Bayesian solutions to some decision problems in crop management and variety choice. Chris Theobald, Adrian Roberts and Mike Talbot

Transcription

1 Bayesian solutions to some decision problems in crop management and variety choice Chris Theobald, Adrian Roberts and Mike Talbot Biomathematics and Statistics Scotland (BioSS) Acknowledgements: research partly funded by the UK Home-Grown Cereals Authority and the Scottish Executive Environment & Rural Affairs Department

2 Bayesian statistics Reverend Thomas Bayes (1701?-1761) Bayes Theorem (Laplace, 1774) Bayesian probability (Fisher, 1950) Decision making under uncertainty (Chicago and Harvard, 1950s): coherent decision making requires a probability distribution for the unknown parameters a utility function We specify a prior distribution for the unknown parameters; combining this with the data gives a posterior distribution. 2

3 Reasons for using the Bayesian approach Complex statistical models can be fitted without largesample approximations Hierarchical models (ie random effects) can be readily incorporated Expert knowledge can be included in analyses This approach provides a procedure for making decisions under uncertainty: choose the decision which maximizes the posterior expected utility, that is the expected utility given the prior distribution and the data 3

4 Some decision problems for cereal growers 1. Which variety of a crop to grow 2. How much seed to sow 3. How much fertilizer to apply 4. How to modify the answers to these questions when the quality of the crop affects its price 4

5 Optimum seed rates for wheat: Introduction 1. Seed rate = number of seeds sown per m 2 2. Higher rates cost more and produce more plants, but they are more likely to be flattened by wind and rain 3. Analysis of earlier experiments suggested that recommended seed rates were too high 4. The conventional analysis estimates a separate optimum rate for each environment 5. Optimum rate may depend on model used for dependence of yield on seed rate, and on any treatments applied etc 5

6 Yield versus seed rate for all environments and varieties Yield (t/ha) Cadenza Claire Haven Soissons Spark Seed rate (seeds per square metre)

7 Optimum seed rates: Dose-yield functions Dose-yield functions relate yield y to seed rate x, and assume zero expected yield at x = 0. Inverse-quadratic function: x E(y x) = β 0 + β 1 x + β 2 x2 (x > 0). Tends to 0 for large x. Exponential-plus-linear function: more widely used, but tends to or for large x, and can give infinite optimum rates. 7

8 Optimum seed rates: Interpretable parameters To allow expert knowledge to be included, parameters in the dose-yield functions are chosen to be easy to interpret: γ maximum expected yield δ seed rate giving maximum expected yield η logit of ratio of expected yield at 2 δ to that at δ The inverse-quadratic function becomes γ δ x E(y x, γ, δ, η) = δ x + 2 e η (x δ) 2 (x > 0). 8

9 Optimum seed rates: Bayesian procedure 1. Consider predicting yield at a target location in a new season 2. Define utility function as value of crop cost of seed and any treatments 3. Use interpretable parameters to model yield variation in whole data set: parameters may depend on environment, variety, treatment and covariates such as sowing date 4. Choose a prior distribution which incorporates knowledge of the crop 9

10 Optimum seed rates: Bayesian procedure 5. Choose dose-yield function 6. Calculate posterior expected utility for combinations of seed rate, variety, treatments and covariates 7. Identify optimum rate, and possibly optimum variety or treatment 8. Repeat calculations with different dose-yield function 9. No point estimation is required! 10

11 Optimum seed rates: UK winter wheat, Phase 1 Phase 1 ( ) used 2 sites seed rates 20, 40, 80, 160, 320, varieties sowing dates from September to December All three interpretable parameters are assumed to depend on environment, variety and sowing date 11

12 Optimum seed rates: UK winter wheat, Phase 1 Prior distributions for expectations, variance components and regression coefficients are based on UK experience Posterior expected utilities etc calculated using WinBUGS software 12

13 Prior expected utilities with and without seed cost Inverse-quadratic function Exponential-plus-linear function Prior expected utility ( /ha) Seed cost ignored Seed cost included Prior expected utility ( /ha) Seed cost ignored Seed cost included Seed rate (seeds per square metre) Seed rate (seeds per square metre)

14 Posterior expected utilities for Phase I data (Inverse-quadratic) 30 September 30 October Posterior expected utility ( /ha) Posterior expected utility ( /ha) Cadenza Haven Soissons Spark 550 Cadenza Haven Soissons Spark Seed rate (seeds per square metre) Seed rate (seeds per square metre)

