Deployment & Documentation & Stats
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Build Status & Coverage & Maintainability & License
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PyOD is a comprehensive and scalable Python toolkit for detecting outlying objects in
multivariate data. This exciting yet challenging field is commonly referred as
Outlier Detection <https://en.wikipedia.org/wiki/Anomaly_detection>
_
or Anomaly Detection <https://en.wikipedia.org/wiki/Anomaly_detection>
_.
PyOD includes more than 30 detection algorithms, from classical LOF (SIGMOD 2000) to
the latest COPOD (ICDM 2020). Since 2017, PyOD has been successfully used in numerous academic researches and
commercial products [#Gopalan2019PIDForest]_ [#Li2019MADGAN]_ [#Wang2020adVAE]_ [#Zhao2019LSCP].
It is also well acknowledged by the machine learning community with various dedicated posts/tutorials, including
Analytics Vidhya <https://www.analyticsvidhya.com/blog/2019/02/outlier-detection-python-pyod/>
,
KDnuggets <https://www.kdnuggets.com/2019/02/outlier-detection-methods-cheat-sheet.html>
,
Towards Data Science <https://towardsdatascience.com/anomaly-detection-for-dummies-15f148e559c1>
,
Computer Vision News <https://rsipvision.com/ComputerVisionNews-2019March/18/>
, and
awesome-machine-learning <https://github.com/josephmisiti/awesome-machine-learning#python-general-purpose>
.
PyOD is featured for:
numba <https://github.com/numba/numba>
_ and joblib <https://github.com/joblib/joblib>
_.API Demo\ :
.. code-block:: python
# train the COPOD detector
from pyod.models.copod import COPOD
clf = COPOD()
clf.fit(X_train)
# get outlier scores
y_train_scores = clf.decision_scores_ # raw outlier scores
y_test_scores = clf.decision_function(X_test) # outlier scores
Citing PyOD\ :
PyOD paper <http://www.jmlr.org/papers/volume20/19-011/19-011.pdf>
_ is published in
JMLR <http://www.jmlr.org/>
_ (machine learning open-source software track).
If you use PyOD in a scientific publication, we would appreciate
citations to the following paper::
@article{zhao2019pyod,
author = {Zhao, Yue and Nasrullah, Zain and Li, Zheng},
title = {PyOD: A Python Toolbox for Scalable Outlier Detection},
journal = {Journal of Machine Learning Research},
year = {2019},
volume = {20},
number = {96},
pages = {1-7},
url = {http://jmlr.org/papers/v20/19-011.html}
}
or::
Zhao, Y., Nasrullah, Z. and Li, Z., 2019. PyOD: A Python Toolbox for Scalable Outlier Detection. Journal of machine learning research (JMLR), 20(96), pp.1-7.
Key Links and Resources\ :
View the latest codes on Github <https://github.com/yzhao062/pyod>
_Execute Interactive Jupyter Notebooks <https://mybinder.org/v2/gh/yzhao062/pyod/master>
_Anomaly Detection Resources <https://github.com/yzhao062/anomaly-detection-resources>
_Table of Contents\ :
Installation <#installation>
_API Cheatsheet & Reference <#api-cheatsheet--reference>
_Model Save & Load <#model-save--load>
_Implemented Algorithms <#implemented-algorithms>
_Algorithm Benchmark <#algorithm-benchmark>
_Quick Start for Outlier Detection <#quick-start-for-outlier-detection>
_Quick Start for Combining Outlier Scores from Various Base Detectors <#quick-start-for-combining-outlier-scores-from-various-base-detectors>
_How to Contribute <#how-to-contribute>
_Inclusion Criteria <#inclusion-criteria>
_Installation ^^^^^^^^^^^^
It is recommended to use pip for installation. Please make sure the latest version is installed, as PyOD is updated frequently:
.. code-block:: bash
pip install pyod # normal install pip install --upgrade pyod # or update if needed pip install --pre pyod # or include pre-release version for new features
Alternatively, you could clone and run setup.py file:
.. code-block:: bash
git clone https://github.com/yzhao062/pyod.git cd pyod pip install .
Note on Python 2.7\ :
The maintenance of Python 2.7 will be stopped by January 1, 2020 (see official announcement <https://github.com/python/devguide/pull/344>
)
To be consistent with the Python change and PyOD's dependent libraries, e.g., scikit-learn, we will
stop supporting Python 2.7 in the near future (dates are still to be decided). We encourage you to use
Python 3.5 or newer for the latest functions and bug fixes. More information can
be found at Moving to require Python 3 <https://python3statement.org/>
.
