Factorize the Joint Distribution According to the Families

Abstract

Diversity in metacommunities is traditionally viewed to consist of the variety inside communities (\(\alpha\)) that is complemented by the differences between communities (\(\beta\)) and then as to effect in the total diverseness (\(\gamma\)) of the metacommunity. This perception of the partitioning of diversity, where \(\beta\) is a function of \(\gamma\) and \(\alpha\) (usually \(\beta =\gamma /\alpha\) with all components specified every bit effective numbers), has several drawbacks, amongst which are (1)\(\alpha\) is an average that can be taken over communities in many means, (2) complete differentiation amid communities cannot always exist uniquely inferred from \(\blastoff\) and \(\gamma\), (3) dissimilar interpretations of \(\beta\) equally effective number of communities (e.g., distinct or monomorphic) are possible, depending on the choice of ideal situations to which the respective constructive numbers refer, and (4) associations betwixt types (species, genotypes, etc.) and community affiliations of individuals are not explicitly covered by \(\alpha\) and \(\gamma\). Detail (4) deserves special regard when quantifying metacommunity diversity. It is argued that this requires consideration of the joint distribution of blazon-community combinations together with its diversity (joint diversity) and its constituent components: type and community affiliation. The quantification of both components tin can be affected past their association as realized in the joint distribution. It is shown that under this perception, the articulation multifariousness can exist factorized into a leading and an associated component, where the first characterizes the minimum number of communities required to obtain the observed joint diversity given the observed type distribution, and the 2nd specifies the constructive number of types represented in the minimally required number of communities. Multiplication of the 2 yields the joint diversity. Interchanging the roles of customs and type, one arrives at the dual factorization with leading minimum number of types and associated effective number of communities. The two dual factorizations are unambiguously divers for all measures of multifariousness and tin be used, for instance, to indicate structural characteristics of metacommunities, such as blazon differentiation amongst communities and associated type polymorphism. The information proceeds of the factorization approach is pointed out in comparison with the classical and more contempo modified approaches to partitioning total blazon diverseness into diversity within and between communities. The apply of factorization in analyses of latent community subdivision is indicated.

Introduction

Diversity in metacommunities is traditionally viewed to consist of the diversity within communities that is complemented past the differences betwixt communities so as to result in the total diversity of the metacommunity (among the offset is Whittaker 1960). In established terminology, these 3 levels of variation are denoted as \(\alpha\)- (within communities), \(\beta\)- (between communities) and \(\gamma\)-diversity (total metacommunity). This view concentrates on trait (more often than not genetic or species) diverseness and the way it is distributed in a metacommunity with the focus on possibilities of partitioning the full trait diverseness into a component roofing the diversity within communities and a component reflecting the differences between communities.

As was pointed out past a number of authors, the crucial component is \(\blastoff\)-diversity (for an overview, see, e.g., Chiu et al. 2014). Besides uniqueness problems (a nearly indefinite number of "averages" of diversities within communities be that fulfill the division criterion; Gregorius 2014), it is known for its failure to account for aspects of differentiation among communities in combination with \(\gamma\)-diversity (run into, east.thou., Jost et al. 2010). The latter in plow implies ambiguity in interpretation of \(\beta\)-variety to reflect differences between communities when considered every bit a office of \(\alpha\) and \(\gamma\)-variety only. It was shown by Gregorius (2010) that the reason lies primarily in the neglect of effects on variety that are explicitly due to community affiliation and its association with the trait. This phenomenon was indicated in earlier piece of work, e.g., by Routledge (1979), Jost (2008), or Tuomisto (2010).

The metacommunity context

Indeed, the label of metacommunity members past their community affiliation is rarely an explicit object of studies of variation or multifariousness. Numbers and sizes of communities, for instance, are not expressly considered as diversity generating factors. However, communities may be characterized in different ways, many of which refer to functional characteristics, environmental weather, geographic region and the like, all of which apparently specify other properties for which individuals can exist distinguished and brandish diversity. Hence, metacommunity variety is intrinsically a joint kind of variety that encompasses both a distinguished trait and community affiliation as flanking sources of diversity. When problems of ecological stability are studied with respect to ecological diversity, the obvious principal subject of assay is neither species (or genetic) nor ecology diversity alone but rather the combination of both.

The effect of this combination on (joint) diverseness becomes apparent through the associations between the 2 components "trait" (genetic, species, etc.) and "customs affiliation" (surround, social rank, etc.). Associations rank between complete clan (where, e.one thousand., communities are completely differentiated for their trait states or types, for short) and the absence of association (where, e.yard., types show equal distribution within all communities). Specially complete association underlines the role played by the "flanking" or "marginal" distributions, since in this instance, the joint variety equals the overall (marginal) trait diversity (when differentiation of communities for types is complete) or the diversity of communities (when differentiation of types for customs amalgamation is complete, see Gregorius 2010). The marginal distributions therefore ready bounds that are essential for the assessment of articulation diversities.

Consequently, in the present paper, attempts to decompose, or factor, diverseness into components will be targeted toward the joint diversity, where the marginal distributions are considered every bit interacting components. This contrasts with the classical forms of division diverseness which focus on the type diversity in the total metacommunity as compared with the type diverseness within the constituent communities. To chronicle to common approaches equally summarized, e.chiliad., in the above-cited piece of work of Jost (2008) and Gregorius (2016), appropriate effective numbers for the components of diversity and their interaction in producing the target variety volition be derived. Hereby, ambivalence inherent in the conception of effective numbers volition be taken special care of and an unambiguous alternative volition be presented.

The differentiation problem

Such ambivalence includes reference to effective numbers of communities in terms of their "distinctness" and thus differentiation especially in defining \(\beta\)-diversity (see, eastward.k., Jost 2007). Intrinsically, community differentiation addresses differences in blazon distribution between communities without implying whatever statements nearly the diversities realized in the communities (Gregorius 2016). To avoid misconception, a stardom is fabricated by occasionally calculation the describing word "compositional" to differentiation, i.e., compositional differentiation. Albeit, the compositional extremes of consummate absenteeism and complete presence of differences in type distribution between communities bear upon relations among components of multifariousness besides, however, without determining the sizes of the components.

