1. 458 Fitting models to data – III (More on Maximum Likelihood Estimation) Fish 458, Lecture 10

2. 458 A Cod Example (model assumptions) The catch is taken in the middle of the year. The catch-at-age and M are known exactly. We can therefore compute all the numbers-at- age given those for the oldest age:

3. 458 A Cod Example (data assumptions) We have survey data for ages 2-14 (the catch data start in 1959): A trawl survey index (1983-99) – surveys are conducted at the end of January and at the end of March. A gillnet index (1994-98) – surveys are conducted at the start of the year. We need to account for when the surveys occur (because fishing mortality can be very high). We assume that the age-specific indices are log- normally distributed about the model predictions (indices can’t be negative) and  is assumed to differ between the two survey series but to be the same for each age within a survey index.

4. 458 Calculation details – the model Terminal numbers-at-age The “terminal” numbers-at-age determine the whole N matrix Oldest-age Ns Most-recent- year Ns (year y max )

5. 458 Calculation details – the likelihood The likelihood function:

6. 458 Fitting this Model The parameters: We reduce the number of parameters that are included in the Solver search by using analytical solutions for the qs and the  s.

7. 458 Analytical Solution for q-I Being able to find analytical solutions for q and  is a key skill when fitting fisheries population dynamics models.

8. 458 Analytical Solution for q-II Repeat this calculation for

9. 458 The Binomial Distribution The density function : Z is the observed number of outcomes; N is the number of trials; and p is the probability of the event happening on a given trial. This density function is used when we have observed a number of events given a fixed number of trials (e.g. annual deaths in a population of known size). Note that the outcome, Z, is discrete (an integer between 0 and N).

10. 458 The Multinomial Distribution Here we extend the binomial distribution to consider multiple possible events: Note: We use this distribution when we age a sample of the population / catch (N is the sample size) and wish to compare the model prediction of the age distribution of the population / catch with the sample.

11. 458 An Example of The Binomial Distribution-I 10 animals in each of 17 size-classes have been assessed for maturity. Fit the following logistic function to these data.

12. 458 An Example of The Binomial Distribution-II We should assume a binomial distribution (because each animal is either mature or immature). The likelihood function is: The negative log-likelihood function is: is the number mature in size-class i

13. 458 An Example of The Binomial Distribution-III

14. 458 An Example of The Binomial Distribution-III An alternative to the binomial distribution is the normal distribution. The negative log-likelihood function for this case is: Why is the normal distribution inappropriate for this problem?

15. 458 The Beta distribution The density function: The mean of this distribution is:

16. 458 The Shapes of the Beta Distribution

17. 458 Recap Time To apply Maximum Likelihood we: Find a model for the underlying process. Identify how the data relate to this model (i.e. which error / sampling distribution to use). Write down the likelihood function. Write down the negative log-likelihood. Minimize the negative log-likelihood.