xewma.arl(spc)
xewma.arl()所属R语言包：spc
Compute ARLs of EWMA control charts
计算连串长度的EWMA控制图
译者：生物统计家园网 机器人LoveR
描述----------Description----------
Computation of the (zero-state) Average Run Length (ARL)
计算（零状态）的平均运行长度（ARL）
用法----------Usage----------
xewma.arl(l,c,mu,zr=0,hs=0,sided="one",limits="fix",q=1,r=40)
参数----------Arguments----------
参数：l
smoothing parameter lambda of the EWMA control chart.
平滑的EWMA控制图参数的lambda。
参数：c
critical value (similar to alarm limit) of the EWMA control chart.
临界值（报警限值）的EWMA控制图。
参数：mu
true mean.
真正的意思。
参数：zr
reflection border for the one-sided chart.
反射边界的片面的图表。
参数：hs
so-called headstart (give fast initial response).
所谓的领先地位提供快速初始响应。
参数：sided
distinguish between one- and two-sided EWMA control chart by choosing "one" and "two", respectively.
区分和双面EWMA控制图的选择"one"和"two"，分别。
参数：limits
distinguish between different control limits behavior.
区分不同的控制限制行为。
参数：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.
改变点的位置。对于q=1和μ=μ_0和μ=μ_1，零状态连串长度通常在控制和控制的情况下，分别计算。对于q>1和μ!=0有条件的延误，也就是E_q(L-q+1|L≥q)，将确定。请注意，MU0 = 0的是，implicitely固定。
参数：r
number of quadrature nodes, dimension of the resulting linear equation system is equal to r+1 (one-sided) or r (two-sided).
正交的节点的数量，所得的线性方程系统的维数是等于r+1（单面）或r（两面）。
Details
详细信息----------Details----------
In case of the two-sided chart with fixed control limits and of the one-sided chart, xewma.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. If limits is not "fix", then the method presented in Knoth (2003) is utilized. Note that for one-sided EWMA charts (sided="one"), only "vacl" and "stat" are deployed, while for two-sided ones (sided="two") also "fir", "both" (combination of "fir" and "vacl"), and "Steiner" are implemented. For details see Knoth (2004).
在双面固定的控制限制和片面的图图的情况下，xewma.arl决定相关的ARL通过的Nystroem方法基于高斯积分方程通过数值求解的平均运行长度（ARL）勒让德正交。 limits如果是不是"fix"，然后在Knoth（2003）提出的方法是利用。请注意，为片面的EWMA图（sided="one"），只有"vacl"和"stat"的部署，而双面的（sided= "two"）"fir"，"both"（组合"fir"和"vacl"），和"Steiner"来实现。有关详细信息，请参阅Knoth（2004年）。
值----------Value----------
Returns a single value which resembles the ARL.
返回一个单一的值，类似于ARL。
（作者）----------Author(s)----------
Sven Knoth
参考文献----------References----------
K.-H. Waldmann (1986), Bounds for the distribution of the run length of geometric moving average charts, Appl. Statist. 35, 151-158.
S. V. Crowder (1987), A simple method for studying run-length distributions of exponentially weighted moving average charts, Technometrics 29, 401-407.
J. M. Lucas, M. S. Saccucci (1990), Exponentially weighted moving average control schemes: Properties and enhancements, Technometrics 32, 1-12.
S. Chandrasekaran, J. R. English, R. L. Disney (1995), Modeling and analysis of EWMA control schemes with variance-adjusted control limits, IIE Transactions 277, 282-290.
T. R. Rhoads, D. C. Montgomery, C. M. Mastrangelo (1996), Fast initial response scheme for exponentially weighted moving average control chart, Quality Engineering 9, 317-327.
S. H. Steiner (1999), EWMA control charts with time-varying control limits and fast initial response, Journal of Quality Technology 31, 75-86.
