Fitting models to data

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

• Fish 458, Lecture 10

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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

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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 cant be negative) and ? is
  assumed to differ between the two survey series
  but to be the same for each age within a survey
  index.

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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
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Calculation details the likelihood

• The likelihood function

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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.

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Analytical Solution for q-I
Being able to find analytical solutions for q and
? is a key skill when fitting fisheries
population dynamics models.
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Analytical Solution for q-II
Repeat this calculation for
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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.

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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.
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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
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An Example of The Binomial Distribution-III
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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?

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The Beta distribution

• The density function
• The mean of this distribution is

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The Shapes of the Beta Distribution
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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.