Solutions to the Base-Age Variant Models

Chris J Cieszewski


Self-referencing models predict the value of Y at age t as a function of both t and a snapshot
observation of Y = Y 0 at t = t 0 , which implicitly integrates the entire environment affecting the development
of Y . Common examples of such models are site-dependent height over age models, or site index models,
hereafter referred to as site models. These models are often developed using pooled cross-sectional and
longitudinal data and describe families of multiple curve shapes.
It is advantageous to formulate these models as algebraic difference equations, which can be referred to as
“dynamic equations,” with their reference variables describing the environment or site quality. For example,
in height modeling, site models predict height as a function of age and a height at a base-age known as the
site index.
The base-age specific modeling ideology suggests that curves generated by these models are unique to a
particular selection of base-age, at which the input data or site index is defined during the estimation
of model parameters. Base-age variant models are designed to capture some of the patterns of curves
corresponding to different base-ages through a single formula. The curves generated by this approach vary
with base-ages and with various methods in which the models can be applied.
However, the available base-age variant models have been limited in their usage to avoid inconsistent
predictions and cannot be considered equations in the algebraic sense since they can show that 1 = 0. To
address this issue, I present a mathematical approach that leads to the derivation of a new type of proper
base-age invariant equations, which can be applied in various alternative ways for the same purpose as the
base-age variant models, but without creating mathematical inconsistencies.


Site models; site index modeling; GADA models; self-referencing functions; base-age invariant; base-age variant; path-invariant.

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