Implementing the MAR-1 Algorithm
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Implementing the MAR-1 Algorithm
A conceptual walkthrough
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Steps for Implementing MAR-1 models Ecological decisions
Data formatting Data transformations Model selection Estimated MAR-1 parameter given
model Model fit metrics, model fit diagnostics Stability properties Bootstrap confidence intervals
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Step 1: Make ecological decisions What interactions are you interested in?
Do you have enough data?
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Step 1: Make ecological decisions
Insects
Plants
Birds
Are you missing major players in your community?
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Step 1: Make ecological decisions
VariateCovariate
Make apriori decisions to simplify the foodweb based on ecological knowledge about the system.
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Step 1: Make ecological decisions
rainfall temperature hunting pressure date
years since last fire rabies prevalence storm frequency
road density
What are the important abiotic covariates?
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Step 1: Complex system reduced to a smaller set of variates and covariates to address the interactions of interest
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Step 2: Format the data
No missing values
One column for each variate and covariate
One row for each time step
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Step 3: Transform the data
LN
Counts for species should be transformed by the natural logarithm.
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Step 3: Transform the data
LN
You may need to transform by z-scores also if the data are on very different scales.
Z
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Step 3: Specify the relationship between the co-variates and the rates of growth
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Covariate value
Rainfall
Rodents
Insects
)ln)1(exp( ,,1, ttjijtiti cunbann
Spp covariates would be ln transformed to be consistent with the MAR framework
The relationship for biotic covariates is system specific
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Step 3: Transform the co-variates
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The transformation for the abiotic covariates is determined by the relationship between covariates and spp growth rates. The best transformation is not necessarily a natural logarithm.
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Step 4: Choose the Model Selection Method Compare a specified set of candidate models
testing a set of particular hypotheses want to use a restricted set of prior models (over
which the best is picked or over which models are averaged).
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Step 4: Choose the Model Selection Method Compare a specified set of candidate models
testing a set of particular hypotheses want to use a restricted set of prior models (over
which the best is picked or over which models are averaged).
Search over the set of all possible models constrain models by known interactions rank models by a model selection metric (such as
AIC or BIC) select the best fit model or use a model average
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Step 4: Choose a model comparison approach Compare a specified set of candidate models
Hawks Foxes Lizards Snakes
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Spars
e m
odel
Full m
odel
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Search over all models and choose the best
Step 4: Choose a model comparison approach
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A. Begin with a proposed model (randomly selected)
1 = included interaction0 = excluded interaction
Step 4: Choose a model comparison approach
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Step 5: Iterate through the possible models (CLS method)
A. Begin with a proposed model (randomly selected)
B. Perform a CLS regression on the proposed model to get values for the included interactions
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Step 5: Iterate through the possible models (CLS method)
C. Use this model to compute the Akaike or Bayesian Information Criterion for that model (AIC or BIC)
A. Begin with a proposed model (randomly selected)
B. Perform a CLS regression on the proposed model to get values for the included interactions
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Step 5: Iterate through the possible models (CLS method)D. Randomly change one matrix element to its opposite
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Step 5: Iterate through the possible models (CLS method)
D. Randomly change one matrix element to its opposite
E. Re-run the CLS regression, find the AIC or BIC, and compare
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Step 5: Iterate through the possible models (CLS method)
G. If the new model has a lower AIC/BIC, keep it. Otherwise, keep the old one.
D. Randomly change one matrix element to its opposite
E. Re-run the CLS regression, find the AIC or BIC, and compare
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Step 5: Iterate through the possible models (CLS method)
G. If the new model has a lower AIC/BIC, keep it. Otherwise, keep the old one.
D. Randomly change one matrix element to its opposite
E. Re-run the CLS regression, find the AIC or BIC, and compare
H. Repeat hundreds of times, until you have the model that generates the lowest possible AIC
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Step 5: Iterate through the possible models (CLS method)I. The search procedure finds the model that best
explains the data with the lowest AIC or BIC
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Step 6: Compute stability propertiesOnce you know the interaction matrix, and the covariance matrix, you can compute the stability properties of the stationary distribution X
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Step 6: Compute stability properties Variance of the stationary distribution
eigenvalues det(B)2/p
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Step 6: Compute stability properties Variance of the stationary distribution
eigenvalues det(B)2/p
Return time to the stationary distribution max(B)
max(BB)
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Step 6: Compute stability properties Variance of the stationary distribution
eigenvalues det(B)2/p
Return time to the stationary distribution max(B) max(BB)
Reactivity of the stationary distribution max(BB)-1 -tr()/tr(V)
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Step 7: Bootstrap re-sampling To obtain CIs on the parameter estimates, one
can perform bootstrap re-sampling.
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Step 7: Bootstrap re-sampling To obtain CIs on the parameter estimates, one
can perform bootstrap re-sampling
Basically, you scramble up the Et matrices To create a bootstrapped E time series From which you create a bootstrapped X time
series
XXtt = = AA + + BB XXt t --1 1 + + CC UUt-1t-1 + + EEtt
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Step 7: Bootstrap re-sampling To obtain CIs on the parameter estimates, one
can perform bootstrap re-sampling
Repeat thousands of times to create thousands of bootstrapped data sets
From each bootstrapped data set, one re-estimates the A, B, and C matrices to get bootstrapped confidence intervals
XXtt = = AA + + BB XXt t --1 1 + + CC UUt-1t-1 + + EEtt
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Step 7: Bootstrap estimates example
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0.25 (0.20,0.30)
0.79 (0.59, 0.99)
0.55 (0.43, 0.67)
0.61 (0.54, 0.68)-0.51 (-0.62, -0.40)
0.09 (0.07, 0.11)0.22 (0.16, 0.28)
00 0
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0.05 (0.01, 0.09)
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Means with 95% CI ranges (made up example)
Can obtain CIs on A and C matrices and on all the stability metrics also in the same way.
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Data Transformations
Type of model selection
Manually specify model
Search over all model possibilities
Estimate A, B, & Cmatrices
Estimate stability metrics, model fit diagnostics, and model selection metrics
Bootstrap to obtain bootstrapped CIs
Use simulation to test robustness of assumptions
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That’s a lot of work. It would be nice if there was a computer program that could do it all for you….
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That’s a lot of work. It would be nice if there was a computer program that could do it all for you….
There is! There is! (We will show it to you after lunch (We will show it to you after lunch and use it in the hands-on section.)and use it in the hands-on section.)