GADA - A Simple Method for Derivation of Dynamic Equation

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:

20. 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.

22. 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.

23. Seemingly Different Definitions

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

25. Parsimonious Reductions of Parameters

26. 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.