On ARL-unbiased c-charts for i.i.d. and INAR(1) Poisson counts

Transcription

1 On ARL-unbiased c-charts for iid and INAR(1) Poisson counts Manuel Cabral Morais (1) with Sofia Paulino (2) and Sven Knoth (3) (1) Department of Mathematics & CEMAT IST, ULisboa, Portugal (2) IST, ULisboa, Portugal (3) Department of Economics and Social Sciences Helmut Schmidt University, Germany IST, ULisboa Lisbon, February 17, 2016

2 Introduction The iid case The INAR(1) case Final thoughts Agenda 1 Introduction Initial thoughts and motivation 2 The iid case Revisiting the c-chart Quantile based control limits ARL-unbiased c charts for λ 3 The INAR(1) case Definitions and properties ARL-unbiased c chart for λ/(1 β) 4 Final thoughts On ARL-unbiased c-charts for iid and INAR(1) Poisson counts

3 Introduction The iid case The INAR(1) case Final thoughts Initial thoughts and motivation Processes of counts of nonconformities arise frequently in SPC Their marginal distribution is usually assumed to be Poisson We often deal with autocorrelated count processes but falsely assume serial independence, namely while planning a control chart to monitor the process mean On ARL-unbiased c-charts for iid and INAR(1) Poisson counts

4 Introduction The iid case The INAR(1) case Final thoughts Initial thoughts and motivation Dealing with a nonnegative, discrete and asymmetrical (eg right-skewed) distribution (eg Poisson) can prevent us to: set a pre-specified in-control ARL; deal with a positive lower control limit; be able to quickly detect small and moderate decreases in the process mean; define an ARL-unbiased control chart (Pignatiello et al, 1995; Acosta-Mejía, 1999) in the sense that it takes longer, in average, to trigger a false alarm than to detect any shifts in the process mean On ARL-unbiased c-charts for iid and INAR(1) Poisson counts

5 Introduction The iid case The INAR(1) case Final thoughts Revisiting the c-chart The c chart is one of the most popular procedures to control the expected number of defects (λ) in a sample of n units Control statistic: total number of defects in the t th sample, X t Distribution: X t indep X Poisson(λ), t N Target mean: λ 0 Process mean: λ = λ 0 + δ 3 σ control limits: LCL = max {0, λ 0 3 } λ 0 (δ is the magnitude of the shift) UCL = λ λ 0 Trigger a signal and deem the process out-of-control at sample t if X t [LCL, UCL] Performance measure run length: RL(δ) Geometric( ξ(δ) = P[X [LCL, UCL] δ] ) ARL(δ) = 1/ξ(δ) On ARL-unbiased c-charts for iid and INAR(1) Poisson counts

6 Introduction The iid case The INAR(1) case Final thoughts Revisiting the c-chart If λ 0 9 then LCL = 0 and the chart triggers false alarms more frequently than valid signals in the presence of any decrease in λ For λ 0 > 9, the ARL function of a c chart with 3 σ control limits attains its maximum value at» 1 δ UCL! UCL LCL+1 (λ 0) = argmax δ ( λ0,+ )ARL(δ) = λ0 < 0 (LCL 1)! argmax λ 0 On ARL-unbiased c-charts for iid and INAR(1) Poisson counts

7 Introduction The iid case The INAR(1) case Final thoughts Revisiting the c-chart Some variants of the c chart c charts with 3 σ control limits may behave poorly when it comes to the detection of decreases in λ Alternative approaches (Aebtarm & Bouguila, 2011): transforming data; standardizing data; optimizing control limits Best overall c chart (optimal control limits) Proposed by Ryan & Schwertman (1997), as defined by Aebtarm & Bouguila (2011) control limits are obtained by linear regression based on a table of the best c chart limits for several values of λ 0: LCL = λ λ 0 ; UCL = λ λ 0 On ARL-unbiased c-charts for iid and INAR(1) Poisson counts

