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.. "auto_examples/svm/plot_rbf_parameters.py"
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.. only:: html
.. note::
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Click :ref:`here `
to download the full example code or to run this example in your browser via Binder
.. rst-class:: sphx-glr-example-title
.. _sphx_glr_auto_examples_svm_plot_rbf_parameters.py:
==================
RBF SVM parameters
==================
This example illustrates the effect of the parameters ``gamma`` and ``C`` of
the Radial Basis Function (RBF) kernel SVM.
Intuitively, the ``gamma`` parameter defines how far the influence of a single
training example reaches, with low values meaning 'far' and high values meaning
'close'. The ``gamma`` parameters can be seen as the inverse of the radius of
influence of samples selected by the model as support vectors.
The ``C`` parameter trades off correct classification of training examples
against maximization of the decision function's margin. For larger values of
``C``, a smaller margin will be accepted if the decision function is better at
classifying all training points correctly. A lower ``C`` will encourage a
larger margin, therefore a simpler decision function, at the cost of training
accuracy. In other words ``C`` behaves as a regularization parameter in the
SVM.
The first plot is a visualization of the decision function for a variety of
parameter values on a simplified classification problem involving only 2 input
features and 2 possible target classes (binary classification). Note that this
kind of plot is not possible to do for problems with more features or target
classes.
The second plot is a heatmap of the classifier's cross-validation accuracy as a
function of ``C`` and ``gamma``. For this example we explore a relatively large
grid for illustration purposes. In practice, a logarithmic grid from
:math:`10^{-3}` to :math:`10^3` is usually sufficient. If the best parameters
lie on the boundaries of the grid, it can be extended in that direction in a
subsequent search.
Note that the heat map plot has a special colorbar with a midpoint value close
to the score values of the best performing models so as to make it easy to tell
them apart in the blink of an eye.
The behavior of the model is very sensitive to the ``gamma`` parameter. If
``gamma`` is too large, the radius of the area of influence of the support
vectors only includes the support vector itself and no amount of
regularization with ``C`` will be able to prevent overfitting.
When ``gamma`` is very small, the model is too constrained and cannot capture
the complexity or "shape" of the data. The region of influence of any selected
support vector would include the whole training set. The resulting model will
behave similarly to a linear model with a set of hyperplanes that separate the
centers of high density of any pair of two classes.
For intermediate values, we can see on the second plot that good models can
be found on a diagonal of ``C`` and ``gamma``. Smooth models (lower ``gamma``
values) can be made more complex by increasing the importance of classifying
each point correctly (larger ``C`` values) hence the diagonal of good
performing models.
Finally, one can also observe that for some intermediate values of ``gamma`` we
get equally performing models when ``C`` becomes very large. This suggests that
the set of support vectors does not change anymore. The radius of the RBF
kernel alone acts as a good structural regularizer. Increasing ``C`` further
doesn't help, likely because there are no more training points in violation
(inside the margin or wrongly classified), or at least no better solution can
be found. Scores being equal, it may make sense to use the smaller ``C``
values, since very high ``C`` values typically increase fitting time.
On the other hand, lower ``C`` values generally lead to more support vectors,
which may increase prediction time. Therefore, lowering the value of ``C``
involves a trade-off between fitting time and prediction time.
We should also note that small differences in scores results from the random
splits of the cross-validation procedure. Those spurious variations can be
smoothed out by increasing the number of CV iterations ``n_splits`` at the
expense of compute time. Increasing the value number of ``C_range`` and
``gamma_range`` steps will increase the resolution of the hyper-parameter heat
map.
.. GENERATED FROM PYTHON SOURCE LINES 79-81
Utility class to move the midpoint of a colormap to be around
the values of interest.
.. GENERATED FROM PYTHON SOURCE LINES 81-96
.. code-block:: default
import numpy as np
from matplotlib.colors import Normalize
class MidpointNormalize(Normalize):
def __init__(self, vmin=None, vmax=None, midpoint=None, clip=False):
self.midpoint = midpoint
Normalize.__init__(self, vmin, vmax, clip)
def __call__(self, value, clip=None):
x, y = [self.vmin, self.midpoint, self.vmax], [0, 0.5, 1]
return np.ma.masked_array(np.interp(value, x, y))
.. GENERATED FROM PYTHON SOURCE LINES 97-101
Load and prepare data set
-------------------------
dataset for grid search
.. GENERATED FROM PYTHON SOURCE LINES 101-108
.. code-block:: default
from sklearn.datasets import load_iris
iris = load_iris()
X = iris.data
y = iris.target
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Dataset for decision function visualization: we only keep the first two
features in X and sub-sample the dataset to keep only 2 classes and
make it a binary classification problem.
.. GENERATED FROM PYTHON SOURCE LINES 112-118
.. code-block:: default
X_2d = X[:, :2]
X_2d = X_2d[y > 0]
y_2d = y[y > 0]
y_2d -= 1
.. GENERATED FROM PYTHON SOURCE LINES 119-123
It is usually a good idea to scale the data for SVM training.
We are cheating a bit in this example in scaling all of the data,
instead of fitting the transformation on the training set and
just applying it on the test set.
