1. SAMO 2007, 21 Jun 2007 Davood Shahsavani and Anders Grimvall Linköping University

2. SAMO 2007, 21 Jun 2007 Motivation Many applications of computer code models require repeated model runs for different sets of inputs Response surface methodologies can:  Help to learn about the model  Facilitate the development of computationally cheap decision support tools

3. SAMO 2007, 21 Jun 2007 The two steps in extracting response surfaces from computationally expensive computer-code models Step 1:  Choose a suitable design of the computer experiment Step 2:  Choose an interpolation method that enables accurate prediction of the model output at previously untried inputs

4. SAMO 2007, 21 Jun 2007 Features of currently used techniques for extracting reponse surfaces  The design criteria favour regular fractional factorial designs or an almost uniform coverage of the input domain  The interpolation is usually based on a global model for the entire input domain

5. SAMO 2007, 21 Jun 2007 Study objective Extraction of response surfaces whose curvature varies strongly over the input domain  Our designs are space-filling and have a particularly good coverage of regions in which the response surface is rough or strongly nonlinear  We fit local models to the responses computed for subsets of design points

6. SAMO 2007, 21 Jun 2007 Splitting procedure

7. SAMO 2007, 21 Jun 2007 Design points after two splits

8. SAMO 2007, 21 Jun 2007 A sequential design algorithm for box-shaped input domains 1. Initiate the design algorithm by selecting a (slightly extended) corner-centre design 2. Start a loop in which the input domain is split into sub-boxes, and new corners and centres are added to the design –A measure of roughness or nonlinearity is used to determine which box that shall be split into two halves –A direction criterion is used to determine in which direction the selected sub-box shall be split

9. SAMO 2007, 21 Jun 2007 Quantifying the roughness of a response surface  The integrated roughness R f (D) of a function f (x 1, ….x p ) in a subset D of the input domain is usually defined as  The integrated roughness of a second order polynomial is  We estimate the integrated roughness of any function by computing where are estimated parameters in a fitted polynomial

10. SAMO 2007, 21 Jun 2007 A roughness measure that focuses on integrated absolute errors in linear predictors  Consider a box in which the i th side has length h i  We estimate the roughness of the response surface by computing

11. SAMO 2007, 21 Jun 2007 Possible splits of a box

12. SAMO 2007, 21 Jun 2007 Splitting direction Suppose that the sub-box D* shall be split, and let be the polynomial fitted to data in or on the border of that box Then we split the selected sub-box along the k th coordinate where

13. SAMO 2007, 21 Jun 2007 Design points for a simple response surface f(x 1, x 2 ) = x 1 5 + x 2 5

14. SAMO 2007, 21 Jun 2007 The INCA-N model ( Integrated Nitrogen in Catchments ) Model parameters: Initial conditions Nitrogen transformation rates Hydrogeological parameters Daily weather data INCA - N 20 40 60 80 100 120 11121 20 40 60 80 100 120 11121 Daily estimates of water discharge and NO 3 and NH 4 concentrations in river water Average annual riverine load of inorganic nitrogen

15. SAMO 2007, 21 Jun 2007 Examples of response surfaces produced by the INCA-N model Average annual nitrogen loss Denitrification rate Max. nitrate uptake rate Plant nitrate uptake rate

16. SAMO 2007, 21 Jun 2007 Design points for the INCA-N model Denitrification rate Plant nitrate uptake rate Average annual nitrogen loss

17. SAMO 2007, 21 Jun 2007 Design points for the INCA-N model

18. SAMO 2007, 21 Jun 2007 Interpolation  Local interpolation is preferably based on simple models  Constant or linear predictors can be too simplistic for nonlinear deterministic functions  We started by fitting quadratic polynomials to minimal neighborhoods of the point at which the response shall be predicted

19. SAMO 2007, 21 Jun 2007 Prediction errors for the function f(x 1, x 2, x 3, x 4, x 5 ) = x 1 5 + x 2 5 + 0.1(x 3 + x 4 + x 5 ) Interpolation from 1024 (4 5 ) design points

20. SAMO 2007, 21 Jun 2007 More carefully selected local neigbourhoods  Determine the minimal number of design points needed to produce a full rank design matrix for the polynomial regression  Add a fixed number of design points  Use the shape of the sub-box surrounding the new point to define a suitable local distance measure

