Anne Chao Ncholas J Goell C seh lzabeh L ander K Ma Rober K Colwell and Aaron M llson 03 Rarefacon and erapolaon wh ll numbers: a framewor for samplng and esmaon n speces dversy sudes cology Monographs Append : Rarefacon and erapolaon of ll numbers for ncdence daa All our dervaons for he resuls n able of he man e for ncdence daa are generally parallel o hose for abundance daa bu some modfcaons are reured In hs Append we only sech he necessary modfcaons For ncdence daa our model s based on he followng bnomal produc model for he observed ncdence-based speces freuences : see b n he man e y y P y y As defned n he man e he ll numbers Δ for he model of ncdence daa are epressed as 0 hese are he ll numbers based on he relave ncdences = For any sample of sze defne he ncdence freuency coun as he number of speces deeced n eacly samplng uns For he reference sample of sze we us use for noaonal smplcy e = he epeced value of can be epressed as: ] = 0 In parcular ] s he epeced number of undeeced speces n samples 0 0 In he reference sample we defne as he oal number of ncdences n he samples ere s an observable varable n he reference sample Defne as he epeced oal number of ncdences for samplng uns and we have: ] 3 ere s an unobservable parameer and mus be esmaed from he reference sample
As dscussed n he man e for abundance daa we defne he epeced dversy Δ for samplng uns as he ll numbers based on he epeced ncdence freuency couns whch are formed by averagng ou ncdence couns for samplng uns uppose a random sample of samplng uns are aen from he enre assemblage hen we oban a se of ncdence freuency couns for hs sample { ; = } Afer an nfne number of such samples have been aen he average of for each = ends o ] derved n he freuency couns epeced n samplng uns consss of he freuency couns { ]; = } wh he epeced oal ncdences ] Noe ha for a se of samplng uns he relave ncdences of speces are smply here are ] such speces here are ] such speces here are ] such speces hus we can oban he epeced dversy Δ for samplng uns can be any posve neger no necessarly resrced o < as ] ] ] 4 Rarefacon refers o he case where < whereas erapolaon refers o he case > hroughou he paper and appendces he heorecal formulas for rarefacon and erapolaon of ll numbers for he model of ncdence daa refer o 4 All heorecal formulas for = 0 and and n general for order > are provded n able of he man e he frs column For fndng esmaors n he rarefacon par we us replace he parameer n 4 by an esmaor see below and he epeced ncdence couns ] by her esmaors gven n he followng proposon Proposon : nder a bnomal produc model b n he man e he mnmum varance unbased esmaor for he epeced ncdence freuency coun ] s < 5 a ere 0 f a < b We use hs convenonal defnon hroughou hs Append he proof b s parallel o ha n Proposon D of Append D and s hus omed Rarefaconrapolaon for peces Rchness = 0
A smlar proof as n Proposon D Append D shows ha he rarefacon esmaor 0 s dencal o he radonal sample-based rarefacon funcon ha s 0 ~ sample obs 0 nce our esmaor for he ncdence freuency couns are vald only for < hey can be used only for rarefacon bu no for erapolaon he erapolaon esmaor for speces 0 rchness for he epeced number of speces D n a sample of sze 0 s shown n C4 and also n able of he man e Rarefaconrapolaon for hannon dversy = he heorecal formula of dversy of order = for a rarefed sample of sze s ep log ] 6 ere we need an esmaor for ] nce an unbased esmaor of he oal ncdence probables s hs mples from 3 ha an unbased esmae of for any s Replacng by and ] by gven n 5 we oban he rarefacon esmaor: ep log < For he erapolaon of ll number of = here s no unbased esmaor as > we adop an approach smlar o he one used for abundance daa For ncdence daa defne as he rue enropy n he assemblage: log Also defne as he epeced enropy for he reference sample of sze e log 3
4 Chao e al 03 recenly obaned a nearly unbased esmaor for he enropy under he model of ncdence daa: sample log 0 7 where ] log 0 r r A r A A and A = + ] hus an esmaor of he hannon dversy s ep sample Afer some epansons we can oban he followng wo appromaon formulas: As wh he abundance daa we assume ha here s a lnear relaonshp n he heorecal enropy funcon: w w We can hen solve for he parameer w o oban w o fnd an esmaor for we subsue and by sample Ĥ gven n 7 and log respecvely hen we oban he followng esmaor for he epeced enropy of sze + : sample 8
5 As he augmened sample sze ends o nfny he erapolaed formula 8 ends o he enropy esmaor sample Ĥ n 7 For esmang he erapolaed dversy of = we us ae he eponenal funcon of he erapolaed enropy ] ep 9 Rarefaconrapolaon for mpson dversy = For = he heorecal formula for any sample sze s ] 0 Replacng by and ] by we oban our proposed rarefacon esmaor: 0 Applyng our general formula 0 o an augmened sample sze of + we oban he followng epeced dversy of order : ] he denomnaor n he above formula can be smplfed o ] ]
6 In he above formula we subsue and respecvely by her unbased esmaors ] and and oban a nearly unbased esmaor for he erapolaed dversy D : 0 As ends o nfny we oban he followng nearly unbased esmaor for he asympoc dversy: 3 Rarefaconrapolaon for ll number of order > In he heorecal formula Δ gven n 4 of any order we can replace by and ] by o oban our proposed rarefacon esmaor: For erapolaon we le be he rlng number of he second nd defned by he coeffcen n he epanson where denoes he fallng facoral funcon Also le V be a bnomal random varable wh parameer + and probably π hen ] V
7 V ] he las eualy follows from a momen propery of a bnomal dsrbuon wh parameer + and probably π : V Replacng and respecvely wh her unbased esmaors and Good 953 we oban he proposed nearly unbased predcor for > as shown n able of he man e: As ends o nfny he nearly unbased esmaor for he asympoc dversy ] for s: 4 hs esmaor can also obaned by nong ha ] s an unbased esmaor of Good 953 and s an unbased esmaor for A replcaon prncple and s generalzaon for he model of ncdence daa Proposon : A replcaon prncple for he model of ncdence daa Assume Assemblage consss of K replcaes of Assemblage ach replcae has he same number of speces and he same speces ncdence probables as Assemblage bu wh compleely dfferen unue speces n each replcae A sample of samplng uns s aen from Assemblage hen number of samplng uns needed n Assemblage o aan he same epeced sample coverage s appromaely K and he epeced dversy of any order 0 n Assemblage for he sample wh sandardzed coverage s appromaely K mes of ha n Assemblage Proposon 3: A generalzaon of he replcaon prncple dscussed n Proposon If Assemblage s unambguously K mes more dverse han Assemblage e for all 0 ll number of order of Assemblage s K mes ha of Assemblage hen n he coverage-based
sandardzaon he epeced dversy of any order 0 n Assemblage s appromaely K mes of ha n Assemblage he proof for hese wo proposons s generally parallel o ha for abundance daa Proposons D4 and D5 n Append D and hus s omed LIRAR CID Chao A Wang and L Jos 03 nropy and speces accumulaon curve: a nearly unbased enropy esmaor va dscovery raes of new speces nder revson Mehods n cology and voluon Good I J 953 he populaon freuences of speces and he esmaon of populaon parameers Bomera 40:37-64 8