.. GENERATED FROM PYTHON SOURCE LINES 123-130
.. code-block:: default
from sklearn.preprocessing import StandardScaler
scaler = StandardScaler()
X = scaler.fit_transform(X)
X_2d = scaler.fit_transform(X_2d)
.. GENERATED FROM PYTHON SOURCE LINES 131-137
Train classifiers
-----------------
For an initial search, a logarithmic grid with basis
10 is often helpful. Using a basis of 2, a finer
tuning can be achieved but at a much higher cost.
.. GENERATED FROM PYTHON SOURCE LINES 137-154
.. code-block:: default
from sklearn.svm import SVC
from sklearn.model_selection import StratifiedShuffleSplit
from sklearn.model_selection import GridSearchCV
C_range = np.logspace(-2, 10, 13)
gamma_range = np.logspace(-9, 3, 13)
param_grid = dict(gamma=gamma_range, C=C_range)
cv = StratifiedShuffleSplit(n_splits=5, test_size=0.2, random_state=42)
grid = GridSearchCV(SVC(), param_grid=param_grid, cv=cv)
grid.fit(X, y)
print(
"The best parameters are %s with a score of %0.2f"
% (grid.best_params_, grid.best_score_)
)
.. rst-class:: sphx-glr-script-out
Out:
.. code-block:: none
The best parameters are {'C': 1.0, 'gamma': 0.1} with a score of 0.97
.. GENERATED FROM PYTHON SOURCE LINES 155-157
Now we need to fit a classifier for all parameters in the 2d version
(we use a smaller set of parameters here because it takes a while to train)
.. GENERATED FROM PYTHON SOURCE LINES 157-167
.. code-block:: default
C_2d_range = [1e-2, 1, 1e2]
gamma_2d_range = [1e-1, 1, 1e1]
classifiers = []
for C in C_2d_range:
for gamma in gamma_2d_range:
clf = SVC(C=C, gamma=gamma)
clf.fit(X_2d, y_2d)
classifiers.append((C, gamma, clf))
.. GENERATED FROM PYTHON SOURCE LINES 168-172
Visualization
-------------
draw visualization of parameter effects
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.. code-block:: default
import matplotlib.pyplot as plt
plt.figure(figsize=(8, 6))
xx, yy = np.meshgrid(np.linspace(-3, 3, 200), np.linspace(-3, 3, 200))
for k, (C, gamma, clf) in enumerate(classifiers):
# evaluate decision function in a grid
Z = clf.decision_function(np.c_[xx.ravel(), yy.ravel()])
Z = Z.reshape(xx.shape)
# visualize decision function for these parameters
plt.subplot(len(C_2d_range), len(gamma_2d_range), k + 1)
plt.title("gamma=10^%d, C=10^%d" % (np.log10(gamma), np.log10(C)), size="medium")
# visualize parameter's effect on decision function
plt.pcolormesh(xx, yy, -Z, cmap=plt.cm.RdBu)
plt.scatter(X_2d[:, 0], X_2d[:, 1], c=y_2d, cmap=plt.cm.RdBu_r, edgecolors="k")
plt.xticks(())
plt.yticks(())
plt.axis("tight")
scores = grid.cv_results_["mean_test_score"].reshape(len(C_range), len(gamma_range))
.. image-sg:: /auto_examples/svm/images/sphx_glr_plot_rbf_parameters_001.png
:alt: gamma=10^-1, C=10^-2, gamma=10^0, C=10^-2, gamma=10^1, C=10^-2, gamma=10^-1, C=10^0, gamma=10^0, C=10^0, gamma=10^1, C=10^0, gamma=10^-1, C=10^2, gamma=10^0, C=10^2, gamma=10^1, C=10^2
:srcset: /auto_examples/svm/images/sphx_glr_plot_rbf_parameters_001.png
:class: sphx-glr-single-img
.. GENERATED FROM PYTHON SOURCE LINES 196-204
Draw heatmap of the validation accuracy as a function of gamma and C
The score are encoded as colors with the hot colormap which varies from dark
red to bright yellow. As the most interesting scores are all located in the
0.92 to 0.97 range we use a custom normalizer to set the mid-point to 0.92 so
as to make it easier to visualize the small variations of score values in the
interesting range while not brutally collapsing all the low score values to
the same color.
.. GENERATED FROM PYTHON SOURCE LINES 204-220
.. code-block:: default
plt.figure(figsize=(8, 6))
plt.subplots_adjust(left=0.2, right=0.95, bottom=0.15, top=0.95)
plt.imshow(
scores,
interpolation="nearest",
cmap=plt.cm.hot,
norm=MidpointNormalize(vmin=0.2, midpoint=0.92),
)
plt.xlabel("gamma")
plt.ylabel("C")
plt.colorbar()
plt.xticks(np.arange(len(gamma_range)), gamma_range, rotation=45)
plt.yticks(np.arange(len(C_range)), C_range)
plt.title("Validation accuracy")
plt.show()
.. image-sg:: /auto_examples/svm/images/sphx_glr_plot_rbf_parameters_002.png
:alt: Validation accuracy
:srcset: /auto_examples/svm/images/sphx_glr_plot_rbf_parameters_002.png
:class: sphx-glr-single-img
.. rst-class:: sphx-glr-timing
**Total running time of the script:** ( 0 minutes 4.712 seconds)
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