S. Knoth (2003), EWMA schemes with non-homogeneous transition kernels, Sequential Analysis 22, 241-255.
S. Knoth (2004), Fast initial response features for EWMA Control Charts, Statistical Papers 46, 47-64.
参见----------See Also----------
xcusum.arl for zero-state ARL computation of CUSUM control charts and xewma.ad for the steady-state ARL.
xcusum.arl零状态CUSUM控制图的ARL计算和xewma.ad的稳态ARL的。
实例----------Examples----------
## Waldmann (1986), one-sided EWMA[＃沃达迈（1986年），片面的EWMA]
l <- .75
round(xewma.arl(l,2*sqrt((2-l)/l),0,zr=-4*sqrt((2-l)/l)),digits=1)
l <- .5
round(xewma.arl(l,2*sqrt((2-l)/l),0,zr=-4*sqrt((2-l)/l)),digits=1)
## original values are 209.3 and 3907.5 (in Table 2).[＃原始值是209.3和3907.5（在表2中）。]
## Waldmann (1986), two-sided EWMA with fixed control limits[＃沃达迈（1986年），双面EWMA有固定的控制范围内]
l <- .75
round(xewma.arl(l,2*sqrt((2-l)/l),0,sided="two"),digits=1)
l <- .5
round(xewma.arl(l,2*sqrt((2-l)/l),0,sided="two"),digits=1)
## original values are 104.0 and 1952 (in Table 1).[＃原始值分别为104.0和1952年（表1）。]
## Crowder (1987), two-sided EWMA with fixed control limits[＃克罗德（1987年），双面EWMA有固定的控制范围内]
l1 <- .5
l2 <- .05
c <- 2
mu <- (0:16)/4
arl1 <- sapply(mu,l=l1,c=c,sided="two",xewma.arl)
arl2 <- sapply(mu,l=l2,c=c,sided="two",xewma.arl)
round(cbind(mu,arl1,arl2),digits=2)
## original results are (in Table 1)[＃原来的结果（表1）]
## 0.00 26.45 127.53[＃0.00 26.45 127.53]
## 0.25 20.12 43.94[＃0.25 20.12 43.94]
## 0.50 11.89 18.97[＃0.50 11.89 18.97]
## 0.75 7.29 11.64[＃0.75 7.29 11.64]
## 1.00 4.91 8.38[＃1.00 4.91 8.38]
## 1.25 3.95* 6.56[＃1.25 3.95 * 6.56]
## 1.50 2.80 5.41[＃1.50 2.80 5.41]
## 1.75 2.29 4.62[＃1.75 2.29 4.62]
## 2.00 1.94 4.04[＃2.00 1.94 4.04]
## 2.25 1.70 3.61[＃2.25 1.70 3.61]
## 2.50 1.51 3.26[＃2.50 1.51 3.26]
## 2.75 1.37 2.99[＃2.75 1.37 2.99]
## 3.00 1.26 2.76[＃3.00 1.26 2.76]
## 3.25 1.18 2.56[＃3.25 1.18 2.56]
## 3.50 1.12 2.39[＃3.50 1.12 2.39]
## 3.75 1.08 2.26[＃3.75 1.08 2.26]
## 4.00 1.05 2.15 (* -- in Crowder (1987) typo!?). [＃4.00 1.05 2.15（* - 克罗德（1987）错字！）。]
## Lucas/Saccucci (1990)[＃卢卡斯/ Saccucci的（1990）]
## two-sided EWMA[＃双面EWMA]
## with fixed limits[＃固定的限制]
l1 <- .5
l2 <- .03
c1 <- 3.071
c2 <- 2.437
mu <- c(0,.25,.5,.75,1,1.5,2,2.5,3,3.5,4,5)
arl1 <- sapply(mu,l=l1,c=c1,sided="two",xewma.arl)
arl2 <- sapply(mu,l=l2,c=c2,sided="two",xewma.arl)
round(cbind(mu,arl1,arl2),digits=2)
