XIII. Sampling Models, 3: Introduction to time-varying...

GEOS 33001/EVOL 33001 8 November 2007 Page 1 of 20

XIII. Sampling Models, 3: Introduction totime-varying models (forward problems)

1 Basic framework

(see Foote 2000, Paleobiology Supplement to 26(4):74-102, Foote 2001, Paleobiology 27:796[erratum], and Foote 2003, Journal of Geology 111:125-148, 752-753 [erratum])

1.1 Let there be n time intervals, each characterized by a set oforigination, extinction, and sampling rates: pi, qi, and ri.

1.2 Use time series of p and q to predict true time series of NbL,NFt, NFL, and Nbt (and thus Nb and Nt).

1.2.1 These quantities all scale to Nbi.

1.2.2 Nb for continuous-turnover model (q.v.) in which origination andextinction occur at constant per-capita rate within an interval:

1. Let the age of the bottom boundary be at time t = x.


2. Let N0 = 1 at t = 0.

3. Let pt and qt be time-specific rates.

4. Then

Nx = exp[ ∫ x

0

(pt − qt) dt].

5. Or, if we divide time into intervals and assume constant p and q within an interval(while still varying among intervals):

Nbi = exp[ i−1∑

j=1

(pj − qj)],

if the rates pi and qi are expressed per lineage per interval. If instead they areexpressed per lineage-million-years, and interval durations are given by ∆ti, then wehave:

Nbi = exp[ i−1∑

j=1

(pj − qj)∆tj].

1.2.3 Nb for pulsed-turnover model (q.v.) in which originations are all at startof interval and extinctions at end of interval (so all lineages extendthroughout the interval):

1. In this model, P is the number of new lineages produced per lineage extant at thestart of the interval. (So number of originations is equal to Nb · P and total intervaldiversity is Nb[1 + P ].)

2. In this model, Q is the extinction probability. (So number of extinctions is equal toNb[1 + P ]Q).

3. Thus

Nbi =i−1∏j=1

(1 + Pj)(1−Qj).

1.3 Use time series of p, q, and r to determine samplingprobabilities for given time intervals.

Probability of being sampled in a given interval of time (either before a reference point,after a reference point, or between two reference points) is obtained as the integral, over allpossible durations, of the probability of having a certain duration multiplied by theprobability of being sampled given that duration.


Journal of Geology E r r a t u m 753

Table 1. Expressions Used to Calculate Survivorship Probabilities

Quantity/model Expression

:Nbt•C �qN eb•P N (1 � q)b:NbL

•C �qN (1 � e )b•P N qb:NFt

CC p�q �pN e (1 � e )bPP N p(1 � q)bCP p �pN e (1 � q)(1 � e )bPC �qN peb:NFL

CCa if ,�qN (e � p � 1) p p qb

if(p�q) �qqe � (p � q)e � p

N p ( qb p � q

PP N pqbCP pN q(e � 1)bPC �qN p(1 � e )b

PA(i):•C n k�1 k�1 n n�S q �q �S r �S q �S rm k m k kmpi�1 mpi�1 kpi�1 kpi�1� e (1 � e ) 1 � e [1 � P (k)] � e 1 � e [1 � P (n)][( ) ]{ ( ) } ( ){ ( ) }( )DFbL A

kpi�1

•P n k�1 nk�1 n�S r �S rm kmpi�1 kpi�1� � 1 � q (q ) 1 � e [1 � P (k)] � � (1 � q ) 1 � e [1 � P (n)]{ ( ) } { ( ) }( )m k DFbL k A( ) [ ]kpi�1 mpi�1 kpi�1

PB(i):C• i�1 i�1 i�1 i�1 i�1�S p �p �S r �S p �S rm k m k kmpk�1 mpk�1 kp1 kp1� e (1 � e ) 1 � e [1 � P (k)] � e 1 � e [1 � P (1)][( ) ]{ ( ) } ( ){ ( ) }( )DFFt B

kp1

P•i�1 i�1 i�1i�1 i�11 p 1k �S r �S rm kmpk�1 kp1� � 1 � e [1 � P (k)] � � 1 � e [1 � P (1)]{ ( ) } { ( ) }DFFt B( )( ) ( )( )kp1 mpk�1 kp11 � p 1 � p 1 � pm k k

:PDFbt•• �r1 � e

:PDFbL•Cb �(q�r) �q[r � qe ]/(q � r) � e

�q1 � e•P �r1 � e

:PDFFtC•c �(p�r) �p[r � pe ]/(p � r) � e

�p1 � eP• �r1 � e

:PDFFLCCd

if ,�p �(p�r)N p r 1 � e p[1 � e ]b � � p p q2{ }N p � r p (p � r)FL

if(p�q) �(q�r) (p�r)N pr[e � 1] pqe [e � 1]b �q p� � e (e � 1) p ( q{ }N (q � r)(p � q) (p � r)(q � r)FL

