Monitoring and data filtering I. Classical Methods

Transcription

1 Monitoring and data filtering I. Classical Methods Advanced Herd Management Dan Børge Jensen, IPH Dias 1

2 Outline Framework and Introduction Shewart Control chart Basic principles Examples: milk yield and daily gain Alarms ---- Break Moving Average Control Chart EWMA Control Chart Monitoring autocorrelation Model for autocorrelation UseEWMA ---- Break & exercises Dias 2

3 Framework Dias 3

4 Class Question: what sgoingon, and whatto do? (2 minutes) Average daily weight gain, Pig herds Results from 2 herds Gain (g) Quarter Expected Herd A Herd B Dias 4 Is the conclusion the same in both herds?

5 Introduction(2/3) So far Control: compare key figures (k) with expected results κ= θ+ e s + e o Deviation: see if significant from a statistical point of view If deviation: adjustement plan or/and implementation Problem: we assume that results can be evaluated without considering results from the previous period Dias 5

6 Introduction (3/3) Key figures regarded as a time series of observations, treated as a whole How to model the results? κ t = θ+ e st + e ot = θ+ ν t κ t : observedvalueof the keyfigure θ: true underlying value e st : sample error(biologicalvariation) e ot : observation error(observation method) Dias 6

7 The Shewart Control Chart: basic principles (1/2) Sample quality characteristic Upper Control Limit (UCL) Center Line Lower Control Limit (LCL) κ 1, κ 2,... κ n θ' Sample number, or time Dias 7 Here: all the points fall inside the CL. Process in control

8 The Shewart Control Chart: basic principles (2/2) Center line = target value CL = θ Determination of the control limits UCL t = θ + a σ t LCL t = θ -a σ t Usuallydistance parameter a = 2 or 3 If a = 2 : 2-sigma controllimit Wetest the hypothesish 0 : θ =θ a = 2 correspondsto approx. 5% precisionlevel Dias 8

9 Example7.1: milkyield I llshow you! Target value: CL = θ = kg for firstlactation Overall herd SD over 24 weeks: 490 kg milk N.Cows at beginning: 275 Control limits: UCL t = θ + a σ t LCL t = θ -a σ t Standard deviation calculated according to number of cows behind the average Dias 9

10 Example 1: milk yield Shewart control chart, 2-sigma CL Dias 10

11 Example 1: milk yield Shewart control chart, 2-sigma CL Dias 11

12 Control and warning limits (1/3) UCL and LCL determinedby a(e.g. a=2 <-> p=0.05) Choiceof significancelevel/ distance parameter: tradeoff between number of False Positives and False Negatives! Possible Scenarios: System HAS changed Low a Alarm No Alarm True Positive High a False Negative Type II Error System has NOT changed False Positive True Negative Dias 12 Type I Error

13 Class Questions(5 minutes): If you have a HIGH observation frequency (e.g. every houror every second) which sort of error should you MINIMIZE? And why? If you have a very LOW observation frequency (e.g. every quarteror every year) which sort of error should you MINIMIZE? And why? Alarm No Alarm System HAS changed True Positive False Negative System has NOT changed False Positive True Negative Dias 13

14 Control and warning limits (2/3) Sampling Frequency The more frequent κis calculated, the higher a shouldbe Average Run Length ARL=1/q ARL: expected number of obs between 2 out-of-control alarms. q: the probability of an arbitrary point exceeding the control limits Average Time to Signal ATS=ARL/ν v: sampling frequency, defined as observations per time unit Dias 14

15 Control and warning limits (2.1/3) Example: Processin control q= p a = 2 p = 0.05 Quaterly obs ARL 0 = 1/q = 1/p = 1/0.05 = 20 Obs/Alarm ATS 0 = ARL 0 /ν Two obs per second ATS 0 = ARL 0 /ν Dias 15

16 Control and warning limits (3/3) Whatis the costof a False Negative? Whatis the costof a False Positive? Alternative: useof warninglimits Dias 16

17 Dias 17 5 Minute Break

18 Pattern detection Whatdo wedetect? - Level change, outliers, increase in variation (control limits) - Trend (increase, decrease), cyclic pattern, autocorrelation Rules of thumb: 1- One point outside the control limits 2- Two out of three consecutive points outside the warning limits 3-Fourout of fiveconsecutivepoints at a distance of more than 1σ from the expected level 4-Eightconsecutivepoints on the same side of the expectedlevel Dias 18

19 Illustration of pattern detection From Example 1! Rule 4 Dias 19

20 Example 2: daily gain of growing pigs Target value: θ = CL = 775 g Precision estimates(σ) Random sampling: 20.2 g Control limits: UCL = aσ, a = 2 LCL = 775 aσ, a = 2 Dias 20

21 Example 2: daily gain of growing pigs Shewart control chart, 2-sigma CL Dias 21

22 Example 2: daily gain of growing pigs Processout of control8 obsout of 16 Seasonnal variation is to be expected in slaughter pig production If there is an expected pattern: use of other monitoring techniques to takeit intoaccount e.g. other classical techniques(presented next) or state space models (chapter 8) If no expected pattern: further analysis/ intervention Dias 22

23 Moving Average Control Charts (1/2) The moving average is the average of the most recent n observations M t κ t n+ 1 + K+ κt 1 + κt ( n) =, n t n with variance 2 σ n E.g. n = 4 M7(n) = 750 M6(n) = 744 M5(n) = 756 M4(n) = 763 Dias 23

24 Moving Average Control Charts (2/2) Using n=4, a=3 What can we conclude? Dias 24

25 Exponentially Weighted Moving Average control charts (1/3) The EWMA is a weighted average of allobservations until now z t λκ with variance, for large t, λ = t + ( 1 ) zt 1 2 σ z t 2 λ σ 2 λ The most recent observations are always given highest weights The EWMA control chart is built the same way as the Shewart control chart Dias 25

26 Exponentially Weighted Moving Average control charts (2/3) First lactation, a=2, λ=0.68 Dias 26

27 Exponentially Weighted Moving Average control charts (3/3) Choice of lambda: Small values favor detection of small shifts of θ! Can take time to detect : small lambda = low weight to new obs Shewart control chart is suggested for detecting large shifts Combination of EWMA + Shewart for both small and large shifts Dias 27

28 Check for autocorrelation - Milk Yield example Present versus previous observation Positive autocorrelation Sample autocorrelation First lactation First lactation milk yields First lactation milk yields Dias 28

29 A model for autocorrelation Model Predict next obs. Errors = Observed- predicted e e,..., e t 1, 2 Forecast error Variance Dias 29 Forecast error Control chart!

30 Control chart correlated data The point of a model: IF everything is fine THEN things progress as expected Therefore: IF things progress UN-expectedly THEN Something is wrong! Dias 30

31 EWMA for autocorrelated data Use EWMA as one-step-ahead predictor for autocorrelated data e t ˆκ t+1 z t Choose λ by minimizing the sum of the squared forecast errors = κt zt 1 t i= 1 2 e i The variance of the forecast errors is calculated as Dias 31 σ 2 e t = i = 1 t e 2 i

32 Concluding remarks Wehave shiftedfocusfrom observinga keyfigureκat time t to an entiretime series (κ 1, κ 2,... κ t ) We tried to detect changes in process(alarms) - Raw data (Shewart control chart) - Averaged data (Moving/ Exponentially Moving Average) We observed autocorrelation: model, e.g. EWMA We observed seasonality Next time: other, slightly more advanced methods for modelling! Dias 32

33 Dias 33 Break and exercises, until 5 PM