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Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA [email protected]
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Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA [email protected]

Apr 30, 2018

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Page 1: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

Introduction to (Bayesian) Isotope Mixing Models

Eric Ward

Northwest Fisheries Science Center Seattle WA, USA

[email protected]

Page 2: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

Isotopes & mixing models

• Overview of Bayesian jargon • Introduction to stable isotope mixing models

– Assumptions, data, etc

• Examples: • Using an informative prior • Accounting for hierarchical variation

– Grey wolves in B.C. • Dealing with prior information / small samples

– Trout in Pacific NW freshwater lakes • Including covariates • Including relative abundance data

Page 3: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

Bayesian / frequentist differences

• Frequentist / likelihood methods assume parameters are fixed, and the data are random

• Bayesians assume that the data are fixed, and the parameters are random

• (all the parameters are probability distributions)

Page 4: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

Bayesian Jargon

• Prior: π(θ) – this is a probability distribution representing prior knowledge

• Likelihood: L(x|θ) – this is the likelihood of your data given the parameter

• Posterior = Pr(θ|x) ≈ π(θ)×L(x|θ)

Page 5: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

Prior

Posterior

Likelihood

= how much we ‘learn’ by observing the data

Page 6: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

Choosing priors for mixing models

• Uninformative – Univariate

• Uniform(0,1) or Beta(1,1) = “flat priors”

– Multivariate • Dirichlet (1,1,1,…) = flat priors on each proportion

• Informative – Univariate

• Beta (3, 2), Beta (1, 8), etc.

– Multivariate • Dirichlet (3,2,3), Dirichlet (2,1,1)

Page 7: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

Stable Isotopes (in brief) • Isotope: atomic elements come in different flavors (# neutrons)

• Stable isotopes are naturally occurring isotopes of elements. • Examples of stable isotopes:

• Nitrogen (15N/14N) • Carbon (13C/12C) • Sulfur (34S/32S) • Strontium, Oxygen, etc.

• Can quantify ratio of isotopes using mass spectrometry

• Isotope ratios (15N/14N) are conserved in predictable ways through

trophic transition!

• Mixing models are a useful tool for inferring the relative contributions of different source items to a mixture

Page 8: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

Mixing Models • If we know:

– The isotope signatures of the things you eat

– Your isotopic signature

• We can estimate the proportions of different food items in your diet

Red paint white paint

pink paint (the mixture!)

Page 9: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

“When you look at the isotopic ratios, we are corn chips with legs” -T. Dawson, UC Berkeley

Page 10: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

C3 v C4 plants in human diet

d13C

C4 (Corn) C3 (Grasses, veg. crops)

USA

Page 11: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

What data do we need to use mixing models?

• Isotopic signatures of consumers • Isotopic signatures of sources

– Either individual points, or means and variances

• Values for TEFs / fractionation (means & vars) – Ideally from experiments – But almost always from the literature

• Post, D.M. 2002. Using stable isotopes to estimate trophic position: models, methods, and assumptions. Ecology, 83:703-718.

• This is one of the BIG weaknesses of mixing models

• Optional external information: gut contents, observations, etc

Page 12: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

C

Isot

ope

1 (e

.g. N

)

Isotope 2 (e.g. C)

Predator data and the isotope signatures of 4 prey items: the raw data

Consumer signatures trophic level higher

Page 13: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

Step 1: Correct source means and variances for fractionation / TEFs

Moore & Semmens (2008)

These represent the sample variances

These represent the sample means

Page 14: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

Illustration of correction • Sources without & with TEFs: shifted and

more variable

Page 15: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

Step 2: Identify potential problems

Isotope 1

(1-p) = 66% p = 33%

• (In this 1st example, there’s nothing wrong)

• In 1-dimension, this is just a weighted average (X1 and X2 are known):

u = p*X1 + (1-p)*X2

Source 1 Source 2

Consumer

6 9 8

Page 16: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

• Q: What is the value of mixing proportion p?

1-dimensional mixing model

• What if our consumer had a different value?

???? Source 2 Source 1

6 9 2

Isotope 1

• A: No positive value exists!! – In general, this means our data are inconsistent with

the assumptions of the model (missing source?)

Page 17: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

1-dimensional mixing model

Isotope 1

p2 p1

• Similarly, what if we had data for a new source?

