Abstract
BackgroundThe Chapman-Richards distribution is developed as a special case of the equilibrium solution to the McKendrick-Von Foerster equation. The Chapman-Richards distribution incorporates the vital rate assumptions of the Chapman-Richards growth function, constant mortality and recruitment into the mathematical form of the distribution. Therefore, unlike ‘assumed’ distribution models, it is intrinsically linked with the underlying vital rates for the forest area under consideration.MethodsIt is shown that the Chapman-Richards distribution can be recast as a subset of the generalized beta distribution of the first kind, a rich family of assumed probability distribution models with known properties. These known properties for the generalized beta are then immediately available for the Chapman-Richards distribution, such as the form of the compatible basal area-size distribution. A simple two-stage procedure is proposed for the estimation of the model parameters and simulation experiments are conducted to validate the procedure for four different possible distribution shapes.ResultsThe simulations explore the efficacy of the two-stage estimation procedure; these cover the estimation of the growth equation and mortality—recruitment derives from the equilibrium assumption. The parameter estimates are shown to depend on both the sample size and the amount of noise imparted to the synthetic measurements. The results vary somewhat by distribution shape, with the smaller, noisier samples providing less reliable estimates of the vital rates and final distribution forms.ConclusionsThe Chapman-Richards distribution in its original form, or recast as a generalized beta form, presents a potentially useful model integrating vital rates and stand diameters into a flexible family of resultant distributions shapes. The data requirements are modest, and parameter estimation is straightforward provided the minimal recommended sample sizes are obtained.
Highlights
The Chapman-Richards distribution is developed as a special case of the equilibrium solution to the McKendrick-Von Foerster equation
Growth Estimation There is little new per se in our results regarding the estimation of the CR growth function by nonlinear least squares, though the extensive use of the constrained approach in a Monte Carlo setting, which was wellbehaved overall, may be somewhat novel
The focus is not on the efficacy of the constrained nonlinear least squares (CNLS) approach in general, rather the intent is to demonstrate the effects of the assumed noise levels on the estimation of the CR growth parameters by comparing the perturbed results to the population
Summary
The Chapman-Richards distribution is developed as a special case of the equilibrium solution to the McKendrick-Von Foerster equation. De Liocourt (1898) developed a reverse J-shaped model to fit the distribution of diameters in a “normal selection forest.”. In addition to Fourier series, Meyer (1930) discusses several graphical methods for fitting “frequency curves” dating back to the 1880’s; he mentions other mathematical methods which include the normal distribution. In his own application, Meyer (1930) used the Charlier system There has been a long history of foresters who have recognized the need to characterize diameter distribution in some mathematical (including graphical) form for use in management
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