[2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

-

May 30, 2018

+

Nov 1, 2018


@@ -109,44 +114,129 @@ end of tocinfo -->

Support Vector Machines, overarching aims

+

+A Support Vector Machine (SVM) is a very powerful and versatile +Machine Learning model, capable of performing linear or nonlinear +classification, regression, and even outlier detection. It is one of +the most popular models in Machine Learning, and anyone interested in +Machine Learning should have it in their toolbox. SVMs are +particularly well suited for classification of complex but small-sized or +medium-sized datasets. + +

+The basic mathematics relies on the definition of hyperplanes and the definition of a margin which separates +classes (in case of classification problems) of variables. It is also used for regression problems. + +

+With SVMs we distinguish between hard margin and soft margins. The latter introduces a so-called softening parameter to be discussed below. +We distringuish also between linearn and non-linear approaches. + +

+These notes will be updated shortly with more material + +

+ + +

SVMs, basics with scikit-learn

+ +

+The following Scikit-Learn code loads the iris dataset, scales the features, and then trains a linear SVM +model (using the LinearSVC class with C = 0.1 and the hinge loss function, described shortly) to detect +Iris-Virginica flowers.

import numpy as np
-from sklearn.svm import SVR
-import matplotlib.pyplot as plt
-
-# Generate sample data
-X = np.sort(5*np.random.rand(40,1), axis=0)
-y = X**3
-y=y.ravel()
-
-# Add noise to targets
-X[::4] +=3*(0.5 - np.random.rand(1))
-y[::5] += 50 * (0.5 - np.random.rand(8))
-
-plt.plot(X,y, 'g^')
-
-#SVR Fit
-svr_poly = SVR(kernel='poly', C=1e3, degree=3)
-y_poly = svr_poly.fit(X, y).predict(X)
-
-# Plots
-z = np.arange(0, 5, 0.1)
-t = z**3
-fig = plt.figure()
-ax = fig.add_subplot(111)
-plt.plot(z,z**3, 'r--', label='Cubic Function with No Noise')
-lw = 2
-plt.scatter(X, y, color='darkorange', label='Gaussian Cubic Noise')
-plt.plot(X, y_poly, color='green', lw=lw, label='Polynomial model')
-plt.xlabel('data')
-plt.ylabel('target')
-plt.title('Cubic Gaussian Distribution')
-plt.legend()
-plt.show()
+from sklearn import datasets
+from sklearn.pipeline import Pipeline
+from sklearn.preprocessing import StandardScaler
+from sklearn.svm import LinearSVC
+iris = datasets.load_iris()
+X = iris["data"][:, (2, 3)] # petal length, petal width
+y = (iris["target"] == 2).astype(np.float64) # Iris-Virginica
+svm_clf = Pipeline((
+("scaler", StandardScaler()),
+("linear_svc", LinearSVC(C=1, loss="hinge")),
+))
+svm_clf.fit(X_scaled, y)
 

+Alternatively, you could use the SVC class, using SVC(kernel="linear", C=1), but it is much slower, +especially with large training sets, so it is not recommended. Another option is to use the SGDClassifier +class, with SGDClassifier(loss="hinge", alpha=1/(m*C)). This applies regular Stochastic +Gradient Descentto train a linear SVM classifier. It does not converge as fast as the +LinearSVC class, but it can be useful to handle huge datasets that do not fit in memory (out-of-core +training), or to handle online classification tasks. + +

+ + +

SVMs, adding polynomial features

+ +

+Although linear SVM classifiers are efficient and work surprisingly well in many cases, many datasets +are not even close to being linearly separable. One approach to handling nonlinear datasets is to add more +features, such as polynomial features. In some cases this can result in a linearly +separable dataset. + +

+ + +

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"))
+))
+polynomial_svm_clf.fit(X, y)
+
+

+ + +

SVMs, polynomials and kernels

+ +

+Adding polynomial features is simple to implement and can work great with all sorts of Machine Learning +algorithms (not just SVMs), but at a low polynomial degree it cannot deal with very complex datasets, +and with a high polynomial degree it creates a huge number of features, making the model too slow. +Fortunately, when using SVMs you can apply an almost miraculous mathematical technique called the +kernel trick discussed during the lectures. It makes it possible to get the same result as if you added many +polynomial features, even with very high-degree polynomials, without actually having to add them. So +there is no combinatorial explosion of the number of features since you don’t actually add any features. +This trick is implemented by the SVC class. + +