Required Dependencies\ :
Optional Dependencies (see details below)\ :
Warning 1\ :
PyOD has multiple neural network based models, e.g., AutoEncoders, which are
implemented in both PyTorch and Tensorflow. However, PyOD does NOT install DL libraries for you.
This reduces the risk of interfering with your local copies.
If you want to use neural-net based models, please make sure Keras and a backend library, e.g., TensorFlow, are installed.
Instructions are provided: neural-net FAQ <https://github.com/yzhao062/pyod/wiki/Setting-up-Keras-and-Tensorflow-for-Neural-net-Based-models>
_.
Similarly, models depending on xgboost, e.g., XGBOD, would NOT enforce xgboost installation by default.
Warning 2\ :
Running examples needs matplotlib, which may throw errors in conda
virtual environment on mac OS. See reasons and solutions mac_matplotlib <https://github.com/yzhao062/pyod/issues/6>
_.
Warning 3\ :
PyOD contains multiple models that also exist in scikit-learn. However, these two
libraries' API is not exactly the same--it is recommended to use only one of them
for consistency but not mix the results. Refer Differences between sckit-learn and PyOD <https://pyod.readthedocs.io/en/latest/issues.html>
_
for more information.
API Cheatsheet & Reference ^^^^^^^^^^^^^^^^^^^^^^^^^^
Full API Reference: (https://pyod.readthedocs.io/en/latest/pyod.html). API cheatsheet for all detectors:
Key Attributes of a fitted model:
Note \ : fit_predict() and fit_predict_score() are deprecated in V0.6.9 due to consistency issue and will be removed in V0.8.0. To get the binary labels of the training data X_train, one should call clf.fit(X_train) and use clf.labels_, instead of calling clf.predict(X_train).
Fast training and prediction: it is possible to train and predict with
a large number of detection models in PyOD by leveraging SUOD framework [#Zhao2021SUOD].
See SUOD Paper <https://www.andrew.cmu.edu/user/yuezhao2/papers/21-mlsys-suod.pdf>
and repository <https://github.com/yzhao062/SUOD>
_.
Model Save & Load ^^^^^^^^^^^^^^^^^
PyOD takes a similar approach of sklearn regarding model persistence.
See model persistence <https://scikit-learn.org/stable/modules/model_persistence.html>
_ for clarification.
In short, we recommend to use joblib or pickle for saving and loading PyOD models.
See "examples/save_load_model_example.py" <https://github.com/yzhao062/pyod/blob/master/examples/save_load_model_example.py>
_ for an example.
In short, it is simple as below:
.. code-block:: python
from joblib import dump, load
# save the model
dump(clf, 'clf.joblib')
# load the model
clf = load('clf.joblib')
Implemented Algorithms ^^^^^^^^^^^^^^^^^^^^^^
PyOD toolkit consists of three major functional groups:
(i) Individual Detection Algorithms :
=================== ================ ====================================================================================================== ===== ======================================== Type Abbr Algorithm Year Ref =================== ================ ====================================================================================================== ===== ======================================== Linear Model PCA Principal Component Analysis (the sum of weighted projected distances to the eigenvector hyperplanes) 2003 [#Shyu2003A]_ Linear Model MCD Minimum Covariance Determinant (use the mahalanobis distances as the outlier scores) 1999 [#Hardin2004Outlier]_ [#Rousseeuw1999A]_ Linear Model OCSVM One-Class Support Vector Machines 2001 [#Scholkopf2001Estimating]_ Linear Model LMDD Deviation-based Outlier Detection (LMDD) 1996 [#Arning1996A]_ Proximity-Based LOF Local Outlier Factor 2000 [#Breunig2000LOF]_ Proximity-Based COF Connectivity-Based Outlier Factor 2002 [#Tang2002Enhancing]_ Proximity-Based CBLOF Clustering-Based Local Outlier Factor 2003 [#He2003Discovering]_ Proximity-Based LOCI LOCI: Fast outlier detection using the local correlation integral 2003 [#Papadimitriou2003LOCI]_ Proximity-Based HBOS Histogram-based Outlier Score 2012 [#Goldstein2012Histogram]_ Proximity-Based kNN k Nearest Neighbors (use the distance to the kth nearest neighbor as the outlier score) 2000 [#Ramaswamy2000Efficient]_ Proximity-Based AvgKNN Average kNN (use the average distance to k nearest neighbors as the outlier score) 2002 [#Angiulli2002Fast]_ Proximity-Based MedKNN Median kNN (use the median distance to k nearest neighbors as the outlier score) 2002 [#Angiulli2002Fast]_ Proximity-Based