PP �r1 � e

CPe p �rr(e � 1) � p(e � 1)pp(r � p)(e � 1)

PCf q �rr(e � 1) � q(e � 1)qq(r � q)(e � 1)

Note. In the two-character code for model, the first character denotes origination and the second extinction; ,C p continuous. A bullet means the expression applies to either model for the corresponding process. Nb is the true standing diversityP p pulsed

at the start of the interval; because all relevant numbers scale to Nb, this can be arbitrarily set to unity.a Foote 2000a, eqq. (6b) and (6c).b Foote 2000a, eq. (27b).c Foote 2000a, eq. (28b).d Foote 2000a, eqq. (29b) and (29c).e Let z represent time within an interval of duration t, where and are the beginning and end of the interval, respectively.z p 0 z p tBy assumption, there is no extinction until the end of the interval. Thus, the density of origination at time z is equal to pz pte /(e �

(cf. Foote 2001a, eq. [3]). Because all lineages originating within the interval extend to the end, the probability of preservation,1)given origin at z and extinction at t, is equal to . It is necessary to integrate the density of origination times the probability�r(t�z)1 � eof preservation over all values of z. Thus, , which is equal to the expression in the table once ttpt pz �r(t�z)P p [1/(e � 1)] e [1 � e ]dz∫0DFFLis set to unity.f Derived as in the foregoing footnote, with origination and extinction reversed.


1.3.1 Use time series of p and r to predict probability PBi that a taxon extantat start of an interval i will be sampled sometime before the interval.

1.3.2 Use time series of q and r to predict probability PAi that a taxon extantat start of an interval i will be sampled sometime before the interval.

1.3.3 Use values of pi, qi, and ri for a given interval to predict the probabilitythat a taxon will be sampled during an interval.

This depends on whether it is, in reality (i.e. prior to sampling), a member of thecategories NbL, NFt, NFL, or Nbt. The corresponding probabilities are denoted PD|bL, PD|Ft,PD|FL, and PD|bt, where the subscript i has been omitted for clarity.

1.4 Let XbL, XFt, XFL, and Xbt be the observed numbers of taxa ineach of the four categories.

Note that a taxon observed to be in the bt-category must have been so in reality, but, forexample, and FL-taxon could in reality have been in any of the four categories prior tosampling.

1.5 Determine the expected numbers of observed taxa X:

XbL =NbLPBPD|bL

+ NbtPBPD|bt(1− PA),XFt =NFtPAPD|Ft

+ Nbt(1− PB)PD|btPA,XFL =NFLPD|FL

+ NbL(1− PB)PD|bL+ NFtPD|Ft(1− PA)+ Nbt(1− PB)PD|bt(1− PA),

and

Xbt =NbtPBPA

and of course we have a number of derived quantities :

Xb =XbL + Xbt

Xt =XFt + Xbt

XF =XFt + XFL

XL =XbL + XFL

Xtot =XbL + XFt + XFL + Xbt

etc.


1.6 Use the resulting X-values to predict patterns of first andlast appearance, apparent rates of origination andextinction, etc.

1.6.1 Example:

qapparent = − ln[XbtXb

]2 Risk models

2.1 Foregoing equations depend on assumed risk model, i.e. thedistribution of origination and extinction within an interval.

2.2 Some possible risk models include:

2.2.1 Constant risk

Note that diversity changes exponentially through a time interval.

2.2.2 Exponentially distributed risk:

Let q be the probability of extinction in a given fine time increment. ThenPr(q ≤ x) = 1− e−x/q̄, where q̄ is the mean of q.

• Implementing in R: q


2.2.4 Pr(q ≤ x) = ek ln(x), where k = q̄/(1− q̄) (Foote 1994, eq. 2).

This was designed to yield episodic pulses like the kill curve, but to be scaled in terms ofany arbitrary q̄ (and therefore applicable, in principle, at any taxonomic level).

• To generate n random extinction rates from this distribution with mean q.mean:q


All models have same long-term mean!A: constantB: exponentialC: Pr(q ≤ x) = ek ln(x)D: Kill Curve


3 Some general results

3.1 Edge effects

3.1.1 Diversity (total and boundary-crossing) artificially depressed nearbeginning or end of time series.

3.1.2 Singletons artificially inflated near beginning and end.

3.1.3 For per-capita rate, origination artificially inflated near beginning, andextinction artificially inflated near end.

