Department of Chemistry and Department of Physics, University of Illinois Chicago, Chicago, Illinois 60607, USA.
J Chem Phys. 2023 May 21;158(19). doi: 10.1063/5.0151614.
Power law distributions are widely observed in chemical physics, geophysics, biology, and beyond. The independent variable x of these distributions has an obligatory lower bound and, in many cases, also an upper bound. Estimating these bounds from sample data is notoriously difficult, with a recent method involving O(N3) operations, where N denotes sample size. Here I develop an approach for estimating the lower and upper bounds that involve O(N) operations. The approach centers on calculating the mean values, x̂min and x̂max, of the smallest x and the largest x in N-point samples. A fit of x̂min or x̂max as a function of N yields the estimate for the lower or upper bound. Application to synthetic data demonstrates the accuracy and reliability of this approach.
幂律分布广泛存在于化学物理、地球物理、生物等领域。这些分布的自变量 x 具有强制性的下限，在许多情况下，也具有上限。从样本数据中估计这些边界值非常困难，最近的一种方法涉及 O(N3) 操作，其中 N 表示样本大小。在这里，我开发了一种涉及 O(N) 操作的估计上下限的方法。该方法的核心是计算 N 点样本中最小 x 和最大 x 的平均值 x̂min 和 x̂max。将 x̂min 或 x̂max 拟合为 N 的函数可以得到下限或上限的估计值。对合成数据的应用证明了该方法的准确性和可靠性。
