Monitoring and data filtering III. The Kalman Filter and its relation with the other methods

Transcription

1 Monitoring and data filtering III. The Kalman Filter and its relation with the other methods Advanced Herd Management Cécile Cornou, IPH Dias 1

2 Gain (g) Before this part of the course Compare key figures (k) with expected results κ = θ + e s + e o Results from 2 herds Quarter Expected Herd A Herd B Dias 2

3 The methods we looked at In part I of the course (1/3) Key figures regarded as a time series of observations, treated as a whole κ = θ + e s + e o κ t = θ + e st + e ot = θ + v t I - Shewart Control Chart - Plot our Time Series of Observations : raw observations - Plot our Target Value (θ ) - Plot of Control Limits (UCL & LCL) Not so good... for our type of data... Dias 3

4 The methods we looked at In part I of the course (2/3) II - Shewart Control Chart - Plot a Moving Average of our TS - Plot our Target Value (θ ) - Plot of Control Limits (UCL & LCL) III - Shewart Control Chart - Plot a EWMA of our TS - Plot our Target Value (θ ) - Plot of Control Limits (UCL & LCL) Still not so good... for our type of data Here: milk yield Dias 4

5 The methods we looked at In part I of the course (3/3) IV Autocorrelated data - Model the data - Calculate a prediction for next obs. - Plot the prediction errors V EWMA for autocorrelated data - Use EWMA as one-step-ahead predictor - Plot the prediction errors Here we use 0 as Center Line and see how error terms are distributed Dias 5

6 The methods we looked at In part II of the course (1/4) A Simple DLM Before: κ t = θ + e st + e ot = θ + v t Time series Y t = (y 1,..., y n ) Observation equation: y t = t + v t, v t» N(0, V t ) Like before: v t = e s + e o The symbol t is the underlying true value at time t. System equation: t = t-1 + w t, w t» N(0, W t ) The true value is not any longer assumed to be constant. Dias 6

7 The methods we looked at In part II of the course (2/4) The Kalman Filter ( t-1 D t-1 )» N(m t-1, C t-1 ) ( t D t )» N(m t, C t ) The updating equations of the Kalman Filter are used for stepwise calculation of m t and C t DLM and EWMA DLM EWMA m t AY t t ( 1 At ) mt 1 z t t ( 1 ) zt 1 As the model adapts to the data, A t converges and becomes A The calculation of the mean of the underlying level for the simple DLM looks very similar to the calculation of the EWMA Dias 7

8 The methods we looked at In part II of the course (3/4) The different models - Simple DLM - DLM with a trend 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 ) t = ( 1,, m ) is a vector of parameters describing the system at time t. Can include: Level, Trend, Seasonality, Periodicity,... Structure + External information According to the data observed, the use of DLM allows many modeling possibilities Dias 8

9 The methods we looked at In part II of the course (4/4) Monitoring methods used with DLM Monitor the forecast errors (e t = Y t f t ) as we did with autocorrelated data in part I NO need for expected value V-mask Tabular cusum Can look at the different components of the models (trends, seasons) Dias 9

10 Further examples Ex 1. Monitoring activity level A simple DLM Monitoring deviations by mean of V-mask and Tabular cusum Ex. 2. Monitoring sows activity types in farrowing house Use of a MPKF of class I Dias 10

11 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 11

12 Dias 12 Outdoor facilities

13 3 D accelerometer The large pen Animals and Housing 5 sows in group of days Activity Measurements Acceleration in 3 dimensions Four measurements per second Transfer PC via Blue Tooth Video Recordings Four cameras used as web cam ACTIVITY Oestrus Detection Golden standard Detect whether activity pattern changes at onset of oestrus Dias 13

14 Definition of the DLM Use hourly averages of the length of the acceleration vector Y t = acc = (acc x2 + acc y2 + acc z2 ) Observation equation: k t = θ t + v t, v t» N(0, V t ) System equation: θ t = θ t-1 + w t, w t» N(0, W t ) V t = V= 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 14

15 Illustration Model Cusum V-mask Tabular Cusum Dias 15

16 Example 2. Monitoring sows activity types Farrowing house Assumption Sow s behaviour is affected by physiological state / illness 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 See whether we can automatically detect the onset of farrowing Dias 16

17 Dias 17 The Farrowing House

18 Data Collected Farrowing house Farrowing Dias 18

19 Time series and activity types Activity types Extracts from time series of acceleration are associated to 5 activity types High Active (HA) Medium Active (MA) Passive Lying Side 1 (L1) Passive Lying Side 2 (L2) Passive Lying Sternally (Ls) 3 dimensions: X,Y, Z Learning data set: 10 minutes of each activity type Estimate the model parameters X Y Z Dias 19

20 Dynamic Linear Model Dynamic Linear Models (DLMs) combined with Kalman Filter (KF) 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. Multivariate DLM: each observation is a 3-dimensional vector V (3 x 3) and W (3 x 3), characteristic of each activity, are estimated by EM algorithm 5 DLMs (5 activities) Y t x y z Dias 20

21 Multi Process Kalman Filter MPKF 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 Dias 21

22 Activity Classification Farrowing 2 days before farrowing Feeding: 7.15, 12.00, Active Dias 22 Lying side 1 Lying sternally Lying side 2

23 Activity Classification Farrowing Farrowing day Feeding / Rooting / Nesting Active Dias 23 Lying side 1 Lying sternally Lying side 2

24 Activity Classification Farrowing Percentage of activity types d-3 to d+1 Dias 24

25 Activity Classification Farrowing Sum of 2 min HA (red), MA (orange) and Passive (blue) / hour Farrowing Cusum of HA t HA t-24 Dias 25

26 The multi process kalman filter (MPKF) (1/2) Multi Process Models Class I ex: Classifying sows activities in Farrowing house A single out of a range of possible DLMs is viewed as appropriate at all time for describing the entire time series Parameters : V and W We try to find out which model is it at time t? Dias 26

27 The multi process kalman filter (MPKF) (2/2) Multi Process Models Class II ex: Monitoring Somatic Cell Counts / Boar visits No single DLM as adequate for all time: the possibility that different models are appropriate at different times is explicitly recognised and modelled through different defining parameters Parameters : V and W and Π (values for Π define fixed prior probabilities) We try to find out normal evolution, outlier, level shift We need to go back f.x. two, three steps to check which model is the right one Dias 27

28 Sum up of sum up We increased the complexity of our models and monitoring methods to try to adapt to the complexity of monitoring living beings Methods here based on a whole time series of observations -> need for automatic registration -> potential on-line monitoring From part I to part II, we made our model dynamics, so the true underlying level is no more constant Dias 28