Fitting models to data

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Dec 1, 2013 (3 years and 6 months ago)

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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