README.rst

Python Outlier Detection (PyOD)

Deployment & Documentation & Stats & License

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Read Me First

Welcome to PyOD, a versatile 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, the most comprehensive anomaly detection benchmark paper. The fully open-sourced ADBench compares 30 anomaly detection algorithms on 57 benchmark datasets.
• Learn more about anomaly detection @ Anomaly Detection Resources
• PyOD on Distributed Systems: you could also run PyOD on databricks.

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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 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 researches and commercial products with more than 17 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.
• High Performance & Efficiency, leveraging numba and joblib for JIT compilation and parallel processing.
• Fast Training & Prediction, achieved through the SUOD framework [48].

Outlier Detection with 5 Lines of Code:

# 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 & 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
• API Cheatsheet & Reference
• ADBench Benchmark and Datasets
• Model Save & Load
• Fast Train with SUOD
• Thresholding Outlier Scores
• Implemented Algorithms
• Quick Start for Outlier Detection
• How to Contribute
• Inclusion Criteria

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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:

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

conda install -c conda-forge pyod

Alternatively, you could clone and run setup.py file:

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)
• keras/tensorflow (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)optional

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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) [33].

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.

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ADBench Benchmark and Datasets

We just released a 45-page, the most comprehensive ADBench: Anomaly Detection Benchmark [15]. The fully open-sourced ADBench compares 30 anomaly detection algorithms on 57 benchmark datasets.

The organization of ADBench is provided below:

For a simpler visualization, we make the comparison of selected models via compare_all_models.py.

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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:

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.

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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 [48]. See SUOD Paper and SUOD example.

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 abritrary value can be replaced with tested techniques for seperating inliers and outliers. Refer to PyThresh for a more in depth look at thresholding.

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

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

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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	[28]
Probabilistic	ABOD	Angle-Based Outlier Detection	2008	[22]
Probabilistic	FastABOD	Fast Angle-Based Outlier Detection using approximation	2008	[22]
Probabilistic	COPOD	COPOD: Copula-Based Outlier Detection	2020	[27]
Probabilistic	MAD	Median Absolute Deviation (MAD)	1993	[19]
Probabilistic	SOS	Stochastic Outlier Selection	2012	[20]
Probabilistic	QMCD	Quasi-Monte Carlo Discrepancy outlier detection	2001	[11]
Probabilistic	KDE	Outlier Detection with Kernel Density Functions	2007	[24]
Probabilistic	Sampling	Rapid distance-based outlier detection via sampling	2013	[40]
Probabilistic	GMM	Probabilistic Mixture Modeling for Outlier Analysis		[1] [Ch.2]
Linear Model	PCA	Principal Component Analysis (the sum of weighted projected distances to the eigenvector hyperplanes)	2003	[39]
Linear Model	KPCA	Kernel Principal Component Analysis	2007	[18]
Linear Model	MCD	Minimum Covariance Determinant (use the mahalanobis distances as the outlier scores)	1999	[16] [35]
Linear Model	CD	Use Cook's distance for outlier detection	1977	[10]
Linear Model	OCSVM	One-Class Support Vector Machines	2001	[38]
Linear Model	LMDD	Deviation-based Outlier Detection (LMDD)	1996	[6]
Proximity-Based	LOF	Local Outlier Factor	2000	[8]
Proximity-Based	COF	Connectivity-Based Outlier Factor	2002	[41]
Proximity-Based	(Incremental) COF	Memory Efficient Connectivity-Based Outlier Factor (slower but reduce storage complexity)	2002	[41]
Proximity-Based	CBLOF	Clustering-Based Local Outlier Factor	2003	[17]
Proximity-Based	LOCI	LOCI: Fast outlier detection using the local correlation integral	2003	[31]
Proximity-Based	HBOS	Histogram-based Outlier Score	2012	[12]
Proximity-Based	kNN	k Nearest Neighbors (use the distance to the kth nearest neighbor as the outlier score)	2000	[34]
Proximity-Based	AvgKNN	Average kNN (use the average distance to k nearest neighbors as the outlier score)	2002	[5]
Proximity-Based	MedKNN	Median kNN (use the median distance to k nearest neighbors as the outlier score)	2002	[5]
Proximity-Based	SOD	Subspace Outlier Detection	2009	[23]
Proximity-Based	ROD	Rotation-based Outlier Detection	2020	[4]
Outlier Ensembles	IForest	Isolation Forest	2008	[29]
Outlier Ensembles	INNE	Isolation-based Anomaly Detection Using Nearest-Neighbor Ensembles	2018	[7]
Outlier Ensembles	DIF	Deep Isolation Forest for Anomaly Detection	2023	[43]
Outlier Ensembles	FB	Feature Bagging	2005	[25]
Outlier Ensembles	LSCP	LSCP: Locally Selective Combination of Parallel Outlier Ensembles	2019	[47]
Outlier Ensembles	XGBOD	Extreme Boosting Based Outlier Detection (Supervised)	2018	[46]
Outlier Ensembles	LODA	Lightweight On-line Detector of Anomalies	2016	[32]
Outlier Ensembles	SUOD	SUOD: Accelerating Large-scale Unsupervised Heterogeneous Outlier Detection (Acceleration)	2021	[48]
Neural Networks	AutoEncoder	Fully connected AutoEncoder (use reconstruction error as the outlier score)		[1] [Ch.3]
Neural Networks	VAE	Variational AutoEncoder (use reconstruction error as the outlier score)	2013	[21]
Neural Networks	Beta-VAE	Variational AutoEncoder (all customized loss term by varying gamma and capacity)	2018	[9]
Neural Networks	SO_GAAL	Single-Objective Generative Adversarial Active Learning	2019	[30]
Neural Networks	MO_GAAL	Multiple-Objective Generative Adversarial Active Learning	2019	[30]
Neural Networks	DeepSVDD	Deep One-Class Classification	2018	[36]
Neural Networks	AnoGAN	Anomaly Detection with Generative Adversarial Networks	2017	[37]
Neural Networks	ALAD	Adversarially learned anomaly detection	2018	[45]
Graph-based	R-Graph	Outlier detection by R-graph	2017	[44]
Graph-based	LUNAR	LUNAR: Unifying Local Outlier Detection Methods via Graph Neural Networks	2022	[13]

