# Taylor's law

Taylor's power law is an empirical law in ecology that relates the variance of the number of individuals of a species per unit area of habitat to the corresponding mean by a power law relationship. It is named after the ecologist who first proposed it in 1961, Lionel Roy Taylor (1924–2007). Taylor's original name for this relationship was the law of the mean. The name Taylor's law was coined by Southwood in 1966.

## Definition

This law was originally defined for ecological systems, specifically to assess the spatial clustering of organisms. For a population count $Y$ with mean $\mu$ and variance $\operatorname {var} (Y)$, Taylor's law is written
    $\operatorname {var} (Y)=a\mu ^{b},$
where a and b are both positive constants. Taylor proposed this relationship in 1961, suggesting that the exponent b be considered a species specific index of aggregation. This power law has subsequently been confirmed for many hundreds of species.
Taylor's law has also been applied to assess the time dependent changes of population distributions. Related variance to mean power laws have also been demonstrated in several non-ecological systems:
* cancer metastasis
* the numbers of houses built over the Tonami plain in Japan.
* measles epidemiology
* HIV epidemiology,
* the geographic clustering of childhood leukemia
* blood flow heterogeneity
* the genomic distributions of single-nucleotide polymorphisms (SNPs)
* gene structures
* in number theory with sequential values of the Mertens function and also with the distribution of prime numbers
* from the eigenvalue deviations of Gaussian orthogonal and unitary ensembles of random matrix theory

## History

The first use of a double log-log plot was by Reynolds in 1879 on thermal aerodynamics. Pareto used a similar plot to study the proportion of a population and their income.
The term variance was coined by Fisher in 1918.

### Biology

Pearson in 1921 proposed the equation (also studied by Neyman)
    $s^{2}=am+bm^{2}$
Smith in 1938 while studying crop yields proposed a relationship similar to Taylor's. This relationship was
    $\log V_{x}=\log V_{1}+b\log x\,$
where V$_x$ is the variance of yield for plots of x units, V$_1$ is the variance of yield per unit area and x is the size of plots. The slope (b) is the index of heterogeneity. The value of b in this relationship lies between 0 and 1. Where the yield are highly correlated b tends to 0; when they are uncorrelated b tends to 1.
Bliss in 1941, Fracker and Brischle in 1941 and Hayman & Lowe in 1961 also described what is now known as Taylor's law, but in the context of data from single species.
Taylor's 1961 paper used data from 24 papers, published between 1936 and 1960, that considered a variety of biological settings: virus lesions, macro-zooplankton, worms and symphylids in soil, insects in soil, on plants and in the air, mites on leaves, ticks on sheep and fish in the sea.; the b value lay between 1 and 3. Taylor proposed the power law as a general feature of the spatial distribution of these species. He also proposed a mechanistic hypothesis to explain this law.
Initial attempts to explain the spatial distribution of animals had been based on approaches like Bartlett's stochastic population models and the negative binomial distribution that could result from birth–death processes. Taylor's explanation was based the assumption of a balanced migratory and congregatory behavior of animals. His hypothesis was initially qualitative, but as it evolved it became semi-quantitative and was supported by simulations.
Many alternative hypotheses for the power law have been advanced. Hanski proposed a random walk model, modulated by the presumed multiplicative effect of reproduction. Hanski's model predicted that the power law exponent would be constrained to range closely about the value of 2, which seemed inconsistent with many reported values.
Anderson et al formulated a simple stochastic birth, death, immigration and emigration model that yielded a quadratic variance function. As a response to this model Taylor argued that such a Markov process would predict that the power law exponent would vary considerably between replicate observations, and that such variability had not been observed.
Kemp reviewed a number of discrete stochastic models based on the negative binomial, Neyman type A, and Polya–Aeppli distributions that with suitable adjustment of parameters could produce a variance to mean power law. Kemp, however, did not explain the parameterizations of his models in mechanistic terms. Other relatively abstract models for Taylor's law followed.
Statistical concerns were raised regarding Taylor's law, based on the difficulty with real data in distinguishing between Taylor's law and other variance to mean functions, as well the inaccuracy of standard regression methods.
Taylor's law has been applied to time series data, and Perry showed, using simulations, that chaos theory could yield Taylor's law.
Taylor's law has been applied to the spatial distribution of plants and bacterial populations As with the observations of Tobacco necrosis virus mentioned earlier, these observations were not consistent with Taylor's animal behavioral model.
A variance to mean power function had been applied to non-ecological systems, under the rubric of Taylor's law. A more general explanation for the range of manifestations of the power law a hypothesis has been proposed based on the Tweedie distributions, a family of probabilistic models that express an inherent power function relationship between the variance and the mean.
Several alternative hypotheses for the power law have been proposed. Hanski proposed a random walk model, modulated by the presumed multiplicative effect of reproduction. Hanski's model predicted that the power law exponent would be constrained to range closely about the value of 2, which seemed inconsistent with many reported values. Anderson et al formulated a simple stochastic birth, death, immigration and emigration model that yielded a quadratic variance function. The Lewontin Cohen growth model. is another proposed explanation. The possibility that observations of a power law might reflect more mathematical artifact than a mechanistic process was raised. Variation in the exponents of Taylor's Law applied to ecological populations cannot be explained or predicted based solely on statistical grounds however. Research has shown that variation within the Taylor's law exponents for the North Sea fish community varies with the external environment, suggesting ecological processes at least partially determine the form of Taylor's law.

