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| In [[probability]] and [[statistics]], the '''Tweedie distributions''' are a family of [[probability distribution]]s which include the purely continuous [[normal distribution|normal]] and [[gamma distribution|gamma]] distributions, the purely discrete scaled [[Poisson distribution]], and the class of mixed compound Poisson–gamma distributions which have positive mass at zero, but are otherwise continuous.<ref name="t84">{{cite conference
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| |last=Tweedie |first=M.C.K.
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| |year=1984
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| |title=An index which distinguishes between some important exponential families
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| |booktitle=Statistics: Applications and New Directions
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| |conference=Proceedings of the Indian Statistical Institute Golden Jubilee International Conference
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| |editor1-first=J.K. |editor1-last=Ghosh
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| |editor2-first=J |editor2-last=Roy
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| |pages=579–604
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| |publisher=Indian Statistical Institute |location=Calcutta
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| |mr=786162
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| }}</ref> For any [[random variable]] ''Y'' that obeys a Tweedie distribution, the [[variance]] var(''Y'') relates to the [[mean]] E(''Y'') by the power law,
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| : <math>\text{var}\,(Y) = a[\text{E}\,(Y)]^p ,</math>
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| where ''a'' and ''p'' are positive constants.
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| The Tweedie distributions were named by Bent Jørgensen<ref>{{cite journal|last=Jørgensen, B|title=Exponential dispersion models|journal=[[Journal of the Royal Statistical Society]], Series B|year=1987|volume=49|issue=2|pages=127–162|jstor=2345415}}</ref> after [[Maurice Tweedie]], a statistician and medical physicist at the [[University of Liverpool]], UK, who presented the first thorough study of these distributions in 1984.<ref name="t84"/><ref>{{cite journal
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| |last=Smith |first=C.A.B. |authorlink=Cedric Smith (statistician)
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| |year=1997
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| |title=Obituary: Maurice Charles Kenneth Tweedie, 1919-96
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| |journal=[[Journal of the Royal Statistical Society]]: Series A
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| (Statistics in Society)
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| |volume=160 |issue=1 |pages=151–154 |doi=10.1111/1467-985X.00052
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| }}</ref>
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| ==Examples==
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| The Tweedie distributions include a number of familiar distributions as well as some unusual ones, each being specified by the [[Domain (mathematical analysis)|domain]] of the index parameter. We have the
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|
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| *[[normal distribution]], ''p=0'',
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| *[[Poisson distribution]], ''p=1'',
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| *compound Poisson–gamma distribution, ''1<p<2'',
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| *[[gamma distribution]], ''p=2'',
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| *positive [[stable distribution]]s, ''2<p<3'',
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| *[[inverse Gaussian distribution]], ''p=3'',
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| *positive stable distributions, ''p>3'', and
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| *extreme stable distributions, ''p''={{math|<VAR>∞</VAR>}}.
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| For 0<''p''<1 no Tweedie model exists.
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| ==Definitions==
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| Tweedie distributions are a special case of [[exponential dispersion model]]s, a class of models used to describe error distributions for the [[generalized linear model]].<ref
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| name="Jørgensen-1997">{{cite book
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| | author = Jørgensen, Bent
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| | year = 1997
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| | title = The theory of dispersion models
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| | publisher = Chapman & Hall
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| | isbn = 978-0412997112
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| }}</ref> The term exponential dispersion model refers to the [[Exponential family|exponential form]] that these models take, evident from the canonical equation used to describe the distribution ''P<sub>λ,θ</sub>'' of the random variable ''Z'' on the [[Measure (mathematics)|measurable sets]] ''A'',
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| : <math>P_{\lambda,\theta}(Z\in A)=\int_{A} \exp[\theta \cdot z-\lambda\kappa(\theta)]\cdot \nu_\lambda\, (dz)</math>,
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| with the interrelated [[Measure (mathematics)|measures]] ''ν<sub>λ</sub>''. ''θ'' is the [[Exponential family|canonical parameter]]; the cumulant function is
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| : <math>\kappa(\theta)=\lambda^{-1}\log\int e^{\theta z}\cdot \nu_\lambda\, (dz)</math>;
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| ''λ'' is the index parameter; and ''z'' the canonical statistic. This equation represents a family of exponential dispersion models ''ED<sup>*</sup>(θ,λ)'' that are completely determined by the parameters ''θ'' and ''λ'' and the cumulant function.
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| ===Additive exponential dispersion models===
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| The models just described are additive models with the property that the distribution of the sum of independent random variables,
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| : <math>Z_+ = Z_1 +\ldots+ Z_n</math>,
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| for which ''Z<sub>i</sub>~ED<sup>*</sup>(θ,λ<sub>i</sub>)'' with fixed ''θ'' and various ''λ'' are members of the family of distributions with the same ''θ'',
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| : <math>Z_+ \sim ED^*(\theta,\lambda_1+\ldots+\lambda_n)</math>.
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| ===Reproductive exponential dispersion models===
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| A second class of exponential dispersion models exists designated by the random variable
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| : <math>Y=Z/\lambda \sim ED(\mu,\sigma^2)</math>,
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| where ''σ<sup>2</sup>=1/λ'', known as reproductive exponential dispersion models. They have the property that for ''n'' independent random variables ''Y<sub>i</sub>~ED(μ,σ<sup>2</sup>/w<sub>i</sub>)'', with weighting factors ''w<sub>i</sub>'' and
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| : <math>w= \sum_{i=1}^n w_i</math>,
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| a weighted average of the variables gives, | |
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| : <math>w^{-1}\sum_{i=1}^n w_iY_i \sim ED(\mu,\sigma^2/w)</math>.
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| For reproductive models the weighted average of independent random variables with fixed ''μ'' and ''σ<sup>2</sup>'' and various values for ''w<sub>i</sub>'' is a member of the family of distributions with same ''μ'' and ''σ<sup>2</sup>''.