+ + +

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)
+
+

+This code trains an SVM classifier using a 3rd-degree polynomial kernel. + +

+ + +

SVMs, regression

+ +

+You can use Scikit-Learn’s LinearSVR class to perform linear SVM Regression. The following code +shows how to use the regression option (more material to come) +

+ + +

from sklearn.svm import LinearSVR
+svm_reg = LinearSVR(epsilon=1.5)
+svm_reg.fit(X, y)
+
+

+To tackle nonlinear regression tasks, you can use a kernelized SVM model diff --git a/doc/pub/svm/html/svm-reveal.html b/doc/pub/svm/html/svm-reveal.html index 17a17f1ce..1782b55ba 100644 --- a/doc/pub/svm/html/svm-reveal.html +++ b/doc/pub/svm/html/svm-reveal.html @@ -132,7 +132,7 @@ td.padding {

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

 
-

May 30, 2018

+

Nov 1, 2018


@@ -145,43 +145,130 @@ td.padding {

Support Vector Machines, overarching aims

+

+A Support Vector Machine (SVM) is a very powerful and versatile +Machine Learning model, capable of performing linear or nonlinear +classification, regression, and even outlier detection. It is one of +the most popular models in Machine Learning, and anyone interested in +Machine Learning should have it in their toolbox. SVMs are +particularly well suited for classification of complex but small-sized or +medium-sized datasets. + +

+The basic mathematics relies on the definition of hyperplanes and the definition of a margin which separates +classes (in case of classification problems) of variables. It is also used for regression problems. + +

+With SVMs we distinguish between hard margin and soft margins. The latter introduces a so-called softening parameter to be discussed below. +We distringuish also between linearn and non-linear approaches. + +

+These notes will be updated shortly with more material +

+ + +
+

SVMs, basics with scikit-learn

+ +

+The following Scikit-Learn code loads the iris dataset, scales the features, and then trains a linear SVM +model (using the LinearSVC class with C = 0.1 and the hinge loss function, described shortly) to detect +Iris-Virginica flowers.

import numpy as np
-from sklearn.svm import SVR
-import matplotlib.pyplot as plt
-
-# Generate sample data
-X = np.sort(5*np.random.rand(40,1), axis=0)
-y = X**3
-y=y.ravel()
-
-# Add noise to targets
-X[::4] +=3*(0.5 - np.random.rand(1))
-y[::5] += 50 * (0.5 - np.random.rand(8))
-
-plt.plot(X,y, 'g^')
-
-#SVR Fit
-svr_poly = SVR(kernel='poly', C=1e3, degree=3)
-y_poly = svr_poly.fit(X, y).predict(X)
-
-# Plots
-z = np.arange(0, 5, 0.1)
-t = z**3
-fig = plt.figure()
-ax = fig.add_subplot(111)
-plt.plot(z,z**3, 'r--', label='Cubic Function with No Noise')
-lw = 2
-plt.scatter(X, y, color='darkorange', label='Gaussian Cubic Noise')
-plt.plot(X, y_poly, color='green', lw=lw, label='Polynomial model')
-plt.xlabel('data')
-plt.ylabel('target')
-plt.title('Cubic Gaussian Distribution')
-plt.legend()
-plt.show()
+from sklearn import datasets
+from sklearn.pipeline import Pipeline
+from sklearn.preprocessing import StandardScaler
+from sklearn.svm import LinearSVC
+iris = datasets.load_iris()
+X = iris["data"][:, (2, 3)] # petal length, petal width
+y = (iris["target"] == 2).astype(np.float64) # Iris-Virginica
+svm_clf = Pipeline((
+("scaler", StandardScaler()),
+("linear_svc", LinearSVC(C=1, loss="hinge")),
+))
+svm_clf.fit(X_scaled, y)
 
+

+Alternatively, you could use the SVC class, using SVC(kernel="linear", C=1), but it is much slower, +especially with large training sets, so it is not recommended. Another option is to use the SGDClassifier +class, with SGDClassifier(loss="hinge", alpha=1/(m*C)). This applies regular Stochastic +Gradient Descentto train a linear SVM classifier. It does not converge as fast as the +LinearSVC class, but it can be useful to handle huge datasets that do not fit in memory (out-of-core +training), or to handle online classification tasks. +