### Physics

In the physics literature Taylor's law has been referred to as fluctuation scaling. Eisler et al, in a further attempt to find a general explanation for fluctuation scaling, proposed a process they called impact inhomogeneity in which frequent events are associated with larger impacts. In appendix B of the Eisler article, however, the authors noted that the equations for impact inhomogeneity yielded the same mathematical relationships as found with the Tweedie distributions.
Another group of physicists, Fronczak and Fronczak, derived Taylor's power law for fluctuation scaling from principles of equilibrium and non-equilibrium statistical physics. Their derivation was based on assumptions of physical quantities like free energy and an external field that caused the clustering of biological organisms. Direct experimental demonstration of these postulated physical quantities in relationship to animal or plant aggregation has yet to be achieved, though. Shortly thereafter, an analysis of Fronczak and Fronczak's model was presented that showed their equations directly lead to the Tweedie distributions, a finding that suggested that Fronczak and Fronczak had possibly provided a maximum entropy derivation of these distributions.

### Mathematics

Taylor's law has been shown to hold for prime numbers not exceeding a given real number. This result has been shown to hold for the first 11 million primes. If the Hardy–Littlewood twin primes conjecture is true then this law also holds for twin primes.

## The Tweedie hypothesis

About the time that Taylor was substantiating his ecological observations, MCK Tweedie, a British statistician and medical physicist, was investigating a family of probabilistic models that are now known as the Tweedie distributions. As mentioned above, these distributions are all characterized by a variance to mean power law mathematically identical to Taylor's law.
The Tweedie distribution most applicable to ecological observations is the compound Poisson-gamma distribution, which represents the sum of N independent and identically distributed random variables with a gamma distribution where N is a random variable distributed in accordance with a Poisson distribution. In the additive form its cumulant generating function (CGF) is:
    $K_{b}^{*}(s;\theta ,\lambda )=\lambda \kappa _{b}(\theta )\left[\left(1+{s \over \theta }\right)^{\alpha }-1\right],$
where κ$_b$(θ) is the cumulant function,
    $\kappa _{b}(\theta )={\frac {\alpha -1}{\alpha }}\left({\frac {\theta }{\alpha -1}}\right)^{\alpha },$
the Tweedie exponent
    $\alpha ={\frac {b-2}{b-1}},$
s is the generating function variable, and θ and λ are the canonical and index parameters, respectively.
These last two parameters are analogous to the scale and shape parameters used in probability theory. The cumulants of this distribution can be determined by successive differentiations of the CGF and then substituting s=0 into the resultant equations. The first and second cumulants are the mean and variance, respectively, and thus the compound Poisson-gamma CGF yields Taylor's law with the proportionality constant
    $a=\lambda ^{1/(\alpha -1)}.$
The compound Poisson-gamma cumulative distribution function has been verified for limited ecological data through the comparison of the theoretical distribution function with the empirical distribution function. A number of other systems, demonstrating variance to mean power laws related to Taylor's law, have been similarly tested for the compound Poisson-gamma distribution.
The main justification for the Tweedie hypothesis rests with the mathematical convergence properties of the Tweedie distributions. The Tweedie convergence theorem requires the Tweedie distributions to act as foci of convergence for a wide range of statistical processes. As a consequence of this convergence theorem, processes based on the sum of multiple independent small jumps will tend to express Taylor's law and obey a Tweedie distribution. A limit theorem for independent and identically distributed variables, as with the Tweedie convergence theorem, might then be considered as being fundamental relative to the ad hoc population models, or models proposed on the basis of simulation or approximation.
This hypothesis remains controversial; more conventional population dynamic approaches seem preferred amongst ecologists, despite the fact that the Tweedie compound Poisson distribution can be directly applied to population dynamic mechanisms.
One difficulty with the Tweedie hypothesis is that the value of b does not range between 0 and 1. Values of b < 1 are rare but have been reported.

