self start for the reparameterized Beta growth function with four parameters
Self starter for Beta Growth function with parameters w.max, lt.m, ldt
bg4rp(time, w.max, lt.e, ldtm, ldtb) SSbg4rp(time, w.max, lt.e, ldtm, ldtb)
time |
input vector (x) which is normally ‘time’, the smallest value should be close to zero. |
w.max |
value of weight or mass at its peak |
lt.e |
log of the time at which the maximum weight or mass has been reached. |
ldtm |
log of the difference between time at which the weight or mass reaches its peak and half its peak. |
ldtb |
log of the difference between time at which the weight or mass reaches its peak and when it starts growing |
For details see the publication by Yin et al. (2003) “A Flexible Sigmoid Function of Determinate Growth”.
This is a reparameterization of the beta growth function (4 parameters) with guaranteed constraints, so it is expected to
behave numerically better than SSbgf4.
Reparameterizing the four parameter beta growth
ldtm = log(t.e - t.m)
ldtb = log(t.m - t.b)
t.e = exp(lt.e)
t.m = exp(lt.e) - exp(ldtm)
t.b = (exp(lt.e) - exp(ldtm)) - exp(ldtb)
The form of the equation is:
w.max * (1 + (exp(lt.e) - time)/exp(ldtm)) * ((time - (exp(lt.e) - exp(ldtb)))/exp(ldtb))^(exp(ldtb)/exp(ldtm))
This is a reparameterized version of the Beta-Growth function in which the parameters are unconstrained, but they are expressed in the log-scale.
bg4rp: vector of the same length as x (time) using the beta growth function with four parameters
require(ggplot2) set.seed(1234) x <- 1:100 y <- bg4rp(x, 20, log(70), log(30), log(20)) + rnorm(100, 0, 1) dat <- data.frame(x = x, y = y) fit <- nls(y ~ SSbg4rp(x, w.max, lt.e, ldtm, ldtb), data = dat) ## We are able to recover the original values exp(coef(fit)[2:4]) ggplot(data = dat, aes(x = x, y = y)) + geom_point() + geom_line(aes(y = fitted(fit)))
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