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Efficient Design and Analysis of Markovian-type Control Charts Prof. Lianjie Shu Email: [email protected] Faculty of Business Administration University of Macau 1 Outline • Markovian-type Control Charts • Accurate Evaluation • Efficient Design and Sensitivity Analysis • Steady-State Analysis • The Use of Probability Limits for Control Charts 2 Markovian-type Control Charts • The charting statistic at time t can be generalized as Yt g (Yt 1 , X t ), where X t obeservation at time t It alarms at ta min{t ,Yt h} or ta min{t ,| Yt | h} 3 Markovian-type Control Charts • It represents an important class of control charts in Statistical Process Control • It includes many control charts as special cases – The standard exponentially weighted moving average (EWMA) and cumulative sum (CUSUM) charts – Recent variations such as adaptive EWMA charts, adaptive CUSUM charts, and Bayesian control charts. 4 Example: CUSUM Chart The upper-sided CUSUM charting statistics: Yt max(0, Yt 1 X t k ), where k a reference value with popular choice of k / 2 a prespecified mean shift for early detection h a decision interval It alarms at ta min{t , Yt h} 5 Example: EWMA The EWMA charting statistics Qt X t (1 )Qt 1 , where 0< 1 and Q0 0 . It alarms when ta min{t ,| Qt | h} When 1, the EWMA chart reduces to the Shewhart chart. 6 Example: Adaptive EWMA chart The adaptive EWMA statistics: (Huber 1981) Qt Qt 1 hu ( et ), where et X t Qt 1 e (1 ) , e and hu ( e) e, | e | e (1 ) , e . Note: 1. When et 0, Qt (1 w( et ))Qt 1 w( et ) X t , where w( et ) hu ( et ) et . 2. When hu ( e) e, the AEWMA estimator reduces to the EWMA estimator. 3. When =0 or =1, hu ( e) e and Qt X t (Shewhart Statistic). 7 Example: Adaptive CUSUM cha • The Adaptive CUSUM Chart (Sparks 2000; Shu and Jiang 2006; Shu et al. 2008) Z t max{0, Z t1 ( X t ˆt / 2) / h(ˆt / 2)}, ˆ max( , Q ), t min t where Qt (EWMA) estimate of the current process mean min minimum mean shift specified for early detection h( k ) operating function that relates the decision interval of the upper-CUSUM chart to k , given an in-control ARL. The ACUSUM chart signals when Z t c, where c is a constant close to 1. 8 Advantages of Markovian-type Control Charts • Simplicity: recursive form • The average run length can be formulated as a Markov chain or an integral equation. This allows us to evaluate the charting performance without running a large number of simulations. 