SHORT COMMUNICATION. Diversity partitions in 3-way sorting: functions, Venn diagram mappings, typical additive series, and examples

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1 COMMUNITY ECOLOGY 7(): 53-57, /$0.00 Akadémiai Kiadó, Budapest DOI: 0.556/ComEc SHORT COMMUNICATION Diversity partitions in 3-way sorting: unctions, Venn diagram mappings, typical additive series, and examples L. Orlóci Department o Biology, University o Western Ontario, London, Canada N6A 5B7. lorloci@uwo.ca Keywords: Diversity analysis, Entropy, Factorial design, Heilongjiang lora, Inormation divergence, Partition unctions, Vegetation. Abstract: In this paper, logarithmic unctions are described based on which the total diversity o a collection can be partitioned into components speciic to actorial eects. The standard statistical modus operandi o testing hypotheses in a actorial design is applied, only the test criteria changed. The actor identities are chosen according to stated hypothesis, observations are made and sorted, the associated entropy and inormation quantities are calculated, and probabilistic tests o signiicance are perormed regarding main eects and interaction terms. The basics are presented in the main text. The partition unctions, their Venn diagram mappings, and a complete printout rom the application program DIVPART.EXE are collected in a separate Appendix accessible with this article on the publisher s website. Introduction Logarithm based entropy and inormation quantities are the diversity scalars. Entropy is scalar or main eects and inormation or interaction eects. When diversity is analysed based on the main eects/interaction model, the actors associated with the eects enter the analysis as the sorting criteria. Thus, the basis o the analysis is a contingency vector (sorting observations by the states o one actor), a contingency table (sorting by the states o two actors) or a solid (sorting by the states o three or more actors). In this communication, I consider the case o three actors, ormulate in all 34 diversity partitions or main eects, conditional eects, interactions, and conditional interactions, and give Venn diagram mappings to guide the choice o additive sequences o the partition unctions. I point out the dierence between proper as opposed to improper sums o partitions, give a worked numerical example, and introduce the application program DIVPART.EXE, which perorms the rather tedious computations automatically. The irst part o this communication introduces the basics unctions. This is ollowed by presentation o a numerical example in the second part, and a discussion o salient points in the third. Key reerences that can lead readers to others depending on the detail o interest conclude the main text. An appendix available through the Internet is attached to the paper. It contains sections on elementary symbolism and some numerical identities, practice run o DIVPART.EXE, including complete set o results printed by the program, and list o partition unctions with their Venn diagram mappings. DIVPART.EXE and sample data ile are downloadable rom the author s webpage ecologia.urgs.br/~lorloci/ at the link o the author s downloadable iles in the DIVPART older. Basic entropy and inormation unctions We designate as A, B, C the three actors whose numbers o states are a, b, c by which the elementary observations are sorted. An elementary observation in the example given is a three-valued vector that identiies the climax type (A), unctional type (B), and loristic domain or lora o origin (C) o a given species. A typical case taken rom Orlóci and He (996) is Abies nephrolepis. This is one o the 646 species in Li s (993) list rom the Heilong Jiang lora in China which we coded, implying a climatic climax tree rom the Boreal-Subalpine lora. The 3-dimensional sorting (AxBxC design) shown in Table is based on this code.

