A comparison between derivative and numerical optimization methods used for diameter distribution estimation

Abstract

ABSTRACT Modeling diameter distribution of forest stands requires suitable function(s) with the appropriate parameter estimation methods. To date, the parameters of most familiar functions in forestry have been estimated with derivative methods such as moment, percentiles, conditional maximum likelihood, etc. with little emphasis on numerical optimization. Therefore, this study compares the suitability of derivative and numerical methods to estimate diameter distribution of forest stands. The derivative and optimization methods were used to fit six commonly used functions in forestry including Weibull (2 and 3-parameter), Johnson’s SB, beta, generalized beta and gamma. The data comprised 303 and 96 permanent sample plots temperate forest (Eucalyptus globulus and Pinus radiata, respectively) and 65 temporary sample plots from tropical forest (Gmelina arborea). Different indices such as Kolmogorov–Smirnov, Cramer-von Mises and mean squared error were used to assess the methods. The results show that the derivative method by moments provided the best fit for beta and Johnson’s SB distributions. The optimization methods (“lifereg” and “optim”) were more suitable for the Weibull and gamma distributions. Both methods were appropriate for the generalized beta distribution. Abbreviation: PW2P: Percentile Weibull two-parameter; MW2P: Moments Weibull two-parameter; MLLW2P: Maximum likelihood “lifereg” Weibull two-parameter; MLOW2P: Maximum likelihood “optim” Weibull two-parameter; PW3P: Percentile Weibull three-parameter; MW3P: Moments Weibull three-parameter; MLLW3P: Maximum likelihood “lifereg” Weibull three-parameter; MLOW3P: Maximum likelihood “optim” Weibull three-parameter; CMLJ: Conditional maximum likelihood Johnson’s SB; MJ Moments Johnson’s SB; MLOJ: Maximum likelihood “optim” Johnson’s SB; MB: Moments beta; MLOB Maximum likelihood “optim” beta; MGB: Moments generalized beta; MLOGB Maximum likelihood “optim” generalized beta; MG2P: Moments gamma two-parameter; MLOG2P: Maximum likelihood “optim” gamma two-parameter