3.1.4 Other rate measures affected at both ends.

3.1.5 Edge effect decays exponentially; edge barely felt after about two taxonlengths.


3.2 Smearing of rate anomalies (Signor-Lipps Effect) anddiversity peaks

3.2.1 Like edge effect, smearing is exponential

3.2.2 Note that, with per-capita rate metrics, only apparent extinction isaffected by peak in extinction.

• With other metrics, both extinction and origination affected.

• Similarly for origination peak.


3.2.3 NB: Empirical scaling of p, q, and r suggests that many genera haveoffset between true origination (extinction) and first (last) appearanceof a full stage or more.

FIGURE 1. Expected offset between stage of extinction and stage of last appearance. Curves portray the probability,given extinction in a specified stage j, that genus is extant and sampled in stage i (i � j) but not sampled after stagei. Probabilities are normalized so that only genera sampled at least once are included. Origination and extinctionrates (p and q) are chosen so that there is no net diversity change within a stage and so that the number of newgenera originating in a stage is 30% of the starting diversity. Thus, for the continuous model, this means that p �q � 0.3, and for the pulsed model, p � 0.3 and q � 0.3/(1 � 0.3) � 0.23. (See Foote 2003a for discussion of rates inthe context of the two models.) For a given sampling rate, the continuous model yields a larger offset. Models aredifficult to distinguish at very low or very high sampling rates, but they are distinct at intermediate sampling rates,which are empirically realistic. Probabilities are based on equations presented in Foote (2003a,b).


3.3 Diversity and rate measures that include singletons areespecially problematic

3.3.1 Distorted in general

3.3.2 Also tending to induce spurious correlation between apparent p and q.

3.3.3 Not even monotonically related to sampling rate.


3.4 Sampling peak

3.4.1 Induces spurious peak in origination and extinction coinciding with thesampling peak...

3.4.2 ...as well as spurious trough in extinction (before the peak) and spurioustrough in origination (after the peak).


3.4.3 Pull of the Recent as special case of sampling peak: (nearly) completesampling of living biota.


3.5 Sampling trough

3.5.1 Induces spurious trough in origination and extinction coinciding withthe sampling trough...

3.5.2 ...as well as spurious peak in extinction (before the trough) and spuriouspeak in origination (after the trough).

0 5 10 15 20

0.0

0.2

0.4

0.6

Effect of short−lived change in sampling rate

Time interval (arbitrary units)

Per

−ca

pita

rat

e

Sampling True extinctionApparent extinction


3.6 Long-term change in sampling rate

3.6.1 Weak effect on apparent rates

Most taxa don’t live long enough to feel the change.

3.6.2 Diversity, by contrast, strongly affected.


4 Patterns of first and last appearance

Standard forward and backward survivorship equations modified to take incomplete andvariable sampling into consideration.

4.1 Probability of LO in interval j, given FO in interval i

4.1.1 Denote this probability P→ij

4.1.2 For i = j, P→ij is the expected number of singletons divided by theexpected total number of first appearances.

(Tacitly assumes large numbers, since, in general, expectation of a ratio not equal to ratioof expectations). Thus

P→ii =XFL

XFL + XFt

4.1.3 For j > i, in words:

P→ij is obtained by integrating or summing, over all possible durations, (the probability ofremaining extant at least through interval j)×(the probability of being sampled in intervalj)×(the probability of not being sampled after interval j), all normalized to the probabilityof being sampled at least once after interval i.

1. For continuous-turnover model

P prijj�1

�S q �q �qk j jkpi�1[(1 � P )e ]{(1 � e )P (j) � e P (j)[1 � P (j)]}rii DFbL DFbt A ,P (i)A

(5a)


2. For pulsed-turnover model

P prijj�1(1 � P )[� (1 � q )]{q P (j) � (1 � q )P [1 � P (j)]}rii k j DFbL j DFbt Akpi�1

,P (i)A

(5b)

4.2 Probability of FO in interval i, given LO in interval j

4.2.1 Denote this probability P←ij

4.2.2 For i = j, P←ij is the expected number of singletons divided by theexpected total number of last appearances.

Thus

P←jj =XFL

XFL + XbL

4.2.3 For i < j, in words:

P←ij is obtained by integrating or summing, over all possible backward durations, (theprobability of having originated already by interval i)×(the probability of being sampled ininterval i)×(the probability of not being sampled before interval i), all normalized to theprobability of being sampled at least once after before j.


1. For continuous-turnover model

P pRijj�1

�S p �p �pk i ikpi�1[(1 � P )e ]{(1 � e )P (i) � e P (i)[1 � P (i)]}Rjj DFFt DFbt B ,P (j)B

(6a)

2. For pulsed-turnover model

P pRij

j�1(1 � P )[� 1/(1� p )]{p /(1 � p )P (i) � 1/(1� p )P (i)[1� P (i)]}Rjj k i i DFFt i DFbt Bkpi�1,

P (j)B

(6b)

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