**1. Introduction**

Even though investments in cryptocurrencies were initially viewed as risky bets, increased participation by individuals as well as institutions have been transforming those views, as the cryptocurrency market is now perceived as a new asset class for many investors. With these revolutions, there have been numerous studies with the overall objective of understanding the cryptocurrency market [1–3] or, more specifically, cryptocurrencies as investment assets. Much effort has been put into analyzing diversification effects and evaluating cryptocurrencies as an asset class [4–8]. Others have focused on diversification across cryptocurrencies [9] and cross-correlation among themselves [10,11]. Predicting price movement of cryptocurrencies using social media data [12], the economic and political uncertainty of the crypto market [13,14], and liquidity of cryptocurrencies [15,16] have also been studied. Due to large swings in cryptocurrencies, analyses also focus on the volatility of the market. Many studies have investigated risk factors of cryptocurrencies [17–20] and examined models for volatility forecasting [21,22], including forecasts of daily value-at-risk [23].

In this article, the risk and volatility of the cryptocurrency market are further examined but from a macro view of observing anomaly scores of market movements. The analysis is based on viewing cryptocurrencies through anomaly scores measured with Mahalanobis distance and its robust variations. The analysis has two significant contributions. First, while cryptocurrencies are generally understood as being more volatile compared to traditional assets, observing anomaly scores provides a standardized framework for identifying unlikely or outlier events, where anomaly calculations incorporate mean, variance, and correlations. Furthermore, anomaly scores are analyzed through cryptocurrency returns as well as risk factors, and robust formulations are proposed to handle extreme outliers in cryptocurrencies. Second, anomaly scores can enhance portfolio management and scenario

**Citation:** Bae, G.; Kim, J.H. Observing Cryptocurrencies through Robust Anomaly Scores. *Entropy* **2022**, *24*, 1643. https://doi.org/ 10.3390/e24111643

Academic Editors: Stanisław Drozd˙ z,˙ Marcin W ˛atorek and Jarosław Kwapie ´n

Received: 27 September 2022 Accepted: 9 November 2022 Published: 11 November 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

analysis of cryptocurrencies. Anomaly scores can act as an indicator of abnormal market conditions, and they can also portray a statistical picture of historical events that provide a medium for measuring historical likelihood as well as estimated likelihood of future scenarios. Performing scenario analyses using risk factors allows a more intuitive and rational interaction with the crypto market. Overall, the analysis provides a practical example of analyzing the crypto market from a more macro perspective that is a valuable complement to volatility analysis of cryptocurrencies.
