Likelihood, entropy and species diversity; some comparisons in a Sumatran forest

In attempts to conserve the species diversity of trees in tropical forests, monitoring of diversity in inventories is essential. For effective monitoring it is crucial to be able to make meaningful comparisons between different regions, or comparisons of the diversity of a region at different times. Many species diversity measures have been defined, including the well-known abundance and entropy measures. All such measures share a number of problems in their effective practical use. However, probably the most problematic is that they cannot be used to meaningfully assess changes, since thay are only concerned with the number of species or the proportions of the population/sample which they constitute. A natural (though simplistic) model of a species frequency distribution is the multinomial distribution. It is shown that the likelihood analysis of samples from such a distribution are closely related to a number of entropy-type measures of diversity. Hence a comparison of the species distribution on two plots, using the multinomial model and likelihood methods, leads to generalised cross-entropy as the LRT test statistic of the null that the species distributions are the same. Data from 30 contiguous plots in a forest in Sumatra are analysed using these methods. Significance tests between all pairs of plots yield extremely low p-values, indicating strongly that it ought to been "Obvious" that the observed species distributions are different on different plots. In terms of how different the plots are, and how these differences vary over the whole study site, a display of the degrees of freedom of the test, (equivalent to the number of shared species) seems to be the most revealing indicator, as well as the simplest.

Bivariate distribution modeling with tree diameter and height data

The Logit-Logistic (LL), Johnson's SB, and the Beta (GBD) are flexible four-parameter probability distribution models in terms of the (skewness-kurtosis) region covered, and each has been used for modeling tree diameter distributions in forest stands. This article compares bivariate forms of these models in terms of their adequacy in representing empirical diameter-height distributions from 102 sample plots. Four bivariate models are compared: SBB, the natural, well-known, and much-used bivariate generalization of SB; the bivariate distributions with LL, SB, and Beta as marginals, constructed using Plackett's method (LL-2P, etc.). All models are fitted using maximum likelihood, and their goodness-of-fits are compared using minus log-likelihood (equivalent to Akaike's Information Criterion, the AIC). The performance ranking in this case study was SBB, LL-2P, GBD-2P, and SB-2P

The Richit-Richards family of distributions and its use in forestry

Johnson's SB and the logit-logistic are four-parameter distribution models that may be obtained from the standard normal and logistic distributions by a four-parameter transformation. For relatively small data sets, such as diameter at breast height measurements obtained from typical sample plots, distribution models with four or less parameters have been found to be empirically adequate. However, in situations in which the distributions are complex, for example in mixed stands or when the stand has been thinned or when working with aggregated data, then distribution models with more shape parameters may prove to be necessary. By replacing the symmetric standard logistic distribution of the logit-logistic with a one-parameter “standard Richards” distribution and transforming by a five-parameter Richards function, we obtain a new six-parameter distribution model, the “Richit-Richards”. The Richit-Richards includes the “logit-Richards”, the “Richit-logistic”, and the logit-logistic as submodels. Maximum likelihood estimation is used to fit the model, and some problems in the maximum likelihood estimation of bounding parameters are discussed. An empirical case study of the Richit-Richards and its submodels is conducted on pooled diameter at breast height data from 107 sample plots of Chinese fir (Cunninghamia lanceolata (Lamb.) Hook.). It is found that the new models provide significantly better fits than the four-parameter logit-logistic for large data sets.