9 Issue I Accurate Evaluation of Average Run Length (ARL) 10 Performance Measure • Run Length: alarm time/stopping time • Average run length (ARL): expected value of run length ARL=E(RL) • In-control ARL: ARL0=E(RL| δ=0) • Out-of-control ARL: ARL1=E(RL| δ≠0) 11 Evaluation Methods – Monte Carlo Simulation: feasible but may not be the most efficient – Markov Chain – Integral Equation • Often Gaussian-Legendre (GL) quadrature applied to evaluate the integral equation in SPC • The GL method often produce fast and accurate ARL results when the integration kernel is smooth but unreliable results when the integration kernel is not smooth. • Non-smooth integration kernels could be due to non-normal distributions 12 More Accurate Methods • The piecewise collocation (PWC) method To approximate a function based on a linear combination of some Chebyshev polynomials • The Clenshaw-Curtis (CC) quadrature (Clenshaw and Curtis 1960) Basic idea: it can separate the kernel function into two parts: continuous part and discontinuous part and then evaluate the discontinuous one individually. 13 Example 1: ARL of CUSUM under Gamma Distributions 14 Example 1: ARL of CUSUM under Gamma Distributions 15 Example 1: ARL of CUSUM under Gamma Distributions • The CUSUM charting statistics • The ARL integral equation 16 Example 1: ARL of CUSUM under Gamma Distributions 17 Example 2: ARL of AEWMA Control Chart under Normal Distributions The AEWMA Statistic: Gt Gt 1 ( et ) (1 w( et ))Gt 1 w( et ) X t , where et X t Gt 1 =forecast error The Huber's score function (Huber 1981): e (1 ) R, e R hu ( e) e, | e | R e (1 ) R, e R. 0.8 Weight ( et )=score function ( et ) w( et ) weight function et R=3.5 R=2.5 1 0.6 0.4 0.2 0 -10 -8 -6 -4 -2 0 e 2 4 6 18 8 10 Example 2: ARL of AEWMA Control Chart under Normal Distributions • The ARL integral equation 19 (a) λ=0.01, R=3 (b) λ=0.05, R=3 20 Some Publications • • • • • Su, Y., Shu, L., and Tsui, K.-L., “Adaptive EWMA Procedures for Monitoring Processes Subject to Linear Drifts”, Computational Statistics and Data Analysis, 55, 2819-2829, 2011. Huang, W., Shu, L., and Jiang, W., “Evaluation of Exponentially Weighted Moving Variance Control Chart Subject to Linear Drifts”, Computational Statistics and Data Analysis, 56, 4278-4289, 2012. Shu, L., Huang, W., Su, Y., and Tsui, K.-L., “Computation of Run Length Percentiles of CUSUM Control Charts under Changes in Variances”, Journal of Statistical Computation and Simulation, 83, 1238-1251, 2013 Huang, W., Shu, L., Jiang, W., and Tsui, K.-L., “Evaluation of Run-Length Distribution for CUSUM Charts under Gamma Distributions”, IIE Transactions, 981-994, 2013. Huang, W., Shu, L., Su, Y. “An Accurate Evaluation of Adaptive Exponentially Weighted Moving Schemes”, IIE Transactions 46, 457-469, 2014 21 Issue II Efficient Design and Sensitivity Analysis 22 Design Criteria • Statistical Design: To optimize the out-of-control ARL, given a fixed in-control ARL • Economical Design: To minimize the cost/penalty function • Simulations are used to obtain the optimal parameters for control charts. Can a control chart be designed in a more efficient manner? 