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 ApproachDerive data directly from measured data Derive data from scientific understanding Combined approach Type of modelDescriptive, 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 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 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 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 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. 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. 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) 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

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 First try to find a theorethical base for the model (r)(r) (θ)(θ) (T)(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 )

Example - stepwise fitting of a multiplicative function Select a period when you expect no limitation from T or θ

Example - stepwise fitting of a multiplicative function Select a period when you expect no limitation from θ

Example - stepwise fitting of a multiplicative function The remaining deviation should be explained by θ

Example - stepwise fitting of a multiplicative function The modelled was controlled by applying it for the callibration year And validated against a different year