nlraa (version 1.2)

SSbg4rp: self start for the reparameterized Beta growth function with four parameters

Description

Self starter for Beta Growth function with parameters w.max, lt.e, ldtm, ldtb

Usage

bg4rp(time, w.max, lt.e, ldtm, ldtb)
SSbg4rp(time, w.max, lt.e, ldtm, ldtb)

Value

bg4rp: vector of the same length as x (time) using the beta growth function with four parameters

Arguments

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

Details

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.

Examples

# \donttest{
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)))
# }