xcusum.arl | R Documentation |
Computation of the (zero-state) Average Run Length (ARL) for different types of CUSUM control charts monitoring normal mean.
xcusum.arl(k, h, mu, hs = 0, sided = "one", method = "igl", q = 1, r = 30)
k |
reference value of the CUSUM control chart. |
h |
decision interval (alarm limit, threshold) of the CUSUM control chart. |
mu |
true mean. |
hs |
so-called headstart (give fast initial response). |
sided |
distinguish between one-, two-sided and Crosier's modified
two-sided CUSUM scheme by choosing |
method |
deploy the integral equation ( |
q |
change point position. For q=1 and μ=μ_0 and μ=μ_1, the usual zero-state ARLs for the in-control and out-of-control case, respectively, are calculated. For q>1 and μ!=0 conditional delays, that is, E_q(L-q+1|L≥ q), will be determined. Note that mu0=0 is implicitely fixed. |
r |
number of quadrature nodes, dimension of the resulting linear
equation system is equal to |
xcusum.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.
Returns a vector of length q
which resembles the ARL and the sequence of conditional expected delays for
q
=1 and q
>1, respectively.
Sven Knoth
A. L. Goel, S. M. Wu (1971), Determination of A.R.L. and a contour nomogram for CUSUM charts to control normal mean, Technometrics 13, 221-230.
D. Brook, D. A. Evans (1972), An approach to the probability distribution of cusum run length, Biometrika 59, 539-548.
J. M. Lucas, R. B. Crosier (1982), Fast initial response for cusum quality-control schemes: Give your cusum a headstart, Technometrics 24, 199-205.
L. C. Vance (1986), Average run lengths of cumulative sum control charts for controlling normal means, Journal of Quality Technology 18, 189-193.
K.-H. Waldmann (1986), Bounds for the distribution of the run length of one-sided and two-sided CUSUM quality control schemes, Technometrics 28, 61-67.
R. B. Crosier (1986), A new two-sided cumulative quality control scheme, Technometrics 28, 187-194.
xewma.arl
for zero-state ARL computation of EWMA control charts
and xcusum.ad
for the steady-state ARL.
## Brook/Evans (1972), one-sided CUSUM ## Their results are based on the less accurate Markov chain approach. k <- .5 h <- 3 round(c( xcusum.arl(k,h,0), xcusum.arl(k,h,1.5) ),digits=2) ## results in the original paper are L0 = 117.59, L1 = 3.75 (in Subsection 4.3). ## Lucas, Crosier (1982) ## (one- and) two-sided CUSUM with possible headstarts k <- .5 h <- 4 mu <- c(0,.25,.5,.75,1,1.5,2,2.5,3,4,5) arl1 <- sapply(mu,k=k,h=h,sided="two",xcusum.arl) arl2 <- sapply(mu,k=k,h=h,hs=h/2,sided="two",xcusum.arl) round(cbind(mu,arl1,arl2),digits=2) ## results in the original paper are (in Table 1) ## 0.00 168. 149. ## 0.25 74.2 62.7 ## 0.50 26.6 20.1 ## 0.75 13.3 8.97 ## 1.00 8.38 5.29 ## 1.50 4.75 2.86 ## 2.00 3.34 2.01 ## 2.50 2.62 1.59 ## 3.00 2.19 1.32 ## 4.00 1.71 1.07 ## 5.00 1.31 1.01 ## Vance (1986), one-sided CUSUM ## The first paper on using Nystroem method and Gauss-Legendre quadrature ## for solving the ARL integral equation (see as well Goel/Wu, 1971) k <- 0 h <- 10 mu <- c(-.25,-.125,0,.125,.25,.5,.75,1) round(cbind(mu,sapply(mu,k=k,h=h,xcusum.arl)),digits=2) ## results in the original paper are (in Table 1 incl. Goel/Wu (1971) results) ## -0.25 2071.51 ## -0.125 400.28 ## 0.0 124.66 ## 0.125 59.30 ## 0.25 36.71 ## 0.50 20.37 ## 0.75 14.06 ## 1.00 10.75 ## Waldmann (1986), ## one- and two-sided CUSUM ## one-sided case k <- .5 h <- 3 mu <- c(-.5,0,.5) round(sapply(mu,k=k,h=h,xcusum.arl),digits=2) ## results in the original paper are 1963, 117.4, and 17.35, resp. ## (in Tables 3, 1, and 5, resp.). ## two-sided case k <- .6 h <- 3 round(xcusum.arl(k,h,-.2,sided="two"),digits=1) # fits to Waldmann's setup ## result in the original paper is 65.4 (in Table 6). ## Crosier (1986), Crosier's modified two-sided CUSUM ## He introduced the modification and evaluated it by means of ## Markov chain approximation k <- .5 h <- 3.73 mu <- c(0,.25,.5,.75,1,1.5,2,2.5,3,4,5) round(cbind(mu,sapply(mu,k=k,h=h,sided="Crosier",xcusum.arl)),digits=2) ## results in the original paper are (in Table 3) ## 0.00 168. ## 0.25 70.7 ## 0.50 25.1 ## 0.75 12.5 ## 1.00 7.92 ## 1.50 4.49 ## 2.00 3.17 ## 2.50 2.49 ## 3.00 2.09 ## 4.00 1.60 ## 5.00 1.22 ## SAS/QC manual 1999 ## one- and two-sided CUSUM schemes ## one-sided k <- .25 h <- 8 mu <- 2.5 print(xcusum.arl(k,h,mu),digits=12) print(xcusum.arl(k,h,mu,hs=.1),digits=12) ## original results are 4.1500836225 and 4.1061588131. ## two-sided print(xcusum.arl(k,h,mu,sided="two"),digits=12) ## original result is 4.1500826715.
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