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| The Tweedie exponential dispersion models are both additive and reproductive; we thus have the ''duality transformation''
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| : <math>Y \mapsto Z=Y/\sigma^2</math>.
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| ===Scale invariance===
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| A third property of the Tweedie models is that they are [[Scale invariance|scale invariant]]: For a reproductive exponential dispersion model ''ED(μ,σ<sup>2</sup>)'' and any positive constant ''c'' we have the property of closure under scale transformation,
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| : <math>c ED(\mu,\sigma^2)=ED(c\mu,c^{2-p}\sigma^2)</math>,
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| where the index parameter ''p'' is a real-valued unitless constant. With this transformation the new variable ''Y’=cY'' belongs to the family of distributions with fixed ''μ'' and ''σ<sup>2</sup>'' but different values of ''c''. | |
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| ===The Tweedie power variance function===
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| To define the variance function for exponential dispersion models we make use of the mean value mapping, the relationship between the canonical parameter ''θ'' and the mean ''μ''. It is define by the function
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| : <math>\tau(\theta)=\kappa^\prime(\theta)=\mu</math>.
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| The [[Natural exponential family|variance function]] ''V(μ)'' is constructed from the mean value mapping,
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| : <math>V(\mu)=\tau^\prime[\tau^{-1}(\mu)]</math>.
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| Here the minus exponent in ''τ<sup> -1</sup>(μ)'' denotes an inverse function rather than a reciprocal. The mean and variance of an additive random variable is then E''(Z)=λμ'' and var''(Z)=λV(μ).
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| Scale invariance implies that the variance function obeys the relationship
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| ''V(μ)=μ<sup> p</sup>.<ref
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| name="Jørgensen-1997">{{cite book
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| | author = Jørgensen, Bent
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| | year = 1997
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| | title = The theory of dispersion models
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| | publisher = [Chapman & Hall]
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| | isbn = 978-0412997112
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| }}</ref>
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| ===The Tweedie cumulant generating functions===
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| The properties of exponential dispersion models give us two [[differential equation]]s.<ref
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| name="Jørgensen-1997">{{cite book
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| | author = Jørgensen, Bent
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| | year = 1997
| |
| | title = The theory of dispersion models
| |
| | publisher = [Chapman & Hall]
| |
| | isbn = 978-0412997112
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| }}</ref> The first relates the mean value mapping and the variance function to each other,
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| : <math>\frac{\partial \tau^{-1}(\mu)}{\partial \mu}= \frac{1}{V(\mu)}</math>. | |
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| The second shows how the mean value mapping is related to the [[cumulant|cumulant function]],
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| : <math>\frac{\partial \kappa(\theta)}{\partial \theta}=\tau(\theta)</math>.
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| These equations can be solved to obtain the cumulant function for different cases of the Tweedie models. A cumulant generating function (CGF) may then be obtained from the cumulant function. The additive CGF is generally specified by the equation
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| : <math>K^*(s)=\log[\text{E}(e^{sZ})]=\lambda[\kappa(\theta+s)-\kappa(\theta)]</math>,
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| and the reproductive CGF by
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| : <math>K(s)=\log[\text{E}(e^{sY})]=\lambda[\kappa(\theta+s/\lambda)-\kappa(\theta)]</math>, | |
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| where ''s'' is the generating function variable.
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| The cumulant functions for specific values of the index parameter ''p'' are<ref
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| name="Jørgensen-1997">{{cite book
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| | author = Jørgensen, Bent
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| | year = 1997
| |
| | title = The theory of dispersion models
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| | publisher = [Chapman & Hall]
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| | isbn = 978-0412997112
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| }}</ref>
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| : <math>\kappa_p(\theta) = \begin{cases} \dfrac{\alpha-1}{\alpha} \left(\dfrac{\theta}{\alpha-1}\right)^\alpha
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| & \quad p \ne 1,2 \\ -\log(-\theta) & \quad p = 2 \\ e^\theta & \quad p = 1 \end{cases}
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| </math>,
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| where ''α'' is the Tweedie exponent
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| : <math>\alpha=\dfrac{p-2}{p-1}</math>.
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| For the additive Tweedie models the CGFs take the form,
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| : <math>K^*_p(s;\theta,\lambda) = \begin{cases} \lambda\kappa_p(\theta)[(1+s/\theta)^\alpha-1]
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| & \quad p \ne 1,2 \\ -\lambda \log(1+s/\theta) & \quad p = 2 \\ \lambda e^\theta (e^s -1) & \quad p = 1 \end{cases}
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| </math>,
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| and for the reproductive models, | |
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| : <math>K_p(s;\theta,\lambda) = \begin{cases} \lambda\kappa_p(\theta)\left \{ [1+s/(\theta \lambda)]^\alpha-1 \right \}
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| & \quad p \ne 1,2 \\ -\lambda \log[1+s/(\theta \lambda)] & \quad p = 2 \\ \lambda e^\theta (e^{s/\lambda} -1) & \quad p = 1 \end{cases}
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| </math>.
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| The additive and reproductive Tweedie models are conventionally denoted by the symbols ''Tw<sup>*</sup><sub>p</sub>(θ,λ)'' and ''Tw<sub>p</sub>(θ,σ<sup>2</sup>)'', respectively.
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| The first and second derivatives of the CGFs, with ''s=0'', yields the mean and variance, respectively. One can thus confirm that for the additive models the variance relates to the mean by the power law,
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| : <math>\mathrm{var} (Z)\propto \mathrm{E}(Z)^p</math>.