23 Gradient-based Design • Motivation Note that the ARL of Markovian-type control charts can be formulated as a Markov chain or an integral equation. Therefore, it is possible to approximate the ARL gradients from the ARL integral equation or the Markov chain model. • Advantages of the gradient-based method More accurate More efficient Facilitates the sensitivity analysis 24 Case 1: A Gradient Approach for Efficient Design of EWMA Charts The EWMA charting statistics Qt X t (1 )Qt 1 , where 0< 1 and Q0 z. It alarms when ta min{t ,| Qt | h} Define L( , h; z, ) E (ta |Y0 z, = Design Criterion: min. L( , h; z, ) ,h s.t. L( , h; z,0) ARL0 25 Problem Formulation minL( , h; z, ) ,h subject to L( , h; z,0) = ARL0 h = ( ) min g ( ; ) L( , ( ); z, ) 26 Problem Solving: Conventional Method • Grid Search Grid Search Procedure: (Lucas and Saccucci 1990) (i) Change λ with a particular step size (say 0.01) over the interval (0,1], starting from λ=0.01; (ii) Calculate the corresponding control limits for each λ in the in-control case; (iii) Select λ that provides the smallest out-ofcontrol ARL at shift size of δ. 27 Problem Solving: A New Approach • The first-order condition at the optimal λ that minimizes g(λ;δ) is g’ (λ;δ)=0. 28 Estimate of 𝐿𝜆 (𝜆, ℎ; 𝑧, 𝛿) [u1, u2 ,..., uN ]: Gauss-Legendre abscissas on [-h,h] [ w1, w2 ,..., wN ]: Gauss-Legendre weights on [-h,h] 29 Estimate of 𝐿𝜆 (𝜆, ℎ; 𝑧, 𝛿) 30 Estimate of 𝐿ℎ (𝜆, ℎ; 𝑧, 𝛿) 31 32 33 Case 2: A Gradient Approach for Efficient Design of CUSUM Charts Under Random Shifts • Design criterion under random shift size – – The ARL based design criterion has limited applicability in practice; When the shift size δ is assumed to be random, a more reasonable performance measure is based on the expected weighted average run length (EWARL) EWARL = E [ ( ) L(k , h; z , )] = b ( ) L(k , h; z, ) g ( )d , a where L(k , h; z , ) ARL of the CUSUM chart with design parameters k and h, conditioned on the initial value Y0 z and shift size of ( ) weight function on the ARL g ( ) random distribution of over the range [a, b] 34 Problem Formulation • Two types of weights 1.Expected quality loss (EQL) in Chen and Chen (2007) () = 1 2 2.Expected relative ARL (ERARL) (Ryu et al. 2010; Zhao et al. 2005; and Han et al. 2007): () = 1/L opt (/2) 35 Problem Formulation b min G(k , h) = a ( ) L(k , h; z, ) g ( )d k ,h subject to L(k , h; z,0) = ARL 0 h = ( k ) min(k ) = G (k , (k )) = k b ( ) L(k , (k ); z, ) g ( )d a 36 Problem Solving • Find Θ’(k) 37 Problem Solving • Derivations of the ARL gradients w.r.t. the design parameters k and h 38 •The simulation results are obtained from Ryu et al. (2010) in which k is searched in [0.25,2] or [0.25,1] with grid size 0.01. For each k, the EWARL is simulated based on 1,000 randomly generated δs . 