Applications
Fitting
Plant structure fitting - Development
In this section, fitting of the growth development process parameters for continuous growth is first described in its simplest case (constant growth)
Fitting of the single Bernoulli process
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Let us assume that, on an axis developing continuously, the appareance of a new phytomer follows a simple constant law.
The shape of the phytomer number distribution produced over a given period is classically a bell shape.
Fitting this distribution by a binomial law B(N,b) makes it possible to retrieve the period of development N (expressed in growth cycles) and the Bernoulli parameter b.
We should, however, consider the stability of this parameter b, studying the expected value- variance relation, assumed to be linear, if the process follows a binomial distribution.
In such a case, parameter and age fitting is easily defined from the distribution expected value X and the variance V, since
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b = 1 - V / X
and
N = X / b
Fitting can be performed at different stages of growth or at a specific time (for instance at the end of growth for annual crops).
Example. Fitting of a cotton tree
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The following example involvesn the main stems of 50 cotton trees. Two approaches are considered.
Method 1: following expected values and variance relations at several growth stages
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If the development process is a Bernoulli process, the mean and variance are proportional to the
Bernoulli parameter (V = (1 - b). X).
The expected value increase is of course obvious when considering growth, thus several growth stages are necessary to build a significant data set.
In this example, 6 growth steps where studied on a population of 50 cotton trees .
Their respective expected values and variances are given below:
| Expected value (X) | 0, | 4.15, | 4.88, | 7.30, | 10.12, | 17.70, | 20.68 |
| Variance (V) | 0, | 0.38, | 1.05, | 0.58, | 1.12, | 3.46, | 3.49 |
A linear regression (see "../../P3_Tools/Tool_lreg_001.html ) leads to :
V = 0.18 X (r=0.96) and thus gives b = 0.82 (+/- 0.04)
Method 2: analysing the distribution of the phytomer number at a given growth stage
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At the end of growth, the number of phytomers is recorded on each plant main axis.
The expected value and variance are estimated to define the Bernoulli probability.
| Number of internodes | 16, | 17, | 18, | 19, | 20, | 21, | 22, | 23, | 24 |
| Number of plants | 1, | 4, | 2, | 4, | 6, | 16, | 12, | 2, | 3 |
The expected value is X = 20.68 and variance V=3.42
Leading to
b = 1 - (3.42 / 20.68) = 0.83
Defining the number of growth cycles
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The number of growth cycles is simply derived from the definition of the expected value, thus giving:
N = X / b
At the crop stage (X=20,68), methods 1 and 2 give the following results, respectively:
N = 20.68 / 0.82 = 25.22 and N = 20.68 / 0.83 = 24.86
The number of growth cycles is thus N = 25.