MBRS detectability talk

Introduction Mosquitoes/WNV Turtle surveys Meta- stuff References

Detectability in ecological systems:two nonstandard examples

Ben Bolker, McMaster UniversityDepartments of Mathematics & Statistics and Biology

Math Bio Research Seminar

3 October 2014

Ben Bolker Math Bio Research Seminar

Detectability


Acknowledgements

Money NSF, NSERC

Computational resources SHARCnet

Data and discussions Aaron Berk, Alan Bolten, Karen Bjorndal,Leonid Bogachev, Ethan Bolker, Ira Gessel, MarmKilpatrick


Detectability


Outline

1 Introduction

2 Mosquitoes/WNV

3 Turtle surveys

4 Meta- stuff


Detectability


Detectability in ecological problems

ecological sampling is imperfect;individuals may vary in detectability

sometimes it matterssometimes it’s unidentifiable

sampling designs(e.g. capture-mark-recapture)

statistical methods(MLE, Bayesian MCMC)

relevance in other fields of mathbio?


Detectability


Introductory meta- stuff

Working on problems:the “Pacala method”

http://weedactivist.com/2013/04/26/reinventing-the-wheel/


Detectability

http://weedactivist.com/2013/04/26/reinventing-the-wheel/


Outline

1 Introduction

2 Mosquitoes/WNV

3 Turtle surveys

4 Meta- stuff


Detectability


The problem

American Robins /mosquitoes /West Nile virus

genotyped blood meals(one per mosquito)

what can we tellabout the robinpopulation from thesedata?size, heterogeneity?

Turdus migratoriusallaboutbirds.org

Culex spp.alamel.free.fr

WNV (Wikipedia) Marm Kilpatrick


Detectability

http://allaboutbirds.org

http://alamel.free.fr


Mathematical framework

occupancy spectrum:S = {si}, i = 0, . . . , imax =# of birds sampled by i mosquitoes∑

si = B,∑

isi = M

V is the (unordered) occupancy:e.g. for B = 4, M = 5:

V = {{0, 1, 1, 3}} ↔ S = {1, 2, 0, 1}

s0 = “missing mass”

(how) can we estimate B?

birds mosquitoes


Detectability


Occupancy spectrum

Maxwell-Boltzmann statistics

define the multinomial coefficient

M(S) ≡ (∑

si)!∏si !

.

then the likelihood of the occupancy spectrum is

P(S|B,M) =1

BMM(S)M(V )

zeros are unobserved;use s0 = B − K where K (total birds observed) ≡∑

i>0 si


Detectability


Maximum likelihood estimation

Log-likelihood as a function of B is

L = C −M log B + log B!− log(B − K )!

we know M (# of mosquitoes) and K (# of birds represented)

→ K is a sufficient statistic for estimating B

apply standard MLE machinery


Detectability


Likelihood estimation

20 50 1001819202122232425

Total number of birds (B)

nega

tive

log-

likel

ihoo

d(L

)

1819202122232425

for K = 16, M = 20:

B̂ = 41CI={21,119}


Detectability


Reasons to like maximum likelihood estimation

consistent and asymptotically Normal(= unbiased for large data sets)

asymptotically efficient(= most statistically powerful unbiased estimator for large datasets)

. . . a universal “Swiss Army Knife”. When it can dothe job, it’s rarely the best tool for the job but it’srarely much worse than the best (at least for largesamples). [Steve Ellner]


Detectability


Simulation results: bias and mean squared error

B: 32 B: 56 B: 100 B: 178 B: 316 B: 562 B: 1000

−1.00−0.75−0.50−0.25

0.000.25

0.0

0.2

0.4

0.6

0.8

stat: biasstat: M

SE

10 20 10 20 5010 20 50 10020 50 100 50 10020050 100200 500100 200 500Number of mosquitoes

method

MLE

Strong negative bias for small B/very small M,slight positive bias ≈ 20% for intermediate samples


Detectability


Good-Turing estimators

alternative approach:count doublets, W =

∑vi · (vi − 1): set observed=expected

and solve for B̂:

B̂ = 1 +12

√1 + 4M(M − 1)/W

Related (loosely) to Good-Turing estimators (Good, 1979)(estimated frequency distribution of codebook pages)

the Pacala method:if you’re reinventing important wheelsyou’re on the right track!


Detectability

http://www.newswise.com/articles/view/501440/


Estimator comparison

B: 32 B: 56 B: 100 B: 178 B: 316 B: 562 B: 1000

−1.00−0.75−0.50−0.25

0.000.25

0.0

0.2

0.4

0.6

0.8

stat: biasstat: M

SE

10 20 10 20 5010 20 50 10020 50 100 50 10020050 100200 500100 200 500Number of mosquitoes

method

MLE

doublets

Doublet method works (much) better:largely suppresses positive bias


Detectability


a bit of data

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Baltimore Foggy_Bottom The_Mall

10

100

1000

2008 2010 2004 2005 2006 2008 2011 2004 2005year

Est

. bird

pop

ulat

ion

(N == K)

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FALSE

TRUE


Detectability


Conclusions & open questions

Conclusions

doublet estimator is better(bias/MSE),reasonable for M > 10− 20

estimates effectivepopulation size —exactly what we want forvector-borne diseasemodels!

