1/15/2020 Department of Veterinary and Animal Sciences Monte Carlo - PDF document

1/15/2020 Department of Veterinary and Animal Sciences Monte Carlo Simulation II Anders Ringgaard Kristensen and Dan Børge Jensen Department of Veterinary and Animal Sciences Outline 1. Summary of Monday’s lecture 2. The SimFlock model • User interface • State variables • Decision variables State of nature • Hyper distribution • Output variables 3. Use of the simulation model • Running simulation jobs • Results and interpretation 4. Simulation as a decision support tool Advanced Quantitative Methods in Herd Management Slide 2 Department of Veterinary and Animal Sciences Summary from Monday Simulation models, two (main) types: 1. Deterministic: no randomness – same input, same output 2. Stochastic: random sampling – same input, different output 1. Random sampling is done using random number generation with an appropriate distribution function State-of-nature: (Basically) a collection of all the information that describes • the system, you are trying to model Uncertainty of the state-of-nature: Each parameter of the s.o.n is specified through a • distribution instead of a value. Such a distribution is called a hyper distribution. • The parameters of a hyper distribution are called hyper • parameters. The SimBatch model A simulation model implemented in R • Fairly straight forward – you could make one yourself! • 1

1/15/2020 The SimFlock Model User interface – visible objects Small holder farms in Africa All birds and eggs present in the flock shown. States of the birds can be investigated Demo Department of Veterinary and Animal Sciences SimFlock: Elements – where are they? Decision rule Θ State of nature Φ 0 Hyper distribution p ( Φ 0 = φ 0 ) State variables Φ s 1 … Φ sT Output variables Ω Θ: thetha, upper case Φ: phi, upper case Ω : Omega, upper case Advanced Quantitative Methods in Herd Management Slide 6 2

1/15/2020 Department of Veterinary and Animal Sciences SimFlock: An object oriented model The farmer, birds, and eggs are represented as Breeding animals objects in the model! Hens & Cocks Eggs Infertile Household consumption Chicks Dead Growers Pullets Cockerels Market Department of Veterinary and Animal Sciences SimFlock: State variables of the objects The state variables of day i are the states of the individual birds and eggs on that day: • Eggs: • Fertilized/not fertilized Specific state variables: • Birds: • Age Cocks: No further states • Weight Chicks and growers: Growth state • Growth potential • Full grown weight Pullet: Age at “puberty”, Growth state • Laying capacity • Gender Cockerel: Age at “puberty”, Growth state • … Hen: Behavior, Laying capacity, • Farmer: State in cycle, • Needs meat? Days since transition in cycle, Eggs at incubating Eggs in nest, Fertile eggs in nest There are millions of state variables in a simulation run. Department of Veterinary and Animal Sciences SimFlock: Decision variables Built-in decisions (farmer icon): • Intended flock size: • Hens • Cocks • Egg removing policy • Days from start laying • Season • Policy for buying breeding birds: • Hens • Cocks Other decisions modeled through expected effects (e.g. on mortality). Advanced Quantitative Methods in Herd Management Slide 9 3

1/15/2020 5 Minute Break Department of Veterinary and Animal Sciences Distribution of state of nature: Main problem It is difficult to specify the distribution of the state of nature. Solution: • We use hyper distributions • The hyper parameters are stimated from production data from 30 flocks in Zimbabwe. • Easy, if parameters are independent • Difficult if they interact For a systematic description of the approach used in the SimFlock model, reference is made to Kristensen & Pedersen (2003) – link at the homepage. Advanced Quantitative Methods in Herd Management Slide 11 Department of Veterinary and Animal Sciences Example: Mortality in SimFlock It is expected that the mortalities of different bird groups in the same flock are correlated – this should be included in the model! Mortality is represented as survival rates p . If we observe N birds over a given period and count the number n that survive, then n is binomially distributed with parameters p and N with p ~ n/N. If other factors influence p (e.g. the bird group) we can express the effect in a logistic model (standard tool for dealing with binomially distributed data) Advanced Quantitative Methods in Herd Management Slide 12 4