(ii) Outlier Ensembles & Outlier Detector Combination Frameworks:

Type	Abbr	Algorithm	Year	Ref
Outlier Ensembles	FB	Feature Bagging	2005	[25]
Outlier Ensembles	LSCP	LSCP: Locally Selective Combination of Parallel Outlier Ensembles	2019	[47]
Outlier Ensembles	XGBOD	Extreme Boosting Based Outlier Detection (Supervised)	2018	[46]
Outlier Ensembles	LODA	Lightweight On-line Detector of Anomalies	2016	[32]
Outlier Ensembles	SUOD	SUOD: Accelerating Large-scale Unsupervised Heterogeneous Outlier Detection (Acceleration)	2021	[48]
Outlier Ensembles	INNE	Isolation-based Anomaly Detection Using Nearest-Neighbor Ensembles	2018	[7]
Combination	Average	Simple combination by averaging the scores	2015	[2]
Combination	Weighted Average	Simple combination by averaging the scores with detector weights	2015	[2]
Combination	Maximization	Simple combination by taking the maximum scores	2015	[2]
Combination	AOM	Average of Maximum	2015	[2]
Combination	MOA	Maximization of Average	2015	[2]
Combination	Median	Simple combination by taking the median of the scores	2015	[2]
Combination	majority Vote	Simple combination by taking the majority vote of the labels (weights can be used)	2015	[2]

(iii) Outlier Detection Score Thresholding Methods:

