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Stochastic modelling implementation

Stochastic structural modelling can easily be implemented on the dualscale automaton
A substructure strategy can be kept for efficient stochastic simulations
Structural factorization can thus also be kept, still allowing formal structure mathematical numbering.
Stochastic automaton

Principles

The dualscale automaton can easily simulate stochastic transitions for both micro and macroscales.
The principle is that each transition carries a probability of each process (development, mortality, branching) occurring.
Be proceeding in this way, implementation respects the modelling principles, inspired from renewal theory.
Continuous growth

The Bernoulli process can be straightforwardly applied to microstate (phytomers) transitions.
And of course the rhythm ratio and the viability process too.
In practice the rhythm ratio defines first whether or not a Bernoulli trial needs to be performed; then viability c is tested; if the terminal bud is still alive, development is finally tested (choosing a random number compared to the probability b).
Applying the process growth cycle by growth cycle, the simulated axis can be encoded by single "1" and "0" codes, respectively standing for phytomer occurrence and rest.
Rhythmic growth
In theory, the same approach could be implemented for both micro and macroscales in the case of rhythmic growth.
However, it is often more efficient to defined the production of the axis for the whole growth unit (the macrostate), and then distribute it (proportionally among microstates).
First, a random number K of phytomers is defined from the binomial law B(N,b), where N stands for the number of effective growth cycles (after a potential rhythm ratio filter).
The corresponding macrostate thus holds K phytomers, with those mapped on the microstate sequence defining the macrostate.
The same approach applies to polycyclism, with preformed and neoformed parts. In this case the K definition from the binomial law B(N,p) is replaced by Kp + Kn where Kp stands for the number of phytomers in the preformed part (also derived from a binomial law) and Kn stands for the number of phytomers in the neoformed part (derived from a bionomial or a negative binomial law).
As a result, the construction process leads to a similar output. The simulated axis is encoded by single "1" and "0" codes, respectively standing for phytomer occurrence and rest describing the growth unit sequence. Usually, according to their botanical definition, simulated growth units are separeted from each the other by rest sequences (list of "0").
Branching

Each microstate may carry several whorls of lateral buds of different physiological ages.
Each physiological age branching is tested, i.e. the delay expressed in the growth cycle is estimated.
When coupling is modelled, branching simulation is potentially controlled by the branching results of the previous phytomer.
Stochastic dual scale automaton (Images X. Xhao, LiamaCASIA and P. de Reffye, CIRAD)
 The dual scale automaton transitions are controlled by probabilities.
At microscale level, transitions are controlled by the development and the viability probabilities
b and c applied to the first microstate sequence, while b', and c' apply to the first microstate to second microstate sequence.
In the GreenLab model implementation those parameters are identical within macrostates (b=b' and c=c')
Branching probabilities (including delays) are processed by the lower transitions (dotted arrows).
Stochastic substructures

The use of substructures can also be extended to the stochastic case.
Each deterministic substructure is replaced by a set of a limited number of substructures, as representative of the substructure distribution for the various phytomer typologies (in terms of expected value and variance).
In a first step, the different substructure sets are built, starting from the older physiological ages to younger.
At each branching, or physiological mutation, the substructures are chosen from ones already created.
For practical reasons, the number of representatives is fixed at the same value, in order to minimize storage and construction costs.
Simulating stochastic substructures (Images H.P. Yan, M.G. Kang, LiamaCASIA and P. de Reffye, CIRAD)
 In this example each substructure group has five stochastic representatives.
Each representative is built using the higher substructure groups.
Each substructure group shows appropriate statistical properties (the sample's expected value and variance fit the theoretical values)