When treating effective numbers of "distinct" communities in diverseness analyses, information technology is therefore crucial to make clear that these numbers do non specify a number of completely differentiated communities realized in a metacommunity (compare the example of Table 2, where communities are clearly differentiated among each other for their types, merely none of the communities is completely singled-out). They rather make a statement about how many communities an assemblage of completely differentiated communities (the ideal situation) it would require to generate the observed diversity. On the other hand, ideal situations can be chosen in different means, and this may pb to different interpretations of the aforementioned formal definition of \(\beta\)-variety. An example is provided in the paper of Gregorius (2016), where it is argued that the perception of \(\beta\) as an effective number of distinct communities can exist modified into a "number of effectively monomorphic communities."

Measures of variety

More fundamentally, it might be worth emphasizing that most measures of diverseness have a biological interpretation that is not explicitly specified in terms of the number of types involved in the biological process. Simpson'due south diversity in its version as the probability of sampling without replacement 2 individuals of unlike blazon, for instance, finds a biological interpretation in terms of species interactions depending on the frequency of interspecific encounters. The number of species participating in the encounters is not of primary involvement here. Yet, the measure out allows for determination of an effective number by equating the observed diversity to the diversity resulting for a number of equally frequent types and solving for this number. This relates to a more than elementary level of interpreting effective numbers in that it touches on the concept of diversity.

It applies to all admissible diverseness measures and it informs most the number of types "finer" participating in the biological process. Without explicit reference to the underlying, biologically motivated diverseness measure, such effective numbers make only abstruse numerical and calibration-gratis statements that provide limited information on biologically relevant characteristics. Rényi-diversities (come across lesser of Table 1), which are admissible variety measures that equal their effective numbers, are an example. They may be obtained as an effective number from quite a variety of master variety measures (Jost 2007; Gregorius 2009, 2014) ^{Footnote one}. Diversity effective numbers simply transform the interpretable diversity measure into the number of types effectively determining the original diversity measure.

Table 1 Notations

Full size table

The wide range of diversity measures with substantial biological estimation likewise justifies to not insist on specific properties frequently stipulated for variety measures and especially for their constructive numbers (such as the replication principle, run into, east.k., Ricotta and Szeidl 2009). A typical category of variety measures whose effective numbers exercise not comply with the replication principle is characterized by the to a higher place-mentioned probability models of non-random encountering (come across, east.g., Patil and Taillie 1982, or Gregorius 2009). Another category that has wide application in physics and economy relates to non-independent information content.

To allow for this range, diversity measures are considered open-door if they fulfill non more than than the basic evenness criterion (diversity increases as the difference in frequency betwixt ii types decreases while the sum of their frequencies remains the same). The criterion is also known as the principle of transfers (run across, e.g., Patil and Taillie 1982). The major results obtained in the nowadays paper do not require more than the evenness criterion and are therefore not restricted to special types or categories of diversity measures (even though translation into Rényi-diversities will exist provided in each case for habitual reasons).

Factorization of joint diversity

In the following, it will exist shown that within this wide scope of measuring diversity, joint diversities allow for multiplicative decomposition (factorization) of their constructive numbers into components relating to the marginal variables "trait" and "community amalgamation." Herewith, 1 component (the leading component) is distinguished past realizing a minimization benchmark. Every bit practical to community amalgamation equally leading component, for example, the criterion determines the minimum number of communities required to realize the observed joint diversity given the observed trait distribution. The criterion does not rely on the concept of effective numbers at the first, just it is amenable to an interpretation in terms of such numbers. The minimally required number lays the basis for a unique factorization of the joint diversity into the leading (e.g., community amalgamation) and the associated marginal component (trait). The relation of the factorization to a more recently proposed method of partition variety is demonstrated that avoids the arbitrariness inherent in averaging diversities within communities (Chiu et al. 2014).

Number of communities for given marginal blazon frequencies

Peculiarly for large numbers of individuals, their types can be distributed over several numbers of communities without affecting the overall blazon frequencies. As a result, the joint diversity should increase roughly with the number of communities which the individuals are assigned to. However, if the assignment is such that individuals of the same type never appear in different communities (consummate community differentiation), the articulation diversity equals the overall (marginal) blazon diversity (\(v_{TC}=v_T\)), since the number of different type-community combinations equals the number of types. Thus, each customs is represented in only one type, and the joint diverseness is not affected by such assignments to communities.

Plainly, this changes only later on individuals of the same type are assigned to different communities, so that the number of different type-community combinations increases with the number of communities that are represented in each type. Moreover, the more than even a type is distributed over communities the larger the joint diversity by the evenness benchmark. For a given number of communities, the maximum diverseness is and so reached if in each blazon all communities are represented and are then at equal proportions. At this end, types are non differentiated for their customs affiliations, and in outcome communities are not differentiated for their types. While during this equalization process the type frequencies remain the aforementioned, an initial customs differentiation is gradually lost.

In this context, the diversity of the observed articulation frequency array \({{{\varvec{\mathsf{{ q}}}}}}=\{q_{i,j }\}\) is to be related to the diversity of an assortment \({{{\varvec{\mathsf{{ q}}}}}}'\) of the form \({{{\varvec{\mathsf{{ q}}}}}}'={{{\varvec{\mathsf{{ q}}}}}}^N_C\) with \(q'_{i,j}=p_i/N\) (see Table 1). As indicated to a higher place, the \(q'_{i,j}\)southward reverberate the situation where at that place is no differentiation for types amid communities, and communities have equal shares within each blazon and consequently in the total metacommunity. The (joint) diversity of this modified assortment, i.e., \(5({{{\varvec{\mathsf{{ q}}}}}}^N_C)\), will be denoted past\(v_{TC}'\) for the moment. For \(N=N_C\), it then follows from the evenness criterion that \(v_{TC}\le v_{TC}'\).

\(v_{TC}'\) is a strictly increasing function of North for given marginal type frequencies, and it starts with \(v_T\) for \(N=1\) and has a unique solution \(North=N_C^{\circ }\) for which \(v_{TC}'=v_{TC}\). Since \(v({{{\varvec{\mathsf{{ q}}}}}}^{N_C}_C )\ge v_{TC}\), it always holds that \(N_C^{\circ }\le N_C\). Note that for every bit sized communities, \(N_C^{\circ }=N_C\) does not hold when communities differ for their type frequencies.

