Generalized -means in GLMs with applications to the outbreak of COVID-19 in the United States

Generalized $$ -means can be combined with any similarity or dissimilarity measure for clustering. Using the well known likelihood ratio or $$ -statistic as the dissimilarity measure, a generalized $$ -means method is proposed to group generalized linear models (GLMs) for exponential family distributions. Given the number of clusters $$ , the proposed method is established by the uniform most powerful unbiased (UMPU) test statistic for the comparison between GLMs. If $$ is unknown, then the proposed method can be combined with generalized liformation criterion (GIC) to automatically select the best $$ for clustering. Both AIC and BIC are investigated as special cases of GIC. Theoretical and simulation results show that the number of clusters can be correctly identified by BIC but not AIC. The proposed method is applied to the state-level daily COVID-19 data in the United States, and it identifies 6 clusters. A further study shows that the models between clusters are significantly different from each other, which confirms the result with 6 clusters.