8 Introduction The iid case The INAR(1) case Final thoughts Quantile based control limits Control limits based on quantiles can be an appropriate alternative to the 3 σ limits, but the discrete character of the Poisson distribution makes it difficult to achieve the pre-specified probability of false alarm α = Quantile based control limits Requiring that P(X [LCL, UCL] δ = 0) α and setting α = α LCL + α UCL leads to obtention of LCL and UCL such that: P(X < LCL δ = 0) α LCL ; P(X > UCL δ = 0) α UCL Thus, the quantile based LCL (resp UCL) is the largest (resp smallest) nonneg integer satisfying the 1st (resp 2nd) condition Performance measure: 1 ARL(δ) = ξ(δ) = P(X [LCL, UCL] δ) ; ARL(0) α 1 On ARL-unbiased c-charts for iid and INAR(1) Poisson counts

9 Introduction The iid case The INAR(1) case Final thoughts ARL-unbiased-c-chart-Table1Figure1nb Quantile based control limits Example 1: Comparing ARL performance (λ 0 = 8) Out[525]= Out[529]= 1 Unlike the 3 σ limits, the quantile based limits lead to a c-chart with an ARL function not as biased as the one of the R&S c-chart Out[530]= 18 Out[531]= ARL( ) Out[532]= Figure: ARL(δ) for: 3 σ control limits, [0, 16] (dotted lines); R&S control limits, [1, 17] (dashed lines); quantile based control limits, [1, 18] (thin solid lines; α = 00027, α LCL = α UCL ) On ARL-unbiased c-charts for iid and INAR(1) Poisson counts

10 Introduction The iid case The INAR(1) case Final thoughts ARL-unbiased c charts for λ Inspired by UMPU (Uniformly Most Powerful Unbiased) tests (Lehmann, 1986, pp ) and randomized tests (rarely used in SPC when dealing with discrete distributions), we defined a c chart, with quantile based control limits, that triggers a signal with: probability one if the sample number of defects is below LCL or above UCL; probabilities γ LCL and γ UCL if the sample number of defects is equal to LCL and UCL, resp We are basically considering H 0 : λ = λ 0 vs H 1 : λ λ 0 1 if x < LCL or x > UCL γ LCL if x = LCL φ(x) = P(Reject H 0 X = x) = γ UCL if x = UCL 0 if LCL < x < UCL On ARL-unbiased c-charts for iid and INAR(1) Poisson counts

11 Introduction The iid case The INAR(1) case Final thoughts ARL-unbiased c charts for λ The randomization probabilities satisfy { E λ0 [φ(x )] = α (prob of a false alarm = α) (γ LCL, γ UCL ) : E λ0 [X φ(x )] = α E λ0 (X ) (unbiased ARL) The solution of this system of linear equations: γ LCL = where a = P λ0 (LCL), de bf ad bc c = LCL P λ0 (LCL), and b = P λ0 (UCL) e = α 1 + UCL x=lclp λ0 (x) γ LCL = d = UCL P λ0 (UCL) f = (α 1) E λ0 (X ) + UCL x=lclx P λ0 (x) af ce ad bc, On ARL-unbiased c-charts for iid and INAR(1) Poisson counts

12 Introduction The iid case The INAR(1) case Final thoughts ARL-unbiased c charts for λ In order to rule out pairs of control limits leading to (γ LCL, γ UCL ) (0, 1) 2, the potentially useful (LCL, UCL) are restricted to the following grid of non-negative integer numbers: {(LCL, UCL) : L min LCL L max, U min UCL U max } The search for admissible values for (γ LCL, γ UCL ) starts with (LCL, UCL) = (L min, U min ) and stops as soon as an admissible solution is found L min = n max F 1 (max{0, F (U min 1) 1 + α}), G 1 o (max{0, G(U min 1) 1 + α}) n o 1 1 L max = min F (α), G (α) n U min = max F 1 (1 α), G 1 o (1 α) n 1 U max = min F (min{1, F (Lmax ) + 1 α}), G 1 o (min{1, G(L max ) + 1 α}) F (x) = P λ0 (X x) G(x) = 1 X x i =0 λ ξ Pp 0 (X = i) 0 F 1 (α) = min{x N 0 : F (x) α} F 1 (α) = min{x N 0 : F (x) > α} G 1 (α) = min{x N 0 : G(x) α} G 1 (α) = min{x N 0 : G(x) > α} On ARL-unbiased c-charts for iid and INAR(1) Poisson counts

