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 {\displaystyle Y} with mean μ {\displaystyle \mu } and variance var ( Y ) {\displaystyle \operatorname {var} (Y)} , Taylor's law is written
var ( Y ) = a μ b , {\displaystyle \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 = a m + b m 2 {\displaystyle 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 {\displaystyle \log V_{x}=\log V_{1}+b\log x\,}
where Vx is the variance of yield for plots of x units, V1 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. Adrienne W. 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 her 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 ; θ , λ ) = λ κ b ( θ ) [ ( 1 + s θ ) α − 1 ] , {\displaystyle K_{b}^{*}(s;\theta ,\lambda )=\lambda \kappa _{b}(\theta )\left[\left(1+{s \over \theta }\right)^{\alpha }-1\right],}
where κb(θ) is the cumulant function,
κ b ( θ ) = α − 1 α ( θ α − 1 ) α , {\displaystyle \kappa _{b}(\theta )={\frac {\alpha -1}{\alpha }}\left({\frac {\theta }{\alpha -1}}\right)^{\alpha },}
the Tweedie exponent
α = b − 2 b − 1 , {\displaystyle \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 = λ 1 / ( α − 1 ) . {\displaystyle 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 = a m i b , {\displaystyle s_{i}^{2}=am_{i}^{b},}
where si2 is the variance of the density of the ith sample, mi 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 {\displaystyle \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 {\displaystyle c} , the exponent ( b {\displaystyle b} ) remains unchanged. To see this let y = cx. Then
μ 1 = E ( x ) {\displaystyle \mu _{1}=\operatorname {E} (x)}
μ 2 = E ( y ) = E ( c x ) = c E ( x ) = c μ 1 {\displaystyle \mu _{2}=\operatorname {E} (y)=\operatorname {E} (cx)=c\operatorname {E} (x)=c\mu _{1}}
σ 1 2 = E ( ( x − μ 1 ) 2 ) {\displaystyle \sigma _{1}^{2}=\operatorname {E} ((x-\mu _{1})^{2})}
σ 2 2 = E ( ( y − μ 2 ) 2 ) = E ( ( c x − c μ 1 ) 2 ) = c 2 E ( ( x − μ 1 ) 2 ) = c 2 σ 1 2 {\displaystyle \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
σ 1 2 = a μ 1 b {\displaystyle \sigma _{1}^{2}=a\mu _{1}^{b}}
and in the rescaled variable (y) it is
σ 2 2 = c 2 σ 1 2 = c 2 a μ 1 b = c 2 − b a ( c μ 1 ) b = c 2 − b a μ 2 b {\displaystyle \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, σ 2 2 {\displaystyle \sigma _{2}^{2}} is still proportional to μ 2 b {\displaystyle \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 = f − φ + ( f − φ ) 2 − 4 r 2 f φ 2 r f {\displaystyle b={\frac {f-\varphi +{\sqrt {(f-\varphi )^{2}-4r^{2}f\varphi }}}{2r{\sqrt {f}}}}}
where r {\displaystyle r} is the Pearson moment correlation coefficient between log ( s 2 ) {\displaystyle \log(s^{2})} and log m {\displaystyle \log m} , f {\displaystyle f} is the ratio of sample variances in log ( s 2 ) {\displaystyle \log(s^{2})} and log m {\displaystyle \log m} and φ {\displaystyle \varphi } is the ratio of the errors in log ( s 2 ) {\displaystyle \log(s^{2})} and log m {\displaystyle \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 ) {\displaystyle \log(s^{2})} and log m {\displaystyle \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 = c n d m b , {\displaystyle s^{2}=cn^{d}m^{b},}
where s2 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 {\displaystyle S=V+P}
Assuming the validity of Taylor's law, we have
V = a m b {\displaystyle V=am^{b}}
Because in the Poisson distribution the mean equals the variance, we have
P = m {\displaystyle P=m}
This gives us
S = V + P = a m b + m {\displaystyle 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.
