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Mothematical Modeling: Temporal and Spatial Models of Moth Distribution at the H.J. Andrews Experimental Forest. - .

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slide1

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)

acknowledgements
Acknowledgements

Dr. Dietterich, CS Professor

Dr. Wong, CS Professor

Steven Highland, Geosciences PhD Candidate

Jorge Ramirez, Math Professor

Dan Sheldon, CS Post-doc

Julia Jones, Geosciences Professor

Rebecca Hutchinson, CS Post-doc

JavierIllan, PhD, Post-doc

studying climate change lepidoptera
Studying Climate Change: Lepidoptera
  • Why are Lepidoptera are good indicator of climate change?
  • Past studies on Lepidoptera
    • Woiwod 1996: Detecting the effects of climate change on Lepidoptera
    • Dewar and Watt 1992: Predicted changes in the synchrony of larval emergence and budburst under climatic warming
research questions
Research Questions

How is vegetation related to moth species distribution and composition?

How does climate affect moth phenology?

study site
Study Site

H.J. Andrews Experimental Forest

http://andrewsforest.oregonstate.edu/about.cfm?topnav=2

vegetation surveying methods
Vegetation Surveying: Methods
  • GPS coordinates
  • Walked out 30m and 100m radius in all directions
  • Presence/absence of 71 species of known host plants
moth trapping methods
Moth Trapping: Methods
  • Moth Trapping
    • 9 sites selected
    • Equipment used
    • Moth preservation
methods
Methods
  • Moth Identification
moth trapping results
Moth Trapping Results

Semiothissignaria

Perooccidentalis

overview is vegetation a good predictor of moth species presence absence
Overview: Is vegetation a good predictor of moth species presence/absence?
  • Develop software tools for exploring/analyzing data
  • Run generalized boosted regression models (GBMs) for each moth species
  • Create GIS layers for the predicted locations of each moth species
software tasks for data exploration
Software Tasks for Data Exploration
  • Format data
  • Compare the similarities and differences between sites, moths and vegetation
  • Discover correlations between vegetation and moth species
  • Calculate marginal probabilities of plant occurrences
  • Visualize results
measuring similarity hamming distance
Measuring Similarity: Hamming Distance
  • Hamming distance is the number of co-variates that differ between sample sets
  • Smaller number means sets are more similar
marginal probabilities
Marginal Probabilities
  • Using the vegetation data collected at 20 sites, generate marginal probabilities for plants occurrences

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 correlation analysis cca
Canonical Correlation Analysis (CCA)

Canonical correlations analysis aims at highlighting correlations between two data sets

Gives us a way of making sense of cross-covariance matrices

Allows ecologists to relate the abundance of species to environmental variables

Using CCA we analyzed our vegetation data and moth data

slide18

X-correlation:

Highlights any correlations among only moth species

(422x422)

Y-correlation:

Highlights any correlations among only plant species

(71x71)

Cross-correlation:

Highlights any correlations between both data sets

(71x422)

generalized boosted regression models gbms
Generalized Boosted Regression Models (GBMs)
  • Regression analysis allows us to explore the relationships between individual moth presence/absence (dependent variable) and various characteristics of each site (independent variables)
  • The goal is to minimize the loss function, which represents the loss associated with an estimate being different from the true value
  • Basis functions are an element of a set of vectors that, in linear combination, can represent every vector in a given vector space
  • Every function can be represented as a linear combination of basis function
  • Boosting is the process of iteratively adding basis functions in a greedy fashion so that each additional basis function further reduces the selected loss function
  • The model is run several times with different values for the tuning parameters to determine the best values
validating the gbm
Validating the GBM
  • All available regressors are used in the model, meaning that the choice of independent variables is not supported by theory
  • The standard approach to validating models is to split the data into a training and a test data set
  • The model is fit on the training data, then used to make predictions on the test data
  • This ensures that the model is generalizableand not overfit
running the model
Running the Model
  • Ran the model for individual moth species using all 256 trap sites at HJA, using moth trapping data collected from 2004 to 2008
  • Did not include vegetation data, since we only collected it at 20 sites
  • The GBM lays a grid over the Andrews forest and calculates the predicted probability of the moth species being present for each grid cell
slide24

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

degree day curve
Degree Day Curve

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

slide28

Multi-Linear Regression Analysis

Find Coefficients

Each Trap_ID will have two sets of coefficients (Maximum and Minumum)

predicting daily temp
Predicting Daily Temp
  • Linear Interpolation
    • Fill gaps in the daily temperature data

In goes the trap_ID,

start_date and end_date

Out comes the min and max for the given day(s)

the problem
The Problem
  • Year-round distribution of moths
  • Limited observation points
  • Unseen, unmeasurable data
    • Catching probabilities
    • Total moth population
example flight times
Example: Flight times

t1

t2

t3

Consider 3 trapping times and 4 associated intervals, and moths with flight times as follows

I0

I1

I2

I3

example distribution
Example: Distribution

t1

t2

t3

This gives us a distribution table:

I0

I1

I2

I3

example distribution1
Example: Distribution

t1

t2

t3

This gives us a distribution table:

I0

I1

I2

I3

example distribution2
Example: Distribution

t1

t2

t3

This gives us a distribution table:

I0

I1

I2

I3

example distribution3
Example: Distribution

t1

t2

t3

This gives us a distribution table:

I0

I1

I2

I3

example distribution4
Example: Distribution

t1

t2

t3

This gives us a distribution table:

I0

I1

I2

I3

example distribution5
Example: Distribution

This gives us a distribution table:

example con t
Example con’t

This gives us a distribution table … and flight counts

example con t1
Example con’t

This gives us a distribution table … and flight counts

example con t2
Example con’t

This gives us a distribution table … and flight counts

example flight counts
Example: Flight Counts

When trapping moths, all we see is flight counts

Given flight counts, we want to predict moth distribution

maximum likelihood model
Maximum Likelihood Model

Maximize Prob (Data | Parameters)

Data = Moth trapping

moths trapped: f=(f1, f2, … fT)

times trapped: t=(t1, t2, … tT)

maximum likelihood model1
Maximum Likelihood Model

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

moth distribution

tj

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

probability table
Probability Table

Emergence Interval

Death Interval

multinomial distribution
Multinomial Distribution

All moths fall into one of the probability squares

Moths have a multinomial distribution

Approximate this with a multivariate Gaussian (or normal)

approximation error
Approximation Error

What is the error associated with this approximation?

approximated as m!=s(m)

Error of

likelihood
Likelihood
  •  ={µE, σE, µS, σS}
likelihood surface
Likelihood surface

Log Loss

µs

µe

21 19 17 15 13 11 9 7 5 3 1

results
Results
  • SemiothisaSignaria
  • Trap 38B
  • 2005
results1
Results

R2 =0.23

p<0.01

how is vegetation related to moth species distribution and composition1
How is vegetation related to moth species distribution and composition?

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

how does climate affect moth phenology1
How does climate affect moth phenology?

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