• Q: How would this affect our estimate of p1?

Source 1 Source 2

Consumer

6 9 8

Source 3

3

p3

Page 18: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

Estimates of p[1] and p[3] are inversely correlated

Page 19: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

Another diagnostic: posterior distributions of estimated source contributions have multiple modes (translation: predator either eats a lot, or none of item X) - The posteriors of p[1] and p[3] will both look like this!

Page 20: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

1-dimensional mixing model

Isotope 1

p2 p1

• Similarly, what if we had data for a new source?

• Q: How would this affect our estimate of p1?

• A: Sources 1/3 totally confounded – Solution: (1) combine sources, or (2) include data from

another isotope / dimension, (3) informative priors – e.g. stomach contents, (4) bounds on parameters

Source 1 Source 2

Consumer

6 9 8

Source 3

3

p3

Page 21: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

Geometry in 2D can also be a problem

• Phillips and Gregg (2003)

Page 22: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

2D examples of bad geometry

• Always plot your data!

• Are consumers within source polygon? This tends to be a dealbreaker!

• Are any estimates potentially confounded? Uncertainty generally increases with # sources (Phillips & Gregg 2003)

Page 23: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

Step 3: Running a mixing model

• (assuming you get past step 2) – Bad geometry = bad results!

• What are the basic assumptions? (1) all consumers have the same diet (2) all sources are equally available (in terms of biomass) (3) no prey missing (4) source parameters known exactly (large sample sizes, > 10)

Page 24: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

Isot

ope

1

Isotope 2

Predator data and the isotope signatures of 4 prey items (corrected for fractionation)

f = [20%, 50%, 20% 10%]

Page 25: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

Isot

ope

1

Isotope 2

f = [20%, 50%, 20% 10%]

There are many probable f vectors!!! MCMC will give us 1000s!

Page 26: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

MixSIR: A Bayesian Framework For Isotopic Mixing Models

• Deal with any number of sources

• Incorporate prior information (e.g. gut contents)

http://www.ecologybox.org

Page 27: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

Source: MixSIR manual, ecologybox.org

Page 28: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

All the data together

Page 29: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

What does MixSIR do?

• Implements the SIR algorithm – Sampling Importance Resampling

• Generates 1000s of independent samples from the posterior distribution of estimated source contributions (you specify # of samples)

• All vectors have to sum to 1!

Page 30: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

In practice (Moore & Semmens 2008)

• Samples from MixSIR

• Uninformative (flat) priors

• Fishes look slightly problematic. Multiple modes??

Page 31: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

• Solution: informative prior from gut contents

Page 32: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

Other examples

• Individual and group variation

• Uncertainty in sources

Page 33: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

Coping with hierarchical variation in stable isotope mixing models

Semmens, B.X.*, E.J. Ward*, J.W. Moore, and CT. Darimont. 2009. Quantifying inter- and intra-population niche variability using hierarchical Bayesian stable isotope mixing models. PLoS One. http://dx.plos.org/10.1371/journal.pone.0006187. * Equal authorship

Darimont Darimont Darimont Darimont Darimont

Page 34: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

Step 3: Running a mixing model

• (assuming you get past step 2) – Bad geometry = bad results!

• What are the basic assumptions? (1) all consumers have the same diet (2) all sources are equally available (in terms of biomass) (3) no prey missing (4) source parameters known exactly (large sample sizes, > 10)

Page 35: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

No uncertainty in prey proportions…

• This assumes that all predators eat the same thing.

• Different animals may have different diets!

f = [20%, 50%, 20% 10%]

Page 36: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

Isot

ope

1

Isotope 2

Instead of this…

f = [20%, 50%, 20% 10%]

Page 37: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

Isot

ope

1

Isotope 2

Consumers look like this…

f = [20%, 50%, 20% 10%]

Page 38: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

Application: Coastal Gray Wolves

Outer Islands

Inner Islands

Mainland

Pack 1

Pack n

Individual 1

Pack n

… N = 9 N = 64

REGION PACK INDIVIDUAL

Darimont

Darimont

Page 39: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

more terrestrial

more marine

d13C (0/00)

d15N

(0 /00

)

low high

low

hi

gh

Measure of resource use - niche

Mea

sure

of r

esou

rce

use

- nic

he

Diet of coastal wolves….