The most characteristic property of \(N_C^{\circ }\) becomes credible when considering joint distributions which share their marginal type frequencies and accept equal joint diversities. All of these distributions have the same\(N_C^{\circ }\), but they may differ in their\(N_C\) values. By the above result, all of these\(N_C\) values are equal to or greater than\(N_C^{\circ }\). Hence, the conclusion that for the observed marginal blazon frequencies, \(N_C^{\circ }\) is the minimum number of communities required to obtain the observed joint diversity. In an earlier attempt of finding effective numbers of communities based on joint diversities (Gregorius 2010, section ten), this minimization characteristic escaped notice.

In a strict sense, however, \(N_C^{\circ }\) should exist a number of communities and therefore should be a natural number that makes \({{{\varvec{\mathsf{{ q}}}}}}_C^{N_C^{\circ }}\) a proper articulation frequency array to which the diversity measure out v can be practical. Every bit was shown by Gregorius (2014, Appendix 2) the diversity of compatible frequency arrays tin always be monotonically extended to real numbers. To maintain the relationship to natural numbers, it is, however, essential to consider the largest natural number Due north for which \(v({{{\varvec{\mathsf{{ q}}}}}}^{N}_C)\le v_{TC}\) and to realize that necessarily \(N\le N_C^{\circ }<N+1\) for this numberN.

For Rényi-diversities (where \(v=v^e\)), \(5({{{\varvec{\mathsf{{ q}}}}}}^N_C)=N\cdot five({{{\varvec{\mathsf{{ q}}}}}}_T)=N\cdot v_T\), which, by setting \(N\cdot v_T=v_{TC}\), yields \(N_C^{\circ }=v_{TC}/v_T\). This family of variety measures now allows \(v_{TC}/v_T\) to be interpreted as specifying the minimum number of communities required to realize the observed joint diversity when retaining the observed marginal type frequencies.

Since both, \(N_C^{\circ }\) and the marginal effective number \(v_C^eastward\) of communities refer to numbers of communities, comparison betwixt the two quantities is in demand. At least in the example provided for Rényi-diversities in Table 2, it becomes apparent that, in accord with the minimum feature of \(N_C^{\circ }\), distinctly fewer communities are required to obtain the observed joint diversity that are effectively represented in the marginal community limerick (\(N_C^{\circ }\ll v_C^e\)). However, more than analysis might be desirable to assess the share in this reduction that is due to the comparatively loftier proportion of type-community combinations (5/12) non represented in the example.

Table 2 A fictitious numerical example demonstrating for several orders a of Rényi-diversity the diverseness components \(v_T\), \(v_C\), \(v_{TC}\), the minimally required numbers of communities (\(N_C^{\circ }\)) and types (\(N_T^{\circ }\)) as well as their associated effective numbers of types (\(N_{T<C}^{east}\)) and communities (\(N_{C<T}^{eastward}\)), respectively, and the dual \(\beta\)-diversities \(N_C/N_C^{\circ }\) and \(N_T/N_T^{\circ }\)

Full size tabular array

Interpretation of \(N_C^{\circ }\) every bit an constructive number

Despite its definition as minimally required number of communities, \(N_C^{\circ }\) can also be conceived of as an effective number on the basis of the in a higher place demonstrations. According to its full general concept (encounter, due east.chiliad., Gregorius 2016, Appendix A), conclusion of an effective number requires specification of an ideal situation and a feature variable associable with a target variable. In the present example, the target variable is the number of communities, the platonic situation is specified by the joint frequency arrays \({{{\varvec{\mathsf{{ q}}}}}}^N_C\) with naturalN, and the characteristic variable is fabricated up of 2 components, with the primary component given past the articulation diversity and the secondary component given by the blazon frequencies. From the above demonstrations, it follows that for each observed joint frequency array \({{\varvec{\mathsf{{ q}}}}}\), there exists an ideal array \({{{\varvec{\mathsf{{ q}}}}}}^N_C\), with N extendable to a existent value that implies equality betwixt the observed and the ideal array for each of the two characteristic variables. Moreover, whatever two ideal distributions with equal feature variable are identical for their target variable \(N_C\). Hence, the number \(N_C^{\circ }\) can be conceived of equally both the result of a minimization process and an constructive number.

To emphasize the difference from other effective numbers of communities like \(\beta\)-diverseness, it seems appropriate at this place to recall that they do non result from an optimization procedure. Instead, \(\beta\)-diversity is claimed to make structural statements apropos the (effective) number of "singled-out" communities (which birthday is critical equally mentioned to a higher place). In dissimilarity, \(N_C^{\circ }\) cannot be used to directly quantify structural aspects such every bit distinctness or its contrary. In detail, \(N_C^{\circ }\) is not aimed at providing an constructive number of "undifferentiated" communities as i might be tempted to conclude from its derivation. The fact that \(N_C^{\circ }=1\) only for \(v_{TC}=v_T\), which holds only for completely differentiated communities and irrespective of the blazon diversity inside the communities, makes this evident. At the same time, nevertheless, this fact indicates possible relations of \(N_C^{\circ }\) to structural aspects as volition be elaborated in the 5th section.

A feature of \(N_C^{\circ }\) that distinguishes it from nearly common approaches to effective numbers is to be seen in its ideal state of affairs and characteristic variable, which explicitly include information on the observed information in terms of the marginal type frequencies. In common approaches, the platonic reference is usually almost completely bathetic from the observation past assumptions like consummate community differentiation, equal diversities for all communities including uniform type distributions, and equal community sizes. Obviously, the higher the caste of such abstraction, the more bug arise in relating the interpretation of the resulting effective numbers to observed properties. Constructive numbers of "singled-out" communities are an example, as is mentioned above. Since \(N_C^{\circ }\) straight quantifies a belongings of the observed frequency array without drawing on an idealized array, its perception as minimally required number should be given preference over its estimation as an effective number of communities.

