Instructor Notes

What is parametric modeling and what are its limitations for modeling and forecasting?

• Model with specific mathematical form
• Don’t know the true mathematical form, which can lead to incorrect fits/conclusions

What are some example of non-linear dynamics in ecology?

• Species interactions influencing responses to environmental factors

What is Empirical Dynamic Modeling and how is it different from parametric modeling?

• Equation free approach
• Model arbitrarily complex F
• Can capture the complexity of a multiple species/state system in a single species/state time-series
• Does this based on Taken’s Theorem
  • Resconstruct a shadow of the real system from single time-series
  • Instead of relying on:

\[x_i(t+1) = F_i\left(x_1(t), x_2(t), \dots, x_d(t)\right)\]

• The system dynamics can be represented as a function of a single variable and its lags

\[x_i(t+1) = G_i\left(x_i(t), x_i(t-1), \dots, x_i(t-(E-1))\right)\]

• \(E\) is the embedding dimension which defines how far back in time we go

Describe what is happening with the Lorenz attractor (from the video)?

• Simple model of complex population dynamics
• Shows the abundances of three species at large number of points in time
• Individual population time series can be shown as the time-series for a single axis

How does EDM address some of the limits of parametric modeling for forecasting ecological systems?

• Doesn’t specificy a specific equation
• Determines what should happen next based on what happen next in the past when the system was in a similar configuration

What are some potential the downsides of this approach?

• Assumes the system is behaving like it used to
• Difficult to interpret

Why is the Fraser River Sockeye Salmon fishery useful for comparing the EDM approach to parametric modeling approaches.

• Well studied system
• Established parametric approaches
• Long time-series with key co-variates

What is parametric approach that the EDM is compared to?

• Ricker population growth

How are forecasts from the Ricker model and EDM model compared?

• Ability to predict held out data from the end of the time-series

How are environmental factors incorporated into EDM?

How do the different modeling approaches compare in this system?

• EDM forecasts perform better in general
• Improvement is modest when just using population time-series
• Improvement is larger when incorporating environmental drivers

Can EDM help infer the causal links among variables?

• In this case it suggests that the environmental drivers that are thought to be important to population dynamics do matter even though they were missed by the parametric approach
• By avoiding the limitations of specifying a parametric model this approach may provide better causal inference in some cases