Keywords
Macroecology, neutral theory, maximum entropy, symmetry
This article is included in the Ecology and Global Change gateway.
Macroecology, neutral theory, maximum entropy, symmetry
“It was Einstein who radically changed the way people thought about nature, moving away from the mechanical viewpoint of the nineteenth century toward the elegant contemplation of the underlying symmetry [invariance] principles of the laws of physics in the twentieth century” (ref. 1, p. 153).
Ecologists have been interested in species abundance distributions (SADs) since the classic papers by Fisher2 and Preston3. Two major patterns have been identified depending on the size of the community. In a large community, abundances often follow the log series distribution4. Specifically, the probability that a species has a population size of n individuals follows pn/n. Communities differ only in their average population size, described by the parameter, p. At smaller spatial scales, the species abundance pattern often follows a skewed lognormal (a random variable is lognormally distributed when its logarithm is normally distributed)5,6.
It is intriguing that the species abundance distribution follows these simple patterns irrespective of the particular group (birds, insects, mammals) and region considered. Other ecological patterns also follow simple probability distributions7–9. Those patterns have attracted a lot of attention. Why does the variability and complexity of biology reduce to such a small range of simple distributions? How can we understand the relations between complex processes and simple patterns?
Approaches such as Harte’s9 maximum entropy formalism and Hubbell’s5 neutral theory have attempted to explain the generality of the log series and skewed lognormal patterns in species abundance distributions. Maximum entropy describes probability distributions that are maximally random subject to satisfying certain constraints10–12. This approach has a long tradition in physics, both in statistical mechanics and information theory. An early maximum entropy approach in ecology derived the biomass pattern of populations13–15.
Neutral theory derives probability distributions by assuming that all individuals are equivalent16. Variation arises by random processes acting on the mechanistically identical individuals. Put another way, the mechanistic processes are “neutral” apart from random processes. Both maximum entropy and neutral theory have been shown to provide a good fit to the empirical patterns of species abundance distributions. In this article, we subsume these two different ways of understanding the log series and skewed lognormal patterns with a more general perspective based on the concept of invariance17.
Invariance can be defined as the property by which a system remains unchanged under some transformation. For example, a circle is the same (invariant) before and after rotation (Figure 1a). In ecology, pattern often depends on the ways in which form remains invariant to changes in measurement. Some patterns retain the same form after uniformly stretching or shrinking the scale of measurement (Figure 2b). Measures of length provide a common example of stretch invariance. One can measure lengths equivalently in millimeters or centimeters without loss of information. As we will see, that kind of invariance often determines the form of observed pattern.
To give another example, consider the common and widely familiar pattern of the normal distribution. By the central limit theorem, when independent random variables are added, their properly normalized sum tends toward a normal distribution, even when the component variables themselves are not normally distributed. The central limit theorem and the normal distribution are often considered as unique aspects of pattern that stand apart from other commonly observed patterns.
The invariance perspective that we promote shows how the normal distribution is in fact a specific example of a wider framework in which to understand the commonly observed patterns of nature. In particular, the normal distribution arises from the rotational invariance of the circle18. For two variables, x and y, with a given squared length, x2 + y2 = r2, all combinations of the variables with the same radius, r, lie along the circumference of a circle (Figure 1a). When each combination is equally likely, the rotationally invariant radius is sufficient to describe the probability pattern.
It is this rotational invariance that gives the particular mathematical form of the normal distribution, in which the average squared radius sets the variance of the distribution. By this perspective, the mathematical forms of all commonly observed distributional patterns express their unique invariances18.
The perspective of invariance was the basis for most of the great conceptual advances of physics in the twentieth century1. For example, Gell-Mann’s pioneering theoretical work on the fundamental particles of nature derived from invariance (symmetry) properties that unified understanding of known particles and predicted new particles such as quarks, which were subsequently observed. By contrast, general aspects of invariance have not been used consistently as the fundamental basis for understanding patterns in ecology. One exception concerns scale invariance, which is often discussed in ecology19–21. But scale invariance is typically limited to special kinds of patterns rather than forming a unified approach to diverse patterns.
The point of this paper is that invariance is the most general way in which to understand commonly observed patterns. Species abundance distributions provide an excellent illustration. For species abundances, we show that maximum entropy and neutral models can succeed in certain cases because they derive from invariance principles. However, maximum entropy and neutrality are often difficult to interpret because they hide their underlying basis in invariance.