+ + +
+

SVMs, adding polynomial features

+ +

+Although linear SVM classifiers are efficient and work surprisingly well in many cases, many datasets +are not even close to being linearly separable. One approach to handling nonlinear datasets is to add more +features, such as polynomial features. In some cases this can result in a linearly +separable dataset. + +

+ + +

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"))
+))
+polynomial_svm_clf.fit(X, y)
+
+
+ + +
+

SVMs, polynomials and kernels

+ +

+Adding polynomial features is simple to implement and can work great with all sorts of Machine Learning +algorithms (not just SVMs), but at a low polynomial degree it cannot deal with very complex datasets, +and with a high polynomial degree it creates a huge number of features, making the model too slow. +Fortunately, when using SVMs you can apply an almost miraculous mathematical technique called the +kernel trick discussed during the lectures. It makes it possible to get the same result as if you added many +polynomial features, even with very high-degree polynomials, without actually having to add them. So +there is no combinatorial explosion of the number of features since you don’t actually add any features. +This trick is implemented by the SVC class. + +

+ + +

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)
+
+

+This code trains an SVM classifier using a 3rd-degree polynomial kernel. +

+ + +
+

SVMs, regression

+ +

+You can use Scikit-Learn’s LinearSVR class to perform linear SVM Regression. The following code +shows how to use the regression option (more material to come) +

+ + +

from sklearn.svm import LinearSVR
+svm_reg = LinearSVR(epsilon=1.5)
+svm_reg.fit(X, y)
+
+

+To tackle nonlinear regression tasks, you can use a kernelized SVM model

diff --git a/doc/pub/svm/html/svm-solarized.html b/doc/pub/svm/html/svm-solarized.html index 9e4adf363..f6107c8cf 100644 --- a/doc/pub/svm/html/svm-solarized.html +++ b/doc/pub/svm/html/svm-solarized.html @@ -34,10 +34,11 @@ div { text-align: justify; text-justify: inter-word; } @@ -63,51 +64,136 @@ end of tocinfo -->
[2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

-

May 30, 2018

+

Nov 1, 2018












Support Vector Machines, overarching aims

+

+A Support Vector Machine (SVM) is a very powerful and versatile +Machine Learning model, capable of performing linear or nonlinear +classification, regression, and even outlier detection. It is one of +the most popular models in Machine Learning, and anyone interested in +Machine Learning should have it in their toolbox. SVMs are +particularly well suited for classification of complex but small-sized or +medium-sized datasets. + +

+The basic mathematics relies on the definition of hyperplanes and the definition of a margin which separates +classes (in case of classification problems) of variables. It is also used for regression problems. + +

+With SVMs we distinguish between hard margin and soft margins. The latter introduces a so-called softening parameter to be discussed below. +We distringuish also between linearn and non-linear approaches. + +

+These notes will be updated shortly with more material + +

+









+ +

SVMs, basics with scikit-learn

+ +

+The following Scikit-Learn code loads the iris dataset, scales the features, and then trains a linear SVM +model (using the LinearSVC class with C = 0.1 and the hinge loss function, described shortly) to detect +Iris-Virginica flowers.

import numpy as np
-from sklearn.svm import SVR
-import matplotlib.pyplot as plt
-
-# Generate sample data
-X = np.sort(5*np.random.rand(40,1), axis=0)
-y = X**3
-y=y.ravel()
-
-# Add noise to targets
-X[::4] +=3*(0.5 - np.random.rand(1))
-y[::5] += 50 * (0.5 - np.random.rand(8))
-
-plt.plot(X,y, 'g^')
-
-#SVR Fit
-svr_poly = SVR(kernel='poly', C=1e3, degree=3)
-y_poly = svr_poly.fit(X, y).predict(X)
-
-# Plots
-z = np.arange(0, 5, 0.1)
-t = z**3
-fig = plt.figure()
-ax = fig.add_subplot(111)
-plt.plot(z,z**3, 'r--', label='Cubic Function with No Noise')
-lw = 2
-plt.scatter(X, y, color='darkorange', label='Gaussian Cubic Noise')
-plt.plot(X, y_poly, color='green', lw=lw, label='Polynomial model')
-plt.xlabel('data')
-plt.ylabel('target')
-plt.title('Cubic Gaussian Distribution')
-plt.legend()
-plt.show()
+from sklearn import datasets
+from sklearn.pipeline import Pipeline
+from sklearn.preprocessing import StandardScaler
+from sklearn.svm import LinearSVC
+iris = datasets.load_iris()
+X = iris["data"][:, (2, 3)] # petal length, petal width
+y = (iris["target"] == 2).astype(np.float64) # Iris-Virginica
+svm_clf = Pipeline((
+("scaler", StandardScaler()),
+("linear_svc", LinearSVC(C=1, loss="hinge")),
+))
+svm_clf.fit(X_scaled, y)
 

