Monitoring and data filtering II. Dynamic Linear Models

Transcription

1 Monitoring and data filtering II. Dynamic Linear Models Advanced Herd Management Cécile Cornou, IPH Dias 1

2 Program for monitoring and data filtering Friday 26 (morning) - Lecture for part I : use of control charts - Exercise 1 with R Tuesday 28 (today) - Lecture for part II : Dynamic Linear Models (DLMs) - Exercise 2 with R - Introduction to mandatory report Tuesday 28 (afternoon) - Thomas N. Madsen presents an application of DLM - Exercises (and MR?) Friday 03 (morning) - Kalman Filter and its relation to other techniques (sum up of the methods introduced) - Mandatory report Dias 2

3 Outline Introduction to the DLM (West and Harrison, chapter 2) Updating equations: Kalman Filter Discount factor as an aid to choose W Incorporate external information: Intervention General form of the DLM Examples Concluding remarks Dias 3

4 Introduction to the DLM A Simple DLM Time series Y t = (y 1,..., y n ) Observation equation: y t = µ t + v t, v t N(0, V t ) t t t t t Like before: v t = e s + e o The symbol µ t is the underlying true value at time t. System equation: µ t = µ t-1 + w t, w t N(0, W t ) The true value is not any longer assumed to be constant. A fair assumption in animal production Basically, we wish to detect large changes in µ t Dias 4

5 Introduction to the DLM A DLM with a trend Time series Y t = (y 1,..., y n ) Observation equation: y t = µ t + v t, v t N(0, V t ) t t t t t System equation: µ t = µ t-1 + β t-1 + w 1t, w 1t N(0, W 1t ) β t = β t-1 + w 2t, w 2t N(0, W 2t ) Dias 5

6 Introduction to the DLM Updating equations: Kalman Filter (a) Posterior for µ t-1 : (µ µ t-1 D t-1 ) N(m t-1, C t-1 ) (b) Prior for µ t : (µ t D t-1 ) N(m t-1, R t ) R t = C t-1 + W t (c) 1-step forecast: (Y t D t-1 ) N(f t, Q t ) f t = m t-1 Q t = R t + V t (d) Posterior for µ t : (µ t D t ) N(m t, C t ) m t = m t-1 + A t.e t C t = A t.v t A t = R t / Q t e t = Y t f t Initial Information: (µ 0 D 0 ) N(m 0, C 0 ) Dias 6

7 Introduction to the DLM Discount factor as an aid to choosing W t To run the model (assume with constant parameters) we need: m 0, C 0, V, W Discount factor can be used if W is unknown we know that W is a fixed proportion of C (West & Harrison 2.4.2) R t = C t-1 + W R t = C t-1 / δ Typically 0.8 < δ < 1 Dias 7

8 Introduction to the DLM Incorporate external information: intervention Types of external information: Known effect, experienced before (ex: change in breed for which we know the different performances) We want the model to adapt to the new known conditions 2. Unknown effect (ex: wave of heat, introduction of new animals in a group) We want the model to adapt to the new unknown conditions 3. Unknown effect we want to measure (ex: change of feed composition, new veterinary treatments) We want to measure the effect of a volontary change Dias 8

9 Introduction to the DLM Intervention - 1. Known effect We want the model to adapt to the new known conditions Ex: Kurrit example from West and Harrison (2.3.2) Estimated mean after change: 286 (vs. 143) Expected change = 143 ( ) Uncertainty: from 80 (pessimistic) to 200 (optimistic) σ = 30 ( = (200 80) / 4) : 4 st.dev (95% interval) Variance associated (Uncertainty) = 30 2 = 900 (ω 10 D 9, S 9 ) N (143, 900) Revised prior: (µ t D t-1 ) N(m t-1, R t ) (µ 10 D 9, S 9 ) N (286, 920) m 9 = 286 and R 10 = C 9 + W 10 = = 920 Dias 9

10 Introduction to the DLM Intervention - 2. Unknown effect (1/2) We want the model to adapt to the new unknown conditions Ex 1: wave of heat We can not adjust because we do not know the exact effect Ex 2: introduction of new animals in a group Consider a method aimed to detect oestrus by monitoring animal behaviour Incoming animals may modify the behaviour of the group Here, intervention aimed to increase model adaptation to new behaviour so to avoid alarms due to a known event Dias 10

11 Introduction to the DLM Intervention - 2. Unknown effect (2/2) In practice we can temporarily reduce the value of the discount factor so the evolution variance increases We put more weight on the new observations and forget about the past See also eating rank Dias 11