## original results are (in Table 3)[＃原来的结果（表3）]
## 0.00 500. 500.[＃0.00 500。 500。]
## 0.25 255. 76.7[＃0.25 255。 76.7]
## 0.50 88.8 29.3[＃0.50 88.8 29.3]
## 0.75 35.9 17.6[＃0.75 35.9 17.6]
## 1.00 17.5 12.6[＃1.00 17.5 12.6]
## 1.50 6.53 8.07[＃1.50 6.53 8.07]
## 2.00 3.63 5.99[＃2.00 3.63 5.99]
## 2.50 2.50 4.80[＃2.50 2.50 4.80]
## 3.00 1.93 4.03[＃3.00 1.93 4.03]
## 3.50 1.58 3.49[＃3.50 1.58 3.49]
## 4.00 1.34 3.11[＃4.00 1.34 3.11]
## 5.00 1.07 2.55.[＃5.00 1.07 2.55。]
## with fir feature[＃杉木功能]
l1 <- .5
l2 <- .03
c1 <- 3.071
c2 <- 2.437
hs1 <- c1/2
hs2 <- c2/2
mu <- c(0,.5,1,2,3,5)
arl1 <- sapply(mu,l=l1,c=c1,hs=hs1,sided="two",limits="fir",xewma.arl)
arl2 <- sapply(mu,l=l2,c=c2,hs=hs2,sided="two",limits="fir",xewma.arl)
round(cbind(mu,arl1,arl2),digits=2)
## original results are (in Table 5)[＃原来的结果（表5）]
## 0.0 487. 406.[＃0.0 487。 406。]
## 0.5 86.1 18.4[＃0.5 86.1 18.4]
## 1.0 15.9 7.36[＃1.0 15.9 7.36]
## 2.0 2.87 3.43[＃2.0 2.87 3.43]
## 3.0 1.45 2.34[＃3.0 1.45 2.34]
## 5.0 1.01 1.57.[＃5.0 1.01 1.57。]
## Chandrasekaran, English, Disney (1995)[＃Chandrasekaran，英语，迪士尼（1995年）]
## two-sided EWMA with fixed and variance adjusted limits (vacl)[双面EWMA固定和方差调整后的限制（VACL）]
l1 <- .25
l2 <- .1
c1s <- 2.9985
c1n <- 3.0042
c2s <- 2.8159
c2n <- 2.8452
mu <- c(0,.25,.5,.75,1,2)
arl1s <- sapply(mu,l=l1,c=c1s,sided="two",xewma.arl)
arl1n <- sapply(mu,l=l1,c=c1n,sided="two",limits="vacl",xewma.arl)
arl2s <- sapply(mu,l=l2,c=c2s,sided="two",xewma.arl)
arl2n <- sapply(mu,l=l2,c=c2n,sided="two",limits="vacl",xewma.arl)
round(cbind(mu,arl1s,arl1n,arl2s,arl2n),digits=2)
## original results are (in Table 2)[＃原始的结果（表2）]
## 0.00 500. 500. 500. 500.[＃0.00 500。 500。 500。 500。]
## 0.25 170.09 167.54 105.90 96.6[＃0.25 170.09 167.54 105.90 96.6]
## 0.50 48.14 45.65 31.08 24.35[＃0.50 48.14 45.65 31.08 24.35]
## 0.75 20.02 19.72 15.71 10.74[＃0.75 20.02 19.72 15.71 10.74]
## 1.00 11.07 9.37 10.23 6.35[＃1.00 11.07 9.37 10.23 6.35]
## 2.00 3.59 2.64 4.32 2.73.[＃2.00 3.59 2.64 4.32 2.73。]
## The results in Chandrasekaran, English, Disney (1995) are not[＃Chandrasekaran，英语，迪士尼（1995）]
## that accurate. Let us consider the more appropriate comparison[＃准确。让我们考虑更恰当的比较]
c1s <- xewma.crit(l1,500,sided="two")
c1n <- xewma.crit(l1,500,sided="two",limits="vacl")
c2s <- xewma.crit(l2,500,sided="two")
c2n <- xewma.crit(l2,500,sided="two",limits="vacl")
mu <- c(0,.25,.5,.75,1,2)
arl1s <- sapply(mu,l=l1,c=c1s,sided="two",xewma.arl)
arl1n <- sapply(mu,l=l1,c=c1n,sided="two",limits="vacl",xewma.arl)