2 54 Orlóci Rényi s (96) generalised entropy o order serves as a scalar or the entropy portion o total diversity in the AxBxC design. This unction is written as a b c HABC (,, ) ln p p i j k The basic deinition o the inormation portion o diversity is scaled by Rényi s (96) inormation o order a b c p IABC (,, ) ln q q i j k The partition unctions corresponding to cases in between are listed in the Appendix. The ollowing are the symbolic data elements: a,b,c the number o states o actors A, B, C based on which the observations are sorted a characteristic element in the three-way solid, a count or requency. The version o DIVPAR.EXE presented in the paper takes the data in the orm ij., i.k,.jk, i..,.j.,..k, marginal totals. Dots in the subscripts indicate summation over the speciic dimension whose subscript is replaced by a dot. Inspection o the Venn diagrams in Figure a,b reveals that H(A,B,C) represents the total diversity (entropy in the design) attributable to the three-actor eects, and that I(ABC) represents the most speciic term in the threeactor interaction partitions called shared or mutual inormation. The letters A, B, C in the Venn diagrams identiy the area segments proportional to the actor eects. Perect proportionality is implied that requires in all cases a b, ij. i. k. jk i... j. to approach the limit. In practice, this requirement can be approximated rather closely by setting equal to almost, say The elegant solution involves rewriting the equations or the case o approaching (Orlóci 99), but this solution would leave the algorithm applicable only to order one o. Example The source data set is Li s (993). The species o the original data set were sorted according to 3 climax types (A), 5 unctional types (B) and 3 lora types (C). Table contains the joint requencies. The numerical values o selected partitions are given in Table and or all 34 partitions in Table 3. The computations were perormed by application program DIVPART.EXE. The Appendix contains a training run o DIVPART.EXE including the complete printout. The source code o DIVPART.EXE is in Kemény and Kurtz s Tru Basic. What we can see in the numbers o Table is the extreme inluence o inheritance as a contributor to diversity, outweighing by ar the eect o the other sorting actors on the regional scale. But when taking the sorting actors other than inheritance, lora element dominates. To be noted is that the sum on line o Table, which is an imperect sum because the terms repeat portions o the interaction inormation to the tune o bits in the manner o Figure o the elementary partitions. The sum on line 6 is a perect sum in the sense o Joint Conditional + Interaction + Mutual Principal marginal Interaction Mutual Table 3 shows results or the 34 partition unctions over six orders o. Order one is the basis o perect sums Total entropy 3-way interaction Figure. A Venn diagram representation o total entropy H(A,B,C) (a) and the 3-way interaction inormation I(ABC) (b). A, B and C are actor identities rom which the diversity eect is assumed to derive, depending in magnitude on the distribution o requencies over the actors states a, b, c by which the observations are sorted. The diversity partition unctions and numerical examples are given in the main text and in the Appendix. Entropy Inormation Figure. Venn diagram mappings o elementary partitions by type rom which all partitions can be constructed. The partition unctions are listed in the Appendixwith unction number labels used in this graph. Numerical results are given in the main text. The Appendix contains the complete list o partition unctions and Venn diagram mappings.

3 Diversity partitions 55 Table. Three-way sorting o 646 species recorded in Heilongjiang lora o China. The source list is Li s (993). The sorting is ours (Orlóci and He 996, Orlóci et al. 00). The numerical results are summarised in Table and Table 3. The Appendixcontains the partition unctions, Venn diagram mappings, and the sample run o the application program DIVPART.EXE. Table. Selected diversity partitions or the sorting model adapted in Table. Partitions unctions are given in the Appendix. Numerical results are listed or 34 partitions in Table 3. The percentages in the table are relative to total diversity. All values are in bits. See the main text or urther details regarding rounding errors and additivity. * Percent o H(A,B,C), ** 646 is the number o species in the Li (993) records. when the logic o the Venn diagrams in Figure is ollowed. Higher order partitions are noted or increased stability in the sense discussed by Orlóci et al. (00). The proo o a sum being perect or imperect should come rom the logic o the algebra rather than the numerical values. The numerical values are not completely reliable since they are aected by rounding errors that can be quite substantial.

