Mothematical Modeling: - PowerPoint Presentation

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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)

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Acknowledgements

Dr. Dietterich

, CS Professor Dr. Wong, CS ProfessorSteven Highland, Geosciences PhD CandidateJorge

Ramirez, Math ProfessorDan Sheldon, CS Post-docJulia Jones, Geosciences Professor Rebecca Hutchinson, CS Post-doc

Javier Illan, PhD, Post-doc

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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 LepidopteraDewar and Watt 1992: Predicted changes in the synchrony of larval emergence and budburst under climatic warming

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Research Questions

How is vegetation related to moth species distribution and composition?

How does climate affect moth phenology?

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Study Site

H.J. Andrews Experimental Forest

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

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How is vegetation related to moth species distribution and composition?

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Vegetation Surveying: Methods

GPS coordinates

Walked out 30m and 100m radius in all directions

Presence/absence of 71 species of known host plants

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Moth Trapping: Methods

Moth Trapping

9 sites selected

Equipment usedMoth preservation

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Methods

Moth Identification

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Moth Trapping Results

Semiothis

signaria

Pero occidentalis

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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

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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

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Measuring Similarity: Hamming Distance

Hamming distance is the number of co-

variates

that differ between sample sets

Smaller number means sets are more similar

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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?

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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 matricesAllows ecologists to relate the abundance of species to environmental variables

Using CCA we analyzed our vegetation data and moth data

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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)

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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

Slide20

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

generalizable

and not

overfit

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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

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Visualizing GBM Results

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How does climate affect moth phenology?

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

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Degree Day CurveUse 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 modelProduce degree day curves for any trap site

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Find Coefficients

Each

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

Multi-Linear Regression Analysis

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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)

Slide30

Temporal Distribution of Moths

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The ProblemYear-round distribution of moths

Limited observation pointsUnseen, unmeasurable

dataCatching probabilitiesTotal moth population

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Example: Flight timesConsider 3 trapping times and 4 associated intervals, and moths with flight times as follows

t

1

t

2

t

3

I

0

I

3

I

2

I

1

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Example: Distribution

This gives us a distribution table:

I

0

I

1

I

2

I

3

I

0

0

0

0

0

I

1

0

0

0

0

I

2

0

0

0

0

I

3

0

0

0

0

t

1

t

2

t

3

I

0

I

3

I

2

I

1

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Example: Distribution

This gives us a distribution table:

I

0

I

1

I

2

I

3

I

0

0

0

0

0

I

1

0

1

0

0

I

2

0

0

0

0

I

3

0

0

0

0

t

1

t

2

t

3

I

0

I

3

I

2

I

1

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Example: Distribution

This gives us a distribution table:

I

0

I

1

I

2

I

3

I

0

0

1

0

0

I

1

0

1

0

0

I

2

0

0

0

0

I

3

0

0

0

0

t

1

t

2

t

3

I

0

I

3

I

2

I

1

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Example: Distribution

This gives us a distribution table:

I

0

I

1

I

2

I

3

I

0

0

1

0

0

I

1

0

1

0

1

I

2

0

0

0

0

I

3

0

0

0

0

t

1

t

2

t

3

I

0

I

3

I

2

I

1

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Example: Distribution

This gives us a distribution table:

I

0

I

1

I

2

I

3

I

0

0

1

1

0

I

1

0

1

0

1

I

2

0

0

0

0

I

3

0

0

0

0

t

1

t

2

t

3

I

0

I

3

I

2

I

1

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Example: Distribution

This gives us a distribution table:

I

0

I

1

I

2

I

3

I

0

1

2

4

1

I

1

0

2

3

3

I

2

0

0

1

2

I

3

0

0

0

1

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Example con’t

I

0

I

1

I

2

I

3

I

0

1

2

4

1

I

1

0

2

3

3

I

2

0

0

1

2

I

3

0

0

0

1

7

f

1

This gives us a distribution table

… and flight counts

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Example con’t

I

0

I

1

I

2

I

3

I

0

1

2

4

1

I

1

0

2

3

3

I

2

0

0

1

2

I

3

0

0

0

1

7

11

f

1

f

2

This gives us a distribution table

… and flight counts

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Example con’t

I

0

I

1

I

2

I

3

I

0

1

2

4

1

I

1

0

2

3

3

I

2

0

0

1

2

I

3

0

0

0

1

7

11

6

f

1

f

2

f

3

This gives us a distribution table

… and flight counts

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Example: Flight CountsWhen trapping moths, all we see is flight counts

Given flight counts, we want to predict moth distribution

7

116

f

1

f

2

f

3

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Maximum Likelihood Model

Maximize Prob (Data | Parameters)Data = Moth trapping

moths trapped: f=(f1, f2, … f

T) times trapped: t=(t1, t2, … tT)

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Maximum Likelihood Model

Parameters = probability distribution of emergence time and life span Emergence and life span assumed to be Gaussian with parameters µE

, σE, µS, σS

Emergence ~ N(µE, σ

E)Life Span ~ N(µS, σ

S

)

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Moth DistributionUse distributions to calculate

p(j,k), the probability of a moth emerging in interval j

and dying in interval k

t

j

r

s

d

t

k

t

k+1

t

j+1

I

j

I

k

…

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Calculating Probabilities

Slide47

Probability Table

Emergence Interval

Death Interval

I

0

I

1

…

I

T

I

0

P(0,0)

P(0,1)

…

P(0,T)

I

1

P(1,0)

P(1,1)

…

P(1,T)

I

2

…

…

…

I

3

P(T,1)

P(T,2)

…

P(T,T)

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Multinomial DistributionAll moths fall into one of the probability squares

Moths have a multinomial distribution

Approximate this with a multivariate Gaussian (or normal)

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Approximation ErrorWhat is the error associated with this approximation?

approximated as m!=

s(m)Error of

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Likelihood

 ={µE, σE, µS, σS}

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Likelihood surface

Log Loss

µ

e

µ

s

21 19 17 15 13 11 9 7 5 3 1

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Results

Semiothisa

Signaria

Trap 38B

2005

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Results

R

2 =0.23p<0.01

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Results

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Synthetic Data

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Model Limitations: The “hidden” population and sample size

Trap 13B

n=9

n=28

n=87

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Model Limitations:Sample Size

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Estimating “Hidden” Moth Population

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How is vegetation related to moth species distribution and composition?

CCA and Hamming distance shows a strong correlation between vegetation and moth speciesFor the Future:

Vegetation surveys at other trap sites would help improve the performance of the model

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How does climate affect moth phenology?

Moth emergence shows a strong correlations to the local temperatureFor the future: incorporating the degree day curves we calculated for each site will make the model more robust

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Questions?