spc (version 0.4.0)

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", 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 (give fast initial response).

sided

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

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 ARL.

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.

References

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

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

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 <- 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", 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)
  }
  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)
    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 standarization 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 woth 5 digits
print(cbind(G.lambda, G.h.new, G.h))

# (iv) calculate the new medians
MEDIAN <- round(XEWMA.Q(G.lambda, G.h.new/sqrt(G.lambda/(2-G.lambda)), 0, .5, sided="two"), digits=1)
print(cbind(G.lambda, MEDIAN))