Monitoring and data filtering II. Dynamic Linear Models
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1 Ouline Monioring and daa filering II. Dynamic Linear Models (Wes and Harrison, chaper 2 Updaing equaions: Kalman Filer Discoun facor as an aid o choose W Incorporae exernal informaion: Inervenion General form of he DLM Advanced Herd Managemen Cécile Cornou, IPH Examples Concluding remarks Dias 1 Dias 2 A Simple DLM Time series Y = (y 1,..., y n A DLM wih a rend Time series Y = (y 1,..., y n Observaion equaion: y = µ + v, v N(0, V Observaion equaion: y = µ + v, v N(0, V Like before: v = e s + e o The symbol µ is he underlying rue value a ime. Sysem equaion: µ = µ -1 + w, w N(0, W Sysem equaion: µ = µ -1 + β -1 + w 1, w 1 N(0, W 1 β = β -1 + w 2, w 2 N(0, W 2 The rue value is no any longer assumed o be consan. A fair assumpion in animal producion Basically, we wish o deec large changes in µ Dias 3 Dias 4 Updaing equaions: Kalman Filer (a Poserior for µ -1 : (µ -1 D -1 N(m -1, C -1 (b Prior for µ : (µ D -1 N(m -1, R where R = C -1 + W (c 1-sep forecas: (Y D -1 N(f, Q where f = m -1 and Q = R + V (d Poserior for µ : (µ D N(m, C Discoun facor as an aid o choosing W To run he model (assume wih consan parameers we need: m 0, C 0, V, W Discoun facor can be used if W is unknown we know ha W is a fixed proporion of C (Wes & Harrison R = C -1 + W R = C -1 / δ Typically 0.8 < δ < 1 wih m = m -1 + A.e and C = A.V where A = R / Q and e = Y - f Iniial Informaion: (µ 0 D 0 N(m 0, C 0 Dias 5 Dias 6 1
2 Incorporae exernal informaion: inervenion Types of exernal informaion: Known effec, experienced before (ex: change in breed for which we know he differen performances We wan he model o adap o he new known condiions Inervenion - 1. Known effec We wan he model o adap o he new known condiions Ex: Kurri example from Wes and Harrison (2.3.2 Esimaed mean afer change: 286 (vs. 143 Expeced change = 143 ( Dias 7 2. Unknown effec (ex: wave of hea, inroducion of new animals in a group We wan he model o adap o he new unknown condiions 3. Unknown effec we wan o measure (ex: change of feed composiion, new veerinary reamens We wan o measure he effec of a volonary change Uncerainy: from 80 (pessimisic o 200 (opimisic Dias 8 σ = 30 ( = ( / 4 : 4 s.dev (95% inerval Variance associaed (Uncerainy = 30 2 = 900 (ω 10 D 9, S 9 N (143, 900 Revised one-sep ahead forecas: (µ D -1 N(m -1, R (µ 10 D 9, S 9 N (286, 920 m 9 = 286 and R 10 = C 9 + W 10 = = 920 Inervenion - 2. Unknown effec (1/2 We wan he model o adap o he new unknown condiions Ex 1: wave of hea We can no adjus because we do no know he exac effec Ex 2: inroducion of new animals in a group Consider a mehod aimed o deec oesrus by monioring animal behaviour Incoming animals may modify he behaviour of he group Inervenion - 2. Unknown effec (2/2 In pracice we can emporarily reduce he value of he discoun facor so he evoluion variance increases We pu more weigh on he new observaions and forge abou he pas Here, inervenion aimed o increase model adapaion o new behaviour so o avoid alarms due o a known even See also eaing rank Dias 9 Dias 10 Inervenion - 3. Unknown effec We wan o measure he effec of a volonary change Ex: A new feed is used and we wan o esimae he associaed change in daily gain We know ha he new feed is used from ime τ (0 < τ < n y = µ + λ I + v, v N(0, V µ = µ -1 + w, w N(0, W Wih: I : inervenion effec ha we wan o measure λ = 0 when < τ λ = 1 when > τ The general DLM Generalisaion from he 1.order pol. Model Simple, mos widely used DLM Marix noaion allows o presen he DLM in a general form and o rea more complex cases Three examples of applicaion Monioring aciviy level Monioring aciviy ypes (MPKF Monioring eaing behaviour Dias 11 Dias 12 2