Factorization of joint diversity into its blazon and community component

Diversity measures exercise not distinguish between the quality of the entities for which they are obtained. Thus, for a specified diversity measure, its values may encompass the same range when applied to the distribution of a single qualitative variable or to multiple variables. For multiple variables, the entities are but combinations of states of the involved variables. The same applies to the respective diversity effective numbers of entities. For the nowadays example of the ii variables "blazon" and "community amalgamation," at that place may exist blazon-customs combinations with goose egg frequency in the articulation distribution, fifty-fifty though all marginal type and community frequencies are positive (for a fictitious instance see Table 2). By illustration with the variety effective number of a unmarried variable, for determination of the joint diversity effective number \(v_{TC}^due east\), only numbers of every bit frequent combinations affair that do not exceed the number of combinations with positive frequency. The involved marginal numbers of types and communities do non explicitly enter the calculations.

Hence, joint diversities \(v_{TC}\) and their effective numbers \(v_{TC}^e\) do not depend on the numbers of states represented in the two marginal variables (type and community). On the other manus, the marginal variables gear up the upper limit to the joint variety in that \(v_{TC}^e\le N_T\cdot N_C\) with equality only if all of the \(N_T\cdot N_C\) potential(!) combinations appear at equal (positive) frequencies. Therefore, particularly if several potential combinations have zippo frequency, the question arises equally to the number of types \(N_T'\), say, and the number of communities \(N_C'\), say, which actually or effectively participate in creating the observed joint multifariousness. Clearly, \(N_T'\le N_T\) and \(N_C'\le N_C\).

Participation in turn requires specification of the style according to which the numbers contribute to the joint distribution. Every bit indicated in a higher place, the full potential of contributions is always realized under contained and equal participation of the marginal variables in the joint frequency array. Setting this as the ideal situation on which effective numbers are based, one arrives at a joint variety the effective number of which (ideally) equals the product \(N_T'\cdot N_C'\). The components of the product are the natural candidates for any endeavor of factorizing the joint diversity. Factorization must therefore exist performed for the constructive number of the joint diversity under consideration

Consequently, equating the product with the observed joint diversity effective number \(v_{TC}^e\), i.e., \(v_{TC}^due east=N_C'\cdot N_T'\), would set the range inside which the numbers \(N_C'\) and \(N_T'\) were allowed to vary. While the solution of this equality always exists and is unique with respect to the production \(N_C'\cdot N_T'\) of existent numbers, uniqueness of the two components is not achieved until i of them is determined via desirable properties. Determination of any of the two factors then settles the other, and both must unambiguously refer to the marginal variables.

The latter fact asks for distinction betwixt a leading and an associated component of the factorization, where the leading component reflects the desirable backdrop. The to a higher place demonstrations propose C as a leading component, for which the number \(N_C^{\circ }\) of communities reflects a desirable property via its minimization characteristic. This justifies the selection of \(N_C'=N_C^{\circ }\), which in plough determines the "associated effective number" \(N_T'\) of types as \(v_{TC}^e/N_C^{\circ }\). The notation \(N_{T<C}^{due east}:=v_{TC}^e/N_C^{\circ }\) volition be used to emphasize the associative character of this effective number too as the community-centered perspective of the factorization. One then arrives at the multiplicative decomposition (or factorization) of joint diversity in the form

$$\begin{aligned} v_{TC}^e=N_C^{\circ }\cdot N_{T<C}^{e}\end{aligned}$$

Indeed, \(N_{T<C}^{e}\) is to exist addressed as an effective number, since its specification (every bit opposed to \(N_C^{\circ }\)) depends on idealizing assumptions on joint distributions.

For Rényi-diversities (see bottom of Tabular array 1), where \(N_C^{\circ }=v_{TC}^e/v_T^e\), one obtains \(N_{T<C}^{e}=v_{TC}^east/N_C^{\circ }=v_T^due east\). Here, the associated effective number of types equals the marginal type diversity and is therefore independent of the (effective) number of communities equally well as their associations with the types. Effects of associations are therefore completely covered past\(N_C^{\circ }\).

This holding of Rényi-diversities demand not utilize to other measures of diversity such as given in Introduction. To make up one's mind the limit for the associated component \(N_{T<C}^{e}\), let N over again exist the largest natural number with \(five({{{\varvec{\mathsf{{ q}}}}}}^N_C)\le v_{TC}\). Then, \(N\le N_C^{\circ }<North+1\) and \(v({{{\varvec{\mathsf{{ q}}}}}}^N_C)\le v_{TC}<five({{{\varvec{\mathsf{{ q}}}}}}^{N+1}_C)\). The smallest possible value for \(N_C^{\circ }\) given \(v_{TC}\) would be \(N_C^{\circ }=N\), for which one obtains \(N_{T<C}^{due east}\cdot N_C^{\circ }=v_{TC}^east=v^e({{{\varvec{\mathsf{{ q}}}}}}^N_C)\le North\cdot N_T\le N_C^{\circ }\cdot N_T\), from which \(N_{T<C}^{eastward}\le N_T\) follows. Since \(v_{TC}^east=N_{T<C}^{east}\cdot N_C^{\circ }\), the smallest possible value for \(N_C^{\circ }\) specifies the largest possible value for \(N_{T<C}^{e}\) for given \(v_{TC}^eastward\), and this implies that always \(N_{T<C}^{east}\le N_T\). In summary

$$\begin{aligned} N_C^{\circ }\le N_C \hbox { and } N_{T<C}^{e}\le N_T \finish{aligned}$$

which concurs with intuitive expectations and conceptual reasoning.

To appreciate the data provided by the associated effective number of types it is helpful to retrieve that joint frequency arrays for which \(N_C^{\circ }\) is realized are characterized by communities of equal size and blazon distributions that are the same within each community and therefore equal the marginal blazon distribution. One thus expects \(N_{T<C}^{eastward}\) to be given by \(v_T^e\) as is true for Rényi-diversities. Yet, for other categories of diversity measures such every bit addressed in Introduction, this does not employ (checked numerically by the writer for the assortment in Table ii). An caption is suggested by the divergence in the biological reasoning of the categories referring to random and not-random encounters of community members. Besides relating to marginal type diversities, the associated effective number of types is apparently sensitive to constructional differences in categories of diversity measures.

More than practically relevant backdrop will be treated later on with reference to structural characteristics of articulation distributions. For the fourth dimension beingness, information technology may suffice to recall that \(N_C^{\circ }\) refers to a hypothetical array of joint frequencies that shares its marginal blazon distribution and variety with the observed joint distribution. Therefore, \(N_{T<C}^{eastward}\) can exist understood to quantify the effective number of types represented in the minimally required number of communities.

