Matrix models using fine size classes and their application to the population dynamics of tree species: Bayesian non‐parametric estimation

Abstract

Matrix models have been widely used to investigate the population dynamics of plant species. To make use of this method, we first divide individuals into groups and estimate transition probabilities per pair of groups. When a continuous variable, such as plant size, is used for grouping, there is often a trade-off: if the class intervals are narrow each group will only include a small number of samples, but if the intervals are wider, this may obscure some changes. This paper introduces a new matrix model in which we no longer have to divide individuals into arbitrarily defined size classes. The methodology is based on the Bayesian non-parametric binary regression. We first divide the data into 'very fine' intervals. For estimating transition probabilities in a 'large' matrix, we do not use the observed transition rate per class directly, but we smooth neighboring observed rates and select the most appropriate degree of smoothing using an information criterion called the Akaike Bayesian Information Criterion (ABIC). Our approach is illustrated using longterm forest monitoring data from an old-growth, warm-temperate evergreen forest, in which we examined the population dynamics of four evergreen subcanopy tree species. Transition probabilities allowed us to represent d.b.h.-related growth and mortality patterns graphically, and matrix analysis provided stable size distributions, reproductive values and elasticity that vary smoothly for trees of different sizes. The quantitative approach makes it possible to determine characteristic patterns of population dynamics for qualitatively similar species.

Multinomial logit estimation of a diameter growth matrix model of two Mediterranean pine species in Spain

“…Division into size classes inevitably involves some subjectivity. A diameter class width similar to ours was used by Shimatani et al (2007) for forests on Tsushina Island. In other forest types, López et al (2008) estimated population growth rates with respect to the width of the diameter classes using matrix models for beech (Fagus sylvatica L.) stands in Navarra (Spain).…”

“…As we can see, different width of size-classes were used in previous studies, while our proposal used very fine classes and calculated a transition probability for every pair of classes. In population dynamics of tree species in Japanese forests, matrix models constructed based on Bayesian non-parametric estimation used also fine diameter classes (Shimatani, Kubota et al, 2007). In summary, many authors argue that narrow categories (1.3-3 cm DBH) should be used in tree matrix models to obtain the best estimations of tree age, population growth rate and elasticity.…”

“…Matrix models normally require grouping trees into size classes that are usually based on diameter at breast height (Picard, Bar-Hen et al, 2003;Shimatani, Kubota et al, 2007). Diameter increment (Lowell and Mitchell, 1987) is a useful proxy for tree biomass growth and can be predicted by probabilistic models based on multinomial distribution.…”

Matrix Dimensions Bias Demographic Inferences: Implications for Comparative Plant Demography

“…This is a nontrivial issue because matrix dimension varies a great deal, from two (Sohn and Policansky 1977) to 24 (Meagher 1982). Although several algorithms have been proposed to determine optimum criteria to establish cutoffs for classes in constructing the original matrices (Vandermeer 1978;Moloney 1986;Shimatani et al 2007), thus offering some standardization, most plant demographers still classify systems on the basis of the specific biology of the species and the question being asked (Caswell 2001).…”

Growth Model of Remmant Stands in Selectively Logged Forest, Papua

“…Variasi atau lebarnya rentang nilai R² tersebut menunjukkan bahwa peranan peubah bebas dalam menerangkan upgrowth mungkin bersifat spesifik untuk setiap lokasi. Disisi lain, lebar kelas diameter dapat mempengaruhi model pertumbuhan diameter (Escalante et al, 2011;Shimatani, Kubota, Araki, Aikawa, & Manabe, 2007).…”