# The invariances of power law size distributions

Steven A. Frank  Department of Ecology and Evolutionary Biology, University of
California, Irvine, CA 92697–2525 USA<EMAIL_ADDRESS>
###### Abstract

Size varies. Small things are typically more frequent than large things. The
logarithm of frequency often declines linearly with the logarithm of size.
That power law relation forms one of the common patterns of nature. Why does
the complexity of nature reduce to such a simple pattern? Why do things as
different as tree size and enzyme rate follow similarly simple patterns? Here
I analyze such patterns by their invariant properties. For example, a common
pattern should not change when adding a constant value to all observations.
That shift is essentially the renumbering of the points on a ruler without
changing the metric information provided by the ruler. A ruler is shift
invariant only when its scale is properly calibrated to the pattern being
measured. Stretch invariance corresponds to the conservation of the total
amount of something, such as the total biomass and consequently the average
size. Rotational invariance corresponds to pattern that does not depend on the
order in which underlying processes occur, for example, a scale that
additively combines the component processes leading to observed values. I use
tree size as an example to illustrate how the key invariances shape pattern. A
simple interpretation of common pattern follows. That simple interpretation
connects the normal distribution to a wide variety of other common patterns
through the transformations of scale set by the fundamental invariances.

Keywords: Measurement; maximum entropy; information theory; statistical
mechanics; extreme value distributions.

Preprint of published version: Frank, S. A. 2016. The invariances of power law
size distributions. F1000Research 5:2074, doi:10.12688/f1000research.9452.2.
Published under a Creative Commons CC BY 4.0 license.

## Introduction

The size of trees follows a simple pattern. Small trees are more frequent than
large trees. The logarithm of frequency declines linearly with the logarithm
of size1. Log-log linearity defines a power law pattern. Power laws are among
the most common patterns in nature2.

Power laws arise by aggregation over a multiplicative process, such as growth.
Many processes in nature apply a recursive repetition of a simple
multiplicative transformation, with some randomness2.

Aggregation over a random multiplicative process often erases all information
except the average logarithm of the multiplications 3, 4. That average
determines the slope of the power law line. In the case of tree size, we must
also account for the fact that trees cannot grow to the sky. The upper bound
on growth causes the frequencies of the largest trees to drop below the power
law line.

That simple view of aggregation and the regularity of power laws contrasts
with an alternative view. By the alternative view, the great regularity of a
power law pattern suggests that there must be a very specific and particular
underlying generative process. If the pattern of tree size is so regular, then
some specific process of trees must have created that regularity.

To support the simple view of aggregation and regularity, I show that a normal
distribution contains the same information as a power law size distribution.
The distributions differ only in the scaling used to measure the distance of
random variations in size from the most common size5.

The normal distribution calls to mind the great regularity in pattern that
arises solely from the aggregation of an underlying stochastic process.
Stochasticity and aggregation alone are sufficient to explain the regularity6.
There is no need to invoke a detailed generative process specific to trees.
Given the observed power law of sizes, maybe all we can reasonably say is that
growth is a stochastic multiplicative process and that trees do not grow to
the sky.

The trees provide an example of deeper principles about pattern and process in
biology. What exactly are those principles? How can we use those principles to
gain insight into biological problems?

To start on those questions, the next section presents an example of tree size
data. Those data follow a power law with an upper bound on size. I show that
those data also match almost exactly to a normal distribution when scaled with
respect to a natural metric of growth.

The normal distribution and the power law pattern express the same underlying
relation between pattern and process. That underlying relation arises from a
few simple invariance principles. I introduce those invariance principles and
how those principles shape the common patterns of nature5.

Figure 1: (A) Tree size, $z=d^{2}$, in which the squared diameter, $d^{2}$, is
proportional to the cross sectional area of the stem, and $d$ ranges over
approximately $11$–$2800$mm. The green line shows great regularity of pattern
as a power law over the range that covers almost all probability. The largest
trees, beyond the green power law line, comprise only a small fraction of all
trees, because of the logarithmic scaling of frequency. (B) The blue line is
$\log q_{z}=\log k-\lambda{T}_{z}$, with ${T}_{z}=\log(1+az)+\gamma z$, and
parameters $\lambda=1.06$, $a=0.004$, and $\gamma=7\times 10^{-7}$, with $\log
k$ shifting curve height and total probability. (C) The fitted blue line in
panel B is a classic normal distribution with variance $1/2\lambda$ when
plotted as $q_{z}\propto e^{-\lambda{T}_{z}}$ versus $\pm\sqrt{{T}_{z}}$, with
respect to $z$ as a positive parameter. In this plot, the metric is shifted so
that the most common type associates with a value ${T}_{z}=0$. Data
approximated from Figure 4 in Farrior et al. 1

## Tree size

Figure 1A shows the distribution of tree size in a tropical forest1. Most of
the trees lie along the green power law. The largest trees, beyond the line,
comprise only a small fraction of all trees, because of the logarithmic
scaling of frequency.

The blue curve in Fig. 1B closely fits the observed pattern. That curve
expresses the natural metric for variation in tree size, $z$, as

${T}_{z}=\log(1+az)+\gamma z.$ (1)

This metric relates size to a logarithmic term for multiplicative growth plus
a linear term for an upper bound on size. There is no additional information
in the fitted curve beyond this natural metric.

The normal distribution in Fig. 1C expresses exactly the same information
about the distribution of tree sizes as the fitted curve in Fig. 1B. The
normal distribution follows from the expression of size variation in terms of
the natural metric, ${T}_{z}$. I derive these conclusions in the following
sections.

## Natural metrics

The pattern of tree size can be understood by considering ${T}_{z}$ as a
natural metric for size. A natural metric expresses a shift and stretch
invariant scale for an observed probability pattern 5. Shift, by adding a
constant to a natural metric, does not change observed pattern. Stretch, by
multiplying the metric by a constant, does not change pattern.

Ideally, a natural metric also expresses the relation between underlying
process and observed pattern. However, we can be right about the proper
natural description of observed pattern but wrong about its underlying cause.
It is important to distinguish description from causal interpretation.

The next section describes the natural metric for tree size with respect to
the fundamental invariances of shift and stretch. I discuss the panels of Fig.
1 as simple expressions of the natural metric. The following sections consider
how to interpret natural metrics, the description of observed pattern, and the
analysis of underlying process. The presentation here extends the underlying
abstract theory to the interpretation and intuitive understanding of empirical
pattern. Technical details can be found in the cited articles.