The dual analogue: the type-centered perspective

The apparent asymmetry in the higher up handling of the community and type component per se asks for consideration of its reversal. In other words, a transition from the community-centered to a type- or trait-centered perspective is suggested. Given the marginal distribution of community affiliations (community sizes), one would and so aim at finding the minimum number of types required to obtain the observed joint multifariousness in the first place. In this style, the contribution of types to the ecological multifariousness of a metacommunity is assessed on the ground of the observed sectionalization of the metacommunity into its elective communities. This washed, the associated effective number of communities is to be specified. Since customs determinants are ofttimes considered to be less dynamic than the traits of the community members existing on or adapting to these determinants, this may exist the more relevant perspective to be taken in many ecological studies.

In essence, this perspective only marks the dual analogue of the previous community-centered arroyo to the decomposition of joint diverseness into community and type components. It therefore results from merely reverting the roles ofT andC. All of the conclusions obtained and then far apply analogously. Thus, one beginning determines the minimum number \(N_T^{\circ }\) of types required to obtain the observed joint diversity as the solution \(Due north=N_T^{\circ }\) for which \(v({{{\varvec{\mathsf{{ q}}}}}}_T^Due north)=v_{TC}\). This is followed by specifying the associated effective number \(N_{C<T}^{due east}\) of communities from

$$\begin{aligned} v_{TC}^eastward=N_{C<T}^{e}\cdot N_T^{\circ }\end{aligned},$$

i.e., \(N_{C<T}^{e}=v_{TC}^due east/N_T^{\circ }\). Now, \(N_T^{\circ }\le N_T\) and \(N_{C<T}^{east}\le N_C\).

For Rényi-multifariousness, \(N_T^{\circ }=v_{TC}^e/v_C^eastward\), and thus, \(N_{C<T}^{e}=v_C^due east\). It is worth noting that none of the two components resembles whatsoever of the effective numbers involved in the classical segmentation of \(\gamma\)- into \(\alpha\)- and \(\beta\)-diversity.

Terminal remarks

The minimum number \(N_C^{\circ }\) of communities required to obtain the observed joint multifariousness is a parameter that provides no direct information on structural characteristics relating, for example, to the distribution of variety over communities. Still, \(N_C^{\circ }\) can be transformed into an indicator of structural characteristics under the community-centered perspective when considering the deviation of \(N_C^{\circ }\) from\(N_C\). To see this, consider that \(N_C^{\circ }=N_C\) is equivalent to \(five({{{\varvec{\mathsf{{ q}}}}}}_C^{N_C})=v_{TC}\), which in turn is equivalent to \({{{\varvec{\mathsf{{ q}}}}}}_C^{N_C}={{{\varvec{\mathsf{{ q}}}}}}\). The latter follows direct from the specification of \(N_C^{\circ }\) via equalization of the community representations within the types (run across as well third paragraph in the second section). Therefore, \(N_C^{\circ }=N_C\) merely if the same communities are represented at equal frequencies within each type. From a more familiar perspective, this is equivalent to "\(N_C^{\circ }=N_C\) only if communities are not differentiated for their type compositions and are equally sized." Think that at the other extreme, \({N_C^{\circ }=1}\) simply if communities are completely differentiated for their type compositions irrespective of the customs sizes.

The latter suggests consideration of \(N_C^{\circ }\) as indicating community differentiation provided the land of the absence of community differentiation includes equal community sizes to brand the absence of differentiation "complete." This inclusion makes sense when variable customs sizes can be argued to have an ecological touch on. In fact, it is hard to conceive a community ecological scenario in which community sizes and their variability are irrelevant. Maintaining this extended concept of differentiation, it follows that

\(N_C^{\circ }\) varies from 1 to \(N_C,\) while structural characteristics vary from complete differentiation to consummate absence of differentiation amidst communities.

When compared with the observed number of communities, \(N_C^{\circ }\) thus displays its twofold nature equally a minimum number of communities and as an indicator of the amount of differentiation amidst communities.

Label of the structural aspects encoded in \(N_{T<C}^{east}\) must be traced back to \(N_C^{\circ }\) because of the "leading" function of the latter quantity. For case, \(N_{T<C}^{eastward}=1\) is equivalent to \(N_C^{\circ }=v_{TC}^e\), and this excludes whatever variation in types within and betwixt communities. The metacommunity therefore consists of a single type (is monomorphic), which is in accord with an constructive number of types equal to ane. To appraise the other extreme of \(N_{T<C}^{east}\), i.eastward., \(N_{T<C}^{e}=N_T\), let North again be the largest natural number with \(5({{{\varvec{\mathsf{{ q}}}}}}^N_C)\le v_{TC}\) so that \(v^e({{{\varvec{\mathsf{{ q}}}}}}_C^Northward)\le N\cdot N_T\le N_C^{\circ }\cdot N_T=v_{TC}^e\). In the example of \(N_C^{\circ }=N\), the chain of inequalities implies that \(v_{TC}^east=v^e({{{\varvec{\mathsf{{ q}}}}}}_C^N)\le N\cdot N_T=v_{TC}^e\) and \(v^east({{{\varvec{\mathsf{{ q}}}}}}_C^N)=Due north\cdot N_T\), with the consequence that the marginal type distribution must be uniform. Since for given \(v_{TC}\) and marginal type frequencies, whatever modify in \(N_C^{\circ }\) goes along with a change in\(N_{T<C}^{e}\) but not inNorth, one infers that generally \(N_{T<C}^{e}=N_T\) merely if the marginal blazon distribution is uniform. Therefore,

while \(N_{T<C}^{e}\) varies from 1 to \(N_T,\) the marginal type frequencies start with monomorphism and terminate with maximum polymorphism (where all types are equally frequent).

This result is obvious for Rényi-diversities, since in that location e'er \(N_{T<C}^{e}=v^e_T\).

The structural characteristics inherent in the factorization of joint multifariousness illustrate more vividly the conceptual advantages of the present approach, which focuses on the numbers of communities and types that essentially decide metacommunity diversity. By "essential," 2 bug are addressed, factors and their interaction. The factors are specified by the minimally required number of communities (which depends on their differences in type composition) and the associated number of types (effectively represented in the minimally required number of communities). Their interaction is "orthogonal" in the sense of an contained operation in producing the articulation diversity.