Notes The origin of the slope (b) in this regression remains unclear. Two hypotheses have been proposed to explain it. One suggests that b arises from the species behavior and is a constant for that species. The alternative suggests that it is dependent on the sampled population. Despite the considerable number of studies carried out on this law (over 1000), this question remains open. It is known that both a and b are subject to change due to age-specific dispersal, mortality and sample unit size. This law may be a poor fit if the values are small. For this reason an extension to Taylor's law has been proposed by Hanski which improves the fit of Taylor's law at low densities.
Extension to cluster sampling of binary data A form of Taylor's law applicable to binary data in clusters (e.q., quadrats) has been proposed. In a binomial distribution, the theoretical variance is
var bin = n p ( 1 − p ) , {\displaystyle {\text{var}}_{\text{bin}}=np(1-p),}
where (varbin) is the binomial variance, n is the sample size per cluster, and p is the proportion of individuals with a trait (such as disease), an estimate of the probability of an individual having that trait. One difficulty with binary data is that the mean and variance, in general, have a particular relationship: as the mean proportion of individuals infected increases above 0.5, the variance deceases. It is now known that the observed variance (varobs) changes as a power function of (varbin). Hughes and Madden noted that if the distribution is Poisson, the mean and variance are equal. As this is clearly not the case in many observed proportion samples, they instead assumed a binomial distribution. They replaced the mean in Taylor's law with the binomial variance and then compared this theoretical variance with the observed variance. For binomial data, they showed that varobs = varbin with overdispersion, varobs > varbin. In symbols, Hughes and Madden's modification to Tyalor's law was
var obs = a ( var bin ) b . {\displaystyle {\text{var}}_{\text{obs}}=a({\text{var}}_{\text{bin}})^{b}.}
In logarithmic form this relationship is
log ( var obs ) = log a + b log ( var bin ) . {\displaystyle \log({\text{var}}_{\text{obs}})=\log a+b\log({\text{var}}_{\text{bin}}).}
This latter version is known as the binary power law. A key step in the derivation of the binary power law by Hughes and Madden was the observation made by Patil and Stiteler that the variance-to-mean ratio used for assessing over-dispersion of unbounded counts in a single sample is actually the ratio of two variances: the observed variance and the theoretical variance for a random distribution. For unbounded counts, the random distribution is the Poisson. Thus, the Taylor power law for a collection of samples can be considered as a relationship between the observed variance and the Poisson variance. More broadly, Madden and Hughes considered the power law as the relationship between two variances, the observed variance and the theoretical variance for a random distribution. With binary data, the random distribution is the binomial (not the Poisson). Thus the Taylor power law and the binary power law are two special cases of a general power-law relationships for heterogeneity. When both a and b are equal to 1, then a small-scale random spatial pattern is suggested and is best described by the binomial distribution. When b = 1 and a > 1, there is over-dispersion (small-scale aggregation). When b is > 1, the degree of aggregation varies with p. Turechek et al. have shown that the binary power law describes numerous data sets in plant pathology. In general, b is greater than 1 and less than 2. The fit of this law has been tested by simulations. These results suggest that rather than a single regression line for the data set, a segmental regression may be a better model for genuinely random distributions. However, this segmentation only occurs for very short-range dispersal distances and large quadrat sizes. The break in the line occurs only at p very close to 0. An extension to this law has been proposed. The original form of this law is symmetrical but it can be extended to an asymmetrical form. Using simulations the symmetrical form fits the data when there is positive correlation of disease status of neighbors. Where there is a negative correlation between the likelihood of neighbours being infected, the asymmetrical version is a better fit to the data.
Applications Because of the ubiquitous occurrence of Taylor's law in biology it has found a variety of uses some of which are listed here.
