Fontanelli Oscar, Miramontes Pedro, Yang Yaning, Cocho Germinal, Li Wentian
Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México, México, DF, México.
Bioinformatics Group and Interdisciplinary Center for Bioinformatics, University of Leipzig, Leipzig, Germany.
PLoS One. 2016 Sep 22;11(9):e0163241. doi: 10.1371/journal.pone.0163241. eCollection 2016.
Although Zipf's law is widespread in natural and social data, one often encounters situations where one or both ends of the ranked data deviate from the power-law function. Previously we proposed the Beta rank function to improve the fitting of data which does not follow a perfect Zipf's law. Here we show that when the two parameters in the Beta rank function have the same value, the Lavalette rank function, the probability density function can be derived analytically. We also show both computationally and analytically that Lavalette distribution is approximately equal, though not identical, to the lognormal distribution. We illustrate the utility of Lavalette rank function in several datasets. We also address three analysis issues on the statistical testing of Lavalette fitting function, comparison between Zipf's law and lognormal distribution through Lavalette function, and comparison between lognormal distribution and Lavalette distribution.
尽管齐普夫定律在自然和社会数据中广泛存在，但人们经常遇到这样的情况：排序数据的一端或两端偏离幂律函数。此前我们提出了贝塔秩函数，以改善对不遵循完美齐普夫定律的数据的拟合。在此我们表明，当贝塔秩函数中的两个参数具有相同值时，即拉瓦莱特秩函数，可以解析地推导出概率密度函数。我们还通过计算和分析表明，拉瓦莱特分布虽然与对数正态分布不完全相同，但大致相等。我们在几个数据集中说明了拉瓦莱特秩函数的效用。我们还讨论了关于拉瓦莱特拟合函数统计检验的三个分析问题、通过拉瓦莱特函数对齐普夫定律和对数正态分布进行比较，以及对数正态分布和拉瓦莱特分布之间的比较。