The factorization reminds of the common approaches to sectionalization diversity in such a fashion that the total (metacommunity) blazon diverseness \(\gamma\) results as a product of the blazon variety within communities (\(\blastoff\)) with a quantity referred to as (type) "diversity" between communities (\(\beta\)), i.e., \(\gamma =\alpha \cdot \beta\) (encounter, e.yard., Jost 2007). This parallelism asks for pointing out potential relationships between the two approaches with special reference to the diversity within (\(\blastoff\)) and between (\(\beta\)) communities equally these practice not explicitly announced in the factorization.

\(\alpha\)- and \(\beta\)-multifariousness

Chiu et al. (2014, eq. (half-dozen)) introduced an alternative concept of \(\alpha\)-diverseness for Rényi-diversities (Hill numbers) that relies on joint diversities and is termed "the effective number of species per assemblage" (types per community). It is treated within the frame of diversity partitioning. To show how their concept relates to the nowadays factorization of articulation diversity, note that for Rényi-diversities \(v^due east({{{\varvec{\mathsf{{ q}}}}}}^N_C )=v_T^e\cdot N\), and this is fix equal to \(v_{TC}^due east\) to obtain \(N_C^{\circ }=v_{TC}^e/v_T^due east\). Since \(N_C^{\circ }\le N_C\), 1 arrives at \(v_{TC}^e/N_C\le v_T^eastward\), where \(v_{TC}^e/N_C\) is argued by Chiu et al. (2014) to replace the classical versions of \(\alpha\)-diversity, i.e.,

$$\begin{aligned} \alpha =\alpha '=v_{TC}^due east/N_C \end{aligned}$$

Since \(v_{TC}^e\le N_T\cdot N_C\), one has \(\alpha '\le N_T\), which is meaningful. However, if all communities are monomorphic, \(v_{TC}^e=v_C^e\), and this allows for \(\blastoff '=v_C^eastward/N_C<one\). The interpretation of \(\alpha '\) every bit "number of species per assemblage" should therefore not be misunderstood to indicate some kind of boilerplate of the diversities within communities (as applies to classical \(\blastoff\)), since such averages are always greater or equal to 1 for effective numbers. Nonetheless, every bit is demonstrated in Chao and Chiu (2016), \(\alpha '\) has a broader application including variance decomposition approaches.

In particular, \(v_T^e\) equals \(\gamma\)-diversity, so that \(\alpha '\le \gamma\) and \(\beta =\beta '=\gamma /\alpha '=N_C\cdot v_T^e/v_{TC}^e\). For Rényi-diversity \(N_C^{\circ }=v_{TC}^due east/v_T^e\), which yields

$$\begin{aligned} \beta '=N_C/N_C^{\circ }, \end{aligned}$$

and thus reveals the direct connection to the present factorization approach. It follows that \(\beta '\le N_C\) with equality just for complete customs differentiation, as it comes close to conventional views of sectionalisation total diversity (\(\gamma\)) into its component within (\(\blastoff\)) and between (\(\beta\)) communities. The authors refer to their \(\beta '\) as an "effective number of...completely distinct assemblages."

Transferring the full general properties of \(N_C^{\circ }\) to \(\beta '=N_C/N_C^{\circ }\), it becomes apparent that only in the complete absence of community differentiation (in the extended sense of equality of relative type frequencies amidst communities and equal community sizes) is\(\beta '=one\). Variability in community sizes e'er implies\(\beta '>1\), and \(\beta '\) assumes its maximum of\(N_C\) for complete customs differentiation simply. Chiu et al. (2014) arrived at a similar result for the special case of Rényi-diversities, where complete absence of community differentiation appears as "all assemblages are identical in species accented abundances" (p.26).

Considering these relationships, information technology is tempting to accept the definition of \(\alpha '=v_{TC}^due east/N_C\) by Chiu et al. (2014) as a full general approach (applying to all diversity measures), and see how this would fit into the common \(\blastoff\)-\(\beta\)-\(\gamma\) frame. Indeed, replacing \(\gamma\) (the marginal multifariousness effective number of types) by \(N_{T<C}^{e}\) (the associated effective number of types) one obtains \(\beta '=\gamma /\alpha '=N_{T<C}^{e}/(v_{TC}^e/N_C)=N_{T<C}^{e}/(N_{T<C}^{e}\cdot N_C^{\circ }/N_C)=N_C/N_C^{\circ }\). Again \(\gamma \ge \alpha '\) with equality, however, only in the complete absence of community differentiation.

Replacement of \(\gamma\) by \(N_{T<C}^{e}\) is justified in the first place by the observation that in the classical approach to partitioning diversity (relying on averaged diversities within communities and where \(\gamma =v_T^e\)), effects of type-community associations as taken account of in the joint diverseness are not explicitly considered. On the other mitt, the example of Rényi-diversity (where \(N_{T<C}^{eastward}=v_T^e\)) confirms the existence of families of diversity measures, for which furnishings of association can exist captured in 1 component (\(N_C^{\circ }\)), while the other component (\(N_{T<C}^{e}\)) maintains its appearance as a marginal diversity. This characterizes the relationships between the nowadays approach of factorizing joint diverseness and the common \(\alpha\)-\(\beta\)-\(\gamma\) arroyo to the partition of (type-)diversity.

The fact that the thus generalized \(\beta '\) varies between 1 and \(N_C\) and, because it equals \(N_C\) but for completely differentiated communities, over again reminds of the habit to interpret \(\beta\) equally an effective number of distinct communities. Yet, the above general reasoning brought forth against quantifying differentiation in terms of numbers of distinct communities is not remedied by the present \(\beta '\). The numerical example in Tabular array 2 demonstrates that each of the individual communities may exist clearly differentiated from the others without any of them beingness completely distinct. In the common interpretation, the values of \(\beta '\) in Tabular array 2 would advise that more than one-half of the three communities (\(\beta ' >1.7\)) are "completely distinct." Distinctness of two among three communities, all the same, implies that all three are singled-out, which contradicts\({\beta '<3}\).

