Welcome to PyOD, a comprehensive but easy-to-use Python library for detecting anomalies in multivariate data. Whether you're tackling a small-scale project or large datasets, PyOD offers a range of algorithms to suit your needs.

For time-series outlier detection, please use TODS <https://github.com/datamllab/tods>_.
For graph outlier detection, please use PyGOD <https://pygod.org/>_.
Performance Comparison & Datasets: We have a 45-page, comprehensive anomaly detection benchmark paper <https://openreview.net/forum?id=foA_SFQ9zo0>. The fully open-sourced ADBench <https://github.com/Minqi824/ADBench> compares 30 anomaly detection algorithms on 57 benchmark datasets.
Learn more about anomaly detection at Anomaly Detection Resources <https://github.com/yzhao062/anomaly-detection-resources>_
PyOD on Distributed Systems: you can also run PyOD on databricks <https://www.databricks.com/blog/2023/03/13/unsupervised-outlier-detection-databricks.html>_.

About PyOD ^^^^^^^^^^

PyOD, established in 2017, has become a go-to Python library for detecting anomalous/outlying objects in multivariate data. This exciting yet challenging field is commonly referred to as Outlier Detection <https://en.wikipedia.org/wiki/Anomaly_detection>_ or Anomaly Detection <https://en.wikipedia.org/wiki/Anomaly_detection>_.

PyOD includes more than 50 detection algorithms, from classical LOF (SIGMOD 2000) to the cutting-edge ECOD and DIF (TKDE 2022 and 2023). Since 2017, PyOD has been successfully used in numerous academic research projects and commercial products with more than 22 million downloads <https://pepy.tech/project/pyod>. 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>, and Towards Data Science <https://towardsdatascience.com/anomaly-detection-for-dummies-15f148e559c1>.

PyOD is featured for:

Unified, User-Friendly Interface across various algorithms.
Wide Range of Models, from classic techniques to the latest deep learning methods in PyTorch.
High Performance & Efficiency, leveraging numba <https://github.com/numba/numba>_ and joblib <https://github.com/joblib/joblib>_ for JIT compilation and parallel processing.
Fast Training & Prediction, achieved through the SUOD framework [#Zhao2021SUOD]_.

Outlier Detection with 5 Lines of Code:

.. code-block:: python

# Example: Training an ECOD detector
from pyod.models.ecod import ECOD
clf = ECOD()
clf.fit(X_train)
y_train_scores = clf.decision_scores_  # Outlier scores for training data
y_test_scores = clf.decision_function(X_test)  # Outlier scores for test data

Selecting the Right Algorithm: Unsure where to start? Consider these robust and interpretable options:

ECOD <https://github.com/yzhao062/pyod/blob/master/examples/ecod_example.py>_: Example of using ECOD for outlier detection
Isolation Forest <https://github.com/yzhao062/pyod/blob/master/examples/iforest_example.py>_: Example of using Isolation Forest for outlier detection

Alternatively, explore MetaOD <https://github.com/yzhao062/MetaOD>_ for a data-driven approach.

Citing PyOD:

PyOD paper <http://www.jmlr.org/papers/volume20/19-011/19-011.pdf>_ is published in Journal of Machine Learning Research (JMLR) <http://www.jmlr.org/>_ (MLOSS 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.

For a broader perspective on anomaly detection, see our NeurIPS papers ADBench: Anomaly Detection Benchmark Paper <https://arxiv.org/abs/2206.09426>_ and ADGym: Design Choices for Deep Anomaly Detection <https://arxiv.org/abs/2309.15376>_::

@article{han2022adbench,
    title={Adbench: Anomaly detection benchmark},
    author={Han, Songqiao and Hu, Xiyang and Huang, Hailiang and Jiang, Minqi and Zhao, Yue},
    journal={Advances in Neural Information Processing Systems},
    volume={35},
    pages={32142--32159},
    year={2022}
}

