Mothematical Modeling: Temporal and Spatial Models of Moth Distribution at the H.J. Andrews Experimental Forest.  .
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Erin Childs (Pomona College) , Andrew Calderon (Heritage University), Evan Goldman (Bard College, Boston University), Molly O’Neill (Lehigh University), Clay Showalter (Evergreen University), with the help of Olivia Poblacion (Oregon State University)
Dr. Dietterich, CS Professor
Dr. Wong, CS Professor
Steven Highland, Geosciences PhD Candidate
Jorge Ramirez, Math Professor
Dan Sheldon, CS Postdoc
Julia Jones, Geosciences Professor
Rebecca Hutchinson, CS Postdoc
JavierIllan, PhD, Postdoc
How is vegetation related to moth species distribution and composition?
How does climate affect moth phenology?
If huckleberry (VAHU) is found at a site, what is the probability of finding thimbleberry (RUPA) but not licorice root (!LIGR) at that site?
Canonical correlations analysis aims at highlighting correlations between two data sets
Gives us a way of making sense of crosscovariance matrices
Allows ecologists to relate the abundance of species to environmental variables
Using CCA we analyzed our vegetation data and moth data
Highlights any correlations among only moth species
(422x422)
Ycorrelation:
Highlights any correlations among only plant species
(71x71)
Crosscorrelation:
Highlights any correlations between both data sets
(71x422)
Thermal Climate of the H.J. Andrews Experimental Forest
PRISM estimated mean monthly maximum and minimum temperature maps showing topographic effects of radiation and sky view factors. Provided by Jonathan W. Smith
Use a linear regression model to interpolate the degree for a given trap site for specific days of a year
Parameterize temperature in order to later be included in the temporal model
Produce degree day curves for any trap site
MultiLinear Regression Analysis
Find Coefficients
Each Trap_ID will have two sets of coefficients (Maximum and Minumum)
In goes the trap_ID,
start_date and end_date
Out comes the min and max for the given day(s)
t1
t2
t3
Consider 3 trapping times and 4 associated intervals, and moths with flight times as follows
I0
I1
I2
I3
This gives us a distribution table:
This gives us a distribution table … and flight counts
This gives us a distribution table … and flight counts
This gives us a distribution table … and flight counts
When trapping moths, all we see is flight counts
Given flight counts, we want to predict moth distribution
Maximize Prob (Data  Parameters)
Data = Moth trapping
moths trapped: f=(f1, f2, … fT)
times trapped: t=(t1, t2, … tT)
Emergence ~ N(µE, σE)
Life Span ~ N(µS, σS)
Parameters = probability distribution of emergence time and life span
Emergence and life span assumed to be Gaussian with parameters µE, σE, µS, σS
tk
tk+1
tj+1
r
s
d
Moth Distribution…
Ik
Ij
Use distributions to calculate p(j,k), the probability of a moth emerging in interval j and dying in interval k
All moths fall into one of the probability squares
Moths have a multinomial distribution
Approximate this with a multivariate Gaussian (or normal)
What is the error associated with this approximation?
approximated as m!=s(m)
Error of
CCA and Hamming distance shows a strong correlation between vegetation and moth species
For the Future: Vegetation surveys at other trap sites would help improve the performance of the model
Moth emergence shows a strong correlations to the local temperature
For the future: incorporating the degree day curves we calculated for each site will make the model more robust