In that location is actually no need to invoke numbers of distinct communities. The present \(\beta '\) merely reverts the minimum characteristic of \(N_C^{\circ }\) so that \(\beta '\) specifies the maximally possible number of communities that realize the observed joint diverseness given the observed trait distribution. The maximum becomes 1 (\(\beta '=1\)) in the consummate absence of differentiation (including equal community sizes) and reaches its largest value (\(\beta '=N_C\)) for complete differentiation. Even in the case of complete differentiation, it would be questionable to address \(\beta '=N_C\) as an effective number of "distinct communities," since this interpretation ignores unequal community sizes so that the aforementioned number of communities may employ to the case where one of 10 communities covers 99% of the metacommunity or where all 10 communities have equal share.

To allow for comparison across information sets, normalizations of \(\beta '\) such every bit \((\beta '-ane)/(N_C-1)\) or \((N_C-N_C^{\circ })/(N_C-one)\) may exist required. Their interpretation is analogous, with the difference that the latter normalization is \(N_C^{\circ }\) times the quondam. The two indices are of the general type where the realized value of a variable is relativized with respect to its minimum and maximum value. In the beginning case, the variable is \(\beta '\) in the 2d\(N_C^{\circ }\). Both indices can be interpreted as indices of dissimilarity amongst the communities (for further such indices of the Sørensen, Jaccard, or Morisity type, see, e.g., Chao et al. 2019, Table 1). The absenteeism of blazon diversities or constructive numbers of types in all of these indices is conspicuous. It makes credible that the information on type distributions that is relevant in the assessment of structural aspects of community differentiation inherent in joint diversities can be summarized in a single quantity, namely\(N_C^{\circ }\). Nevertheless, remember that \(N_C^{\circ }\) is provisional on the observed type frequencies.

The type-centered perspective

Transition to the dual counterpart shifts the focus to the type variable as leading factor with associated effective number of communities. The unsaid modify from the community-centered to the type-centered perspective concentrates on the minimum number \(N_T^{\circ }\) of types that is required to produce the observed metacommunity diversity measured past its articulation diversity and given the observed community sizes. To appreciate the significance of this analogue, it is once more helpful to have a await at the structural information that tin can be extracted from \(N_T^{\circ }\) by considering the difference between \(N_T\) and \(N_T^{\circ }\).

Now, \(N_T^{\circ }=1\) only if all types are completely differentiated for their community affiliations, which is tantamount to monomorphism of all communities. Herewith, different communities may be monomorphic for the same blazon or for dissimilar types. Conversely, \(N_T^{\circ }\) reaches its maximum of \(N_T\) only in the absenteeism of differentiation of types for their customs affiliations (type differentiation) together with equal frequencies of all types, i.due east., for "complete" absence of blazon differentiation. Thus, as \(N_T^{\circ }\) moves from 1 to \(N_T\), structural characteristics outset with consummate blazon differentiation and thus monomorphism of communities and tend toward complete absence of type differentiation.

In an coordinating way, i concludes that the associated number \(N_{C<T}^{e}\) of communities equals i if only one community exists, and it equals\(N_C\) only if all communities have equal size.

These demonstrations propose application of the type-centered perspective to the to a higher place-treated method of partitioning diversity proposed past Chiu et al. (2014). \(\gamma\) then becomes \(\gamma _d=N_{C<T}^{e}\), \(\alpha '\) becomes \(\alpha _d=v_{TC}^e/N_T\), and \(\beta '\) becomes \(\beta _d=N_T/N_T^{\circ }\). For Rényi-diversity, the corresponding quantities are \(\gamma _d=v_C^east\), \(\alpha _d=v_{TC}^e/N_T\), and \(\beta _d=N_T\cdot v_C^eastward/v_{TC}^eastward\). Now, \(\beta _d\) equals the maximally possible number of types that realize the observed joint multifariousness given the observed community sizes.

The unfamiliar type-centered perspective can be transferred into more familiar perceptions by considering that for given customs sizes, the joint diversity decreases, and thus, \(\beta _d\) increases, as the communities become less polymorphic. The evolutionary and ecological connotations and so become apparent when realizing the consequences, for example, of competitive exclusion, migrate under isolation, endemism or specialization. All of these are characterized by tendencies toward monomorphism. The opposite, i.e., the complete absence of monomorphism and thus complete polymorphism, implies equal type frequencies in addition to the absence of community differentiation. "Complete" absenteeism of community differentiation is not relevant here, since community sizes are arbitrarily fixed observations.

In a more general context, systems of social or biochemical interaction that imply self- or cantankerous-incompatibility within, betwixt or across communities may directly influence type polymorphisms and their associations with communities. Herewith, self-incompatibility equally coded past Southward-allele systems or realized in heterotypic mating or other heterotypic preferences, for example, tend to stabilize, balance and increase polymorphism, while systems of cantankerous-incompatibility (such as heterozygote disadvantage, homotypic mating or other homotypic preferences, including inbreeding) destabilize polymorphisms and by this imply tendencies toward monomorphism.

Since in self-incompatibility systems, heterotypic combinations are preferred irrespective of customs affiliations, potential differences in customs affiliation betwixt types tend to exist equalized. This state of affairs is characterized past small \(\beta _d\)-values (\(N_T^{\circ }\) close to \(N_T\)) implying high polymorphism within communities. In the same manner, cross-incompatibility tends to heighten differences in community affiliation between types and so equally to prevent incompatible contacts within the same customs. \(\beta _d\)-values are big in this case and imply low polymorphism within communities. From a generalized incompatibility betoken of view, \(\beta _d\) can therefore exist conceived of as reflecting effects on type diversity equally are exerted past the continuum of compatibility reactions ranging between consummate self- and cantankerous-incompatibility.

Comparison between the two perspectives

It remains to demonstrate how the 2 perspectives are continued. Since both perspectives rest on the same joint diversity \(v_{TC}\), one obtains \(N_{C<T}^{e}/N_{T<C}^{e}=N_C^{\circ }/N_T^{\circ }\), so that the ratio betwixt numbers of communities and numbers of types is the same for associated effective numbers and for minimum numbers. Amongst other relationships between numbers of types and numbers of communities, such equally \(\gamma /\gamma _d=N_T^{\circ }/N_C^{\circ }\) and \(\blastoff '/\blastoff _d=N_T/N_C\), it is interesting to see how the same numbers differ between perspectives. In this case, comparisons are to be made between minimum and associated numbers. Taking \(N_C^{\circ }\) and \(N_{C<T}^{due east}\), a brief glance at the values in Table 2 informs us that the minimum numbers are all smaller than the associated numbers. The aforementioned holds for the number of types \(N_T^{\circ }\) and \(N_{T<C}^{e}\).

