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GADA: A Method for Dynamic Equation Derivation

This paper presents the Generalized Algebraic Difference Approach (GADA) for deriving dynamic equations in forest growth modeling. It discusses the methodology for direct use of initial conditions, base age invariance, and biologically interpretable bases. The approach involves identifying suitable models, solving for site parameters, and formulating implicitly defined equations. The GADA offers more flexibility, generality, and parsimony compared to traditional forestry approaches. It allows for modeling forest growth with polymorphism and variable asymptotes, integrating biological theories into unified laws. The paper also emphasizes the parsimonious nature of dynamic equations and the benefits of implicit models over explicit ones.

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GADA: A Method for Dynamic Equation Derivation

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  1. GADA - A Simple Method for Derivation of Dynamic Equation Chris J. Cieszewski and Ian Moss

  2. Variables of Interest: • Height (of trees, people, etc.); • Volume, Biomass, Carbon, Mass, Weight; • Diameter, Basal Area, Investment; • Number of Trees/Area, Population Density; • other ...

  3. Definitions of Dynamic Equations • Equations that compute Y as a function of a sample observation of Y and another variable such as t. • Examples: Y = f(t,Yb), Y = f(t,t0,Y0), H = f(t,S); • Self-referencing functions (Northway 1985); • Initial Condition Difference Equations; • other ...

  4. Example of real data

  5. Basic Rules of Use • 1. When on the line: follow the line; • 2. When between the lines interpolate new line; and • 3. Go to 1.

  6. Examples of curve shape patterns

  7. The Objective: • A methodology for models with: • direct use of initial conditions • base age invariance • biologically interpretable bases • polymorphism and variable asymptotes

  8. The Algebraic Difference Approach (Bailey and Clutter 1974) • 1) Identification of suitable model: • 2) Choose and solve for a site parameter: • 3) Substitute the solution for the parameter:

  9. The Generalized Algebraic Difference Approach (Cieszewski and Bailey 2000) • Consider an unobservable Explicit site variable describing such factors as, the soil nutrients and water availability, etc. • Conceptualize the model as a continuous 3D surface dependent on the explicit site variable • Derive the implicit relationship from the explicit model

  10. Stages of the Model Conceptualization:

  11. The Other Examples

  12. The GADA • 1) Identification of suitable longitudinal model: • 2) Definition of model cross-sectional changes: • 3) Finding solution for the unobservable variable: • 4) Formulation of the implicitly defined equation:

  13. A Traditional Example • 1) Identification of suitable longitudinal model: • 2) Anamorphic model (traditional approach): • 3) Polymorphic model with one asymptote (t.a.):

  14. Proposed Approach (e.g., #1) • 1) Identification of suitable longitudinal model: • 2) Def. #1: • 3) Solution: • 4) The implicitly defined model:

  15. Proposed Approach (e.g., #2) • 1) Identification of suitable longitudinal model: • 2) Def. #2: • 3) Solution: • 4) The implicitly defined model:

  16. Proposed Approach (e.g., #3) • 1) Identification of suitable longitudinal model: • 2) Def. #3: • 3) Solution: • 4) The implicitly defined model:

  17. Proposed Approach (e.g., #4) • 1) Identification of suitable longitudinal model: • 2) Def. #4: • 3) Solution: • 4) The implicitly defined model:

  18. 1) Conclusions • Dynamic equations with polymorphism and variable asymptotes described better the Inland Douglas Fir data than anamorphic models and single asymptote polymorphic models. • The proposed approach is more suitable for modeling forest growth & yield than the traditional approaches used in forestry.

  19. 2) Conclusions • The dynamic equations are more general than fixed base age site equations. • Initial condition difference equations generalize biological theories and integrate them into unified approaches or laws.

  20. Seemingly Different Definitions

  21. 3) Conclusions • Derivation of implicit equations helps to identify redundant parameters. • Dynamic equations are in general more parsimonious than explicit growth & yield equations.

  22. Parsimonious Reductions of Parameters

  23. Final Summary • In comparison to explicit equations the implicit equations are • more flexible; • more general; • more parsimonious; and • more robust with respect applied theories. • The proposed approach allows for derivation of more flexible implicit equations than the other currently used approaches.

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