## The metric of tree size: affine invariance

The data1 in Fig. 1 arose from measurements of trunk diameter, $d$. I sought a
natural metric based on $d$ that describes the data in a shift and stretch
invariant manner5.

How does one find a shift and stretch invariant natural metric that matches an
observed pattern? In practice, one uses the extensive underlying theory and
prior experience in what often works3, 7, 8, 4. I achieved an excellent fit to
the observed tree size data in Fig. 1B based on the metric, ${T}_{z}$, in eqn
1. I summarize the steps by which I arrived at that metric.

The data form a probability distribution. Probability patterns have a generic
form. Measurements, $z$, relate to the associated probability, $q_{z}$. The
natural metric, ${T}_{z}$, transforms measurements such that the probability
pattern has the exponential form

$q_{z}=ke^{-\lambda{T}_{z}},$ (2)

in which adjusts the stretch of ${T}_{z}$, and $k$ adjusts the total
probability to be one.

Probability patterns in the exponential form are shift and stretch invariant
with respect to the metric, ${T}_{z}$. In particular, the affine
transformation of shift and stretch, ${T}_{z}\mapsto\alpha+\beta{T}_{z}$, is
exactly compensated by adjustments of $k$ and , leaving the probability
pattern invariant.

Intuitively, we can think of affine invariance as defining a ruler that is
linear in the metric, ${T}_{z}$. In a linear ruler, it does not matter where
we put the zero point. The information in measurement depends only on the
distance from where we set zero to where the observation falls along the
ruler. That independence of the starting point is shift invariance.

Similarly, if we uniformly stretch or shrink the ruler, we still get the same
information about the relative values of different measurements. All we have
to do is multiply all measurements by a single number to recover exactly the
same distances along the original ruler. The metric ${T}_{z}$ provides
information that is stretch invariant.

To fit the data of Fig. 1A, we have to find the matching affine invariant
metric, ${T}_{z}$, for probability expressed in the exponential form of eqn 2.

## The metric of tree size: scale

Most natural metrics are simple combinations of linear, logarithmic, and
exponential scaling8, 4. For example, in the metric ${T}_{z}=\log z+\gamma z$,
the logarithmic term dominates when $z$ is small, and the linear term
dominates when $z$ is large. The metric scales in a log-linear way. Change in
scale with magnitude often occurs in natural metrics.

Roughly speaking, the linear, logarithmic, and exponential scales correspond
to addition, multiplication, and exponentiation. Those arithmetic operations
are the three primary ways by which quantities combine. One can think of
numbers combining additively, multiplicatively or exponentially at different
magnitudes, depending on the way in which process changes with magnitude.

Small trees tend to grow multiplicatively, and large trees tend to scale
linearly as they approach an upper size limit. Farrior et al. 1 used
logarithmic scaling at small magnitudes and linear scaling at large
magnitudes. However, they did not express a metric that smoothly changed the
proportion of the two scalings with magnitude. Instead, they switched from log
to linear scaling at some transition point.

The observed data fit roughly to a pure log-linear metric, ${T}_{z}=\log
z+\gamma z$, with $z=d$ as tree diameter. I obtained a better fit by modifying
this metric in two ways to obtain the expression in eqn 1.

First, I used the square of the diameter, $z=d^{2}$, which is proportional to
the cross sectional area of the trunk at the point of measurement. Various
intuitive reasons favor area rather than diameter as a measure of size and
growth. However, I ultimately chose area because it fit the data.

Second, I replaced $\log z$ by $\log(1+az)$. On a pure log scale, $\log z$
explodes to negative infinity as $z$ approaches zero. In application to
positive data, such as size, it almost always makes sense to use $\log(1+az)$.
This expression becomes smaller in magnitude as $z$ declines. The parameter
$a$ scales the rate of change with respect to the point of origin.

Size distributions often follow the metric, ${T}_{z}=\log(1+az)+\gamma z$. Of
course, not all distributions follow that pattern. But one can use it as a
default. When observations depart from this default, the particular
differences can be instructive.

## Interpretation of natural metrics

The natural metric of a probability pattern transforms observed values on the
scale $z$ into probability values on the scale ${T}_{z}$. Through the natural
metric, the particular pattern on the observed scale, $z$, becomes a universal
probability pattern in the natural metric, ${T}_{z}$.

One can understand the intuitive basis of natural metrics by considering the
properties of the universal probability scale. Probability patterns are often
discussed with words such as _information_ or _entropy_ 9. Those words have
various technical and sometimes conflicting definitions. But all approaches
share essential intuitive concepts.

_Surprise_ expresses the intuition10. Rare events are more surprising than
common events. Suppose a particular size, $z$, occurs in one percent of the
population, and another size, $z^{\prime}$, occurs in two percent of the
population. We will be more surprised to see $z$ than $z^{\prime}$. How much
more surprised?

Surprise is relative. We should be equally surprised by comparing
probabilities of $0.01$ versus $0.02$ and $0.0001$ versus $0.0002$. Each
contrast compares one event against another that is twice as common.

What is a natural metric of probability that captures these intuitive notions
of surprise? For probability, $q_{z}$, the surprise is defined as

${\cal{S}}_{z}=-\log q_{z}.$ (3)

We compare events $z$ and $z^{\prime}$ by taking the difference

${\cal{S}}_{z}-{\cal{S}}_{z^{\prime}}=\log q_{z^{\prime}}-\log
q_{z}=\log\frac{q_{z^{\prime}}}{q_{z}}.$

This natural metric, ${\cal{S}}_{z}$, leads to affine invariant comparisons of
surprise values. In the affine transformation,
${\cal{S}}\mapsto\alpha+\beta{\cal{S}}$, the shift cancels in the difference
${\cal{S}}_{z}-{\cal{S}}_{z^{\prime}}$. The stretch causes a constant change
in length independently of location, so the metric retains the same
information at all magnitudes of the scale.

The relation between the universal metric of probability, ${\cal{S}}_{z}$, and
the natural metric for a particular observed scale, ${T}_{z}$, follows from
the exponential form for probability5 in eqn 2. From that exponential form, we
can write ${\cal{S}}_{z}=\lambda{T}_{z}-\log k$. Because ${\cal{S}}_{z}$ is
shift invariant, we can ignore the constant $\log k$ term, yielding

${\cal{S}}_{z}=\lambda{T}_{z}.$

The natural metric, ${T}_{z}$, transforms an observed scale, $z$, into the
universal metric of probability patterns, ${\cal{S}}_{z}$. The fitted curve in
Fig. 1B is a plot of ${\cal{S}}_{z}=\lambda{T}_{z}$ versus $\log z$.

