1. Seminar 3 - Inverse modeling
C. D. Canham
Case Study 4:
Data requirements, limitations, and challenges:
Inverse modeling of seed and seedling dispersal

2. Approaches to Estimation of Seed and Seedling Dispersal Functions
Seminar 3 - Inverse modeling
C. D. Canham
Approaches to Estimation of Seed and Seedling Dispersal Functions
Direct sampling around isolated trees (David Greene)
Develop mechanistic models with directly measurable parameters (Ran Nathan)
Inverse modeling using likelihood methods and neighborhood models (Eric Ribbens, Jim Clark, and a rapidly growing community of practitioners…)

3. Seminar 3 - Inverse modeling
C. D. Canham
The questions…
What are the shapes of the dispersal functions?
How does fecundity vary as a function of tree size?
What other factors determine the spatial distribution of seeds and seedlings around parent trees?
Wind direction (anisotropy)
Secondary dispersal
Density and distance - dependent seed predation and pathogens
Substrate conditions
Light levels

4. The basic approach: field methods
Seminar 3 - Inverse modeling
C. D. Canham
The basic approach: field methods
Map the distribution of potential parent trees within a stand
Sample the density of seeds or seedlings at mapped locations within the stand
Measure any additional features at the location of the seed traps or seedling quadrats

5. Seminar 3 - Inverse modeling
C. D. Canham

6. Seminar 3 - Inverse modeling
C. D. Canham
The Probability Model
Observations consist of counts
Assume the counts are either Poisson or Negative Binomial distributed
Poisson PDF:
Where x = observed density (integer), and l = predicted density (continuous)

7. Seminar 3 - Inverse modeling
C. D. Canham
Negative Binomial PDF
“shape” of the PDF controlled by both the expected mean (m) and a “shape” parameter (k)
As k varies, the distribution can vary from over- to under-dispersed (i.e. variance > or < mean)
This formulation would probably never be used, because taking the log to get log likelihoods converts to an equation with ln(gamma) terms, and these are much easier to compute…
This is the notation for the gamma function…

8. The basic “scientific” model
Seminar 3 - Inverse modeling
C. D. Canham
The basic “scientific” model
Seed rain at a given location is the sum of the input of N parent trees, with the input from any given tree a function of the:
Size (typically DBH) and
Distance to the parent

9. How does total seed production vary with tree size?
Seminar 3 - Inverse modeling
C. D. Canham
How does total seed production vary with tree size?
Common assumption: seed production is a function of DBH2 (following Ribbens et al. 1994)
where a = 2, and STR = total standardized seed production of a 30 cm DBH tree
Is this a reasonable assumption?
Is it supported by either independent data or theory?

10. How does seed rain vary with distance from a parent tree?
Seminar 3 - Inverse modeling
C. D. Canham
How does seed rain vary with distance from a parent tree?
Two basic classes of functions are commonly used*:
Monotonically declining (negative exponential):
Lognormal:
As written, both of these functions are bounded between 0 and 1, so they have the nice feature of acting like a fractional scalar…
*See Greene et al. (2004), J. Ecol. for a discussion…

11. Seminar 3 - Inverse modeling
C. D. Canham
One more trick…
Normalizing the dispersal function [g(dist)] so that STR is in meaningful units…
Where h is the “arcwise” (i.e. 360o) integration of the dispersal function

12. Seminar 3 - Inverse modeling
C. D. Canham
So, the basic scientific models…
Lognormal form:
Exponential form:

13. Seminar 3 - Inverse modeling
C. D. Canham
The Scientific Models

14. Anisotropy: does direction matter?
Seminar 3 - Inverse modeling
C. D. Canham
Anisotropy: does direction matter?
For the lognormal dispersal function:
Incorporate effect of direction from source tree on modal dispersal distance1:
Angle = angle from tree to trap
Delta = direction the wind is blowing from
Xp = magnitude of displacement from mean dispersal distance
1Staelens, J., L. Nachtergale, S. Luyssaert, and N. Lust A model of wind-influenced leaf litterfall in a mixed hardwood forest. Canadian Journal of Forest Research.

15. Shape of the wind direction effect
Seminar 3 - Inverse modeling
C. D. Canham
Shape of the wind direction effect
When would this matter?
(just to increase goodness of fit and improve parameter estimation?)

16. Potential Dataset Limitations
Seminar 3 - Inverse modeling
C. D. Canham
Potential Dataset Limitations
Censored data: not all parents are accounted for
Insufficient variation in predicted values: parents are too uniformly distributed
Two different populations treated as one: not all potential parents actually produce seeds
Lack of independence: spatial autocorrelation among nearby samples

17. Beware of simplifying assumptions in your model...
Seminar 3 - Inverse modeling
C. D. Canham
Beware of simplifying assumptions in your model...

