lns2ewma.arl {spc}

Compute ARLs of EWMA ln S^2 control charts (variance charts)

Description

Computation of the (zero-state) Average Run Length (ARL) for different types of EWMA control charts (based on the log of the sample variance S^2) monitoring normal variance.

Usage

lns2ewma.arl(l,cl,cu,sigma,df,hs=NULL,sided="upper",r=40)

Arguments

l

smoothing parameter lambda of the EWMA control chart.

cl

lower control limit of the EWMA control chart.

cu

upper control limit of the EWMA control chart.

sigma

true standard deviation.

df

actual degrees of freedom, corresponds to subsample size (for known mean it is equal to the subsample size, for unknown mean it is equal to subsample size minus one.

hs

so-called headstart (enables fast initial response) – the default value (hs=NULL) corresponds to the in-control mean of ln S^2.

sided

distinguishes between one- and two-sided two-sided EWMA-S^2 control charts by choosing "upper" (upper chart with reflection at cl), "lower" (lower chart with reflection at cu), and "two" (two-sided chart), respectively.

r

dimension of the resulting linear equation system: the larger the better.

Details

lns2ewma.arl determines the Average Run Length (ARL) by numerically solving the related ARL integral equation by means of the Nystroem method based on Gauss-Legendre quadrature.

Value

Returns a single value which resembles the ARL.

Author(s)

Sven Knoth

References

S. V. Crowder and M. D. Hamilton (1992), An EWMA for monitoring a process standard deviation, Journal of Quality Technology 24, 12-21.

S. Knoth (2005), Accurate ARL computation for EWMA-S^2 control charts, Statistics and Computing 15, 341-352.

See Also

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

Examples

lns2ewma.ARL <- Vectorize("lns2ewma.arl", "sigma")

## Crowder/Hamilton (1992)
## moments of ln S^2
E_log_gamma <- function(df) log(2/df) + digamma(df/2)
V_log_gamma <- function(df) trigamma(df/2)
E_log_gamma_approx <- function(df) -1/df - 1/3/df^2 + 2/15/df^4
V_log_gamma_approx <- function(df) 2/df + 2/df^2 + 4/3/df^3 - 16/15/df^5

## results from Table 3 ( upper chart with reflection at 0 = log(sigma0=1) )
## original entries are (lambda = 0.05, K = 1.06, df=n-1=4)
# sigma   ARL
# 1       200
# 1.1      43
# 1.2      18
# 1.3      11
# 1.4       7.6
# 1.5       6.0
# 2         3.2

df <- 4
lambda <- .05
K <- 1.06
cu <- K * sqrt( lambda/(2-lambda) * V_log_gamma_approx(df) )

sigmas <- c(1 + (0:5)/10, 2)
arls <- round(lns2ewma.ARL(lambda, 0, cu, sigmas, df, hs=0, sided="upper"), digits=1)
data.frame(sigmas, arls)

## Knoth (2005)
## compare with Table 3 (p. 351)
lambda <- .05
df <- 4
K <- 1.05521
cu <- 1.05521 * sqrt( lambda/(2-lambda) * V_log_gamma_approx(df) )

## upper chart with reflection at sigma0=1 in Table 4
## original entries are
# sigma   ARL_0    ARL_-.267
# 1       200.0    200.0
# 1.1      43.04    41.55
# 1.2      18.10    19.92
# 1.3      10.75    13.11
# 1.4       7.63     9.93
# 1.5       5.97     8.11
# 2         3.17     4.67

M <- -0.267
cuM <- lns2ewma.crit(lambda, 200, df, cl=M, hs=M, r=60)[2]
arls1 <- round(lns2ewma.ARL(lambda, 0, cu, sigmas, df, hs=0, sided="upper"), digits=2)
arls2 <- round(lns2ewma.ARL(lambda, M, cuM, sigmas, df, hs=M, sided="upper", r=60), digits=2)
data.frame(sigmas, arls1, arls2)