Power law and entropy analysis of catastrophic phenomena

Catastrophic events, such as wars and terrorist attacks, tornadoes and hurricanes, earthquakes, tsunamis, floods and landslides, are always accompanied by a large number of casualties. The size distribution of these casualties has separately been shown to follow approximate power law (PL) distributions. In this paper, we analyze the statistical distributions of the number of victims of catastrophic phenomena, in particular, terrorism, and find double PL behavior. This means that the data sets are better approximated by two PLs instead of a single one. We plot the PL parameters, corresponding to several events, and observe an interesting pattern in the charts, where the lines that connect each pair of points defining the double PLs are almost parallel to each other. A complementary data analysis is performed by means of the computation of the entropy. The results reveal relationships hidden in the data that may trigger a future comprehensive explanation of this type of phenomena.

Double power laws, fractals and self-similarity

Power law (PL) distributions have been largely reported in the modeling of distinct real phenomena and have been associated with fractal structures and self-similar systems. In this paper, we analyze real data that follows a PL and a double PL behavior and verify the relation between the PL coefficient and the capacity dimension of known fractals. It is to be proved a method that translates PLs coefficients into capacity dimension of fractals of any real data.

Leis de pot��ncia em economia e finan��as

Nesta disserta����o aborda-se a aplica����o de Leis de Pot��ncia (LPs), tamb��m designadas de Leis de Pareto ou Leis de Zipf, a dados econ��micos. As LPs s��o distribui����es estat��sticas amplamente usadas na compreens��o de sistemas naturais e artificiais. O aparecimento das LPs deve-se a Vilfredo Pareto que, no s��culo XIX, publicou o manual de economia pol��tica,���Cours d���Economie Politique���. Nesse manual refere que grande parte da economia mundial segue uma LP, em que 20% da popula����o re��ne 80% da riqueza do pa��s. Esta propriedade carateriza uma vari��vel que segue uma distribui����o de Pareto (ou LP). Desde ent��o, as LPs foram aplicadas a outros fen��menos, nomeadamente a ocorr��ncia de palavras em textos, os sobrenomes das pessoas, a varia����o dos rendimentos pessoais ou de empresas, o n��mero de v��timas de inunda����es ou tremores de terra, os acessos a s��tios da internet, etc. Neste trabalho, �� estudado um conjunto de dados relativos ��s fortunas particulares ou coletivas de pessoas ou organiza����es. Mais concretamente s��o analisados dados recolhidos sobre as fortunas das mulheres mais ricas do mundo, dos homens mais ricos no ramo da tecnologia, das fam��lias mais ricas, das 20 mulheres mais ricas da Am��rica, dos 400 homens mais ricos da Am��rica, dos homens mais ricos do mundo, dos estabelecimentos mais ricos do mundo, das empresas mais ricas do mundo e dos pa��ses mais ricos do mundo, bem como o valor de algumas empresas no mercado de a����es. Os resultados obtidos revelam uma boa aproxima����o de parte desses dados a uma LP simples e uma boa aproxima����o pelos restantes dados a uma LP dupla. Observa-se, assim, diferencia����o na forma de crescimento das fortunas nos diferentes casos estudados. Como trabalho futuro, procurar-se-�� analisar estes e outros dados, utilizando outras distribui����es estat��sticas, como a exponencial ou a lognormal, que possuem comportamentos semelhantes �� LP, com o intuito de serem comparados os resultados. Um outro aspeto interessante ser�� o de encontrar a explica����o anal��tica para as vantagens da aproxima����o de dados econ��micos por uma LP simples vs por uma LP dupla.

Double power laws, fractals and self-similarity

Power law (PL) distributions have been largely reported in the modeling of distinct real phenomena and have been associated with fractal structures and self-similar systems. In this paper, we analyze real data that follows a PL and a double PL behavior and verify the relation between the PL coefficient and the capacity dimension of known fractals. It is to be proved a method that translates PLs coefficients into capacity dimension of fractals of any real data.

Casualties distribution in human and natural hazards

Catastrophic events, such as wars and terrorist attacks, big tornadoes and hurricanes, huge earthquakes, tsunamis, floods, and landslides, are always accompanied by a large number of casualties. The size distribution of these casualties have separately been shown to follow approximate power law (PL) distributions. In this paper, we analyze the number of victims of catastrophic phenomena, in particular, terrorism, and find double PL behavior. This means that the data set is better approximated by two PLs instead of one. We have plotted the two PL parameters corresponding to all terrorist events occurred in every year, from 1980 to 2010. We observe an interesting pattern in the chart, where the lines, that connect each pair of points defining the double PLs, are roughly aligned to each other.