13 Introduction General::stop : Further output The of iid LessEqual::nord case will be suppressed The during INAR(1) thiscase calculation Final thoughts ARL-unbiased GreaterEqual::nord c charts for: λinvalid comparison with ComplexInfinity attempted General::stop : Further output of GreaterEqual::nord will be suppressed during this calculation Example 2: Comparing ARL performance (λ 0 = 8) When 0=8 and =00027 we get LCLunbiased=1, UCLunbiased=18, L= , U= , in-control ARL=37037, relative bias= ARL( ) Figure: ARL(δ) for: c chart with quantile based control limits (thin solid lines); ARL-unbiased c chart with [LCL, UCL] = [1, 18] & randomization prob (γ LCL, γ UCL ) = ( , ) (α = 00027; thick solid lines) Randomization increases the prob of triggering a signal, thus smaller ARL values, yet a reasonable in-control ARL (1/ ) On ARL-unbiased c-charts for iid and INAR(1) Poisson counts

14 Introduction The iid case The INAR(1) case Final thoughts ARL-unbiased c charts for λ ARL-unbiased designs Table: Limits of the search grid, quantile based control limits and randomization prob of the ARL-unbiased c charts, for λ 0 = 005, 01(01)1, 2 20, α = λ 0 L min L max LCL U min U max UCL γ LCL γ UCL On ARL-unbiased c-charts for iid and INAR(1) Poisson counts

15 Introduction The iid case The INAR(1) case Final thoughts Time series of counts Arise in areas such as industry the monthly number of accidents in a manufacturing plant and the number of defects per sample have to be controlled When the time series consists only of small integer numbers, ARMA processes are of limited use, namely because the multiplication of an integer-valued rv by a real constant may lead to a non-integer rv A possible way out is to replace the scalar multiplication by a random operation, such as the binomial thinning operation On ARL-unbiased c-charts for iid and INAR(1) Poisson counts

16 Introduction The iid case The INAR(1) case Final thoughts Definitions and properties Binomial thinning operation Let X be a discrete rv with range N 0 and β a scalar in (0, 1) Then the binomial thinning operation on X results in another rv defined as follows: X β X = Y i, i=1 where {Y i : i N} is a sequence of iid Bernoulli(β) rv, independent of X β X arises from X by binomial thinning On ARL-unbiased c-charts for iid and INAR(1) Poisson counts

17 Introduction The iid case The INAR(1) case Final thoughts Definitions and properties INAR(1) process {X t : t Z} is said to be a first-order integer-valued autoregressive process if X t = β X t 1 + ɛ t, where: β (0, 1); represents the binomial thinning operator; {ɛ t : t Z} be a sequence of nonnegative integer-valued iid rv, with mean µ ɛ and variance σ 2 ɛ ; ɛ t and X t 1 are assumed to be independent rv; all thinning operations are performed independently of each other and of {ɛ t : t Z}; the thinning operations at time t are independent of {, X t 2, X t 1 } On ARL-unbiased c-charts for iid and INAR(1) Poisson counts

18 Introduction The iid case The INAR(1) case Final thoughts Definitions and properties Poisson INAR(1) process If ɛ t iid Poisson(λ), t Z, then {X t = β X t 1 + ɛ t : t Z} is said to be a Poisson INAR(1) process It is a second order weakly stationary process such that X t Poisson (λ/(1 β)), t Z Its transition probability matrix, P = [p i j ] i,j = [P(X t = j X t 1 = i)] i,j, has entries given by p i j p i j (λ, β) ix = P(β X t 1 = m X t 1 = i) P(ɛ t = j m) = m=0 min{i,j} X m=0! i β m (1 β) i m e λ λ j m, i, j N0 m (j m)! On ARL-unbiased c-charts for iid and INAR(1) Poisson counts