Our unifying invariance analysis clarifies why seemingly different conceptual approaches have the same consequences for pattern. Similarly, seemingly different biological processes may often lead to the same observed pattern, because those different processes share a common basis in invariance. That deeper understanding suggests a more insightful way to think about alternative mechanistic models. It also suggests the kinds of empirical tests that may differentiate between alternative causal processes.
This manuscript is organized as follows. First, we highlight key theoretical results for species abundance distributions. Second, we review how invariance defines probability patterns in a general way18,22,23. The log series distribution24 and the gamma-lognormal distribution for species abundances follow directly from the universal invariance expression of probability patterns. Third, we show that maximum entropy and neutrality can easily be understood as special examples of invariance principles. Finally, we discuss the broad role of invariance in the analysis of ecological pattern.
This article develops two key theoretical results. We highlight those results before starting on the general overview of invariance and pattern.
First, we present a simple maximum entropy derivation of the log series pattern. We show that constraining the average abundance per species is sufficient when analyzing randomness and entropy on the proper demographic scale25.
The simplicity of our maximum entropy derivation contrasts with Harte’s more complicated maximum entropy model9,26. Harte had to assume an additional unnecessary constraint on energy usage. He required that unnecessary constraint because he evaluated randomness on the scale of measured abundances rather than on the scale of demographic process. This will be made explicit below.
We use this result to demonstrate that maximum entropy is the outcome of deeper underlying principles of invariance and pattern. By working at the deeper level of invariance, one obtains a simpler and more powerful understanding of pattern.
The second result shows that Hubbell’s5 neutral model is the simple expression of three basic invariances. Hubbell’s full range of log series and skewed lognormal (zero sum multinomial) results follows immediately from those three underlying invariances.
The three invariances correspond to a maximum entropy model that constrains the average abundance of species and the average and variance of the demographic processes influencing abundance. The three invariances lead to a simple gamma-lognormal distribution that matches the neutral theory pattern for species abundances25. The gamma-lognormal is a product of the standard gamma and lognormal distributions.
This section reviews how invariance considerations lead to the log series distribution24. We delay discussion of the gamma-lognormal until the later section on Hubbell’s neutral model.
We can rewrite almost any probability distribution as
in which T(z) ≡ Tz is a function of the variable, z, and k and λ are constants. For example, Student’s t-distribution, usually written as
can be written in the form of Equation 1 with λ = (ν + 1)/2 and Tz = log(1 + z2/ν).
The probability pattern, qz, is invariant to a constant shift, Tz ↦ a + Tz, because we can write the transformed probability pattern in Equation 1 as
with k = kae–λa (Figure 2a). We express k in this way because k adjusts to satisfy the constraint that the total probability be one. In other words, conserved total probability implies that the probability pattern is shift invariant with respect to Tz18.
Now consider the consequences if the average of some value over the distribution qz is conserved. For example, the average of z is the mean, µ, and the average of (z – µ)2 is the variance. A constraint causes the probability pattern to be invariant to a multiplicative stretching (or shrinking), Tz ↦ bTz, because
with λ = λbb (Figure 2b). We specify λ in this way because λ adjusts to satisfy the constraint of conserved average value. Thus, invariant average value implies that the probability pattern is stretch invariant with respect to Tz.
Conserved total probability and conserved average value cause the probability pattern to be invariant to an affine transformation of the Tz scale, Tz ↦ a + bTz, in which “affine” means both shift and stretch.
The affine invariance of probability patterns with respect to Tz induces significant structure on the form of Tz and the associated form of probability patterns. Understanding that structure provides insight into probability patterns and the processes that generate them18,22,23.
In particular, Frank and Smith22 showed that the invariance of probability patterns to affine transformation, Tz ↦ a + bTz, implies that Tz satisfies the differential equation
in which w(z) is a function of the variable z. The solution of this differential equation expresses the scaling of probability patterns in the generic form
in which, because of the affine invariance of Tz, we have added and multiplied by constants to obtain a convenient form, with Tz → w as β → 0.
By writing Tz in this way, w expresses a purely shift-invariant aspect of the fundamental affine-invariant scale, because the shift transformation w ↦ a + w multiplies Tz by a constant, and probability pattern is invariant to constant multiplication of Tz. Thus, Equation 2 dissects the anatomy of a probability pattern (Equation 1) into its component invariances.
With this expression for Tz, we may write probability patterns generically as
This form has the advantage that w(z) expresses the shift-invariant structure of a probability pattern. Most of the commonly observed probability patterns have a simple form for w23,27. That simplicity of the shift-invariant scale suggests that focus on w provides insight into common patterns.
