spc (version 0.7.1)

pois.cusum.crit.L0L1: Compute the CUSUM k and h for given in-control ARL L0 and out-of-control ARL L1, Poisson case

Description

Computation of the reference value k and the alarm threshold h for one-sided CUSUM control charts monitoring Poisson data, if the in-control ARL L0 and the out-of-control ARL L1 are given.

Usage

pois.cusum.crit.L0L1(mu0, L0, L1, sided="upper", OUTPUT=FALSE)

Value

Returns a data frame with results for the denominator m of the rational approximation, km as (integer) enumerator of the reference value (approximation), the corresponding out-of-control mean mu1, the final approximation k of the reference value, the threshold values hm (integer) and h (=hm/m), and the randomization constant gamma (the target in-control ARL is exactly matched).

Arguments

mu0

in-control Poisson mean.

L0

in-control ARL.

L1

out-of-control ARL.

sided

distinguishes between "upper" and "lower" CUSUM designs.

OUTPUT

controls whether iteration details are printed.

Author

Sven Knoth

Details

pois.cusum.crit.L0L1 determines the reference value k and the alarm threshold h for given in-control ARL L0 and out-of-control ARL L1 by applying grid search and using pois.cusum.arl() and pois.cusum.crit(). These CUSUM design rules were firstly (and quite rarely afterwards) used by Ewan and Kemp. In the Poisson case, Rossi et al. applied them while analyzing three different normal approximations of the Poisson distribution. See the example which illustrates the validity of all these approaches.

References

W. D. Ewan and K. W. Kemp (1960), Sampling inspection of continuous processes with no autocorrelation between successive results, Biometrika 47 (3/4), 363-380.

K. W. Kemp (1962), The Use of Cumulative Sums for Sampling Inspection Schemes, Journal of the Royal Statistical Sociecty C, Applied Statistics 11(1), 16-31.

G. Rossi, L. Lampugnani and M. Marchi (1999), An approximate CUSUM procedure for surveillance of health events, Statistics in Medicine 18(16), 2111-2122.

See Also

pois.cusum.arl for zero-state ARL and pois.cusum.crit for threshold h computation.

Examples

## Table 1 from Rossi et al. (1999) -- one-sided CUSUM
La <- 500 # in-control ARL
Lr <- 7 # out-of-control ARL
m_a <- 0.52 # in-control mean of the Poisson variate
if (FALSE) kh <- xcusum.crit.L0L1(La, Lr, sided="one")
# kh <- ...: instead of deploying EK1960, one could use more accurate numbers
EK_k <- 0.60 # EK1960 results in
EK_h <- 3.80 # Table 2 on p. 372
eZR <- 2*EK_h # reproduce normal ooc mean from reference value k
m_r <- 1.58 # EK1960 Table 3 on p. 377 for m_a = 0.52
R1 <- round( eZR/sqrt(m_a) + 1, digits=2)
R2 <- round( ( eZR/2/sqrt(m_a) + 1 )^2, digits=2)
R3 <- round(( sqrt(4 + 2*eZR/sqrt(m_a)) - 1 )^2, digits=2)
RS <- round( m_r / m_a, digits=2 )
if (FALSE) K_hk <- pois.cusum.crit.L0L1(m_a, La, Lr) # 'our' 'exact' approach
K_hk <- data.frame(m=1000, km=948, mu1=1.563777, k=0.948, hm=3832, h=3.832, gamma=0.1201901)
# get k for competing means mu0 (m_a) and mu1 (m_r)
k_m01 <- function(mu0, mu1) (mu1 - mu0) / (log(mu1) - log(mu0))
# get ooc mean mu1 (m_r) for given mu0 (m_a) and reference value k
m1_km0 <- function(mu0, k) {
  zero <- function(x) k - k_m01(mu0,x)
  upper <- mu0 + .5
  while ( zero(upper) > 0 ) upper <- upper + 0.5
  mu1 <- uniroot(zero, c(mu0*1.00000001, upper), tol=1e-9)$root
  mu1
}
K_m_r <- m1_km0(m_a, K_hk$k)
RK <- round( K_m_r / m_a, digits=2 )
cat(paste(m_a, R1, R2, R3, RS, RK, "\n", sep="\t"))