A modified growth function with interpretable parameters applied to the age–height relationship of individual trees

Abstract

Growth functions frequently used in forestry have in common that among the model parameters to be estimated, only the asymptote is expressed in the dimensions of the input data. By contrast, parameters determining rate and shape of the curve often exhibit indefinite scales. This might cause problems in specifying adequate starting values and in parameter interpretation. We present a mathematical derivation to obtain a modified growth function based on the four-parameter Richards function. Two of the rate and shape parameters were replaced by new parameters directly related to the growth process: time of maximum growth and maximum growth rate. Both the original model and its modified form were fitted to individual-tree height–age data from the National Forest Inventory in Germany. The modified function has several advantages: (i) easier interpretability of model parameters, (ii) easier specification of starting values, (iii) improved linear behavior allowing for more reliable asymptotic inferences and for better convergence, and (iv) reduced correlation between model parameters. As a further benefit, the presented model allows for deriving biologically interpretable forms of the Gompertz function, the von Bertalanffy function, and the logistic function. Based on the results, we suggest using the modified function provided for further applications in growth modeling.