Python Outlier Detection (PyOD)

Deployment & Documentation & Stats & License

---

News: We just released a 45-page, the most comprehensive anomaly detection benchmark paper. The fully open-sourced ADBench compares 30 anomaly detection algorithms on 57 benchmark datasets.

For time-series outlier detection, please use TODS. For graph outlier detection, please use PyGOD.

PyOD is the most comprehensive and scalable Python library for detecting outlying objects in multivariate data. This exciting yet challenging field is commonly referred as Outlier Detection or Anomaly Detection.

PyOD includes more than 40 detection algorithms, from classical LOF (SIGMOD 2000) to the latest ECOD (TKDE 2022). Since 2017, PyOD has been successfully used in numerous academic researches and commercial products with more than 10 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 APIs, detailed documentation, and interactive examples across various algorithms.
• Advanced models, including classical distance and density estimation, latest deep learning methods, and emerging algorithms like ECOD.
• Optimized performance with JIT and parallelization using numba and joblib.
• Fast training & prediction with SUOD [46].

Outlier Detection with 5 Lines of Code:

# train an ECOD detector
from pyod.models.ecod import ECOD
clf = ECOD()
clf.fit(X_train)

# get outlier scores
y_train_scores = clf.decision_scores_  # raw outlier scores on the train data
y_test_scores = clf.decision_function(X_test)  # predict raw outlier scores on test

Personal suggestion on selecting an OD algorithm. If you do not know which algorithm to try, go with:

• ECOD: Example of using ECOD for outlier detection
• Isolation Forest: Example of using Isolation Forest for outlier detection

They are both fast and interpretable. Or, you could try more data-driven approach MetaOD.

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.

If you want more general insights of anomaly detection and/or algorithm performance comparison, please see our NeurIPS 2022 paper ADBench: Anomaly Detection Benchmark Paper:

@inproceedings{han2022adbench,
    title={ADBench: Anomaly Detection Benchmark},
    author={Songqiao Han and Xiyang Hu and Hailiang Huang and Mingqi Jiang and Yue Zhao},
    booktitle={Neural Information Processing Systems (NeurIPS)}
    year={2022},
}

Key Links and Resources:

• View the latest codes on Github
• Anomaly Detection Resources

Table of Contents:

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

---

Installation

It is recommended to use pip or conda for installation. Please make sure the latest version is installed, as PyOD is updated frequently:

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 3.6+
• joblib
• matplotlib
• numpy>=1.19
• numba>=0.51
• scipy>=1.5.1
• scikit_learn>=0.20.0
• six
• statsmodels

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)
• pandas (optional, required for running benchmark)
• suod (optional, required for running SUOD model)
• xgboost (optional, required for XGBOD)
• pythresh to use thresholding

Warning: PyOD has multiple neural network based models, e.g., AutoEncoders, which are implemented in both Tensorflow and PyTorch. However, PyOD does NOT install these deep learning 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 these deep learning libraries are installed. Instructions are provided: neural-net FAQ. Similarly, models depending on xgboost, e.g., XGBOD, would NOT enforce xgboost installation by default.

---

API Cheatsheet & Reference

Full API Reference: (https://pyod.readthedocs.io/en/latest/pyod.html). API cheatsheet for all detectors:

• fit(X): Fit detector. y is ignored in unsupervised methods.
• decision_function(X): Predict raw anomaly score of X using the fitted detector.
• predict(X): Predict if a particular sample is an outlier or not using the fitted detector.
• predict_proba(X): Predict the probability of a sample being outlier using the fitted detector.
• predict_confidence(X): Predict the model's sample-wise confidence (available in predict and predict_proba) [32].

Key Attributes of a fitted model:

• decision_scores_: The outlier scores of the training data. The higher, the more abnormal. Outliers tend to have higher scores.
• labels_: The binary labels of the training data. 0 stands for inliers and 1 for outliers/anomalies.

---

ADBench Benchmark

We just released a 45-page, the most comprehensive ADBench: Anomaly Detection Benchmark [14]. The fully open-sourced ADBench compares 30 anomaly detection algorithms on 57 benchmark datasets.

The organization of ADBench is provided below:

The comparison of selected models is made available below (Figure, compare_all_models.py, Interactive Jupyter Notebooks). For Jupyter Notebooks, please navigate to "/notebooks/Compare All Models.ipynb".

---

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.

