Growth model of the reared sea urchin Paracentrotus ... - SciViews

Finally, ∆D∞ should follow a normal distribution, as size distributions
when approaching asymptotic maximum size are normal or close to
normal (see Fig. 28A, t > 1500 days, and also Grosjean et al, 1996, see
Part III):
with µ D∞
Part IV: A growth model with intraspecific competition
∆D ( τ) ∼ N ( µ , σ )
(34)
∞ ∆D ∆D
∆ being the mean and σ ∆D∞
normal distribution of ∆D∞(τ).
∞ ∞
Replacing eqs. 32-34 into eq. 31, we get:
−k1⋅t' 1+ e
D'( t', τ) =∆D
( τ)
1 −s⋅(1 −τ) ⋅e
being the standard deviation of the
∞ −k2⋅t' (35)
which links curves for all quantiles 0 < τ < 1 and has 5 parameters to be
estimated: k1, k2, s, µ D∞
∆ and σ ∆ D∞
. It includes individual variations into
the model and is a kind of 3D-surface that envelops data (see Fig. 33). For
this reason, it will be called an 'envelope model'.
The quantile regression method is modified as follows. Considering
that every individual present in the batch is measured at each sampling
time, unconditional quantiles at each size distribution can be regarded as
estimators of conditional quantiles τ at corresponding time t' in eq. 35
(note that this is fundamentally different than the previous quantile
regression method in eqs. 21-22 where unconditional quantiles were not
used at all in the regression). Estimators of τ, noted ˆ τ , are then calculated
as:
ˆ τ =
i
nt' ( i )
∑
j=
1
( D'j t'i < D'i)
I ( )
nt' ( )
i
(36)
where t'i is t' corresponding to the i th observation, n(t'i) is the total number
of individuals measured at time t'i, D'j(t'i) is the j th observation among all
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