39 40 Some Publications • • • Shu, L., Huang, W., and Jiang, W., “A Novel Gradient Approach for Optimal Design and Sensitivity Analysis of EWMA Control Charts”, Naval Research Logistics, 61, 223-237, 2014 . Huang, W., and Shu, L. “A Gradient Approach for Efficient Design of CUSUM Charts under Uncertainty”, under revision with Journal of Quality Technology, 2014. Huang, W. and Shu, L., “A Gradient Approach to Efficient Design of Multivariate EWMA Control Charts”, working paper, 2014. 41 Issue III Steady-State Analysis of Control Charts 42 Two Types of Steady-State Distribution (i) Cyclical steady-state distribution – It assumes that the charting statistic was reset to its initial state whenever a signal occurs, e.g., in industrial quality control (ii) Conditional steady-state distribution – It assumes that the charting statistic would reach its stationary distribution, conditioned that no signal is given before the change point. – It is common in medical settings. For example, it is unlikely that a hospital would close or suspend the treatment of patients even if there is a deterioration of performance suspected (Gandy and Lau 2013) 43 Two Types of Steady-State Distribution (i) Cyclical steady-state distribution – It assumes that the charting statistic was reset to its initial state whenever a signal occurs, e.g., in industrial quality control (ii) Conditional steady-state distribution – It assumes that the charting statistic would reach its stationary distribution, conditioned that no signal is given before the change point. – It is common in medical settings. For example, it is unlikely that a hospital would close or suspend the treatment of patients even if there is a deterioration of performance is suspected (Gandy and Lau 2013) 44 The Conditional Distribution of the CUSUM Charting Statistics • Assumption: 𝐻0 : 𝑋𝑡 ~𝑁 0,1 • The one-sided upper CUSUM chart: 𝑉. 𝑆. 𝐻1 : 𝑋𝑡 ~𝑁 𝜇, 1 + 𝑆𝑡+ = max(0, 𝑆𝑡−1 + 𝑋𝑡 − 𝑘), where 𝑘 is the reference value. 45 The Conditional Distribution of the CUSUM Charting Statistics • Define Φ1𝑡 (𝑢; 𝑧) as the probability that 𝑆𝑡+ ≤ 𝑢 (0 ≤ 𝑢 ≤ ℎ) conditioned that no signal is triggered before and at time step t and 𝑆0+ = 𝑧, i.e., Φ1𝑡 (𝑢; 𝑧) = 𝑃𝑟(𝑆𝑡+ ≤ 𝑢|𝑆0+ = 𝑧, 𝑆𝑗+ ∈ 𝐼𝑗 , 𝑗 = 1, … , 𝑡), where 𝐼𝑗 = [0, ℎ] denotes the in-control region at time step j. • Define 𝜑𝑡1 𝑢; 𝑧 = 𝑑 Φ1𝑡 𝑢; 𝑧 /𝑑𝑢 • Then 𝛽𝑡 = 𝛼𝑡 where 𝛽𝑡 = 𝑃𝑟(𝑆𝑡+ ≤ 𝑢|𝑆0+ = 𝑧, 𝑆𝑗+ ∈ 𝐼𝑗 , 𝑗 = 1, … , 𝑡 − 1) and 𝛼𝑡 = 𝑃𝑟(𝑆𝑡+ ∈ [0, ℎ]|𝑆0+ = 𝑧, 𝑆𝑗+ ∈ 𝐼𝑗 , 𝑗 = 1, … , 𝑡 − 1) Φ1𝑡 (𝑢; 𝑧) 46 The Conditional Distribution of the CUSUM Charting Statistics • When t=1 47 The Conditional Distribution of the CUSUM Charting Statistics • When 𝑡 ≥ 2 48 The cyclical Distribution of the CUSUM Charting Statistics • Define Φ𝑡2 (𝑢; 𝑧) as the probability that 𝑆𝑡+ ≤ 𝑢 (0 ≤ 𝑢 ≤ ℎ) conditioned on 𝑆0+ = 𝑧, i.e., Φ𝑡2 (𝑢; 𝑧) = 𝑃𝑟(𝑆𝑡+ ≤ 𝑢|𝑆0+ = 𝑧) • Define 𝜑𝑡2 𝑢; 𝑧 = 𝑑 Φ𝑡2 𝑢; 𝑧 /𝑑𝑢 • When 𝑡 = 1 Φ𝑡2 (𝑢; 𝑧) = 𝑃𝑟(0 ≤ 𝑆1+ ≤ 𝑢|𝑆0+ = 𝑧) = 𝑃𝑟(𝑋1 + 𝑧 − 𝑘 > ℎ)+𝑃𝑟(𝑋1 + 𝑧 − 𝑘 ≤ 𝑢) = 1 − 𝐹(ℎ + 𝑘 − 𝑧)+𝐹(𝑢 + 𝑘 − 𝑧) 49 The cyclical Distribution of the CUSUM Charting Statistics • When 𝑡 ≥ 2 50 The stationary distribution of the charting statistic (i) The probability distribution of the CUSUM charting statistics converges as t goes to infinity; (ii) The stationary distribution of the CUSUM charting statistics is independent of its initial value. 51 52 Computation of the steady-state ARL (SSARL) The SSARL is a weighted average of the zero-state ARL LSS L(u )d (u ) where (u) is the stationary distribution of the charting statistic or LSS P(ui ) L(ui ) 53 54 Issue IV The Use of Probability Limits for Control Charts 55 Motivation (i) The use of probability limits is a very general idea and enables us to control the false alarm rate at each step at a desired level. However, the use of constant control limits fail to do this. (ii) The use of probability limits can facilitate the design of control chart in the case with varying sample sizes. In this case, the pattern of the varying sample sizes need to be assumed before simulating the (fixed) control limits. Instead, the use of probability can relax this assumption. 56 The conventional CUSUM chart unde varying sample sizes • Assume the observations are sampling from a normal distribution with mean 𝜇 and variance 𝜎 2 . • Let 𝑛𝑡 be the sample size at time t, and 𝑋𝑡 be the subgroup mean. • The standardized sample mean is 𝑍𝑡 = • 𝑋𝑡 − 𝜇0 𝜎/ 𝑛𝑡 The log-likelihood ratio for testing the hypotheses: 𝐻0 : 𝜇 = 𝜇0 and 𝐻1 : 𝜇 = 𝜇1 𝐿𝑡 = 𝑙𝑜𝑔 𝑓1 (𝑋𝑡 ) 𝑓0 (𝑋𝑡 ) = 𝑛𝑡 (𝜇1 −𝜇0 ) 𝑋𝑡 −𝜇0 ( 𝜎 𝜎/ 𝑛𝑡 − 𝜇1 −𝜇0 ) 𝜎/ 𝑛𝑡 57 The conventional CUSUM chart unde varying sample sizes • The CUSUM chart based on the standardized observations (SD-CUSUM): 𝑆1,𝑡 = max(0, 𝑆1,𝑡−1 + 𝑍𝑡 − 𝑘𝑡 ), 𝜇 −𝜇0 . 𝑛𝑡 1 where 𝑘𝑡 is the reference value 𝑘𝑡 = 2𝜎/ • The CUSUM chart based on the log-likelihood ratio (Arnold and Reynolds 2001) 𝑆2,𝑡 = max{0, 𝑆2,𝑡−1 + 𝑛𝑡 𝑍𝑡 − 𝑘𝑡 }, which was referred to as the generalized CUSUM chart, denoted as GLRCUSUM. 