To understand the log series, we must consider the relation n = er between the observed pattern of abundances, n, and the processes, r. Here, r represents the total of all proportional processes acting on abundance24.
A proportional process simply means that the number of individuals or entities affected by the process increases in proportion to the number currently present, n. Demographic processes, such as birth and death, act proportionally.
The sum of all of the proportional processes on abundance over some period of time is
Here, m(t) is a proportional process acting at time t to change abundance. Birth and death typically occur as proportional processes. The value of r = log n is the total of the m values over the total time, τ. For simplicity, we assume n0 = 1.
The log series follows as a special case of the generic probability pattern in Equation 3. To analyze abundance, focus on the process scale by letting the variable of interest be z ≡ r, with the key shift-invariant scale as simply the process variable itself, w(r) = r. Then Equation 3 becomes
in which qrdr is the probability of a process value, r, in the interval r + dr.
Using w(r) = r sets the the shift-invariant scale as the variable itself. Substituting this simplest form for the shift-invariant scale into the canonical equation for common probability patterns in Equation 3 yields the simplest generic expression of probability pattern as Equation 4.
We can generalize the relation between abundance and process, n = er, by writing nβ = eβr, which uses an additional parameter β to allow comparison with the canonical form of probability distributions in the previous subsection. When we focus on standard models of species abundances, we use β = 1.
We can change from the process scale, r, to the abundance scale, n, by noting that β log n = βr, and so, for any β, we have r = log n. Thus, we can use the substitutions r ↦ log n and dr ↦ n–1dn in Equation 4, yielding the identical probability pattern expressed on the abundance scale
The value of k always adjusts to satisfy the constraint of invariant total probability, and the value of λ always adjusts to satisfy the constraint of invariant average value.
For proportional processes and species abundances, β = 1, as noted above. For that value of β, we obtain the log series distribution24
replacing n – 1 by n in the exponential term which, because of affine invariance, describe the same probability pattern. The log series is often written with e–λ = p, and thus qn = kpn/n. One typically observes discrete values n = 1, 2, …. See https://doi.org/10.5281/zenodo.2597895 for the general relation between discrete and continuous distributions. The continuous analysis here is sufficient to understand pattern.
We can also write the log series on the process scale, r, from Equation 4, as 24
This form shows that the log series is the simplest expression of generic probability patterns in Equation 3. The log series arises from β = 1, associated with n = er, and from the base shift-invariant scale as w ≡ r for proportional processes, r.
This subsection summarizes a few technical points about invariance. These technical points provide background for our simpler and more general derivation in the following section of maximum entropy models for species abundances. Those previous models focused only on abundances, n, without considering the underlying process scale, r.
We begin with invariance on the process scale, r. On that scale, the log series in Equation 7 is the pure expression of additive shift invariance to r and lack of multiplicative stretch invariance to r. For example, note in Equation 7 that an additive change, r ↦ r + a, is compensated by a change in λ to maintain the overall invariance, whereas a multiplicative change, r ↦ br, cannot be compensated by a change in one of the constants. For example, if r is net reproductive rate, then an improvement in the environment that adds a constant to everyone’s reproductive rate does not alter the log series pattern. By contrast, multiplying reproductive rates by a constant does alter pattern.
To understand the parameter, β, from Equation 2, consider that
in which β is the relative curvature of the measurement scale for abundance, n, with respect to the scale for process, r. The relative curvature is β = T″/T′, with the primes denoting differentiation with respect to r27.
For the log series, the curvature of β = 1 describes the amount of bending of the abundance scale, n = er, with respect to multiplying the process scale, r, by a constant—the departure from stretch invariance.
The simple invariances with respect to process, r, become distorted and more difficult to interpret when we focus only on the observed scale for abundance, n, associated with the log series in Equation 6. In that form of the distribution, the canonical scale is
In this expression, purely in terms of abundances, the log-arithmic term dominates when n is small, and the linear term dominates when n is large. Thus, the scale changes from stretch but not shift invariant at small magnitudes to both shift and stretch invariant at large magnitudes24. Without the simple insight provided by the process scale, r, we are left with a complicated and nonintuitive pattern that is separated from its simple cause. That difficulty has led to unnecessary complications in maximum entropy theories of pattern.
Pueyo et al. developed a simple alternative approach for deriving the log series distribution that combines invariance and maximum entropy25. In their derivation, the average value of n is a maximum entropy constraint, and the equivalent of our r variable is considered as an invariant Bayesian prior in the sense of Jaynes12. Previous publications describe the differences between our invariance approach and the invariant prior maximum entropy approach of Pueyo et al.18,22,28,29.
