Semi-mechanistic modelling in Nonlinear Regression: a case study by Katarina Domijan 1, Murray Jorgensen 2 and Jeff Reid 3 1 AgResearch Ruakura 2 University of Waikato 3 Crop and Food Research

Introduction  Reid (2002) developed a crop response model  genetic algorithm  no measures of confidence for individual parameter estimates

General structure  non-linear model  semi-mechanistic  parts of the model that would ideally be mechanistic are replaced by empirically estimated functions  relatively complex  (26 parameters to be estimated)  generality of application  challenging to fit

General structure water stress, plant density, quantity of light etc yield under ideal nutrient/pH conditions (maximum yield) nutrient supply (N, P, K, Mg) observed yield These effects are assumed to act multiplicatively and independently of each other and the nutrient effects

General structure – maize data Crop hybrid

General structure - nutrients Model structure is the same for all nutrients Each nutrient supply is assumed to have: a minimum value (the crop yield is zero) and an optimal value (further increases cause no additional yield) Reid (2002) defines a scaled nutrient supply index which is: = 0 at the minimum nutrient value and = 1 at the optimal nutrient value In soil added as fertilizer nutrient supply efficiency factor 1 efficiency factor 2 Proportion of the optimum amount of nutrient supply

General structure - nutrients For each nutrient, the effect of scaled nutrient supply index (x) on yield is modelled using the family of curves: N opt where: γ =shape parameter

General structure - combining nutrients The combined scaled yield is given by: (or 0 if this is negative). Note:  are scaled yields corrected for unavailability of the respective nutrients  Nutrient stresses are assumed to affect yield independently of each other Soil pH  Treated as if it were an extra nutrient  Only stress due to low pH is modelled and not stress due to excessive pH if the effects of a particular stressor are known to be absent for a set of data, then

just 2 nutrients (N and K) Scaled yield General structure - combining nutrients

Data  model was tested for maize crops grown in the North Island between 1996 and 1999  data was collated from 3 different sources of measurements  experimental and commercial crops  12 sites  6 hybrids  84 observations

Genetic Algorithms  stochastic optimization tools that work on “Darwinian” models of population biology  don’t need requirement of differentiability!  relatively robust to local minima/maxima  don’t need initial values  have no indication of how well the algorithm has performed  convergence to a global optimum in a fixed number of generations?  slow to move from an arbitrary point in the neighbourhood of the global optimum to the optimum point itself  no measure of confidence for individual parameters

Our approaches:  simplifying the model:  1 nutrient (N)  9 parameters  simulated data  combining GA with derivative based methods:  common methods (Gauss-Newton, Levenberg- Marquardt)  AD MODEL BUILDER  obtain CI’s:  gradient information  likelihood methods

Correlation of Parameter Estimates: g Nm Np d b e1 e2 E.1 Nmin B 1 Nopt.. 1 delta 1 beta, 1 eta1. 1 eta2 1 E.n1, 1 E.n2, + Simulated data - simple model  investigate the structure of the correlation matrix  generated so it mimics the “real” data as much as possible  large n (300), small residual variance (0.01) N min and γ N are highly correlated! (blank) , * B Key:

Complete model n=50000, σ 2 = Key: (blank) , * B0.95-1

N min Levenberg-Marquardt algorithm  Maize data  use GA estimates as starting values  simple model:  multicollinearity  parameter Nmin tends to –ve  reparametrization

 complete model:  (again) biological restrictions (N min, K min =0)  problems with equations which are constant for ranges of values (eg scaled yields)  replace nondiffentiable functions (pH, water stress)  some stressors held constant (P, Mg) N opt constant Levenberg-Marquardt algorithm

 2 nutrients (N and K) + stress due to low pH + water and population stresses  12 parameters

Profile likelihood CI’s N opt estimate Wald CI Likelihood CI Approach outlined in Bates and Watts (1989) Assess validity of the linear approximation to the expectation surface

Profile likelihood CI’s Estimation surface seems to be nonlinear with respect to most of the parameters in the model Especially E N1 and pH c -> one sided CIs Better estimates of uncertainty than linear approx. results

AD Model builder  automatic differentiation  faster  observed information matrix (better se’s)  we run into the same problems as with L-M  requires model to be differentiable  good initial values

In the end... CI’s are too wide to be of ‘practical’ use e.g. for parameter N opt (optimum amt of N supply per tonne of maximum yield) : but in the ‘maize dataset’ N supply per tonne of maximum yield varies between 6 and 54 Problems of  nonidentifiability  correlated estimates  poor precision of estimation in certain directions These phenomena are not clearly distinguished in nonlinear setting L-M and ADMB estimate 95% LINEAR APPROX. CI’s 95% PROFILE LIKELIHOOD CI’s 95% CI’s (ADMB) (3.80, 33.35)(11.67, 31.45)(11.32, 25.83)

Recommendations  do more experimentation - collect more information about parameters  particularly ‘approximately nonidentifiable’ parameters  replace all nondifferentiable equations in the model with smooth versions  bootstrapping  global optimum?