1. Quiz 2

2. Schaefer model with index of abundance
Constant: index assumed linearly proportional to biomass
Unfished biomass = K
Observed index value
Predicted index value
Schaefer MB (1954) Some aspects of the dynamics of the population important to the management of the commercial marine fisheries. Inter-American Tropical Tuna Commission Bulletin 1(2):25-56

3. Simulated lobster CPUE data a one-way trip
5 Lobster simulation.xlsx, sheet “Simulate”

4. r = 0.05, K = 40271, lnSSQ = 0.902 r = 0.6, K = 9993, lnSSQ = 0.326
r = 1.0, K = 6355, SSQ = 0.359
5 Lobster simulation.xlsx, sheet “Simulate”

5. r = 0.05, K = 40271, lnSSQ = 0.902
r = 0.6, K = 9993, lnSSQ = 0.326
r = 0.20, K = 22550, lnSSQ = 0.611
r = 1.0, K = 6355, SSQ = 0.359

6. “Lobster” model fits All the models fit the index data very well
But terminal harvest rates (u2005) range from 0.18 to >1
We need auxiliary information about harvest rates
Expert knowledge: from length data it is clear than almost all lobster are caught, thus harvest rates are around 0.7 in recent years.
Solution: add a term (u2005 – 0.7)2 to the SSQ

7. Unequal weighting How to weight multiple data sources
How to make probabilistic statements about the results

8. Lobster lessons learned
A one-way trip is not very informative
Harvest and growth are confounded
We can add outside information by including new terms in the SSQ

9. Summary of SSQ fitting Make the predicted close to the observed!
A simple approach that can be applied to simple or very complex models
Find the hypothesis that comes closest to the data
Also find competing hypotheses that fit data nearly as well
We always want to understand the fit of competing hypotheses

10. Next steps Models often have many sources of data
Move to likelihood that provides a logical method of weighting alternative data sources
Likelihood also lets us make more probabilistic statements about competing fits

11. Nonlinear function minimization

12. Readings Hilborn and Mangel, Ecological detective, chapter 11
Numerical Recipes 3rd Edition: The Art of Scientific Computing, chapter 11 mostly online:

13. Warning
Some of the material in this lecture looks complex and very scary
But implementing the equations is intuitive and good practice
The algebra is presented for completeness, and we won’t deal with it outside the lecture
You can master this subject!

14. The general problem
To find a maximum or minimum in a multi-dimensional surface

15. Hill climbing in reverse
Conceptually we want to walk down hill until every direction is up, to minimize the objective function (e.g. sum of squares)

17. How to avoid false summits
Make sure you are on a true summit
Look in all directions
In Solver restart from solution to make sure it has converged
See if other starting points end up at the same place
Start Solver from a number of places, and see if they all go to the same place

18. Multiple starting points

19. Basic approach: gradient method
Start with a guess about x
See which direction is down (calculate the slope dy/dx)
Look at the slope and the change in slope (1st and 2nd derivatives) to guess about how far we have to go until we reach the bottom
Move that direction
Start over again

20. Newton’s minimization method Ecological detective p. 267
We want to find the minimum value of f(x)
Isaac Newton

21. Newton’s Method:
6 Newtons method.xlxs, sheet Newton derivatives

22. Lambda = 1.9
Lambda = 0.2
6 Newtons method.xlxs, sheet Newton derivatives

23. 6 Newtons method.xlxs, sheet Newton derivatives

24. Intuition Function minimum implies first derivative is zero
The second derivative is the rate of change of first derivative
Divide first derivative by second derivative to get number of units of x to jump to find where first derivative is zero
Set lambda < 1 to prevent overshooting

25. Basic theory
If the curve is quadratic (as it is in linear models), then the second derivative is uniform over the entire range of x and you can jump right to the minimum
If the curve is not quadratic, then the derivatives change with x and you have to iterate
6 Newtons method.xlxs, sheet Newton derivatives

26. Numerical derivatives
6 Newtons method.xlxs, sheet Newton numerical

27. 6 Newtons method.xlxs, sheet Newton numerical

28. Why numerical derivatives?
Many real-life functions and models do not have easy formulae from which derivatives can be directly calculated.
But it is always easy to change the parameters by a small number (Δ) and rerun the function to get a close approximation to the numerical derivative.

29. Golden section search Numerical recipes; Ecological detective p. 271-3
One parameter only, no derivatives or differentiation required
Minimum must be within bounds; much faster than bisection search
Algorithm:
Select lower bound L and upper bound U containing the answer
Select two new points x1 and x2:
Calculate the function at those points
Replace U or L with one of the new points as appropriate to narrow the bounds containing the minimum
Select another new point to replace the one chosen as a new bound, and repeat
Choice of x1 and x2 based on “golden” ratio

30. Golden section search Step 1 L = 0 x1 x2 U = 1 Step 2 L = x1 x2 x3

31. Golden section search
6 Golden search.xlxs