3 Modeling of he variable Dynamic Linear Models (DLMs combined wih Kalman Filer (KF Le Y = (y 1,, yn be a vecor of key figures observed a ime. Le θ = (θ 1,, θm be a vecor of parameers describing he sysem a ime. General form of he DLM Observaion Equaion: Y = F θ + ν, ν ~ N(0,V Sysem Equaion: θ = G θ -1 + ω, ω ~ N(0,W Monioring Deviaions from he model V-mask (parameers d and Ψ Applied on he cumulaive sum (cusum of he sandardized errors u = e / Q C = u = u + c = 1 1 DLM combined wih Kalman Filer: esimae he underlying sae vecor θ by is mean vecor m and is variance-covariance marix C. Elemens from KF used in monioring deviaions: f : One sep forecas mean e : One sep forecas error (e = Y f Q : One sep forecas variance Tabular Cusum (parameers K and H Creae a cusum: accumulae u i, using a reference value (K Alarm when cusum exceeds a decision inerval (H Dias 13 Dias 14 Example 1. Monioring aciviy level Conex Developmen of Group housing in EU resuls of Council Direcive 2001/88/EEC Difficulies idenifying and accessing individual sow Oesrus Deecion Oesrus Deecion I (from day 4 BPT (3 x / day in he maing secion Sow no inseminaed Transfered o gesaion secion Idea Sore daa in a chip and ransmi info o he farmer s PC Sensor in he chip allows o monior aciviy of he sow Assumpion Body Aciviy of sows is expeced o change around he onse of oesrus Objecive Develop an auomaed oesrus deecion mehod for group housed sows using sows acceleraion measuremens Mehod Use of Dynamic Linear Models o model he sows aciviy Use of conrol mehods ha deecs model deviaions a he onse of oesrus Dias 15 Weaning Oesrus Deecion II (from day 21 BPT (3 x / day in he gesaion secion d0 Dias 16 Golden Sandard Deec wheher aciviy paern changes a onse of oesrus BPT I Transfer d7 d10 Acceleraion measuremens d21 BPT II d30 Daa collecion Definiion of he DLM Place, Animals, Housing and Feeding 1 producion herd, March sows in group of days Aciviy Measuremens Acceleraion in 2 and 3 dimensions Four measuremens per second Transfer PC via Blue Tooh Video Recordings Four cameras used as web cam Oesrus Deecion Golden sandard Deec wheher aciviy paern changes a onse of oesrus Use hourly averages of he lengh of he acceleraion vecor µ = 0 V = unknow and consan Y = acc = (acc x2 + acc y2 + acc z2 θ F' = ( 1,0 W = 0 (In normal condiion: no change in aciviy Model iniialized by mean of Reference Analysis G = I Model observaions (Y weighed by number of observaions per hour Missing observaion: e =0 Dias 17 Dias 18 3
4 Illusraion Model Example 2. Monioring sows aciviy ypes Assumpion Daily Anoesrus Cusum Sow s behaviour is affeced by physiological sae / illness Oesrus: increase in aciviy Lameness: walking Daily Oesrus V-mask Acceleromeer: measured any ime / during whole reproducive cycle Objecive Tabular Cusum Develop a mehod ha auomaically classify sows aciviy ypes Model seleced aciviy ypes using DLM Classify each aciviy ype using a Muli Process Kalman Filer Dias 19 Dias 20 Time series and aciviy ypes Acceleraion daa Aciviy ypes Exracs from ime series of acceleraion are associaed o five aciviy ypes Feeding (FE Rooing (RO Walking (WA Lying sernally (LS Lying Laerally (LL 3 dimensions: X,Y, Z ACC = (acc x2 + acc y2 + acc z2 Two daa ses Aciviy filled whole daa se / no overlapping Learning daa se: 10 minues of each aciviy ype Esimae he model parameers Tes daa se: 10 x 2 minues of each aciviy ype Implemen he classificaion mehod X Y Z ACC Dias 21 Dias 22 Modeling each aciviy ype Classificaion mehod Use averages per second of acceleraion daa µ θ = s c 2π 2π ' = 1,sin,cos T T F G = I Model includes a periodic movemen cyclic componens V and W: esimaed using he EM algorihm Learning daa se 20 DLMs 5 aciviies x 4 axes (X, Y, Z, ACC Muli Process Kalman Filer of class I A ime : Each DLM is analysed using he updaing equaions of he Kalman Filer: One sep forecas mean f One sep forecas variance Q Poserior Probabiliies are esimaed for each DLM p ( i φ ( i p 1( i Dias 23 Dias 24 4
5 Classificaion mehod Illusraion: walking aciviy Resuls FE: bes recognized WA: axis Z beer RO: slow recogniion LL: axis Y beer LS: well recognized Perspecives Applicaion? Acive vs. Passive Illness Paruriion Example 3. Modeling Eaing Behaviour Assumpion Sow s feeding behaviour is affeced by oesrus and illness Currenly: lis of sows ha have no eaen is used o idenify individuals Objecive Develop a mehod ha auomaically deec oesrus, lameness and oher healh disorders for sows fed by ESF Model feeding behaviour (Feeding rank using DLM Deec deviaions by mean of conrol char Axis z Sed og dao (Indsæ --> Diasnummer Dias 25 Dias 26 Daa collecion Modeling of he variable Place, Housing and Feeding 3 producion herds, January 2005 January 2006 Herds 1 and 2: dynamic groups Herd 3: saic groups Elecronic Sow Feeders (ESF Regisraion ESF Oesrus (BPT Lameness and Healh disorders Use daily Feeding Rank (Y µ = β θ F' = ( 1,0 V assumed unknow and consan W esimaed by discoun facor Missing observaion : e =0 Exernal informaion: subgroup of sows eners or leaves a group or boh : Lower discouning G 1 = Dias 27 Dias 28 Deecion mehod Opimizaion of V-mask parameers for 3 condiions i oesrus ii lameness iii Oher healh disorders Crieria Sensiiviy of a leas 50% Number of FP is minimum Illusraion - Inervenion Subgroup in Subgroup in + ou Individual eaing rank Model forecas Increase Adapive Coefficien Dias 29 Dias 30 5
6 g g g g Resuls Example Daily Gain (from firs lecure i Oesrus deecion Sensiiviy ranges from 59 o 75% (vs. Lis of sows: 9 o 20% Marices Specificaion ii Lameness and iii Oher healh disorders Sensiiviy ranges from 41 o 70% (vs. Lis of sows: 22 o 39% Too many false alarms Perspecives Include Seasonal componens Include oher variables: e.g. ear base emperaure, aciviy Mulivariae model Dias 31 Dias 32 Example Daily Gain (from firs lecure Daily gain, slaugher pigs Observed gain Daily gain, slaugher pigs Prediced gain Seasons Daily gain, slaugher pigs Observed gain Prediced gain Level Season 1 Season 2 Season 3 Season Daily gain, slaugher pigs 2. kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral kvaral 01 Trend Errors Concluding remarks Differens Models were presened Simple local level model DLM in is general form Examples The general form of he model allow o include cyclic paern (as for eaing aciviy, daily gain Thomas Nejsum Madsen will presen an approach based on sine funcions o incorporae a diurnal paern. No necessarily as graphs auomaic alarms (as V mask. Many handles o adjus dangerous Always combine wih your knowledge on animal producion Season 1 Season 2 Season 3 Season 4 Forecas error Lower limi Upper limi Dias 33 Dias 34 6
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