*4.2. Anomaly Scores of Cryptocurrencies*

The first set of anomaly scores were computed directly from the top cryptocurrencies without the use of factors. In order to resolve sensitivity in MD, shrinkage estimators of the mean vector [26] and covariance matrix [27] were used. Shrinkage estimators combine unbiased estimators, such as sample mean, with another component with more structure. A shrinkage estimator for the vector of expected return can be expressed as:

$$
\hat{\mu}\_{shrink} = (1 - a)\hat{\mu} + a\mu\_0 1
$$

where *<sup>μ</sup>*<sup>ˆ</sup> <sup>∈</sup> <sup>R</sup>*<sup>n</sup>* is the sample mean, *<sup>μ</sup>*<sup>0</sup> <sup>∈</sup> <sup>R</sup> is the shrinkage target, 1 <sup>∈</sup> <sup>R</sup>*<sup>n</sup>* is the vector of ones, and *α* is the shrinkage intensity. Similarly, a shrinkage estimator for the covariance matrix of returns can be written as:

$$
\hat{\Sigma}\_{shrink} = (1 - a)S + a\Sigma\_0,
$$

where *<sup>S</sup>* <sup>∈</sup> <sup>R</sup>*n*×*<sup>n</sup>* is the sample covariance matrix and <sup>Σ</sup><sup>0</sup> <sup>∈</sup> <sup>R</sup>*n*×*<sup>n</sup>* is the target. In our analysis, the shrinkage target *μ*<sup>0</sup> was set as the expected return of the portfolio with lowest risk (minimum variance portfolio) and Σ<sup>0</sup> was set as a scaled identity matrix. Even though the sample estimates can be sensitive to the estimation period, shrinking them toward a shrinkage target improves robustness [40].

**Figure 1.** Historical annualized standard deviation (30-day rolling).

**Figure 2.** Historical values of Crypto Volatility index.

These shrinkage estimators are frequently applied in portfolio optimization to mitigate sensitivity in the performance of optimal allocations [41,42]. Figure 3 shows anomaly scores when mean and covariance are estimated from the entire period (from January 2018 to February 2022) or from only the last 104 weeks (from March 2000 to February 2020).

Several observations are noteworthy in Figure 3. For each of the seven figures, anomaly scores are not sensitive to the estimation period, and the results are very similar between the scores based on the market condition during January 2018 to February 2022 and the condition during March 2000 to February 2022. This clearly shows the strength of using shrinkage estimators (in contrast, Figure A2 demonstrates the high sensitivity of using nonrobust MD for measuring anomaly scores). Moreover, a comparison of the seven graphs in Figure 3 shows that the overall trend and spikes in anomaly scores are fairly robust to the choice of weekly return calculations. For all seven graphs, high anomaly scores are cited between late 2020 and mid-2021, followed by a short spike from around October to November of 2021. Even though there are spikes between late 2018 and mid-2019, the overall anomaly scores are relatively low from the beginning of 2018 until late 2020.

**Figure 3.** *Cont*.

**Figure 3.** Anomaly scores from top cryptocurrencies (with shrinkage estimators).

When compared with the volatility measures from Section 4.1, anomaly scores show that the high volatility periods during early-to-mid 2021 are also reflected in the anomaly scores. However, more importantly, the market movement during March to May of 2020 was rather *normal*, whereas the condition from October to November of 2021 was *abnormal*. It must be clarified that a *normal* period based on anomaly scores does not necessarily reflect a less volatile period. Since anomaly scores show squared distances from the mean that are standardized by the covariance matrix, a cryptocurrency with high volatility on average will not necessarily have a large anomaly score simply because it deviates much from the mean. This is the key reason why anomaly scores are not a substitute for market volatility but an essential complement for analyzing market movements. For example, high anomaly scores from October to November 2021 were caused by a large spike in Decentraland (MANA), which increased more than five times in less than two months. Further analysis shows that the high anomaly was not only a result of large returns but also due to changes in cross-correlation that were captured by MD. In fact, this was a period when metaverse cryptocurrencies were soaring and anomaly scores were able to capture this new wave in the market.
