GEOS 36501/EVOL 33001 3 February 2012 Page 1 of 26IX. Sampling Models, 1: Introduction tohomogeneous sampling models1 Basic framework for homogeneous sampling models1.1 r is the per-capita rate of sampling per lineage-million-years.1.2 “Sampling” means the joint incidence of preservation,exposure, collection, identification, etc.1.3 Generally consider the relevance only of extincti on r ate q(tacitly assume p ≈ q and tr uncati on e ffect s negl ig ibl e).This restriction can be fairly easily relaxed.1.4 Assume all rates temporally and taxonomically homogeneous(for now).1.5 It will often be useful to deal with “dimensionless rates” and“dimensionless time”.Let q = 1, express r in terms of multiples of q, and express time in terms of multiples of1/q (expected mean duration).GEOS 36501/EVOL 33001 3 February 2012 Page 2 of 262 Long-term average properties of a set of taxa(See Foote, 1997, Paleobiology 23:278-300.)2.1 For a lineage with duration T , model sampling as a Poissonprocess with parameter rT .Thus:• Pr(lineage never sampled)= e−rT(Poisson probability of zero successes.)• Pr(lineage sampled at least once)= 1 − e−rTGEOS 36501/EVOL 33001 3 February 2012 Page 3 of 26• Pr(lineage sampled exactly once)= rT e−rT(Poisson probability of exactly onesuccess.)• Pr(observed range=t [t > 0])= r2(T − t)e−r(T −t).This is actually a density, not aprobability, and is derived as follows:f(t|T) =ZT −t0[re−ry][re−r(T −y−t)] dy,where– the first term within the integral is the density function for the first samplingevent from the beginning of the duration– the second term is the density function for the first event from the end of theduration ([T − y − t] is the distance from the end working backward)and– the upper limit of integration is the longest that a true duration (T ) can be ifthe range is t.• If sampling rate varies, substitute e−RT0rxdxfor e−rT.2.2 These probabilities are then integrated over the entiredistribution of durations (exponential or otherwise).2.2.1 Example with untruncated expo nential distribution• The probability that a randomly chosen lineage will never be sampled is equal to:R∞0qe−qT· e−rTdT = q/(r + q).• Pr(sampled at least once)=R∞0qe−qT· (1 − e−rT) dT = r/(r + q). This is the expectedprortion of lineages sampled, which has also been referred to as paleontologicalcompleteness.• Pr(sampled exactly once)=R∞0qe−qT· rT e−rTdT = qr/(r + q)2. These are thesingletons.• Pr(observed range=t [t > 0])=R∞0qe−qT· r2(T − t)e−r(T −t)dT = qr2e−qt/(r + q)2.(NB: This is actually a density.)• By the number of times sampled, we mean the number of distinguishablestratigraphic horizons at which it is sampled, not the actual number of fossils orlocalities. Thus, a singleton may be sampled numerous times, but if these are all,within the limits of resolution, at a single time horizon, this counts as being sampledonce.GEOS 36501/EVOL 33001 3 February 2012 Page 4 of 262.3 If we want to focus on distribution of stratigraphic ranges oflineages that are actually sampled, then we normalize all theprobabilities by overall probability of sampl ing (r/ [r + q] inthe exponential case). Thus:• Pr(singleton)= q/(r + q)• Pr(range=t [t > 0])= qre−qt/(r + q) (NB: This is actually a density.)• The non-singleton part of this distribution is exponential with parameter q. Thus,even though sampling is complete and therefore ranges are truncated, if we ignoresingletons we expect shape of distribution to reflect true extinction rate accurately.• Alroy’s data on mammal species seem to bear this out.GEOS 36501/EVOL 33001 3 February 2012 Page 5 of 26Solid line: all points. Dashed line: singletons omitted.Note excess of singletons over exponential model.●●●●●●●●●●●●●●●●0.010.11.0101000 5 10 15 20Duration (m.y.)Cumulative percent surviving (log scale)A●●●●●●●●●●●●●●● ●0.010.11.0101000 5 10 15 20Duration (m.y.)Percent surviving (log scale)B2.4 Comments on excess of singletons• Implication of these models is that the frequency of singletons is overestimated in rawstratigraphic range data.• This is not to say that any individual singleton taxon is an artifact; some taxa inreality did endure just one time interval.• Frequency of singletons depends on both q and r; thus this frequency is a good guideto comparing r of two or more groups only if the groups have (roughly) the same q.– Example: Kidwell’s comparison between calcitic and aragonitic bivalve generayields similar frequency of singletons for the two (2005, Science 307:914).– Assuming similar q for calcitic and aragonitic taxa, this implies similar r for thetwo groups.• In much of (q, r) parameter space, the probability that a taxon was in realitysingleton, given that it is sampled in only one interval, is comparatively low.GEOS 36501/EVOL 33001 3 February 2012 Page 6 of 260.0 0.5 1.0 1.5 2.00.0 0.2 0.4 0.6 0.8 1.0Discrete time modelq (instantaneous rate per lineage per interval)Pr(truly confined to single interval)●●●●●●●●●●●●●●●●●●●0.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.0R (sampling probability per interval)Pr(sampled as singleton | sampled at least once)●●●●●●●●●●●●●●●●●●●●●q=0.1q=0.2q=0.5q=1.0q=2.0●●●●●●●●●●●●●●●●●●●0.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.0R (sampling probability per interval)Pr(truly confined to single interval | sampled as singleton)●●●●●●●●●●●●●●●●●●●●●q=0.1q=0.2q=0.5q=1.0q=2.0GEOS 36501/EVOL 33001 3 February 2012 Page 7 of 262.5 Density distribution can also be modified to accom modatefinite window o f observation (truncation of record at startand end).We did this previously when looking at taxon age distributions, but we glossed over thedetails.• Key is to establish density function for times of origination and density for times ofextinction, given times of origination.• Let w be length of window of observation.• Assume diversity at t = 0 is equal to n0= 1 (this does not matter; everything scalesto n0).• Total progeny in window of time w:Mw= 1 +Zw0pe(p−q)tdt= 1 + pw if p = q=q − pe(p−q)wq − pif p 6= q• Origination density at time x:g(x) =pe(p−q)xMwif x > 0=1Mwif x = 0• Extinction density at elapsed time z after origin (assuming origin at time x):f(z) =