19 Introduction The iid case The INAR(1) case Final thoughts Definitions and properties c-chart for the mean of a Poisson INAR(1) process Control statistic: X t, t N Target value: λ 0 /(1 β 0 ) k σ control limits: LCL = max { 0, λ 0 /(1 β 0 ) k } λ 0 /(1 β 0 ) UCL = λ 0 /(1 β 0 ) + k λ 0 /(1 β 0 ), where k is a positive constant (Weiss, 2009, p 419) Parameters: λ = λ 0 + δ λ or β = β 0 + δ β, where δ λ and δ β are the magnitude of shifts in λ and β On ARL-unbiased c-charts for iid and INAR(1) Poisson counts

20 Introduction The iid case The INAR(1) case Final thoughts Definitions and properties Run length Let: y = LCL and x = UCL ; Then RL u (λ, β) = min{t : X t < y or X t > x X 0 = u} be the RL of the c-chart, conditional on X 0 = u (u {x,, y}), λ and β ARL u (λ, β) = e u [I Q(λ, β)] 1 1, (1) where e u : (u y + 1) th vector of the orthogonal basis for R (x y+1), Q(λ, β) = [p lk (λ, β)] x l,k=y, I: identity matrix with rank (x y + 1), 1: column-vector with (x y + 1) ones In addition, overall ARL function (Weiss and Testik, 2009): x ARL(λ, β) = 1 + ARL u (λ, β) P[X 1 (λ, β) = u] ARL-biased chart! u=y On ARL-unbiased c-charts for iid and INAR(1) Poisson counts

21 Introduction The iid case The INAR(1) case Final thoughts ARL-unbiased c chart for λ/(1 β) The ARL-unbiased c chart for the mean of a Poisson INAR(1) process triggers a signal at sample t with: probability one if the count of nonconformities, X t, is beyond the control limits L and U; probability γ L (resp γ U ) if X t is equal to L (resp U) Since the control statistics are dependent rv, we can no longer: explicitly define a search grid for the non-negative integers L and U in the terms of the false alarm rate; obtain (γ L, γ U ) by solving a system of linear equations On ARL-unbiased c-charts for iid and INAR(1) Poisson counts

22 Introduction The iid case The INAR(1) case Final thoughts ARL-unbiased c chart for λ/(1 β) RL of the ARL-unbiased c chart Randomizing the emission of a signal means considering Q(λ, β, γ L, γ U ) equal to 2 3 (1 γ L ) p L L p L L+1 p L U 1 (1 γ U ) p L U (1 γ L ) p L+1 L p L+1 L+1 p L+1 U 1 (1 γ U ) p L+1 U (1 γ L ) p U 1 L p U 1 L+1 p U 1 U 1 (1 γ U ) p U 1 U 5 (1 γ L ) p U L p U L+1 p U U 1 (1 γ U ) p U U The associated overall ARL, ARL(λ, β, γ L, γ U ): 1 + (1 γ L ) ARL L (λ, β, γ L, γ U ) P[X 1(λ, β) = L] + X U 1 u=l+1arl u (λ, β, γ L, γ U ) P[X 1(λ, β) = u] + (1 γ U ) ARL U (λ, β, γ L, γ U ) P[X 1(λ, β) = U], where ARL u (λ, β, γ L, γ U ) is obtained from (1), by taking y = L, x = U and Q(λ, β, γ L, γ U ) instead of Q(λ, β) The procedure to obtain L, U, γ L, and γ U involves a nested secant rule, etc On ARL-unbiased c-charts for iid and INAR(1) Poisson counts

23 Introduction The iid case The INAR(1) case Final thoughts ARL-unbiased c chart for λ/(1 β) Example 3: Comparing ARL performance (λ 0 = 3, β 0 = 05) µ ARL(λ,β 0 ) Figure: ARL(λ, β 0 ) for: c chart with 3 σ control limits (thin solid lines); ARL-unbiased c chart, [LCL, UCL] = [0, 15] & (γ L, γ U ) = ( , ) (ARL = 3704; thick solid lines) λ On ARL-unbiased c-charts for iid and INAR(1) Poisson counts