+Alternatively, you could use the SVC class, using SVC(kernel="linear", C=1), but it is much slower, +especially with large training sets, so it is not recommended. Another option is to use the SGDClassifier +class, with SGDClassifier(loss="hinge", alpha=1/(m*C)). This applies regular Stochastic +Gradient Descentto train a linear SVM classifier. It does not converge as fast as the +LinearSVC class, but it can be useful to handle huge datasets that do not fit in memory (out-of-core +training), or to handle online classification tasks. + +

+









+ +

SVMs, adding polynomial features

+ +

+Although linear SVM classifiers are efficient and work surprisingly well in many cases, many datasets +are not even close to being linearly separable. One approach to handling nonlinear datasets is to add more +features, such as polynomial features. In some cases this can result in a linearly +separable dataset. + +

+ + +

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"))
+))
+polynomial_svm_clf.fit(X, y)
+
+

+









+ +

SVMs, polynomials and kernels

+ +

+Adding polynomial features is simple to implement and can work great with all sorts of Machine Learning +algorithms (not just SVMs), but at a low polynomial degree it cannot deal with very complex datasets, +and with a high polynomial degree it creates a huge number of features, making the model too slow. +Fortunately, when using SVMs you can apply an almost miraculous mathematical technique called the +kernel trick discussed during the lectures. It makes it possible to get the same result as if you added many +polynomial features, even with very high-degree polynomials, without actually having to add them. So +there is no combinatorial explosion of the number of features since you don’t actually add any features. +This trick is implemented by the SVC class. + +

+ + +

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)
+
+

+This code trains an SVM classifier using a 3rd-degree polynomial kernel. + +

+









+ +

SVMs, regression

+ +

+You can use Scikit-Learn’s LinearSVR class to perform linear SVM Regression. The following code +shows how to use the regression option (more material to come) +

+ + +

from sklearn.svm import LinearSVR
+svm_reg = LinearSVR(epsilon=1.5)
+svm_reg.fit(X, y)
+
+

+To tackle nonlinear regression tasks, you can use a kernelized SVM model diff --git a/doc/pub/svm/html/svm.html b/doc/pub/svm/html/svm.html index 497df0d1e..cf2fe755b 100644 --- a/doc/pub/svm/html/svm.html +++ b/doc/pub/svm/html/svm.html @@ -39,10 +39,11 @@ div { text-align: justify; text-justify: inter-word; } @@ -68,51 +69,136 @@ end of tocinfo -->

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

-

May 30, 2018

+

Nov 1, 2018












Support Vector Machines, overarching aims

+

+A Support Vector Machine (SVM) is a very powerful and versatile +Machine Learning model, capable of performing linear or nonlinear +classification, regression, and even outlier detection. It is one of +the most popular models in Machine Learning, and anyone interested in +Machine Learning should have it in their toolbox. SVMs are +particularly well suited for classification of complex but small-sized or +medium-sized datasets. + +

+The basic mathematics relies on the definition of hyperplanes and the definition of a margin which separates +classes (in case of classification problems) of variables. It is also used for regression problems. + +

+With SVMs we distinguish between hard margin and soft margins. The latter introduces a so-called softening parameter to be discussed below. +We distringuish also between linearn and non-linear approaches. + +

+These notes will be updated shortly with more material + +

+









+ +

SVMs, basics with scikit-learn

+ +

+The following Scikit-Learn code loads the iris dataset, scales the features, and then trains a linear SVM +model (using the LinearSVC class with C = 0.1 and the hinge loss function, described shortly) to detect +Iris-Virginica flowers.

