On the ability of the benford’s law to detect earthquakes and discriminate seismic signals

• Autores: Jordi Díaz Cusí, Josep Gallart Muset, Mario Ruiz Fernández
• Localización: Seismological Research Letters, ISSN 0895-0695, ISSN-e 1938-2057, Vol. 86, Nº. 1, 2015, págs. 192-201
• Idioma: inglés
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• Resumen

  • Benford�s law (BL), also known as the first-digit law, is an intriguing pattern in data sets, first discovered more than a century ago then forgotten and rediscovered in the twentieth century. This law establishes that the frequency of occurrence of the first digit in many real-world data sets is not uniform, but instead favors the first digits. In particular, the occurrence of numbers starting with 1 and 2 is close to 30% and 18%, respectively, whereas numbers starting with 8 or 9 are close to 5%. This law was first formulated in the late nineteenth century by the Canadian�American astronomer and mathematician Simon Newcomb, who noticed that library books of logarithms were more thumbed in the first pages, those containing numbers starting with 1. He proposed a law stating that the probability of a digit N being the first digit of a number was equal to log(N+1)-log(N) (Newcomb, 1881).

    This law remained unnoticed for the scientific community for more than 50 years. In 1938, it was rediscovered by the engineer Frank Benford, who presented data collections from up to 20 different sources, including financial data, population of cities, or averages of the American baseball league, each of them satisfying the expected proportionality (Benford, 1938). His work became popular, and the rule finally took his name. Benford extended the rule to any arbitrary base of the logarithm and confirmed that it applies further than the leading digit.

    The law is commonly expressed as Embedded Image(1)in which P(d) is the probability of occurrence of the first significant digit d, considering this digit as the leftmost nonzero value and disregarding any negative sign or decimal point (d=1,2,�8,9). This implie...