SOD Subspace Outlier Detection 2009 [#Kriegel2009Outlier]_ Proximity-Based ROD Rotation-based Outlier Detection 2020 [#Almardeny2020A]_ Probabilistic ABOD Angle-Based Outlier Detection 2008 [#Kriegel2008Angle]_ Probabilistic COPOD COPOD: Copula-Based Outlier Detection 2020 [#Li2020COPOD]_ Probabilistic FastABOD Fast Angle-Based Outlier Detection using approximation 2008 [#Kriegel2008Angle]_ Probabilistic MAD Median Absolute Deviation (MAD) 1993 [#Iglewicz1993How]_ Probabilistic SOS Stochastic Outlier Selection 2012 [#Janssens2012Stochastic]_ Outlier Ensembles IForest Isolation Forest 2008 [#Liu2008Isolation]_ Outlier Ensembles Feature Bagging 2005 [#Lazarevic2005Feature]_ Outlier Ensembles LSCP LSCP: Locally Selective Combination of Parallel Outlier Ensembles 2019 [#Zhao2019LSCP]_ Outlier Ensembles XGBOD Extreme Boosting Based Outlier Detection (Supervised) 2018 [#Zhao2018XGBOD]_ Outlier Ensembles LODA Lightweight On-line Detector of Anomalies 2016 [#Pevny2016Loda]_ Neural Networks AutoEncoder Fully connected AutoEncoder (use reconstruction error as the outlier score) [#Aggarwal2015Outlier]_ [Ch.3] Neural Networks VAE Variational AutoEncoder (use reconstruction error as the outlier score) 2013 [#Kingma2013Auto]_ Neural Networks Beta-VAE Variational AutoEncoder (all customized loss term by varying gamma and capacity) 2018 [#Burgess2018Understanding]_ Neural Networks SO_GAAL Single-Objective Generative Adversarial Active Learning 2019 [#Liu2019Generative]_ Neural Networks MO_GAAL Multiple-Objective Generative Adversarial Active Learning 2019 [#Liu2019Generative]_ =================== ================ ====================================================================================================== ===== ========================================
(ii) Outlier Ensembles & Outlier Detector Combination Frameworks:
=================== ================ ===================================================================================================== ===== ======================================== Type Abbr Algorithm Year Ref =================== ================ ===================================================================================================== ===== ======================================== Outlier Ensembles Feature Bagging 2005 [#Lazarevic2005Feature]_ Outlier Ensembles LSCP LSCP: Locally Selective Combination of Parallel Outlier Ensembles 2019 [#Zhao2019LSCP]_ Outlier Ensembles XGBOD Extreme Boosting Based Outlier Detection (Supervised) 2018 [#Zhao2018XGBOD]_ Outlier Ensembles LODA Lightweight On-line Detector of Anomalies 2016 [#Pevny2016Loda]_ Combination Average Simple combination by averaging the scores 2015 [#Aggarwal2015Theoretical]_ Combination Weighted Average Simple combination by averaging the scores with detector weights 2015 [#Aggarwal2015Theoretical]_ Combination Maximization Simple combination by taking the maximum scores 2015 [#Aggarwal2015Theoretical]_ Combination AOM Average of Maximum 2015 [#Aggarwal2015Theoretical]_ Combination MOA Maximization of Average 2015 [#Aggarwal2015Theoretical]_ Combination Median Simple combination by taking the median of the scores 2015 [#Aggarwal2015Theoretical]_ Combination majority Vote Simple combination by taking the majority vote of the labels (weights can be used) 2015 [#Aggarwal2015Theoretical]_ =================== ================ ===================================================================================================== ===== ========================================
(iii) Utility Functions:
=================== ====================== ===================================================================================================================================================== ======================================================================================================================================
Type Name Function Documentation
=================== ====================== ===================================================================================================================================================== ======================================================================================================================================
Data generate_data Synthesized data generation; normal data is generated by a multivariate Gaussian and outliers are generated by a uniform distribution generate_data <https://pyod.readthedocs.io/en/latest/pyod.utils.html#module-pyod.utils.data.generate_data>