To interpret a scale, it is useful to think about what happens along each
increment of the scale. Define $\textrm{d}{\cal{S}}_{z}$ and
$\textrm{d}{T}_{z}$ as small increments along the scales at the point
associated with $z$. Then

$\textrm{d}{\cal{S}}_{z}=\lambda\textrm{d}{T}_{z},$

which means that the scales ${\cal{S}}_{z}$ and ${T}_{z}$ change in the same
way at all magnitudes of $z$, with as the constant of proportionality in the
translation from one scale to the other.

How do small increments in the natural metric, $\textrm{d}{T}_{z}$, relate to
increments in the observed values, $\textrm{d}z$? If we assume that ${T}_{z}$
increases with $z$, and define
${T}_{z}^{\prime}=\textrm{d}{T}_{z}/\textrm{d}z$ as the derivative (slope) of
${T}_{z}$ with respect to $z$, then

$\textrm{d}{\cal{S}}_{z}=\lambda{T}_{z}^{\prime}\textrm{d}z.$

Here ${T}_{z}^{\prime}$ transforms increments along the observable scale,
$\textrm{d}z$, into increments along the universal scale of probability
pattern, $\textrm{d}{\cal{S}}_{z}$. All of the information that relates
observation to probability pattern is summarized by the natural metric,
${T}_{z}$.

## Generative process: generic vs particular

What underlying generative process leads to an observed pattern? We must
separate two aspects. Generic aspects arise from general properties of
aggregation, measurement and scale that apply to all problems. Particular
aspects arise from the special attributes of each problem.

Confusing generic and particular aspects leads to the greatest
misunderstandings of pattern and process3, 4. For example, the observed
pattern in Fig. 1 perfectly expresses generic properties. Aggregation leads to
the normal distribution by the central limit theorem (Fig. 1C). The natural
metric of size, ${T}_{z}$, relates the normal distribution to power law and
exponential scaling in Fig. 1A,B, when probability is plotted with respect the
logarithm of the observed values, $z$.

In the tree size data, simple generic properties account for all of the
observed pattern. I do not mean that there is nothing particular about trees
or that we cannot study how ecological processes influence tree size. I mean
that we must not confuse the generic for the particular in our strategy of
inference 3, 6, 11.

This article focuses on generic aspects of pattern. The following sections
discuss those generic aspects in more detail.

## The normal distribution and generic pattern

One often observes great regularity in probability patterns. Tree size follows
a power law with an upper bound. Other measurements, such as height, weight,
and enzymatic rate, also express regularity, but with different patterns.

A single underlying quantity captures the generic regularity in seemingly
different patterns. That underlying quantity is the average distance of
observations from the most common type6. The key is to get the correct measure
of distance, which is the natural metric.

The normal distribution is a pure expression of the generic regularity in
probability patterns. In the normal distribution, the variance is the average
distance of fluctuations from the mean.

In the normal distribution, the natural metric is the squared deviation from
the mean, ${T}_{z}=z^{2}$. Here, $z$ is the observed deviation from the mean,
and ${T}_{z}$ is the natural metric for distance. The normal distribution
follows from the standard expression of probability patterns in eqn 2,
repeated here with $v=k$, as

$q_{z}=ve^{-\lambda{T}_{z}}.$ (4)

The average of the squared deviations, ${T}_{z}=z^{2}$, is the average
distance of fluctuations from the most common type, which is the definition of
the variance, 2. We can express the parameters in terms of the variance

$\lambda=\frac{1}{2{}^{2}}\mskip
80.0muv=\sqrt{\frac{\lambda}{\pi}}=\frac{1}{\sqrt{2\pi{}^{2}}},$ (5)

from which we derive the commonly written form for the normal distribution as

$q_{z}=\frac{1}{\sqrt{2\pi{}^{2}}}\,e^{-z^{2}/2{}^{2}}.$ (6)

The normal distribution is universally known but rarely understood.
Interpreting the powerful generic aspect of probability patterns often reduces
to correctly reading this equation.

The standard expression for the normal distribution in eqn 6 seems obscure. By
understanding that eqn 4 expresses the same information in a much more general
and broadly applicable way, we learn to read the simple generic aspect of
common pattern. The key arises from the relation between the natural metric,
${T}_{z}$, and the measurement scale, $z$, used to express the pattern.

## Metrics of probability and measurement

This section discusses key aspects of the natural metric transformations,
${T}_{z}$, of the underlying measurements, $z$. The understanding of
probability pattern arises from these key aspects of the natural metric.

Suppose that two observers measure the same pattern. One uses a ruler that
follows the scale, $z$. Another has a logarithmic ruler that returns
logarithmic values, $\log z$, for the same underlying values. The two
observers do not know that they are using different scales.

When the two observers plot their data, each will see a different probability
pattern. The plot of $q_{z}$ versus $z$ differs from the plot of $q_{z}$
versus $\log z$.

Similarly, two observers may see different patterns of human size if they
measure different things. Suppose one observer measures femur length, the
other measures cross sectional area of the chest. The probability patterns of
femur and chest size differ. But the different patterns reflect the same
information about the underlying size variation in the population.

What is the best way to find the relation between different observed values
and the common underlying information about variation? Often, the natural
metric for each observed scale provides the universally comparable scale for
probability pattern. That universally comparable scale can be used to express
variation as a normal distribution.

When an observed probability pattern matches the normal distribution, then the
variance summarizes all of the information in the pattern6. We can write the
variance, 2, which is the average of the squared distance for fluctuations
from the mean, as

${}^{2}=\left\langle z^{2}\right\rangle_{z}$

in which the angle brackets denote the average value of $z^{2}$, and the
subscript $z$ means that the average is taken with respect to the underlying
scale, $z$.

The great generality of the normal distribution arises from a broader concept
of the average distance of fluctuations from a central location

${}^{2}=\left\langle z^{2}\right\rangle_{z}\mskip 20.0mu\longrightarrow\mskip
20.0mu\tilde{\sigma}^{2}=\left\langle T\mskip 1.0mu\right\rangle_{\sqrt{T}}.$
(7)

The left shows the standard definition of the variance as the average squared
distance from a central location. The right generalizes that notion of average
squared distance by using the average of the natural metric, ${T}_{z}$, in
which the average is taken with respect to the square root of the natural
metric, $\sqrt{{T}_{z}}$. Here, ${T}_{z}$ is shifted so that the most common
type associates with ${T}_{z}=0$, and the metric expresses fluctuations from
the most common type 5.

