1 Observation error Second finding: Randomness The true daily gains - - PDF document

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The Kalman filter

• and other methods

Outline Filtering techniques applied to monitoring of daily gain in slaughter pigs:

• Introduction
• Basic monitoring
• Shewart control charts
• DLM and the Kalman filter
• Simple case
• Seasonality
• Online monitoring
• Used as input to decision support

”E-kontrol”, slaughter pigs

Quarterly calculated production results Presented as a table A result for each of the most recent quarters and aggregated Sometimes comparison with expected (target) values Offered by two companies:

• Dansk Landbrugsrådgivning,

• AgroSoft A/S

One of the most important key figures: Average daily gain

Average daily gain, slaughter pigs We have:

• 4 quarterly results
• 1 annual result
• 1 target value

How do we interpret the results? Question 1: How is the figure calculated?

How is the figure calculated? The basic principles are:

• Total (live) weight of pigs delivered:

xxxx

• Total weight of piglets inserted:

−xxxx

• Valuation weight at end of the quarter:

+xxxx

• Valuation weight at beginning of the quarter:

−xxxx

• Total gain during the quarter

yyyy

Daily gain = (Total gain)/(Days in feed) Registration sources?

• * Slaughter house – rather precise
• ** Scale – very precise
• *** ??? – anything from very precise to very uncertain

* ** *** ***

First finding: Observation error All measurements are encumbered with uncertainty (error), but it is most prevalent for the valuation weights. We define a (very simple) model: κ = τ + eo , where:

• κ is the calculated daily gain (as it appears in the report)
• τ is the true daily gain (which we wish to estimate)
• eo is the observation error which we assume is normally distributed N(0, σo2)

The structure of the model (qualitative knowledge) is the equation The parameters (quantitative knowledge) is the value of σo (the standard deviation of the observation error). It depends on the observation method.

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Observation error κ = τ + eo , eo ∼ N(0, σo

2)

What we measure is κ What we wish to know is τ The difference between the two variables is undesired noise We wish to filter the noise away, i.e. we wish to estimate τ from κ

τ κ

Second finding: Randomness The true daily gains τ vary at random. Even if we produce under exactly the same conditions in two successive quarters the results will differ. We shall denote the phenomenon as the “sample error”. We have, τ = θ + es, where

• es is the sample error expressing random variation. We assume es ∼ N(0, σs

• θ is the underlying permanent (and true) value

This supplementary qualitative knowledge should be reflected in the stucture of the model: κ = τ + eo = θ + es + eo The parameters of the model are now: σs og σo

Sample error and measurement error

What we measure is κ What we wish to know is θ The difference between the two variables is undesired noise:

• Sample noise
• Observation noise

We wish to filter the noise away, i.e. we wish to estimate θ from κ

θ κ τ

The model is necessary for any meaningful interpretation of calculated production results. The standard deviation on the sample error, σs , depends on the natural individual variation between pigs in a herd and the herd size. The standard deviation of the observation error, σo , depends on the measurement method of valuation weights. For the interpretation of the calculated results, it is the total uncertainty, σ , that matters (σ2 = σs

Competent guesses of the value of σ using different observation methods (1250 pigs):

• Weighing of all pigs: σ = 3 g
• Stratified sample: σ = 7 g
• Random sample: σ = 20 g
• Visual assessment: σ = 29 g

The model in practice: Preconditions

Different observation methods

θ κ τ κ κ κ

σ = 3 g σ = 7 g σ = 20 g σ = 29 g

The model in practice: Interpretation Calculated daily gain in a herd was 750 g, whereas the expected target value was 775 g. Shall we be worried? It depends on the observation method! A lower control limit (LCL) is the target minus 2 times the standard deviation, i.e. 775 – 2σ Using each of the 4 observation methods, we

• btain the following LCLs:
• Weighing of all pigs: 775 g – 2 x 3 g = 769
• Stratified sample: 775 g – 2 x 7 g = 761
• Random sample: 775 g – 2 x 20 g = 735
• Visual assessment: 775 g – 2 x 29 g = 717

