The bivariate Power-Normal and the bivariate Johnson’s System bounded distribution in forestry, including height curves. Supplementary material.

Abstract

This document contain supplementary material concerning two published papers: Mønness, E. 2011b. The Power-Normal Distribution: Application to forest stands. Canadian Journal of Forest Research 41(4): 707-714. doi: 10.1139/X10-246. Mønness, E 2014. The bivariate Power-Normal and the bivariate Johnson’s System bounded distribution in forestry, including height curves. Canadian Journal of Forest Research. 2015 Supplementary material to the above article; SAS programs and computing details are found in Mønness, E. 2011a. The Power-Normal Distribution and Johnsons System bounded distribution: Computing details and programs. Available from http://hdl.handle.net/11250/133513 3] . From published abstracts: The Power-Normal (PN) distribution, originated from the inverse Box-Cox transformation, is presented and some possibilities in forest research are explored. The Power-Normal achieve shapes, by a Skewness * Kurtosis value, common to diameters and heights of forest stands. The estimation of the parameters by maximum likelihood is straightforward with good numerical properties. The shapes achieved by PN are very diverse even with only three parameters: The Johnson System bounded distribution (SB), also used in forestry, can encounter numerical problems with maximum likelihood estimation. The PN distribution is seen to give good estimates of diameter and height distributions, judged by the Kolmogorov-Smirnov statistic and visual inspection. It seems to perform better than the SB, especially on heights. A bivariate diameter and height distribution yields a unified model of a forest stand. The bivariate Johnson’s System bounded distribution and the bivariate power-normal distribution are explored. The power-normal originates from the well-known Box-Cox transformation. As evaluated by the bivariate Kolmogorov-Smirnov distance, the bivariate power-normal distribution seems to be superior to the bivariate Johnson’s System bounded distribution. The conditional median height given the diameter is a possible height curve and is compared with a simple hyperbolic height curve. Evaluated by the height deviance, the hyperbolic function yields the best height prediction. A close second is the curve generated by a bivariate power-normal distribution. Johnson’s System bounded distributions suffer from the sigmoid shape of the association between height and diameter. The data is from Vestjordet, E. 1977. Precommercial thinning of young stands of Scots Pine and Norway Spruce I: Data stability, dimension distribution etc. Medd. Nor. inst. skogforsk 33(9): 1-436.