# 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
(2007) [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] Direct importance estimation with model selection and its application to covariate shift adaptation. In Proceedings of the 21 st International Conference on Neural Information Processing Systems, Vancouver, BC, Canada, 3-6 December 2007; Volume 20. * [PERSON] et al. (2013) [PERSON]; [PERSON]; [PERSON] [PERSON]; [PERSON] [PERSON]; [PERSON] [PERSON] Relative density-ratio estimation for robust distribution comparison. _Neural Comput._**2013**, _25_, 1324-1370. [CrossRef]
* [PERSON] et al. (2009) [PERSON]; [PERSON]; [PERSON] [PERSON] A least-squares approach to direct importance estimation. _[PERSON]. [PERSON]. Learn. Res._**2009**, _10_, 1391-1445. * [PERSON] et al. (2018) [PERSON]; [PERSON]; [PERSON]; [PERSON] Generative Ratio Matching Networks. _arXiv_**2018**, arXiv:1806.00101. * [PERSON] and [PERSON] (2021) [PERSON]; [PERSON] [PERSON] Non-Negative Bregman Divergence Minimization for Deep Direct Density Ratio Estimation. In Proceedings of the 38 th International Conference on Machine Learning, Virtual, 18-24 July 2021; Proceedings of Machine Learning Research; Volume 139, pp. 5320-5333. * [PERSON] et al. (2022) [PERSON]; [PERSON]; [PERSON] Unsupervised Anomaly Detection by Robust Density Estimation. In Proceedings of the AAAI Conference on Artificial Intelligence, Online, 22 February-1 March 2022; Volume 36, pp. 4101-4108. [CrossRef]
* [PERSON] et al. (2024) [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] Multivariate Time Series Change-Point Detection with a Novel Pearson-like Scaled Bregman Divergence. _Stats_**2024**, \(7\), 462-480. [CrossRef] [PubMedMed]
* [PERSON] et al. (2012) [PERSON]; [PERSON]; [PERSON] [PERSON] _Density Ratio Estimation in Machine Learning_; Cambridge University Press: Cambridge, UK, 2012. [CrossRef]
* [PERSON] et al. (2011) [PERSON]; [PERSON]; [PERSON]; [PERSON] [PERSON]; [PERSON] [PERSON] Statistical outlier detection using direct density ratio estimation. _Knowl. Inf. Syst._**2011**, _26_, 309-336. [CrossRef]
* [PERSON] et al. (2011) [PERSON]; [PERSON] [PERSON]; [PERSON]; [PERSON] Direct density-ratio estimation with dimensionality reduction via least-squares hetero-distributional subspace search. _Neural Netw._**2011**, _24_, 183-198. [CrossRef] [PubMedMed]
* [PERSON] et al. (2013) [PERSON] [PERSON]; [PERSON]; [PERSON]; [PERSON] [PERSON]; [PERSON] [PERSON] Direct divergence approximation between probability distributions and its applications in machine learning. _J. Comput. Sci. Eng._**2013**, \(7\), 99-111. [CrossRef]
* [PERSON] et al. (2013) [PERSON]; [PERSON] [PERSON]; [PERSON] Change-Point Detection in Time-Series Data by Relative Density-Ratio Estimation. _Neural Netw._**2013**, _37_, 396-411. [CrossRef]
* [PERSON] and [PERSON] (2012) [PERSON]; [PERSON] [PERSON] On Bregman distances and divergences of probability measures. _IEEE Trans. Inf. Theory_**2012**, _58_, 1277-1288. [CrossRef]
* [PERSON] et al. (2013) [PERSON]; [PERSON]; [PERSON] Computational complexity of kernel-based density-ratio estimation: A condition number analysis. _Mach. Learn._**2013**, _90_, 431-460. [CrossRef]
* [PERSON] et al. (2001) [PERSON]; [PERSON]; [PERSON] Benchmark Repository for Machine Learning. 2001. Available online: [[https://zenodo.org/records/18110](https://zenodo.org/records/18110)]([https://zenodo.org/records/18110](https://zenodo.org/records/18110)) (accessed on 11 December 2024). * [PERSON] et al. (2023) [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] [PERSON] A novel f-divergence based generative adversarial imputation method for scRNA-seq data analysis. _PLoS ONE_**2023**, _18_, e0292792. [CrossRef]
* [PERSON] et al. (2025) [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] [PERSON] Bidirectional f-Divergence-Based Deep Generative Method for Imputing Missing Values in Time-Series Data. _Stats_**2025**, \(8\), 7. [CrossRef]
* [PERSON] (2018) [PERSON]; [PERSON] [PERSON] PBLR: An accurate single cell RNA-seq data imputation tool considering cell heterogeneity and prior expression level of dropouts. _bioRxiv_**2018**, _379883_.