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| ==The Tweedie convergence theorem==
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| The Tweedie exponential dispersion models are fundamental in statistical theory consequent to their roles as foci of [[convergence in distribution|convergence]] for a wide range of statistical processes. Jørgensen ''et al'' proved a theorem that specifies the asymptotic behaviour of variance functions known as the Tweedie convergence theorem".<ref>{{cite journal|last=Jørgensen, B|coauthors=Martinez, JR & Tsao, M|title=Asymptotic behaviour of the variance function|journal=Scandinavian Journal of Statistics|year=1994|volume=21|pages=223–243}}</ref> This theorem, in technical terms, is stated thus:<ref
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| name="Jørgensen-1997">{{cite book
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| | author = Jørgensen, Bent
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| | year = 1997
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| | title = The theory of dispersion models
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| | publisher = [Chapman & Hall]
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| | isbn = 978-0412997112
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| }}</ref> The unit variance function is regular of order ''p'' at zero (or infinity) provided that ''V(μ)~c<sub>0</sub>μ<sup>p</sup>'' for ''μ'' as it approaches zero (or infinity) for all real values of ''p'' and ''c<sub>0</sub> >0''. Then for a unit variance function regular of order ''p'' at either zero or infinity and for
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| : <math>p \notin (0,1)</math> ,
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| for any <math>\mu>0</math>, and <math> \sigma^2>0</math> we have | |
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| : <math>c^{-1} ED(c\mu,\sigma^2c^{2-p}) \rightarrow Tw_p(\mu,c_0 \sigma^2)</math>
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| as <math>c \downarrow 0</math> or <math>c \rightarrow \infty</math>, respectively, where the convergence is through values of ''c'' such that ''cμ'' is in the domain of ''θ'' and ''c<sup>p-2</sup>/σ<sup>2</sup>'' is in the domain of ''λ''. The model must be infinitely divisible as ''c<sup>2-p</sup>'' approaches infinity.<ref
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| name="Jørgensen-1997">{{cite book
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| | author = Jørgensen, Bent
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| | year = 1997
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| | title = The theory of dispersion models
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| | publisher = [Chapman & Hall]
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| | isbn = 978-0412997112
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| }}</ref>
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| In nontechnical terms this theorem implies that any exponential dispersion model that asymptotically manifests a variance-to-mean power law is required to have a variance function that comes within the [[Attractor|domain of attraction]] of a Tweedie model. Almost all distribution functions with finite cumulant generating functions qualify as exponential dispersion models and most exponential dispersion models manifest variance functions of this form. Hence many probability distributions have variance functions that express this asymptotic behavior, and the Tweedie distributions become foci of convergence for a wide range of data types.<ref name=Kendal2011b>{{cite doi|10.1103/PhysRevE.84.066120}}</ref>
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| ==The Tweedie models and Taylor’s power law==
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| [[Taylor's 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.<ref name=Taylor1961>Taylor LR (1961) Aggregation, variance and the mean. ''Nature'' 189, 732–735</ref> For the population count ''Y'' with mean ''µ'' and variance var(''Y''), Taylor’s law is written,
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| : <math>\text{var}\,(Y) = a\mu^p</math>,
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| where ''a'' and ''p'' are both positive constants. Since L. R. Taylor described this law in 1961 there have been many different explanations offered to explain it, ranging from animal behavior,<ref name="Taylor1961" /> a [[random walk]] model,<ref name=Hanski1980>Hanski I (1980) Spatial patterns and movements in coprophagous beetles. ''Oikos'' 34, 293-310</ref> a [[birth-death process|stochastic birth, death, immigration and emigration model]],<ref name=Anderson1961>Anderson RD, Crawley GM & Hassell M (1982) Variability in the abundance of animal and plant species. ''Nature'' 296, 245–248</ref> to a consequence of equilibrium and non-equilibrium [[statistical mechanics]].<ref name=Fronczak2010>Fronczak A & Fronczak P (2010) Origins of Taylor’s power law for fluctuation scaling in complex systems. ''Phys Rev E'' 81, 066112</ref> No consensus exists as to an explanation for this model.
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| Since Taylor’s law is mathematically identical to the variance-to-mean power law that characterizes the Tweedie models, it seemed reasonable to use these models and the Tweedie convergence theorem to explain the observed clustering of animals and plants associated with Taylor’s law.<ref name=Kendal2002>Kendal WS (2002) Spatial aggregation of the Colorado potato beetle described by an exponential dispersion model. ''Ecological Modelling'' 151, 261–269</ref><ref name=Kendal2004>Kendal WS (2004) Taylor’s ecological power law as a consequence of scale invariant exponential dispersion models. ''Ecol Complex'' 1, 193–209</ref> The majority of the observed values for the power-law exponent ''p'' have fallen in the interval (1,2) and so the Tweedie compound Poisson–gamma distribution would seem applicable. Comparison of the [[empirical distribution function]] to the theoretical compound Poisson–gamma distribution has provided a means to verify consistency of this hypothesis.<ref name="Kendal2002" />
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| Whereas conventional models for Taylor’s law have tended to involve ''[[ad hoc]]'' animal behavioral or [[population dynamics|population dynamic]] assumptions, the Tweedie convergence theorem would imply that Taylor’s law results from a general mathematical convergence effect much as how the [[central limit theorem]] governs the convergence behavior of certain types of random data. Indeed, any mathematical model, approximation or simulation that is designed to yield Taylor’s law (on the basis of this theorem) is required to converge to the form of the Tweedie models.<ref name=Kendal2011b>Kendal WS & Jørgensen BR (2011) Tweedie convergence: a mathematical basis for Taylor's power law, ''1/f'' noise and multifractality. ''Phys. Rev E'' 84, 066120</ref>
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| ==The double power law==
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| The [[eponym]] Taylor's power law has been applied to a wide range of data that manifests a variance-to-mean power function. However, subtle mathematical differences exist between the [[transformation (function)|transformational]] properties of some of these data. A double power law, which includes Taylor’s original law, has been proposed to describe these differences.<ref name=Jørgensen2011>Jørgensen B, Martinez JR & Demetrio CGB (2011) Self-similarity and Lamperti convergence for families of stochastic processes. ''Lith Math J'' 51, 342–362</ref> For a population count drawn from an area of size ''t'' with mean abundance per unit area ''µ'', and where
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| : <math>\text{E}\,[Y(\mu;t)] = t\mu\,\!</math>,
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| we have for the double power law:
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| : <math>\text{var}\,[Y(\mu;t)] = a\mu^pt^{2-d}\,\!</math>.