Open questions

confidence intervals,K == M estimates fordoublets

estimate coverage?

estimating heterogeneity/subtler effects ofheterogeneity on diseasedynamics?

combining data frommultiple sites & years


Detectability


Outline

1 Introduction

2 Mosquitoes/WNV

3 Turtle surveys

4 Meta- stuff


Detectability


Green turtles at Tortuguero

green turtles(Chelonia mydas)at Tortuguero, Costa Rica

data fromCarr/Bjorndal/Bolten

survey data: 1971–present;renesting interval data:1955–present

estimate detectionprobability,recover 1955-1970population size estimates?

Sea Turtle Conservancy /http://www.conserveturtles.org


Detectability

http://www.conserveturtles.org


data

1970s 1980s 1990s

200

400

600

20 40 60 20 40 60 20 40 60Renesting interval (days)

Cou

nts

(squ

are-

root

scal

e)


Detectability


Fit by convolution

true distribution of inter-nesting intervals F (t ,θ)distribution of turtles observed on their second nestingattempt is pF ,where p is the detection probabilitydistribution of nth-nesting-interval times:n-fold convolution, F n ≡ F ∗ F ∗ F ∗ . . . ∗ Fprobability of detecting after n intervals is geometric,p(1− p)n−1

overall distribution observed is

F ∗ =∑

n

p(1− p)n−1F n(θ)

obst ∼ NegBinom(F ∗(t))


Detectability


Brute force approach

make F a discrete distribution with support from days 7–18

θ is just 11 parameters describing the distribution(constraints: 0 < Fi < 1,

∑Fi = 1)

use distr package in R for numerical convolution calculationsbrute-force convolution calculation

(various MCMC/latent-variable strategies also possible,but probably slower)


Detectability


Yearly renesting interval estimates

0.0

0.1

0.2

0.3

9 12 15 18day

prop

ortio

n


Detectability


Prediction for 1971

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0.1

0.2

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20 40 60Renesting interval (days)

Pro

port

ion


Detectability


Estimated detection probabilities

0.00

0.25

0.50

0.75

1.00

1960 1970 1980 1990Year

Est

.de

tect

ion

prob

abili

ty(p̂)


Detectability


Raw and adjusted counts

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1960 1970 1980 1990Year

Tota

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nts

variable●

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countadjcount


Detectability


calibration

0.25

0.50

0.75

1.00

1960 1970 1980 1990year

dete

ctio

n pr

obab

ility model

fn_dnbinomfn_dnbinom1fn_dpois

methodBFGSL−BFGS−BNelder−Mead


Detectability


Conclusions & open questions

Conclusions

detection probability≈ 60–70%(highly variable)

seems to recapture

Open questions

check calibration on moderndata

smoother renesting-intervalcurve?


Detectability


Outline

1 Introduction

2 Mosquitoes/WNV

3 Turtle surveys

4 Meta- stuff


Detectability


Cross-citation study

who cares about math biology?

more specifically, what is the information flowfrom MB to bio (or math) and vice versa?

extract information from ISI Journal Citation Report(thanks to Aaron Berk)

find top 100 cited/citing journals for:(Bull Math Biol, J Theor Biol, Theor Popul Biol, J Math Biol,Math Biosci, PLoS Comput Biol)


Detectability


Ordination of (1/(1+avg cites))

