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Update Richards vignette: parameters, fig width, eqnarray #17

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Dec 5, 2024
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Update Richards vignette: parameters, fig width, eqnarray #17

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f449135
Update Richards vignette: parameters, fig width, eqnarray
arni-magnusson Dec 4, 2024
c9c1baa
Update Richards vignette: parameters
arni-magnusson Dec 5, 2024
616df33
Update Richards: no risk of b becoming zero
arni-magnusson Dec 5, 2024
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63 changes: 28 additions & 35 deletions website/qmds/richards_growth_curve.qmd
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Original file line number Diff line number Diff line change
Expand Up @@ -30,13 +30,18 @@ library(r4ss)

### Parameter definitions

The Richards growth model has four parameters,

$L_1$: length at age $A_1$\
$L_2$: length at age $A_2$\
$k$: growth coefficient\
$b$: shape parameter\

and three input variables:

$t$: age\
$L1$: Minimum length\
$L2$: Maximum length\
$k$: von Bertalanffy growth coefficient parameter\
$b$: Richards growth parameter\
$A1$: Fixed age for first size-at-age\
$A2$: Fixed age for second size-at-age
$A_1$: young age selected by user, for the estimation of $L_1$\
$A_2$: old age selected by user, for the estimation of $L_2$

### SS3 Code

Expand Down Expand Up @@ -130,7 +135,7 @@ Compare three growth curves: length-at-age reported in the Stock
Synthesis output, predicted lengths using the `ss3_code()` function, and
predicted lengths using the `schnute()` function.

```{r, fig.width=6, fig.height=8, fig.align="center", out.width="50%"}
```{r, fig.width=6, fig.height=8, fig.align="center", out.width="60%"}
t <- 0:20
plot(Len_Beg~Age_Beg, model$endgrowth, subset=Sex==1, ylim=c(0,80),
xlab="age", ylab="length", type="l", lwd=1.5, col=2)
Expand Down Expand Up @@ -165,7 +170,7 @@ lines are drawn using the actual von Bertalanffy and Gompertz functions,
and the circles are drawn using the Richards curve with the appropriate
values for the $b$ parameter.

```{r, fig.width=6, fig.height=8, fig.align="center", out.width="50%"}
```{r, fig.width=6, fig.height=8, fig.align="center", out.width="60%"}
plot(NA, xlim=range(t), ylim=c(0,80), xlab="age", ylab="length")
lines(t, von_bertalanffy(t, L1, L2, k, A1, A2), ylim=c(0,80), lwd=2, col=2)
points(t, schnute(t, L1, L2, k, 1, A1, A2), col=2)
Expand All @@ -179,7 +184,7 @@ legend("topleft", c("von Bertalanffy (b=1)","Gompertz (b near 0)"),

### Richards growth curves with different $k$ and $b$ combinations {#sec-b_combinations}

```{r, fig.width=6, fig.height=6, fig.align="center", out.width="55%"}
```{r, fig.width=6, fig.height=6, fig.align="center", out.width="60%"}
x <- seq(0, 20, by=0.1)
L1 <- 5
L2 <- 80
Expand Down Expand Up @@ -229,8 +234,8 @@ The one value of $b$ that is not allowed when using the Richards growth
curve in Stock Synthesis is $b\!=\!0$. This also holds for the R
functions `ss3_code()` and `schnute()` used in this vignette. When
estimating $b$ as a floating-point number, there is effectively no risk
of the parameter having the value of precisely zero, as long as the
initial value is non-zero. To use a Gompertz growth curve, the $b$
of the parameter becoming precisely zero during estimation, as long as
the initial value is non-zero. To use a Gompertz growth curve, the $b$
parameter can be fixed at a small value such as 0.0001, as demonstrated
in @sec-special.

Expand Down Expand Up @@ -267,39 +272,27 @@ $$L_{\infty R} ~=~ L_1^b \;+\; \frac{L_2^b\,-\,L_1^b}{1-e^{-k(A_2-A_1)}}$$
We proceed by replacing $L_{\infty R}$ with its definition and then
gradually simplify the equation:

$$
L_t ~=~ \bigg[\,L_1^b \;+\; \frac{L_2^b-L_1^b}{1-e^{-k(A_2-A_1)}} \;+\;
\begin{eqnarray*}
L_t &=& \bigg[\,L_1^b \;+\; \frac{L_2^b-L_1^b}{1-e^{-k(A_2-A_1)}} \;+\;
\left(L_1^b\,-\,\Big[L_1^b \,+\,
\frac{L_2^b-L_1^b}{1-e^{-k(A_2-A_1)}}\Big]\right)
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e^{-k(t-A_1)}\,\bigg]^{1/b}
$$

$$
L_t ~=~ \bigg[\,L_1^b \;+\; \frac{L_2^b-L_1^b}{1-e^{-k(A_2-A_1)}} \;+\;
e^{-k(t-A_1)}\,\bigg]^{1/b}\\[4ex]
L_t &=& \bigg[\,L_1^b \;+\; \frac{L_2^b-L_1^b}{1-e^{-k(A_2-A_1)}} \;+\;
\left(L_1^b\,-\, L_1^b \,-\, \frac{L_2^b-L_1^b}{1-e^{-k(A_2-A_1)}}
\right) e^{-k(t-A_1)}\,\bigg]^{1/b}
$$

$$
L_t ~=~ \bigg[\,L_1^b \;+\; \frac{L_2^b-L_1^b}{1-e^{-k(A_2-A_1)}} \;-\;
\frac{L_2^b-L_1^b}{1-e^{-k(A_2-A_1)}}\, e^{-k(t-A_1)}\,\bigg]^{1/b}
$$

$$
L_t ~=~ \bigg[\,L_1^b \;+\; (L_2^b-L_1^b)\,\frac{1}{1-e^{-k(A_2-A_1)}} \;-\;
(L_2^b-L_1^b)\,\frac{e^{-k(t-A_1)}}{1-e^{-k(A_2-A_1)}}\,\bigg]^{1/b}
$$

$$
L_t ~=~ \left[\:L_1^b\;+\;(L_2^b-L_1^b)\,
\right) e^{-k(t-A_1)}\,\bigg]^{1/b}\\[4ex]
L_t &=& \bigg[\,L_1^b \;+\; \frac{L_2^b-L_1^b}{1-e^{-k(A_2-A_1)}} \;-\;
\frac{L_2^b-L_1^b}{1-e^{-k(A_2-A_1)}}\, e^{-k(t-A_1)}\,\bigg]^{1/b}\\[4ex]
L_t &=& \bigg[\,L_1^b \;+\; (L_2^b-L_1^b)\,\frac{1}{1-e^{-k(A_2-A_1)}} \;-\;
(L_2^b-L_1^b)\,\frac{e^{-k(t-A_1)}}{1-e^{-k(A_2-A_1)}}\,\bigg]^{1/b}\\[4ex]
L_t &=& \left[\:L_1^b\;+\;(L_2^b-L_1^b)\,
\frac{1-e^{-k(t-A_1)}}{1-e^{-k(A_2-A_1)}}\,\right]^{1/b}
$$
\end{eqnarray*}


```{r}
schnute <- function(t, L1, L2, k, b, A1, A2)
{
(L1^b + (L2^b-L1^b) * ((1-exp(-k*(t-A1))) / (1-exp(-k*(A2-A1)))))^(1/b)
(L1^b + (L2^b-L1^b) * (1-exp(-k*(t-A1))) / (1-exp(-k*(A2-A1))))^(1/b)
}
```

Expand Down