Generalized Taylor's Law for Dependent and Heterogeneous Heavy-Tailed Data
A new study extends Taylor's law, which links variance and mean in data, to heavy-tailed and dependent datasets where mean and variance can be infinite. The authors introduce a new covariance condition and prove the law holds for time series and network data, including real-world networks like Wikipedia and Epinions. Their findings show the log-variance versus log-mean relationship remains valid, even with infinite moments, and the scaling exponent can reveal structural patterns in complex systems.
What it examines
This paper studies Taylor's law, which links variance to mean, in data with heavy tails, dependence, and heterogeneity. The authors generalize existing results to cover time series, network data, and cases with infinite mean and variance, using probabilistic limits, mixing conditions, and simulations to support their findings.
What it concludes
The results show Taylor's law holds for dependent, heterogeneous, and network data with heavy tails, even when mean and variance are infinite. This has practical uses in analyzing social networks, finance, and risk management. Future work may extend these results to more complex network structures and relax independence assumptions.
Evidence objects
The study introduces a new covariance condition, Condition A(p), and proves Taylor's law applies under weak dependence and mixing, making it robust for time series, network data, and complex systems like social networks.
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Researchers extend Taylor's law, showing its mean-variance scaling holds even for heavy-tailed data where both mean and variance can be infinite, challenging previous statistical assumptions and broadening its real-world relevance.
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Extensive simulations and analysis of Wikipedia, Epinions, and DBpedia networks reveal a few nodes dominate activity, and the log-variance versus log-mean relationship remains meaningful, even when population moments are infinite.
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This paper innovatively generalizes Taylors law to dependent, heterogeneous, heavy-tailed data, extending its applicability to infinite mean/variance distributions and network structures. Employing Karamatas theorem and a probabilistic approach, it offers novel insights for Quantitative Risk Management, making it compelling for modeling extreme events and financial data with dependencies.
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Raw abstract and provenance
Abstract: Taylor's law, also known as fluctuation scaling in physics and the power-law variance function in statistics, is an empirical pattern widely observed across fields including ecology, physics, finance, and epidemiology. It states that the variance of a sample scales as a power function of the mean of the sample. We study generalizations of Taylor's law in the context of heavy-tailed distributions w… ▽ More Taylor's law, also known as fluctuation scaling in physics and the power-law variance function in statistics, is an empirical pattern widely observed across fields including ecology, physics, finance, and epidemiology. It states that the variance of a sample scales as a power function of the mean of the sample. We study generalizations of Taylor's law in the context of heavy-tailed distributions with infinite mean and variance. We establish the probabilistic limit and analyze the associated convergence rates. Our results extend the existing literature by relaxing the i.i.d. assumption to accommodate dependence and heterogeneity among the random variables. This generalization enables application to dependent data such as time series and network-structured data. We support the theoretical developments by extensive simulations, and the practical relevance through applications to real network data. △ Less
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