4 56 Orlóci Table 3. Numerical values o entropy and inormation partitions, corresponding to the partition unctions listed in the Appendix. All values are in bits. This table shows results over six orders o. Zero order results register the state richness o the sorting actors individually and jointly. Order one is the basis o perect sums when the logic o the Venn diagrams in Figure is ollowed. Higher order partitions are noted or increased stability (Orlóci et al. 00). See the main text or urther details regarding rounding errors and additivity. Discussion This communication is a ollow up on two o our earlier papers on biodiversity (Orlóci and He 996, Orlóci et al. 00) with novel contents. We reer readers to the original publications or a broader list o relevant reerences, discussions, and typical examples. The novel materials are presented in both main text and Appendix, including partition unctions and their Venn diagram mappings. While the paper limits itsel to the case o 3 actors, the idea o diversity partitions is general and applicable to any number o actors. But since the number o partitions grows as a product with the number o actors considered, beyond a point the large number o possible partitions makes it impractical to derive all o them. But to deine them all is really unnecessary, considering that well planned research is ocused on a limited number o hypotheses about a limited number o actor interactions. So the practical thing to do is to derive diversity partitions that correspond to the speciic hypotheses to be tested. The derivation o the partitions that it a priori stated hypotheses is the irst step. The next step is the derivation o probability distributions or the partitions to acilitate the test o the hypotheses. I do not discuss this aspect since I have nothing novel to present in that regard. Kullback (959) is a undamental source or arguments that proo that when the regularity conditions that he speciies exist, twice the inormation o order will have an asymptoti-

5 Diversity partitions 57 cally chi-squared distribution with given degrees o reedom that I give in the Appendix. My preerence is, however, or empirical distributions derived in the spirit o some Monte Carlo procedure. In this regard reerence is made to Pillar (996, 999) or ecological context and to Edgington (987) or theory. The introduction o the application program DIVPART.EXE is among our objectives. The version available ree o charge or downloading (source address takes data rom a three-dimensional contingency table like in Table. The number o states is not limited. The program can be used or the analysis o two dimensional contingency tables, but with dummy numbers or third dimension. I we were to use it or such a case, with actors limited to A and B in Table, we would interpret only the terms in Tables and 3 that have the A and B labels. Acknowledgenents. I received support rom the CNPq and UFRGS o Brasil, NSERC and UWOo Canada, and UH at Manoa o the United States during tenure o the general project. Reerences Edgington, E.S Randomization Tests. nd ed. Marcel Dekker, New York. Kullback, S Inormation Theory and Statistics. Wiley, New York. Li, J The Forests o Heilongjiang. Northeast Forestry University Press, Harbin, China. Orlóci, L. 99. Entropy and Inormation. Ecological Computations Series, Vol. 3. SPB Academic Publishing, The Hague. Orlóci, L. and X. S. He The entropy structure o biodiversity. Bull. Bot. Res. (NFU, Harbin, China) 6: Orlóci, L., M. Anand. and V.D. Pillar. 00. Biodiversity analysis: issues, concepts, techniques. Community Ecol. 3:7-36. Pillar, V. D A randomisation-based solution or vegetation classiication and homogeneity testing. Coenoses :9-36. Pillar, V.D How sharp are classiications? Ecology 80: Rényi, A. 96. On measures o entropy and inormation. In: J. Neyman (ed.), Proceedings o the 4th Berkeley Symposium on Mathematical Statistics and Probability. University o Caliornia Press, Berkeley, pp Appendix Received October 4, 006 Revised October 3, 0006 Accepted November 7, 006 The Appendix is downloadable rom the Publishers web site o this issue o Community Ecology at address

6 Appendix Please read the main text or details and important acts about the materials resented in this appendix.. Elementary symbols and some numerical identities F AB F AC F BC F AB F AC Principle marginal totals: F A.. [ ] [ ] F [ ]. B [ ] [ ] F.. C Characteristic elementary vectors: [ ] [ ] F B [ ] [ 0 6 3] F 33C Practice run o DIVPART.EXE

