Power law scaling models have been used to understand the complexity of systems as diverse as cities, neurological activity, and rainfall and lightning. In the scaling framework, power laws and standard linear regression methods are widely used to estimate model parameters with assumed normality and fixed variance. Generalized linear models (GLM) can accommodate a wider range of distributions where the chosen distribution must meet the assumptions of the data to prevent model bias. We present a widely applicable Bayesian generalized logistic regression (BGLR) framework to more flexibly model a continuous real response addressing skew and heteroscedasticity. The Generalized Logistic Distribution (GLD) was selected to flexibly model skewed continuous data. This resulted in a nonlinear posterior distribution which may not have an analytical solution which can be solved numerically with Markov Chain Monte Carlo (MCMC) methods. We compared the BGLR model to standard and Bayesian normal models having fixed and varying variance when fitting power laws to 759 days of COVID-19 data. The BGLR yielded information beyond existing methods about the evolution of skew and skedasticity while revealing parameter bias of widely used methods. The BGLR flexibly modelled the complex characteristics necessary for an improved understanding of the propagation and dynamics of this infectious disease. The model is generally applicable and can be used as a template for modeling complexity with other distributions.

翻译：幂律标度模型已被用于理解从城市、神经活动到降雨和闪电等多样系统的复杂性。在标度框架中，幂律和标准线性回归方法被广泛用于估计模型参数，其中假设正态性和固定方差。广义线性模型可以适应更广泛的分布，但所选分布必须满足数据假设以避免模型偏差。我们提出了一种广泛适用的贝叶斯广义逻辑回归框架，以更灵活地建模连续实值响应，解决偏斜和异方差问题。选择广义逻辑分布来灵活建模偏斜连续数据，这导致后验分布是非线性的，可能没有解析解，但可以通过马尔可夫链蒙特卡洛方法进行数值求解。我们将BGLR模型与固定和变动方差下的标准及贝叶斯正态模型进行了比较，拟合了759天COVID-19数据的幂律。BGLR揭示了现有方法无法提供的关于偏斜和波动性演化的信息，同时揭示了广泛使用方法的参数偏差。BGLR灵活建模了理解这种传染病传播和动态所需复杂特征。该模型具有普遍适用性，可作为使用其他分布建模复杂性的模板。