CEDA Theory Session

1
CEDA Theory Session
FMSP Stock Assessment Tools Training
Workshop Mangalore College of Fisheries 20th
-24th September 2004
2
What CEDA does

• The CEDA software analyses catch, effort and
  abundance data to provide estimates of
• Unexploited or initial stock size (K or N1)
• Catchability (q) or power of fishing
• Current stock size/biomass (performance
  indicator)
• MSY (reference point - DRP models only).
• CEDA also has a facility to project stock sizes
  into the future under various scenarios.

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CEDA Theory - Contents

• Analysis of catch, effort and abundance data
• CEDA data requirements
• Models available in CEDA
• No recruitment
• Indexed Recruitment models
• Deterministic Recruitment / Production (DRP)
  Models
• Choice of appropriate models
• Model outputs and use
• Guide to fitting models (Theory Session 2)
• Error Models
• Residual Plots
• Outliers
• Influential Points
• Sensitivity Analysis
• Confidence Limits and Bootstrapping
• Summary

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Analysis of Catch, Effort and Abundance Data

• The models used in CEDA are designed to mimic the
  changes over time (the dynamics) in the total
  numbers or total biomass of an exploited fish
  stock.
• For each model, it is assumed that you have
  historical data on the total catches that have
  been taken, and on an index of relative abundance
  (e.g. CPUE).
• Given these, using CEDA, you can then estimate
  the historical population abundances and the
  associated fishery parameters.
• You will also be able to predict how a stock will
  react to different future effort or catch
  scenarios and thus be able to manage and plan the
  fisheries better.

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Analysis of Catch, Effort and Abundance Data
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CEDA Data Requirements (1/2)

• All the models in CEDA assume that the data
  refers to a single discrete stock, i.e. a
  population without any significant immigration
  from that population or emigration to other
  population not covered by the data.
• Comprehensive record of total catch for the whole
  time period to be analysed, with no gaps.
• A Good relative index of population size or
  abundance. The two most common types of index are
  research survey data and commercial CPUE data.
  There can be advantages and disadvantages to
  both....

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CEDA Data Requirements (2/2)

• Research Survey Data
• Good measure of population size, but low sample
  size and low frequency therefore causing high
  variability.
• Commercial CPUE
• Cheap to collect, lower variability, but is it a
  good measure of population size?

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Potential problems with CPUE data

• Way in which effort is measured.
• Target switching.
• Changes in fishing power.
• Sequential depletion.

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CEDA Model Types

• Types of models available
• 1. No recruitment
• 2. Indexed Recruitment
• 3. Deterministic Recruitment / Production Models
• Constant recruitment
• Schaefer production model
• Fox production model
• Pella-Tomlinson production model
• The correct choice of model is vital and depends
  on the pattern of recruitment to the fishery and
  the data available

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No Recruitment Model

• These are typically used for within-season
  assessments of short-lived, annual species such
  as squid and shrimp.
• Also useful for the analysis of experimental
  fishing, where you fish for a short period in one
  location and assume that there was no recruitment
  during the period.
• Assumes that there is one pulse of recruitment at
  the start of the series and nothing after this
  point. (I.e. no recruitment within the data
  set)
• Constant natural mortality (M) is assumed
  throughout.
• Requires catch and abundance index in numbers.
• (can use mean weight in each week to convert
  catch in weight)

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Problems with the No Recruitment Model

• With the no recruitment model the assumption is
  made that the spatial distribution is constant.
  Therefore problems occur for within-year
  analyses where migration occurs, e.g.
• - Juveniles moving to / from nursery grounds
• - Adults moving to / from spawning
• - Migration of the adult stock
• This will affect the catchability or CPUE series,
  and make the estimates of population size
  unreliable.

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Indexed recruitment model

• Typical of small pelagic species such as sardines
  and anchovies.
• Need an index of relative recruitment that is
  proportional to the number of recruits in each
  year. This is the only method in CEDA that can
  cope with substantial inter-annual variability in
  recruitment. How to get this index?
• - Larval or juvenile survey data
• - Catch data from another fishery operating in
  the same area, but catching smaller animals
• - Length-frequency data.
• Assumes constant M (Natural Mortality rate).
• Requires catches in numbers.

