Paul: Do you know enough about Mixing Models to present it to the class?
yes Paul: Good. Paul: Do you know enough about Mixing Models to present it to the class? no Paul: Well, time to learn.
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.
Mixing Models Concentration Dependence & Incorporating Uncertainty
Mixing Models Concentration Dependence & Incorporating Uncertainty
Minimum Convex Hull
50% 50% 50% 50% 50% 50% 50% 50% Linear Mixing Model Along the Hull, all but 2 diet sources can be eliminated
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
Assumptions
Assumptions Linearly Independent Equations
Assumptions Linearly Independent Equations Variations in C & N isotopes are typically generated by different processes, so this is okay
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
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
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
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
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
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
More Assumptions
More Assumptions C & N isotopes from all dietary sources must be completely homogenized
More Assumptions C & N isotopes from all dietary sources must be completely homogenized Routing is not taking place
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!
What is Concentration Dependence? vs.
C:N C:N C:N 50%
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
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
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
Rearrange equations, then Plug & Chug Still okay- a system of 3 equations and 3 unknowns F = A -1 B
Nonlinearity: fairly easy to imagine: High [N] Low [N]
Does a bear Does it make a difference?
Incorporating Uncertainty
Incorporating Uncertainty Interpreting results
Incorporating Uncertainty Across Populations, Individuals Interpreting results
Incorporating Uncertainty Across Populations, Individuals Process Errors Interpreting results
Incorporating Uncertainty Across Populations, Individuals Process Errors Fractionation Errors (Trophic, Tissue) Interpreting results
Incorporating Uncertainty Across Populations, Individuals Process Errors Fractionation Errors (Trophic, Tissue) Natural Variability Interpreting results
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
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
The Model Calculate probability distributions for fi Via a Bayesian paradigm and numerical analysis
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
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
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
Likelihood of fq given prior information is calculated Finally: unnormalized posterior probability:
Great- for a random draw. Now what? Use a Sampling-Importance-Resampling algorithm to converge to the correct posterior distribution...
Does it work?
Over variable uncertainty
The End.
The End.