---

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 [46]. 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)

---

Implemented Algorithms

PyOD toolkit consists of three major functional groups:

(i) Individual Detection Algorithms :

Type	Abbr	Algorithm	Year	Ref
Probabilistic	ECOD	Unsupervised Outlier Detection Using Empirical Cumulative Distribution Functions	2022	[27]
Probabilistic	ABOD	Angle-Based Outlier Detection	2008	[21]
Probabilistic	FastABOD	Fast Angle-Based Outlier Detection using approximation	2008	[21]
Probabilistic	COPOD	COPOD: Copula-Based Outlier Detection	2020	[26]
Probabilistic	MAD	Median Absolute Deviation (MAD)	1993	[18]
Probabilistic	SOS	Stochastic Outlier Selection	2012	[19]
Probabilistic	KDE	Outlier Detection with Kernel Density Functions	2007	[23]
Probabilistic	Sampling	Rapid distance-based outlier detection via sampling	2013	[39]
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	[38]
Linear Model	KPCA	Kernel Principal Component Analysis	2007	[17]
Linear Model	MCD	Minimum Covariance Determinant (use the mahalanobis distances as the outlier scores)	1999	[15] [34]
Linear Model	CD	Use Cook's distance for outlier detection	1977	[10]
Linear Model	OCSVM	One-Class Support Vector Machines	2001	[37]
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	[40]
Proximity-Based	(Incremental) COF	Memory Efficient Connectivity-Based Outlier Factor (slower but reduce storage complexity)	2002	[40]
Proximity-Based	CBLOF	Clustering-Based Local Outlier Factor	2003	[16]
Proximity-Based	LOCI	LOCI: Fast outlier detection using the local correlation integral	2003	[30]
Proximity-Based	HBOS	Histogram-based Outlier Score	2012	[11]
Proximity-Based	kNN	k Nearest Neighbors (use the distance to the kth nearest neighbor as the outlier score)	2000	[33]
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	[22]
Proximity-Based	ROD	Rotation-based Outlier Detection	2020	[4]
Outlier Ensembles	IForest	Isolation Forest	2008	[28]
Outlier Ensembles	INNE	Isolation-based Anomaly Detection Using Nearest-Neighbor Ensembles	2018	[7]
Outlier Ensembles	FB	Feature Bagging	2005	[24]
Outlier Ensembles	LSCP	LSCP: Locally Selective Combination of Parallel Outlier Ensembles	2019	[45]
Outlier Ensembles	XGBOD	Extreme Boosting Based Outlier Detection (Supervised)	2018	[44]
Outlier Ensembles	LODA	Lightweight On-line Detector of Anomalies	2016	[31]
Outlier Ensembles	SUOD	SUOD: Accelerating Large-scale Unsupervised Heterogeneous Outlier Detection (Acceleration)	2021	[46]
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	[20]
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	[29]
Neural Networks	MO_GAAL	Multiple-Objective Generative Adversarial Active Learning	2019	[29]
Neural Networks	DeepSVDD	Deep One-Class Classification	2018	[35]
Neural Networks	AnoGAN	Anomaly Detection with Generative Adversarial Networks	2017	[36]
Neural Networks	ALAD	Adversarially learned anomaly detection	2018	[43]
Graph-based	R-Graph	Outlier detection by R-graph	2017	[42]
Graph-based	LUNAR	LUNAR: Unifying Local Outlier Detection Methods via Graph Neural Networks	2022	[12]

(ii) Outlier Ensembles & Outlier Detector Combination Frameworks:

Type	Abbr	Algorithm	Year	Ref
Outlier Ensembles	FB	Feature Bagging	2005	[24]
Outlier Ensembles	LSCP	LSCP: Locally Selective Combination of Parallel Outlier Ensembles	2019	[45]
Outlier Ensembles	XGBOD	Extreme Boosting Based Outlier Detection (Supervised)	2018	[44]
Outlier Ensembles	LODA	Lightweight On-line Detector of Anomalies	2016	[31]
Outlier Ensembles	SUOD	SUOD: Accelerating Large-scale Unsupervised Heterogeneous Outlier Detection (Acceleration)	2021	[46]
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) 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

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):

---

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
• 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, 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

[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]	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.

[12]	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.

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

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

[15]	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.

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

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

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

[19]	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.

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

[21]	(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.

[22]	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.

[23]	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.

[24]	(1, 2) Lazarevic, A. and Kumar, V., 2005, August. Feature bagging for outlier detection. In KDD '05. 2005.

[25]	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.

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

[27]	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.

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

[29]	(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.

[30]	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.

[31]	(1, 2) Pevný, T., 2016. Loda: Lightweight on-line detector of anomalies. Machine Learning, 102(2), pp.275-304.

[32]	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.

[33]	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.

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

[35]	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.

[36]	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.

[37]	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.

[38]	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.

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

[40]	(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.

[41]	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.

[42]	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.

[43]	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.

[44]	(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.

[45]	(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.

[46]	(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).