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Problems with the indexed recruitment model

• The indexed recruitment model can sometimes
  produce unreliable estimates of population size,
  e.g. when
• - The recruitment index is not a good measure
  of relative annual recruitment.
• - There is little annual variability in
  recruitment.
• It is possible to use an indirect estimate of
  recruitment such as a measure of upwelling, but
  this is not usually recommended.

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Deterministic Production Models

• Deterministic - Describes a system whose time
  evolution can be predicted exactly.
• Production In this case production refers to
  the net production in terms of yield after
  recruitment, growth, migration (into and out of
  the population) and natural mortality have been
  taken into account.
• Models assume constant carrying capacity (K)
• Two types - Constant Recruitment
  Production Models

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Constant Recruitment Model

• First described by Allen (1966) with respect to
  whales. It has also been used for some reef fish
  species.
• Assumes that the stock started in deterministic
  equilibrium and that annual recruitment (in
  numbers) is constant and independent of stock
  size!
• This assumption may seem a little strange, but
  for many stocks it appears that recruitment
  declines only when the adult stock has been
  reduced to rather low levels (e.g. around 20 of
  unexploited levels).
• The method is not suitable for use when there is
  substantial inter-annual variation in recruitment.

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Production Models

• Deterministic Production Models work on the basis
  that the net change in biomass of a population
  from one year to the next is a result of
• the catch taken during the current year, and
• the stock production during the current year.
• The stock production combines the effects of
  recruitment to the population, growth and natural
  mortality, and it is assumed to be a
  deterministic function of current or recent stock
  sizes.

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Production Models

• Three production models are included in CEDA
• Schaefer Production Model
• Fox Production Model
• Pella-Tomlinson Production Model

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Schaefer Production Model (1/2)

• The Schaefer production model (Schaefer, 1954)
  assumes that there is a symmetrical relationship
  between stock size and production (and yield),
  which is a function of the unexploited population
  size (or carrying capacity) K, and the intrinsic
  growth rate r.
• For Schaefer models, the sustainable yield curves
  are symmetrical and they all have a maximum ( the
  maximum sustainable yield, or MSY) which occurs
  at a biomass of K/2.
• In order to obtain reliable estimates of r and K,
  data must be available for a wide range of stock
  sizes (giving good contrast)

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Schaefer Production Model (2/2)
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Fox Production Model (1/2)

• The Fox production model (Fox, 1970) is
  essentially similar to the Schaefer model, in
  that stock production is again related to r and
  K.
• However, the relationship between stock size and
  production has a somewhat different form, being
  much flatter to the right of the peak, rather
  than symmetrical.
• The position and height of the peak in production
  are again determined by r and K, and the data
  requirements for reliable estimation of these
  parameters are similar to those for the Schaefer
  model.

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Fox Production Model (2/2)
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Pella-Tomlinson Model (1/2)

• The Pella-Tomlinson generalised production model
  (Pella and Tomlinson, 1969) specifies a
  relationship similar in mathematical form to the
  Schaefer model.
• The difference between the two is that the
  Pella-Tomlinson model has an extra parameter, z,
  which allows the symmetry of the Schaefer model
  to be distorted.
• When z1, the Pella-Tomlinson and Schaefer models
  are identical, with the peak occurring at K/2
  when zlt1, the peak occurs to the left of K/2 as
  z tends to zero, the shape (but not the height)
  of the function approaches that of the Fox model.
  When zgt1, the peak occurs to the right of K/2.

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Pella-Tomlinson Model (2/2)
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Time Lags in Production Models

• Often fish do not recruit into the fishery at age
  0. Many fish have a long juvenile phase where
  they are not fished as part of the exploited
  stock.
• This means that the effects of a much lower (or
  higher) adult stock size in one year, in terms of
  subsequent lower (or higher) recruitment, may not
  be apparent for a number of years.
• To account for these cases, CEDA allows you to
  incorporate a time lag L into the DRP models,
  linking biomass production with the stock size L
  years ago.
• NOTE, however, that recruitment is only one part
  of production, the other parts of which
  definitely happen during the current year.
  Normally, we recommend time lags are not used.

