1 A comprehensive example Sow replacement in practice At every - - PDF document

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Animal replacement models

Anders Ringgaard Kristensen

Outline

Short survey of animal replacement models A 4-level model for optimal feeding level, fattening policy and slaughtering of organic steers (BKN):

• Model structure
• Decisions being optimized

A 3-level model for optimal replacement of sows

• Model structure
• Integration with Dynamic Linear Model

Animal replacement models: Breeding animals

Dairy cows:

• Traits:
• Age (lactation number and stage)
• Milk yield
• Pregnancy status
• Diseases
• Decisions
• Replace
• Inseminate
• Treat (for diseases)

Sows

• More or less like cows (except milk yield …)

Dairy heifers Slaughter pigs Bull calves Steers Traits considered:

• Age
• Weight
• Pregnancy status (heifers …)

Decisions considered:

• Feeding level
• Insemination (heifers)
• Slaughter (pigs, steers)

Animal replacement models: Growing animals

Sow model: Litter size and Markov property

Litter size at next parity depends on all previous litter size observations. In other words, the Markov property is not satisfied.

Circumventing the Markov violation Define the state as the combined value of present and previous litter size. If each may be “Low”, “Average”, “High” (9 combinations):

• Low, Low
• Low, Average
• Low, High
• Average, Low
• Average, Average
• Average, High
• High, Low
• High, Average
• High, High

Only a computational problem – the “curse of dimensionality” once again.

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A comprehensive example A sow replacement model

• Developed for practical use at prototype level
• 3 levels: Decisions at 2 levels
• Estimation at herd level
• Bayesian updating (Dynamic linear model)
• Decision making

Sow replacement in practice

At every weaning or return to oestrus it is decided whether or not to cull the sow. The decision is based on a prediction of the future performance of the sow. The prediction is compared to the expected performance of a gilt. Only based on existing registrations.

Prediction of sow performance

Based on

• Individual conditions
• Age
• Litter size results (all)
• Re-matings
• Herd specific conditions
• Litter size profile of the herd
• Level of involuntary replacement
• Piglet mortality
• Prices
• Feed intake
• Standard conditions
• Weight of sows

What do we know about litter size in sows? The profile for a herd:

• It is lower for a gilt than for a second parity sow
• A maximum is reached at parity 3, 4, 5
• It is decreasing for older sows
• Some herds produce at a higher level than others

Within herd:

• Some sows produce at a higher level than others
• The repeatability over parities is rather low
• Large random variation

A litter size model Toft & Jørgensen (2002) suggested the following litter size model meeting the demands mentioned on the previous slide: Yit = µt + Mi(t) + εit , where

• Yit is the litter size of sow i at parity t
• µt = -θ1 exp(-(t2-1)θ2) + θ3 - θ4t is the litter size profile of the

herd.

• Mi(t) ∼ N(0, σ2) is the effect of sow i at parity t
• εit ∼ N(0, τ2) is random variation (noise)

The sow effect is auto correlated over parities:

• Cov(Mi(t), Mi(t+u)) = exp(-uα)σ2, in other words
• Mi(t) = ρMi(t-1) + ηit , where
• ρ = exp(-α)
• ηit ∼ N(0, (1-a2)σ2)

Described by 7 herd specific parameters: θ1, θ2, θ3, θ4, τ, σ, α

7 9 11 13 15 1 2 3 4 5 6 7 8 9 10 11 12 Parity Litter size

Litter size profile for a specific herd

Slope = θ4 Intercept = θ3 Curve: θ2 θ1

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Litter size profile for a specific herd

Estimated from herd data Herd level productivity Censoring:

• Only the best sows are kept
• Profile will be biased if simple

averages are used

• A maximum likelihood

estimation technique is used

Between-herd-variation, 8 Danish herds

Between-herd-variation, 7 selected Spanish herds

Between-herd-variation, “all” Spanish herds

The Markov property Let in be the state at stage n The Markov property is satisfied if and only if

• P(in+1| in, in-1, … , i1) = P (in+1| in)
• In words: The distribution of the state at next stage depends
• nly on the present state – previous states are not relevant.

This property is crucial in Markov decision processes.