15 Posterior expected utilities for Phase I data (Exponential-plus-linear) 30 September 30 October Posterior expected utility ( /ha) Posterior expected utility ( /ha) Cadenza Haven Soissons Spark 550 Cadenza Haven Soissons Spark Seed rate (seeds per square metre) Seed rate (seeds per square metre)

16 Optimum seed rates: UK winter wheat, Phase 1 Inverse-quadratic and exponential-plus-linear functions indicate lower seed rates than are currently recommended Dependence on sowing date: seed rate for maximum yield increases by 2% per day maximum expected yield reduces by 12 kg/ha per day 16

17 Examining the fit of dose- Optimum seed rates: yield functions Inverse-quadratic function indicates higher optimum rates than exponential-plus-linear (by about 30%) We compare the mean yield at each seed rate with the posterior distribution of the predicted mean yield under each model: the inverse-quadratic model fits well the exponential-plus-linear over-estimates yields at 80 seeds per m 2 and under-estimates them at 20, 320 and

18 Optimum seed rates: UK winter wheat, Phase 2 Phase 2 ( ) used a single variety with 7 sites from southern England to northern Scotland 4 treatments, varied in one environment at a time: rotational position slug treatment time of nitrogen application use of plant growth regulator (PGR) The interpretable parameters are now assumed to depend on environment, treatment and latitude 18

19 Dependence of posterior expected utility on seed rate and treatments for inverse-quadratic model and Edinburgh latitude 700 Posterior expected utility ( /ha) Standard treatments Early nitrogen Late nitrogen No slug treatment Sibutol Secur slug treatment Second/third wheat Second/third wheat + Latitude No PGR PGR at tillering Seed rate (seeds per square metre)

20 Optimum seed rates: Robustness of optima We can examine the effect on the posterior expected utility (and hence on the optima) of modifying the prior distribution including interaction between sowing date and variety in the maximum-yield parameter allowing quadratic dependence on latitude allowing residual variance to differ between trials None of these changes has a substantial effect on optimum rates 20

21 Optimum fertilizer nitrogen rates Similar procedure to finding optimum seed rates We have to consider nitrogen already in the soil: assume that yield depends on total nitrogen We can use information on soil nitrogen at the target site from a soil test or knowledge of previous crop 21

22 Dependence of posterior expected utility on applied nitrogen for three spring barley varieties, without and with soil nitrogen estimate of 40 kg/ha Posterior expected utility ( /ha) Posterior expected utility ( /ha) Georgie Midas Sundance Georgie Midas Sundance Applied nitrogen level (kg/ha) Applied nitrogen level (kg/ha)

23 Optimum fertilizer nitrogen rate when crop price depends on quality The price paid for a crop may depend on its quality: Wheat grain with high nitrogen content may be better for making bread Barley grain with low nitrogen content is more suitable for making whisky Here the utility function depends on both yield and grain nitrogen 23

24 Price for barley as a function of grain nitrogen Price ( /tonne) Grain nitrogen (%)

25 Optimum fertilizer nitrogen rate when crop price depends on grain nitrogen We model the dependence of yield and grain nitrogen on total nitrogen Grain nitrogen increases with applied nitrogen level: we assume a logistic model 25

26 Posterior expected utilities for two winter barley varieties with and without premium price for low grain nitrogen Posterior expected utilities ( /ha) Pipkin, premium price Puffin, premium price Pipkin, basic price Puffin, basic price Applied nitrogen level (kg/ha)

27 Some conclusions 1. Statistical decision problems should be solved using a probability distribution for the unknown parameters a utility function 2. Possible decisions are compared by using posterior expected utility 3. For crop-management decisions, the utility should relate to a future site 27

28 Some conclusions 4. Series of trials require a combined analysis using hierarchical models 5. Non-linear hierarchical models are required for the dependence of crop yield on seed rate and fertilizer level 6. Decision-making is improved by incorporating expert knowledge about variation in crop yield: using prior distributions for interpretable parameters allows us to achieve this 28

29 References Fienberg, S.E. (2006) When did Bayesian inference become Bayesian? Bayesian Analysis 1, Theobald, C.M. and Talbot, M. (2002) The Bayesian choice of crop variety and fertilizer dose. Applied Statistics 51, Theobald, C.M. and Talbot, M. (2004) Bayesian selection of fertilizer level when crop price depends on quality. Computational Statistics and Data Analysis 47, Theobald, C.M., Roberts, A.M.I., Talbot, M. and Spink, J.H. (2006) Estimation of economically optimum seed rates for winter wheat from series of trials. To appear in Journal of Agricultural Science (available in pdf from 29