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Stochastic or deterministic single-tree models: is there any difference in growth predictions?

Abstract

Introduction

Deterministic single-tree models are commonly used in forestry. However, there is evidence that stochastic events may interact with the nonlinear mechanisms that underlie forest growth. As a consequence, stochastic and deterministic simulations could yield different results for the same single-tree model and the same initial conditions. This hypothesis was tested in this study.

Material and methods

We used a single-tree growth model that can be implemented either stochastically or deterministically. Two data sets of 186 and 342 plots each were used for the comparisons. For each plot, the simulations were run on a 100-year period using 10-year growth steps. Three different response variables were compared.

Results

The results showed that there were differences between the predictions from stochastic and deterministic simulations for some response variables and that randomness alone could not explain these differences. In the case of deterministic simulations, the fact that predictions are reinserted into the model at each growth step is a concern. These predictions are actually random variables and their transformations may result in biased quantities. Forest growth modellers should be aware that deterministic simulations may not correspond to the mathematical expectation of the natural dynamics.

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Acknowledgements

The authors are thankful to Mrs. Isabelle Auger of the Natural Resources and Wildlife Ministry of Quebec-Forest Research Branch (Ministère des Ressources Naturelles et de la Faune du Québec-Direction de la Recherche Forestière) and to two anonymous reviewers for their helpful comments on a preliminary version of this paper.

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Correspondence to Mathieu Fortin.

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Handling Editor: Daniel Auclair

Appendix

Appendix

Random vector drawn from a multivariate normal distribution

Let us consider the vector ε as a random vector that follows a multivariate normal distribution with mean μ and covariance V, i.e., ɛ∼MVN(μ, V). The Cholesky decomposition provides the lower triangular matrix A that satisfies the condition V = AA T. If y is a vector of independent standard normal random variates (i.e., all independently and normally distributed with mean 0 and variance 1), the sum μ + Ay yields a random vector from the desired multivariate distribution.

For example, let us consider the bivariate case where:

$$ {\mathbf{\mu }} = \left[ {\begin{array}{*{20}{c}} 0 \\ 0 \\ \end{array} } \right]\;{\text{and}}\;{\mathbf{V}} = \left[ {\begin{array}{*{20}{c}} {0.700} & {0.500} \\ {0.500} & {2.000} \\ \end{array} } \right] $$

The Cholesky decomposition of V yields:

$$ {\mathbf{A}} = \left[ {\begin{array}{*{20}{c}} {0.837} & {0.000} \\ {0.598} & {1.282} \\ \end{array} } \right] $$

Now, let us draw a random vector of independent standard normal variates where:

$$ {\mathbf{y}} = {(0.246,{ } - {1}{.976)}^{\text{T}}} $$

The sum of μ + Ay, i.e.,

$$ \left[ {\begin{array}{*{20}{c}} 0 \\ 0 \\ \end{array} } \right] + \left[ {\begin{array}{*{20}{c}} {0.837} & {0.000} \\ {0.598} & {1.282} \\ \end{array} } \right] \times \left[ {\begin{array}{*{20}{c}} {0.246} \\ { - 1.976} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}{c}} {0.206} \\ { - 2.386} \\ \end{array} } \right] $$

yields a vector that follows a MVN distribution with mean μ and covariance V.

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Fortin, M., Langevin, L. Stochastic or deterministic single-tree models: is there any difference in growth predictions?. Annals of Forest Science 69, 271–282 (2012). https://doi.org/10.1007/s13595-011-0112-0

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Keywords

  • Growth modelling
  • Monte Carlo simulation
  • Bias
  • Single-tree models
  • Stochastic
  • Deterministic