Type	Abbr	Algorithm	Documentation
Kernel-Based	AUCP	Area Under Curve Percentage	AUCP
Statistical Moment-Based	BOOT	Bootstrapping	BOOT
Normality-Based	CHAU	Chauvenet's Criterion	CHAU
Linear Model	CLF	Trained Linear Classifier	CLF
cluster-Based	CLUST	Clustering Based	CLUST
Kernel-Based	CPD	Change Point Detection	CPD
Transformation-Based	DECOMP	Decomposition	DECOMP
Normality-Based	DSN	Distance Shift from Normal	DSN
Curve-Based	EB	Elliptical Boundary	EB
Kernel-Based	FGD	Fixed Gradient Descent	FGD
Filter-Based	FILTER	Filtering Based	FILTER
Curve-Based	FWFM	Full Width at Full Minimum	FWFM
Statistical Test-Based	GESD	Generalized Extreme Studentized Deviate	GESD
Filter-Based	HIST	Histogram Based	HIST
Quantile-Based	IQR	Inter-Quartile Region	IQR
Statistical Moment-Based	KARCH	Karcher mean (Riemannian Center of Mass)	KARCH
Statistical Moment-Based	MAD	Median Absolute Deviation	MAD
Statistical Test-Based	MCST	Monte Carlo Shapiro Tests	MCST
Ensembles-Based	META	Meta-model Trained Classifier	META
Transformation-Based	MOLL	Friedrichs' Mollifier	MOLL
Statistical Test-Based	MTT	Modified Thompson Tau Test	MTT
Linear Model	OCSVM	One-Class Support Vector Machine	OCSVM
Quantile-Based	QMCD	Quasi-Monte Carlo Discrepancy	QMCD
Linear Model	REGR	Regression Based	REGR
Neural Networks	VAE	Variational Autoencoder	VAE
Curve-Based	WIND	Topological Winding Number	WIND
Transformation-Based	YJ	Yeo-Johnson Transformation	YJ
Normality-Based	ZSCORE	Z-score	ZSCORE

(iV) 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

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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

Computer Vision News (March 2019): Python Open Source Toolbox for Outlier Detection

"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.

1. Initialize a kNN detector, fit the model, and make the prediction.

   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]

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

   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)

3. See a sample output & visualization.

   On Training Data:
   KNN ROC:1.0, precision @ rank n:1.0
   
   On Test Data:
   KNN ROC:0.9989, precision @ rank n:0.9

   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):

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Reference

[1]	(1, 2) Aggarwal, C.C., 2015. Outlier analysis. In Data mining (pp. 237-263). Springer, Cham.

[2]	(1, 2, 3, 4, 5, 6, 7) Aggarwal, C.C. and Sathe, S., 2015. Theoretical foundations and algorithms for outlier ensembles.ACM SIGKDD Explorations Newsletter, 17(1), pp.24-47.

[3]	Aggarwal, C.C. and Sathe, S., 2017. Outlier ensembles: An introduction. Springer.

[4]	Almardeny, Y., Boujnah, N. and Cleary, F., 2020. A Novel Outlier Detection Method for Multivariate Data. IEEE Transactions on Knowledge and Data Engineering.

[5]	(1, 2) 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.

[6]	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).

[7]	(1, 2) 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.

[8]	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.

[9]	Burgess, Christopher P., et al. "Understanding disentangling in beta-VAE." arXiv preprint arXiv:1804.03599 (2018).

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

[11]	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.

[12]	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.

[13]	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.

[14]	Gopalan, P., Sharan, V. and Wieder, U., 2019. PIDForest: Anomaly Detection via Partial Identification. In Advances in Neural Information Processing Systems, pp. 15783-15793.

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

[16]	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.

[17]	He, Z., Xu, X. and Deng, S., 2003. Discovering cluster-based local outliers. Pattern Recognition Letters, 24(9-10), pp.1641-1650.

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

[19]	Iglewicz, B. and Hoaglin, D.C., 1993. How to detect and handle outliers (Vol. 16). Asq Press.

[20]	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.

[21]	Kingma, D.P. and Welling, M., 2013. Auto-encoding variational bayes. arXiv preprint arXiv:1312.6114.

[22]	(1, 2) Kriegel, H.P. and Zimek, A., 2008, August. Angle-based outlier detection in high-dimensional data. In KDD '08, pp. 444-452. ACM.

[23]	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.

[24]	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.

[25]	(1, 2) Lazarevic, A. and Kumar, V., 2005, August. Feature bagging for outlier detection. In KDD '05. 2005.

[26]	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.

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

[28]	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.

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

[30]	(1, 2) 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.

[31]	Papadimitriou, S., Kitagawa, H., Gibbons, P.B. and Faloutsos, C., 2003, March. LOCI: Fast outlier detection using the local correlation integral. In ICDE '03, pp. 315-326. IEEE.

