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https://github.com/yzhao062/pyod

A Comprehensive and Scalable Python Library for Outlier Detection (Anomaly Detection)
https://github.com/yzhao062/pyod

anomaly anomaly-detection autoencoder data-analysis data-mining data-science deep-learning fraud-detection machine-learning neural-networks novelty-detection out-of-distribution-detection outlier-detection outlier-ensembles outliers python python3 unsupervised-learning

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A Comprehensive and Scalable Python Library for Outlier Detection (Anomaly Detection)

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README

        

Python Outlier Detection (PyOD)
===============================

**Deployment & Documentation & Stats & License**

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-----

Read Me First
^^^^^^^^^^^^^

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 `_.

* **For graph outlier detection**, please use `PyGOD `_.

* **Performance Comparison & Datasets**: We have a 45-page, comprehensive `anomaly detection benchmark paper `_. The fully `open-sourced ADBench `_ compares 30 anomaly detection algorithms on 57 benchmark datasets.

* **Learn more about anomaly detection** at `Anomaly Detection Resources `_

* **PyOD on Distributed Systems**: you can also run `PyOD on databricks `_.

----

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 `_ or `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 `_. It is also well acknowledged by the machine learning community with various dedicated posts/tutorials, including `Analytics Vidhya `_, `KDnuggets `_, and `Towards Data Science `_.

**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 `_ and `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 `_: Example of using ECOD for outlier detection
- `Isolation Forest `_: Example of using Isolation Forest for outlier detection

Alternatively, explore `MetaOD `_ for a data-driven approach.

**Citing PyOD**:

`PyOD paper `_ is published in `Journal of Machine Learning Research (JMLR) `_ (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 `_ and `ADGym: Design Choices for Deep Anomaly Detection `_::

@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 `_. 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 `_ [#Han2022ADBench]_.
The fully `open-sourced 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 `_\.

.. 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 `_ for clarification.

In short, we recommend to use joblib or pickle for saving and loading PyOD models.
See `"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 `_
and `#88 `_
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 `_
and `SUOD example `_.

.. 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 `_ 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 `_.

----

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

PyOD toolkit consists of four major functional groups:

**(i) Individual Detection Algorithms** :

=================== ================== ====================================================================================================== ===== ========================================
Type Abbr Algorithm Year Ref
=================== ================== ====================================================================================================== ===== ========================================
Probabilistic ECOD Unsupervised Outlier Detection Using Empirical Cumulative Distribution Functions 2022 [#Li2021ECOD]_
Probabilistic ABOD Angle-Based Outlier Detection 2008 [#Kriegel2008Angle]_
Probabilistic FastABOD Fast Angle-Based Outlier Detection using approximation 2008 [#Kriegel2008Angle]_
Probabilistic COPOD COPOD: Copula-Based Outlier Detection 2020 [#Li2020COPOD]_
Probabilistic MAD Median Absolute Deviation (MAD) 1993 [#Iglewicz1993How]_
Probabilistic SOS Stochastic Outlier Selection 2012 [#Janssens2012Stochastic]_
Probabilistic QMCD Quasi-Monte Carlo Discrepancy outlier detection 2001 [#Fang2001Wrap]_
Probabilistic KDE Outlier Detection with Kernel Density Functions 2007 [#Latecki2007Outlier]_
Probabilistic Sampling Rapid distance-based outlier detection via sampling 2013 [#Sugiyama2013Rapid]_
Probabilistic GMM Probabilistic Mixture Modeling for Outlier Analysis [#Aggarwal2015Outlier]_ [Ch.2]
Linear Model PCA Principal Component Analysis (the sum of weighted projected distances to the eigenvector hyperplanes) 2003 [#Shyu2003A]_
Linear Model KPCA Kernel Principal Component Analysis 2007 [#Hoffmann2007Kernel]_
Linear Model MCD Minimum Covariance Determinant (use the mahalanobis distances as the outlier scores) 1999 [#Hardin2004Outlier]_ [#Rousseeuw1999A]_
Linear Model CD Use Cook's distance for outlier detection 1977 [#Cook1977Detection]_
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 (Incremental) COF Memory Efficient Connectivity-Based Outlier Factor (slower but reduce storage complexity) 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]_
Outlier Ensembles IForest Isolation Forest 2008 [#Liu2008Isolation]_
Outlier Ensembles INNE Isolation-based Anomaly Detection Using Nearest-Neighbor Ensembles 2018 [#Bandaragoda2018Isolation]_
Outlier Ensembles DIF Deep Isolation Forest for Anomaly Detection 2023 [#Xu2023Deep]_
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]_
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]_
Neural Networks DeepSVDD Deep One-Class Classification 2018 [#Ruff2018Deep]_
Neural Networks AnoGAN Anomaly Detection with Generative Adversarial Networks 2017 [#Schlegl2017Unsupervised]_
Neural Networks ALAD Adversarially learned anomaly detection 2018 [#Zenati2018Adversarially]_
Neural Networks AE1SVM Autoencoder-based One-class Support Vector Machine 2019 [#Nguyen2019scalable]_
Neural Networks DevNet Deep Anomaly Detection with Deviation Networks 2019 [#Pang2019Deep]_
Graph-based R-Graph Outlier detection by R-graph 2017 [#You2017Provable]_
Graph-based LUNAR 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 `_
Data generate_data_clusters Synthesized data generation in clusters; more complex data patterns can be created with multiple clusters `generate_data_clusters `_
Stat wpearsonr Calculate the weighted Pearson correlation of two samples `wpearsonr `_
Utility get_label_n Turn raw outlier scores into binary labels by assign 1 to top n outlier scores `get_label_n `_
Utility precision_n_scores calculate precision @ rank n `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 `_

**KDnuggets**: `Intuitive Visualization of Outlier Detection Methods `_, `An Overview of Outlier Detection Methods from PyOD `_

**Towards Data Science**: `Anomaly Detection for Dummies `_

`"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 `_.

#. 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 `_\ ):

.. 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.

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