Seminar 3 Data requirements, limitations, and challenges: Inverse modeling of seed and seedling...
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Transcript of Seminar 3 Data requirements, limitations, and challenges: Inverse modeling of seed and seedling...
Seminar 3
Data requirements, limitations, and challenges:
Inverse modeling of seed and seedling dispersal
Likelihood Methods in Forest Ecology
October 9th – 20th , 2006
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…)
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
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
Poisson PDF
0.0
0.1
0.2
0.3
0 5 10 15 20 25 30
x
Pro
b(x
)
m = 2.5
m = 5
m = 10
The Probability Model
Observations consist of counts Assume the counts are either Poisson
or Negative Binomial distributed
Poisson PDF:
!)(
x
exprob
x
Where x = observed density (integer), and = predicted density (continuous)
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)
nk
km
m
k
m
!n)k(
)nk()nX(p
1
This is the notation for the gamma function…
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
)()1
i
N
ii distgsizef
2/m(# rain Seed
How does total seed production vary with tree size?
Common assumption: seed production is a function of DBH2 (following Ribbens et al. 1994)
30iDBH
STRi tree of production seed
where = 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?
How does seed rain vary with distance from a parent tree?
Two basic classes of functions are commonly used*:
- Monotonically declining (negative exponential):
- Lognormal:
2)ln(
21
)(
0
b
i
i
X
Xdist
edistg
*See Greene et al. (2004), J. Ecol. for a discussion…
idistedistg
)(
One more trick…
Normalizing the dispersal function [g(dist)] so that STR is in meaningful units…
idistedistg
1)(
Where is the “arcwise” (i.e. 360o) integration of the dispersal function
2)ln(
21
1
0.30)
0
1
b
i
n
i
i
X
Xdist
edbh
STR /m(# rain seed 2
Lognormal form:
Exponential form:
i
n
i
i diste
dbhSTR
/m(# rain seed 2
1
0.30)
1
So, the basic scientific models…
The Scientific Models
0.0
0.5
1.0
1.5
2.0
2.5
0 10 20 30 40
Distance
Lognormal
Exponential (apha = 2)
Anisotropy: does direction matter?
ip angleXXX cos(00
• For the lognormal dispersal function:
• Incorporate effect of direction from source tree on modal dispersal distance1:
1Staelens, J., L. Nachtergale, S. Luyssaert, and N. Lust. 2003. A model of wind-influenced leaf litterfall in a mixed hardwood forest. Canadian Journal of Forest Research.
2)ln(
21
1
0.30)
0
1
b
i
n
i
i
X
Xdist
edbh
STR /m(# rain seed 2
Shape of the wind direction effect
When would this matter? (just to increase goodness of fit and improve parameter estimation?)
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
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
0 10 20 30 40 50
Distance From 50 cm Parent Tree (m)
See
dlin
g D
ensi
ty (
#/m
2) Theoretical
25 m radius
50 m radius
100 m radius
Effects of Search Radius
Theoretical 25 m radius 50 m radius 100 m radius
Likelihood -528.79 -493.59 -492.55r2 0.83 0.84 0.84slope 0.96 0.99 1.00mean dispersal 24.59 28.64 27.95 25.21STR 500 1449.51 487.26 462.77alpha 2 1.63 2.17 2.10X0 10 15.25 11.60 11.45Xb 0.6 0.50 0.59 0.56
0
0.5
1
1.5
2
2.5
3
3.5
0 10 20 30 40 50
Distance from a 50 cm Parent Tree (m)
See
d R
ain
(#/m
2)
Theoretical
25 m radius
50 m radius
100 m radius
50 m edge
100 m edge
Sampling along the edge...
Theoretical 25 m radius 50 m edge
Likelihood -528.79 -560.23r2 0.83 0.71slope 0.96 1.04mean dispersal 24.59 28.64 23.95STR 500.00 1449.51 4384.73alpha 2.00 1.63 0.68X0 10.00 15.25 13.09Xb 0.60 0.50 0.52
What if all of the trees are uniformly spaced?
This produces relatively similar neighborhoods for all observations…
Random vs. strategic sampling…
Theoretical 10/ha 50/ha 100/ha 500/har2 0.14 0.25 0.65 0.45mean dispersal distance 36.83 24.51 30.99 39.93 29.39STR 1000 1551.22 1241.98 1723.11 1402.88alpha 2 1.47 1.81 1.53 1.62X0 15 14.01 16.30 12.41 15.20Xb 0.6 0.54 0.55 0.71 0.53
0.0
0.5
1.0
1.5
2.0
2.5
0 25 50 75
Distance from a 50 cm DBH tree
See
d r
ain
(#/
m2)
Theoretical
density = 10/ha
density = 50/ha
density = 100/ha
density = 500/ha
Goodness of Fit – Sites with Different Densities
Predicted Seed Rain
0 50 100 150 200 250 300
Obs
erve
d S
eed
Rai
n
0
50
100
150
200
250
300
10/ha50/ha100/ha500/ha
What if all of the trees are the same size?