_
Data generate_data_clusters Synthesized data generation in clusters; more complex data patterns can be created with multiple clusters generate_data_clusters <https://pyod.readthedocs.io/en/latest/pyod.utils.html#pyod.utils.data.generate_data_clusters>
_
Stat wpearsonr Calculate the weighted Pearson correlation of two samples wpearsonr <https://pyod.readthedocs.io/en/latest/pyod.utils.html#module-pyod.utils.stat_models.wpearsonr>
_
Utility get_label_n Turn raw outlier scores into binary labels by assign 1 to top n outlier scores get_label_n <https://pyod.readthedocs.io/en/latest/pyod.utils.html#module-pyod.utils.utility.get_label_n>
_
Utility precision_n_scores calculate precision @ rank n precision_n_scores <https://pyod.readthedocs.io/en/latest/pyod.utils.html#module-pyod.utils.utility.precision_n_scores>
_
=================== ====================== ===================================================================================================================================================== ======================================================================================================================================
Algorithm Benchmark ^^^^^^^^^^^^^^^^^^^
The comparison among of implemented models is made available below
(\ Figure <https://raw.githubusercontent.com/yzhao062/pyod/master/examples/ALL.png>
\ ,
compare_all_models.py <https://github.com/yzhao062/pyod/blob/master/examples/compare_all_models.py>
\ ,
Interactive Jupyter Notebooks <https://mybinder.org/v2/gh/yzhao062/pyod/master>
_\ ).
For Jupyter Notebooks, please navigate to "/notebooks/Compare All Models.ipynb".
.. image:: https://raw.githubusercontent.com/yzhao062/pyod/master/examples/ALL.png :target: https://raw.githubusercontent.com/yzhao062/pyod/master/examples/ALL.png :alt: Comparision_of_All
A benchmark is supplied for select algorithms to provide an overview of the implemented models.
In total, 17 benchmark datasets are used for comparison, which
can be downloaded at ODDS <http://odds.cs.stonybrook.edu/#table1>
_.
For each dataset, it is first split into 60% for training and 40% for testing. All experiments are repeated 10 times independently with random splits. The mean of 10 trials is regarded as the final result. Three evaluation metrics are provided:
Check the latest benchmark <https://pyod.readthedocs.io/en/latest/benchmark.html>
. You could replicate this process by running
benchmark.py <https://github.com/yzhao062/pyod/blob/master/notebooks/benchmark.py>
.
Quick Start for Outlier Detection ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
PyOD has been well acknowledged by the machine learning community with a few featured posts and tutorials.
Analytics Vidhya: An Awesome Tutorial to Learn Outlier Detection in Python using PyOD Library <https://www.analyticsvidhya.com/blog/2019/02/outlier-detection-python-pyod/>
_
KDnuggets: Intuitive Visualization of Outlier Detection Methods <https://www.kdnuggets.com/2019/02/outlier-detection-methods-cheat-sheet.html>
, An Overview of Outlier Detection Methods from PyOD <https://www.kdnuggets.com/2019/06/overview-outlier-detection-methods-pyod.html>
Towards Data Science: Anomaly Detection for Dummies <https://towardsdatascience.com/anomaly-detection-for-dummies-15f148e559c1>
_
Computer Vision News (March 2019): Python Open Source Toolbox for Outlier Detection <https://rsipvision.com/ComputerVisionNews-2019March/18/>
_
"examples/knn_example.py" <https://github.com/yzhao062/pyod/blob/master/examples/knn_example.py>
_
demonstrates the basic API of using kNN detector. It is noted that the API across all other algorithms are consistent/similar.
More detailed instructions for running examples can be found in examples directory <https://github.com/yzhao062/pyod/blob/master/examples>
_.
#. Initialize a kNN detector, fit the model, and make the prediction.
.. code-block:: python
from pyod.models.knn import KNN # kNN detector
# train kNN detector
clf_name = 'KNN'
clf = KNN()
clf.fit(X_train)
# get the prediction label and outlier scores of the training data
y_train_pred = clf.labels_ # binary labels (0: inliers, 1: outliers)
y_train_scores = clf.decision_scores_ # raw outlier scores
# get the prediction on the test data
y_test_pred = clf.predict(X_test) # outlier labels (0 or 1)
y_test_scores = clf.decision_function(X_test) # outlier scores
#. Evaluate the prediction by ROC and Precision @ Rank n ([email protected]).
