Taylor’s law of fluctuation scaling for semivariances and higher moments of heavy-tailed data

We generalize Taylor’s law for the variance of light-tailed distributions to many sample statistics of heavy-tailed distributions with tail index α in (0, 1), which have infinite mean. We show that, as the sample size increases, the sample upper and lower semivariances, the sample higher moments, the skewness, and the kurtosis of a random sample from such a law increase asymptotically in direct proportion to a power of the sample mean. Specifically, the lower sample semivariance asymptotically scales in proportion to the sample mean raised to the power 2, while the upper sample semivariance asymptotically scales in proportion to the sample mean raised to the power (2−α)/(1−α)>2. The local upper sample semivariance (counting only observations that exceed the sample mean) asymptotically scales in proportion to the sample mean raised to the power (2−α2)/(1−α). These and additional scaling laws characterize the asymptotic behavior of commonly used measures of the risk-adjusted performance of investments, such as the Sortino ratio, the Sharpe ratio, the Omega index, the upside potential ratio, and the Farinelli–Tibiletti ratio, when returns follow a heavy-tailed nonnegative distribution. Such power-law scaling relationships are known in ecology as Taylor’s law and in physics as fluctuation scaling. We find the asymptotic distribution and moments of the number of observations exceeding the sample mean. We propose estimators of α based on these scaling laws and the number of observations exceeding the sample mean and compare these estimators with some prior estimators of α.

Application of the laser-induced fluorescence to study local characteristics of a gas-liquid flow in rectangular microchannel

The Laser-Induced Fluorescence (LIF) method was used to characterize liquid phase distribution in rectangular slit microchannel with cross-section 200×1205 μm for horizontal gas-liquid flow. Ethanol and nitrogen were used as working liquid and gas accordingly. The feature of this study is an application of hydraulic focusing cross-junction mixer for obtaining elongated bubble and transition flows in the microchannel with a high aspect ratio. Using LIF measurements for elongated bubble and transition flows the liquid film distributions were obtained for different distances from the bubble top and average liquid film thickness was compared with the prediction according to Taylor’s law.

Taylor's Law Detects and Interprets Temporal Trends of the Spatial Distribution of Covid-19 Daily Infection Density Across Italian Provinces

Background: Taylor’s law, which formulates a power-law relationship between the variance and the mean, is used to assess the convergence over time to the spatial uniformity of the density of new infections in Italian provinces during the Covid-19 pandemic.Methods: Using the frequency of the number of new daily COVID 19 infections in the 110 Italian provinces during the period from 25 February 2020 to 15 March 2021, we estimated Taylor's Law using a system of simultaneous equations in five time sub-periods.Results: The infection in Italy initially manifested itself almost exclusively in a few provinces in the north and was very intense. Within a month, the infection spread to the whole country, with varying intensity between provinces. Subsequently, partly as a result of the introduction of containment measures or their relaxation, the infection manifested itself in different ways depending on both the time frame and the geographical area.Conclusions: Our results show that Taylor’s law fits well the temporal dynamics of infection density and can therefore be considered as a useful tool to predict the spatial variability of the Covid-19 infection density. Its slope parameter reflects the temporal correlation of infection density and the effect of nationwide lockdown on the disease spread through the lens of two stochastic population growth models. Our finding suggests that Taylor's law may be used to monitor the Covid-19 outbreak development and to predict the spatial spread of the infection density in the Italian provinces in successive pandemic waves of Covid-19.

A macroecological perspective on the fluctuations of exploited fish populations

The natural variability of fish populations is increased by exploitation, but the specific mechanisms driving this variability are still debated. We propose a macroscopic approach combining the size-density relationship and Taylor’s law to predict the temporal variance of exploited and unexploited fish populations. Using information from 11 years of fishery-independent abundance surveys, we showed that the body-size dependence of the variance of exploited (targeted) and unexploited (non-targeted or bycatch) fish populations can be accurately predicted. Targeted fish populations showed a variability that was 2 orders of magnitude greater than that of non-targeted fish populations. Such variability was explained solely by the higher relative abundance of the former, regardless of their specific trophic position, while aggregated community fluctuation was lower in a high trophic position group. This study showed the usefulness of the macroscopic approach to predict fish variability and fishing effect in the whole community. This approach is complementary to other modeling strategies and seems to be useful in tackling the problem of variability in population fluctuations of exploited fish, particularly in cases where specific details of the interacting species are lacking.

Direct Scaling of Measure on Vortex Shedding through a Flapping Flag Device in the Open Channel around a Cylinder at Re∼103: Taylor’s Law Approach

The problem of vortex shedding, which occurs when an obstacle is placed in a regular flow, is governed by Reynolds and Strouhal numbers, known by dimensional analysis. The present work aims to propose a thin films-based device, consisting of an elastic piezoelectric flapping flag clamped at one end, in order to determine the frequency of vortex shedding downstream an obstacle for a flow field at Reynolds number Re∼103 in the open channel. For these values, Strouhal number obtained in such way is in accordance with the results known in literature. Moreover, the development of the voltage over time, generated by the flapping flag under the load due to flow field, shows a highly fluctuating behavior and satisfies Taylor’s law, observed in several complex systems. This provided useful information about the flow field through the constitutive law of the device.