18. Seminar 3 - Inverse modeling
C. D. Canham
Parameter Estimation – Varying a

19. What is the minimum size of a reproductive adult?
Seminar 3 - Inverse modeling
C. D. Canham
What is the minimum size of a reproductive adult?
Most studies have arbitrarily assumed that all adults over a low minimum size (10 – 15 cm DBH) contribute seeds.
One approach – estimate the minimum (don’t assume it)
How could we determine the effective minimum reproductive size?

20. Parent size and seedling production in a Puerto Rican rainforest
Seminar 3 - Inverse modeling
C. D. Canham
Parent size and seedling production in a Puerto Rican rainforest
Source: Uriarte et al. (2005) J. Ecology

21. Seminar 3 - Inverse modeling
C. D. Canham
Scaling reproductive output to tree size: Maximum likelihood parameter estimates
Species a min. size (cm)
Casearia arborea
Dacryodes excelsa NA
Guarea guidonia
Inga laurina
Manilkara bidentata
Prestoea acuminata
Schefflera morototoni
Sloanea berteriana
Tabebuia heterophylla
Source: Uriarte, M., C. D. Canham, J. Thompson, J. K. Zimmerman, and N. Brokaw.
2005. Seedling recruitment in a hurricane-driven tropical forest: light limitation, density-dependence and the spatial distribution of parent trees. Journal of Ecology 93:

22. Should there be an “intercept” in the model?
Seminar 3 - Inverse modeling
C. D. Canham
Should there be an “intercept” in the model?
Allowing for long-distance dispersal via a “bath” term:
Where b is an average input of seeds even when there are no parents in the neighborhood…

23. For seedlings: does light influence germination?
Seminar 3 - Inverse modeling
C. D. Canham
For seedlings: does light influence germination?
Lopt
Lhi = slope away from Lopt
Llo = slope to Lopt
0 < M(GLI) < 1

24. Seminar 3 - Inverse modeling
C. D. Canham
Light Availability

25. Is there evidence of density dependence in seedling establishment?
Seminar 3 - Inverse modeling
C. D. Canham
Is there evidence of density dependence in seedling establishment?
Add yet another multiplier... C δ
DD Effect (0-1)
Conspecific seedling density

26. Negative Conspecific Density Dependence
Seminar 3 - Inverse modeling
C. D. Canham
Negative Conspecific Density Dependence

27. Seminar 3 - Inverse modeling
C. D. Canham
Spatial autocorrelation
Spatial autocorrelation

28. Dealing with spatial autocorrelation among observations…
Seminar 3 - Inverse modeling
C. D. Canham
Dealing with spatial autocorrelation among observations…
Remember - the formula for calculating log-likelihood assumes that observations are independent…
We have been conditioned to assume that two observations taken at locations close together are likely to be not independent (a legacy of Stuart Hurlbert)
Moran’s I and other indices of spatial autocorrelation
Use blackboard to define independence in statistical terms… when dependent, P(A&B)=P(A)*P(B|A)
How do you determine whether this is true?

29. A critical distinction…
Seminar 3 - Inverse modeling
C. D. Canham
A critical distinction…
Remember – the issue is whether the residuals (the error terms in the probability model) are independent. NOT whether the raw observations are…
If your scientific model “explains” why two nearby observations have similar values, then the fact that they are similar is NOT evidence of lack of independence*…
*despite assertions to the contrary in some papers on the subject

30. So, examine your residuals for spatial autocorrelation
Seminar 3 - Inverse modeling
C. D. Canham
So, examine your residuals for spatial autocorrelation
A “best-case” species…
Moran’s I
Distance class (m)
Examples from a study of seedling recruitment in a New Zealand rainforest (data from Elaine Wright)

31. Seminar 3 - Inverse modeling
C. D. Canham
Another species…
A worse case…
Moran’s I
Distance class (m)

32. Causes and consequences of fine-scale spatial autocorrelation…
Seminar 3 - Inverse modeling
C. D. Canham
Causes and consequences of fine-scale spatial autocorrelation…
The causes are probably legion:
Many trees don’t produce seed in any given mast year,
Many factors can cluster input of seeds or survival of seedlings
The consequences are important but not fatal:
Generally very little bias in parameter estimates themselves,
But estimates of the variance of the parameters will be biased (low)
Do the thought experiment or test this with real data – what would happen if you duplicated some observations in the dataset and then redid the analysis?