UTADA: Unified theory of the algebraic differences approaches—derivation of dynamic site equations from direct yield-site relationships

Abstract

Dynamic-equation-based self-referencing models of the form Y = f(t, t0, y0) describe changes in Y as a function of a longitudinal variable t and an unobservable cross-sectional variable X, which is implicitly represented by a known snapshot observation of Y, y0, at an arbitrary value of t, t0. The unobservable variable X denotes the environment potential, or site, which cannot be directly measured or precisely defined due to its extreme complexity and variability. While elusive and difficult in handling, X is the most critical variable of the site equations due to its disproportionate impact on the modeled dynamics. All traditional approaches to such modeling are predominantly based on a detailed analysis of primarily longitudinal relationships Y = u(t), which subsequently, to be helpful in practice, are modified into the self-referencing forms, thus incidentally accounting for the site impacts. All the former approaches devote little to no effort to explicitly model the cross-sectional relationships governed by the unobservable variable X.

I hereby present a proof of a concept for a novel approach to derivation of the dynamic-equation-based self-referencing models that unifies the modeling efforts of defining the yield and site relationships equally, by focusing primarily on direct mathematical formulations describing the theory of the yield-site relationships. This approach considers the variable t only in the secondary analysis, adding it to the framework through modifications of the final model parameters. Despite the somewhat elusive nature of exploring the unobservable variable properties of the site, the new approach appears to be highly empowering by analyzing simple and direct yet more robust relationships between Y and X as opposed to those between Y and t. The self-referencing dynamic site equations derived through this approach have all the desirable properties of site models, such as the base-age-invariance, path-invariance, and a high degree of flexibility with complex polymorphism and variable asymptotes.