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| The constant ''d''{{math|<VAR>∈</VAR>}} [0,1] has been identified as a [[Fractal dimension#Role of scaling|fractal exponent]]. The factor ''aµ''<sup>'' p''</sup> represents the original Taylor’s law, a function of the mean abundance per unit area; the last term ''t''<sup>2-''d''</sup> describes how the power law scales with the enumerative bin size. This last term implies a statistically [[self-similarity|self-similar]] scaling of the spatial distribution of items of interest as the size of the bin changes. It is this second portion of the double power law that underlies the variance-to-mean power law reported in systems like regional blood flow heterogeneity,<ref name=Kendal2001>Kendal WS (2001) A stochastic model for the self-similar heterogeneity of regional organ blood flow. ''Proc Natl Acad Sci U S A'' 98, 837-841</ref> the genomic distribution of SNPs<ref name=Kendal2003>Kendal WS (2003) An exponential dispersion model for the distribution of human single nucleotide polymorphisms" ''Mol Biol Evol'' 20 579-590</ref> and genes,<ref name=KendalGenes>{{cite journal|last=Kendal|first=WS|title=A scale invariant clustering of genes on human chromosome 7|journal=BMC Evol Biol|year=2004|volume=4|pages=3|doi=10.1186/1471-2148-4-3}}</ref> and [[number theory|number theoretic]] examples.<ref name="Kendal2011b" /><ref name=Kendal2011a>Kendal WS & Jørgensen B (2011) Taylor's power law and fluctuation scaling explained by a central-limit-like convergence. ''Phys. Rev. E'' 83,066115</ref>
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| ==Tweedie convergence and 1/''f'' noise==
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| [[Pink noise]], or 1/''f'' noise, refers to a pattern of noise characterized by a power-law relationship between its intensities ''S(f)'' at different frequencies ''f'',
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| : <math>S(f)\propto 1/f^{ \gamma}</math>,
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| where the dimensionless exponent ''γ''{{math|<VAR>∈</VAR>}} [0,1]. It is found within a diverse number of natural processes.<ref name=Dutta1981>Dutta P & Horn PM (1981) Low frequency fluctuations in solids: ''1/f'' noise. ''Rev Mod Phys'' 53,497-516</ref> Many different explanations for ''1/f'' noise exist, a widely held hypothesis is based on [[Self-organized criticality]] where dynamical systems close to a [[Critical point (thermodynamics)|critical point]] are thought to manifest [[scale-invariance|scale-invariant]] spatial and/or temporal behavior.
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| In this subsection a mathematical connection between ''1/f'' noise and the Tweedie variance-to-mean power law will be described. To begin, we first need to introduce [[self-similar process]]es: For the sequence of numbers
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| : <math>Y=(Y_i :i=0,1,2,...,N)</math>
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| with mean
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| : <math>\hat{\mu}=\text{E}(Y_i)</math>,
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| deviations
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| : <math>y_i = Y_i - \hat{\mu} </math>,
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| variance
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| : <math>\hat{\sigma}^2=\text{E}(y_i^2)</math>,
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|
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| and autocorrelation function
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| : <math>r(k)=\text{E}(y_i,y_{i+k})/\text{E}(y_i^2)</math>
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| with lag ''k'', if the [[autocorrelation]] of this sequence has the long range behavior
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| : <math>r(k)\sim k^{-d} L(k) </math>
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| as ''k''{{math|<VAR>→∞</VAR>}} and where ''L(k)'' is a slowly varying function at large values of ''k'', this sequence is called a self-similar process.<ref name=Leland1994>Leland WE, Taqqu MS, Willinger W & Wilson DV (1994) On the self-similar nature of ethernet traffic. ''IEE/ACM Trans Networking'' 2, 1-15</ref>
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| The '''method of expanding bins''' can be used to analyze self-similar processes. Consider a set of equal-sized non-overlapping bins that divides the original sequence of ''N'' elements into groups of ''m'' equal-sized segments (''N/m'' is integer) so that new reproductive sequences, based on the mean values, can be defined:
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| : <math>Y_i^{(m)}=(Y_{im-m+1}+...+Y_{im})/m</math>.
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| The variance determined from this sequence will scale as the bin size changes such that
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| : <math>\text{var}[Y^{(m)}]=\hat{\sigma}^2 m^{-d}</math>
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|
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| if and only if the autocorrelation has the limiting form<ref name=Tsybakov1997>Tsybakov B & Georganas ND (1997) On self-similar traffic in ATM queues: definitions, overflow probability bound, and cell delay distribution. ''IEEE/ACM Trans Networking'' 5, 397–409</ref> | |
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| : <math>\lim_{k \to \infty}r(k)/k^{-d} = (2-d)(1-d)/2</math>.
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| One can also construct a set of corresponding additive sequences
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| : <math>Z_i^{(m)} = mY_i^{(m)}</math>,
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| based on the expanding bins,
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| : <math>Z_i^{(m)}=(Y_{im-m+1}+...+Y_{im})</math>.
