chapter 8. Estimation within-population models: application
and τ true = 100. The choice of the covariance matrix actually corresponds to Σtrue = diag{(ηtrue/20)2} and that of the precision amounts to a multiplicative normal observation noise with standard deviation σtrue = (τ true)−1/2 = 0.1. Data were simulated for 24 individuals: this was done on a single process and then broadcasted to all others, the underlying motivation being that all processes obviously need to have the same experimental data for the different individuals. For the i-th individual, a set of true individual parameters was first sampled from the true population distribution:
θtrue ∼ N (ηtrue, Σtrue) (8.4)
and then used for the simulation of true leaf areas (av )1:ν max,1:ns:T (hidden states) and their corresponding
noised values (a˜v)1:ν max,1:ns:T (observations) where:v i,n v i,n (1 + ξv,n) with ξv ∼ N (0, (τ true)−1) . (8.5)
The areas of all leaves were observed on every day for 21 days which, taking into account that the phyllochron ϕ is fixed and given the rate of appearance of each leaf, amounts to observing 16 leaves over the whole growth and a number of observations of ni = 136 for each individual i, i.e. a total number of observations
of ntot = ∑N ni = 3264. We denote by τv = (tv , . . . , tv v ) the timeline for the v-th leaf and by
α˜v = (a˜v , . . . , a˜v v ) the corresponding observations on leaf area at these time steps. The vector of all
concatenated observations thus reads:
νmax yi = (α˜1, . . . , α˜νmax ) ∈ Rni with ni = ∑ nv. (8.6)
The calculations were performed on as many processes as individuals, i.e. on np = 24 processes. For repro- ducibility issues, the seed of the random number generator was fixed so that the same data was generated all the time. However, for the estimation part and as soon as the experimental data is loaded on the different processes, the seed is reset to a random value on each process.
The flattened vector of experimental data for all individuals is denoted as Y = (y1, . . . , yN ) ∈ Rntot . Examples of simulated data is displayed on Figures 8.1 and 8.2. On Figure 8.1, the leaf areas for each of the first eight leaves are displayed for the 24 individuals and on Figure 8.2 are displayed some examples of typical growth curves for the first eight leaves of several individuals. Despite the rather low population covariance used, there is still a high variability in the leaf areas. This is easily seen from Figure 8.2 when comparing, for instance, the 6th and 8th individuals, the latter having maximum leaf areas twice as high as the former.
8.1.2 Initialization of the prior distributions
Defining the prior distributions of the population parameters is of crucial importance. A first step consists in obtaining realistic values for the individual parameters θi. This can be done rather easily by performing a GLS procedure on each individual i. These first sets of parameters can be used to compute reasonable estimates of the population mean vector η and covariance matrix Σ. A first estimate of the individual set is therefore independently computed
for all these reasons, we will first investigate the case of synthetic data simulated directly from the GreenLab model. This ensures that there is no uncertainty related to either the plant model or the image analysis algorithm and that the Bayesian parameter estimation method effectively works.
8.1 Simulated data
Data were simulated using the GreenLab model presented in Section 2.4. The control variables were taken to be the same as for the experimental conditions described in Section 7.1. The number of hours per day was set to ns = 8h, and the daily temperature to tn = 21◦C. The photosynthetically active radiation was taken to be rn = 2.52 10−5MJ • cm2 • h−1. Each leaf was attributed a specific time of appearance: the first two leaves appeared at n = 1h, the third and fourth leaves appeared at n = 24h. The time of appearance for the fifth leaf was set to n5 = 40h. After that, the 5 + i-th leaf, for i > 1, appears at time n5+i = n5 + i ϕ, where ϕ = 12h is the phyllochron. We focus on the estimation of the parameters θ = (e, µ1, µ2), even though more parameters will be estimated in Sections 8.1.8 and 8.2.
8.1.1 Simulation of synthetic data
The dimension of the estimation problem is therefore d = 3. All the other parameters were fixed at constant values throughout both the simulation of data and the estimation procedure. These values were:
µ = 3.150 • 10+00,
s = 5.000 • 10+00,
k = 7.000 • 10−01,
σ1 = 4.593 • 10−01,
σ2 = 3.991 • 10−01,
(8.1)
ρ2 = 7.801 • 10−02,
q0 = 3.807 • 10−05.
The parameters varying in the population are given true values for their mean and covariance matrix. In the present case:
and:
1.558 • 10−03
ηtrue = +00
5.390 • 10+00
6.069 • 10−09 0 0
(8.2)
Σtrue =0 5.131 • 10−02 0
(8.3)
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