arl2s <- sapply(mu,l=l2,c=c2s,sided="two",xewma.arl)
arl2n <- sapply(mu,l=l2,c=c2n,sided="two",limits="vacl",xewma.arl)
round(cbind(mu,arl1s,arl1n,arl2s,arl2n),digits=2)
## which demonstrate the abilities of the variance-adjusted limits[＃展现的能力方差调整的限制]
## scheme more explicitely.[＃计划更明确地定义。]
## Rhoads, Montgomery, Mastrangelo (1996)[＃罗兹，蒙哥马利，马斯特兰赫洛（1996）]
## two-sided EWMA with fixed and variance adjusted limits (vacl),[双面EWMA固定和方差调整后的限制（VACL），]
## with fir and both features[＃与杉木，这两个功能]
l <- .03
c <- 2.437
mu <- c(0,.5,1,1.5,2,3,4)
sl <- sqrt(l*(2-l))
arlfix <- sapply(mu,l=l,c=c,sided="two",xewma.arl)
arlvacl <- sapply(mu,l=l,c=c,sided="two",limits="vacl",xewma.arl)
arlfir <- sapply(mu,l=l,c=c,hs=c/2,sided="two",limits="fir",xewma.arl)
arlboth <- sapply(mu,l=l,c=c,hs=c/2*sl,sided="two",limits="both",xewma.arl)
round(cbind(mu,arlfix,arlvacl,arlfir,arlboth),digits=1)
## original results are (in Table 1)[＃原来的结果（表1）]
## 0.0 477.3* 427.9* 383.4* 286.2*[＃0.0 477.3 * 427.9 * 383.4 * 286.2 *]
## 0.5 29.7 20.0 18.6 12.8[＃0.5 29.7 20.0 18.6 12.8]
## 1.0 12.5 6.5 7.4 3.6[＃1.0 12.5 6.5 7.4 3.6]
## 1.5 8.1 3.3 4.6 1.9[＃1.5 8.1 3.3 4.6 1.9]
## 2.0 6.0 2.2 3.4 1.4[＃2.0 6.0 2.2 3.4 1.4]
## 3.0 4.0 1.3 2.4 1.0[＃3.0 4.0 1.3 2.4 1.0]
## 4.0 3.1 1.1 1.9 1.0[＃4.0 3.1 1.1 1.9 1.0]
## * -- the in-control values differ sustainably from the true values![＃* - 在控制值不同，可持续的真正价值！]
## Steiner (1999)[＃施泰纳（1999）]
## two-sided EWMA control charts with various modifications[＃双面EWMA控制图进行各种修改]
## fixed vs. variance adjusted limits[＃固定与方差调整后的限制]
l <- .05
c <- 3
mu <- c(0,.25,.5,.75,1,1.5,2,2.5,3,3.5,4)
arlfix <- sapply(mu,l=l,c=c,sided="two",xewma.arl)
arlvacl <- sapply(mu,l=l,c=c,sided="two",limits="vacl",xewma.arl)
round(cbind(mu,arlfix,arlvacl),digits=1)
## original results are (in Table 2)[＃原始的结果（表2）]
## 0.00 1379.0 1353.0[＃0.00 1379.0 1353.0]
## 0.25 135.0 127.0[＃0.25 135.0 127.0]
## 0.50 37.4 32.5 [＃0.50 37.4 32.5]
## 0.75 20.0 15.6[＃0.75 20.0 15.6]
## 1.00 13.5 9.0[＃1.00 13.5 9.0]
## 1.50 8.3 4.5[＃1.50 8.3 4.5]
## 2.00 6.0 2.8[＃2.00 6.0 2.8]
## 2.50 4.8 2.0[＃2.50 4.8 2.0]
## 3.00 4.0 1.6[＃3.00 4.0 1.6]
## 3.50 3.4 1.3[＃3.50 3.4 1.3]
## 4.00 3.0 1.1.[＃4.00 3.0 1.1。]
## fir, both, and Steiner's modification[＃杉，都和施泰纳的修改]
l <- .03
cfir <- 2.44
cboth <- 2.54
cstein <- 2.55
hsfir <- cfir/2
hsboth <- cboth/2*sqrt(l*(2-l))
mu <- c(0,.5,1,1.5,2,3,4)