Maximum entropy describes probability distributions that are maximally random subject to satisfying certain constraints10–12. In Equation 1, with the generic description for distributions as
maximum entropy interprets this form as the expression of maximum randomness with respect to the scale z, subject to the constraint that the average of Tz is fixed23.
This section begins with a maximum entropy derivation for the log series based on our separation between the scales of process, r, and observed abundance, n.
We then discuss Harte’s9,26 alternative maximum entropy derivation of the log series. Harte’s derivation emphasizes mechanistic aspects of energy constraints rather than our emphasis on the different scales of process and abundance.
The log series in Equation 7 is
Here, T = er = n. This distribution expresses maximum entropy with respect to the process scale, r. The constraint is the ecological limitation on average abundance
in which 〈·〉r denotes average value with respect to the process scale, r.
In this case, process values, r, are maximally random, subject to the ecological constraint that limits abundance, n. Thus, maximizing entropy with respect to the process scale, r, subject to a constraint on the observed pattern scale, n, leads immediately to the log series.
Relating the process scale, r, to the scale of ecological constraint, n, often makes sense. Typically, environmental perturbations associate with changes in demographic variables, such as birth and death rates. Such demographic factors typically act proportionally on populations, consistent with our interpretation of r as the aggregate of proportionally acting processes. The perturbations, acting on demographic variables, associate the process scale with the scale of randomness.
In contrast with the process scale of perturbation and randomness for the demographic variables, the scale of constraint naturally arises with respect to a limit on the number of individuals, n. Thus, randomness happens on the r scale and constraint happens on the n scale.
It is, of course, possible to formulate alternative models in which randomness and constraint happen on scales that differ from our interpretation. Different formulations are not intrinsically correct or incorrect. Instead, they express different assumptions about the relations between process, randomness, and invariance. The next section considers an alternative formulation.
Harte developed comprehensive maximum entropy models of ecological pattern. He tested those theories with the available data. His work synthesizes many aspects of ecological pattern9.
For species abundances, Harte9,26 analyzed maximum randomness with respect to the scale of abundance values, n. Maximum entropy derivations commonly evaluate randomness on the same scale as the observations. In this case, with observations for the probabilities of abundances, pn, entropy on the same scale is the sum or integral of –pn log pn.
However, there is no a priori reason to suppose that the scale of observation is the same as the scale of randomness. The fact that observation, randomness, and process may occur on different scales often makes maximum entropy models difficult to develop and difficult to interpret. For example, we may observe the probabilities of abundances, pn, but randomness may be maximized on the scale of process, as the sum or integral of –pr log pr.
In the final part of this section, we argue that invariance provides a truer path to the natural scale of analysis and to the mechanistic processes that generate pattern than does maximum entropy. Before comparing invariance and maximum entropy, it is useful to sketch the details of Harte’s maximum entropy model for species abundances.
The simplest maximum entropy model analyzes entropy with respect to abundance, n, subject to a constraint on the average abundance, ⟨n⟩. That analysis yields an exponential distribution
The exponential pattern differs significantly from the observed log series pattern. Thus, maximizing entropy with respect to the scale of abundance, n, and constraining the average abundance is not sufficient.
From our invariance perspective, it is natural to think of the scale of randomness in terms of dr, the scale of proportional processes, rather than in terms of dn, the scale of abundance. Maximizing randomness with respect to dr leads directly to the log series, as shown in the previous section.
Harte did not consider the distinction between the exponential and log series patterns with respect to the scale of randomness. Instead, to go from the default exponential pattern of maximum entropy to the log series, his maximum entropy analysis required additional assumptions. He proceeded in the following way.
Suppose that the total quantity of some variable, 𝜖, is constrained to be constant over all individuals of all species. The average value per individual is ⟨𝜖⟩. It does not matter what the variable 𝜖 is. All that matters is that the constraint exists. Harte assumed that 𝜖 is energy, but that assumption is unnecessary with regard to the species abundance distribution.
The value 𝜖 is distributed over individuals independently of their species identity. Thus, the variable δ|n = n𝜖 is the total value in a species with n individuals, with average value ⟨δ|n⟩ = n⟨𝜖⟩.
The joint distribution of n and δ is
The explicit form of this joint distribution can be obtained by maximizing entropy subject to the constraints on the average abundance per species, ⟨n⟩, and the average total value in a species with n individuals, ⟨δ|n⟩, yielding
We obtain the form presented by Harte26 using the equivalence δ = n𝜖, yielding
The species abundance distribution is obtained by
Noting that ∫ e–λ′n𝜖 = 1/λ′n, and absorbing the constant λ′ into k, we obtain the log series for the species abunance distribution
Harte’s maximum entropy derivation of the log series assumes joint constraints of abundance, n, and some auxiliary variable, 𝜖, which he labeled as energy. He evaluated entropy on the scales of n and 𝜖.
