# Average Run Length (ARL) Get the ARL `getRL` ```r getARL( n, m, theta = NULL, Ftheta = NULL, dist, mu, sigma, dist.par = c(0, 1, 1), chart, chart.par, scoring = "Z", Chi2corrector = "None", replicates = 10000, isParallel = TRUE, print.RL = FALSE, progress = FALSE, calibrate = FALSE, arl0 = 370, alignment = "unadjusted", constant = NULL, absolute = FALSE, isFixed = FALSE, rounding.factor = NULL ) ``` ## Arguments - `n`: scalar. Subroup size - `m`: scalar. Reference sample size - `theta`: scalar. Value corresponig with the `Ftheta` quantile. - `Ftheta`: scalar. Quantile of the data distribution. The values that take are between (0,1). - `dist`: character string. Select from: * "Uniform: Continuous Uniform distribution . * "Normal": Normal distribution (default). * "Normal2": Squared Normal distribution (also known as Chi-squared). * "DoubleExp": Double exponential distribution (also known as Laplace distribution). * "DoubleExp2": Double exponential squared distribution from a `DoubleExp(0,1)`. * "LogNormal": Lognormal distribution. * "Gamma": Gamma distribution. * "Weibull": Weibull distribution. * "t": Student-t distribution. - `mu`: vector. Two elements, the first one is the mean of the reference sample and the second one is the mean of the monitoring sample. - `sigma`: vector. Two elements, the first one is the sd of the reference sample and the second one is the sd of the monitoring sample. - `dist.par`: vector. Distribution parameters. `c(par.a, par.b)`. Default `c(0,1)`. - `chart`: character string. Selected type of chart. Three options are available: Shewhart, CUSUM, EWMA - `chart.par`: vector. The size depends on the selected chart: - **Shewhart scheme:**: is `c(k)`, where `k` comes from $UCL = mu + k\sigma, LCL = mu - k\sigma.$ - **CUSUM scheme:**: is `c(k, h, t)` where `k` is the reference value and `h` is the control limit, and `t` is the type of the chart (1:positive, 2:negative, 3:two sides) - **EWMA scheme:**: is `c(lambda, L)`, where `lambda` is the smoothing constant and `L` multiplies standard deviation to get the control limit - `scoring`: character string. If "Z" (normal scores) (default). If "Z-SQ" (normal scores squared). - `Chi2corrector`: character string. Only when scoring is Z-SQ. Select from * "approx: Z^2*(m + 1 + 1.3)/(m+1). * "exact": Z^2/mean(Z). * "none": Z^2. If "approx" () (default). If "exact" (normal scores squared). - `replicates`: scalar. Number of replicates to get the ARL - `isParallel`: logical. If `TRUE` the code runs in parallel according to the number of cores in the computer,otherwise the code runs sequentially. Default `TRUE`. - `print.RL`: logical. If `TRUE` return the vectors of RL for each iteration. - `progress`: logical. If `TRUE` it shows the progress in the console. - `calibrate`: logical. If `TRUE` the RL is limit to 10 times the target ARL. - `arl0`: scalar. Expected value of the RL. Default `370`. - `alignment`: character string. Aligment of the data `X` and `Y`. Select from * "unadjusted": nothing is sustracte from `X` and `Y` (default). * "overallmean": overall mean is sustracted from `X` and `Y`. * "overallmedian": overall median is sustracted from `X` and `Y`. * "samplemean": mean from corresponding group (`X` and `Y`) is sustracted from its corresponing vector. * "samplemedian": median from corresponding group (`X` and `Y`) is sustracted from its corresponing vector. * "referencemean": mean from `Y` is subtracted from `X` and `Y`. * "referencemedian": median from `Y` is subtracted from `X` and `Y`. * "constantvalue": a constant value is subtracted from `X` and `Y`. - `constant`: scalar. Only used when the `alignment` is selected "constantvalue". Default `NULL`. - `absolute`: logical. If `TRUE`, the absolute aligned values are obtained. (Default `FALSE`) - `isFixed`: logical. If `TRUE` the reference sample does not update, otherwise the reference sample is updated whenever the batch is in control. - `rounding.factor`: scalar. positive value that determine the range between two consecutive rounded values. ## Returns Multiple output. Select by `output$` * `ARL`: scalar. Average Run Length for the `RL`s of all the `replicates`. * `SDRL`: scalar. Standard Deviation Run Length for the `RL` in all the `replicates`. * `MRL`: bolean. Median Run Length for the `RL`s of all the `replicates`. * `QRL`: vector. It retrieve the quantiles (0.05, 0.1, 0.2, 0.25, 0.5, 0.75, 0.8, 0.9, 0.95) for all the `RL`s. ## Examples ```r n <- 5 # subgroup size m <- 100 # reference-sample size dist <- "Normal" mu <- c(0, 0) # c(reference sample mean, monitoring sample mean) sigma <- c(1, 1) # c(reference sample sd, monitoring sample sd) #### Normal distribution parameters dist.par <- c(0, 1) # c(location, scale) #### Other Parameters replicates <- 2 print.RL <- TRUE isParallel <- FALSE calibrate <- FALSE progress <- TRUE arl0 <- 370 #### Control chart parameters chart <- "Shewhart" chart.par <- c(3) shewhart <- getARL(n, m, theta = NULL, Ftheta = NULL, dist, mu, sigma, dist.par = dist.par, chart = chart, chart.par = chart.par, print.RL = print.RL, replicates = replicates, isParallel = isParallel, calibrate = calibrate, arl0 = arl0 ) chart <- "CUSUM" chart.par <- c(0.25, 4.4181, 3) cusum <- getARL(n, m, theta = NULL, Ftheta = NULL, dist, mu, sigma, dist.par = dist.par, chart = chart, chart.par = chart.par, print.RL = print.RL, replicates = replicates, isParallel = isParallel, calibrate = calibrate, arl0 = arl0 ) chart <- "EWMA" chart.par <- c(0.2, 2.962) shewhart <- getARL(n, m, theta = NULL, Ftheta = NULL, dist, mu, sigma, dist.par = dist.par, chart = chart, chart.par = chart.par, print.RL = print.RL, replicates = replicates, isParallel = isParallel, calibrate = calibrate, arl0 = arl0 ) ```

getARL function