SmartSVM

SmartSVM is a Python package that implements the methods from Fast Meta-Learning for Adaptive Hierarchical Classifier Design by Gerrit J.J. van den Burg and Alfred O. Hero. The package contains functions for estimating the Bayes error rate (BER) using the Henze-Penrose divergence and a hierarchical classifier called SmartSVM. See the Usage documentation below for more details.

Installation

SmartSVM is available on PyPI and can be installed easily with:

pip install smartsvm

Usage

In the paper the main focus is on the accurate Bayes error estimates and the hierarchical classifier SmartSVM. These will therefore be of most interest to users of the SmartSVM package. Below we briefly explain how to use these functions.

Citing

If you use this package in your research, please cite the paper using the following BibTex entry:

@article{van2017fast,
  title={Fast Meta-Learning for Adaptive Hierarchical Classifier Design},
  author={Gerrit J.J. van den Burg and Alfred O. Hero},
  journal={arXiv preprint arXiv:1711.03512},
  archiveprefix={arXiv},
  year={2017},
  eprint={1711.03512},
  url={https://arxiv.org/abs/1711.03512},
  primaryclass={cs.LG}
}

Bayes error estimates

Error estimation is implemented three functions:

• hp_estimate for the Henze-Penrose estimator of the Bayes error rate. This can be used as:

  >>> import numpy as np
  >>> from smartsvm import hp_estimate
  >>> X1 = np.random.multivariate_normal([-1, 0], [[1, 0], [0, 1]], 100)
  >>> X2 = np.random.multivariate_normal([1, 0], [[1, 0], [0, 1]], 100)
  >>> hp_estimate(X1, X2) # with normalization
  >>> hp_estimate(X1, X2, normalize=False) # without normalization

• compute_error_graph and compute_ovr_error respectively compute the complete weighted graph of pairwise BER estimates or the One-vs-Rest BER for each class. They have a similar interface:

  >>> import numpy as np
  >>> from smartsvm import compute_error_graph, compute_ovr_error
  >>> from sklearn.datasets import load_digits
  >>> digits = load_digits(5)
  >>> n_samples = len(digits.images)
  >>> X = digits.images.reshape((n_samples, -1))
  >>> y = digits.target
  >>> G = compute_error_graph(X, y, n_jobs=2, normalize=True)
  >>> d = compute_ovr_error(X, y, normalize=True)

SmartSVM Classifier

SmartSVM is an adaptive hierarchical classifier which constructs a classification hierarchy based on the Henze-Penrose estimates of the Bayes error between each pair of classes. The classifier is build on top of Scikit-Learn and can be used in the exact same way as other sklearn classifiers:

>>> import numpy as np
>>> from smartsvm import SmartSVM
>>> from sklearn.datasets import load_digits
>>> digits = load_digits(10)
>>> n_samples = len(digits.images)
>>> X = digits.images.reshape((n_samples, -1))
>>> y = digits.target
>>> clf = SmartSVM()
>>> clf.fit(X, y)
>>> clf.predict(X)

By default, the SmartSVM classifier uses the Linear Support Vector Machine (LinearSVC) as the underlying binary classifier for each binary subproblem in the hierarchy. This can easily be changed with the binary_clf parameter to the class constructor, for instance:

>>> from sklearn.tree import DecisionTreeClassifier
>>> clf = SmartSVM(binary_clf=DecisionTreeClassifier)
>>> clf.fit(X, y)
>>> clf._get_binary()
DecisionTreeClassifier(class_weight=None, criterion='gini',
        max_depth=None, max_features=None, max_leaf_nodes=None,
        min_impurity_decrease=0.0, min_impurity_split=None,
        min_samples_leaf=1, min_samples_split=2,
        min_weight_fraction_leaf=0.0, presort=False, random_state=None,
        splitter='best')

You may optionally add parameters for the classifier through the clf_params parameter. This should be a dict with the parameters to the binary classifier, as follows:

>>> clf = SmartSVM(binary_clf=DecisionTreeClassifier, clf_params={'criterion': 'entropy'})
>>> clf.fit(X, y)
>>> clf._get_binary()
DecisionTreeClassifier(class_weight=None, criterion='entropy',
        max_depth=None, max_features=None, max_leaf_nodes=None,
        min_impurity_decrease=0.0, min_impurity_split=None,
        min_samples_leaf=1, min_samples_split=2,
        min_weight_fraction_leaf=0.0, presort=False, random_state=None,
        splitter='best')

Finally, it's possible to retrieve probability estimates for the classes if the underlying classifier supports the predict_proba method:

>>> from sklearn.svm import SVC
>>> clf = SmartSVM(binary_clf=SVC, clf_params={"probability": True})
>>> clf.fit(X, y)
>>> prob = clf.predict_proba(X)
>>> import pandas as pd
>>> df = pd.DataFrame(prob)
>>> df
                 0             1             2      ...
0     9.999997e-01  1.716831e-18  2.677824e-13      ...
1     1.000000e-07  9.956408e-01  1.035589e-09      ...
2     2.595652e-05  1.452011e-02  9.722321e-01      ...

For more information about parameters to SmartSVM, see the API documentation here.

Known Limitations

The Henze-Penrose estimator of the Bayes error rate is based on construction of the Euclidean minimal spanning tree. The current algorithm for this in the SmartSVM package uses an adaptation of Whitney's algorithm. This is not the fastest way to construct a minimal spanning tree. The Fast Euclidean Minimal Spanning Tree algorithm by March et al., would be a faster option but this makes it more difficult to construct orthogonal MSTs. Incorporating this algorithm into the SmartSVM package is considered a topic for future work.

References

The main reference for this package is:

• G.J.J. van den Burg and A.O. Hero - Fast Meta-Learning for Adaptive Hierarchical Classifier Design (2017).

The theory of the Henze-Penrose estimator is developed in:

• V. Berisha, A. Wisler, A.O. Hero, A. Spanias - Empirically Estimable Classification Bounds Based on a Nonparametric Divergence Measure (2016).
• V. Berisha, A.O. Hero - Empirical Non-Parametric Estimation of the Fisher Information (2015).