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Python Outlier Detection (PyOD)

Deployment & Documentation & Stats & License

PyPI version

Anaconda version

Documentation status

GitHub stars

GitHub forks

Downloads

testing

Coverage Status

Maintainability

License

PyOD is a comprehensive and scalable Python toolkit 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 30 detection algorithms, from classical LOF (SIGMOD 2000) to the latest COPOD (ICDM 2020) and SUOD (MLSys 2021). Since 2017, PyOD has been successfully used in numerous academic researches and commercial products12. It is also well acknowledged by the machine learning community with various dedicated posts/tutorials, including Analytics Vidhya, KDnuggets, Towards Data Science, Computer Vision News, and awesome-machine-learning.

PyOD is featured for:

  • Unified APIs, detailed documentation, and interactive examples across various algorithms.
  • Advanced models, including classical ones from scikit-learn, latest deep learning methods, and emerging algorithms like COPOD.
  • Optimized performance with JIT and parallelization when possible, using numba and joblib.
  • Fast training & prediction with SUOD3.
  • Compatible with both Python 2 & 3.

Outlier Detection with 5 Lines of Code:

# train the COPOD detector
from pyod.models.copod import COPOD
clf = COPOD()
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

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.

Key Links and Resources:

Table of Contents:

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 2.7, 3.5, 3.6, or 3.7
  • combo>=0.0.8
  • joblib
  • numpy>=1.13
  • numba>=0.35
  • scipy>=0.19.1
  • scikit_learn>=0.20.0
  • statsmodels

Optional Dependencies (see details below):

  • combo (optional, required for models/combination.py and FeatureBagging)
  • keras (optional, required for AutoEncoder, and other deep learning models)
  • matplotlib (optional, required for running examples)
  • pandas (optional, required for running benchmark)
  • suod (optional, required for running SUOD model)
  • tensorflow (optional, required for AutoEncoder, and other deep learning models)
  • xgboost (optional, required for XGBOD)

Warning 1: PyOD has multiple neural network based models, e.g., AutoEncoders, which are implemented in both PyTorch and Tensorflow. However, PyOD does NOT install DL 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 Keras and a backend library, e.g., TensorFlow, are installed. Instructions are provided: neural-net FAQ. Similarly, models depending on xgboost, e.g., XGBOD, would NOT enforce xgboost installation by default.

Warning 2: PyOD contains multiple models that also exist in scikit-learn. However, these two libraries' API is not exactly the same--it is recommended to use only one of them for consistency but not mix the results. Refer Differences between scikit-learn and PyOD for more information.

API Cheatsheet & Reference

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

  • fit(X): Fit detector.
  • 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)4.

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.

Fast training and prediction: it is possible to train and predict with a large number of detection models in PyOD by leveraging SUOD framework5. See SUOD Paper and repository.

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 framework6. 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 2021 7
Probabilistic ABOD Angle-Based Outlier Detection 2008 8
Probabilistic FastABOD Fast Angle-Based Outlier Detection using approximation 2008 9
Probabilistic COPOD COPOD: Copula-Based Outlier Detection 2020 10
Probabilistic MAD Median Absolute Deviation (MAD) 1993 11
Probabilistic SOS Stochastic Outlier Selection 2012 12
Linear Model PCA Principal Component Analysis (the sum of weighted projected distances to the eigenvector hyperplanes) 2003 13
Linear Model MCD Minimum Covariance Determinant (use the mahalanobis distances as the outlier scores) 1999 1415
Linear Model OCSVM One-Class Support Vector Machines 2001 16
Linear Model LMDD Deviation-based Outlier Detection (LMDD) 1996 17
Proximity-Based LOF Local Outlier Factor 2000 18
Proximity-Based COF Connectivity-Based Outlier Factor 2002 19
Proximity-Based (Incremental) COF Memory Efficient Connectivity-Based Outlier Factor (slower but reduce storage complexity) 2002 20
Proximity-Based CBLOF Clustering-Based Local Outlier Factor 2003 21
Proximity-Based LOCI LOCI: Fast outlier detection using the local correlation integral 2003 22
Proximity-Based HBOS Histogram-based Outlier Score 2012 23
Proximity-Based kNN k Nearest Neighbors (use the distance to the kth nearest neighbor as the outlier score) 2000 24
Proximity-Based AvgKNN Average kNN (use the average distance to k nearest neighbors as the outlier score) 2002 25
Proximity-Based MedKNN Median kNN (use the median distance to k nearest neighbors as the outlier score) 2002 26
Proximity-Based SOD Subspace Outlier Detection 2009 27
Proximity-Based ROD Rotation-based Outlier Detection 2020 28
Outlier Ensembles IForest Isolation Forest 2008 29
Outlier Ensembles FB Feature Bagging 2005 30
Outlier Ensembles LSCP LSCP: Locally Selective Combination of Parallel Outlier Ensembles 2019 31
Outlier Ensembles XGBOD Extreme Boosting Based Outlier Detection (Supervised) 2018 32
Outlier Ensembles LODA Lightweight On-line Detector of Anomalies 2016 33

Outlier Ensembles Neural Networks

SUOD AutoEncoder

SUOD: Accelerating Large-scale Unsupervised Heterogeneous Outlier Detection (Acceleration) Fully connected AutoEncoder (use reconstruction error as the outlier score)

2021

34 35 [Ch.3]
Neural Networks VAE Variational AutoEncoder (use reconstruction error as the outlier score) 2013 36
Neural Networks Beta-VAE Variational AutoEncoder (all customized loss term by varying gamma and capacity) 2018 37
Neural Networks SO_GAAL Single-Objective Generative Adversarial Active Learning 2019 38
Neural Networks MO_GAAL Multiple-Objective Generative Adversarial Active Learning 2019 39
Neural Networks DeepSVDD Deep One-Class Classification 2018 40

