Paul: Do you know enough about Mixing Models to present it to the class?

Transcription

1 Paul: Do you know enough about Mixing Models to present it to the class?

2 yes Paul: Good. Paul: Do you know enough about Mixing Models to present it to the class? no Paul: Well, time to learn.

3 yes Paul: Good. Paul: Do you know enough about Mixing Models to present it to the class? no Mixing Models. Paul: Well, time to learn.

4 Mixing Models Concentration Dependence & Incorporating Uncertainty

5 Mixing Models Concentration Dependence & Incorporating Uncertainty

6

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8 Minimum Convex Hull

9 50% 50% 50% 50% 50% 50% 50% 50% Linear Mixing Model Along the Hull, all but 2 diet sources can be eliminated

10 3 unknowns 3 equations All is right with the world. Analytical Mixing Models can only work if you us n isotope ratios to investigate n + 1 sources

11 Assumptions

12 Assumptions Linearly Independent Equations

13 Assumptions Linearly Independent Equations Variations in C & N isotopes are typically generated by different processes, so this is okay

14 Assumptions Linearly Independent Equations Variations in C & N isotopes are typically generated by different processes, so this is okay All potential dietary items are included

15 Assumptions Linearly Independent Equations Variations in C & N isotopes are typically generated by different processes, so this is okay All potential dietary items are included No more than n + 1 sources

16 Assumptions Linearly Independent Equations Variations in C & N isotopes are typically generated by different processes, so this is okay All potential dietary items are included No more than n + 1 sources Low variance

17 Assumptions Linearly Independent Equations Variations in C & N isotopes are typically generated by different processes, so this is okay All potential dietary items are included No more than n + 1 sources Low variance Pred must fall in mixing space

18 Assumptions Linearly Independent Equations Variations in C & N isotopes are typically generated by different processes, so this is okay All potential dietary items are included No more than n + 1 sources Low variance Pred must fall in mixing space Food sources must be different

19 Assumptions Linearly Independent Equations Variations in C & N isotopes are typically generated by different processes, so this is okay All potential dietary items are included No more than n + 1 sources Low variance Pred must fall in mixing space Food sources must be different Fractionations must be accounted for

20 More Assumptions

21 More Assumptions C & N isotopes from all dietary sources must be completely homogenized

22 More Assumptions C & N isotopes from all dietary sources must be completely homogenized Routing is not taking place

23 More Assumptions C & N isotopes from all dietary sources must be completely homogenized Routing is not taking place C:N ratios of all sources are equal!

24 What is Concentration Dependence? vs.

25 C:N C:N C:N 50%

26 C:N C:N C:N 50% Because has a higher [N] than Just a little will heavily bias N Nonlinear Isotopic ratios are carried along for the ride

27 Introduce Concentration Dependence for each Isotope Weight each isotopic input by the concentration unique to each source Fractions of assimilated BIOMASS of X, Y, Z in mix M

28 Model assumes that constribution of a food source to a consumer is proportional to the assimilated biomass * elemental concentration Fractional Contributions for each Element Last source (Z) is not independent: Can be solved by 1-(fx + fy): allows for reduction

29 Rearrange equations, then Plug & Chug Still okay- a system of 3 equations and 3 unknowns F = A -1 B

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31 Nonlinearity: fairly easy to imagine: High [N] Low [N]

32 Does a bear Does it make a difference?

33 Incorporating Uncertainty

34 Incorporating Uncertainty Interpreting results

35 Incorporating Uncertainty Across Populations, Individuals Interpreting results

36 Incorporating Uncertainty Across Populations, Individuals Process Errors Interpreting results

37 Incorporating Uncertainty Across Populations, Individuals Process Errors Fractionation Errors (Trophic, Tissue) Interpreting results

38 Incorporating Uncertainty Across Populations, Individuals Process Errors Fractionation Errors (Trophic, Tissue) Natural Variability Interpreting results

39 Incorporating Uncertainty Across Populations, Individuals Process Errors Fractionation Errors (Trophic, Tissue) Natural Variability Interpreting results These are not prob. distrib. Histograms are uninformative Only establishes ranges of possibilities

40 Incorporating Priors If there is known dietary data out there, it would make sense to inform isotopic data with other sources Gut contents Foraging Observations

41 The Model Calculate probability distributions for fi Via a Bayesian paradigm and numerical analysis

42 Randomly generate q proportional source contributions fq Sum of fi prey in each vector1 Derive isotopic distributions for the proposed mixture by solving for proposed means and st. dev fq = 0.1, 0.1, 0.8

43 Take a random proportion for prey i Multiply that proportion by the true values for prey i Result = proposed MIX based on the random proportion

44 Likelihood is determined by calculating the product of the likelihoods of each individual mixture isotope value Across each Xkj = j th isotope of k th mix true mix For each isotope

45 Likelihood of fq given prior information is calculated Finally: unnormalized posterior probability:

46 Great- for a random draw. Now what? Use a Sampling-Importance-Resampling algorithm to converge to the correct posterior distribution...

47 Does it work?

48 Over variable uncertainty

49 The End.

50 The End.