Monitoring and data filtering II. Dan Jensen IPH, KU

Transcription

1 Monitoring and data filtering II Dan Jensen IPH, KU

2 Outline Introduction to Dynamic Linear Models (DLM) - Conceptual introduction - Difference between the Classical methods and DLM - A very simple DLM and the Kalman filter - Break (5 minutes) Appliction examples using the simple DLM - Break (5 minutes) General form of the DLM Appliction examples using the general DLM Concluding remarks and Exercises

3 The basics of a DLM Dynamic i.e. non-static, adaptive TRUE VALUE Estimation Observed value Forecasted value Uncertainties

4 The basics of a DLM Linear Current value = Previous value + Trend

5 The basics of a DLM Model i.e. we can make forecasts! Alarm Decision Effect estimation: system: support: If tried I staythis the new course, feedhow mixture. will my production look over Doesthe it make nextmy fewproduction years? look better or worse, Therefore: after we strip away the observational noise? How much will it better? look if I change to a faster growing breed? IF everything is fine THEN things progress as expected IF things progress UN-expectedly THEN Something is wrong!

6 Classical methods compared to DLM In Chapter 7: Time series: k1,, kt Model: Moving average EWMA Control charts: test if θ = θ Fundemental assumption: θ is constant over time Here: Model: (Observation equation) Notice: the underlying mean,, can change over time! (System equation)

7 First Order Univariate Polynomial DLM A very simple DLM Updated with each time step!

8 Updating the (simple) DLM: the Kalman filter Prior information: 1-step forecast information: Adaptation information Filtered (posterior) information Prior mean Prior variance Forecast Forecast variance Forecast error Adaptive coefficient Filtered mean Filtered variance

9 Relation to the EWMA EWMA: Filtered mean: When and :

10

11 Incorporate external information Incorporate external information: intervention Types of external information: 1. Known effect, experienced before (ex: new breed with a known different performance) 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 voluntary change

12 Incorporate external information Intervention - 1. Known effect We want the model to adapt to the new known conditions Ex. 8.4, p. 88: Productivity in broilers / reference weight 38 days V=10000 g 2 (i.e. SD of 333 g) Until batch 10: Ross 208; m 10 = 1883 g, W= 100 g 2 From batch 11: Ross 308,» N (µ, W ) where µ = 70 and W = 100 Revised prior: (θ t D t -1 )» N(m t -1 + µ, R t ), R t = C t -1 + W t + W m 11 = m = = 1953 And R 11 = C 10 + W 11 + W = = 1401

13 Incorporate external information Intervention - 2. Unknown effect We want the model to adapt to the new unknown conditions Ex 1: broiler example with no prior information ~ N (0, 20000) for t = 11 In practice we temporarily apply a larger system variance W

14 Incorporate external information Intervention - 2. Unknown effect Productivity in broilers / reference weight Intervention, prior knowlegde ~ N (70, 100) Intervention, no prior knowlegde ~ N (0, 20000) No intervention Change in production should be modeled as intervention If any prior information is available: use it!

15 Incorporate external information Intervention - 3. Estimation of effect Productivity in broilers / reference weight Smoothed mean + 75 g Intervention, no prior knowlegde ~ N (0, 20000) Smoothing Retrospective analysis

16 Specification of variance components Discount factor (δ)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 δ can be used if W is unknown - 0 < δ < 1 we know that W is a fixed proportion of C R t = C t-1 + W R t = C t-1 / δ For a process in control we use the value of delta that minimize the sum of the squares of the forecast errors e t

17 Specification of variance components Daily gain example High value of delta: small system (evolution) variance W, slow adaptation to new information Low value of delta: very adaptive model NB: lower delta can be used for modeling intervention!

18

19 The general Dynamic Linear Model Definition In a general DLM, observations may be multivariate (i.e. vectors) Let Y t = (y 1,, y n ) be a vector of key figures observed at time t. 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 ) F t is the design matrix: extracts expected observations from θ t G t is the system matrix: describe how θ t changes over time 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.

20 A linear growth model Parameter vector Design matrix Observation equation k t = F t θ t + ν t, ν t ~ N(0,V t ) System matrix Covariance matrix System equation θ t = G t θ t-1 + ω t, ω t ~ N(0,W t ) Modeling patterns with a DLM = 0 1 F t = t t θ θ θ = t G = t t t W W W

21 Modeling patterns with a DLM A linear growth model: example of daily feed intake Parameter vector System matrix θ t G t 600 = 25 1 = What is the feed intake at t+1? θ t+1 = G t θ t + ω t Design matrix F t 1 = 0 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 )

22 Updating the general DLM Prior information: 1-step forecast information: Adaptation information Filtered (posterior) information Prior mean Prior variance Forecast Forecast variance Adaptive coefficient Forecast error Filtered mean Filtered variance

23 Modeling patterns with a DLM A linear growth model: example of daily feed intake Specification of the priors m 0 Observational variance: V = (i.e. 300 g standard deviation) Evolution variance: delta = = 25 C =

24 Monitoring Deviations from the model First of all: use standardized errors: Then: Control chart, with alarm limits Or V-mask (parameters d and Ψ) Applied on the cumulative sum (cusum) of the standardized error: C t t = ut = ut + ct t= 1 1 Or: Others, e.g. Bayesian networks, neural networks, etc.

25 Concluding remarks Differents Models were presented Simple local level model DLM in its general form Examples The general form of the model allows us to include patterns (e.g. cyclic patterns for for drinking activity, linear patterns for daily gain) Not necessarily as graphs automatic alarms (as V-mask, MPKF) Many handles to adjust dangerous Always combine with your knowledge on animal production