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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, 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 ≈ = = = = 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) = () 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 = () = () ( ) Select a period when you expect no limitation from T or θ 1.6 1.4 1.2 ( ) 1 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 = () = () () Select a period when you expect no limitation from θ 3 2.5 2 () =1 1.5 1 0.5 0 0 5 10 15 Temperature (˚C) 20 25 Example - stepwise fitting of a multiplicative function = () = () () The remaining deviation should be explained by θ 1.4 1.2 1 () () 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