58 The conventional CUSUM chart unde varying sample sizes • The weighted CUSUM (WCUSUM) based on a geometric weight (Yashchin 1989): – The weighted standardized CUSUM (WS1-CUSUM) is based on 𝑆3,𝑡 = max(0, 𝛾𝑆3,𝑡−1 + 𝑍𝑡 − 𝑘𝑡 ), where 0 < 𝛾 ≤ 1 is a discount factor. – The weighted GLR-CUSUM chart (WGLR1-CUSUM) is based on 𝑆4,𝑡 = max{0, 𝛾𝑆4,𝑡−1 + 𝑛𝑡 𝑍𝑡 − 𝑘𝑡 }, 59 The conventional CUSUM chart unde varying sample sizes • The weighted CUSUM (WCUSUM) based on a robust weight function from Huber’s score function (Shu et al. 2011) – The weighted standardized CUSUM (WS2-CUSUM) is based on 𝑆5,𝑡 = max{0, 𝑆5,𝑡−1 + 𝑤ℎ𝑢 (𝑒𝑡 ) 𝑍𝑡 − 𝑘𝑡 }, where 𝑒𝑡 = 𝐿𝑡 −𝐸(𝐿𝑡 |𝜇=𝜇0 ) 𝑣𝑎𝑟(𝐿𝑡 |𝜇=𝜇0 ) = 𝑍𝑡 and – The weighted GLR-CUSUM chart (WGLR2-CUSUM) is based on 𝑆6,𝑡 = max{0, 𝑆6,𝑡−1 + 𝑤ℎ𝑢 (𝑒𝑡 ) 𝑛𝑡 𝑍𝑡 − 𝑘𝑡 }, 60 The CUSUM Chart with Probability Limits • The use of a fixed control limit will lead to different false alarm rate at each time step. Instead, the use of probability limits can maintain a desired conditional false alarm rate at each time step. 61 The CUSUM Chart with Probability Limits • Define 𝛼𝑡 as the false alarm rate at time t conditioned that there is no alarms before time t. • Similar to Margavio et al. (1995), define 𝐻𝑡 𝛼𝑡 as the threshold of the GLR-CUSUM chart at time t for controlling the desired false alarm rate of 𝛼𝑡 . Namely, 𝑃𝑟0 𝑆2,1 > 𝐻1 𝛼1 = 𝛼1 𝑃𝑟0 𝑆2,𝑡 > 𝐻𝑡 𝛼𝑡 |𝑆2,𝑖 ≤ 𝐻𝑖 𝛼𝑖 , 𝑖 = 1,2, … , 𝑡 − 1 = 𝛼𝑡 , 𝑡 = 2,3, … . • The in-control run length distribution is given by 𝑛 𝑃𝑟0 𝑅𝐿 > 𝑛 = (1 − 𝛼𝑖 ) 𝑓𝑜𝑟 𝑛 ≥ 1. 𝑖=1 • When 𝛼𝑖 maintains the same over time, the in-control run length distribution would exactly follow a geometric distribution. 62 The CUSUM Chart with Probability Limits • Define Φ𝑖,𝑡 (𝑥) as the in-control (i=0) and out-of-control distribution (i=1) of the GLR-CUSUM charting statistic, conditioned on no signal before time t, i.e., Φ𝑖,𝑡 𝑥 = 𝑃𝑟𝑖 (𝑆2,𝑡 ≤ 𝑥|𝑆2,𝑗 ∈ 𝐼𝑗 , 𝑗 = 1, … , 𝑡 − 1), where 𝐼𝑗 = [0, 𝐻𝑗 𝛼𝑗 ] denotes the in-control region at time step j. • Similarly, define Φ𝑖,𝑡 (𝑥) as the probability distribution of the GLRCUSUM charting statistics, conditioned on the event that there is no signal at and before time step t, i.e., Φ𝑖,𝑡 𝑥 = 𝑃𝑟𝑖 𝑆2,𝑡 ≤ 𝑥 𝑆2,𝑗 ∈ 𝐼𝑗 , 𝑗 = 1, … , 𝑡 = Φ𝑖,𝑡 𝑥 Φ𝑖,𝑡 𝐻𝑡 𝛼𝑡 for 𝑥 ∈ 𝐼𝑡 63 Determination of the Probability Limi • Define Φ𝑖,𝑡 (𝑥) as the in-control (i=0) and out-of-control distribution (i=1) of the GLR-CUSUM charting statistic, conditioned on no signal before time t, i.e., Φ𝑖,𝑡 𝑥 = 𝑃𝑟𝑖 (𝑆2,𝑡 ≤ 𝑥|𝑆2,𝑗 ∈ 𝐼𝑗 , 𝑗 = 1, … , 𝑡 − 1), where 𝐼𝑗 = [0, 𝐻𝑗 𝛼𝑗 ] denotes the in-control region at time step j. • Similarly, define Φ𝑖,𝑡 (𝑥) as the probability distribution of the GLRCUSUM charting statistics, conditioned on the event that there is no signal at and before time step t, i.e., Φ𝑖,𝑡 𝑥 = 𝑃𝑟𝑖 𝑆2,𝑡 ≤ 𝑥 𝑆2,𝑗 ∈ 𝐼𝑗 , 𝑗 = 1, … , 𝑡 = Φ𝑖,𝑡 𝑥 Φ𝑖,𝑡 𝐻𝑡 𝛼𝑡 for 𝑥 ∈ 𝐼𝑡 64 Determination of the Probability Limits • Note that Φ0,𝑡 𝐻𝑡 𝛼𝑡 = 1 −𝛼𝑡 • To search for 𝐻𝑡 𝛼𝑡 , one can employ Newton’s method Φ0,𝑡 𝐻𝑡𝑘−1 𝛼𝑡 + 𝛼𝑡 − 1 𝐻𝑡𝑘 𝛼𝑡 = 𝐻𝑡𝑘−1 𝛼𝑡 − 𝜑0,𝑡 𝐻𝑡𝑘−1 𝛼𝑡 where 𝜑0,𝑡 𝑥 = [Φ0,𝑡 𝐻𝑡 𝛼𝑡 ]′ . 