Python Outlier Detection (PyOD) =============================== **Deployment & Documentation & Stats & License** |badge_pypi| |badge_anaconda| |badge_docs| |badge_stars| |badge_forks| |badge_downloads| |badge_testing| |badge_coverage| |badge_maintainability| |badge_license| |badge_benchmark| .. |badge_pypi| image:: https://img.shields.io/pypi/v/pyod.svg?color=brightgreen :target: https://pypi.org/project/pyod/ :alt: PyPI version .. |badge_anaconda| image:: https://anaconda.org/conda-forge/pyod/badges/version.svg :target: https://anaconda.org/conda-forge/pyod :alt: Anaconda version .. |badge_docs| image:: https://readthedocs.org/projects/pyod/badge/?version=latest :target: https://pyod.readthedocs.io/en/latest/?badge=latest :alt: Documentation status .. |badge_stars| image:: https://img.shields.io/github/stars/yzhao062/pyod.svg :target: https://github.com/yzhao062/pyod/stargazers :alt: GitHub stars .. |badge_forks| image:: https://img.shields.io/github/forks/yzhao062/pyod.svg?color=blue :target: https://github.com/yzhao062/pyod/network :alt: GitHub forks .. |badge_downloads| image:: https://pepy.tech/badge/pyod :target: https://pepy.tech/project/pyod :alt: Downloads .. |badge_testing| image:: https://github.com/yzhao062/pyod/actions/workflows/testing.yml/badge.svg :target: https://github.com/yzhao062/pyod/actions/workflows/testing.yml :alt: Testing .. |badge_coverage| image:: https://coveralls.io/repos/github/yzhao062/pyod/badge.svg :target: https://coveralls.io/github/yzhao062/pyod :alt: Coverage Status .. |badge_maintainability| image:: https://api.codeclimate.com/v1/badges/bdc3d8d0454274c753c4/maintainability :target: https://codeclimate.com/github/yzhao062/Pyod/maintainability :alt: Maintainability .. |badge_license| image:: https://img.shields.io/github/license/yzhao062/pyod.svg :target: https://github.com/yzhao062/pyod/blob/master/LICENSE :alt: License .. |badge_benchmark| image:: https://img.shields.io/badge/ADBench-benchmark_results-pink :target: https://github.com/Minqi824/ADBench :alt: Benchmark ----- 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. .. 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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*. .. [#Nguyen2019scalable] Nguyen, M.N. and Vien, N.A., 2019. Scalable and interpretable one-class svms with deep learning and random fourier features. In *Machine Learning and Knowledge Discovery in Databases: European Conference*, ECML PKDD, 2018. .. [#Pang2019Deep] Pang, Guansong, Chunhua Shen, and Anton Van Den Hengel. "Deep anomaly detection with deviation networks." In *KDD*, pp. 353-362. 2019. .. 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