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| Provided the autocorrelation function exhibits the same behavior, the additive sequences will obey the relationship
| |
| | |
| : <math>\text{var}[Z_i^{(m)}]=m^2 \text{var}[Y^{(m)}]=(\hat{\sigma}^2 /\hat{\mu}^{2-d})\text{E}[Z_i^{(m)}]^{2-d}</math>
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| Since <math>\hat{\mu}</math> and <math>\hat{\sigma}^2</math> are constants this relationship constitutes a variance-to-mean power law, with ''p''=2-''d''.<ref name="Kendal2011b" /><ref name=Kendal2007>Kendal WS (2007) Scale invariant correlations between genes and SNPs on Human chromosome 1 reveal potential evolutionary mechanisms. ''J Theor Biol'' 245, 329-340</ref>
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| The [[Logical biconditional|biconditional]] relationship above between the variance-to-mean power law and power law autocorrelation function, and the [[Wiener–Khinchin theorem]]<ref name=McQuarrie1976>McQuarrie DA (1976) ''Statistical mechanics'' [Harper & Row]</ref> imply that any sequence that exhibits a variance-to-mean power law by the method of expanding bins will also manifest ''1/f'' noise, and vice versa. Moreover, the Tweedie convergence theorem, by virtue of its central limit-like effect of generating distributions that manifest variance-to-mean power functions, will also generate processes that manifest ''1/f'' noise.<ref name="Kendal2011b" /> The Tweedie convergence theorem thus allows provides an alternative explanation for the origin of ''1/f'' noise, based its central limit-like effect.
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| | |
| Much as the [[central limit theorem]] requires certain kinds of random processes to have as a focus of their convergence the [[normal distribution|Gaussian distribution]] and thus express [[white noise]], the Tweedie convergence theorem requires certain non-Gaussian processes to have as a focus of convergence the Tweedie distributions that express ''1/f'' noise.<ref name="Kendal2011b" />
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| ==The Tweedie models and multifractality==
| |
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| From the properties of self-similar processes, the power-law exponent ''p''=2-''d'' is related to the [[Hurst exponent]] ''H'' and the [[fractal dimension]] ''D'' by<ref name="Tsybakov1997" />
| |
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| : <math>D = 2-H = 2 - p/2 </math>.
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| A one-dimensional data sequence of self-similar data may demonstrate a variance-to-mean power law with local variations in the value of ''p'' and hence in the value of ''D''. When fractal structures manifest local variations in fractal dimension, they are said to be [[multifractal system|multifractals]]. Examples of data sequences that exhibit local variations in ''p'' like this include the eigenvalue deviations of the [[Random matrix|Gaussian Orthogonal and Unitary Ensembles]].<ref name="Kendal2011b" /> The Tweedie compound Poisson–gamma distribution has served to model multifractality based on local variations in the Tweedie exponent ''α''. Consequently, in conjunction with the variation of ''α'', the Tweedie convergence theorem can be viewed as having a role in the genesis of such multifractals.
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| The variation of ''α'' has been found to obey the asymmetric [[Laplace distribution|Laplace distribution]] in certain cases. <ref name=Kendal2014>Kendal WS (2014) Multifractality attributed to dual central limit-lie convergence effects. ''Physica A'' 401, 22-33</ref> This distribution has been shown to be a member of the family of geometric Tweedie models <ref name=Jørgensen2011> Jørgensen B, Kokonendji CC (2011) Dispersion models for geometric sums. ''Braz J Probab Stat'' 25, 263-293</ref>, that manifest as limiting distributions in a convergence theorem for geometric dispersion models.
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| ==Applications==
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| ===Regional organ blood flow===
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| Regional organ blood flow has been traditionally assessed by the injection of [[Isotopic labeling|radiolabelled]] [[polyethylene microspheres]] into the arterial circulation of animals, of a size that they become entrapped within the [[microcirculation]] of organs. The organ to be assessed is then divided into equal-sized cubes and the amount of radiolabel within each cube is evaluated by [[liquid scintillation counting]] and recorded. The amount of radioactivity within each cube is taken to reflect the blood flow through that sample at the time of injection. It is possible to evaluate adjacent cubes from an organ in order to additively determine the blood flow through larger regions. Through the work of '''J B Bassingthwaighte''' and others an empirical power law has been derived between the relative dispersion of blood flow of tissue samples (''RD''=standard deviation/ mean)of mass ''m'' relative to reference sized samples:<ref name=Bassingthwaighte1989>Bassingthwaighte JB (1989) Fractal nature of regional myocardial blood flow heterogeneity. ''Circ Res'' 65, 578-590</ref>
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| : <math>RD(m)=RD(m_{ref})\left (\frac{m}{m_{ref}}\right )^{1-D_s}</math>
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| This power law exponent ''D<sub>s</sub>'' has been called a fractal dimension. '''Bassingthwaighte’s power law''' can be shown to directly relate to the variance-to-mean power law. Regional organ blood flow can thus be modelled by the Tweedie compound Poisson–gamma distribution.<ref name="Kendal2001" /> In this model tissue sample could be considered to contain a random (Poisson) distributed number of entrapment sites, each with [[gamma distribution|gamma distributed]] blood flow. Blood flow at this microcirculatory level has been observed to obey a gamma distribution,<ref>Honig CR, Feldstein ML, Frierson JL. 1977. Capillary lengths, anastomoses, and estimated capillary transit times in skeletal muscle. Am J Physiol Heart Circul Physiol 233: H122--H129.</ref> thus providing support for this hypothesis.
