💡 Theory overview

Topological Data Analysis is a recent and fast growing field providing topological and geometric tools to infer features for complex data. See an introduction in [CM21].

Topological Anomaly Detection in this module:

• run a sliding window algorithm and represent each time series window with topological features,

  see Topological Embedding,

• use a MinCovDet algorithm to robustly estimate the data mean and covariance in the embedding space,

  and use these to derive an embedding mahalanobis distance and associated outlier detection procedure, see Elliptic Envelope.

This library is the implementation result of the TADA algorithm introduced in [CLR24].

For more details on the way to produce a temporal topological embedding, please refer to [CLR24].

Elliptic Envelope.

Essentially once you estimate the mean and covariance of a set of vectors, assuming a Gaussian multivariate span you have a natural envelope of said span using the mahalanobis distance. Elliptic Envelope is a sklearn tool that derives that score.

📑 References

[CM21]

F. Chazal, B. Michel. An introduction to Topological Data Analysis: fundamental and practical aspects for data scientists. Frontiers in AI, 2021.

[CLR24] (1,2)

F. Chazal, C. Levrard and M. Royer. Topological Analysis for Detecting Anomalies (TADA) in dependent sequences: application to Time Series. Journal of Machine Learning Research, 2024.