24 Introduction The iid case The INAR(1) case Final thoughts ARL-unbiased c chart for λ/(1 β) Example 3 (cont d) µ ARL(λ 0,β) Inf β Figure: ARL(λ 0, β) for: c chart with 3 σ control limits (thin solid lines); ARL-unbiased c chart, [LCL, UCL] = [0, 15] & (γ L, γ U ) = ( , ) (ARL = 3704; thick solid lines) On ARL-unbiased c-charts for iid and INAR(1) Poisson counts

25 Introduction The iid case The INAR(1) case Final thoughts Iid/INAR(1) Poisson output The ARL-unbiased c charts we derived have in-control ARL equal to the pre-specified and desired value, requires, in average, less time to trigger a signal in the presence of all shifts than to trigger a false alarm, tackle the curse of the null lower control limit and detect decreases in a timely fashion, in contrast to the c charts with 3 σ control limits On-going and future work Morais (2016a) (resp Morais, 2016b) has already derived an ARL-unbiased np (resp geometric) chart and improved what Acosta-Mejía (1999) (resp Zhang et al, 2004) termed as a nearly ARL-unbiased p (resp geometric) chart Derive an ARL-unbiased version of the CUSUM charts for both iid and INAR(1) output A few difficulties will arise with Poisson INAR(1) output On ARL-unbiased c-charts for iid and INAR(1) Poisson counts

26 Introduction The iid case The INAR(1) case Final thoughts Acosta-Mejía, CA (1999) Improved p-charts to monitor process quality IIE Transactions 31, Aebtarm, S and Bouguila, N (2011) An empirical evaluation of attribute control charts for monitoring defects Expert Systems with Applications 38, Knoth, S and Morais, MC (2013) On ARL-unbiased control charts In S Knoth, W Schmid, Frontiers in Statistical Quality Control 11, pp Springer Lehmann, EL (1986) Testing Statistical Hypotheses (2nd edition) Pacific Grove, California: Wadsworth & Brooks/Cole Advanced Books & Software Montgomery, DC (2009) Introduction to Statistical Quality Control (6thedition) New York: John Wiley & Sons, Inc Morais, MC (2012) Real- and Integer-valued Time Series and Quality Control Charts Unpublished report Morais, MC (2016a) An ARL-unbiased np-chart Accepted for publication in Economic Quality Control volume 31, issue 1 (June 2016) Morais, MC (2016b) ARL-unbiased geometric and CCC G control charts In preparation Paulino, S, Morais, MC, Knoth, S (2016) An ARL-unbiased c-chart Accepted for publication in Quality and Reliability Engineering International Paulino, S, Morais, MC, Knoth, S (2015) On ARL-unbiased c-charts for INAR(1) Poisson counts Submitted for publication Pignatiello, J J, Jr, Acosta-Mejía, C A, Rao, BV (1995) The performance of control charts for monitoring process dispersion 4th Industrial Engineering Research Conference, Ryan, TP and Schwertman, NC (1997) Optimal limits for attribute control charts Journal of Quality Technology 29, Weiss, CH (2007) Controlling correlated processes of Poisson counts Quality and Reliability Engineering International 23, Weiss, CH (2009) Categorical Time Series Analysis and Applications in Statistical Quality Control PhD Thesis, Fakultät fur Mathematik und Informatik der Universität Würzburg dissertationde Verlag im Internet GmbH Weiss, CH and Testik, MC (2009) CUSUM monitoring of first-order integer-valued autoregressive processes of Poisson counts Journal of Quality Technology 41, Zhang, L, Govindaraju, K, Bebbington, M and Lai, CD (2004) On the statistical design of geometric control charts Quality Technology & Quantitative Management 2, On ARL-unbiased c-charts for iid and INAR(1) Poisson counts