# 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