Incorporating length information and growth estimation in the Woods Hole Assessment Model (WHAM)
Giancarlo M. Correa1, Cole Monnahan2, Jane Sullivan3, James T. Thorson2, Andre E. Punt1
1University of Washington, Seattle, WA
2Alaska Fisheries Science Center, NOAA, Seattle, WA
3Alaska Fisheries Science Center, NOAA, Juneau, AK
Outline
- State-space assessment models
- The Woods Hole Assessment Model (WHAM)
- Limitations
- New features in WHAM
- Examples
State-space assessment models
Allows separation of both observation and process errors.
Observation equation:
E[Yt∣Xt]=g(Xt,θ)
Process equation:
E[Xt∣Xt−1]=h(Xt−1,θ)
Xt is the unobserved state at time step t and Yt are observations. θ is the vector of all unknown model parameters (fixed effects). Xt treated as random effects (Aeberhard et al. 2018).
State-space assessment models
Why use them? (Aeberhard et al. 2018, Stock and Miller, 2021)
- handle structural breaks, time-varying parameters
- handle missing observations
- can be used to do forecasting
- model complex and nonlinear relationships
- less retrospective bias
The Woods Hole Assessment Model (WHAM)
- Fully state-space age-structured model
- Developed from ASAP3 (built in ADMB)
- Data: catch, indices of abundance, age compositions, empirical weight-at-age
- Environmental covariates (mechanistic approach)
- Written in TMB and R (user friendly!) ( see R package ).
The Woods Hole Assessment Model (WHAM)
The Woods Hole Assessment Model (WHAM)
Random effects in:
- Selectivity parameters
- Natural mortality
- Abundance-at-age
- Catchability
Stock and Miller et al., 2021
The Woods Hole Assessment Model (WHAM)
Yellowtail flounder on the US East Coast (Stock and Miller, 2021):
The Woods Hole Assessment Model (WHAM)
Limitations:
- best practices for random effects use
- length information and growth modeling not available
- single sex, no spatial component, no aging error
- only selectivity-at-age
- only input empirical weight-at-age
Great talk by Brian Stock and Tim Miller on WHAM.
WHAM expansion
Data inputs
- (Marginal) length compositions
- Conditional age-at-length
- Input age-length transition matrix (φf,l,a)
Growth modeling
- Use input age-length transition matrix
To distribute abundance-at-age to abundance-at-age and length.
Growth modeling
- Use von Bertalanffy growth equation:
The mean length-at-age at the start of the year (y=1):
˜Ly,a=L∞+(L1−L∞)exp(−k(a−1))
and when y>1:
˜Ly,a={L1,if a=1˜Ly−1,a−1+(˜Ly−1,a−1−L∞)(exp(−k)−1)otherwise
Growth modeling
- Use von Bertalanffy growth equation:
Then, to calculate the mean length-at-age at any fraction of the year:
Ly,a=˜Ly,a+(˜Ly,a−L∞)(exp(−kfy)−1) fy is the year fraction.
Growth modeling
- Use von Bertalanffy growth equation:
Also, Ly,a and variation of length-at-age ( σa ) are used to calculate the age-length transition matrix (Stock Synthesis - SS - approach):
φy,l,a={Φ(L′min−Ly,aσy,a)for l=1Φ(L′l+1−Ly,aσy,a)−Φ(L′l−Ly,aσy,a)for 1<l<nL1−Φ(L′max−Ly,aσy,a)for l=nL
Growth modeling
- Use von Bertalanffy growth equation:
Random effects on growth parameters can be modeled:
. . .
log(L∞t)=μL∞+δ1,t
log(kt)=μk+δ2,t
log(L1t)=μL1+δ3,t
. . .
t represents year or cohort effects and can be iid or AR1.
Growth modeling
- Estimate mean length-at-age:
For this case, mean length-at-age ( ˜La ) are assumed to be parameters and can be estimated. σa still needed.
. . .
Time variability can be modeled by including random effects:
log(˜Ly,a)=μ˜La+δa,y
δa,y can vary by year and age.