By contrast, our invariance derivation arises from a constraint on abundance plus evaluation of invariance or entropy on the scale r = log n. On that scale, the log series arises in a simple and clear way. There is no need for constraint of a second auxiliary variable.
Without an invariance argument, nothing compels us to analyze with respect to the r scale. Harte, without focus on invariance, followed the most natural approach of using n as the scale for maximization of randomness and for constraint. That approach required an auxiliary constraint on a second scale to arrive at the log series.
Harte’s approach was a major step in unifying the analysis of empirical pattern. But, in retrospect, his approach was unnecessarily complicated.
One might say that Harte’s approach provided a richer theory because it led to predictions about both abundance and energy. However, the data on abundance patterns match very closely to the log series, whereas the data for different proxies of energy vary considerably9.
Our invariance approach strips away the unnecessary auxiliary variable. The invariance theory therefore provides a much simpler way to derive and to understand abundance patterns.
Maximum entropy can be thought of purely as a basic invariance method of analysis. Maximum entropy distributions have the form in Equation 1 as
in which Tz is the affine-invariant scale that defines the probability pattern. Thus, the method of maximum entropy is simply a method for deriving the affine-invariant expression, Tz. In practice, maximum entropy has three limitations.
First, maximum entropy is silent with respect to the proper choice for the scale on which entropy is maximized and the constraints that set the affine-invariant expression, Tz. By contrast, focus on invariance led us to the shift invariance of the process scale, r. That scale provided a much simpler analysis, in which r is the incremental scale with respect to invariance and the measurement scale with respect to entropy.
In other words, maximum entropy is a blind application of the most basic invariance principles, without any guidance about the proper scales for invariance, randomness, and constraint. By contrast, an explicit invariance approach takes advantage of the insight provided by the analysis of invariance.
Second, by focusing on invariance, we naturally obtain the full invariance (symmetry) group expression in Equation 3 as the generic form of probability patterns
That generic expression leads us to a generalization of the log series in Equation 5 as 24
which is a two parameter distribution for abundances with respect to λ and β. The log series is a special case with β = 1.
Third, invariance leads to a deeper understanding of the relation between observed pattern and alternative mechanistic models of process. The following section provides an example.
Here, we analyze Hubbell’s5 neutral model of species abundances in the light of our invariance perspective. With that example in mind, we then discuss more generally how neutral models relate to invariance and maximum entropy.
The strong recent interest in Hubbell’s neutral model follows from the match of the theory to the contrasting patterns of species abundance distributions (SADs) that have been observed at different spatial scales. In the theory, many local island-like communities are connected by migration into a broader metacommunity. Sufficiently large metacommunities follow the log series pattern of species abundances. Each local community follows a distribution that Hubbell called the zero-sum multinomial30, which is similar to a skewed lognormal. As noted by Rosindell et al.6, it is this flexibility of the classic neutral model to reconcile the log series and lognormal distributions that allows it to fit empirical data well31.
Broad consensus suggests that species abundances closely follow the log series pattern at large spatial scales. Extensive data support that conclusion4.
Observed pattern at small spatial scales differs from the log series. Consensus favors a skewed lognormal pattern. The data typically show an excess of rare species, causing a skew relative to the symmetry of the lognormal when plotted on a logarithmic scale.
At small spatial scales, most recent analyses focus on data from a single long-term study of tree species in Panama5,30. Thus, some ambiguity remains about the form and consistency of the actual pattern at small scales. The blue curve of Figure 3 shows Chisholm & Pacala’s30 fit of the neutral theory to the Panama tree data for species abundances at small spatial scales. The gold curve shows the close match to the neutral theory pattern by a simple probability distribution derived from the analysis of invariance.
To obtain the matching distribution derived by invariance, we begin with the canonical form for probability distributions in Equation 3. That canonical form expresses pattern in terms of the shift-invariant scale, w. Next, we need to find the specific form of the scale w that relates this canonical form for probability distributions to the neutral theory. Because the neutral theory derives abundance, n, as an outcome of demographic processes, r, the fundamental shift-invariant scale for neutral theory is expressed in terms of the demographic process variable as
Below, we discuss why this is a natural shift-invariant scale for neutral theory. For now, we focus on the details of the mathematical expressions. Recall that n = er relates measured abundances, n, to the demographic process scale, r. If we assume that β = 1 in Equation 3 and use w from Equation 10, we obtain
with parameters λ, a, and α. We can write this distribution equivalently on the n scale for abundance as
In the second distribution, µ = (a – ã)/2α. Thus, both distributions have the same three parametric degrees of freedom.