@article{jiang2023adgym,
    title={ADGym: Design Choices for Deep Anomaly Detection},
    author={Jiang, Minqi and Hou, Chaochuan and Zheng, Ao and Han, Songqiao and Huang, Hailiang and Wen, Qingsong and Hu, Xiyang and Zhao, Yue},
    journal={Advances in Neural Information Processing Systems},
    volume={36},
    year={2023}
}

Table of Contents:

Installation <#installation>_
API Cheatsheet & Reference <#api-cheatsheet--reference>_
ADBench Benchmark and Datasets <#adbench-benchmark-and-datasets>_
Model Save & Load <#model-save--load>_
Fast Train with SUOD <#fast-train-with-suod>_
Thresholding Outlier Scores <#thresholding-outlier-scores>_
Implemented Algorithms <#implemented-algorithms>_
Quick Start for Outlier Detection <#quick-start-for-outlier-detection>_
How to Contribute <#how-to-contribute>_
Inclusion Criteria <#inclusion-criteria>_

Installation ^^^^^^^^^^^^

PyOD is designed for easy installation using either pip or conda. We recommend using the latest version of PyOD due to frequent updates and enhancements:

.. code-block:: bash

pip install pyod # normal install pip install --upgrade pyod # or update if needed

.. code-block:: bash

conda install -c conda-forge pyod

Alternatively, you can clone and run the setup.py file:

.. code-block:: bash

git clone https://github.com/yzhao062/pyod.git cd pyod pip install .

Required Dependencies:

Python 3.8 or higher
joblib
matplotlib
numpy>=1.19
numba>=0.51
scipy>=1.5.1
scikit_learn>=0.22.0

Optional Dependencies (see details below):

combo (optional, required for models/combination.py and FeatureBagging)
pytorch (optional, required for AutoEncoder, and other deep learning models)
suod (optional, required for running SUOD model)
xgboost (optional, required for XGBOD)
pythresh (optional, required for thresholding)

API Cheatsheet & Reference ^^^^^^^^^^^^^^^^^^^^^^^^^^

The full API Reference is available at PyOD Documentation <https://pyod.readthedocs.io/en/latest/pyod.html>_. Below is a quick cheatsheet for all detectors:

fit(X): Fit the detector. The parameter y is ignored in unsupervised methods.
decision_function(X): Predict raw anomaly scores for X using the fitted detector.
predict(X): Determine whether a sample is an outlier or not as binary labels using the fitted detector.
predict_proba(X): Estimate the probability of a sample being an outlier using the fitted detector.
predict_confidence(X): Assess the model's confidence on a per-sample basis (applicable in predict and predict_proba) [#Perini2020Quantifying]_.

Key Attributes of a fitted model:

decision_scores_: Outlier scores of the training data. Higher scores typically indicate more abnormal behavior. Outliers usually have higher scores.
labels_: Binary labels of the training data, where 0 indicates inliers and 1 indicates outliers/anomalies.

ADBench Benchmark and Datasets ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

We just released a 45-page, the most comprehensive ADBench: Anomaly Detection Benchmark <https://arxiv.org/abs/2206.09426>_ [#Han2022ADBench]. The fully open-sourced ADBench <https://github.com/Minqi824/ADBench> compares 30 anomaly detection algorithms on 57 benchmark datasets.

The organization of ADBench is provided below:

.. image:: https://github.com/Minqi824/ADBench/blob/main/figs/ADBench.png?raw=true :target: https://github.com/Minqi824/ADBench/blob/main/figs/ADBench.png?raw=true :alt: benchmark-fig

For a simpler visualization, we make the comparison of selected models via compare_all_models.py <https://github.com/yzhao062/pyod/blob/master/examples/compare_all_models.py>_.

.. image:: https://github.com/yzhao062/pyod/blob/development/examples/ALL.png?raw=true :target: https://github.com/yzhao062/pyod/blob/development/examples/ALL.png?raw=true :alt: Comparison_of_All

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')

It is known that there are challenges in saving neural network models. Check #328 <https://github.com/yzhao062/pyod/issues/328#issuecomment-917192704>_ and #88 <https://github.com/yzhao062/pyod/issues/88#issuecomment-615343139>_ for temporary workaround.