12 Introduction to the DLM Intervention - 3. Unknown effect We want to measure the effect of a volontary change Ex: A new feed is used and we want to estimate the associated change in daily gain We know that the new feed is used from time τ (0 < τ < n) y t = µ t + λ t I t + v t, v t N(0, V t ) µ t = µ t-1 + w t, w t N(0, W t ) With: I t : intervention effect that we want to measure λ t = 0 when t < τ λ t = 1 when t > τ Dias 12

13 The general DLM Generalisation from the 1.order pol. Model Simple, most widely used DLM Matrix notation allows to present the DLM in a general form and to treat more complex cases Four examples of application Monitoring activity level Monitoring activity types (MPKF) Monitoring eating behaviour Monitoring daily gain Dias 13

14 Modeling of the variable Dynamic Linear Models (DLMs) combined with Kalman Filter (KF) Let Y t = (y 1,, y n ) be a vector of key figures observed at time t. t 1 n Let θ t = (θ 1,, θ m ) be a vector of parameters describing the system at time t. General form of the DLM Observation Equation: Y t = F t θ t + ν t, ν t ~ N(0,V t ) System Equation: θ t = G t θ t-1 + ω t, ω t ~ N(0,W t ) DLM combined with Kalman Filter: estimate the underlying state vector θ t by its mean vector m t and its variance-covariance matrix C t. Dias 14

15 Matrix specification for general DLM (1/3) How do we relate to the local level model? Little note: matrix multiplication Consider matrices A and B, C is the product AB Dias 15

16 Matrix specification for general DLM (2/3) Observation equation θ t is the latent process θ t = ( µ t 0 ) F t is the design matrix F t = ( 1 0 ) Yt = F t θ t + v t 1 µ t 0 + v µ t t + v t 0 Yt = µ t + v t Dias 16

17 Matrix specification for general DLM (3/3) System equation θ t is the latent process θ t = ( µ t 0 ) G t is the system matrix G t = θ t = G t θ t-1 + w t 1 0 µ t w t µ t w t µ t = µ t-1 + w t Dias 17

18 Monitoring Deviations from the model Elements from KF used in monitoring deviations: f t : One step forecast mean e t : One step forecast error (e t = Y t f t ) Q t : One step forecast variance Monitoring methods: V-mask Tabular cusum Multi Process Kalman Filter Dias 18

19 Monitoring Deviations from the model V-mask (parameters d and Ψ) Applied on the cumulative sum (cusum) of the standardized errors u t = e t / Q t = t t C t u t = u t + ct 1 1 t= 1 Tabular Cusum (parameters K and H) Create a cusum: accumulate u i, using a reference value (K) Alarm when cusum exceeds a decision interval (H) Dias 19

20 Example 1. Monitoring activity level Context Development of Group housing in EU results of Council Directive 2001/88/EEC Difficulties identifying and accessing individual sow Idea Store data in a chip and transmit info to the farmer s PC Sensor in the chip allows to monitor activity of the sow Assumption Body Activity of sows is expected to change around the onset of oestrus Objective Develop an automated oestrus detection method for group housed sows using sows acceleration measurements Method Use of Dynamic Linear Models to model the sows activity Use of control methods that detects model deviations at the onset of oestrus Dias 20

21 Oestrus Detection Oestrus Detection I (from day 4) BPT (3 x / day) in the mating section Sow not inseminated Transfered to gestation section Oestrus Detection II (from day 21) BPT (3 x / day) in the gestation section Golden Standard Detect whether activity pattern changes at onset of oestrus Weaning Transfer Acceleration measurements d0 BPT I d7 d10 d21 BPT II d30 Dias 21

22 Data collection Place, Animals, Housing and Feeding 1 production herd, March sows in group of days Activity Measurements Acceleration in 2 and 3 dimensions Four measurements per second Transfer PC via Blue Tooth Video Recordings Four cameras used as web cam Oestrus Detection Golden standard Detect whether activity pattern changes at onset of oestrus Dias 22

23 Definition of the DLM Use hourly averages of the length of the acceleration vector Y t = acc = (acc x2 + acc y2 + acc z2 ) t µ t = 0 θ F' = ( 1,0) t G t = I V t = unknow and constant W t = 0 (In normal condition: no change in activity) Model initialized by mean of Reference Analysis Model observations (Y t ) weighted by number of observations per hour Missing observation: e t =0 Dias 23

24 Illustration Model Cusum V-mask Tabular Cusum Dias 24

25 Example 2. Monitoring sows activity types Daily Anoestrus Assumption Sow s behaviour is affected by physiological state / illness Oestrus: increase in activity Lameness: walking Daily Oestrus Accelerometer: measured any time / during whole reproductive cycle Objective Develop a method that automatically classify sows activity types Model selected activity types using DLM Classify each activity type using a Multi Process Kalman Filter Dias 25

