Observation and process error

1. 1 Observation and process error

2. 2 Readings

3. 3 Observation errorProcess error There are two major random components Random production (process error) Randomness in observation

4. 4 Schnute�s simple example

5. 5 Conclusion What we think about the relation between X and Y depends on whether we think there is error in the process or the observation

6. 6 Logistic growth model

7. 7 Stochastic Logistic Growth Model

8. 8 Process error only

9. 9 The deterministic prediction

10. 10 Process error

11. 11 Likelihood of w�s

12. 12 Process error Estimate r,k,q, sigma Don�t need to assume initial conditions because we assume no observation error B1=I1/q Can only make predictions (easily) when we have a continuous time series of data

13. 13 Observation error only

14. 14 Likelihood of v�s

15. 15 Observation error Estimate r,k,q, sigma, B0 We do need to assume initial conditions or estimate B0 , often we assume B0 =k Can make predictions without a continuous time series of data

16. 16

17. 17 the process for fitting 1. Generate deterministic data 2. Fit model to these data 3. Add observation and process error and fit 4. Now fit the �real data� 5. Compare observation and process error estimates

18. 18 Analytic formula for sd

19. 19 Analytic formula for q

20. 20 Estimating both observation and process error We can try to estimate both the v�s and the w�s, and ?v and ?w. But if we do that we often end up with silly answers, ?v or ?w. becoming close to zero. If we specify one of the two sigmas or their ratio, then we can obtain good estimates This approach is now standard in age-structured population dynamics models

21. 21 Example of an observation and process error analysis

22. 22

23. 23 Summary Observation and process error are almost always present It is often useful to contrast the estimates with observation error to the estimates with process error It is possible to include simultaneous observation and process error