On the left, we average $z^{2}$ with respect to $z$. On the right, we average
${T}_{z}$ with respect to $\sqrt{{T}_{z}}$. The general form on the right-hand
side includes the left-hand side as the special case of ${T}_{z}=z^{2}$.

The key conclusion is that common probability patterns expressed in their
natural metric

$q_{z}=ve^{-\lambda{T}_{z}}$

are normal distributions when plotting $q_{z}$ versus $\pm\sqrt{{T}_{z}}$.

The following sections present examples. Later sections show why the square
root is a natural measurement scale for common probability patterns.

## Natural metrics and generic forms

The tree size data match almost perfectly to the generic normal distribution
(Fig. 1C). I discuss that match in terms of universal properties of the normal
distribution, given in the prior sections.

Tree size variation follows a simple log-linear natural metric, ${T}_{z}$.
That metric and its associated probability pattern

$q_{z}=ke^{-\lambda{T}_{z}}$

closely fit the data. Figure 1B shows the fit when plotting $\log q_{z}$
versus $\log z$. Figure 1C shows that the same observed variation closely fits
a normal distribution when plotting $q_{z}$ versus $\pm\sqrt{{T}_{z}}$.

The generalized variance is the average squared fluctuation of tree size from
the most common type, when squared fluctuations are expressed by the natural
metric, and fluctuations are measured by the square root of the natural
metric. By the generalized notion of the variance in eqn 7, all of the
information in the observed distribution of tree size is contained in the
average distance of fluctuations, measured in the natural metric.

The transformation of data into a normal distribution is sometimes considered
a trivial step in the statistical analysis of significance levels. Here, in
contrast, the natural metric and the associated expression in normal form
provide an essential step in the general understanding of pattern and process.

Later sections discuss why the normal distribution arises as the simple
expression of pattern in relation to natural metrics. Before turning to those
concepts, I present another example.

Figure 2: A pair of common natural metrics related by dimensional inversion,
with generic expression by the normal distribution. (A) The probability
distribution based on the natural metric in eqn 1, with
${T}_{r}=\log(1+ar)+\gamma r$. This plot uses a linear abscissa, compared with
the logarithmic abscissa of Fig. 1A. The curve approximately fits the
enzymatic rate data in Figure 2B of Iversen et al. 12, in which $r$ has units
$S/t$ measured as number of molecules per unit time (seconds). Here, $r$
varies between $0$ and $8$. The approximately fitted parameters are $a=0.5$,
$\gamma=0.05$, and $\lambda=1.6$. (B) The Laplace transform of the upper panel
yields a shifted gamma probability distribution that expresses the identical
information with a natural metric
${T}=(1/\lambda-1)\log(\tau-\gamma\lambda)+\tau/\lambda a$. The inverted
measure has units $t/S$ as time per molecule, varying in the plot between and
$4$. (C) The same probability distribution as in panel A, on a double log
scale over the range of $r$ values $0.6$ to $50$. (D) Both the original
distribution in A and the Laplace inverted distribution in B are normal
distributions when expressed in relation to the square root of their
respective natural metrics, with generalized variance $\tilde{\sigma}^{2}$ in
eqn 7.  Figure 3: The fitted probability distribution for the tree size data
in Fig. 1B. This distribution has the same natural metric as in Fig. 2C, but
with different parameters. The curve is colored to show the change in the
scaling of the natural metric with increasing magnitude as linear (blue),
logarithmic (gold), and linear (green).

## Dimensional inversion and metric pairs

Natural metrics sometimes come in pairs7, 4. For example, rates and
frequencies follow dual metrics. Rates have dimensional units $S/t$, in which
$S$ is a generic size or number unit, and $t$ is a time unit. A growth rate
for trees may be given in terms of the change in size per year. A chemical
reaction rate may be given as the number of molecules produced per unit time.

The inverse of a rate has units $t/S$. That inverse expresses the time to grow
larger or smaller by a particular size unit, or the time to produce a
particular number of molecules.

This section illustrates the common dual metrics for rates and times. The dual
metrics yield different probability patterns that contain exactly the same
underlying information. Each metric takes on the same common normal
distribution form when stochastic fluctuations are measured by the metric
relative to its square root.

To illustrate the dual metrics, I use the measured rates of chemical reactions
for individual enzyme molecules given by Iversen et al. 12. The measurements
produce a probability pattern for the distribution of reaction rates. The
measurements are not sufficiently precise to determine exactly which natural
metric fits the data.

I made an approximate fit to the data by using the natural metric in eqn 1,
which I previously used to fit tree size. My only purpose here is to
illustrate typical aspects of rate and frequency patterns, rather than to
over-analyze the limited data available in this particular study.

Fig. 2A shows the fitted distribution of reaction rates. The rates are in
molecules per second, $r$, with units $S/t$. The colors in the curve express
the change in the scaling relations of the natural metric as magnitude
increases. The natural metric from eqn 1, repeated here with $r=z$, is

${T}_{r}=\log(1+ar)+\gamma r.$

When $r$ is small, linear scaling of ${T}_{r}$ dominates, as shown by the blue
coloring. As $r$ increases, logarithmic scaling dominates, as shown by the
gold coloring. Fig. 2C, covering a greater range of $r$ values, shows that
further increase in $r$ leads to linear dominance of scale, as shown by the
green color. The upper linearity expresses the bound on size or number. Trees
do not grow to the sky. Reaction rates do not become infinitely fast. Fig. 3
shows the tree size data colored by the linear-log-linear transitions.

The probability pattern for rates, $S/t$, has a natural dual pattern expressed
by inverted units for time, $t/S$. We can invert units by the Laplace
transform7, 4. The inversion leads to an altered probability pattern based on
the natural metric

$\lambda{T}=\alpha\log(\tau-d)+\tau/a,$

with $\alpha=1-\lambda$ and $d=\gamma\lambda$. The parameters match the paired
metric, ${T}_{r}$. The common value of shared by the paired distributions
arises from the full expression for probability patterns in eqn 2. The
probability pattern for time, arising from ${T}$, is a gamma distribution
shifted by $d$.