import numpy as np
-from sklearn.svm import SVR
-import matplotlib.pyplot as plt
-
-# Generate sample data
-X = np.sort(5*np.random.rand(40,1), axis=0)
-y = X**3
-y=y.ravel()
-
-# Add noise to targets
-X[::4] +=3*(0.5 - np.random.rand(1))
-y[::5] += 50 * (0.5 - np.random.rand(8))
-
-plt.plot(X,y, 'g^')
-
-#SVR Fit
-svr_poly = SVR(kernel='poly', C=1e3, degree=3)
-y_poly = svr_poly.fit(X, y).predict(X)
-
-# Plots
-z = np.arange(0, 5, 0.1)
-t = z**3
-fig = plt.figure()
-ax = fig.add_subplot(111)
-plt.plot(z,z**3, 'r--', label='Cubic Function with No Noise')
-lw = 2
-plt.scatter(X, y, color='darkorange', label='Gaussian Cubic Noise')
-plt.plot(X, y_poly, color='green', lw=lw, label='Polynomial model')
-plt.xlabel('data')
-plt.ylabel('target')
-plt.title('Cubic Gaussian Distribution')
-plt.legend()
-plt.show()
+from sklearn import datasets
+from sklearn.pipeline import Pipeline
+from sklearn.preprocessing import StandardScaler
+from sklearn.svm import LinearSVC
+iris = datasets.load_iris()
+X = iris["data"][:, (2, 3)] # petal length, petal width
+y = (iris["target"] == 2).astype(np.float64) # Iris-Virginica
+svm_clf = Pipeline((
+("scaler", StandardScaler()),
+("linear_svc", LinearSVC(C=1, loss="hinge")),
+))
+svm_clf.fit(X_scaled, y)
 

+Alternatively, you could use the SVC class, using SVC(kernel="linear", C=1), but it is much slower, +especially with large training sets, so it is not recommended. Another option is to use the SGDClassifier +class, with SGDClassifier(loss="hinge", alpha=1/(m*C)). This applies regular Stochastic +Gradient Descentto train a linear SVM classifier. It does not converge as fast as the +LinearSVC class, but it can be useful to handle huge datasets that do not fit in memory (out-of-core +training), or to handle online classification tasks. + +

+









+ +

SVMs, adding polynomial features

+ +

+Although linear SVM classifiers are efficient and work surprisingly well in many cases, many datasets +are not even close to being linearly separable. One approach to handling nonlinear datasets is to add more +features, such as polynomial features. In some cases this can result in a linearly +separable dataset. + +

+ + +

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"))
+))
+polynomial_svm_clf.fit(X, y)
+
+

+









+ +

SVMs, polynomials and kernels

+ +

+Adding polynomial features is simple to implement and can work great with all sorts of Machine Learning +algorithms (not just SVMs), but at a low polynomial degree it cannot deal with very complex datasets, +and with a high polynomial degree it creates a huge number of features, making the model too slow. +Fortunately, when using SVMs you can apply an almost miraculous mathematical technique called the +kernel trick discussed during the lectures. It makes it possible to get the same result as if you added many +polynomial features, even with very high-degree polynomials, without actually having to add them. So +there is no combinatorial explosion of the number of features since you don’t actually add any features. +This trick is implemented by the SVC class. + +

+ + +

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)
+
+

+This code trains an SVM classifier using a 3rd-degree polynomial kernel. + +

+









+ +

SVMs, regression

+ +

+You can use Scikit-Learn’s LinearSVR class to perform linear SVM Regression. The following code +shows how to use the regression option (more material to come) +

+ + +

from sklearn.svm import LinearSVR
+svm_reg = LinearSVR(epsilon=1.5)
+svm_reg.fit(X, y)
+
+

