# Anomaly Detection in High-Dimensional Time Series Data with Scaled Bregman Divergence
## 2 Materials and Methods
### Algorithm and Computational Complexity Analysis
#### 2.3.1 Anomaly Detection Algorithm
So, although deep learning methods may be better suited for scenarios with abundant computational resources and well-annotated datasets, our method offers a competitive alternative in settings with limited computational capacity or scarce labeled data. Conceptualization, H.G. ; methodology, Y.W., L.Z., T.S., G.B. and H.G. ; formal analysis, Y.W. and L.Z. ; writing, Y.W., L.Z. and H.G. ; funding acquisition, H.G. All authors have read and agreed to the published version of the manuscript. This research was partially supported by the National Institute of General Medical Sciences of the National Institutes of Health under Award Number R15 GM148915 (H.G.) and President's Research Funds (H.G.). The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health. Not applicable. The original Yahoo data presented in the study are available from Yahoo at [[https://webscope.sandbox.yahoo.com/](https://webscope.sandbox.yahoo.com/)]([https://webscope.sandbox.yahoo.com/](https://webscope.sandbox.yahoo.com/)) (accessed on 15 January 2025). The original Biomedical Benchmark data are openly available in Zenodo at [[https://zenodo.org/records/18110](https://zenodo.org/records/18110)]([https://zenodo.org/records/18110](https://zenodo.org/records/18110)) (accessed on 15 January 2025). The R code for the proposed method is available on GitHub: [[https://github.com/yungewang/AnomalyDetection-by-Density-Ratio-Estimation](https://github.com/yungewang/AnomalyDetection-by-Density-Ratio-Estimation)]([https://github.com/yungewang/AnomalyDetection-by-Density-Ratio-Estimation](https://github.com/yungewang/AnomalyDetection-by-Density-Ratio-Estimation)) (accessed on 15 January 2025). The authors declare no conflicts of interest. The following abbreviations are used in this manuscript:

PLSBD Pearson-like scaled Bregman divergence

RuLSIF Relative unconstrained least-squares importance fitting

uLSIF Unconstrained least-squares importance fitting

ROC Receiver operating characteristic

AUC Area under the curve

## References

* (1) [PERSON]; [PERSON]; [PERSON] Anomaly detection for cybersecurity of the substations. _IEEE Trans. Smart Grid_**2011**, \(2\), 865-873. [CrossRef]
* (2) [PERSON]; [PERSON]; [PERSON] Anomaly detection in graphs of bank transactions for anti money laundering applications. _IEEE Access_**2022**, _10_, 47699-47714. [CrossRef]
* (3) [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] An anomaly detection approach to identify chronic brain infarcts on MRI. _Sci. Rep._**2021**, _11_, 7714. [CrossRef] [PubMed]
* (4) [PERSON]; [PERSON]. A comprehensive survey of anomaly detection in banking, wireless sensor networks, social networks, and healthcare. _Intell. Decis. Technol._**2019**, _13_, 229-270. [CrossRef]
* (5) [PERSON]; [PERSON]; [PERSON]; [PERSON] A review of novelty detection. _Signal Process._**2014**, _99_, 215-249. [CrossRef]
* (6) [PERSON]; [PERSON]; [PERSON] A survey of network anomaly detection techniques. _J. Netw. Comput. Appl._**2016**, _60_, 19-31. [CrossRef]
* (7) [PERSON]; [PERSON]; [PERSON] Machine learning techniques for anomaly detection: An overview. _Int. J. Comput. Appl._**2013**, _79_, 33-41. * (8) [PERSON] Cancer outlier detection based on likelihood ratio test. _Bioinformatics_**2008**, _24_, 2193-2199. [CrossRef]
* (9) [PERSON]; [PERSON]; [PERSON] Peakseg: Constrained optimal segmentation and supervised penalty learning for peak detection in count data. In Proceedings of the 32 nd International Conference on Machine Learning, Lille, France, 6-11 July 2015; Volume 37, pp. 324-332. * (10) [PERSON]; [PERSON]; [PERSON]; [PERSON] Sequential (quickest) change detection: Classical results and new directions. _IEEE J. Sel. Areas Inf. Theory_**2021**, \(2\), 494-514. [CrossRef]
* (11) [PERSON]; [PERSON]; [PERSON] [PERSON]; [PERSON] Octopus-IIDS: An anomaly based intelligent intrusion detection system. In Proceedings of the IEEE Symposium on Computers and Communications, Riccione, Italy, 22-25 June 2010; pp. 405-410.