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| ===Cancer metastasis===
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| The "experimental cancer [[metastasis]] assay"<ref name=Fidler1977>{{cite journal|last=Fidler|first=IJ|coauthors=Kripke, M|title=Metastasis results from preexisting variant cells within a malignant tumor|journal=Science|year=1977|volume=197|pages=893–895|bibcode = 1977Sci...197..893F |doi = 10.1126/science.887927 }}</ref> has some resemblance to the above method to measure regional blood flow. Groups of [[syngeneic]] and age matched mice are given intravenous injections of equal-sized aliquots of suspensions of cloned cancer cells and then after a set period of time their lungs are removed and the number of cancer metastases enumerated within each pair of lungs. If other groups of mice are injected with different cancer cell [[Clone (cell biology)|clones]] then the number of metastases per group will differ in accordance with the metastatic potentials of the clones. It has been long recognized that there can be considerable intraclonal variation in the numbers of metastases per mouse despite the best attempts to keep the experimental conditions within each clonal group uniform.<ref name=Fidler1977>Fidler IJ & Kripke M. 1977. Metastasis results from pre-existing variant cells within a malignant tumor" ''Science'' 197: 893--895.</ref> This variation is larger than would be expected on the basis of a [[Poisson distribution]] of numbers of metastases per mouse in each clone and when the variance of the number of metastases per mouse was plotted against the corresponding mean a power law was found.<ref name=Kendal1987>Kendal WS & Frost P (1987) Experimental metastasis: a novel application of the variance-to-mean power function. ''J Natl Cancer Inst'' 79, 1113-1115</ref> | |
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| The variance-to-mean power law for metastases was found to also hold for '''spontaneous murine metastases'''<ref>Kendal WS. 1999. Clustering of murine lung metastases reflects fractal nonuniformity in regional lung blood flow. Invasion Metastasis 18: 285--296.</ref> and for cases series of human metastases.<ref>Kendal WS, Lagerwaard, FJ & Agboola O. 2000. Characterization of the frequency distribution for human hematogenous metastases: evidence for clustering and a power variance function. Clin Exp Metastasis 18: 219--229.</ref>
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| Since hematogenous metastasis occurs in direct relationship to regional blood flow<ref>Weiss L, Bronk J, Pickren JW & Lane WW. 1981. Metastatic patterns and targe organ arterial blood flow. Invasion Metastasis 1: 126--135.</ref> and videomicroscopic studies indicate that the passage and entrapment of cancer cells within the circulation appears analogous to the microsphere experiments<ref>Chambers AF, Groom AC & MacDonald IC. 2002. Dissemination and growth of cancer cells in metastatic sites. Nature Rev Cancer 2: 563--572.</ref> it seemed plausible to propose that the variation in numbers of hematogenous metastases could reflect heterogeneity in regional organ blood flow.<ref>Kendal WS. 2002. A frequency distribution for the number of hematogenous organ metastases. Invasion Metastasis 1: 126--135.</ref>
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| The blood flow model was based on the Tweedie compound Poisson–gamma distribution, a distribution governing a continuous random variable. For that reason in the metastasis model it was assumed that blood flow was governed by that distribution and that the number of regional metastases occurred as a [[Poisson process]] for which the intensity was directly proportional to blood flow. This lead to the description of the Poisson negative binomial (PNB) distribution as a [[discrete probability distribution|discrete equivalent]] to the Tweedie compound Poisson–gamma distribution. The [[probability-generating function|probability generating function]] for the PNB distribution is
| |
| | |
| : <math>G(s)= \exp \left [\lambda \frac {\alpha-1}{\alpha} \left( \frac{\theta} {\alpha-1} \right)^\alpha \left\{ \left(1- \frac{1} {\theta}+ \frac {s} {\theta}\right)^\alpha-1 \right\}\right]</math>.
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| The relationship between the mean and variance of the PNB distribution is then
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| : <math>\text{var}\,(Y) = a\text{E}(Y)^b + \text{E}(Y)</math>,
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| | |
| which, in the range of many experimental metastasis assays, would be indistinguishable from the variance-to-mean power law. For sparse data, however, this discrete variance-to-mean relationship would behave more like that of a Poisson distribution where the variance equaled the mean.
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| ===Genomic structure and evolution===
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| The local density of [[Single-nucleotide polymorphism|Single Nucleotide Polymorphisms]] (SNPs) within the [[human genome]], as well as that of [[gene]]s, appears to cluster in accord with the variance-to-mean power law and the Tweedie compound Poisson–gamma distribution.<ref name="Kendal2003" /><ref name=KendalGenes /> In the case of SNPs their observed density reflects the assessment techniques, the availability of genomic sequences for analysis, and the [[Nucleotide diversity|nucleotide heterozygosity]].<ref>The international SNP map working group. 2001. A map of human genome variation containing 1.42 million single nucleotide polymorphisms" ''Nature'' 409: 928--933.</ref> The first two factors reflect ascertainment errors inherent to the collection methods, the latter factor reflects an intrinsic property of the genome.
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| In the [[Coalescent theory|coalescent model]] of population genetics each genetic locus has its own unique history. Within the evolution of a population from some species some genetic loci could presumably be traced back to a relatively [[Most recent common ancestor|recent common ancestor]] whereas other loci might have more ancient [[Genetic genealogy|genealogies]]. More ancient genomic segments would have had more time to accumulate SNPs and to experience [[Genetic recombination|recombination]]. '''R R Hudson''' has proposed a model where recombination could cause variation in the time to [[Most recent common ancestor|most common recent ancestor]] for different genomic segments.<ref>Hudson RR. 1991. Gene genealogies and the coalescent process. Oxford surveys in evolutionary biology 7: 1--44.</ref> A high recombination rate could cause a chromosome to contain a large number of small segments with less correlated genealogies.
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| Assuming a constant background rate of mutation the number of SNPs per genomic segment would accumulate proportionately to the time to the most recent common ancestor. Current [[population genetics|population genetic theory]] would indicate that these times would be [[gamma distribution|gamma distributed]], on average.<ref>Tavare S, Balding DJ, Griffiths RC & Donnelly P. 1997. Inferring coalescent times from DNA sequence data" ''Genetics'' 145: 505--518.</ref> The Tweedie compound Poisson–gamma distribution would suggest a model whereby the SNP map would consist of multiple small genomic segments with the mean number of SNPs per segment would be gamma distributed as per Hudson’s model.
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| The distribution of genes within the human genome also demonstrated a variance-to-mean power law, when the method of expanding bins was used to determine the corresponding variances and means.<ref name=KendalGenes /> Similarly the number of genes per enumerative bin was found to obey a Tweedie compound Poisson–gamma distribution. This probability distribution was deemed compatible with two different biological models: the '''microarrangement model''' where the number of genes per unit genomic length was determined by the sum of a random number of smaller genomic segments derived by random breakage and reconstruction of protochormosomes. These smaller segments would be assumed to carry on average a gamma distributed number of genes.