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Third finding: Dynamics, time

Daily gain, slaughter pigs

600 650 700 750 800 850 900 950

• 2. quarter 97
• 3. quarter 97
• 4. quarter 97
• 1. quarter 98
• 2. quarter 98
• 3. quarter 98
• 4. quarter 98
• 1. quarter 99
• 2. quarter 99
• 3. quarter 99
• 4. quarter 99
• 1. quarter 00
• 2. quarter 00
• 3. quarter 00
• 4. quarter 00
• 1. quarter 01
• 2. quarter 01

Quarter g

Daily gain in a herd over 4 years. Is this good or bad?

Modeling dynamics

We extend our model to include time. At time n we model the calculated result as follows: κn = τsn + eon = θ + esn + eon Only change from before is that we know we have a new result each quarter. We can calculate control limits for each quarter and plot everything in a diagram: A Shewart Control Chart …

κ1 τ1 θ κ2 τ2 κ3 τ3 κ4 τ4 …

A simple Shewart control chart: Weighing all pigs Daily gain, slaughter pigs

600 650 700 750 800 850 900 950

• 2. kvartal 97
• 3. kvartal 97
• 4. kvartal 97
• 1. kvartal 98
• 2. kvartal 98
• 3. kvartal 98
• 4. kvartal 98
• 1. kvartal 99
• 2. kvartal 99
• 3. kvartal 99
• 4. kvartal 99
• 1. kvartal 00
• 2. kvartal 00
• 3. kvartal 00
• 4. kvartal 00
• 1. kvartal 01
• 2. kvartal 01

Period g Observed gain Expected Upper control limit Lower control limit

Periode

Daily gain, slaughter pigs

600 650 700 750 800 850 900 950

• 2. kvartal 97
• 3. kvartal 97
• 4. kvartal 97
• 1. kvartal 98
• 2. kvartal 98
• 3. kvartal 98
• 4. kvartal 98
• 1. kvartal 99
• 2. kvartal 99
• 3. kvartal 99
• 4. kvartal 99
• 1. kvartal 00
• 2. kvartal 00
• 3. kvartal 00
• 4. kvartal 00
• 1. kvartal 01
• 2. kvartal 01

Period g Observed gain Expected Upper control limit Lower control limit

Simple Shewart control chart: Visual assessment

Periode

Interpretation: Conclusion

Something is wrong! Possible explanations:

• The pig farmer has serious problems with fluctuating daily gains.
• Something is wrong with the model:
• Structure – our qualitative knowledge
• Parameters – the quantitative knowledge (standard

deviations).

More findings: κn = θ + esn + eon The true underlying daily gain in the herd, θ , may change over time:

• Trend
• Seasonal variation

The sample error esn may be auto correlated

• Temporary influences

The observation error eon is obviously auto correlated:

• Valuation weight at the end of Quarter n is the

same as the valuation weight at the start of Quarter n+1

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”Dynamisk e-kontrol”

Developed and described by Madsen & Ruby (2000). Principles:

• Avoid labor intensive valuation weighing.
• Calculate new daily gain every time pigs have been sent to

slaughter (typically weekly)

• Use a simple Dynamic Linear Model to monitor daily gain
• κn = θn + esn + eon = θn + vn , where vn ∼ N(0, σv

• θn = θn-1 + wn, where wn ∼ N(0, σw

• The calculated results are filtered by the Kalman filter in order to

remove random noise (sample error + observation error)

”Dynamisk E-kontrol”, results

Raw data to the left – filtered data to the right Figures from:

• Madsen & Ruby (2000). An application for early detection of growth rate

changes in the slaughter pig production unit. Computers and Electronics in Agriculture 25, 261-270. Still: Results only available after slaughter

The Dynamic Linear Model (DLM) Example

Observation equation

System equation

General, first order

Observation equation

System equation

θ1 κ1 τ1 θ2 κ2 τ2 θ3 κ3 τ3 θ4 κ4 τ4 µ1

Y1

µ2

Y2

µ3

Y3

µ4

Y4

Extending the model

Fnθn is the true level described as a vector product. A general level, θ0n, and 4 seasonal effects θ1n, θ2n, θ3n and θ4n are included in the model. From the model we are able to predict the expected daily gain for next quarter. As long as the forecast errors are small, production is in control (no large change in true underlying level)!