The right-hand exponential term of Equation 12 is a lognormal distribution with parameters µ and σ2 = 1/2α. The remaining terms are a gamma distribution with parameters ã and λ. We call this product of the gamma and lognormal forms the gamma-lognormal distribution.
Figure 3 showed that the gamma-lognormal distribution matches the neutral theory fit for the Panama tree data. Figure 4 shows that the shape of the gamma-lognormal matches the shape of the neutral theory predictions for various mechanistic parameters of the neutral theory.
In summary, the neutral theory distribution appears to be nearly identical to a gamma-lognormal distribution when compared over realistic parameter values. Both distributions have the same three parametric degrees of freedom. Pueyo et al.25 derived the gamma-lognormal by using an invariance argument to obtain the n = er relation as a Bayesian prior for maximum entropy and then using additional constraints in a maximum entropy analysis. They also noted the good fit to Hubbell’s neutral theory. As mentioned above, our invariance analysis and our interpretation of invariance differ from Pueyo et al.’s Jayesian invariant prior approach for maximum entropy.
The constraints on pattern can be seen most clearly by rewriting Equation 11 as
in which is the squared deviation from µ, in which µ is the average value of r. This expression remains a three-parameter distribution because, as noted above, μ = (a–ã)/2α.
With this set of parameters, the affine-invariant scale is
Note that T and w are related by Equation 2. We are using w from Equation 10 and β = 1, as noted below Equation 10. We ignore the extra –1 term in T of Equation 2, because the canonical form of probability distributions is invariant to adding a constant to T. The tilde parameters of the distribution in Equation 13 are interchangeable with the nontilde parameters of the identical distribution in Equation 12. The tilde expressions focus on the invariances that will help us to interpret ecological pattern. The nontilde expressions describe pattern in terms of the classic forms for the gamma and lognormal distributions.
By the standard theory of maximum entropy, qr maximizes entropy on the incremental scale dr subject to a constraint on the average value of the defining affine-invariant scale, ⟨T⟩r. That constraint is the linear combination of three constraints: the average abundance on the process scale, ⟨n = er⟩r, the average demographic process value, ⟨r⟩, and the variance in the demographic process values,
By maximum entropy, all of the information in Hubbell’s mechanistic process theory of neutrality and the matching gamma-lognormal pattern reduces to maximum randomness subject to these three constraints.
However, it is very unlikely that we would have derived the correct form by maximum entropy without knowing the answer in advance. This limitation emphasizes that maximum entropy provides deep insight into process and pattern, but often we need an external theory to guide our choice among various possible maximum entropy formulations.
Put another way, maximum entropy and process oriented theories, such as Hubbell’s model, often work together synergistically to provide deeper insight than either approach alone.
Before turning to invariance and the gamma-lognormal pattern of neutral theory, it is useful to consider some basic properties of invariance and information34,35. In particular, this subsection develops our claim that the affineinvariant scale provides the deepest insights into the relations between pattern and process.
We start by noting that, in the general expression for probability distributions
the affine-invariant scale, Tz, is equivalent to a common expression for the information content in a measurement, z, as
This expression follows from assuming that: information depends on the probability, qz, of observing the measured value and not on the value itself; rarely observed values provide more information than commonly observed values; and the information in two independent measurements is the sum of the information in each measurement. From the general expression for probability distributions
Thus, an incremental change in information is equal to an incremental change in the affine-invariant scale
Equivalently, the change in information with respect to a change in the affine-invariant scale,
is constant at all magnitudes of the measurement, z. Every measured increment on the Tz scale provides the same amount of information about pattern. Constancy of information at all magnitudes is the ideal for a measurement scale. Thus, affine-invariance provides the ideal scale on which to evaluate the pattern in measurements23. Figure 5 illustrates some key properties of the affineinvariant scale.
Information is sometimes thought of as a primary concept. However, it is important to understand that, in this context, information and affine invariance are the same thing. Neither is intrinsically primary.
We prefer to emphasize invariance, because it is an explicit description of the properties that pattern and process must obey17,27,36. Further analysis of invariant properties leads to deeper insight. For example, only through invariance can we obtain the group theory expression for the canonical form of probability patterns (Equation 3).