Length-weight relationship
Optional when empirical weight-at-age not provided:
wl=Ω1lΩ2
Random effects on Ω1 and Ω2 can also be estimated (like growth parameters).
. . .
Then, we can calculate the population weight-at-age at any moment during the year:
ˆwy,a=∑lφy,l,awl
. . .
ˆwy,a can also be fitted to wy,a (empirical weight-at-age)
Selectivity
Originally, these selectivity-at-age functions were available:
age-specific: by age.logistic: increasing logistic.double-logisticdecreasing-logistic
. . .
Variability by year and parameter autocorrelation.
Selectivity
We incorporated one extra option:
double-normal: SS-like (Methot and Wetzel, 2013)
. . .
But also some selectivity-at-length functions:
len-logistic: increasing logistic at lengthlen-decreasing-logisticlen-double-normal
Environmental covariates
Xt (unobserved environmental variable at time t ) can be linked to any parameter presented here (Stock and Miller, 2021).
Two options for the process model:
- Random walk:
Xt+1|Xt∼N(Xt,σ2X)
σ2X is the process variance.
Environmental covariates
- AR(1):
X1∼N(μX,σ2X1−ϕ2X)
Xt∼N(μX(1−ϕX)+ϕXXt−1,σ2X)
μX, σ2X, and ∣ϕX∣<1 are the marginal mean, conditional variance, and autocorrelation parameter.
Environmental covariates
Observations Yt are assumed to be normally distributed with mean Xt and variance σ2Yt:
Yt∣Xt∼N(Xt,σ2Yt)
σ2Yt treated as known or estimated.
Environmental covariates
An environmental covariate can be linked to a state (i.e. parameter):
Pt=Pexp(β1Xt)
P is the base state (parameter) value.
Other links are also available (polynomials). Lags can be modeled.
Examples
Examples
Using SS ( ss3sim, Anderson et al., 2014 ), we simulated data that was then incorporated into WHAM.
- Growth variability
- Conditional age-at-length (CAAL) data
- Length-weight variability
Growth variability
Simulate data:
- catch (one fishery)
- index of abundance (one survey)
- length compositions (fishery and survey)
- temporal variability in asymptotic length ( L∞ ) using PDO as the driver
Growth variability
We implemented three models in WHAM:
- Random effects (by year) on L∞
- Random effects (
iid) on mean length-at-age (LAA) - Include an environmental variable (PDO), linked to L∞
Growth variability
Models:
m1: random effects (by year) on L∞m2: random effects on mean length-at-age (LAA)m3: environmental variable (PDO), linked to L∞
Growth variability
- LAA: mean length-at-age at the start of the year
- Still exploring these differences
CAAL data
Simulate data:
- catch (one fishery)
- index of abundance (one survey)
- length compositions (fishery)
- conditional age-at-length (survey)
- selectivity-at-length
CAAL data
CAAL residuals:
CAAL data
Estimated selectivity-at-length:
Length-weight variability
Simulate data:
- catch (one fishery)
- index of abundance (one survey)
- length compositions (fishery and survey)
- empirical weight-at-age data fitted
- temporal variability in the Ω1 parameter (L-W relationship)
Length-weight variability
Summary
- state-space models more frequent with TMB (mostly in Europe and US East Coast)
- coherent way to include climate indices into assessments (and projections!)
- new tools developed in this project to apply them in the West Coast and Alaska region
- future projects: sex and spatial structure
- interest in organizing a workshop and applications (e.g. GOA pollock)
Thanks for listening
Contact:
gcorrea@uw.edu
giancarlo.correa@noaa.gov
Incorporating length information and growth estimation in the Woods Hole Assessment Model (WHAM) Giancarlo M. Correa 1 , Cole Monnahan 2 , Jane Sullivan 3 , James T. Thorson 2 , Andre E. Punt 1 1 University of Washington, Seattle, WA 2 Alaska Fisheries Science Center, NOAA, Seattle, WA 3 Alaska Fisheries Science Center, NOAA, Juneau, AK