# A Robust and Flexible EM Algorithm for Mixtures of Elliptical Distributions with Missing Data

arXiv: https://arxiv.org/abs/2201.12020 · HTML: https://mlsanity.com/paper/2201.12020/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 54 references that are in the map.
- First posted on arXiv 28 Jan 2022.

## Builds on (most cited first, 9 of 9)

- [A Mixture of Generalized Hyperbolic Distributions](https://mlsanity.com/paper/1305.1036/) (2013)
- [A flexible EM-like clustering algorithm for noisy data](https://mlsanity.com/paper/1907.01660/) (2019)
- [Robust low-rank covariance matrix estimation with a general pattern of missing values](https://mlsanity.com/paper/2107.10505/) (2021)
- [Structured Covariance Matrix Estimation with Missing-Data for Radar Applications via Expectation-Maximization](https://mlsanity.com/paper/2105.03738/) (2021)
- [Mixtures of Generalized Hyperbolic Distributions and Mixtures of Skew-t Distributions for Model-Based Clustering with Incomplete Data](https://mlsanity.com/paper/1703.02177/) (2017)
- [Parameter Estimation of Heavy-Tailed AR Model with Missing Data via Stochastic EM](https://mlsanity.com/paper/1809.07203/) (2018)
- [Provable Subspace Tracking from Missing Data and Matrix Completion](https://mlsanity.com/paper/1810.03051/) (2018)
- [Model-based Clustering with Missing Not At Random Data](https://mlsanity.com/paper/2112.10425/) (2021)
- [Reconstruction of Sentinel-2 Time Series Using Robust Gaussian Mixture Models -- Application to the Detection of Anomalous Crop Development in wheat and rapeseed crops](https://mlsanity.com/paper/2110.11780/) (2021)

## Built on by (most cited first, 2 of 2)

- [ZTFed-MAS2S: A Zero-Trust Federated Learning Framework with Verifiable Privacy and Trust-Aware Aggregation for Wind Power Data Imputation](https://mlsanity.com/paper/2508.18318/) (2025)
- [Signal Recovery Using a Spiked Mixture Model](https://mlsanity.com/paper/2501.01840/) (2025)

## Abstract (arXiv)

This paper tackles the problem of missing data imputation for noisy and non-Gaussian data. A classical imputation method, the Expectation Maximization (EM) algorithm for Gaussian mixture models, has shown interesting properties when compared to other popular approaches such as those based on k-nearest neighbors or on multiple imputations by chained equations. However, Gaussian mixture models are known to be non-robust to heterogeneous data, which can lead to poor estimation performance when the data is contaminated by outliers or follows non-Gaussian distributions. To overcome this issue, a new EM algorithm is investigated for mixtures of elliptical distributions with the property of handling potential missing data. This paper shows that this problem reduces to the estimation of a mixture of Angular Gaussian distributions under generic assumptions (i.e., each sample is drawn from a mixture of elliptical distributions, which is possibly different for one sample to another). In that case, the complete-data likelihood associated with mixtures of elliptical distributions is well adapted to the EM framework with missing data thanks to its conditional distribution, which is shown to be a multivariate t-distribution. Experimental results on synthetic data demonstrate that the proposed algorithm is robust to outliers and can be used with non-Gaussian data. Furthermore, experiments conducted on real-world datasets show that this algorithm is very competitive when compared to other classical imputation methods.