Fitting models to data
Step 5) Express the relationships mathematically in equations
Step 6)
Get values of
parameters
Determine what type of model you will make

functional or mechanistic
Use ”standard” equations if possible
Analyse relationships with a statistical software
Approaches used for different types of mathematical models
Approach
Derive data
directly from
measured data
Derive data
from scientific
understanding
Combined
approach
Type of model
Descriptive,
functional
Mechanistic,
descriptive, non

functional
Predictive,
mechanistic,
functional
Form of
Equation
Which techniques should be used to develop your mathematical model?
Potential
candidate
equations known
Unknown
Known
Complexity
of system
Not
complex
Not
complex
Complex
Availability
of data
Extensive
Limited
Extensive
Limited
Statistical
fitting
Neural
networks
Bayesian
statistics
Parameter
optimisation
Cellular
automata
Simulated
annealing
Evolutionary
algorithm
Form of equation: unknown. System: not complex. Data: Extensive
Neural networks
Input nodes ar set up, analogous to the neural
nodes in the brain
Through a iterative ”training” process different
weights are given to the different connections
in the network
http://
www.oup.com/uk/orc/bin/9780199272068/01stu
dent/weblinks/ch02/smith_smith_box2b.pdf
Potential candidated equations known
Bayesian statistics
(or Bayesian
inference)
Estimates
the probability of
different
hypothesis (candidate models) instead of
rejection of hypothesis which is the more
common approach
http://www.oup.com/uk/orc/bin/9780199272068/01stu
dent/weblinks/ch02/smith_smith_box2c.pdf
Form of equation:
known
. System:
complex but can be
simplified.
Data:
Limited
Cellular automata
For processes that
have a spatial
dimension (2D or 3D)
Equations for the
interaction between
neighboring cells are
fitted
http://www.oup.com/uk/orc/bin/9780199272068/01stu
dent/weblinks/ch02/smith_smith_box2e.pdf
Form of equation:
known
. System: complex. Data: Extensive
Simulated annealing
Simulated
annealing
: used to locate
a good approximation to
the global optimum of a given function in a large search
space
The process is iterated until a satisfactory level of
accuracy is achieved.
http://
www.oup.com/uk/orc/bin/9780199272068/01stu
dent/weblinks/ch02/smith_smith_box2f.pdf
Form of equation: known. System: complex. Data: Extensive
Evolutionary
algorithm
Evolutionary algorithms
: are similar
to
what is
used in
simulated annealing, but instead of mutating parameter
values the rules themselves are
altered.
Fitness
of the rule set is measured in terms of both how
well the model fits the data, and how complex the model
is.
A
simple model, which gives the same results as a complex
one is preferable.
http://
www.oup.com/uk/orc/bin/9780199272068/01stu
dent/weblinks/ch02/smith_smith_box2f.pdf
Form of equation: known. System:
not complex
Parameter optimisation of known equations
Example of curve fitting tools
Excel

Only functions that can be solved analytically
with least square methods
Statistical software, e.g. SPSS
Matlab
Special curve fitting tools, e.g. TabelCurve
Form of equation:
unknown
. System:
not complex
Statistical
fitting
Additive or multiplicative functions
Additive
Y = f(A) + f(B)
Y 0

1
If equal weight: f(A
) 0

0.5,
f(B) 0

0.5
If f(A) = 0 and f(B) = 0 then y = 0
If f(A) = 0 and f(B) =
0.5
then y =
0.5
If f(A) =
0.5
and f(B) = 0 then y =
0.5
If f(A) =
0.5
and f(B) =
0.5
then y =
1
Multiplicative
Y = f(A)
×
f(B)
Y 0

1
If equal weight: f(A
)
0

1, f(B) 0

1
If f(A) = 0 and f(B) = 0 then y = 0
If f(A) = 0 and f(B) = 1
then y =
0
If
f(A) =
1
and f(B) = 0 then y =
0
If f(A) =
1
and f(B) =
1
then y =
1
Additive or multiplicative functions
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.0
0.1
0.2
0.3
0.4
Y
f(B)
f(A)
Additive
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
Y
f(B)
f(A)
Multiplicative
Example

stepwise fitting of a
multiplicative function
Model of transpiration
≈
𝑉𝑃𝐷
𝑘
=
𝑎 𝑖 𝑎𝑖
=
𝑎 𝑦
𝑎
𝑉𝑃𝐷
=
𝑎
𝑎
𝑖 𝑖
𝑘
=
𝑎
Lagergren and Lindroth 2002
Example

stepwise fitting of a multiplicative function
Lagergren and Lindroth 2002
R
e
l
a
t
i
v
e
c
o
n
d
u
c
t
a
n
c
e
Radiation
Vapour preassure deficit
Temperature
Soil water content
0
1
0
1
First try to find a theorethical base for the model
=
𝑚𝑎𝑥
𝑉𝑃𝐷
𝑇
(
𝜃
)
(
r
)
(
θ
)
(
T
)
(
D
VPD
)
Example

stepwise fitting of a multiplicative function
Envelope fitting of the first dependency
(Alternately: Select a period when you expect no limitation from
r
,
T
or
θ
)
Gives:
g
max
and
f
(
D
VPD
)
Conductance
Vapour pressure deficit
Example

stepwise fitting of a multiplicative function
Select a period when you expect no limitation from
T
or
θ
=
𝑚𝑎𝑥
𝑉𝑃𝐷
𝑇
(
𝜃
)
𝑚𝑎𝑢
𝑚𝑎𝑥
(
𝑉𝑃𝐷
)
=
𝑇
(
𝜃
)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0
50
100
150
200
250
300
350
400
450
Radidation (W m

2
)
𝑚𝑎𝑢
𝑚𝑎𝑥
(
𝑉𝑃𝐷
)
Example

stepwise fitting of a multiplicative function
Select a period when you expect no limitation from
θ
=
𝑚𝑎𝑥
𝑉𝑃𝐷
𝑇
(
𝜃
)
𝑚𝑎𝑢
𝑚𝑎𝑥
𝑉𝑃𝐷
(
)
=
𝑇
(
𝜃
)
𝑚𝑎𝑢
𝑚𝑎𝑥
𝑉𝑃𝐷
(
)
𝑇
=
1
0
0.5
1
1.5
2
2.5
3
0
5
10
15
20
25
Temperature (˚C)
Example

stepwise fitting of a multiplicative function
The remaining deviation should be explained by
θ
=
𝑚𝑎𝑥
𝑉𝑃𝐷
𝑇
(
𝜃
)
𝑚𝑎𝑢
𝑚𝑎𝑥
𝑉𝑃𝐷
(
)
𝑇
=
(
𝜃
)
𝑚𝑎𝑢
𝑚𝑎𝑥
𝑉𝑃𝐷
(
)
𝑇
(
𝜃
)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0
0.2
0.4
0.6
0.8
1
Relative extractable water
Example

stepwise fitting of a multiplicative function
The modelled was controlled by applying it for the callibration year
And validated against a different year
0
0.4
0.8
1.2
Transpiration (mm d

1)
Meassured
Modelled
0.0
0.5
1.0
1.5
2.0
2.5
Transpiration (mm d

1)
Meassured
Modelled
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