The latter ascertainment becomes evident when considering the fact that \(N_C^{\circ }/N_{C<T}^{due east}=N_T^{\circ }/N_{T<C}^{due east}=N_C^{\circ }\cdot N_T^{\circ }/v_{TC}^due east\). Since \(N_C^{\circ }\) and \(N_T^{\circ }\) are minimum numbers based on the respective marginal distributions, one expects that \(N_C^{\circ }\cdot N_T^{\circ }\le v_{TC}^east\) holds in general. For Rényi-diversities, where \(N_C^{\circ }\cdot N_T^{\circ }/v_{TC}^e=v_{TC}^e/(v_T^east\cdot v_C^due east)\), it is known that the inequality indeed applies for club \({a=1}\). Even so, the inequality does non extend to orders \({a\ne ane}\) (Gregorius 2010). Hence, in general the quotient is inappropriate for quantifying structural aspects of joint distributions. This confirms that structural characteristics of metacommunities should be assessed separately and independently between the community- and the type-centered perspective.

Indeterminate (latent) customs subdivisions

Often, the spatial distribution of communities does not show a sufficient degree of fragmentation to permit for determination of a unique subdivision into subcommunities. Every bit a consequence, more than i subdivision may go relevant, each of which may show a different partitioning of the same underlying overall blazon diversity. Similarly, different criteria for subdivision, such as the spatial distribution of ecology variables, may be relevant in comparative causal analyses of observed genetic or species variation. In this case, us of each variable give rise to a subdivision into "communities," which in plow enables comparison of the effects of the various variables on the observed blazon variation.

The relevant studies are typically of the kind known from the analysis of latent variables (or factors), which is established especially in population genetics and implemented in methods such as structure (Pritchard et al. 2000), baps (Corander et al. 2003), or geneland (Guillot et al. 2005). When adapted to the nowadays situation, these methods go along from assignments of the observed individuals to hypothetical communities, where the assignments tin can be generated past one model, and a 2d model assembly the community affiliations of the individuals with their types. The issue of each such assignment is evaluated for fulfillment of special qualification criteria. The aim is to find assignments or functions thereof that optimize the qualification (for a demonstration of the general concept of the model-based assay of latent causal factors run into Gregorius 2018).

An consignment of the above kind specifies a articulation distribution of types and community affiliations and therefore allows computation of the articulation diversity together with its factorization into \(N_C^{\circ }\) and \(N_{T<C}^{e}\). Here, the associated effective number of types \(N_{T<C}^{e}\) is of simply secondary concern, since the marginal blazon distribution does not vary with the assignments. Particularly for Rényi-diversities, \(N_{T<C}^{e}\) is unaffected by the assignments, since it equals the marginal blazon diversity. Thus, \(N_C^{\circ }\), \(N_C\) and \(v_{TC}\) retain their significance as indicators of structural characteristics when applied to the assignments. In particular, \(N_C^{\circ }\) increases strictly with \(v_{TC}\) considering of the invariance of the marginal blazon distribution.

In well-nigh cases, the models underlying an analysis of latent factors are based on probability laws that specify likelihoods or posterior probabilities of the assignments. These probabilities serve as the chief qualification criterion that is to be maximized (particularly maximizing likelihood). This allows consideration of the most likely structural characteristics indicated by \(N_C^{\circ }\) and \(N_C\) and their relations (in terms of \(\beta '\), for case). In many applications, still, such as the methods cited above, the probability laws are used in MCMC-algorithms to estimate the expected values of the indicators, instead of the posterior probabilities of assignments as subdivisions of all individuals into communities.

Notes

1. In view of the popularity of Rényi-diversities (encounter Tabular array 1), information technology might exist useful to retrieve the significance of multifariousness measures whose effective numbers need not be Rényi-diversities. In many cases, Rényi-diversities event from diversity measures that reflect models of random encounter (see, east.yard., Patil and Taillie 1982). Encounters can, notwithstanding, be non-random with preferences especially for types other than the own. Such encounter probabilities could exist of the form \((ane-p)^a\) with p as type frequency, where for \(0<a<1\), the encounter would take place more frequently than at random. Tendencies to come across other types more oftentimes than at random are commonly conceived of as cross-preference. Hence, \(Q:=\sum _ip_i\cdot (1-p_i)^a\) is the average cantankerous-preference. It is straightforward to evidence that \(p\cdot (1-p)^a\) is a concave part of p for \(0<a\le 1\), which was demonstrated past Gregorius (2014) to imply validity of the evenness benchmark for Q and by this confirms Q to be a measure of diversity. The effective number of this measure out becomes \(one/(1-Q^{i/a})\), which, except of \({a=ane}\), is not a Rényi-diversity.

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Acknowledgements

Open Access funding provided by Projekt Deal. The comments of an bearding reviewer and Anne Chao helped in relating the contents to further work on the subject and to specify more clearly several of the ideas.

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1. Abteilung Forstgenetik und Forstpflanzenzüchtung, Universität Göttingen, Büsgenweg 2, 37077, Göttingen, Germany

   Hans-Rolf Gregorius

2. Institut für Populations- und ökologische Genetik, Am Pfingstanger 58, 37075, Göttingen, Germany

   Hans-Rolf Gregorius

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Gregorius, Hour. Factorization of joint metacommunity diversity into its marginal components: an alternative to the segmentation of trait diversity. Theory Biosci. 139, 253–263 (2020). https://doi.org/10.1007/s12064-020-00316-4

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• Received: 07 March 2019

• Accustomed: 01 May 2020

• Published: 01 June 2020

• Event Appointment: September 2020

• DOI : https://doi.org/10.1007/s12064-020-00316-4

Keywords

• Factorization
• Articulation diverseness
• Marginal variety
• Effective numbers
• Segmentation of variety
• Differentiation
• Polymorphism

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