(ii) Outlier Ensembles & Outlier Detector Combination Frameworks:

Type Abbr Algorithm Year Ref
Outlier Ensembles Feature Bagging 2005 41
Outlier Ensembles LSCP LSCP: Locally Selective Combination of Parallel Outlier Ensembles 2019 42
Outlier Ensembles XGBOD Extreme Boosting Based Outlier Detection (Supervised) 2018 43
Outlier Ensembles LODA Lightweight On-line Detector of Anomalies 2016 44
Outlier Ensembles SUOD SUOD: Accelerating Large-scale Unsupervised Heterogeneous Outlier Detection (Acceleration) 2021 45
Combination Average Simple combination by averaging the scores 2015 46
Combination Weighted Average Simple combination by averaging the scores with detector weights 2015 47
Combination Maximization Simple combination by taking the maximum scores 2015 48
Combination AOM Average of Maximum 2015 49
Combination MOA Maximization of Average 2015 50
Combination Median Simple combination by taking the median of the scores 2015 51
Combination majority Vote Simple combination by taking the majority vote of the labels (weights can be used) 2015 52

(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

Algorithm Benchmark

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

Comparision_of_All

A benchmark is supplied for select algorithms to provide an overview of the implemented models. In total, 17 benchmark datasets are used for comparison, which can be downloaded at ODDS.

For each dataset, it is first split into 60% for training and 40% for testing. All experiments are repeated 10 times independently with random splits. The mean of 10 trials is regarded as the final result. Three evaluation metrics are provided:

  • The area under receiver operating characteristic (ROC) curve
  • Precision @ rank n (P@N)
  • Execution time

Check the latest benchmark. You could replicate this process by running benchmark.py.

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):

kNN example 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 zhaoy@cmu.edu :)

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. 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.↩

  2. 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).↩

  3. 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).↩

  4. 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.↩

  5. 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).↩

  6. 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).↩

  7. Li, Z., Zhao, Y., Hu, X., Botta, N., Ionescu, C. and Chen, H. G. ECOD: Unsupervised Outlier Detection Using Empirical Cumulative Distribution Functions. arXiv preprint arXiv:2201.00382 (2021).↩

  8. Kriegel, H.P. and Zimek, A., 2008, August. Angle-based outlier detection in high-dimensional data. In KDD '08, pp. 444-452. ACM.↩

  9. Kriegel, H.P. and Zimek, A., 2008, August. Angle-based outlier detection in high-dimensional data. In KDD '08, pp. 444-452. ACM.↩

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

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

  12. 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.↩

  13. 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.↩

  14. 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.↩

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

  16. 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.↩

  17. 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).↩

  18. 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.↩

  19. 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.↩

  20. 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.↩

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

  22. 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.↩

  23. 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.↩

  24. 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.↩

  25. 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.↩

  26. 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.↩

  27. 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.↩

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

  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. Lazarevic, A. and Kumar, V., 2005, August. Feature bagging for outlier detection. In KDD '05. 2005.↩

  31. 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.↩

  32. Zhao, Y. and Hryniewicki, M.K. XGBOD: Improving Supervised Outlier Detection with Unsupervised Representation Learning. IEEE International Joint Conference on Neural Networks, 2018.↩

  33. Pevný, T., 2016. Loda: Lightweight on-line detector of anomalies. Machine Learning, 102(2), pp.275-304.↩

  34. 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).↩

  35. Aggarwal, C.C., 2015. Outlier analysis. In Data mining (pp. 237-263). Springer, Cham.↩

  36. Kingma, D.P. and Welling, M., 2013. Auto-encoding variational bayes. arXiv preprint arXiv:1312.6114.↩

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

  38. 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.↩

  39. 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.↩

  40. 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.↩

  41. Lazarevic, A. and Kumar, V., 2005, August. Feature bagging for outlier detection. In KDD '05. 2005.↩

  42. 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.↩

  43. Zhao, Y. and Hryniewicki, M.K. XGBOD: Improving Supervised Outlier Detection with Unsupervised Representation Learning. IEEE International Joint Conference on Neural Networks, 2018.↩

  44. Pevný, T., 2016. Loda: Lightweight on-line detector of anomalies. Machine Learning, 102(2), pp.275-304.↩

  45. 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).↩

  46. Aggarwal, C.C. and Sathe, S., 2015. Theoretical foundations and algorithms for outlier ensembles.ACM SIGKDD Explorations Newsletter, 17(1), pp.24-47.↩

  47. Aggarwal, C.C. and Sathe, S., 2015. Theoretical foundations and algorithms for outlier ensembles.ACM SIGKDD Explorations Newsletter, 17(1), pp.24-47.↩

  48. Aggarwal, C.C. and Sathe, S., 2015. Theoretical foundations and algorithms for outlier ensembles.ACM SIGKDD Explorations Newsletter, 17(1), pp.24-47.↩

  49. Aggarwal, C.C. and Sathe, S., 2015. Theoretical foundations and algorithms for outlier ensembles.ACM SIGKDD Explorations Newsletter, 17(1), pp.24-47.↩

  50. Aggarwal, C.C. and Sathe, S., 2015. Theoretical foundations and algorithms for outlier ensembles.ACM SIGKDD Explorations Newsletter, 17(1), pp.24-47.↩

  51. Aggarwal, C.C. and Sathe, S., 2015. Theoretical foundations and algorithms for outlier ensembles.ACM SIGKDD Explorations Newsletter, 17(1), pp.24-47.↩

  52. Aggarwal, C.C. and Sathe, S., 2015. Theoretical foundations and algorithms for outlier ensembles.ACM SIGKDD Explorations Newsletter, 17(1), pp.24-47.↩