26 Time series and activity types Activity types Extracts from time series of acceleration are associated to five activity types Feeding (FE) Rooting (RO) Walking (WA) Lying sternally (LS) Lying Laterally (LL) 3 dimensions: X,Y, Z ACC = (acc x2 + acc y2 + acc z2 ) Two data sets Activity filled whole data set / no overlapping Learning data set: 10 minutes of each activity type Estimate the model parameters Test data set: 10 x 2 minutes of each activity type Implement the classification method X Y Z ACC Dias 26

27 Acceleration data Dias 27

28 Modeling each activity type Use averages per second of acceleration data θ t µ t s ct = t F' t 2π 2π = 1,sin t,cos t T T G t = t I Model includes a periodic movement cyclic components V and W: estimated using the EM algorithm Learning data set 20 DLMs 5 activities x 4 axes (X, Y, Z, ACC) Dias 28

29 Classification method Multi Process Kalman Filter of class I At time t: Each DLM is analysed using the updating equations of the Kalman Filter: One step forecast mean f t One step forecast variance Q t Posterior Probabilities are estimated for each DLM p t ( i) φ ( i) p 1( i) t t Dias 29

30 Illustration: walking activity Results FE: best recognized WA: axis Z better RO: slow recognition LL: axis Y better LS: well recognized Perspectives Application? Active vs. Passive Illness Parturition Axis z Dias 30

31 Example 3. Modeling Feeding Behaviour Assumption Sow s feeding behaviour is affected by oestrus and illness Currently: list of sows that have not eaten is used to identify individuals Objective Develop a method that automatically detect oestrus, lameness and other health disorders for sows fed by ESF Model feeding behaviour (Feeding rank) using DLM Detect deviations by mean of control chart Dias 31

32 Data collection Place, Housing and Feeding 3 production herds, January 2005 January 2006 Herds 1 and 2: dynamic groups Herd 3: static groups Electronic Sow Feeders (ESF) Registration ESF Oestrus (BPT) Lameness and Health disorders Dias 32

33 Modeling of the variable Use daily Feeding Rank (Y t ) t µ t = βt θ ' = ( 1,0) F t G t = V t assumed unknow and constant W t estimated by discount factor Missing observation : e t =0 External information: subgroup of sows enters or leaves a group or both : Lower discounting Dias 33

34 Detection method Optimization of V-mask parameters for 3 conditions i) oestrus ii) lameness iii) Other health disorders Criteria Sensitivity of at least 50% Number of FP is minimum Dias 34

35 Illustration - Intervention Subgroup in Subgroup in + out Individual eating rank Model forecast Increase Adaptive Coefficient Dias 35

36 Results i) Oestrus detection Sensitivity ranges from 59 to 75% (vs. List of sows: 9 to 20%) ii) Lameness and iii) Other health disorders Sensitivity ranges from 41 to 70% (vs. List of sows: 22 to 39%) Too many false alarms Perspectives Include other variables: e.g. ear base temperature, activity Multivariate model Dias 36

37 Example Daily Gain (from first lecture) Matrices Specification Include Seasonal components Dias 37

38 Example Daily Gain (from first lecture) Daily gain, slaughter pigs Daily gain, slaughter pigs Trend g kvartal kvartal kvartal kvartal kvartal kvartal kvartal kvartal kvartal kvartal kvartal kvartal kvartal kvartal kvartal kvartal kvartal kvartal kvartal kvartal kvartal kvartal kvartal kvartal kvartal kvartal kvartal kvartal kvartal kvartal kvartal kvartal 01 Period Observed gain Predicted gain Period Observed gain Predicted gain Level Season 1 Season 2 Season 3 Season 4 Seasons Errors Daily gain, slaughter pigs Daily gain, slaughter pigs g g g 2. kvartal kvartal kvartal kvartal kvartal kvartal kvartal kvartal kvartal kvartal kvartal kvartal kvartal kvartal kvartal kvartal kvartal kvartal kvartal kvartal kvartal kvartal kvartal kvartal kvartal kvartal kvartal kvartal kvartal kvartal kvartal kvartal 01 Period Period Season 1 Season 2 Season 3 Season 4 Forecast error Lower limit Upper limit Dias 38

39 Concluding remarks Differents Models were presented Simple local level model DLM in its general form Examples The general form of the model allows to include cyclic patterns (as for eating activity, daily gain) Thomas Nejsum Madsen will present an approach based on sine functions to incorporate a diurnal pattern. Not necessarily as graphs automatic alarms (as V mask). Many handles to adjust dangerous Always combine with your knowledge on animal production Dias 39