The time per molecules pattern in Fig. 2B matches the dual enzyme rate pattern
of molecules per time in Fig. 2A. The dual distributions express the identical
information.

Dimensional inversion associates the various linear-log-linear scales between
the two forms7, 4. The linear, blue component at small magnitude in the upper
panel matches the long blue tail at large magnitude in the lower panel. Put
another way, slow rates, $r$, correspond to long waiting times, .

In the top, the gold logarithmic component for high rates matches the lower
gold component for short waiting times. For very high rates, $r$, we have to
look at Fig. 2C. The upper green linear tail corresponds to the rapid decline
in the probability of observing extremely high rates, associated with the
natural upper bound on rates. The green upper bound on rates matches the green
lower limit on times in Fig. 2B. If extremely rapid rates of reaction, $r$,
are very rare, then no reactions will produce molecules in very short time
periods, . That limitation produces the green shift at small times in Fig. 2B.

The dual natural metrics of rate, ${T}_{r}$, and time, ${T}$, correspond to
similar expressions of the normal distribution5 in Fig. 2D. In general,
different probability patterns expressed in different metrics, $T$, become
normal distributions when fluctuations from the most common value are measured
by $\pm\sqrt{T}$.

Figure 4: Aggregation and asymptotic invariance. The top shows polygons
randomly rotated about their center. Aggregation leads asymptotically to loss
of all information about rotational orientation. A circle purely expresses
that rotational invariance. The bottom shows the aggregate summing of
observations from arbitrary probability distributions. Aggregates combine to
produce normal distributions, purely expressing the loss of all information
except the average distance (variance) from the most common observation. The
normal distribution is invariant to the order in which observations are
combined. Order invariance is similar to rotational invariance (Fig. 5). Thus,
the asymptotic circle and the asymptotic normal distribution express similar
aspects of information loss and invariance.

## Aggregation and asymptotic invariance

Why do tree sizes and enzyme rates match a simple natural metric? Why do a few
simple natural metrics match most of the commonly observed patterns? Part of
the answer arises from the way in which aggregation leads to simple invariant
pattern.

The top rows of Fig. 4 illustrate aggregation and invariance. Each row begins
on the left with two regular polygons, randomly rotated about their center.
Columns to the right add more randomly rotated components. As the random
rotations aggregate, the shape converges asymptotically to an invariant
circular form.

Random rotation causes loss of information about the angle of orientation. In
the aggregate, the asymptotic form is rotationally invariant. In other words,
the circular shape remains invariant no matter how it is rotated. A circle
expresses pure rotational invariance.

The bottom two rows illustrate aggregation and the invariant pattern of the
normal distribution. Each row begins on the left with a probability
distribution. For each distribution, the horizontal axis represents observable
values, and the vertical axis represents the relative probability of each
observed value. I chose the shapes of the distributions to be highly irregular
and to differ from each other.

The second column is the probability distribution for the sum of two randomly
chosen values from the distribution in the left column. The third, fourth, and
fifth columns are, respectively, the sum of four, eight, and 16 randomly
chosen values. The greater the aggregation of randomly chosen values, the more
perfectly the pattern matches a normal distribution. Adding randomly chosen
values often causes an aggregate sum to converge asymptotically to the
invariant normal form.

## Natural metrics and a universal scale

The invariant normal form expresses a universal scale. That universal scale
clarifies the concept of natural metrics. To understand the universal scale,
we begin with the fact that the same pattern can be described in different
ways.

Consider enzyme catalysis. Fluctuations can be measured as the rate of
molecules produced per unit time. Alternatively, fluctuations can be measured
as the interval of time per molecule produced. Fig. 2A, B show the dual
expression of the same underlying information.

The dual measurement scales each have their own natural metric. A natural
metric transforms a particular measurement scale into a universal scale that
expresses the common underlying information. A metric is _natural_ in the
sense that it connects a particular scale of observation to a common universal
scale.

The normal distribution purely expresses the universal scale. Suppose we begin
with different scales of measurement, such as the rate of molecules produced
per unit time and the interval of time per molecule produced. Each scale has
its own distinctive pattern of random fluctuations, as in Fig. 2A, B. When we
transform each scale to its natural, universal metric, ${T}_{z}$, the pattern
of random fluctuations follows the normal distribution (Fig. 2D).

A normal distribution expresses information only about the average distance of
fluctuations from the most commonly observed value. If we measure distance for
different underlying measurements in their natural metrics, then that distance
is the universal form of variance in eqn 7 as

$\tilde{\sigma}^{2}=\left\langle T\mskip 1.0mu\right\rangle_{\sqrt{T}}.$

The generalized variance expresses the average deviation of the natural metric
relative to the square root of the natural metric.

Why is the relation between a natural metric and its square root the universal
measure of scale and also the expression of the normal distribution? The
answer concerns how rotation and aggregation lose information and leave an
invariant pattern (Fig. 4).

The next section discusses rotational invariance and its relation to the
universal scaling of the normal distribution. The following sections return to
tree size and other commonly observed size distributions. The concepts of
rotational invariance and the normal distribution clarify why the natural
metric for tree size, given in eqn 1 as ${T}_{z}=\log(1+az)+\gamma z$, is a
common natural metric for size patterns.

Figure 5: Rotational invariance and natural metrics. A circle expresses a
rotationally invariant radial distance from a central location. A natural
metric can be thought of as a measure of radial distance. Different component
observations that add to the same radial distance define a rotationally
invariant circle.

## Rotational invariance

To understand the universal scale of the normal distribution, we begin with
circles and rotational invariance (Fig. 5). Simple geometric concepts provide
the key to natural metrics, universal scales, and the structure of commonly
observed patterns.

A circle expresses a rotationally invariant radial distance from a central
location. In Euclidean geometry, squared distance is the sum of squared values
along each dimension. Invariant radial distance in two dimensions, $x_{1}$,
and $x_{2}$, may be written as $R^{2}=x_{1}^{2}+x_{2}^{2}$. The points
$\left(x_{1},x_{2}\right)$ at constant radial distance lie along the circle.
The radial distance is rotationally invariant to the angle of orientation. The
circular pattern is also invariant to interchange of the order of $x_{1}$ and
$x_{2}$.

We can think of the rotationally invariant circle as a way to decompose a
given value into components. If we start with any observed value and equate
that value with a radial distance, $R^{2}$, then the observed value is equally
consistent with all points $\left(x_{1},x_{2}\right)$ that satisfy the
circular constraint, $R^{2}=x_{1}^{2}+x_{2}^{2}$.