Fast Train with SUOD ^^^^^^^^^^^^^^^^^^^^

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 SUOD example <https://github.com/yzhao062/pyod/blob/master/examples/suod_example.py>_.

.. code-block:: python

from pyod.models.suod import SUOD

# initialized a group of outlier detectors for acceleration
detector_list = [LOF(n_neighbors=15), LOF(n_neighbors=20),
                 LOF(n_neighbors=25), LOF(n_neighbors=35),
                 COPOD(), IForest(n_estimators=100),
                 IForest(n_estimators=200)]

# decide the number of parallel process, and the combination method
# then clf can be used as any outlier detection model
clf = SUOD(base_estimators=detector_list, n_jobs=2, combination='average',
           verbose=False)

Thresholding Outlier Scores ^^^^^^^^^^^^^^^^^^^^^^^^^^^

A more data-based approach can be taken when setting the contamination level. By using a thresholding method, guessing an arbitrary value can be replaced with tested techniques for separating inliers and outliers. Refer to PyThresh <https://github.com/KulikDM/pythresh>_ for a more in-depth look at thresholding.

.. code-block:: python

from pyod.models.knn import KNN
from pyod.models.thresholds import FILTER

# Set the outlier detection and thresholding methods
clf = KNN(contamination=FILTER())

See supported thresholding methods in thresholding <https://github.com/yzhao062/pyod/blob/master/docs/thresholding.rst>_.

Implemented Algorithms ^^^^^^^^^^^^^^^^^^^^^^

PyOD toolkit consists of four major functional groups:

(i) Individual Detection Algorithms :