1/15/2020 The Logit-transformation Logit 4 3 2 1 Logit(p) 0 0 0,2 0,4 0,6 0,8 1 -1 -2 -3 -4 p The Logit-transformation converts a probability p ∈ [0;1] to a value y ∈ ]- ∞ ; ∞ [. The transformed variable, y , may be used as response variable in “usual” regression analysis etc. Department of Veterinary and Animal Sciences The SimFlock survival rate model logit( p ij ) = µ + α j + F i + ( α F ) ij Where • µ is the intercept • α 1 , α 2 , α 3 , α 4 are the systematic effects of bird groups (i.e. chicks, growers, pullets and cockerels) • F i ~ N(0, σ F ) is the random effect of flock • ( α F ) ij ~ N(0, σ α F ) is the random interaction between flock and bird group. State of nature parameters: p i 1 , p i 2 , p i 3 , p i 4 , i.e. a survival rate for each bird group. Hyper parameters: µ , α 1 , α 2 , α 3 , α 4 , σ F , σ α F – estimated from field data from 30 flocks in Zimbabwe. Department of Veterinary and Animal Sciences Defining the Survival state of nature - sampling from the hyper distribution Draw a random effect of flock F i from N(0, σ F 2 ) Draw 4 random bird/flock interaction values ( α F ) i 1 , ( α F ) i 2 , ( α F ) i 3 , ( α F ) i 4 from N(0, σ α F 2 ) Calculate the 4 logit values ( j = 1, 2, 3, 4) y ij = logit( p ij ) = µ + α j + F i + ( α F ) ij Transform to 4 survival rates ( j = 1, 2, 3, 4) log( p ij /(1- p ij )) = y ij ⇔ p ij = 1/(e - y ij + 1) 5

1/15/2020 Department of Veterinary and Animal Sciences SimFlock: State of nature parameters In SimFlock, a state of nature is described by 42 parameters : • Daily gains of birds, general linear model • Survival rates, logistic model • Full grown weights, normal distribution • Age at puberty, normal distribution • Egg fertilization probability, beta distribution • Egg hatching probability, logistic model • Number of eggs before incubation, normal dist. Each time a parameter is defined, a hyper distribution is specified. Advanced Quantitative Methods in Herd Management Slide 16 Department of Veterinary and Animal Sciences SimFlock: Hyper distribution(s) The hyper distribution of the state of nature is specified through 64 hyper parameters . Most of them estimated from the field data collected in 30 flocks. A state of nature drawn from the hyper distribution represents one (hypothetical) flock. • By drawing e.g. many states of nature we can generate many realistic hypothetical flocks. • Decision rules may have different effects in different flocks. Advanced Quantitative Methods in Herd Management Slide 17 Department of Veterinary and Animal Sciences SimFlock: Output variables A total of 40 are defined: • Realised gain • Realised mortality • Eggs removed • Chickens produced • … Usual technical and economical key figures. Advanced Quantitative Methods in Herd Management Slide 18 6

1/15/2020 10 Minute Break Use of the simulation model Department of Veterinary and Animal Sciences Use of the simulation model System comprehension • Answering “what if” questions General decision support (at population level) • The main purpose of SimFlock Decision support at flock level • Not yet possible Advanced Quantitative Methods in Herd Management Slide 21 7

1/15/2020 Department of Veterinary and Animal Sciences System comprehension Usually carried out under one state of nature Answer questions like: • If we assume the state of nature parameters are Φ 0 = φ 0 what are then the consequences? • What if we could improve the survival rate of chicks? • Vary the survival rate systematically – run simulations and explore the results • etc. Weakness: State of nature parameters are mutually correlated! Advanced Quantitative Methods in Herd Management Slide 22 Department of Veterinary and Animal Sciences General decision support Population level Carried out under multiple states of nature Questions like: • Under what circumstances does it pay to change the decision rule from Θ 1 to Θ 2 ? • Generate multiple states of nature (random flocks) • Run a simulation job under Θ 1 • Run a simulation job under Θ 2 • Identify the states of nature where it pays Advanced Quantitative Methods in Herd Management Slide 23 Department of Veterinary and Animal Sciences Defining a simulation job in SimFlock Create an initial flock Specify: • Number of states of nature (if more than 1) • A question of obtaining a representative sample of flocks from the abstract population. • Number of replications per state of nature • How precise do you want the results for each flock to be? • Mean values • Distribution • Number of days to simulate • A long simulation period will increase the precision • Burn-in days • We want to ignore the effect of the initial flock. Advanced Quantitative Methods in Herd Management Slide 24 8