spc (version 0.4.2)

xewma.q: Compute RL quantiles of EWMA control charts

Description

Computation of quantiles of the Run Length (RL) for EWMA control charts monitoring normal mean.

Usage

xewma.q(l, c, mu, p, zr=0, hs=0, sided="one", limits="fix", q=1, r=40)

Arguments

l

smoothing parameter lambda of the EWMA control chart.

c

critical value (similar to alarm limit) of the EWMA control chart.

mu

true mean.

p

quantile level.

zr

reflection border for the one-sided chart.

hs

so-called headstart (enables fast initial response).

sided

distinguishes between one- and two-sided EWMA control chart by choosing "one" and "two", respectively.

limits

distinguishes between different control limits behavior.

q

change point position. For $q=1$ and $\mu=\mu_0$ and $\mu=\mu_1$, the usual zero-state ARLs for the in-control and out-of-control case, respectively, are calculated. For $q>1$ and $\mu!=0$ conditional delays, that is, $E_q(L-q+1|L\geq)$, will be determine

r

number of quadrature nodes, dimension of the resulting linear equation system is equal to r+1 (one-sided) or r (two-sided).

Value

• Returns a single value which resembles the RL quantile of order q.

Details

Instead of the popular ARL (Average Run Length) quantiles of the EWMA stopping time (Run Length) are determined. The algorithm is based on Waldmann's survival function iteration procedure. If limits is not "fix", then the method presented in Knoth (2003) is utilized. Note that for one-sided EWMA charts (sided="one"), only "vacl" and "stat" are deployed, while for two-sided ones (sided="two") also "fir", "both" (combination of "fir" and "vacl"), and "Steiner" are implemented. For details see Knoth (2004).

References

F. F. Gan (1993), An optimal design of EWMA control charts based on the median run length, J. Stat. Comput. Simulation 45, 169-184.

S. Knoth (2003), EWMA schemes with non-homogeneous transition kernels, Sequential Analysis 22, 241-255.

S. Knoth (2004), Fast initial response features for EWMA Control Charts, Statistical Papers 46, 47-64.

K.-H. Waldmann (1986), Bounds for the distribution of the run length of geometric moving average charts, Appl. Statist. 35, 151-158.

See Also

xewma.arl for zero-state ARL computation of EWMA control charts.

Examples

## Gan (1993), two-sided EWMA with fixed control limits
## some values of his Table 1 -- any median RL should be 500
XEWMA.Q <- Vectorize("xewma.q", c("l", "c"))
G.lambda <- c(.05, .1, .15, .2, .25)
G.h <- c(.441, .675, .863, 1.027, 1.177)
MEDIAN <- ceiling(XEWMA.Q(G.lambda, G.h/sqrt(G.lambda/(2-G.lambda)), 0, .5, sided="two"))
print(cbind(G.lambda, MEDIAN))

## increase accuracy of thresholds

# (i) calculate threshold for given in-control median value by
#     deplyoing secant rule
xewma.c.of.quantile <- function(l, L0, mu, p, zr=0, hs=0, sided="one", mode="integer", r=40) {
  c2 <- 0
  a2 <- 0
  while ( a2 < L0 ) {
    c2 <- c2 + .5
    a2 <- xewma.q(l, c2, mu, p, zr=zr, hs=hs, sided=sided, r=r)
    if ( mode=="integer" ) a2 <- ceiling(a2)
  }
  c1 <- c2 - .5
  a1 <- xewma.q(l, c1, mu, p, zr=zr, hs=hs, sided=sided, r=r)
  a.error <- 1; c.error <- 1
  while ( a.error>1e-6 && c.error>1e-12 ) {
    c3 <- c1 + (L0 - a1)/(a2 - a1)*(c2 - c1)
    a3 <- xewma.q(l, c3, mu, p, zr=zr, hs=hs, sided=sided, r=r)
    if ( mode=="integer" ) a3 <- ceiling(a3)
    c1 <- c2; c2 <- c3
    a1 <- a2; a2 <- a3
    a.error <- abs(a2 - L0); c.error <- abs(c2 - c1)
  }
  c3
}
XEWMA.c.of.quantile <- Vectorize("xewma.c.of.quantile", "l")

# (ii) re-calculate the thresholds and remove the standardization step
L0 <- 500
G.h.new <- XEWMA.c.of.quantile(G.lambda, L0, 0, .5, sided="two")
G.h.new <- round(G.h.new * sqrt(G.lambda/(2-G.lambda)), digits=5)

# (iii) compare Gan's original values and the new ones with 5 digits
print(cbind(G.lambda, G.h.new, G.h))

# (iv) calculate the new medians
MEDIAN <- ceiling(XEWMA.Q(G.lambda, G.h.new/sqrt(G.lambda/(2-G.lambda)), 0, .5, sided="two"))
print(cbind(G.lambda, MEDIAN))