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Choice of Appropriate Models

• Often, only one of the six models described above
  will be suitable for the type of data available.
• If you have weekly data from a species without
  recruitment, then use the no-recruitment model.
• If you have annual data with a recruitment index,
  then use the indexed recruitment model. You may
  also want to try a production model.
• If you only have annual catch and abundance data
  in weight, then you will have to use a production
  model.
• There are two main situations in which you have a
  choice over the model
• Constant recruitment model vs DRP model
• Three alternative DRP models

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Choice of Appropriate Models (2)

• See Section 4.5.1 in draft FAO document

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Choice of Timescale

• If you have monthly data collected over a number
  of years, then would normally aggregate over 12
  month periods.
• However, if noticeable seasonality in catch
  rates, should still aggregate catch data over 12
  month period, but only use CPUE data from months
  with the highest catch rates.
• Remember years do not necessarily start in
  January!

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CEDA Model outputs - Table 4.1, p91
Yes
Yes
Yes
Replacement Yield
RY
Yes
Yes
Yes
F
giving MSY
F
Yes
Yes
Yes
MSY
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Using CEDA outputs (1)

• No recruitment (depletion) model
• For an annual species (squid, shrimp?), estimate
  number remaining as season progresses to ensure
  enough escapement from fishery for remaining
  spawners to produce next years stock (see p92 and
  squid tutorial)
• Estimate stock size each year as performance
  indicator
• Or, estimate recruitment strength at start of
  each season, then input data into an indexed
  recruitment model in CEDA or fit a stock recruit
  relationship to estimate minimum biomass required
  to sustain stock

Section 4.5.3, p92
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Using CEDA outputs (2)

• DRP (deterministic recruitment/production) models
• Estimate MSY-based reference points - MSY, BMSY
  and FMSY or fMSY (see next slide, p93 and tuna
  tutorial)
• Estimate current stock size each year as
  performance indicator for comparison with BMSY -
  above or below?
• Set catch quotas or effort levels based on MSY or
  fMSY, or using projections to allow recovery to
  BMSY over an agreed time scale

Section 4.5.3, p93
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CEDA outputs - DRP models (1)
MSY ( r K / 4 )
Unexploited biomass, K
Size of
Catch
BMSY ( K / 2 )
Biomass / Stock size
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CEDA outputs - DRP models (2)
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(No Transcript)
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CEDA Theory Part 2 - Guide to Fitting Models

• Error Models
• Residual Plots
• Outliers
• Influential Points
• Sensitivity Analysis
• Confidence Intervals

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Error Models

• No data ever fits a model perfectly. Fitting
  involves searching for parameter estimates that
  minimise the discrepancy (called the residual)
  between observed data and the values that would
  be expected if the parameter estimates were
  correct.
• Different error models quantify the discrepancy
  in different ways.
• Three error models used in CEDA
• Least Squares model (normal distribution,
  constant variance)
• Gamma model (skewed distribution, smaller errors
  at smaller catches)
• Log Transform model (even more skewed
  distribution)
• Best model will be identified by residual plots
  ..

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Residual Plots (1/4)

• Once you have tried a particular fit, you will
  obtain both parameter estimates and a range of
  diagnostics, which should help you to determine
  whether your fit was good.
• The most important diagnostics are simply the
  graphs of observed and expected values of catch
  and CPUE. Looking at these will soon reveal
  whether you have a reasonably good fit.
• The residual plots are closely related to the
  catch graph, and their purpose is to enable model
  assumptions to be checked. Two residual plots
  are available
• residuals plotted against the expected catch
• residuals plotted against time.

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Residual Plots (2/4)
The figure below is an example of a good residual
plot. The points observed are scattered evenly
in a horizontal band above and below the zero
residual line and over time.
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Residual Plots (3/4)
The following is an example of a bad residual
plot. This curved shape to the plot shows that
the population model used does not fit the data
correctly. Bad plots can be identified by trends
or runs of individual points on either side of
the zero residual.
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Residual Plots (4/4)
Another bad plot. Here the shape shown by the
points is not evenly distributed. Here smaller
residuals appear at low catch values and higher
residuals at higher values. This type of plot
can occur when the wrong error models have been
used.
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Outliers (1/4)

• An outlier in a data set fitted with a particular
  model is an observation that would be extremely
  unlikely to occur under the best fitting model.
  The main way of detecting outliers is by
  examining the residual plots.
• An outlier is a point that lies a "long way" from
  the x-axis (or 0.5 line on a percentile plot)
  relative to the other points. The definition of a
  "long way" depends on the probability of
  accidentally labeling a perfectly good data point
  as an outlier.
• The occurrence of an outlier indicates that there
  is probably a fault with either the model or the
  data.