.. code-block:: python
from pyod.utils.data import evaluate_print
# evaluate and print the results
print("\nOn Training Data:")
evaluate_print(clf_name, y_train, y_train_scores)
print("\nOn Test Data:")
evaluate_print(clf_name, y_test, y_test_scores)
#. See a sample output & visualization.
.. code-block:: python
On Training Data:
KNN ROC:1.0, precision @ rank n:1.0
On Test Data:
KNN ROC:0.9989, precision @ rank n:0.9
.. code-block:: python
visualize(clf_name, X_train, y_train, X_test, y_test, y_train_pred,
y_test_pred, show_figure=True, save_figure=False)
Visualization (\ knn_figure <https://raw.githubusercontent.com/yzhao062/pyod/master/examples/KNN.png>
_\ ):
.. image:: https://raw.githubusercontent.com/yzhao062/pyod/master/examples/KNN.png :target: https://raw.githubusercontent.com/yzhao062/pyod/master/examples/KNN.png :alt: kNN example figure
Quick Start for Combining Outlier Scores from Various Base Detectors ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Outlier detection often suffers from model instability due to its unsupervised nature. Thus, it is recommended to combine various detector outputs, e.g., by averaging, to improve its robustness. Detector combination is a subfield of outlier ensembles; refer [#Aggarwal2017Outlier]_ for more information.
Four score combination mechanisms are shown in this demo:
#. Average: average scores of all detectors. #. maximization: maximum score across all detectors. #. Average of Maximum (AOM): divide base detectors into subgroups and take the maximum score for each subgroup. The final score is the average of all subgroup scores. #. Maximum of Average (MOA): divide base detectors into subgroups and take the average score for each subgroup. The final score is the maximum of all subgroup scores.
"examples/comb_example.py" illustrates the API for combining the output of multiple base detectors
(\ comb_example.py <https://github.com/yzhao062/pyod/blob/master/examples/comb_example.py>
\ ,
Jupyter Notebooks <https://mybinder.org/v2/gh/yzhao062/pyod/master>
\ ). For Jupyter Notebooks,
please navigate to "/notebooks/Model Combination.ipynb"
#. Import models and generate sample data.
.. code-block:: python
from pyod.models.knn import KNN
from pyod.models.combination import aom, moa, average, maximization
from pyod.utils.data import generate_data
X, y = generate_data(train_only=True) # load data
#. First initialize 20 kNN outlier detectors with different k (10 to 200), and get the outlier scores.
.. code-block:: python
# initialize 20 base detectors for combination
k_list = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, 130, 140,
150, 160, 170, 180, 190, 200]
train_scores = np.zeros([X_train.shape[0], n_clf])
test_scores = np.zeros([X_test.shape[0], n_clf])
for i in range(n_clf):
k = k_list[i]
clf = KNN(n_neighbors=k, method='largest')
clf.fit(X_train_norm)
train_scores[:, i] = clf.decision_scores_
test_scores[:, i] = clf.decision_function(X_test_norm)
#. Then the output scores are standardized into zero mean and unit variance before combination. This step is crucial to adjust the detector outputs to the same scale.
.. code-block:: python
from pyod.utils.utility import standardizer
train_scores_norm, test_scores_norm = standardizer(train_scores, test_scores)
#. Then four different combination algorithms are applied as described above.
.. code-block:: python
comb_by_average = average(test_scores_norm)
comb_by_maximization = maximization(test_scores_norm)
comb_by_aom = aom(test_scores_norm, 5) # 5 groups
comb_by_moa = moa(test_scores_norm, 5)) # 5 groups
#. Finally, all four combination methods are evaluated with ROC and Precision @ Rank n.
.. code-block:: bash
Combining 20 kNN detectors
Combination by Average ROC:0.9194, precision @ rank n:0.4531
Combination by Maximization ROC:0.9198, precision @ rank n:0.4688
Combination by AOM ROC:0.9257, precision @ rank n:0.4844
Combination by MOA ROC:0.9263, precision @ rank n:0.4688
How to Contribute ^^^^^^^^^^^^^^^^^
You are welcome to contribute to this exciting project:
Please first check Issue lists for "help wanted" tag and comment the one you are interested. We will assign the issue to you.
Fork the master branch and add your improvement/modification/fix.
Create a pull request to development branch and follow the pull request template PR template <https://github.com/yzhao062/pyod/blob/master/PULL_REQUEST_TEMPLATE.md>
_
Automatic tests will be triggered. Make sure all tests are passed. Please make sure all added modules are accompanied with proper test functions.
To make sure the code has the same style and standard, please refer to abod.py, hbos.py, or feature_bagging.py for example.