We can break up a given value into $n$ components, $R^{2}=\sum x_{i}^{2}$,
which is the invariant radial distance of a sphere in $n$ dimensions. Changing
the order of the components does not change the radial distance. Rotational
invariance implies order invariance of the component dimensions.

Figure 4 illustrates how aggregation leads to invariant distance. The top two
rows aggregate randomly rotated shapes. Initially, the rows differ, because
they begin with different shapes in different orientations. However, after
adding many shapes, the aggregate patterns converge to the same circular form,
because the order no longer matters in a large sample. The pattern of distance
from the center becomes the same in every direction.

The lower two rows of Fig. 4 show a similar aggregate tendency to an invariant
measure of distance. On the left, the initial patterns differ. As more samples
are added, all information is lost except the average distance of fluctuations
from the center.

The rotational invariance of circles relates to the invariance of average
distance in the normal distribution5. In both cases, the squared distance is
the standard Pythagorean definition of Euclidean geometric distance as the sum
of squares. To see the connection between the rotational invariance of circles
and the average distance of fluctuations in the normal distribution, we begin
with an observed value and consider how it might have arisen by the
aggregation of underlying components.

## Aggregation and natural metrics

Suppose we transform an observed value, $z$, into a natural metric value,
${T}_{z}$. What different aggregations would lead to the same value of
${T}_{z}$? If we think of ${T}_{z}={R}_{z}^{2}$ as a radial distance, we can
evaluate the combinations of underlying values that lead invariantly to the
same radial distance5.

Previously, we partitioned squared radial distance as

${R}_{z}^{2}=\sum x_{i}^{2}.$

We can equate the explicitly squared radial distance to the implicitly squared
natural metric, ${R}_{z}^{2}={T}_{z}$. Similarly, we can equate the explicitly
squared component dimensions to the implicitly squared dimensions, $x^{2}=y$,
or equivalently, $x=\sqrt{y}$. Then ${R}_{z}^{2}={T}_{z}$ can be written as

${T}_{z}=\sum\sqrt{y_{i}}^{\,2}.$

In two dimensions, the points $\left(x_{1},x_{2}\right)$ form a circle with
radius ${R}_{z}$. The points $\pm\left(\sqrt{y}_{1},\sqrt{y}_{2}\right)$ form
an equivalent circle with radius ${R}_{z}=\sqrt{{T}_{z}}$.

To partition a natural metric, ${T}_{z}$, of the observed value, $z$, we can
write each component dimension, $z_{i}$, in its natural metric,
$T(z_{i})=T_{i}=y_{i}$, and thus

${T}_{z}=\sum\sqrt{T_{i}}^{\,2}.$

This equation shows the different component observations of an aggregate that
lead to the same rotationally invariant squared radial distance,
${R}_{z}^{2}={T}_{z}$, or equivalently, distance as ${R}_{z}=\sqrt{{T}_{z}}$.

For the natural metric, ${T}_{z}$, the square root scale, $\sqrt{T}$, is the
natural scale of distance, aggregation, and rotational invariance.

## The normal distribution

The prior section emphasized that the natural metric ${T}_{z}={R}_{z}^{2}$ has
the square root $\sqrt{{T}_{z}}={R}_{z}$ as its natural scale of distance.
This section relates the normal distribution to this association between
natural metrics and radial distance. See Frank 5 for additional details.

We can write the standard form of probability distributions from eqn 2 as

$q_{z}=ke^{-\lambda{T}_{z}}=ke^{-\lambda{R}_{z}^{2}}$ (8)

measured in relation to the incremental scale
$\textrm{d}\sqrt{{T}_{z}}=\textrm{d}{R}_{z}$. Using the expression for the
generalized variance, $\tilde{\sigma}^{2}$, in eqn 7, we have

$1/2\lambda=\tilde{\sigma}^{2}=\left\langle
T\right\rangle_{\sqrt{T}}=\left\langle R^{2}\right\rangle_{R},$

and $k=\sqrt{\lambda/\pi}$. If we shift ${T}_{z}$ so that it is expressed as a
deviation from its minimum value, then for many natural metrics, ${T}_{z}$,
the probability pattern in eqn 8 is a normal distribution with respect to the
incremental scale $\textrm{d}\sqrt{{T}_{z}}=\textrm{d}{R}_{z}$. The
distribution is centered at the minimum of ${T}_{z}$ and has average distance
of fluctuations from the central location as the generalized variance,
$\tilde{\sigma}^{2}$.

Different natural metrics can often be expressed in this normal form. Thus,
the rotationally invariant normal form expresses a universal scale (Fig. 2D).

Rotational invariance often implies invariance with respect to the order of
observations in an aggregate. Order invariance connects the asymptotic
rotational invariance of circles and natural metrics to the asymptotic form of
the normal distribution in Fig. 4. Thus, the normal distribution, expressed in
natural metrics, provides a universal scale for understanding probability
pattern.

## Inductive: observed metric to universal scale

How does one find natural metrics? For tree size and chemical reaction rates,
I began with the observed probability pattern. From those data, I found a
natural metric that fit the observed pattern. In those cases, I chose the
natural metric based on the fact that patterns of size and reaction rate tend
to follow a particular, commonly observed natural metric.

This inductive approach matches a natural metric to a particular problem. The
natural metric can then be used to transform the observed pattern into the
universal scale of the normal distribution. What do we learn by this inductive
fit of a metric and subsequent transformation to the normal form?

We have a good sense of the normal distribution as the outcome of simple
aggregation and its connection to rotational invariance (Fig. 4). Thus, once
we find the proper scaling through the natural metric, we can think of an
observed probability pattern an an expression of the normal form on a
different scale.

For example, we can think of tree size as following a normal distribution when
we express size, $z$, in the natural metric ${T}_{z}=\log(1+az)+\gamma z$. The
normal form follows by expressing ${T}_{z}$ relative to the most common size
as the squared distance of a random fluctuation in relation to the distance,
$\sqrt{{T}_{z}}$.

By recognizing the universal normal form, we can see that different
measurements of the same underlying pattern express the same information. In
Fig. 2, the different probability patterns for rate and time have a common
normal expression. Of course, many patterns that arise from unrelated
processes also have the normal form.

