Python Outlier Detection (PyOD) V2 ================================== **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 well-developed and easy-to-use Python library for detecting anomalies in multivariate data. Whether you are working with a small-scale project or large datasets, PyOD provides a range of algorithms to fit your needs. **PyOD Version 2 is now available** (`Paper `_) [#Chen2024PyOD]_, featuring: * **Expanded Deep Learning Support**: Integrates 12 modern neural models into a single PyTorch-based framework, bringing the total number of outlier detection methods to 45. * **Enhanced Performance and Ease of Use**: Models are optimized for efficiency and consistent performance across different datasets. * **LLM-based Model Selection**: Automated model selection guided by a large language model reduces manual tuning and assists users who may have limited experience with outlier detection. **Additional Resources**: * **NLP Anomaly Detection**: `NLP-ADBench `_ provides both NLP anonaly detection datasets and algorithms * **Time-series Outlier Detection**: `TODS `_ * **Graph Outlier Detection**: `PyGOD `_ * **Performance Comparison & Datasets**: Our 45-page `anomaly detection benchmark paper `_ and `ADBench `_, comparing 30 algorithms on 57 datasets * **PyOD on Distributed Systems**: `PyOD on Databricks `_ * **Learn More**: `Anomaly Detection Resources `_ **Check out our latest research in 2025 on LLM-based anomaly detection** [#Yang2024ad]_: `AD-LLM: Benchmarking Large Language Models for Anomaly Detection `_. ---- 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 `26 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**: If you use PyOD in a scientific publication, we would appreciate citations to the following paper(s): `PyOD 2: A Python Library for Outlier Detection with LLM-powered Model Selection `_ is available as a preprint. If you use PyOD in a scientific publication, we would appreciate citations to the following paper:: @article{zhao2024pyod2, author = {Chen, Sihan and Qian, Zhuangzhuang and Siu, Wingchun and Hu, Xingcan and Li, Jiaqi and Li, Shawn and Qin, Yuehan and Yang, Tiankai and Xiao, Zhuo and Ye, Wanghao and Zhang, Yichi and Dong, Yushun and Zhao, Yue}, title = {PyOD 2: A Python Library for Outlier Detection with LLM-powered Model Selection}, journal = {arXiv preprint arXiv:2412.12154}, year = {2024} } `PyOD paper `_ is published in `Journal of Machine Learning Research (JMLR) `_ (MLOSS track).:: @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]_. * **predict_with_rejection(X)**\ : Allow the detector to reject (i.e., abstain from making) highly uncertain predictions (output = -2) [#Perini2023Rejection]_. **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. .. [#Chen2024PyOD] Chen, S., Qian, Z., Siu, W., Hu, X., Li, J., Li, S., Qin, Y., Yang, T., Xiao, Z., Ye, W. and Zhang, Y., 2024. PyOD 2: A Python Library for Outlier Detection with LLM-powered Model Selection. arXiv preprint arXiv:2412.12154. .. [#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. .. 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