Recommendations as to use It has been recommended based on simulation studies in applications testing the validity of Taylor's law to a data sample that: (1) the total number of organisms studied be > 15 (2) the minimum number of groups of organisms studied be > 5 (3) the density of the organisms should vary by at least 2 orders of magnitude within the sample
Randomly distributed populations It is commonly assumed (at least initially) that a population is randomly distributed in the environment. If a population is randomly distributed then the mean ( m ) and variance ( s2 ) of the population are equal and the proportion of samples that contain at least one individual ( p ) is
p = 1 − e − m {\displaystyle p=1-e^{-m}}
When a species with a clumped pattern is compared with one that is randomly distributed with equal overall densities, p will be less for the species having the clumped distribution pattern. Conversely when comparing a uniformly and a randomly distributed species but at equal overall densities, p will be greater for the randomly distributed population. This can be graphically tested by plotting p against m. Wilson and Room developed a binomial model that incorporates Taylor's law. The basic relationship is
p = 1 − e − m log ( s 2 / m ) ( s 2 / m − 1 ) − 1 {\displaystyle p=1-e^{-m\log(s^{2}/m)(s^{2}/m-1)^{-1}}}
where the log is taken to the base e. Incorporating Taylor's law this relationship becomes
p = 1 − e − m log ( a m b − 1 ) ( a m b − 1 − 1 ) − 1 {\displaystyle p=1-e^{-m\log(am^{b-1})(am^{b-1}-1)^{-1}}}
Dispersion parameter estimator The common dispersion parameter (k) of the negative binomial distribution is
k = m 2 s 2 − m {\displaystyle k={\frac {m^{2}}{s^{2}-m}}}
where m {\displaystyle m} is the sample mean and s 2 {\displaystyle s^{2}} is the variance. If 1 / k is > 0 the population is considered to be aggregated; 1 / k = 0 ( s2 = m ) the population is considered to be randomly (Poisson) distributed and if 1 / k is < 0 the population is considered to be uniformly distributed. No comment on the distribution can be made if k = 0. Wilson and Room assuming that Taylor's law applied to the population gave an alternative estimator for k:
k = m a m b − 1 − 1 {\displaystyle k={\frac {m}{am^{b-1}-1}}}
where a and b are the constants from Taylor's law. Jones using the estimate for k above along with the relationship Wilson and Room developed for the probability of finding a sample having at least one individual
p = 1 − e − m log ( a m b − 1 ) ( a m b − 1 − 1 ) − 1 {\displaystyle p=1-e^{-m\log(am^{b-1})(am^{b-1}-1)^{-1}}}
derived an estimator for the probability of a sample containing x individuals per sampling unit. Jones's formula is
P ( x ) = P ( x − 1 ) k + x − 1 x m k − 1 m k − 1 − 1 {\displaystyle P(x)=P(x-1){\frac {k+x-1}{x}}{\frac {mk^{-1}}{mk^{-1}-1}}}
where P( x ) is the probability of finding x individuals per sampling unit, k is estimated from the Wilson and Room equation and m is the sample mean. The probability of finding zero individuals P( 0 ) is estimated with the negative binomial distribution
P ( 0 ) = ( 1 + m k ) − k {\displaystyle P(0)=\left(1+{\frac {m}{k}}\right)^{-k}}
Jones also gives confidence intervals for these probabilities.
C I = t ( P ( x ) ( 1 − P ( x ) ) N ) 1 / 2 {\displaystyle \mathrm {CI} =t\left({\frac {P(x)(1-P(x))}{N}}\right)^{1/2}}
where CI is the confidence interval, t is the critical value taken from the t distribution and N is the total sample size.
Katz family of distributions Katz proposed a family of distributions (the Katz family) with 2 parameters ( w1, w2 ). This family of distributions includes the Bernoulli, Geometric, Pascal and Poisson distributions as special cases. The mean and variance of a Katz distribution are
m = w 1 1 − w 2 {\displaystyle m={\frac {w_{1}}{1-w_{2}}}}
s 2 = w 1 ( 1 − w 2 ) 2 {\displaystyle s^{2}={\frac {w_{1}}{(1-w_{2})^{2}}}}
where m is the mean and s2 is the variance of the sample. The parameters can be estimated by the method of moments from which we have
w 1 1 − w 2 = m {\displaystyle {\frac {w_{1}}{1-w_{2}}}=m}
w 2 1 − w 2 = s 2