+To tackle nonlinear regression tasks, you can use a kernelized SVM model diff --git a/doc/pub/svm/ipynb/ipynb-svm-src.tar.gz b/doc/pub/svm/ipynb/ipynb-svm-src.tar.gz index 0d17f826b043b270783afba61052c26759098351..d5c9e5f736ac6ce781bd053982b89c3d6500b316 100644 GIT binary patch literal 207 zcmb2|=3r=;cq^KL`R)0GSxklk#|qEe9kn@V^m^UPodNuEEkVDSCidRlVtP?U?EBkT z(ObdQlY@T!kF`saXq&nG=-s|h3yiMZZEvDG9acfAWoe{+t^I9dE7TYk0oZ?(+VqdNX(3vi*NK8IZvref0%u-Om^_ G7#IMoA7MZM literal 207 zcmb2|=3wAW<&S1yetZ6W7L%dC@xt?VM{N#H`kQPc{_3E!;}P+WoRVPEjh<89-7XK! z@UNO2^y_a--5CQOtZ=?7xJGtX(PL#zCMO>(zFCd;DaP=;La`+}E49@4Y#GTw>crnIOY|Z~YmN!GrprObL3n(hM35 F3;>=gUHSk3 diff --git a/doc/pub/svm/ipynb/svm.ipynb b/doc/pub/svm/ipynb/svm.ipynb index 697c1a740..2876278d5 100644 --- a/doc/pub/svm/ipynb/svm.ipynb +++ b/doc/pub/svm/ipynb/svm.ipynb @@ -10,14 +10,35 @@ " \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: **May 30, 2018**\n", + "Date: **Nov 1, 2018**\n", "\n", "Copyright 1999-2018, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n", "\n", "\n", "\n", + "## Support Vector Machines, overarching aims\n", "\n", - "## Support Vector Machines, overarching aims" + "A Support Vector Machine (SVM) is a very powerful and versatile\n", + "Machine Learning model, capable of performing linear or nonlinear\n", + "classification, regression, and even outlier detection. It is one of\n", + "the most popular models in Machine Learning, and anyone interested in\n", + "Machine Learning should have it in their toolbox. SVMs are\n", + "particularly well suited for classification of complex but small-sized or\n", + "medium-sized datasets. \n", + "\n", + "The basic mathematics relies on the definition of hyperplanes and the definition of a **margin** which separates\n", + "classes (in case of classification problems) of variables. It is also used for regression problems.\n", + "\n", + "With SVMs we distinguish between hard margin and soft margins. The latter introduces a so-called softening parameter to be discussed below.\n", + "We distringuish also between linearn and non-linear approaches.\n", + "\n", + "**These notes will be updated shortly with more material**\n", + "\n", + "## SVMs, basics with scikit-learn\n", + "\n", + "The following Scikit-Learn code loads the iris dataset, scales the features, and then trains a linear SVM\n", + "model (using the LinearSVC class with C = 0.1 and the hinge loss function, described shortly) to detect\n", + "Iris-Virginica flowers." ] }, { @@ -28,41 +49,122 @@ }, "outputs": [], "source": [ - "%matplotlib inline\n", - "\n", "import numpy as np\n", - "from sklearn.svm import SVR\n", - "import matplotlib.pyplot as plt\n", + "from sklearn import datasets\n", + "from sklearn.pipeline import Pipeline\n", + "from sklearn.preprocessing import StandardScaler\n", + "from sklearn.svm import LinearSVC\n", + "iris = datasets.load_iris()\n", + "X = iris[\"data\"][:, (2, 3)] # petal length, petal width\n", + "y = (iris[\"target\"] == 2).astype(np.float64) # Iris-Virginica\n", + "svm_clf = Pipeline((\n", + "(\"scaler\", StandardScaler()),\n", + "(\"linear_svc\", LinearSVC(C=1, loss=\"hinge\")),\n", + "))\n", + "svm_clf.fit(X_scaled, y)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Alternatively, you could use the SVC class, using **SVC(kernel=\"linear\", C=1)**, but it is much slower,\n", + "especially with large training sets, so it is not recommended. Another option is to use the SGDClassifier\n", + "class, with **SGDClassifier(loss=\"hinge\", alpha=1/(m*C))**. This applies regular Stochastic\n", + "Gradient Descentto train a linear SVM classifier. It does not converge as fast as the\n", + "LinearSVC class, but it can be useful to handle huge datasets that do not fit in memory (out-of-core\n", + "training), or to handle online classification tasks.