=================== Type Abbr =================== Probabilistic ECOD Probabilistic ABOD Probabilistic FastABOD Probabilistic COPOD Probabilistic MAD Probabilistic SOS Probabilistic QMCD Probabilistic KDE Probabilistic Sampling Probabilistic GMM Linear Model PCA Linear Model KPCA Linear Model MCD Linear Model CD Linear Model OCSVM Linear Model LMDD Proximity-Based LOF Proximity-Based COF Proximity-Based Proximity-Based CBLOF Proximity-Based LOCI Proximity-Based HBOS Proximity-Based kNN Proximity-Based AvgKNN Proximity-Based MedKNN Proximity-Based SOD Proximity-Based ROD Outlier Ensembles IForest Outlier Ensembles INNE Outlier Ensembles DIF Outlier Ensembles FB Outlier Ensembles LSCP Outlier Ensembles XGBOD Outlier Ensembles LODA Outlier Ensembles SUOD Neural Networks AutoEncoder Neural Networks VAE Neural Networks Beta-VAE Neural Networks SO_GAAL Neural Networks MO_GAAL Neural Networks DeepSVDD Neural Networks AnoGAN Neural Networks ALAD Neural Networks AE1SVM Neural Networks DevNet Graph-based R-Graph Graph-based LUNAR =================== ================== ====================================================================================================== ===== ======================================== Algorithm Year Ref ================== ====================================================================================================== ===== ======================================== Unsupervised Outlier Detection Using Empirical Cumulative Distribution Functions 2022 [#Li2021ECOD]_ Angle-Based Outlier Detection 2008 [#Kriegel2008Angle]_ Fast Angle-Based Outlier Detection using approximation 2008 [#Kriegel2008Angle]_ COPOD: Copula-Based Outlier Detection 2020 [#Li2020COPOD]_ Median Absolute Deviation (MAD) 1993 [#Iglewicz1993How]_ Stochastic Outlier Selection 2012 [#Janssens2012Stochastic]_ Quasi-Monte Carlo Discrepancy outlier detection 2001 [#Fang2001Wrap]_ Outlier Detection with Kernel Density Functions 2007 [#Latecki2007Outlier]_ Rapid distance-based outlier detection via sampling 2013 [#Sugiyama2013Rapid]_ Probabilistic Mixture Modeling for Outlier Analysis [#Aggarwal2015Outlier]_ [Ch.2] Principal Component Analysis (the sum of weighted projected distances to the eigenvector hyperplanes) 2003 [#Shyu2003A]_ Kernel Principal Component Analysis 2007 [#Hoffmann2007Kernel]_ Minimum Covariance Determinant (use the mahalanobis distances as the outlier scores) 1999 [#Hardin2004Outlier]_ [#Rousseeuw1999A]_ Use Cook's distance for outlier detection 1977 [#Cook1977Detection]_ One-Class Support Vector Machines 2001 [#Scholkopf2001Estimating]_ Deviation-based Outlier Detection (LMDD) 1996 [#Arning1996A]_ Local Outlier Factor 2000 [#Breunig2000LOF]_ Connectivity-Based Outlier Factor 2002 [#Tang2002Enhancing]_ (Incremental) COF Memory Efficient Connectivity-Based Outlier Factor (slower but reduce storage complexity) 2002 [#Tang2002Enhancing]_ Clustering-Based Local Outlier Factor 2003 [#He2003Discovering]_ LOCI: Fast outlier detection using the local correlation integral 2003 [#Papadimitriou2003LOCI]_ Histogram-based Outlier Score 2012 [#Goldstein2012Histogram]_ k Nearest Neighbors (use the distance to the kth nearest neighbor as the outlier score) 2000 [#Ramaswamy2000Efficient]_ Average kNN (use the average distance to k nearest neighbors as the outlier score) 2002 [#Angiulli2002Fast]_ Median kNN (use the median distance to k nearest neighbors as the outlier score) 2002 [#Angiulli2002Fast]_ Subspace Outlier Detection 2009 [#Kriegel2009Outlier]_ Rotation-based Outlier Detection 2020 [#Almardeny2020A]_ Isolation Forest 2008 [#Liu2008Isolation]_ Isolation-based Anomaly Detection Using Nearest-Neighbor Ensembles 2018 [#Bandaragoda2018Isolation]_ Deep Isolation Forest for Anomaly Detection 2023 [#Xu2023Deep]_ Feature Bagging 2005 [#Lazarevic2005Feature]_ LSCP: Locally Selective Combination of Parallel Outlier Ensembles 2019 [#Zhao2019LSCP]_ Extreme Boosting Based Outlier Detection (Supervised) 2018 [#Zhao2018XGBOD]_ Lightweight On-line Detector of Anomalies 2016 [#Pevny2016Loda]_ SUOD: Accelerating Large-scale Unsupervised Heterogeneous Outlier Detection (Acceleration) 2021 [#Zhao2021SUOD]_ Fully connected AutoEncoder (use reconstruction error as the outlier score) [#Aggarwal2015Outlier]_ [Ch.3] Variational AutoEncoder (use reconstruction error as the outlier score) 2013 [#Kingma2013Auto]_ Variational AutoEncoder (all customized loss term by varying gamma and capacity) 2018 [#Burgess2018Understanding]_ Single-Objective Generative Adversarial Active Learning 2019 [#Liu2019Generative]_ Multiple-Objective Generative Adversarial Active Learning 2019 [#Liu2019Generative]_ Deep One-Class Classification 2018 [#Ruff2018Deep]_ Anomaly Detection with Generative Adversarial Networks 2017 [#Schlegl2017Unsupervised]_ Adversarially learned anomaly detection 2018 [#Zenati2018Adversarially]_ Autoencoder-based One-class Support Vector Machine 2019 [#Nguyen2019scalable]_ Deep Anomaly Detection with Deviation Networks 2019 [#Pang2019Deep]_ Outlier detection by R-graph 2017 [#You2017Provable]_ LUNAR: Unifying Local Outlier Detection Methods via Graph Neural Networks 2022 [#Goodge2022Lunar]_ ================== ====================================================================================================== ===== ========================================

(ii) Outlier Ensembles & Outlier Detector Combination Frameworks:

