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Fitting models to data – III (More on Maximum Likelihood Estimation)

Fitting models to data – III (More on Maximum Likelihood Estimation). Fish 458, Lecture 10. A Cod Example (model assumptions). The catch is taken in the middle of the year. The catch-at-age and M are known exactly.

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Fitting models to data – III (More on Maximum Likelihood Estimation)

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

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

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

  6. 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. 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. Analytical Solution for q-II Repeat this calculation for

  9. 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. 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. 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. 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. An Example of The Binomial Distribution-III

  14. 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. The Beta distribution • The density function: • The mean of this distribution is:

  16. The Shapes of the Beta Distribution

  17. 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.

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