7 The version o DIVPAR.EXE described takes the data in sorted ( ) orm as a string with each element separated by a return (paragraph) character through subscripts k, j and i. In the example: 343 The application program DIVPART.EXE is available ree o charge or downloading on the website in older Diversity partition program and data. The ollowing is printed by the program: Program: biodiversity analysis o a 3-dimensional contingency table. Revised Renyi's generalised entropy and inormation are partitioned into components. The components contain additive terms when the order variable Alpha is approaching. Maximum sorting dimension: 3 (A,B,C). Input data: a x b x c requencies presented in a column string. Alpha starts with 0. An upper limit requested. Your response was 5. Number o partitions: 34 Data ile name: C:\Documents and Settings\Laszlo Orloci\Desktop\Compile diversity prog\dataheilo NGJIANG 3X5X3.TRU Date and time:00609 :6:06 Printda ile name: printda.tru Number o sorting criteria: 3 Number o actor states: a 3 b 5 c 3 Number o iterations: Data and marginal totals: Layers o the a x b x c contingency solid: axb principal marginal totals:

8 axc principal marginal totals: bxc principal marginal totals: a principal marginal totals: b principal marginal totals: c principal marginal totals: NOTE I( ) Renyi's inormation o order alpha H( ) Renyi's entropy o order alpha Richness: nats bits Alpha runs rom 0 to upper limit chosen. All logs in nats. Divide nats by ln to convert into bits. Values in rows in order o Alpha rom zero to upper limit: H(A):** **.4378 **.664 ** ** ** H(B):**.398 ** ** **.6605 ** ** H(C):** **.074 **.8459 ** **.63 ** H(A B):** ** **.4735 **.0563 ** ** H(A C):** **.3476 **.5377 ** ** ** H(B A):**.398 ** ** ** **.4603 ** H(B C):**.398 ** ** **.5979 ** ** H(C A):** ** ** ** **.680 ** H(C B):** **.0345 ** **.6788 **.634 ** H(A,B,C):** ** ** ** ** **.8336 H(A,B):** ** ** ** **.6845 ** H(A,C):** **.4507 ** **.7564 **.573 ** H(B,C):** ** ** **.96 **.0649 ** H(A,B C):** ** ** ** ** ** H(A,C B):** **.6447 **.7984 ** ** ** H(B,C A):** ** **.974 ** **.8747 ** H(A B,C):** **.895 **.9874 ** **.8988 ** H(B A,C):**.398 **.770 ** ** ** ** H(C A,B):** ** **.7039 ** ** ** I(ABC):** 0 ** **.5458 ** ** **.0476 I(AB):** 0 **.0583 **.0785 ** **.3864 ** I(AC):** 0 **.070 **.5006 ** **.3340 ** I(BC):** 0 ** ** ** ** ** I(AB C):** 0 ** **.8899 **.466 ** ** I(AC B}):** 0 **.63 **.545 **.4805 **.5764 ** I(BC A}):** 0 **.8438 **.3067 **.6087 **.9874 ** I(A{B,C}):** 0 **.3584 **.438 ** ** **

9 8 I(B{A,C}):** 0 **.337 ** ** **.6897 ** I(CA,B)** 0 ** ** **.6783 ** ** I(ABC):** 0 **.0558 **.0475 ** **.3453 ** I(A):** 0 **.78 **.3337 ** ** ** I(B):** 0 **.3793 ** ** ** ** I(C):** 0 ** ** ** **.9684 ** I(A,B,C):** 0 **.3467 ** ** ** ** :6:06 3. Complete set o partition unctions A. Entropy o order # : Principal marginal H(A) a H( A) ln p i i p i i.. #: Principal marginal H(B) b H( B) ln p j j p j.. j #3: Principal marginal H(C) c HC ( ) ln p k k p k #4: Equivocation H(A B) # - # b a H( A B).. j ln p i j ( ) j i ij. p i j.. j

10 #5: Equivocation H(A C) # - #3 c a H( A C) ln p i k ( ) k i p i k ik. #6: Equivocation H(B A) # - # a b H( B A) i.. ln p j i ( ) i j p j i ij. i.. #7: Equivocation H(B C) #3 - #3 c b H( B C) ln p j k ( ) k j p j k. jk.. k #8: Equivocation H(C A) # - # a c HC ( A).. ln p i k i p ik. k i ( ) i k i.. #9: Equivocation H(C B) #3- # b c HC ( B).. ln p. jk j k j p k j ( ) j k.. j #0: 3-way joint H(A,B,C) # + # + #3 - #0 a b c H( A, B, C) ln p p i j k

