Transcript 13th NCSx
An Exponentially Weighted Moving Average Control Chart for ZeroTruncated Poisson Processes: A Design and Analytic Framework with Fast Initial Response Feature LEONG, Robert Neil F. Mathematics Department De La Salle University 13th National Convention on Statistics October 3-4, 2016, EDSA Shangri-La Hotel, Mandaluyong City OBJECTIVE OF THE STUDY • To develop an Exponentially-Weighted Moving Average (EWMA) Control Chart for a Zero-Truncated Poisson (ZTP) Process Specifically: a design scheme Leverages on striking a BALANCE between desirable properties of the CuSum (minimax optimality) and the Shewhart (simplicity) charts 13th National Convention on Statistics October 3-4, 2016, EDSA Shangri-La Hotel, Mandaluyong City ZERO-TRUNCATED POISSON PROCESS • Probability Mass Function: • Some Moments: Some Applications: • No. of days until discharge in a hospital following a surgery • *See Plackett (1953) and Cohen (1972) • MLE Estimation: 13th National Convention on Statistics October 3-4, 2016, EDSA Shangri-La Hotel, Mandaluyong City DESIGN OF THE ZTP-EWMA CONTROL CHART • Monitored Statistic: • Control Limits: The same PHASE I and PHASE II (standard control charts) apply! *See Montgomery (2012) *If one-sided upper (lower) charts are desired, THE SAME upper (lower) control limits are formulated (but change L!). 13th National Convention on Statistics October 3-4, 2016, EDSA Shangri-La Hotel, Mandaluyong City AVERAGE RUN LENGTH (ARL) COMPUTATION • ARL – average (discrete) time until the chart signals (i.e., the statistic goes OUTSIDE the control limits) ARL0: ARL when the process is in-control ARL1: ARL when the process is out-of-control Uses a Markov chain approach: *See Brook and Evans (1972) – based on the mean absorption time of the chain 13th National Convention on Statistics October 3-4, 2016, EDSA Shangri-La Hotel, Mandaluyong City ARL COMPUTATIONS General Findings: • longer ARLs for smaller 𝜔’s • exponential growth of ARLs as 𝐿 increases Same for both ARL0 and ARL1. NOTE: ARL0 ∝ 1 / ARL1 • ↑ ARL0 ~ reduced false alarms • ↓ ARL1 ~ increased sensitivity to shifts 13th National Convention on Statistics October 3-4, 2016, EDSA Shangri-La Hotel, Mandaluyong City 𝑳 is chosen to BALANCE! ARL COMPUTATIONS 13th National Convention on Statistics October 3-4, 2016, EDSA Shangri-La Hotel, Mandaluyong City FAST INITIAL RESPONSE (FIR) FEATURE • FIR – “head-start” feature to possibly increase sensitivity without risking too much false alarms one suspects a change before the monitoring scheme is started one wants to detect a possibly ineffective control action • Monitored Statistic: *same as before – with adjusted initialization *See Lucas and Saccucci (1990) • Control Limits: *same as before 13th National Convention on Statistics October 3-4, 2016, EDSA Shangri-La Hotel, Mandaluyong City ARL COMPUTATIONS W/ FIR FEATURE w/ FIR Feature *Entire Table in Full Paper 13th National Convention on Statistics October 3-4, 2016, EDSA Shangri-La Hotel, Mandaluyong City ARL COMPUTATIONS W/ FIR FEATURE *Tabulated results for “Out-of-Control Shifts” for different 𝜆s and 𝜔s are in the full paper. General Findings: • smaller reduction in ARLs as 𝐿 increases • higher variability of ARLs w/ FIR Feature; but are consistently reduced as 𝐿 increases • reduced variability of ARLs for both w/ and w/o FIR Feature as out-of-control shifts in 𝜆 increases FIR Feature (*adjusted initialization) led to QUICKER out-ofcontrol detection at MINIMAL INCREASE in false alarm rates, but at a concerning LOSS OF PRECISION (for small 𝐿). 13th National Convention on Statistics October 3-4, 2016, EDSA Shangri-La Hotel, Mandaluyong City APPLICATION: DURATION OF HOSPITAL ADMISSION PHASE I PHASE II 𝝎 = 𝟎. 𝟏𝟎 • Parameter Estimation: 𝜆𝑀𝐿𝐸 ≈ 4.3 • (ZTP) Goodness-of-Fit Test: p-value ≈ 0.1899 *See Best et al. (2007) 𝑈22 dispersion test 13th National Convention on Statistics October 3-4, 2016, EDSA Shangri-La Hotel, Mandaluyong City APPLICATION: DURATION OF HOSPITAL ADMISSION 13th National Convention on Statistics October 3-4, 2016, EDSA Shangri-La Hotel, Mandaluyong City THANK YOU! 13th National Convention on Statistics October 3-4, 2016, EDSA Shangri-La Hotel, Mandaluyong City