# Western Electric Company Rules (WECO) for Shewhart Control Chart

## Simulation

We simulate normally distributed data as follows:

require(weco);
## Loading required package: weco
set.seed(10000);
n.sim     <- 50000;
sdx       <- 2;
simu.data <- rnorm(n.sim, sd = sdx);
quants    <- c(0.25, 0.5, 0.75);
xmax      <- 3000;
ymax      <- 0.015;

## Single rule

### Rule 1

1 point $$>L$$ standard deviations from center line

l          <- 3;
rst.1      <- weco.rule(rule=1, x=simu.data, l=l, sdx=sdx, mux=0);
simu.arl   <- weco.rl(rst.1);
hist(simu.arl, breaks = 100, freq = F,
xlim=c(0, xmax), ylim=c(0,ymax),
xlab="Running Length", ylab="Probability", main="Rule 1");

### Rule 2

$$K$$ points in a row on the same side of the center line

k          <- 9;
rst.2      <- weco.rule(rule=2, x=simu.data, k=k, sdx=sdx, mux=0);
simu.arl   <- weco.rl(rst.2);
hist(simu.arl, breaks = 100, freq = F,
xlim=c(0, xmax), ylim=c(0,ymax),
xlab="Running Length", ylab="Probability", main="Rule 2");

### Rule 3

$$K$$ points in a row, all increasing or decreasing

k          <- 6;
rst.3      <- weco.rule(rule=3, x=simu.data, k=k, sdx=sdx, mux=0);
simu.arl   <- weco.rl(rst.3);
hist(simu.arl, breaks = 100, freq = F,
xlim=c(0, xmax), ylim=c(0,ymax),
xlab="Running Length", ylab="Probability", main="Rule 3");

### Rule 4

$$K$$ points in a row, alternating up and down

k          <- 14;
rst.4      <- weco.rule(rule=4, x=simu.data, k=k, sdx=sdx, mux=0);
simu.arl   <- weco.rl(rst.4);
hist(simu.arl, breaks = 100, freq = F,
xlim=c(0, xmax), ylim=c(0,ymax),
xlab="Running Length", ylab="Probability", main="Rule 3");

### Rule 5

$$K$$ out $$K+1$$ points out of 2 standard deviations from center line

k          <- 2;
rst.5      <- weco.rule(rule=5, x=simu.data, k=k, l=2, sdx=sdx, mux=0);
simu.arl   <- weco.rl(rst.5);
hist(simu.arl, breaks = 100, freq = F,
xlim=c(0, xmax), ylim=c(0,ymax),
xlab="Running Length", ylab="Probability", main="Rule 5");

### Rule 6

$$K$$ out $$K+1$$ points out of 1 standard deviations from center line

k          <- 4;
rst.6      <- weco.rule(rule=6, x=simu.data, k=k, l=1, sdx=sdx, mux=0);
simu.arl   <- weco.rl(rst.6);
hist(simu.arl, breaks = 100, freq = F,
xlim=c(0, xmax), ylim=c(0,ymax),
xlab="Running Length", ylab="Probability", main="Rule 6");

### Rule 7

$$K$$ points in a row within 1 standard deviations from center line (either side)

k          <- 15;
rst.7      <- weco.rule(rule=7, x=simu.data, k=k, l=1, sdx=sdx, mux=0);
simu.arl   <- weco.rl(rst.7);
hist(simu.arl, breaks = 100, freq = F,
xlim=c(0, xmax), ylim=c(0,ymax),
xlab="Running Length", ylab="Probability", main="Rule 7");

### Rule 8

$$K$$ points in a row > 1 standard deviations from center line (either side)

k          <- 8;
rst.8      <- weco.rule(rule=8, x=simu.data, k=k, l=1, sdx=sdx, mux=0);
simu.arl   <- weco.rl(rst.8);
hist(simu.arl, breaks = 100, freq = F,
xlim=c(0, xmax), ylim=c(0,ymax),
xlab="Running Length", ylab="Probability", main="Rule 8");

## Multiple rules

Multiple rules can be combined as a list.

##rules 1 and 2
lst.rules  <- list(list(1, l=3),
list(2, k=9));
rst.m      <- weco.combine(simu.data, lst.rules=lst.rules);
simu.arl   <- weco.rl(rst.m);
hist(simu.arl, breaks = 100, freq = F,
xlim=c(0, xmax), ylim=c(0,ymax),
xlab="Running Length", ylab="Probability", main="Multiple Rules");

## Plot results

A trace plot may be plotted as follows

plot(rst.m, start=1000, end=1500);

## Graphical user interface

The package provides a graphical user interface based on Shiny, which can be brought up by

run.weco();