\n", "\n", - "# Generate sample data\n", - "X = np.sort(5*np.random.rand(40,1), axis=0)\n", - "y = X**3\n", - "y=y.ravel()\n", + "## SVMs, adding polynomial features\n", "\n", - "# Add noise to targets\n", - "X[::4] +=3*(0.5 - np.random.rand(1))\n", - "y[::5] += 50 * (0.5 - np.random.rand(8))\n", + "Although linear SVM classifiers are efficient and work surprisingly well in many cases, many datasets\n", + "are not even close to being linearly separable. One approach to handling nonlinear datasets is to add more\n", + "features, such as polynomial features. In some cases this can result in a linearly\n", + "separable dataset." + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "from sklearn.datasets import make_moons\n", + "from sklearn.pipeline import Pipeline\n", + "from sklearn.preprocessing import PolynomialFeatures\n", + "polynomial_svm_clf = Pipeline((\n", + "(\"poly_features\", PolynomialFeatures(degree=3)),\n", + "(\"scaler\", StandardScaler()),\n", + "(\"svm_clf\", LinearSVC(C=10, loss=\"hinge\"))\n", + "))\n", + "polynomial_svm_clf.fit(X, y)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## SVMs, polynomials and kernels\n", "\n", - "plt.plot(X,y, 'g^')\n", + "Adding polynomial features is simple to implement and can work great with all sorts of Machine Learning\n", + "algorithms (not just SVMs), but at a low polynomial degree it cannot deal with very complex datasets,\n", + "and with a high polynomial degree it creates a huge number of features, making the model too slow.\n", + "Fortunately, when using SVMs you can apply an almost miraculous mathematical technique called the\n", + "kernel trick discussed during the lectures. It makes it possible to get the same result as if you added many\n", + "polynomial features, even with very high-degree polynomials, without actually having to add them. So\n", + "there is no combinatorial explosion of the number of features since you don’t actually add any features.\n", + "This trick is implemented by the SVC class." + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "from sklearn.svm import SVC\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)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This code trains an SVM classifier using a 3rd-degree polynomial kernel.\n", "\n", - "#SVR Fit\n", - "svr_poly = SVR(kernel='poly', C=1e3, degree=3)\n", - "y_poly = svr_poly.fit(X, y).predict(X)\n", "\n", - "# Plots\n", - "z = np.arange(0, 5, 0.1)\n", - "t = z**3\n", - "fig = plt.figure()\n", - "ax = fig.add_subplot(111)\n", - "plt.plot(z,z**3, 'r--', label='Cubic Function with No Noise')\n", - "lw = 2\n", - "plt.scatter(X, y, color='darkorange', label='Gaussian Cubic Noise')\n", - "plt.plot(X, y_poly, color='green', lw=lw, label='Polynomial model')\n", - "plt.xlabel('data')\n", - "plt.ylabel('target')\n", - "plt.title('Cubic Gaussian Distribution')\n", - "plt.legend()\n", - "plt.show()" + "## SVMs, regression\n", + "\n", + "You can use Scikit-Learn’s LinearSVR class to perform linear SVM Regression. The following code\n", + "shows how to use the regression option (more material to come)" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "from sklearn.svm import LinearSVR\n", + "svm_reg = LinearSVR(epsilon=1.5)\n", + "svm_reg.fit(X, y)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "To tackle nonlinear regression tasks, you can use a kernelized SVM model" ] } ], diff --git a/doc/pub/svm/pdf/svm-beamer-handouts2x3.pdf b/doc/pub/svm/pdf/svm-beamer-handouts2x3.pdf index 3991ef9c0bad873ac052039e5017fc6bfad54683..9ca94c518e87fc256f57755db106fd6c1c7dc27a 100644 GIT binary patch delta 101316 zcmb@tV{~9$w60mPZB=aBs@Qf?72EcXZ5x$}lZtKIwrxAP-*?Wr-97r8?q7HBKkr_9 z{#s+qXTEdIvAV7y=LVqZ$rVM#>6sbWVaccG2bN)(S%{d3>^s!dtiH+;5)8pZ^6o?=KfCTA$NpJ-2B-dAkKOBY5Qx!s7+(RlNIR973=4*ubS zIq=P?rl&3b$qdp+*}_87BGDLx1%y2@6(o|M>z^%9CSnw4<|1NY{x8kWM#RGQUz&xT 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