Transcript A Simple Method for Derivation of Biologically Based

GADA - A Simple Method for
Derivation of Dynamic Equation
Chris J. Cieszewski
and
Ian Moss
Variables of Interest:
– Height (of trees, people, etc.);
– Volume, Biomass, Carbon, Mass, Weight;
– Diameter, Basal Area, Investment;
– Number of Trees/Area, Population Density;
– other ...
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 ...
Example of real data
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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.
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He ig h t
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Age
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a)
Y
Y
a)
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S1
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S1
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t
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t
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a)
Y
a)
Y
Examples
of curve
shape
patterns
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S1
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t
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S1
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t
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The Objective:
• A methodology for models with:
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direct use of initial conditions
base age invariance
biologically interpretable bases
polymorphism and variable asymptotes
The Algebraic Difference Approach
(Bailey and Clutter 1974)
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Y
a)
• 1) Identification of suitable model:
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S1
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X
t
• 2) Choose and solve for a site parameter:
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• 3) Substitute the solution for the parameter:
Y
a)
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S1
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X
t
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
Stages of the Model
Conceptualization:
c)
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a)
Y
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S21
Y
0
d)
S1
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S1
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t
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S1
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a)
S2 1
t
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Y
S1
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S1
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1 00
1 50
t
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t
The Other Examples
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a)
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a)
S1
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S1
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t
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t
d)
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Y
S1
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Y
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S21
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d)
d)
d)
S1
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t
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t
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Y
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Y
0
a)
a)
Y
Y
•
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S21
S21
S21
S1
S1
S1
S1
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1 00
t
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S1
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t
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S1
S1
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t
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S1
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1 00
t
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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:
A Traditional Example
Y
 S
1  e t  ln S
• 1) Identification of suitable longitudinal model:
• 2) Anamorphic model (traditional approach):
• 3) Polymorphic model with one asymptote (t.a.):
Proposed Approach (e.g., #1)
• 1) Identification of suitable longitudinal model:
• 2) Def. #1:
• 3) Solution:
• 4) The implicitly defined model:
Proposed Approach (e.g., #2)
• 1) Identification of suitable longitudinal model:
• 2) Def. #2:
• 3) Solution:
• 4) The implicitly defined model:
Proposed Approach (e.g., #3)
• 1) Identification of suitable longitudinal model:
• 2) Def. #3:
• 3) Solution:
• 4) The implicitly defined model:
Proposed Approach (e.g., #4)
• 1) Identification of suitable longitudinal model:
• 2) Def. #4:
• 3) Solution:
• 4) The implicitly defined model:
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Height
1  e t  ln S
MHGen
MHGen
Anam
Poly
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0
0
0
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d)
c)
Height
Y
 S
b)
a)

MHGen
MHGen
Poly+V-A
Poly+V-A
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0
0
Age (y)
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0
Age (y)
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7
b)
a)
5
1  e t  ln S
Residuals (m)
3
1
0
-1 0
0
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200
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-5
Poly.
Anam.
-7
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d)
c)
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3
Residuals (m)
Y
 S
1
0
-1 0
0
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-5
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Poly+V-A
Poly+V-A
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Age (y)
Age (y)
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.
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Height
1  e t  ln S
MHGen
MHGen
G-ii model
G-i. model
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c)
d)
Sum(Yobs-Ypred)^2=0
Height
Y
b)
a)
 S
MHGen
S-based Aii
S-based Aii
Aii
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0
0
Age (y)
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Age (y)
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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.
Seemingly Different Definitions
3) Conclusions
• Derivation of implicit equations helps to
identify redundant parameters.
• Dynamic equations are in general more
parsimonious than explicit growth & yield
equations.
Parsimonious Reductions of Parameters
Final Summary
• In comparison to explicit equations the
implicit equations are
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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.