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| In the alternative '''gene cluster model''', genes would be distributed randomly within the protochromosomes. Over large evolutionary timescales there would occur [[gene duplication|tandem duplication]], [[mutation|mutations, insertions, deletions]] and [[chromosomal rearrangement|rearrangements]] that could affect the genes through a stochastic [[Birth–death process|birth, death and immigration process]] to yield the Tweedie compound Poisson–gamma distribution.
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| Both these mechanisms would implicate [[Neutral theory of molecular evolution|neutral evolutionary processes]] that would result in regional clustering of genes.
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| ===Random matrix theory===
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| | |
| The '''[[Random matrix|Gaussian unitary ensemble]]''' (GUE) consists of complex [[Hermitian matrix|Hermitian matrices]] that are invariant under [[unitary transformation]]s whereas the '''[[Random matrix|Gaussian orthogonal ensemble]]''' (GOE) consists of real symmetric matrices invariant under [[orthogonal transformation]]s. The ranked [[Eigenvalues and eigenvectors|eigenvalues]] ''E<sub>n</SUB>'' from these random matrices obey '''[[Wigner semicircle distribution|Wigner’s semicircular distribution]]''': For a ''NxN'' matrix the average density for eigenvalues of size ''E'' will be
| |
| | |
| : <math>\bar{\rho}(E)= \begin{cases} \sqrt{2N-E^2}/\pi
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| & \quad \left\vert E \right\vert < \sqrt{2N} \\ 0 & \quad \left\vert E \right\vert > \sqrt{2N} \end{cases} </math>
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| | |
| as ''E''{{math|<VAR>→ ∞ </VAR>}}. Integration of the semicircular rule provides the number of eigenvalues on average less than ''E'',
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| : <math>\bar{\eta}(E) = \frac{1}{2\pi}\left [E\sqrt{2N-E^2}+2N \arcsin \left( \frac{E}{\sqrt{2N}} \right )+ \pi N \right ] </math>.
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| The ranked eigenvalues can be '''unfolded''', or renormalized, with the equation
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| | |
| : <math>e_n = \bar{\eta}(E)=\int \limits_{-\infty}^{E_n}dE^\prime \bar{\rho}(E^\prime) </math>.
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| This removes the trend of the sequence from the fluctuating portion. If we look at the absolute value of the difference between the actual and expected cumulative number of eigenvalues
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| | |
| : <math>\left | \bar{D}_n \right | =\left | n- \bar{\eta}(E_n) \right | </math>
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| | |
| we obtain a sequence of '''eigenvalue fluctuations''' which, using the method of expanding bins, reveals a variance-to-mean power law.<ref name="Kendal2011b" />
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| The eigenvalue fluctuations of both the GUE and the GOE manifest this power law with the power law exponents ranging between 1 and 2, and they similarly manifest 1/''f'' noise spectra. These eigenvalue fluctuations also correspond to the Tweedie compound Poisson–gamma distribution and they exhibit multifractality.<ref name="Kendal2011b" />
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| | |
| ===The distribution of [[prime number]]s===
| |
| | |
| The '''second [[Chebyshev function]]''' ''ψ''(''x'') is given by,
| |
| | |
| :<math> \psi(x) = \sum_{\hat{p}^k\le x}\log \hat{p}=\sum_{n \leq x} \Lambda(n) </math>
| |
| | |
| where the summation extends over all prime powers <math>\hat{p}^k </math> not exceeding ''x'', ''x'' runs over the positive real numbers, and <math>\Lambda(n)</math> is the [[von Mangoldt function]]. The function ''ψ''(''x'') is related to the [[prime-counting function]] ''π''(''x''), and as such provides information with regards to the distribution of prime numbers amongst the real numbers. It is asymptotic to ''x'', a statement equivalent to the [[prime number theorem]] and it can also be shown to be related to the zeros of the [[Riemann zeta function]] located on the critical strip ρ, where the real part of the zeta zero ρ is between 0 and 1. Then ψ expressed for ''x'' greater than one can be written:
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| :<math>\psi_0(x) = x - \sum_\rho \frac{x^\rho}{\rho} - \ln 2\pi - \frac12 \ln(1-x^{-2})</math>
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| | |
| where
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| | |
| : <math>\psi_0(x) = \lim_{\varepsilon \rightarrow 0}\frac{\psi(x-\varepsilon)+\psi(x+\varepsilon)}2.</math>
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| | |
| The [[Riemann hypothesis]] states that the [[root of a function|nontrivial zeros]] of the [[Riemann zeta function]] all have [[real part]] ½. These zeta function zeros are related to the [[prime number theorem|distribution of prime numbers]]. [[Lowell Schoenfeld|'''Schoenfeld''']]<ref>{{cite journal|last=Schoenfeld|first=J|title=Sharper bounds for the Chebyshev functions θ(x) and ψ(x). II|journal=Math Computation|year=1976|volume=30|issue=134|pages=337–360}}</ref> has shown that if the Riemann hypothesis is true then
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| : <math> \Delta(x)=\left\vert \psi(x)-x \right\vert < \sqrt{x} \log^{2}(x)/(8 \pi)</math>
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| | |
| for all <math>x>73.2</math>. If we analyze the Chebyshev deviations Δ'' (n)'' on the integers ''n'' using the method of expanding bins and plot the variance versus the mean a variance to mean power law can be demonstrated.<ref>{{cite journal|last=Kendal|first=WS|title=Fluctuation scaling and 1/f noise: shared origins from the Tweedie family of statistical distributions|journal=J Basic Appl Phys|year=2013|volume=2|pages=40–49}}</ref> Moreover, these deviations correspond to the Tweedie compound Poisson-gamma distribution and they exhibit ''1/f'' noise.