AM_J_PHYSIOL−HEART_C

AM_NAT

ANIM_BEHAV

APPL_MATH_COMPUT

APPL_MATH_MODEL

ASTROPHYS_JBBA−BIOMEMBRANES

BEHAV_ECOLBEHAV_ECOL_SOCIOBIOL

BIOCHEMISTRY−US

BIOINFORMATICS

BIOPHYS_J

BIOSYSTEMS

BMC_BIOINFORMATICS

BMC_EVOL_BIOL

BMC_GENOMICS BMC_SYST_BIOL

B_MATH_BIOLCANCER_RES

CELLCIRC_RES

COMMUN_NONLINEAR_SCI

COMPUT_MATH_APPL

CURR_OPIN_STRUC_BIOL

DISCRETE_CONT_DYN−B

ECOLOGYECOL_LETT

ECOL_MODELEVOLUTION

GENETICS

INT_J_BIOMATH

J_AM_CHEM_SOC

J_APPL_PROBAB

J_BIOL_CHEM

J_BIOL_SYST

J_CHEM_PHYSJ_COMPUT_PHYS

J_EVOLUTION_BIOL

J_EXP_BIOL

J_GEOPHYS_RES

J_IMMUNOL

J_MATH_ANAL_APPL

J_MATH_BIOL

J_MOL_BIOLJ_NEUROPHYSIOLJ_NEUROSCI

J_PHYSIOL−LONDON

J_PHYS_CHEM_B

J_PHYS_CHEM_C

J_THEOR_BIOL

J_VIROL

LANGMUIR

MATH_BIOSCIMATH_BIOSCI_ENG

MATH_COMPUT_MODEL

MATH_MOD_METH_APPL_S

MOL_BIOL_EVOL

MOL_BIOSYST

MOL_ECOL

NATURE

NAT_GENET

NAT_NEUROSCI

NAT_REV_GENET

NEURAL_COMPUT

NEUROIMAGE

NEURON

NONLINEAR_ANAL−REALNONLINEAR_ANAL−THEOR

NONLINEAR_DYNAM

NUCLEIC_ACIDS_RES

OIKOS

PHILOS_T_R_SOC_B

PHYS_BIOL

PHYS_REV_A

PHYS_REV_B

PHYS_REV_EPHYS_REV_LETT

PLOS_COMPUT_BIOL

PLOS_GENET

PLOS_ONE

PROG_BIOPHYS_MOL_BIO

PROTEINSPROTEIN_PEPTIDE_LETT

P_NATL_ACAD_SCI_USA

P_ROY_SOC_B−BIOL_SCI

SCIENCE

SIAM_J_APPL_MATH

SOFT_MATTER

STOCH_PROC_APPL

THEOR_ECOL−NETH

THEOR_POPUL_BIOL

TRENDS_ECOL_EVOL

NMDS axis 1

NM

DS

axi

s 2 cat3

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biologychemistryeco_evo_behavgeneralmathmathbiophysics


Detectability


Biology + math biology only

AM_J_PHYSIOL−HEART_C

AM_NAT

ANIM_BEHAV

BBA−BIOMEMBRANES

BEHAV_ECOLBEHAV_ECOL_SOCIOBIOL

BIOCHEMISTRY−US

BIOINFORMATICS

BIOPHYS_J

BIOSYSTEMS

BMC_BIOINFORMATICS

BMC_EVOL_BIOL

BMC_GENOMICS BMC_SYST_BIOL

B_MATH_BIOL

CANCER_RES

CELLCIRC_RES

CURR_OPIN_STRUC_BIOL

ECOLOGYECOL_LETT

ECOL_MODEL

EVOLUTION

GENETICS

INT_J_BIOMATH

J_BIOL_CHEM

J_BIOL_SYST

J_EVOLUTION_BIOL

J_EXP_BIOL

J_IMMUNOL

J_MATH_BIOL

J_MOL_BIOLJ_NEUROPHYSIOLJ_NEUROSCI

J_PHYSIOL−LONDON

J_THEOR_BIOL

J_VIROL

MATH_BIOSCIMATH_BIOSCI_ENG

MOL_BIOL_EVOL

MOL_BIOSYST

MOL_ECOL

NATURE

NAT_GENET

NAT_NEUROSCI

NAT_REV_GENET

NEURAL_COMPUT

NEUROIMAGE

NEURON

NUCLEIC_ACIDS_RES

OIKOS

PHILOS_T_R_SOC_B

PHYS_BIOL

PLOS_COMPUT_BIOL

PLOS_GENET

PLOS_ONE

PROG_BIOPHYS_MOL_BIO

PROTEINSPROTEIN_PEPTIDE_LETT

P_NATL_ACAD_SCI_USA

P_ROY_SOC_B−BIOL_SCI

SCIENCE

THEOR_ECOL−NETH

THEOR_POPUL_BIOL

TRENDS_ECOL_EVOL

NMDS axis 1

NM

DS

axi

s 2

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biochemistryvirologymolbiocellbioimmunologyneurobiomedicinephysiologybioinformaticsgeneticsevolutioneeecologybehaviorbiologygeneralmathbio


Detectability


Good, IJ, 1979. Biometrika, 66(2):393–396. ISSN 0006-3444. doi:10.2307/2335677. URLhttp://www.jstor.org/stable/2335677.

Platt, JR, 1964. Science, 146:347–353. ISSN 00368075. URL http://links.jstor.org/sici?sici=0036-8075%2819641016%293%3A146%3A3642%3C347%3ASI%3E2.0.CO%3B2-K.


Detectability

http://www.jstor.org/stable/2335677

http://links.jstor.org/sici?sici=0036-8075%2819641016%293%3A146%3A3642%3C347%3ASI%3E2.0.CO%3B2-K

http://links.jstor.org/sici?sici=0036-8075%2819641016%293%3A146%3A3642%3C347%3ASI%3E2.0.CO%3B2-K

MBRS detectability talk

Science

Transcript of MBRS detectability talk