The key is that the structure of commonly observed pattern arises from the
generic processes of aggregation and rotational invariance, when evaluated
with the proper natural metric, rather than from the special attributes of
particular processes. That conclusion is simply the well known principle of
statistical mechanics.

The principle of statistical mechanics is both well known and frequently
ignored in the study of pattern. The reason is that the different scales on
which observed patterns arise tend to obscure the underlying commonality. The
point here is that one can understand natural metrics and universal scales in
a rational way, and thus connect abstract principles to real problems in ways
that have often been missed.

## Deductive: universal scale to predicted metric

The inductively fit metric expresses the essence of an observed pattern. But
the fit does not tell us about the generative process that led to that
particular metric.

Ideally, one would deduce the appropriate natural metric for a problem by
considering the generative process and the necessary invariances that must be
satisfied. For example, tree size must depend on growth processes, and the
consequent probability pattern likely satisfies shift, stretch, and rotational
invariance. However, three difficulties arise.

First, the relations between process, measurement and pattern can be obscure.
For tree size, what is the proper scale on which to measure the consequences
of growth, competition, and other processes? We could use trunk diameter, $d$,
or cross-sectional area, proportional to $d^{2}$, or a fractal exponent of
diameter, $d^{s}$, or another size measure correlated with diameter.

The natural metric is often the scale that aggregates additively, leading to
patterns that tend to be shift, stretch, and rotationally invariant. However,
what we measure may be a complex transformation of that underlying scale.
Inductive fit gets around the problem by describing the pattern and its
associated invariant scale, rather than trying to deduce the processes that
caused the observed pattern.

Second, multiple processes may shape pattern. Different processes may dominate
at different scales. For example, exponential growth may dominate among
smaller trees, whereas a bound on maximum size may dominate among larger
trees. In general, different processes may dominate at different magnitudes.
Predicting the metric that fits observations requires proper combination of
the different underlying processes.

Third, natural metrics express the patterns that arise by loss of information,
subject to a few minimal constraints of invariance. Because aggregation
dissipates information, many seemingly distinct processes will generate the
same observable pattern. Common patterns are common exactly because they match
so many distinctive underlying processes 3. The natural metrics of common
patterns reflect only the similarities of the simple invariances. Most of the
special attributes of different generative processes tend to disappear in the
aggregate.

## Deductive: tree size example

Tree size depends on growth, on limits to maximum size, and on a variety of
other factors. Here, I give a simple introduction to natural metrics that
arise from growth. I do not include bounds on size or other processes. I do
not include difficulties of measurement. In spite of those limitations, this
simplified analysis of growth and natural metrics provides insight into
commonly observed probability patterns.

I begin with the form

$q_{z}=ke^{-\lambda{T}_{z}},$

which is a normal distribution when we measure increments on the square root
scale, $\textrm{d}\sqrt{{T}_{z}}$. The normal distribution arises when we
consider ${T}_{z}$ values to be an aggregate sum of component values.

For tree size, the problem concerns how the aggregation of random growth
increments leads to the observed size. We can split total growth into $t$
increments. Each incremental unit multiplies current size by $e^{g_{i}}$, in
which $g_{i}$ is the growth rate in the $i$th increment. The average growth
per increment is

$g=\frac{1}{t}\sum_{i=1}^{t}g_{i}.$

Total growth is the product of all the growth increments

$\prod e^{g_{i}}=e^{gt}=e^{w},$ (9)

in which $w=gt$ is the sum of the $t$ incremental growth rates.

The variable $w$ provides a natural base scale for growth, because it
expresses the aggregate sum of growth components. The sum is invariant to the
order of the components. Thus, the total of the incremental growth rates can
be thought of as a rotationally invariant radial distance.

Natural metrics arise from shift and stretch (affine) invariance to
transformations of their base values 7, 8, 4. Thus, a natural metric,
$T(w)\equiv{T}_{w}$, for the base scale, $w$, arises from affine invariance to
a generator transformation, $G(w)$, such that

$T\left[G(w)\right]=\alpha+bT(w)$

for some constants and $b$. If we consider

$G(w)=\delta+w$

to be a shift of the growth rates, so that the shape of probability patterns
for size does not depend on adding a constant value to growth rates, then a
natural metric for size with respect to growth is

${T}_{w}=e^{\beta w},$

in which is a positive parameter. This metric remains affine invariant to a
shift of the base scale, $w\mapsto\delta+w$, because

$T\left[G(w)\right]=e^{\beta(\delta+w)}=bT(w)$

for $b=e$. The metric ${T}_{w}$ is perhaps the most generic and important form
of all natural metrics. Its application to growth is a special case of its
underlying generality. I discussed this metric extensively in earlier articles
8, 4. Here, I confine myself to the problem of growth in relation to size.

The natural metric ${T}_{w}$ associates with the probability pattern

$q_{w}=ke^{-\phi T_{w}}=ke^{-\phi e^{\beta w}}$

when measured with respect to the incremental scale, $\textrm{d}{T}_{w}$. If
we wish to express the probability pattern with respect to measurements of
growth rate, on the incremental scale $\textrm{d}w$, note that

$\textrm{d}{T}_{w}=\beta{T}_{w}\textrm{d}w=\beta e^{\beta w}\textrm{d}w,$

yielding the probability pattern when measured with respect to the incremental
base scale, $\textrm{d}w$, as

$q_{w}=ke^{\beta w-\phi e^{\beta w}},$

in which, as always, $k$ adjusts so that the total probability is one.

Suppose we wish to transform from growth, $w$, to size, $z$, in which $w(z)$
expresses growth as a function of size. If $w$ increases with $z$, then we can
write

$\textrm{d}w=w^{\prime}\textrm{d}z,$

in which $w^{\prime}$ is the derivative of $w$ with respect to $z$. The
generic probability pattern becomes

$q_{z}=ke^{-\lambda{T}_{z}}=ke^{\log w^{\prime}+\beta w-\phi e^{\beta w}}$
(10)

with respect to the incremental measurement scale, $\textrm{d}z$.

In the tree size example, $w$ is the aggregate growth rate. Let $z_{0}+z$ be
size, with $z_{0}$ as initial size, and $z$ as the increase in size by growth,
thus

$z_{0}+z=z_{0}e^{w},$ (11)

implying that $w$ as a function of $z$ is

$w=\log(1+az).$ (12)

In this particular derivation, $a=1/z_{0}$. However, one should not interpret
parameters literally. Different generative processes will lead to the same
form, with alternative assumptions about process and parameters. Ultimately,
the invariant properties of the metric capture the essence of common pattern.
This particular derivation is meant only to show one way in which a metric
arises.