The Markov property in sows

Does ”Age” satisfy the Markov property

• Yes!

Does ”Rematings” satisfy the Markov property?

• Yes, almost. The probability of conception hardly depends on

previous results at all.

Does ”Litter size” satisfy the Markov property?

• No, certainly not. Several high (or low) will further increase (or

decrease) our expectations to future litter size. Example:

• Sow 1: 12 – 14 – 16 – 15 – 16 piglets
• Sow 2: 6 – 5 – 7 – 6 – 16 piglets
• We prefer sow 1

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Litter size profile for a specific herd

Individual sows are compared to that

• Sow 1 is below average
• Sow 2 is above average

The difference between the

• bservations and the profile is

due to

• The sow effect Mi(t)
• The random error εit

Sow 1 Sow 2

Litter size prediction for an individual sow

Based on:

• Litter size profile of the herd
• Individual deviations from the litter size profile
• All previous litter sizes must be taken into account,

because we don’t know Mi(t)

Uncertainty about the prediction is taken into account

• Not just “average”

Lacking Markov property: The lacking Markov property must be considered when the state is defined Straight forward solution:

• Define the state as in = (y1, y2, … , yn)
• For a sow in parity 8 this means e.g. 158 = 2.5 x

109 state combinations.

• Prohibitive

From registration to information

From registrations to information

Interpretation

• Registration: Litter size λi = yi
• Data: Λ = {y1, y2, … , yn}
• Processing: Ψ() - Kalman filtering, DLM
• Information: I = Ψ(Λ) = (i1, i2)
• Decision: Θ
• Decision strategy: I → Θ

Ψ(): As little loss of information as possible (preferably none).

Coping with the Markov property

Estimate the litter size parameters θ1, θ2, θ3, θ4, τ, σ, α of the herd in question. These parameters are then considered as known, implying that also the litter size profile µt = -θ1 exp(-(t2-1)θ2) + θ3 - θ4t is known. Define the deviation for a sow as

• It = Yt - µt = M(t) + εt

Define a Dynamic Linear Model: Observation equation

• It = M(t) + εt , εit ∼ N(0, τ2)

System equation:

• Mi(t) = ρMi(t-1) + ηit , ηit ∼ N(0, (1-a2)σ2)

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The Kalman filter (West & Harrison, 1997)

Applying the Kalman filter to a DLM allows us to estimate M(t) ~ N(0, σ2) parity by parity as litter size observations are done. Having observed t parities of a sow, we have that the conditional distribution of M(t) given all previous observations is At each parity, the distribution is updated according to

Prediction

The current estimate mt for M(t) is sufficient for prediction. All information from previous litter size observations is included in that. We have

• E(Yt+1 | Y1, … , Yt) = µt+1 + ρ mt
• Var(Yt+1 | Y1, … , Yt) = Vt+1

The variance only depends on the number of

• bservations – not the values observed.

It is not necessary to remember all observed litter sizes. The current estimate mt for M(t) is sufficient! Thus, we satisfy the Markov property by only one state variable.

Biological parameters

Litter size parameters:

• Herd level estimation
• Censoring

Other assumptions

• Conception rates (parity, remating)
• Piglet mortality (parity)
• Sow weight (parity)
• Involuntary culling (parity)
• Feed intake

Model structure

Founder level

• Stage: Life span of a sow
• State: Dummy
• Decision: Dummy

Child level 1

• Stage: Production cycle from weaning
• State: M(n)
• Decision: Mating method

Model structure

Child level 2

• Stages: ”Mating”, ”Gestation”, ”Suckling”
• State:
• Mating: ”Healthy”, ”Diseased”
• Gestation: ”Pregnant”, ”Infertile”, ”Diseased”
• Suckling: Present litter size
• Decision:
• Mating: Allow m matings, m ∈ {1,…,5}
• Gestation: Dummy
• Suckling: ”Keep”, ”Replace”

Prototype

A plug-in for the MLHMP software Herd interface

• Herd specific parameters and prices
• Reads sow data
• Presents results

Built for use in practice Demonstration

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DLM and MDP

First processing: Monitoring & filtering Second processing: Decision making MDP with DLM integrates both parts