Fitting models to data sum of squares Fish 458

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Fitting models to datasum of squaresFish 458

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Reading

• The Ecological Detective Hilborn and Mangel
  Chapter 5

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Why fit models to data

• Models are hypotheses about nature
• we use the data to see how much support there is
  for competing hypotheses
• does your model fit the data?
• Only in comparison to other hypotheses - no real
  absolute measure of acceptable fit

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What is needed

• Competing models
• data
• goodness of fit criterion
• Algorithm to maximize goodness of fit

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The model, linear regression
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Some data B.C. herring
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Goodness of fit

• every value of a and b makes predictions about
  the length for each individual
• how do we decide what parameters are best
• sum of squares

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The best fitlinear_growth.xls
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Sum of squares surface
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How to find the minimum sum of squares

• direct search (ok for 1-3 parameters)
• algebra (linear regression - linear models)
• non-linear gradient searches

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What are the competing hypotheses?

• Different values of slope or intercept
• We could expand our analysis to include models
  that have shapes other than linear and ask if the
  data support them.

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Schaefer model
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Why log SSQ

• so that predicted 10 vs observed 20
• has same weight as
• predicted 1 vs observed 2
• Again the multiplicative error assumption

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New Zealand Lobster
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Lobster

• The blue dots are the Index (CPUE) divided by q,
  the red line is the model fit and the light blue
  line is the annual exploitation rate
• We fit the CPUE quite well, because that is the
  SSQ criterion
• But not that it implies a very low exploitation
  rate, even though it is known that the
  exploitation rate after 1980 was very high, maybe
  70

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An alternative fit with r fixed at 0.5
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Differences

• We fit the CPUE not as well
• SSQ goes from 1.9 top 3.2
• But now we have much more reasonable estimates of
  fishing mortality rate

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So we have two types of information

• The CPUE series
• The knowledge (from length frequency analysis)
  that exploitation rates are high.

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Problems with SSQ

• how to weight multiple data sources
• how to make probabilistic statements about the
  results

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New Zealand lobster

• previous fit has biomass in 199023,000 tons and
  1990 catch2,770 tons which is a harvest rate of
  12
• but we know from length frequency that the
  exploitation rate was at least 70 through all
  the 1980s
• add additional SSQ term

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equal SSQ weighting
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Equal weighting gives almost no weight to the
exploitation rate data
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unequal weighting
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w10
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Lessons re lobster

• A one-way trip is not very informative
• Harvest and growth are confounded
• We can force the model to have higher
  exploitation rates
• But the logistic model is, in the end, not very
  satisfactory
• In NZ the logistic was rejected and a more
  complex model fitting to length frequency
  allowing for recruitment to be estimated for each
  year is now used

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Extinction risk vs yield

• If our concern is extinction the key question is
  what would be the impact of reducing harvest
• If our concern is yield the key question is what
  would be the impact of reducing harvesting
• You cant get away from the confounding of
  population rate of increase vs harvesting

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Summary SSQ

• Make the predicted close to the observed!
• Is 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

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Next steps

• Many models 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