[32]	(1, 2) Pevný, T., 2016. Loda: Lightweight on-line detector of anomalies. Machine Learning, 102(2), pp.275-304.

[33]	Perini, L., Vercruyssen, V., Davis, J. Quantifying the confidence of anomaly detectors in their example-wise predictions. In Joint European Conference on Machine Learning and Knowledge Discovery in Databases (ECML-PKDD), 2020.

[34]	Ramaswamy, S., Rastogi, R. and Shim, K., 2000, May. Efficient algorithms for mining outliers from large data sets. ACM Sigmod Record, 29(2), pp. 427-438.

[35]	Rousseeuw, P.J. and Driessen, K.V., 1999. A fast algorithm for the minimum covariance determinant estimator. Technometrics, 41(3), pp.212-223.

[36]	Ruff, L., Vandermeulen, R., Goernitz, N., Deecke, L., Siddiqui, S.A., Binder, A., Müller, E. and Kloft, M., 2018, July. Deep one-class classification. In International conference on machine learning (pp. 4393-4402). PMLR.

[37]	Schlegl, T., Seeböck, P., Waldstein, S.M., Schmidt-Erfurth, U. and Langs, G., 2017, June. Unsupervised anomaly detection with generative adversarial networks to guide marker discovery. In International conference on information processing in medical imaging (pp. 146-157). Springer, Cham.

[38]	Scholkopf, B., Platt, J.C., Shawe-Taylor, J., Smola, A.J. and Williamson, R.C., 2001. Estimating the support of a high-dimensional distribution. Neural Computation, 13(7), pp.1443-1471.

[39]	Shyu, M.L., Chen, S.C., Sarinnapakorn, K. and Chang, L., 2003. A novel anomaly detection scheme based on principal component classifier. MIAMI UNIV CORAL GABLES FL DEPT OF ELECTRICAL AND COMPUTER ENGINEERING.

[40]	Sugiyama, M. and Borgwardt, K., 2013. Rapid distance-based outlier detection via sampling. Advances in neural information processing systems, 26.

[41]	(1, 2) Tang, J., Chen, Z., Fu, A.W.C. and Cheung, D.W., 2002, May. Enhancing effectiveness of outlier detections for low density patterns. In Pacific-Asia Conference on Knowledge Discovery and Data Mining, pp. 535-548. Springer, Berlin, Heidelberg.

[42]	Wang, X., Du, Y., Lin, S., Cui, P., Shen, Y. and Yang, Y., 2019. adVAE: A self-adversarial variational autoencoder with Gaussian anomaly prior knowledge for anomaly detection. Knowledge-Based Systems.

[43]	Xu, H., Pang, G., Wang, Y., Wang, Y., 2023. Deep isolation forest for anomaly detection. IEEE Transactions on Knowledge and Data Engineering.

[44]	You, C., Robinson, D.P. and Vidal, R., 2017. Provable self-representation based outlier detection in a union of subspaces. In Proceedings of the IEEE conference on computer vision and pattern recognition.

[45]	Zenati, H., Romain, M., Foo, C.S., Lecouat, B. and Chandrasekhar, V., 2018, November. Adversarially learned anomaly detection. In 2018 IEEE International conference on data mining (ICDM) (pp. 727-736). IEEE.

[46]	(1, 2) Zhao, Y. and Hryniewicki, M.K. XGBOD: Improving Supervised Outlier Detection with Unsupervised Representation Learning. IEEE International Joint Conference on Neural Networks, 2018.

[47]	(1, 2) Zhao, Y., Nasrullah, Z., Hryniewicki, M.K. and Li, Z., 2019, May. LSCP: Locally selective combination in parallel outlier ensembles. In Proceedings of the 2019 SIAM International Conference on Data Mining (SDM), pp. 585-593. Society for Industrial and Applied Mathematics.

[48]	(1, 2, 3, 4) Zhao, Y., Hu, X., Cheng, C., Wang, C., Wan, C., Wang, W., Yang, J., Bai, H., Li, Z., Xiao, C., Wang, Y., Qiao, Z., Sun, J. and Akoglu, L. (2021). SUOD: Accelerating Large-scale Unsupervised Heterogeneous Outlier Detection. Conference on Machine Learning and Systems (MLSys).