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 ofEquation

Which techniques should be used to develop your mathematical

model? Potential candidate equations

known

Unknown Known

Complexity of system

Not comple

x

Not comple

x

Complex

Availability of data

Extensive

Limited Extensive

Limited

Statistical fitting

Neural network

s

Bayesian statistics

Parameter optimisatio

n

Cellular automat

a

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/9780199272068/01student/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/9780199272068/01student/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/9780199272068/01student/weblinks/ch02/smith_smith_box2e.pdf

Form of equation: known. System: complex. Data:

Extensive

Simulated annealingSimulated annealing: used to locate a good approximation to the global optimum of a given function in a large search spaceThe process is iterated until a satisfactory level of accuracy is achieved.

http://www.oup.com/uk/orc/bin/9780199272068/01student/weblinks/ch02/smith_smith_box2f.pdf

Form of equation: known. System: complex. Data:

Extensive

Evolutionary algorithmEvolutionary 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/9780199272068/01student/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

AdditiveY = f(A) + f(B) Y 0-1If equal weight: f(A) 0-0.5, f(B) 0-0.5If f(A) = 0 and f(B) = 0 then y = 0If f(A) = 0 and f(B) = 0.5 then y = 0.5If f(A) = 0.5 and f(B) = 0 then y = 0.5If f(A) = 0.5 and f(B) = 0.5 then y = 1

MultiplicativeY = f(A) × f(B) Y 0-1If equal weight: f(A) 0-1, f(B) 0-1If f(A) = 0 and f(B) = 0 then y = 0If f(A) = 0 and f(B) = 1 then y = 0If f(A) = 1 and f(B) = 0 then y = 0If 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

0.00.1

0.20.3

0.40.5

0.00.1

0.20.3

0.4

Y

f(B)

f(A)

Additive

0.0

0.2

0.4

0.6

0.8

1.0

0.00.2

0.40.6

0.81.0

0.00.2

0.40.6

0.8

Y

f(B)

f(A)

Multiplicative

Example - stepwise fitting of a multiplicative function

Model of transpiration

Lagergren and Lindroth 2002

Example - stepwise fitting of a multiplicative function

Lagergren and Lindroth 2002

Re

lativ

e co

ndu

ctan

ce

Radiation Vapour preassure deficit

Temperature Soil water content

0

1

0

1

First try to find a theorethical base for the model

𝑔=𝑔𝑚𝑎𝑥 𝑓 (𝐷𝑉𝑃𝐷 ) 𝑓 (𝑟 ) 𝑓 (𝑇 ) 𝑓 (𝜃)

(r)

(θ)(T)

(DVPD)

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: gmax and f(DVPD)C

ondu

ctan

ce

Vapour pressure deficit

Example - stepwise fitting of a multiplicative function

Select a period when you expect no limitation from T or θ

𝑔=𝑔𝑚𝑎𝑥 𝑓 (𝐷𝑉𝑃𝐷 ) 𝑓 (𝑟 ) 𝑓 (𝑇 ) 𝑓 (𝜃)

𝑔𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑

𝑔𝑚𝑎𝑥 𝑓 (𝐷𝑉𝑃𝐷)= 𝑓 (𝑟 ) 𝑓 (𝑇 ) 𝑓 (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)

𝑔𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑

𝑔𝑚𝑎𝑥 𝑓 (𝐷𝑉𝑃𝐷)𝑓 (𝑟 )

Example - stepwise fitting of a multiplicative function

Select a period when you expect no limitation from θ

𝑔=𝑔𝑚𝑎𝑥 𝑓 (𝐷𝑉𝑃𝐷 ) 𝑓 (𝑟 ) 𝑓 (𝑇 ) 𝑓 (𝜃)

𝑔𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑

𝑔𝑚𝑎𝑥 𝑓 (𝐷𝑉𝑃𝐷 ) 𝑓 (𝑟 )= 𝑓 (𝑇 ) 𝑓 (𝜃)

𝑔𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑

𝑔𝑚𝑎𝑥 𝑓 (𝐷𝑉𝑃𝐷 ) 𝑓 (𝑟 )𝑓 (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

𝑔=𝑔𝑚𝑎𝑥 𝑓 (𝐷𝑉𝑃𝐷 ) 𝑓 (𝑟 ) 𝑓 (𝑇 ) 𝑓 (𝜃)

𝑔𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑

𝑔𝑚𝑎𝑥 𝑓 (𝐷𝑉𝑃𝐷 ) 𝑓 (𝑟 ) 𝑓 (𝑇 )= 𝑓 (𝜃)

𝑔𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑

𝑔𝑚𝑎𝑥 𝑓 (𝐷𝑉𝑃𝐷 ) 𝑓 (𝑟 ) 𝑓 (𝑇 )𝑓 (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

Tran

spira

tion

(mm

d -1

) Meassured

Modelled

0.0

0.5

1.0

1.5

2.0

2.5

Tran

spira

tion

(mm

d -1

)

Meassured

Modelled