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| ===Other applications===
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| Applications of Tweedie distributions include:
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| * actuarial studies<ref>Haberman, S. and Renshaw, A. E. 1996.
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| Generalized linear models and actuarial science. The Statistician, 45:
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| 407--436.</ref><ref>Renshaw, A. E. 1994.
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| Modelling the claims process in the presence of covariates. ASTIN
| |
| Bulletin 24: 265--286.</ref><ref>Jørgensen, B. and Paes de
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| Souza, M. C. 1994. Fitting Tweedie's compound Poisson model to insurance
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| claims
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| data. Scand. Actuar. J. 1: 69--93.</ref><ref>Haberman, S.,
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| and Renshaw, A. E. 1998.
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| Actuarial applications of generalized linear models. In Statistics in
| |
| Finance, D. J. Hand and S. D. Jacka (eds), Arnold,
| |
| London.</ref><ref>Mildenhall, S. J. 1999. A systematic
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| relationship between minimum bias and generalized linear models. 1999
| |
| Proceedings of the Casualty Actuarial Society 86:
| |
| 393--487.</ref><ref>Murphy, K. P., Brockman, M. J., and Lee,
| |
| P. K. W. (2000). Using generalized linear models to build dynamic
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| pricing systems. Casualty Actuarial Forum, Winter
| |
| 2000.</ref><ref>{{cite journal
| |
| |last=Smyth |first=G.K.
| |
| |last2=Jørgensen |first2=B.
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| |year=2002
| |
| |title=Fitting Tweedie's compound Poisson model to insurance claims data: dispersion modelling
| |
| |journal=ASTIN Bulletin
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| |volume=32 |pages=143–157
| |
| |url=http://www.casact.org/library/astin/vol32no1/143.pdf
| |
| }}</ref>
| |
| * assay analysis <ref>Davidian, M. 1990. Estimation of variance
| |
| functions in assays with possible unequal
| |
| replication and nonnormal data. Biometrika 77:
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| 43--54.</ref><ref>Davidian, M., Carroll, R. J. and Smith, W.
| |
| 1988. Variance functions and the minimum detectable concentration in
| |
| assays. Biometrika 75: 549--556.</ref>
| |
| * survival analysis<ref>Aalen, O. O. 1992. Modelling heterogeneity
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| in survival analysis by the compound Poisson distribution. Ann. Appl.
| |
| Probab. 2: 951--972.</ref><ref>Hougaard, P. , Harvald, B.
| |
| and Holm, N. V. 1992. Measuring the similarities between the lifetimes
| |
| of adult Danish twins born between 1881--1930. Journal of the American Statistical Association
| |
| 87: 17--24.</ref><ref>Hougaard, P. 1986. Survival models for
| |
| heterogeneous populations derived from stable distributions.
| |
| Biometrika, 73: 387--396.</ref>
| |
| * ecology <ref name=Kendal2002>Kendal WS (2002) Spatial aggregation of the Colorado potato beetle described by an exponential dispersion model. ''Ecol Model'' 151, 261–269</ref>
| |
| * analysis of alcohol consumption in British teenagers <ref>Gilchrist, R. and Drinkwater, D. 1999.
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| Fitting Tweedie models to data with probability of zero responses.
| |
| Proceedings of the 14th International
| |
| Workshop on Statistical Modelling, Graz, pp. 207--214.</ref>
| |
| * medical applications <ref name="smyth1996">Smyth, G. K. 1996.
| |
| Regression analysis of quantity data with exact zeros.
| |
| Proceedings of the Second Australia--Japan Workshop on Stochastic Models
| |
| in Engineering, Technology and Management. Technology Management
| |
| Centre, University of Queensland, 572--580.</ref>
| |
| * meteorology and climatology <ref name="smyth1996"/><ref>Hasan, M.M.; Dunn, P.K. (2010) "Two
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| Tweedie distributions that are near-optimal for modelling monthly
| |
| rainfall in Australia", ''International Journal of Climatology'',
| |
| {{doi|10.1002/joc.2162}}</ref>
| |
| * fisheries <ref>Candy, S. G. 2004. Modelling catch and effort
| |
| data using generalized linear models,
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| the Tweedie distribution, random vessel effects and random
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| stratum-by-year effects.
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| CCAMLR Science. 11: 59--80.</ref>
| |
| * [[Mertens function]] <ref name="Kendal2011a" />
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| | |
| ==References==
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| | |
| {{reflist}}
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| | |
| == Further reading ==
| |
| | |
| * Kaas, R. (2005). [http://ucs.kuleuven.be/seminars_events/other/files/3afmd/Kaas.PDF "Compound Poisson distribution and GLM’s – Tweedie’s distribution"]. In ''Proceedings of the Contact Forum "3rd Actuarial and Financial Mathematics Day"'', pages 3–12. Brussels: Royal Flemish Academy of Belgium for Science and the Arts.
| |
| * Ohlsson, E and Johansson, B. (2003) [http://www.math.su.se/matstat/reports/seriea/2003/rep15/report.pdf ''Exact Credibility and Tweedie Models''], University of Stockholm, Research report, October 2003.
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| * Tweedie, M.C.K. (1956). "Some statistical properties of inverse Gaussian distributions". ''Virginia J. Sci. (N.S.)'' 7, 160—165.
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| ==External links==
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| * Tweedie distributions. http://www.statsci.org/s/tweedie.html
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| * Tweedie generalized linear model family. http://www.statsci.org/s/tweedief.html
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| * Examples of use of the model. http://www.sci.usq.edu.au/staff/dunn/Datasets/tech-glms.html#Tweedie
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| * tweeDEseq: R package for RNA-seq data analysis using the Poisson-Tweedie family of distributions. http://bioconductor.org/packages/2.9/bioc/html/tweeDEseq.html
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| {{ProbDistributions|families}}
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| [[Category:Continuous distributions|continuous distributions]]
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| [[Category:Probability distributions]]
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| [[Category:Systems of probability distributions]]
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