Observed and predicted

Daily gain

Blue: Observed Pink: Predicted

Analysis of prediction errors

Daily gain

• 100
• 80
• 60
• 40
• 20

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The last model Dynamic Linear Model Structure of the model (qualitative knowledge):

• Seasonal variation allowed (no assumption about the size).
• The general level as well as the seasonal pattern may change over time.

Are those assumptions correct? Parameters of the model:

• The observation and sample variance and the system variance.

The model learns as observations are done, and adapts to the

• bservations over time.

Seasonal varation may be modeled more sophistically as demonstrated by Thomas Nejsum Madsen in FarmWatch™

Moral

If we wish to analyze the daily gain of a herd you need to:

• Know exactly how the observations are done (and know

the precision).

• Know how it may naturally develop over time.

Without professional knowledge you may conclude anything. Without a model you may interpret the results inadequately. Through the structure of the model we apply our professional knowledge to the problem.

On-line monitoring of slaughter pigs: PigVision Innovation project led by Danish Pig Production:

• Danish Institute of Agricultural Sciences
• Videometer (external assistance)
• Skov A/S
• LIFE, IPH, Production and Health

Continuous monitoring of daily gain while still in herd:

• Dynamic Linear Models
• Chance of interference in the fattening period
• Adaptation of delivery policy

PigVision: Principles

A camera is placed above the pen. In case of movements a series of pictures are recorded and sent to a computer. The computer automatically identifies the pig (by use of a model) and calculates the area (seen from above). If the computer doesn’t belief that a pig has been identified, the picture is ignored. The area is converted to live weight (using a model). Through many pictures, the average weight and the standard deviation are estimated.

What is online weight assessment used for? Continuous monitoring of gain. Collection of evidence about growth capacity (learning) Adaptation of delivery policies depending on:

• Whether the pigs grow fast or slowly
• Whether the uniformity is small or big
• Whether a new batch of piglets is ready
• Prices

Direct advice about pigs to deliver

The decision support model

Technique:

• A hierarchical Markov Decision Process (dynamic programming) with a

Dynamic Linear Model (DLM) embedded. Every week, the average weight and the standard deviation is observed After each observation the parameters of the DLM are opdated using Kalman filtering:

• Permanent growth capacity of pigs, L
• Temporary deviation, e(t)
• Within-pen standard deviation, ρ(t)

Decisions based on (state space):

• Number of pigs left
• Estimated values of the 3 parameters

Decision: Deliver all pigs with live weight bigger than a threshold Uncertainty of knowledge is directly built into the model through the DLM

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On-line weight assessment Pen with n pigs is monitored. No identification of pigs. At any time t we have: The precision 1/σ2 is assumed known

Objectives Given the on-line weight estimates to assign an

• ptimal delivery policy for the pigs in the pen.

Sequential (weekly) decision problem with decisions at two levels:

• Slaughtering of individual pigs (the price is highest in a

rather narrow interval)

• Terminating the batch (slaughter all remaining pigs and

insert a new batch of weaners)

Dynamic linear models

A dynamic linear weight model, I

Known average herd specific growth curve: True weights at time t distributed as:

The scaling factor L

In principle unknown and not directly

• bservable

Initial belief: The belief is updated each time we

• bserve a set of live weights from the

pen. Let L ∼ N(1, σL

2) be the true average

weight Then

Observation & system equation 1

Full observation equation for mean: Auto-correlated sample error (system eq.):

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Observation & system equation 2

Far more information available from the

• bserved live weights

Sample variance not normally distributed. Use the 0.16 sample quantile: The symbol ρ(t) is the standard deviation

• f the observed values. System

equation:

Full equation set

Learning, permanent growth capacity

Learning: Homogeneity (standard deviation)