Tradeoffs between STR and
30iDBH
STRi tree of production seed
0.0
1.0
2.0
3.0
4.0
5.0
0 5 10 15 20
X
Y
xy = a(a=2)
What if not all trees produce seed?
0
0.1
0.2
0.3
0.4
0.5
0.6
0 10 20 30 40 50Distance From 30 cm Parent Tree (m)
See
dlin
g D
ensi
ty (
#/m
2) Theoretical - 50%reproductive
100 m radius
50 m radius
25 m radius
Theoretical 100 m radius 50 m radius 25 m radiusLikelihood -563.80 -564.24 -595.97r2 0.73 0.73 0.73slope 1.00 0.99 0.96MDD 24.59 25.95 31.83 28.36STR 500 444.21 479.13 1445.32alpha 2 2.17 2.27 1.65X0 10 10.70 10.14 15.38Xb 0.6 0.60 0.68 0.49
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 25 50 75
Distance from a 50 cm DBH Parent Tree (m)
See
d R
ain
(#/m
2)
Theoretical
Lognormal - 100 m
Weibull - 100 m
2Dt - 100 m
Theoretical Lognormal Weibull 2DtLikelihood -492.548 -500.289 -499.061r2 0.839 0.819 0.821slope 1.000 0.993 0.998mean dispersal 24.59 25.213 25.278STR 500 462.772 411.536 426.169alpha 2 2.103 2.218 2.183X0 10 11.454Xb 0.6 0.562
Can the data discriminate between different functions?
Beware of simplifying assumptions in your model...
Effect of alpha on fecundity
0
200
400
600
800
1000
0 25 50 75 100DBH (cm)
Tota
l See
d P
rod
uct
ion alpha = 0
alpha = 1
alpha = 2
alpha = 3
...0.30
)1
n
i
i2 dbh STR/m(# rain seed
= 3 = 2 = 1Estimated Theoretical Estimated Theoretical Estimated Theoretical
r2 0.96 0.84 0.83
STR 1056.04 1000 487.26 500 952.89 1000alpha 3.08 3 2.17 2 1.22 1
X0 10.71 10 11.60 10 11.65 10Xb 0.62 0.6 0.59 0.6 0.59 0.6
Parameter Estimation – Varying
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)
min
min
1
0
...0.30
)
dbhdbh
dbhhSTR if dbs
dbhs
i
i
n
i
i
if
/m(# rain seed 2
How could we determine the effective minimum reproductive size?
Parent size and seedling production in a Puerto Rican rainforest
Source: Uriarte et al. (2005) J. Ecology
Scaling reproductive output to tree size:
Maximum likelihood parameter estimates
Species min. size (cm) Casearia arborea 0.14 13.7Dacryodes excelsa 0.51 NAGuarea guidonia 2.06 48.13Inga laurina 2.38 16.39Manilkara bidentata 0.01 44.04Prestoea acuminata 0.15 13.89Schefflera morototoni 3.22 9.61Sloanea berteriana 1.70 11.06Tabebuia heterophylla 0.01 20.93
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 93:291-304.
Should there be an “intercept” in the model?
Allowing for long-distance dispersal via a “bath” term:
...0.30
)1
n
i
idbhSTRb /m(# rain seed 2
Where b is an average input of seeds even when there are no parents in the
neighborhood…
For seedlings: does light influence germination?
0 < M(GLI) < 1Lopt
Lhi = slope away from Lopt
Llo = slope to Lopt
...0.30
*)()1
n
i
idbhSTRGLIm /m(# rain seed 2
Light Availability
Light level (GLI)
0 10 20 30 40 50
Ligh
t m
ultip
lier
0.0
0.2
0.4
0.6
0.8
1.0
Casearia arboreaGuarea guidoniaManilkara bidentataPrestoea acuminataSchefflera morototoniInga laurinaTabebuia heterophylla
Is there evidence of density dependence in seedling
establishment?
Add yet another multiplier...
Seedlings.edPrexpcruitsReActual C
δ
0
0.2
0.4
0.6
0.8
1
1.2
Conspecific seedling density
DD
Eff
ect (
0-1)
Negative Conspecific Density Dependence
Seedling density (#/sq. meter?)
0 20 40 60 80
Sur
viva
l
0.0
0.2
0.4
0.6
0.8
1.0
Casearia arboreaDacryodes excelsaGuarea guidoniaManilkara bidentataPrestoea acuminataSchefflera morotoniTabebuia heterophylla
Spatial autocorrelation
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
How do you determine whether this is true?
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
So, examine your residuals for spatial autocorrelation
Distance class (m)
Mor
an’s
I
Weinmannia racemosa
-0.2
0
0.2
0.4
0.6
0.8
1
0 10 20 30 40 50
Total PredictedTotal ObservedResidual
A “best-case” species…
Examples from a study of seedling recruitment in a New Zealand rainforest (data from Elaine Wright)
Another species…
Nothofagus menziesii
-0.2
0
0.2
0.4
0.6
0.8
1
0 10 20 30 40 50
Distance Class (m)
Mo
ran
's I
Total Predicted
Total Observed
residual
A worse case…
Distance class (m)
Mor
an’s
I
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?