=================== ================ ===================================================================================================== ===== ======================================== Type Abbr Algorithm Year Ref =================== ================ ===================================================================================================== ===== ======================================== Outlier Ensembles FB 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]_ Outlier Ensembles SUOD SUOD: Accelerating Large-scale Unsupervised Heterogeneous Outlier Detection (Acceleration) 2021 [#Zhao2021SUOD]_ Outlier Ensembles INNE Isolation-based Anomaly Detection Using Nearest-Neighbor Ensembles 2018 [#Bandaragoda2018Isolation]_ 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>_ =================== ====================== ===================================================================================================================================================== ======================================================================================================================================

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>_

"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

   # it is possible to get the prediction confidence as well
   y_test_pred, y_test_pred_confidence = clf.predict(X_test, return_confidence=True)  # outlier labels (0 or 1) and confidence in the range of [0,1]

#. Evaluate the prediction by ROC and Precision @ Rank n (p@n).

.. 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

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).

.. [#Bandaragoda2018Isolation] Bandaragoda, T. R., Ting, K. M., Albrecht, D., Liu, F. T., Zhu, Y., and Wells, J. R., 2018, Isolation-based anomaly detection using nearest-neighbor ensembles. Computational Intelligence\ , 34(4), pp. 968-998.

.. [#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).

.. [#Cook1977Detection] Cook, R.D., 1977. Detection of influential observation in linear regression. Technometrics, 19(1), pp.15-18.

.. [#Fang2001Wrap] Fang, K.T. and Ma, C.X., 2001. Wrap-around L2-discrepancy of random sampling, Latin hypercube and uniform designs. Journal of complexity, 17(4), pp.608-624.

.. [#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.

.. [#Goodge2022Lunar] Goodge, A., Hooi, B., Ng, S.K. and Ng, W.S., 2022, June. Lunar: Unifying local outlier detection methods via graph neural networks. In Proceedings of the AAAI Conference on Artificial Intelligence.

.. [#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.

.. [#Han2022ADBench] Han, S., Hu, X., Huang, H., Jiang, M. and Zhao, Y., 2022. ADBench: Anomaly Detection Benchmark. arXiv preprint arXiv:2206.09426.

.. [#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.

.. [#Hoffmann2007Kernel] Hoffmann, H., 2007. Kernel PCA for novelty detection. Pattern recognition, 40(3), pp.863-874.

.. [#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.

.. [#Latecki2007Outlier] Latecki, L.J., Lazarevic, A. and Pokrajac, D., 2007, July. Outlier detection with kernel density functions. In International Workshop on Machine Learning and Data Mining in Pattern Recognition (pp. 61-75). Springer, Berlin, Heidelberg.

.. [#Lazarevic2005Feature] Lazarevic, A. and Kumar, V., 2005, August. Feature bagging for outlier detection. In KDD '05. 2005.

.. [#Li2019MADGAN] Li, D., Chen, D., Jin, B., Shi, L., Goh, J. and Ng, S.K., 2019, September. MAD-GAN: Multivariate anomaly detection for time series data with generative adversarial networks. In International Conference on Artificial Neural Networks (pp. 703-716). Springer, Cham.

.. [#Li2020COPOD] Li, Z., Zhao, Y., Botta, N., Ionescu, C. and Hu, X. COPOD: Copula-Based Outlier Detection. IEEE International Conference on Data Mining (ICDM), 2020.

.. [#Li2021ECOD] Li, Z., Zhao, Y., Hu, X., Botta, N., Ionescu, C. and Chen, H. G. ECOD: Unsupervised Outlier Detection Using Empirical Cumulative Distribution Functions. IEEE Transactions on Knowledge and Data Engineering (TKDE), 2022.

.. [#Liu2008Isolation] Liu, F.T., Ting, K.M. and Zhou, Z.H., 2008, December. Isolation forest. In International Conference on Data Mining\ , pp. 413-422. IEEE.

.. [#Liu2019Generative] Liu, Y., Li, Z., Zhou, C., Jiang, Y., Sun, J., Wang, M. and He, X., 2019. Generative adversarial active learning for unsupervised outlier detection. IEEE Transactions on Knowledge and Data Engineering.

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Python Outlier Detection (PyOD)

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