arlfir <- sapply(mu,l=l,c=cfir,hs=hsfir,sided="two",limits="fir",xewma.arl)
arlboth <- sapply(mu,l=l,c=cboth,hs=hsboth,sided="two",limits="both",xewma.arl)
arlstein <- sapply(mu,l=l,c=cstein,sided="two",limits="Steiner",xewma.arl)
round(cbind(mu,arlfir,arlboth,arlstein),digits=1)
## original values are (in Table 5)[＃原始值（在表5中）]
## 0.0 383.0 384.0 391.0[＃0.0 383.0 384.0 391.0]
## 0.5 18.6 14.9 13.8[＃0.5 18.6 14.9 13.8]
## 1.0 7.4 3.9 3.6[＃1.0 7.4 3.9 3.6]
## 1.5 4.6 2.0 1.8[＃1.5 4.6 2.0 1.8]
## 2.0 3.4 1.4 1.3[＃2.0 3.4 1.4 1.3]
## 3.0 2.4 1.1 1.0[＃3.0 2.4 1.1 1.0]
## 4.0 1.9 1.0 1.0.[＃4.0 1.9 1.0 1.0。]
## SAS/QC manual 1999[＃SAS / QC手册1999年]
## two-sided EWMA control charts with fixed limits[＃双面EWMA控制图与固定的限制]
l <- .25
c <- 3
mu <- 1
print(xewma.arl(l,c,mu,sided="two"),digits=11)
# original value is 11.154267016.[原来的值是11.154267016。]
## Some recent examples for one-sided EWMA charts[＃最近的一些例子片面的EWMA图]
## with varying limits and in the so-called stationary mode[＃有不同的限制和所谓的固定模式]
# 1. varying limits = "vacl"[1。不同的限制=“VACL”]
lambda <- .1
L0 <- 500
## Monte Carlo results (10^9 replicates)[＃蒙特卡罗（10 ^ 9个重复）]
# mu ARL s.e.[亩ARL标准差表示。]
# 0 500.00 0.0160[0 500.00 0.0160]
# 0.5 21.637 0.0006[0.5 21.637 0.0006]
# 1 6.7596 0.0001[1 6.7596 0.0001]
# 1.5 3.5398 0.0001[1.5 3.5398 0.0001]
# 2 2.3038 0.0000[2 2.3038 0.0000]
# 2.5 1.7004 0.0000[2.5 1.7004 0.0000]
# 3 1.3675 0.0000[3 1.3675 0.0000]
zr <- -6
r <- 50
c <- xewma.crit(lambda, L0, zr=zr, limits="vacl", r=r)
Mxewma.arl <- Vectorize(xewma.arl, "mu")
mus <- (0:6)/2
arls <- round(Mxewma.arl(lambda, c, mus, zr=zr, limits="vacl", r=r), digits=4)
data.frame(mus, arls)
# 2. stationary mode, i. e. limits = "stat"[2。固定的模式，i。 E。限制=“STAT”]
## Monte Carlo results (10^9 replicates)[＃蒙特卡罗（10 ^ 9个重复）]
# mu ARL s.e.[亩ARL标准差表示。]
# 0 500.00 0.0159[0 500.00 0.0159]
# 0.5 22.313 0.0006[0.5 22.313 0.0006]
# 1 7.2920 0.0001[1 7.2920 0.0001]
# 1.5 3.9064 0.0001[1.5 3.9064 0.0001]
# 2 2.5131 0.0000[2 2.5131 0.0000]
# 2.5 1.7983 0.0000[2.5 1.7983 0.0000]
# 3 1.4029 0.0000[3 1.4029 0.0000]
c <- xewma.crit(lambda, L0, zr=zr, limits="stat", r=r)
arls <- round(Mxewma.arl(lambda, c, mus, zr=zr, limits="stat", r=r), digits=4)
data.frame(mus, arls)
转载请注明:出自 生物统计家园网(http://www.biostatistic.net)。
注：
注1：为了方便大家学习，本文档为生物统计家园网机器人LoveR翻译而成，仅供个人R语言学习参考使用，生物统计家园保留版权。
注2：由于是机器人自动翻译，难免有不准确之处，使用时仔细对照中、英文内容进行反复理解，可以帮助R语言的学习。
注3：如遇到不准确之处，请在本贴的后面进行回帖，我们会逐渐进行修订。
|
发表于 2012-9-30 14:33:28
