scusum.arl {spc} | R Documentation |
Compute ARLs of CUSUM control charts (variance charts)
Description
Computation of the (zero-state) Average Run Length (ARL)
for different types of CUSUM control charts (based on the sample variance
S^2
) monitoring normal variance.
Usage
scusum.arl(k, h, sigma, df, hs=0, sided="upper", k2=NULL,
h2=NULL, hs2=0, r=40, qm=30, version=2)
Arguments
k |
reference value of the CUSUM control chart. |
h |
decision interval (alarm limit, threshold) of the CUSUM control chart. |
sigma |
true standard deviation. |
df |
actual degrees of freedom, corresponds to subgroup size (for known mean it is equal to the subgroup size, for unknown mean it is equal to subgroup size minus one. |
hs |
so-called headstart (enables fast initial response). |
sided |
distinguishes between one- and two-sided two-sided CUSUM- |
k2 |
In case of a two-sided CUSUM chart for variance the reference value of the lower chart. |
h2 |
In case of a two-sided CUSUM chart for variance the decision interval of the lower chart. |
hs2 |
In case of a two-sided CUSUM chart for variance the headstart of the lower chart. |
r |
Dimension of the resulting linear equation system (highest order of the collocation polynomials times number of intervals – see Knoth 2006). |
qm |
Number of quadrature nodes for calculating the collocation definite integrals. |
version |
Distinguish version numbers (1,2,...). For internal use only. |
Details
scusum.arl
determines the Average Run Length (ARL) by numerically
solving the related ARL integral equation by means of collocation (piecewise Chebyshev polynomials).
Value
Returns a single value which resembles the ARL.
Author(s)
Sven Knoth
References
S. Knoth (2005),
Accurate ARL computation for EWMA-S^2
control charts,
Statistics and Computing 15, 341-352.
S. Knoth (2006),
Computation of the ARL for CUSUM-S^2
schemes,
Computational Statistics & Data Analysis 51, 499-512.
See Also
xcusum.arl
for zero-state ARL computation of CUSUM control charts for monitoring normal mean.
Examples
## Knoth (2006)
## compare with Table 1 (p. 507)
k <- 1.46 # sigma1 = 1.5
df <- 1
h <- 10
# original values
# sigma coll63 BE Hawkins MC 10^9 (s.e.)
# 1 260.7369 260.7546 261.32 260.7399 (0.0081)
# 1.1 90.1319 90.1389 90.31 90.1319 (0.0027)
# 1.2 43.6867 43.6897 43.75 43.6845 (0.0013)
# 1.3 26.2916 26.2932 26.32 26.2929 (0.0007)
# 1.4 18.1231 18.1239 18.14 18.1235 (0.0005)
# 1.5 13.6268 13.6273 13.64 13.6272 (0.0003)
# 2 5.9904 5.9910 5.99 5.9903 (0.0001)
# replicate the column coll63
sigma <- c(1, 1.1, 1.2, 1.3, 1.4, 1.5, 2)
arl <- rep(NA, length(sigma))
for ( i in 1:length(sigma) )
arl[i] <- round(scusum.arl(k, h, sigma[i], df, r=63, qm=20, version=2), digits=4)
data.frame(sigma, arl)