65 Determination of the Probability Limits • When 𝑡 = 1 Φ0,1 𝑥 = 𝐹0 (𝑘1 + 𝜑0,1 𝑥 = 1 𝑓 (𝑘 𝑛1 0 1 𝑥 ) 𝑛1 + 𝑥 ), 𝑛1 where F0 (. ) and f0 (. ) represent the in-control CDF and PDF of the standard normal distribution. Φ0,1 𝑥 = Φ0,1 𝑥 1−𝛼1 for 0 ≤ 𝑥 ∈ 𝐼𝑡 and 𝐻1 𝛼1 = 𝑛1 𝐹0−1 1 −𝛼1 − 𝑛1 𝑘1 66 Determination of the Probability Limits • When 𝑡 ≥ 2 67 Determination of the Probability Limits 68 Out-of-Control Performance • Define 𝑝𝑡 as the probability that run length is larger than t, i.e., 𝑝𝑡 = 𝑃𝑟1 (𝑅𝐿 > 𝑡), + • Define Φ1,𝑡 (𝑥) as the probability that the chart does not signal before time t and the charting statistics less than or equal to x, i.e., + Φ1,𝑡 (𝑥) = 𝑃𝑟1 (𝑆2,𝑡 ≤ 𝑥, 𝑆2,𝑗 ∈ 𝐼𝑗 , 𝑗 = 1, … , 𝑡 − 1), + + • Note that 𝑝𝑡 = Φ1,𝑡 𝐻𝑡 𝛼𝑡 , we need to find Φ1,𝑡 𝐻𝑡 𝛼𝑡 compute 𝑝𝑡 . in order to 69 Out-of-Control Performance • + How to compute Φ1,𝑡 𝐻𝑡 𝛼𝑡 ? – Let 𝑞𝑡 be the probability that the chart does not signal at time t given no signal before time t, i.e., 𝑞𝑡 = Φ1,𝑡 𝐻𝑡 𝛼𝑡 . Then 𝑝𝑡 = 𝑝𝑡−1 𝑞𝑡 = 𝑞1 𝑞2 … 𝑞𝑡 , for t=1, 2, … – In the OC case, – + Based on Φ1,𝑡 𝑥 = Φ1,𝑡 (𝑥) 𝑞1 𝑞2 … 𝑞𝑡−1 , multiplying the above equation by 𝑞1 𝑞2 … 𝑞𝑡−1 leads to + + Φ1,𝑡 𝑥 = 𝐹𝑡 (𝑘𝑡 + (𝑥 − 𝐻𝑡−1 𝛼𝑡−1 )/ 𝑛𝑡 ) Φ1,𝑡−1 𝐻𝑡−1 𝛼𝑡−1 + 𝐻𝑡−1 𝛼𝑡−1 1 𝑛𝑡 0 + + 𝑓𝑡 (𝑘𝑡 + (𝑥 − 𝑦)/ 𝑛𝑡 ) Φ1,𝑡−1 𝑦 𝑑𝑦 70 Out-of-Control Performance • Algorithm for computing the out-of-control ARL 71 Performance Comparisons • Approximation Accuracy 72 Performance Comparisons • Comparison under varying sample sizes 73 Performance Comparison • Comparison with CUSUM charts with fast initial response (FIR) 74 • Comparison with CUSUM charts with fast initial response (FIR) 75 Remarks • An integral equation approach can be developed to determine the probability limits of the CUSUM charts under varying sample sizes. It is more efficient than the Monte Carlos simulation. • Also, an integral equation approach can be employed to analyze the out-of-control performance. The control chart with dynamic control limits is a general idea and can be used to improve the ARL performance of the control charts with a fixed control limit. • The approach discussed here can be generalized any Markoviantype control charts with/without varying sample sizes. 76 Thank You! Prof. Lianjie SHU Email: [email protected] University of Macau Taipa, Macau, China Tel: (853) 8397-4741 homepage: http://www.umac.mo/fba/staff/shulianjie.html 77 Postgraduate Program • Requirements (http://www.umac.mo/grs/en/admissions.php) – – GPA 3/4 or above English • • • • Application Procedures: online – • CET 6 or TOFEL 550 or IELTS 6 https://isw.umac.mo/naweb_grs/faces/index.jspx Studentship – Mop $14,000/month 78