spc (version 0.7.1)

scusum.arl: 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)

Value

Returns a single value which resembles the ARL.

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-\(S^2\) control charts by choosing "upper" (upper chart), "lower" (lower chart), and "two" (two-sided chart), respectively. Note that for the two-sided chart the parameters "k2" and "h2" have to be set too.

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.

Author

Sven Knoth

Details

scusum.arl determines the Average Run Length (ARL) by numerically solving the related ARL integral equation by means of collocation (piecewise Chebyshev polynomials).

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)