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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
0
10
20
30
40
0 50 100 150 200
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.
0
5
10
15
20
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30
35
40
0 20 40 60 80 100 120
Age
He
igh
t
Examples of curve
shape patterns
0
140
0 50 100 150 200
S1
tY
a)
0
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0 50 100 150 200
S1
t
Y
a)
0
140
0 50 100 150 200
S1
t
Y
a)
0
140
0 50 100 150 200
S1
t
Y
a)
The Objective:
• A methodology for models with:– direct use of initial conditions– base age invariance– biologically interpretable bases– polymorphism and variable asymptotes
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:
0
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0 50 100 150 200
S 1
t
Y
a)
0
140
0 50 100 150 200
S 1
t
Y
a)
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:
a)
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0 200
t
Y
0
140
X 0 50 100 150 200
S1
t
Y
a)
0
200
X0 50 100 150
S1
S1
S21
t
Y
c)
0
200
X
0 50 1 00 1 50
S1
S1
S21
t
Y
d)
The Other Examples
•
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0 50 100 150 200
S1
t
Y
a)
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0 50 100 150 200
S1
t
Y
a)
0
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0 50 100 150 200
S1
tY
a)
0
140
0 50 100 150 200
S1
t
Y
a)
0
140
0 50 100 150 200
S1
S1
S21
t
Y
d)
0
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0 50 100 150 200
S1
S1
S21
t
Y
d)
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140
0 50 100 150 200
S1
S1
S21
t
Y
d)
0
140
0 50 100 150 200
S1
S1
S21
t
Y
d)
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
• 1) Identification of suitable longitudinal model:
• 2) Anamorphic model (traditional approach):
• 3) Polymorphic model with one asymptote (t.a.):
Ste
SY
ln1
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:
Ste
SY
ln1
a)
0
200
0
He
igh
tMHGen
Anam
b)
0
0
MHGen
Poly
c)
0
200
0 200Age (y)
He
igh
t
MHGen
Poly+V-A
d)
0
0 200Age (y)
MHGen
Poly+V-A
a)
-7
-5
-3
-1
1
3
5
7
0 200
Re
sid
ua
ls (
m)
Anam.
b)
-7
0
7
0 200
Poly.
c)
-7
-5
-3
-1
1
3
5
7
0 200
Age (y)
Re
sid
ua
ls (
m)
Poly+V-A
d)
-7
0
7
0 200
Age (y)
Poly+V-A
Ste
SY
ln1
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.
b)
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0 200
MHGen
G-ii model
a)
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200
0 200
Hei
gh
t
MHGen
G-i. model
c)
0
200
0 200Age (y)
Hei
gh
t
MHGen
S-based Aii
d)
0
200
0 200Age (y)
S-based Aii
Aii
Sum(Yobs-Ypred)^2=0
Ste
SY
ln1
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 – 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.
