Stochastic Simulation of Forest Regeneration Establishment Using a Multilevel Multivariate Model

Abstract

Abstract Here we present a method for simulating high unexplained variation in establishment of tree seedlings in planted Norway spruce (Picea abies [L.] Karst.) stands. The simulation method is based on an existing hierarchical multilevel multivariate model for establishment of regeneration (Miina, J., and T. Saksa. 2006. New For. 32:265–283). The model includes seven simultaneously estimated models for numbers and heights of tree seedlings on 20-m2 plots within a regeneration area. The variation in plot-level expectations is described by fixed effects and two nested multivariate normal random effects. Conditionally, given the expected values, one of the count variables is underdispersed and four are overdispersed relative to Poisson law. The conditional distributions of the square roots of the height variables are normal. For the conditional joint distribution, the model defines the product-moment cross-correlation matrix, which includes both positive and negative cross-correlations. The first two moments do not, however, uniquely define the marginal distributions of the count variables or the joint distribution of all seven variables. We applied binomial distribution for the underdispersed Poisson variation and negative binomial distribution for the overdispersed Poisson variation. The conditional joint distribution was defined by a multivariate normal copula and the marginal distributions. Despite relatively low cross-correlations among responses, if the cross-correlations among random components of the responses were ignored in stochastic simulations, the amount and costs of precommercial thinning (i.e., cleaning out competing broadleaves) were greatly underestimated. If all random effect components in model parameters were dropped, the error increased markedly.