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Outliers (2/4)
An example of a outlier in a CPUE series is shown
below. The very high CPUE shown at t7 is also
shown in the outlier in the residual plot. The
error here in the data was down to poor data
entry with the effort entered being ten times the
amount that was actually recorded.
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Outliers (3/4)

• Any apparent outliers should be subjected to
  further scrutiny.
• If an outlier occurs with a model that seems to
  fit the other data well, the first task is to
  investigate the offending point or points.
• The problem could be caused by unusual conditions
  at that time or by measurement errors in the
  abundance index, catch data, recruitment index,
  or even the mean weight.

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Outliers (4/4)

• If conditions were anomalous at that time (e.g.
  unusual sea temperature), then the point may be
  excluded from analysis (when you do this, you
  should also check the other data for a similar
  circumstance any other similarly anomalous
  points should be excluded as well, even if they
  do not appear to be outliers).
• Outliers may disappear under different error
  model
• If an outlier still exists and no good reason
  exists for its removal then sensitivity analysis
  can be carried out with reference to that point.

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Influential Points (1/2)

• Influential points are those whose presence or
  absence make a large difference to the results.
• Influential points tend usually, but not
  invariably, to lie near the extremes of the data
  set, i.e. near the lowest and highest stock
  sizes.
• Influential points can often be identified from
  plots of residuals against expected catches they
  will be points corresponding to isolated large or
  small expected catches, usually with small
  residuals.
• If you suspect that a point is influential, you
  can easily check by toggling it out and seeing if
  the parameter estimates change substantially.

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Influential Points (2/2)

• The data for an influential point should be
  carefully scrutinized, just as for a potential
  outlier.
• If there are serious data problems, then the
  point could be dropped from subsequent analysis.
  As for outliers you must have very good reasons
  to exclude the point from analysis.
• If no good reason exists, then sensitivity
  analysis should be used. What are the effects of
  including and excluding the point?
• It is definitely wrong to exclude an influential
  point and then "forget" about it, as one might
  for an outlier you will bias the results and
  will dramatically reduce the precision of your
  estimates.

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Sensitivity Analysis

• Investigate the effect of varying the model
  assumptions when you are uncertain about which is
  correct.
• It is better to be honest about the uncertainty
  in one's results than it is to be wrong!
• Try different assumptions
• - Different models
• - Different data
• - Different error models
• - Including / excluding influential points from
  the model
• - Different values for user-supplied parameters
  (e.g. M)
• Present results of all sensitivity analyses to
  decision makers so that the real uncertainty in
  the results will be clear

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Confidence Intervals

• Should not only look at point estimates, need to
  also consider confidence intervals (CIs), in
  addition to sensitivity tests.
• A CI of a given size, e.g. 95, is the
  probability that the interval contains the true
  value.
• Should form management decisions based on CIs,
  rather than point estimates alone. (Remember
  precautionary reference points)
• Estimate CIs as part of sensitivity analysis.
  Comparing two point values may show them to be
  quite different but the overlap of their CIs may
  be very similar.
• Estimate CIs by bootstrapping.

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Bootstrapping

• Bootstrapping uses the set of discrepancies
  between observed and expected values in the
  original data to simulate new data sets, or
  re-samples. For each re-sampled data set, the
  parameter estimation procedure is repeated.
• Bootstrapping will show any skewness in the
  estimates, but care should be taken with any
  values appearing at 0 and ? as these may be false
  values caused by repeated selection on a
  particular (wrong) value.
• The problem with bootstrapping is that it can be
  a slow process. Less of a problem now with
  higher powered computers but be wary on older
  machines.

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Summary (CEDA Theory)

• What have we covered?
• Analysis of catch, effort and abundance data
• CEDA data requirements
• Models available in CEDA
• No recruitment models
• Constant recruitment models
• Deterministic Recruitment / Production (DRP)
  Models
• Choice of appropriate models
• Use of CEDA outputs
• Guide to fitting models - residual plots etc