11 #: -way joint H(A,B) # + # - # a b H( A, B) ln p ij i j p ij ij. #: -way joint H(A,C) # + #3 - # a c H( A, C) ln p ik i k p ik ik. #3: -way joint H(B,C) # + #3 - #3 b c H( B, C) ln p ik j k p jk. jk #4: Conditional joint H(A,B C) #0 - #3 c a b H( A, B C) ln.. p k ( ) k i j ij k p ij k #5: Conditional joint H(A,C B) #0 - # b a c H( A, C B) ln.. p j ( ) j i k ik j p ik j.. j

12 #6: Conditional joint H(B,C A) #0 - # a b c H( B, C A) ln.. p i ( ) i j k jk i p jk i i.. #7: Equivocation H(A B,C) #0 - #3 b c a H( A B, C) ln. jk pi jk p ( ) i jk j k i. jk #8: Equivocation H(B A,C) #0 - # a c b H( B A, C) ln ik. p jik p ( ) j ik i k j ik. #9: Equivocation H(C A,B) #0 - # a b c HC ( AB, ) ln. p ij ( ) i j k kij p k ij ij. B. Inormation o order #0: 3-way interaction I(ABC) #0 - #7- #8 - #9 - #30 a b c p I( A B C) ln i j k q i... j. q 3 DF abc a b c + p,

13 #: Marginal interaction I(AB) #4 - #30 a b pij I( AB) ln i j q ij DF ab a b + ij. p ij, q ij i... j. #: Marginal interaction I(AC) #5 - #30 a c p I( AC) ln ik i k q ik DF ac a c + p ik. ik, q ik i.. #3: Marginal interaction I(BC) #6 - #30 b c p jk I( BC) ln j k q jk DF bc b c +. jk p jk, q jk.. j #4: Conditional interaction I(AB C) # + #30 c a b p ij k I( AB C) ln p, ( ) ij k k i j q ij k ik.. jk q ij k DF c(ab a b + ) #5: Conditional interaction I(AC B) # + #30 b a c p ik j I( AC B).. j ln p ( ) ik j,.. j i k q ik j.. j ij.. jk q ik j DF b(ac a c + ).. j

14 S#6: Conditional interactions I(BC A) #3 + #30 a b c p jk i I( BC A) i.. ln p ( ) jk i, i j k q jk i i.. ij. i. k q jk i DF a(bc b c + ) i.. #7: Joint interaction I(A{B,C}) # + #5 a b c p I( A{ B, C}) ln p, i j k q i... jk q DF abc b c + #8: Joint interaction I(B{A,C}) # + #6 a b c p I( B{ A, C}) ln p, i j k q.. j ik. q DF abc a c + #9: Joint interaction I(C{A,B}) # + #6 a b c p ICAB ( {, }) ln p, i j k q ij. q DF abc b c + #30: Mutual inormation I(ABC) #4 - # p a b c I( ABC) ln p, j k q ij. i. k. jk q i... j. DF (c )(ab a b + ) (b )(ac a c +) (a )(bc b c + )

15 #3: Marginal eect I(A) a p I( A) ln i i q i DF a- p i i.., q i a #3: Marginal eect I(B) b p j I( B) ln j q j DF b-.. j p j, q j b #33: Marginal eect I(C) c p IC ( ) ln k k q k DF c- p k, q k c #34: Joint eect I(A,B,C) #0 + #3 + #3 + #33 a b c p I( A, B, C) ln i j k q DF abc- p q abc 3. Elementary partitions The graphs below identiy partitions by type o unction or quick reerence to assist the construction o additive sequences o the order one H and I quantities. Entropy Inormation

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