21. SAMO 2007, 21 Jun 2007 Prediction errors for the function f(x 1, x 2, x 3, x 4, x 5 ) = x 1 5 + x 2 5 + 0.1(x 3 + x 4 + x 5 ) Minimal local neighbourhoodExtended local neighbourhood Interpolation from 1024 (4 5 ) design points

22. SAMO 2007, 21 Jun 2007 Predicted values for simple nonlinear functions f(x 1, x 2, x 3, x 4, x 5 ) = x 1 5 + x 2 5 + 0.1(x 3 + x 4 + x 5 ) f(x 1, x 2, x 3, x 4, x 5, x 6, x 7 ) = x 1 5 + x 2 5 + 0.1(x 3 + x 4 + x 5 + x 6 + x 7 ) 1024 (4 5 ) design points 2187 (3 7 ) design points

23. SAMO 2007, 21 Jun 2007 Efficiency of local quadratic approximation of f(x 1, x 2, x 3, x 4, x 5 ) = x 1 5 + x 2 5 + 0.1(x 3 5 + x 4 5 + x 5 5 ) Break down

24. SAMO 2007, 21 Jun 2007 Efficiency of local quadratic approximation of INCA –N Sequential design with extended local neighbourhoods

25. SAMO 2007, 21 Jun 2007 Efficiency of local quadratic approximation of INCA –N Sequential design and a regular grid design

26. SAMO 2007, 21 Jun 2007 Accommodation of correlated inputs and arbitrarily shaped input domains Substitute for where g(x) is the joint probability density of the inputs

27. SAMO 2007, 21 Jun 2007 Main conclusions  Our sequential design automatically adapts to the nonlinear features of the response surface under consideration  Local interpolation using quadratic polynomials performs satisfactorily, provided that local neighbourhoods are selected with care (appropriate size, local distance measure)

28. SAMO 2007, 21 Jun 2007 Other conclusions  Both the design algorithm and the interpolation technique are conceptually simple and computationally cheap  The derived surrogate model forms a good basis for sensitivity analyses and user-friendly decision support tools  Our procedure is particularly suitable for studies of a single output from strongly nonlinear models with 2 to 7 inputs

29. SAMO 2007, 21 Jun 2007 Sensitivity analysis  The input parameters were divided into three groups: initial conditions, hydrogeological parameters, and nitrogen transformation rates  Variance-based sensitivity analyses were first carried out for each group of parameters and then for the six most influential parameters  Although the hydrogeological parameters influenced the timing of the nitrogen losses, the average annual loss was almost exclusively determined by the nitrogen transformation rates

30. SAMO 2007, 21 Jun 2007 Sensitivity indices for the six most influential inputs to the INCA-N model ParameterMain effect (S i ) Total sensitivity index (S t ) Denitrification rate0.200.23 Plant nitrate uptake rate0.460.49 Mineralization rate0.080.12 Immobilization rate0.180.23 Soil moisture deficit0.020.04 Maximum soil retention volume0.030.06

31. SAMO 2007, 21 Jun 2007 Nitrogen transformation parameters ParameterRange x 1 : Denitrification rate (m/day)[0, 0.01] x 2 : Nitrogen fixation (kg N/ha/day)[0, 0.01] x 3 : Plant nitrate uptake rate (kg N/ha/day)[0, 0.05] x 4 : Maximum nitrate uptake rate(kg N/ha/day )[80, 140] x 5 : Mineralization (kg N/ha/day)[ 0, 1] x 6 : Immobilization(m/day)[0, 0.1] x 7 : Plant ammonium uptake rate(kg N/ha/day)[0, 0.05]

32. SAMO 2007, 21 Jun 2007 The INCA-N model Nitrogen transformation rates: 1- Denitrification 2- Plant nitrate uptake rate 3- Max nitrate uptake rate 4- Nitrification 5- Mineralisation 6- Immobilisation 7- Plant ammonium uptake rate Nitrification and all initial conditions and hydrogeological parameters are fixed at the mid value INCA – N Model output Model Inputs Average annual riverine load of inorganic nitrogen