You are also welcome to share your ideas by opening an issue or dropping me an email at [email protected] :)
Inclusion Criteria ^^^^^^^^^^^^^^^^^^
Similarly to scikit-learn <https://scikit-learn.org/stable/faq.html#what-are-the-inclusion-criteria-for-new-algorithms>
_,
We mainly consider well-established algorithms for inclusion.
A rule of thumb is at least two years since publication, 50+ citations, and usefulness.
However, we encourage the author(s) of newly proposed models to share and add your implementation into PyOD for boosting ML accessibility and reproducibility. This exception only applies if you could commit to the maintenance of your model for at least two year period.
Reference ^^^^^^^^^
.. [#Aggarwal2015Outlier] Aggarwal, C.C., 2015. Outlier analysis. In Data mining (pp. 237-263). Springer, Cham.
.. [#Aggarwal2015Theoretical] Aggarwal, C.C. and Sathe, S., 2015. Theoretical foundations and algorithms for outlier ensembles.\ ACM SIGKDD Explorations Newsletter\ , 17(1), pp.24-47.
.. [#Aggarwal2017Outlier] Aggarwal, C.C. and Sathe, S., 2017. Outlier ensembles: An introduction. Springer.
.. [#Almardeny2020A] Almardeny, Y., Boujnah, N. and Cleary, F., 2020. A Novel Outlier Detection Method for Multivariate Data. IEEE Transactions on Knowledge and Data Engineering.
.. [#Angiulli2002Fast] Angiulli, F. and Pizzuti, C., 2002, August. Fast outlier detection in high dimensional spaces. In European Conference on Principles of Data Mining and Knowledge Discovery pp. 15-27.
.. [#Arning1996A] Arning, A., Agrawal, R. and Raghavan, P., 1996, August. A Linear Method for Deviation Detection in Large Databases. In KDD (Vol. 1141, No. 50, pp. 972-981).
.. [#Breunig2000LOF] Breunig, M.M., Kriegel, H.P., Ng, R.T. and Sander, J., 2000, May. LOF: identifying density-based local outliers. ACM Sigmod Record\ , 29(2), pp. 93-104.
.. [#Burgess2018Understanding] Burgess, Christopher P., et al. "Understanding disentangling in beta-VAE." arXiv preprint arXiv:1804.03599 (2018).
.. [#Goldstein2012Histogram] Goldstein, M. and Dengel, A., 2012. Histogram-based outlier score (hbos): A fast unsupervised anomaly detection algorithm. In KI-2012: Poster and Demo Track\ , pp.59-63.
.. [#Gopalan2019PIDForest] Gopalan, P., Sharan, V. and Wieder, U., 2019. PIDForest: Anomaly Detection via Partial Identification. In Advances in Neural Information Processing Systems, pp. 15783-15793.
.. [#Hardin2004Outlier] Hardin, J. and Rocke, D.M., 2004. Outlier detection in the multiple cluster setting using the minimum covariance determinant estimator. Computational Statistics & Data Analysis\ , 44(4), pp.625-638.
.. [#He2003Discovering] He, Z., Xu, X. and Deng, S., 2003. Discovering cluster-based local outliers. Pattern Recognition Letters\ , 24(9-10), pp.1641-1650.
.. [#Iglewicz1993How] Iglewicz, B. and Hoaglin, D.C., 1993. How to detect and handle outliers (Vol. 16). Asq Press.
.. [#Janssens2012Stochastic] Janssens, J.H.M., Huszár, F., Postma, E.O. and van den Herik, H.J., 2012. Stochastic outlier selection. Technical report TiCC TR 2012-001, Tilburg University, Tilburg Center for Cognition and Communication, Tilburg, The Netherlands.
.. [#Kingma2013Auto] Kingma, D.P. and Welling, M., 2013. Auto-encoding variational bayes. arXiv preprint arXiv:1312.6114.
.. [#Kriegel2008Angle] Kriegel, H.P. and Zimek, A., 2008, August. Angle-based outlier detection in high-dimensional data. In KDD '08\ , pp. 444-452. ACM.
.. [#Kriegel2009Outlier] Kriegel, H.P., Kröger, P., Schubert, E. and Zimek, A., 2009, April. Outlier detection in axis-parallel subspaces of high dimensional data. In Pacific-Asia Conference on Knowledge Discovery and Data Mining\ , pp. 831-838. Springer, Berlin, Heidelberg.
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