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, nonfunctional
Predictive,
mechanistic,
functional
Which techniques should be used to develop your mathematical model?
Form of
Equation
Complexity
of system
Availability
of data
Potential
candidate
equations known
Unknown
Not
complex
Limited
Statistical
fitting
Not
complex
Extensive
Bayesian
statistics
Neural
networks
Limited
Cellular
automata
Parameter
optimisation
Known
Complex
Extensive
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
Form of equation: unknown. System: not complex
Statistical fitting
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
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
Multiplicative
Additive
1.0
0.8
0.8
0.6
0.6
Y
Y
1.0
0.4
0.4
f(B
0.0
0.3
0.2
f(A)
0.1
0.0
0.2
1.0
0.8
0.6
0.4
0.2
0.0
)
0.5
0.4
0.3
0.2
0.1
0.0
)
0.2
0.0
0.8
f(B
0.4
0.6
0.4
f(A)
0.2
0.0
Example - stepwise fitting of a
multiplicative function
Model of transpiration
𝐸 ≈ 𝑔𝐷𝑉𝑃𝐷 𝑘
𝐸 = 𝑡𝑟𝑎𝑛𝑠𝑝𝑖𝑟𝑎𝑡𝑖𝑜𝑛
𝑔 = 𝑐𝑎𝑛𝑜𝑝𝑦 𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑎𝑛𝑐𝑒
𝐷𝑉𝑃𝐷 = 𝑣𝑎𝑝𝑜𝑢𝑟 𝑝𝑟𝑒𝑎𝑠𝑠𝑢𝑟𝑒 𝑑𝑒𝑓𝑖𝑐𝑖𝑡
𝑘 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
Lagergren and Lindroth 2002
Example - stepwise fitting of a multiplicative function
First try to find a theorethical base for the model
R e la tive c o n d u c ta n ce
1
0
R a d ia tio n
(r)
Va p o u r p re a ssu re d e ficit
(DVPD)
1
0
Te m p e ra tu re
(T)
S o il w a te r co n te n t
(θ)
𝑔 = 𝑔𝑚𝑎𝑥 𝑓 𝐷𝑉𝑃𝐷 𝑓 𝑟 𝑓 𝑇 𝑓(𝜃)
Lagergren and Lindroth 2002
Example - stepwise fitting of a multiplicative function
Conductance
Envelope fitting of the first dependency
(Alternately: Select a period when you expect no limitation from r, T or θ)
Gives: gmax and f(DVPD)
Vapour pressure deficit
Example - stepwise fitting of a multiplicative function
𝑔 = 𝑔𝑚𝑎𝑥 𝑓 𝐷𝑉𝑃𝐷 𝑓 𝑟 𝑓 𝑇 𝑓(𝜃)
𝑔𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑
= 𝑓 𝑟 𝑓 𝑇 𝑓(𝜃)
𝑔𝑚𝑎𝑥 𝑓(𝐷𝑉𝑃𝐷 )
Select a period when you expect no limitation from T or θ
1.6
1.4
1.2
𝑔𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑
𝑔𝑚𝑎𝑥 𝑓(𝐷𝑉𝑃𝐷 )
1
𝑓 𝑟
0.8
0.6
0.4
0.2
0
0
50
100
150
200
250
300
Radidation (W m-2)
350
400
450
Example - stepwise fitting of a multiplicative function
𝑔 = 𝑔𝑚𝑎𝑥 𝑓 𝐷𝑉𝑃𝐷 𝑓 𝑟 𝑓 𝑇 𝑓(𝜃)
𝑔𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑
= 𝑓 𝑇 𝑓(𝜃)
𝑔𝑚𝑎𝑥 𝑓 𝐷𝑉𝑃𝐷 𝑓(𝑟)
Select a period when you expect no limitation from θ
3
2.5
2
𝑔𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑
𝑔𝑚𝑎𝑥 𝑓 𝐷𝑉𝑃𝐷 𝑓(𝑟)
𝑓 𝑇 =1
1.5
1
0.5
0
0
5
10
15
Temperature (˚C)
20
25
Example - stepwise fitting of a multiplicative function
𝑔 = 𝑔𝑚𝑎𝑥 𝑓 𝐷𝑉𝑃𝐷 𝑓 𝑟 𝑓 𝑇 𝑓(𝜃)
𝑔𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑
= 𝑓(𝜃)
𝑔𝑚𝑎𝑥 𝑓 𝐷𝑉𝑃𝐷 𝑓(𝑟)𝑓 𝑇
The remaining deviation should be explained by θ
1.4
1.2
1
𝑔𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑
𝑔𝑚𝑎𝑥 𝑓 𝐷𝑉𝑃𝐷 𝑓(𝑟)𝑓 𝑇
𝑓(𝜃)
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
Relative extractable water
0.8
1
Example - stepwise fitting of a multiplicative function
The modelled was controlled by applying it for the callibration year
Transpiration (mm d -1)
1.2
Meassured
0.8
Modelled
0.4
0
And validated against a different year
Transpiration (mm d -1)
2.5
2.0
1.5
1.0
0.5
0.0
Meassured
Modelled