By contrast, “information” is just a vague word that associates with underlying invariances. Further analysis of information requires unwinding the definitions to return to the basic invariances.
We turn now to the neutral theory model for abundances at local spatial scales. We showed that all of the information about pattern and process in the neutral theory is captured by the gamma-lognormal pattern in Equation 13 as
which defines the affine-invariant scale in Equation 14 as
On this scale, changes in r provide the same amount of information about pattern at all magnitudes. Shifting the scale by a constant does not change the information about pattern in measurements. In other words, it does not matter where we set the zero point for Tr. Similarly, uniformly stretching or shrinking the scale, Tr, does not change the information in measurements of r.
We can parse the terms of Equation 16 with respect to constraint and invariance. When r is large, the term λer = λn dominates the shape of the distribution in the upper tail, which decays as
for sufficiently large er = n. The smaller the value of λ relative to ã and α, the greater er must be for this pattern to dominate. When λ is relatively large compared with ã and α, this pattern dominates at all magnitudes and leads to the log series.
With respect to constraint, for large values of abundance, n, the constraint on average abundances dominates the way in which altered process influences pattern. With respect to invariance, a process that additively shifts or multiplicatively stretches the er = n values does not alter the pattern in the upper tail. Similarly, pattern is invariant to a process that additively shifts process values, r, but processes that multiplicatively change r alter pattern. Thus, we can evaluate the role of particular processes by considering how they change n or r.
The pattern at small and intermediate values of r depends on the relative sizes of the parameters. If the ãr term dominates, then the constraint, ⟨r⟩, on the average process value is most important. With respect to invariance when ãr dominates, a process that additively shifts or multiplicatively stretches the r values does not alter the pattern in the lower tail. That lower tail is a rising exponential shape, eãr, as in Figure 4c.
When the term is negligible at all magnitudes, the combination of the dominance by ãr in the lower tail, and the dominance by λer in the upper tail, yields the gamma distribution pattern on the abundance scale, n.
Finally, for magnitudes of r at which the term dominates, the constraint, σ2 = ⟨r – µ2⟩, on the variance in process values is most important. In this case, pattern follows a normal distribution, e–α(r–µ)2, on the r scale, which is a lognormal distribution on the abundance scale, n.
When combining numerous process values to obtain an overall net r value, approximate rotational invariance is sufficient for the pattern to be very close to a perfect normal curve (see Introduction). When measuring net squared deviations from the mean, which is the squared radial distance, the pattern is invariant to shift and stretch of the squared radial measures, (r – µ)2.
In practice, the lognormal pattern of abundance dominates when a constraint on r dominates and net values of r obey rotational invariance (symmetry) with respect to the summing up of the individual processes acting on abundance.
Any theory of process that leads to those three basic invariances will follow the gamma-lognormal pattern. The great unsolved puzzle is how specific mechanistic processes combine such that the structure of pattern is fully expressed by these particular invariances of pattern or, equivalently, by constraints on the average values of certain quantities in the context of maximum entropy. Our work opens the way for a more direct attack on this great puzzle by clarifying the anatomy of a pattern, thereby clarifying the puzzle that must be solved.
[J]ust as the physiologist divides the animal world, according to anatomy, into families and classes, so the ornamentist is able to classify all pattern-work according to its structure [invariance]. Like the scientist, he is able even to show the affinity between groups to all appearance dissimilar; and, indeed, to point out how few are the varieties of skeleton upon which all this variety of effect is framed (ref. 37, pp. 3–4). … The fact of the matter is, the characteristic lines of time-honoured patterns are mainly the direct result of the restrictions under which the craftsman was working (ref. 37, p. 47).
Invariances comprise the structural components in the anatomy of pattern. Commonly observed patterns almost always dissect completely into a few simple invariances. Our primary goal has been to introduce into ecological study the anatomy of pattern and the methods of dissection.
Identifying and naming the parts does not tell one how those parts came to be. In fact, common patterns are widespread exactly because so many different underlying mechanistic processes give rise to the same simple invariances.
Roughly speaking, one can think of a common pattern as an attractor. Each different underlying mechanistic process that develops into the generic form traces a distinctive path from some starting point to the generic endpoint of the attractor. All of the different mechanistic processes and starting points that end up at the same attractor form the basin of attraction for that pattern.
Our work characterized the anatomy of pattern—the anatomy of the attractors. The next step requires understanding how various combinations of mechanistic processes lead to one attractor or another. Equivalently, one can think of a mechanistic process as something that transforms inputs into outputs38. Three questions follow. How do particular cascades of input-output transformations ultimately combine to produce overall transformations that associate with simple invariances? What separates some cascades from others with regard to association with different invariances? In other words, how can we assign different mechanistic cascades to one basin of attraction or another?