Darimont

Darimont

Darimont

Darimont

Page 40: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov
Page 41: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov
Page 42: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov
Page 43: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov
Page 44: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov
Page 45: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

DIC Results Region Pack Individual DIC

N N N 1343

Y N N 693

Y Y N 502

N N Y 335

Y N Y 333

Y Y Y 325

Page 46: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov
Page 47: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

How to quantify niche?

• How variable is your isotopic signature?

• How variable are the resources you consume? – Dietary niche, we can output of the hierarchical

mixing model to get this

Page 48: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

d13C (0/00)

d15N

(0 /00

)

low high

low

hi

gh

Measure of resource use - niche

Mea

sure

of r

esou

rce

use

- nic

he

Stable Isotope Analysis (e.g. Bayesian ellipses) •data derived from tissue

Measuring Niche Variation

Page 49: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

standard deviation parameters controlling the variation in diet across three scales

Page 50: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

Other extensions to MixSIR

• Improving uncertainty in sources – Source signatures may be based on a small

number of samples

– We may have data from other systems to inform these

– In addition to treating the proportional contributions as parameters, the source means and variances also become parameters

Page 51: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

Dealing with source uncertainty

Ward, E.J., Semmens, B.X., and D.E. Schindler. 2010. Including source uncertainty and prior information in the analysis of stable isotope mixing models. Environmental Science & Technology, 44(12): 4645-4650

Figure 5: Map of Urban Lakes Gradient locations.

Page 52: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

I Isot

ope

1

Isotope 2

Graphically:

f = [20%, 50%, 20% 10%]

Page 53: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

Design Subject of inference: 7 pred in 6 pristine lakes

One lake has no pred:

2 additional lakes: stocked pred

Page 54: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

Design Subject of inference: 7 fish in 6 pristine lakes

One lake has no fish:

2 additional lakes: Stocked fish

PRIOR

Page 55: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

Estimated proportions

Page 56: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

Other extensions to MixSIR

• Including residual error / uncertainty

– Extra variation not accounted for by your model, or the variation among sources

Page 57: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

SIAR package in R

• Good if you don’t have identifying information about individuals (groups, etc)

• May help estimation if you’re missing a minor source

• Being fused with MixSIR = MixSIAR

Page 58: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

Also in SIAR()

• Concentration dependence:

• MixSIR assumes contribution of C/N is the same for all sources

• Some prey may be C/N enriched or depleted

Page 59: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

Other extensions to MixSIR

• Including covariates

– Consumers might not be assigned to groups, but experience a gradient of some covariate

Page 60: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

This example is 1D, but could easily be extended to 2D

Page 61: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

Other extensions to MixSIR

• Including relative prey availability

– All sources might not be equally available

Page 62: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

• Prey might be isotopically identical, but have very different abundances

Page 63: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

• Density varies by an order of magnitude

• Yeakel et al. re-weight posterior after running MixSIR

Page 64: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

Other extensions to MixSIR

• Ontogenetic diet shifts

– Consumers shift diet as they get older / bigger

– Collaboration w/C. Harvey (in prep)

Page 65: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

Other extensions to MixSIR

• Incorporating movement and the Isoscape

– Isotopic signatures of consumers reflect environment they inhabit

– We can infer where animals have been based on signatures

– Collaboration w/ J. Moore & C. Phillis

Page 66: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

Summary

• Mixing models are a rapidly growing field

• Analyses are exclusively Bayesian – Requires some basic understanding of underlying

methods / techniques

Page 67: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

Where Can I Get These Glorious Tools?

• http://www.ecologybox.org

• Tools include: – R code to run

– Worked examples

– Step-by-step code explanations

Page 68: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

isoecol.blogspot.com

Page 69: Introduction to (Bayesian) Isotope Mixing Models · Introduction to (Bayesian) Isotope Mixing Models Eric Ward Northwest Fisheries Science Center Seattle WA, USA eric.ward@noaa.gov

Acknowledgements

NOAA Seattle • Brice Semmens • Eli Holmes • Mark Scheuerell • Eric Buhle • Tessa Francis UC Santa Cruz • Jon Moore • Chris Darimont UW • Daniel Schindler • Gordon Holtgrieve • Tessa Francis

• Additional collaborators • Andrew Jackson • Stu Bearhop • Andrew Parnell • Rich Inger • Don Phillips