%0 Generic %1 giometto2014sample %A Giometto, Andrea %A Formentin, Marco %A Rinaldo, Andrea %A Cohen, Joel E. %A Maritan, Amos %D 2014 %K exponents finite population variance %T Sample and population exponents of generalized Taylor's law %U http://arxiv.org/abs/1412.5026 %X Taylor's law (TL) states that the variance $V$ of a non-negative random variable is a power function of its mean $M$, i.e. $V=a M^b$. The ubiquitous empirical verification of TL, typically displaying sample exponents $b \simeq 2$, suggests a context-independent mechanism. However, theoretical studies of population dynamics predict a broad range of values of $b$. Here, we explain this apparent contradiction by using large deviations theory to derive a generalized TL in terms of sample and populations exponents $b_jk$ for the scaling of the $k$-th vs the $j$-th cumulant (conventional TL is recovered for $b=b_12$), with the sample exponent found to depend predictably on the number of observed samples. Thus, for finite numbers of observations one observes sample exponents $b_jkk/j$ (thus $b\simeq2$) independently of population exponents. Empirical analyses on two datasets support our theoretical results.

@misc{giometto2014sample, abstract = {Taylor's law (TL) states that the variance $V$ of a non-negative random variable is a power function of its mean $M$, i.e. $V=a M^b$. The ubiquitous empirical verification of TL, typically displaying sample exponents $b \simeq 2$, suggests a context-independent mechanism. However, theoretical studies of population dynamics predict a broad range of values of $b$. Here, we explain this apparent contradiction by using large deviations theory to derive a generalized TL in terms of sample and populations exponents $b_{jk}$ for the scaling of the $k$-th vs the $j$-th cumulant (conventional TL is recovered for $b=b_{12}$), with the sample exponent found to depend predictably on the number of observed samples. Thus, for finite numbers of observations one observes sample exponents $b_{jk}\simeq k/j$ (thus $b\simeq2$) independently of population exponents. Empirical analyses on two datasets support our theoretical results.}, added-at = {2014-12-17T11:18:40.000+0100}, author = {Giometto, Andrea and Formentin, Marco and Rinaldo, Andrea and Cohen, Joel E. and Maritan, Amos}, biburl = {https://www.bibsonomy.org/bibtex/24dd4f21a0f5f04f19ba32edb16cac543/marcogherardi}, description = {Sample and population exponents of generalized Taylor's law}, interhash = {ad3e677ef9e7c93415cf21c3c369d2bb}, intrahash = {4dd4f21a0f5f04f19ba32edb16cac543}, keywords = {exponents finite population variance}, note = {cite arxiv:1412.5026Comment: 41 pages, 10 figures, 6 tables}, timestamp = {2014-12-17T11:18:40.000+0100}, title = {Sample and population exponents of generalized Taylor's law}, url = {http://arxiv.org/abs/1412.5026}, year = 2014 }