If we could answer those questions, then we could predict whether different mechanistic processes lead to the same pattern or to different patterns.
The fact that different processes can attract to the same pattern has been widely discussed in ecology30,39–47. However, that past work typically did not explain common patterns in terms of invariance. Without invariance, one does not have a basis for describing the anatomy of common patterns or the reasons why certain processes attract to a particular pattern and others do not.
Invariance may provide a way to compare different models of process that lead to the same pattern. Among the many complex component processes that may occur in a model, which truly matter? In other words, which component processes shape the defining invariances and which are irrelevant? For the focal component processes of each model that matter, which empirical tests would tell us which of the alternative mechanistic models is the more likely match to natural processes?
The apparent simplicity of invariance can mislead about its ultimate power. For example, probability patterns express a shift and stretch invariant scaling. That affine-invariant scaling provides a constant measure of information at all magnitudes.
Shift and stretch invariance seem almost trivially simple. Yet, by analyzing how repeated transformations of shift and stretch retain invariance, we obtain the most general form that expresses various affine-invariant scales (Equation 2). That affine symmetry group defines the simple, general structure of probability patterns and their uniform measurement scales.
Knowing the general invariant form of probability patterns reveals the relations between different approaches. Invariance provides powerful methods to analyze pattern and process.
To sum up, our invariance approach is not just another one among various alternatives. Rather, it is the only way to relate process to pattern, because the essence of pattern is invariance. Only by understanding what pattern actually is and how it generally arises can one begin to formulate testable hypotheses about mechanism.
Put another way, pattern is always the interaction between, on the one hand, the generic aspects of invariance and scale that arise in all cases and, on the other hand, the particular aspects of biology that operate in each case. Without a clear view of that duality between the generic and the particular, it is easy to mistakenly attribute generic aspects of observed pattern to particular causes. To properly understand the role of specific mechanistic aspects in shaping pattern, one must evaluate pattern simultaneously from the perspectives of the generic and the particular.
All data underlying the results are available as part of the article and no additional source data are required.
The Mathematica code for the analysis and creation of Figure 3 and Figure 4 is available at Zenodo: https://doi.org/10.5281/zenodo.324336433.
License: Creative Commons Attribution 4.0 International license.
SAF initiated the project, did the mathematical analyses, created the figures, and wrote the first draft. JB developed connections to broader issues in ecology, expanded the text to clarify exposition and significance, extended the framing of concepts and the explanation and discussion of key points, and edited the entire manuscript.
SAF completed this work while on sabbatical in the Theoretical Biology group of the Institute for Integrative Biology at Eidgenössische Technische Hochschule (ETH) Zürich.
Previous versions of this article are available on bioRxiv: https://doi.org/10.1101/673590.
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Is the work clearly and accurately presented and does it cite the current literature?
Yes
Is the study design appropriate and is the work technically sound?
Yes
Are sufficient details of methods and analysis provided to allow replication by others?
Yes
If applicable, is the statistical analysis and its interpretation appropriate?
Yes
Are all the source data underlying the results available to ensure full reproducibility?
Yes
Are the conclusions drawn adequately supported by the results?
Yes
References
1. Frank SA, Smith E: A simple derivation and classification of common probability distributions based on information symmetry and measurement scale.J Evol Biol. 2011; 24 (3): 469-84 PubMed Abstract | Publisher Full TextCompeting Interests: No competing interests were disclosed.
Reviewer Expertise: Epidemiology, decision theory, information theory, quantitative biology, policy analysis.
Is the work clearly and accurately presented and does it cite the current literature?
Yes
Is the study design appropriate and is the work technically sound?
Yes
Are sufficient details of methods and analysis provided to allow replication by others?
Yes
If applicable, is the statistical analysis and its interpretation appropriate?
Not applicable
Are all the source data underlying the results available to ensure full reproducibility?
No source data required
Are the conclusions drawn adequately supported by the results?
Yes
References
1. Pueyo S, He F, Zillio T: The maximum entropy formalism and the idiosyncratic theory of biodiversity.Ecol Lett. 2007; 10 (11): 1017-28 PubMed Abstract | Publisher Full TextCompeting Interests: No competing interests were disclosed.
Reviewer Expertise: community ecology, population biology, infectious diseases, biodiversity research, climate change, environmental forcing, stochastic birth-death processes, non-linear interactions, self-organization, complex systems
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