We can use eqn 12 to write the probability pattern of eqn 10 explicitly in
terms of the increase in size by growth, $z$, as

$q_{z}=ke^{(\beta-1)\log(1+az)-\phi(1+az)}=ke^{-\lambda{T}_{z}}$

with respect to the incremental scale, $\textrm{d}z$, yielding

${T}_{z}=\log(1+az)+\gamma(1+az)$ (13)

for $\beta<1$, and dropping constants of proportionality. For certain
parameter combinations, this probability pattern will be similar to the
pattern for the size metric ${T}_{z}=\log(1+az)+\gamma z$.

I presented this derivation to encourage future study. The proper way to
relate general growth processes to invariant probability patterns remains an
open problem.

## Conclusion

Probability patterns often follow a few simple scaling relations. Those
scaling relations define natural metrics. A natural metric transforms
measurements to a universal scale. On the universal scale, the average
distance of random fluctuations from the most commonly observed value defines
a generalized variance. When observed values arise by aggregation of random
processes, that aggregation erases all information except the average
fluctuation, the generalized variance.

Many different probability patterns become a normal distribution when
expressed on the universal scale of natural metrics. The only information in
each distribution is the generalized variance. Transforming the natural metric
distance back to the underlying observed values yields the standard
description for probability pattern on the scale of the observed measurements.

The great regularity of observed patterns, such as power laws, often arises
from the same aspects of aggregation and invariance that lead to the normal
distribution. A power law pattern and a normal distribution may simply be
different transformations of the same underlying pattern.

The transformations arise from measurement and from the invariances that
define scaling relations and natural metrics 7, 8, 4, 5. These key aspects of
scale provide the framework in which to study the relations between pattern
and process.

### Author contributions

SAF did all the research and wrote the article.

### Competing interests

No competing interests were disclosed.

### Grant information

National Science Foundation grant DEB–1251035 supports my research.

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## Appendix: lognormal and power law distributions

The lognormal distribution often fits well to observed patterns. For example,
many size distributions with power law attributes match reasonably well to the
lognormal form. Conceptually, the lognormal distribution is easily understood
by simple analogy with the normal distribution and the central limit theorem.
For these reasons, the lognormal distribution is frequently used in empirical
study.

In this Appendix, I briefly introduce the lognormal distribution and its
relation to power law distributions. I then explain the conceptual limitations
of the lognormal distribution, and why I did not mention the lognormal
distribution in main text. Finally, I show how pure power laws arise from
metrics given in the text.

### Approximate lognormal size distributions

In the last full section of this article, I described a simple model of growth
and consequent size. I split total growth into $t$ increments. Each
incremental unit multiplies current size by $e^{g_{i}}$, in which $g_{i}$ is
the growth rate in the $i$th increment. The average growth per increment is

$g=\frac{1}{t}\sum_{i=1}^{t}g_{i}.$

Total growth is the product of all the growth increments

$\prod e^{g_{i}}=e^{gt}=e^{w},$

in which $w=gt$ is the sum of the $t$ incremental growth rates. Given the
total growth over the increments, we can write final size relative to initial
size from eqn 11 as

$y=\frac{z_{0}+z}{z_{0}}=1+az=e^{w}.$

If growth in each increment, $g_{i}$, is a random variable, then $w$ is the
sum of random growth variables. The sum of the growth variables will sometimes
converge to an approximately normal distribution by the central limit theorem.
The approximation to normality may be close or far off depending on particular
aspects of the growth process.

The lognormal distribution is defined as the exponential of a normal
distribution. Thus, if $w$ is normally distributed, then size, $y=e^{w}$, has
a lognormal distribution of the form

$q_{y}=ke^{-\lambda(\log y-\mu)^{2}-\log y}$

on the incremental scale $\textrm{d}y$, in which $\lambda=1/2{}^{2}$ with 2 as
the variance of the normal distribution in $y$.

Lognormal distributions sometimes match reasonably well to power law patterns.
As the variance in the underlying normal distribution becomes large, becomes
small, and the lognormal distribution becomes approximately

$q_{y}\approx ke^{-\log y}=ky^{-1}.$

This limiting form approximates a power law, because the log-log plot of $\log
q_{y}$ versus $\log y$ is approximately a straight line with a slope of minus
one.

### Conceptual limitations of the lognormal

I did not mention the lognormal distribution in the text because of its
conceptual limitations. The lognormal distribution is identical to the normal
distribution. If a variable, $w$, is normally distributed, then the variable
$y=e^{w}$ has a lognormal distribution. Conceptually, there is no difference
between the normal and lognormal distributions. If a pattern is normal it is
also lognormal, and vice versa. Nothing is gained or lost by using one form or
the other.

In this article, I showed how one can change a wide variety of different
distributions into the normal distribution. My changes truly altered the
patterns of the different distributions, showing their broad conceptual unity.
For example, if one has a gamma distribution

$q_{z}=ke^{-\lambda(\log z+\gamma z)}=ke^{-\lambda{T}_{z}},$

then plotting $q_{z}$ versus $\pm\sqrt{{T}_{z}}$ leads to a normal
distribution. Here, I have related the gamma pattern to the normal form,
relating two distinct probability patterns to each other. A similar approach
works for many different metrics, ${T}_{z}$.

The value in relating distributions in this way arises from two aspects.
First, the common form of distributions in terms of metrics ${T}_{z}$ follows
from the two simple invariances of shift and stretch. Second, the relations
between different distributions and the common normal form arises from the
third invariance of rotation. Those three invariances together provide a
unified framework for understanding commonly observed pattern. The lognormal
does not add to that understanding, because it simply expresses the normal
pattern in a slightly different way, therefore sitting outside of the
conceptual framing that is the topic of this work.

### Pure power laws

The metrics in eqn 1 and eqn 13 include pure power law forms as special cases.
As $\gamma\rightarrow 0$, the metrics become ${T}_{z}\rightarrow\log(1+az)$,
and the distribution becomes

$q_{z}=k(1+az)^{-\lambda},$

which is the classic Lomax or Pareto Type II distribution. That distribution
is a power law for large values of $z$. As $a$ becomes large, the
distributions become a pure power law form

$q_{z}=kz^{-\lambda},$

in which $k$ adjusts to maintain a total probability of one.