Department of Veterinary and Animal Sciences Monitoring methods – revisited Anders Ringgaard Kristensen
Department of Veterinary and Animal Sciences Outline Filtering techniques applied for 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 Slide 2
”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: • AgroSoft A/S • Cloudfarms 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?
Department of Veterinary and Animal Sciences 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 Slide 5
Department of Veterinary and Animal Sciences 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 + e o , where: • κ is the calculated daily gain (as it appears in the report) • τ is the true daily gain (which we wish to estimate) • e o is the observation error which we assume is normally distributed N(0, σ o 2 ) The structure of the model (qualitative knowledge) is the equation The parameters (quantitative knowledge) is the value of σ o (the standard deviation of the observation error). It depends on the observation method. Slide 6
Department of Veterinary and Animal Sciences Observation error κ = τ + e o , e o ~ N(0, σ o 2 ) τ What we measure is κ What we wish to know is τ The difference between the two variables is undesired noise κ We wish to filter the noise away, i.e. we wish to estimate τ from κ Slide 7
Department of Veterinary and Animal Sciences 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 + e s , where • e s is the sample error expressing random variation. We assume e s » N(0, σ s 2 ) • θ is the underlying permanent (and true) value This supplementary qualitative knowledge should be reflected in the stucture of the model: κ = τ + e o = θ + e s + e o The parameters of the model are now: σ s og σ o Slide 8
Department of Veterinary and Animal Sciences Sample error and measurement error What we measure is κ θ What we wish to know is θ The difference between the two variables is undesired noise: τ • Sample noise • Observation noise We wish to filter the noise away, i.e. we wish to estimate κ θ from κ Slide 9
Department of Veterinary and Animal Sciences The model in practice: Preconditions The model is necessary for any meaningful interpretation of calculated production results. The standard deviation on the sample error, σ s , depends on the natural individual variation between pigs in a herd and the herd size. The standard deviation of the observation error, σ o , depends on the measurement method of valuation weights. For the interpretation of the calculated results, it is the total uncertainty, σ , that matters ( σ 2 = σ s 2 + σ ο 2 ) 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 Slide 10
Department of Veterinary and Animal Sciences Different observation methods θ τ κ κ κ κ σ = 3 g σ = 7 g σ = 20 g σ = 29 g Slide 11
Department of Veterinary and Animal Sciences 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 – 2 σ Using each of the 4 observation methods, we obtain 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 Slide 12
Is this good or bad? Daily gain in a herd over 4 years. Third finding: Dynamics, time g 600 650 700 750 800 850 900 950 2. quarter 97 3. quarter 97 4. quarter 97 Daily gain, slaughter pigs 1. quarter 98 2. quarter 98 3. quarter 98 4. quarter 98 1. quarter 99 Quarter 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
Department of Veterinary and Animal Sciences Modeling dynamics We extend our model to include time. At time n we model the calculated result as follows: κ n = τ sn + e on = θ + e sn + e on 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 … θ τ 4 … τ 1 τ 2 τ 3 κ 1 κ 2 κ 3 κ 4 Slide 14
Department of Veterinary and Animal Sciences A simple Shewart control chart: Weighing all pigs Daily gain, slaughter pigs 950 900 850 800 g 750 700 650 600 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 Periode Observed gain Expected Upper control limit Lower control limit Slide 15
Simple Shewart control chart: Visual assessment Slide 16 g 600 650 700 750 800 850 900 950 2. kvartal 97 3. kvartal 97 4. kvartal 97 1. kvartal 98 Daily gain, slaughter pigs Upper control limit Observed gain 2. kvartal 98 3. kvartal 98 4. kvartal 98 Periode 1. kvartal 99 Period 2. kvartal 99 3. kvartal 99 Lower control limit Expected 4. kvartal 99 1. kvartal 00 Department of Veterinary and Animal Sciences 2. kvartal 00 3. kvartal 00 4. kvartal 00 1. kvartal 01 2. kvartal 01
Department of Veterinary and Animal Sciences 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). Slide 17
Department of Veterinary and Animal Sciences More findings: κ n = θ + e sn + e on The true underlying daily gain in the herd, θ , may change over time: • Trend • Seasonal variation The sample errors e sn may be auto correlated • Temporary influences The observation error e on 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 Slide 18
Department of Veterinary and Animal Sciences ”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 • κ n = θ n + e sn + e on = θ n + v n , where v n ~ N(0, σ v 2 ) θ n = θ n -1 + w n , where w n ~ N(0, σ w • 2 ) • The calculated results are filtered by the Kalman filter in order to remove random noise (sample error + observation error) Slide 19
”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
Department of Veterinary and Animal Sciences The Dynamic Linear Model (DLM) Example Observation equation κ n = θ n + v n , v n » N(0, σ v 2 ) System equation θ n = θ n -1 + w n , w n » N(0, σ w 2 ) θ 1 θ 2 θ 3 θ 4 τ 1 τ 2 τ 3 τ 4 κ 1 κ 2 κ 3 κ 4 Slide 21
Extending the model F n θ n is the true level described as a vector product. A general level, θ 0 n , and 4 seasonal effects θ 1 n , θ 2 n , θ 3 n and θ 4 n 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)!
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