## Mathematical formulation

In symbols
    $s_{i}^{2}=am_{i}^{b},$
where s$_i$$^2$ is the variance of the density of the ith sample, m$_i$ is the mean density of the ith sample and a and b are constants.
In logarithmic form
    $\log s_{i}^{2}=\log a+b\log m_{i}$

### Scale invariance

The exponent in Taylor's law is scale invariant: If the unit of measurement is changed by a constant factor $c$, the exponent ($b$) remains unchanged.
To see this let y = cx. Then
    $\mu _{1}=\operatorname {E} (x)$
    $\mu _{2}=\operatorname {E} (y)=\operatorname {E} (cx)=c\operatorname {E} (x)=c\mu _{1}$
    $\sigma _{1}^{2}=\operatorname {E} ((x-\mu _{1})^{2})$
    $\sigma _{2}^{2}=\operatorname {E} ((y-\mu _{2})^{2})=\operatorname {E} ((cx-c\mu _{1})^{2})=c^{2}\operatorname {E} ((x-\mu _{1})^{2})=c^{2}\sigma _{1}^{2}$
Taylor's law expressed in the original variable (x) is
    $\sigma _{1}^{2}=a\mu _{1}^{b}$
and in the rescaled variable (y) it is
    $\sigma _{2}^{2}=c^{2}\sigma _{1}^{2}=c^{2}a\mu _{1}^{b}=c^{2-b}a(c\mu _{1})^{b}=c^{2-b}a\mu _{2}^{b}$
Thus, $\sigma _{2}^{2}$ is still proportional to $\mu _{2}^{b}$ (even though the proportionality constant has changed).
It has been shown that Taylor's law is the only relationship between the mean and variance that is scale invariant.

### Extensions and refinements

A refinement in the estimation of the slope b has been proposed by Rayner.
    $b={\frac {f-\varphi +{\sqrt {(f-\varphi )^{2}-4r^{2}f\varphi }}}{2r{\sqrt {f}}}}$
where $r$ is the Pearson moment correlation coefficient between $\log(s^{2})$ and $\log m$, $f$ is the ratio of sample variances in $\log(s^{2})$ and $\log m$ and $\varphi$ is the ratio of the errors in $\log(s^{2})$ and $\log m$.
Ordinary least squares regression assumes that φ = ∞. This tends to underestimate the value of b because the estimates of both $\log(s^{2})$ and $\log m$ are subject to error.
An extension of Taylor's law has been proposed by Ferris et al when multiple samples are taken
    $s^{2}=cn^{d}m^{b},$
where s$^2$ and m are the variance and mean respectively, b, c and d are constants and n is the number of samples taken. To date, this proposed extension has not been verified to be as applicable as the original version of Taylor's law.

### Small samples

An extension to this law for small samples has been proposed by Hanski. For small samples the Poisson variation (P) - the variation that can be ascribed to sampling variation - may be significant. Let S be the total variance and let V be the biological (real) variance. Then
    $S=V+P$
Assuming the validity of Taylor's law, we have
    $V=am^{b}$
Because in the Poisson distribution the mean equals the variance, we have
    $P=m$
This gives us
    $S=V+P=am^{b}+m$
This closely resembles Barlett's original suggestion.

### Interpretation

Slope values (b) significantly > 1 indicate clumping of the organisms.
In Poisson-distributed data, b = 1. If the population follows a lognormal or gamma distribution, then b = 2.
For populations that are experiencing constant per capita environmental variability, the regression of log( variance ) versus log( mean abundance ) should have a line with b = 2.
Most populations that have been studied have b < 2 (usually 1.5–1.6) but values of 2 have been reported. Occasionally cases with b > 2 have been reported. b values below 1 are uncommon but have also been reported ( b = 0.93 ).
It has been suggested that the exponent of the law (b) is proportional to the skewness of the underlying distribution. This proposal has criticised: additional work seems to be indicated.