Using a chain of models to simulate the distribution between quality clusters of a set of boards coming from a sessile oak (Quercus petraea Liebl.) resource. Validation of the simulations

Abstract

Taking sessile oak as an example, this paper initially presents a method to predict the final production (quantity and quality) coming from a forest resource when two sets of data are available. The data sets are from two models: measured or simulated ring width profiles from pith to bark of the constituent trees as well as a mixed model for the basic wood properties which are used to grade the boards into quality clusters. The second part of the paper contains a validation for the proposed method. Simulations are used to predict two basic wood properties (volumetric swelling coefficient and wood density) in the trees of a forest resource in relation to the ring width profile of each tree. The simulations are used to compute a map of these two basic properties in each plank derived from the trees. A quality index derived from this map of basic wood properties in the boards is then used to allocate the planks to quality clusters. The basic wood properties considered in this paper are modelled with linear mixed models. Since computation of the plank properties or definition of the grading rule can use several properties simultaneously, the models used to simulate the basic properties are joint models. Modelling jointly several properties with a mixed model consists of defining a covariance structure between the random effects of the model. Such a model can be substantial in terms of parameters and computational resources required, thus we compared three kinds of joint models. The simplest one is not quite a joint model but is simply obtained from the juxtaposition of independent models, one for each of the two properties taken into consideration. We also defined a model with a moderate covariance structure between the two properties, and lastly, we used a third model with a full covariance structure. Simulations of volumetric swelling coefficient, wood density and the resulting board grading were carried out with each of these three models. All give results roughly in accordance with the observations, but the two “truly” joint models give better results than the simplest model.

Appendix: Simulating properties according to a linear mixed model

With matrix notations, a linear mixed model can be written as:

$${\mathbf{P}}_{{ij}} = {\mathbf{X}}_{{ij}} {\mathbf{\beta }} + {\mathbf{Z}}_{{ij}} {\mathbf{u}}_{i} + {\mathbf{e}}_{{ij}} $$

where P _ij is the vector of properties for cube j in tree i, X _ij and Z _ij are respectively the incidence matrixes of the fixed and random effects for cube j in tree i, β is the vector of fixed effects, u _i is the vector of random effects for the tree i (that is the “tree effect”), e _ij is the vector of residual effects for cube j in tree i.

Matrixes X _ij and Z _ij are known. They contain the characteristics of the ring width profile (ring width, age from the pith, etc.). Vectors u _i and e _ij have to be simulated according to centred multidimensional normal laws with variance matrixes G and R respectively.

Vector β and matrixes G and R have been estimated and are here considered as known.

The most technical point in the simulation of the properties is how to simulate a normal random vector v according to a given variance matrix V.

Let w be a p-dimensional random normal centred vector with variance matrix I _p , where I _p is the (p*p) identity matrix. The p components of such a vector are independent and can be simulated from p independent simulations of a standard normal random variable, one simulation per component.

Let L be any (p*p) matrix. As a classical statistical result, the vector (Lw) is a centred normal random vector with variance matrix LI _p L’= LL’, where L’ is the transposition of L.

If the matrix L is chosen so that we have LL’=V, the vector v=Lw has a centred normal distribution with variance matrix V.

Since V is a variance matrix, it is a semi-definite positive matrix. For a given V, in general, several matrixes L exist so that LL’=V. One of the different solutions is to choose L as the Cholesky decomposition of V.

To simulate the basic wood properties, one simulation of u _i per tree with the variance matrix G is needed, and one simulation of e _ij per cube with the variance matrix R.

Simulation of standard normal law and matrix operations (addition, multiplication, Cholesky decomposition, etc.) are available in most scientific programming languages, either directly or from routine libraries. If all the parameters of the model are known, there is no difficulty in programming simulation of properties according to a linear mixed model.