# ARL by Monte Carlo simulation

    library(shewhartr)
    library(ggplot2)
    library(dplyr)

The Average Run Length (ARL) of a control chart is the expected number of samples until the chart signals. It is the single most important operating characteristic — it tells you, on average, how quickly the chart catches a real shift, and how often it cries wolf.

Two flavours:

- In-control ARL (\(\mathrm{ARL}_0\)): the expected number of in-control samples until a false alarm. Larger is better. A high \(\mathrm{ARL}_0\) means low false-alarm rate.
- Out-of-control ARL (\(\mathrm{ARL}_1\)): the expected number of samples after a real shift before the chart signals. Smaller is better. A low \(\mathrm{ARL}_1\) means fast detection.

These two are in tension. Adding more rules to a chart sharpens detection (lower \(\mathrm{ARL}_1\)) but increases false alarms (lower \(\mathrm{ARL}_0\)). `shewhart_arl()` quantifies the trade-off.

## Closed-form benchmarks

For the simplest setup — Nelson 1 only, normal data, 3-sigma limits — the ARL is known in closed form. The probability of falling outside 3-sigma is \(2 \cdot \Phi(-3) \approx 0.0027\), so:

\[ \mathrm{ARL}_0 = \frac{1}{0.0027} \approx 370.4. \]

Let’s verify by simulation:

    set.seed(2025)
    shewhart_arl(
      shift = 0,
      rules = "nelson_1_beyond_3s",
      n_sim = 5000,
      max_run = 2000
    )

The estimate should be close to 370 (Monte Carlo error around the closed form).

## Adding rules sharpens detection

What happens if we add Nelson 2 (9 points same side) on top?

    set.seed(2025)
    shewhart_arl(
      shift = 0,
      rules = c("nelson_1_beyond_3s", "nelson_2_nine_same"),
      n_sim = 5000
    )

The ARL_0 drops from ~370 to ~250, meaning false alarms become more common. Whether that’s worth it depends on what we get in detection power. Let’s plot ARL across a grid of shifts:

    shifts <- seq(0, 3, by = 0.25)

    set.seed(2025)
    arl_n1 <- shewhart_arl(shifts, "nelson_1_beyond_3s", n_sim = 2000)
    arl_n12 <- shewhart_arl(shifts,
                            c("nelson_1_beyond_3s", "nelson_2_nine_same"),
                            n_sim = 2000)

    bind_rows(
      arl_n1  |> mutate(rules = "Nelson 1 only"),
      arl_n12 |> mutate(rules = "Nelson 1 + 2")
    ) |>
      ggplot(aes(shift, arl, colour = rules)) +
      geom_line(linewidth = 0.7) +
      geom_ribbon(aes(ymin = arl_lower, ymax = arl_upper, fill = rules),
                  alpha = 0.12, colour = NA) +
      scale_y_log10() +
      scale_colour_manual(
        values = c(`Nelson 1 only` = unname(shewhart_palette("signal")["in_control"]),
                   `Nelson 1 + 2`  = unname(shewhart_palette("family")["memory_based"]))) +
      scale_fill_manual(
        values = c(`Nelson 1 only` = unname(shewhart_palette("signal")["in_control"]),
                   `Nelson 1 + 2`  = unname(shewhart_palette("family")["memory_based"]))) +
      labs(x = "Shift (sigma)", y = "ARL (log scale)",
           title = "Operating characteristics") +
      shewhart_theme()

The gain is largest for small shifts (around 0.5-1 sigma), where Nelson 1 alone takes a long time to fire. For shifts of 2 sigma or more, both rule sets detect within a couple of samples.

## A quantitative comparison: WE-7 vs Nelson 2

The original `Shewhart` package (now `shewhartr`) (v0.1.x) used a Western Electric “7 in a row” rule for phase detection. The default in v1.0 is Nelson 2 (9 in a row). The trade-off:

    set.seed(2025)
    arl_we  <- shewhart_arl(0, "we_seven_same",       n_sim = 5000, max_run = 2000)
    arl_n2  <- shewhart_arl(0, "nelson_2_nine_same",  n_sim = 5000, max_run = 2000)
    arl_we
    arl_n2

The WE rule’s ARL_0 is around 64 (a false phase change every 64 in-control samples on average); Nelson 2’s is around 256 (one every ~256 samples). For most monitoring contexts that is a meaningful difference. The WE rule is easier to teach, but the false-alarm cost is high.

`shewhart_regression()` accepts either via the `phase_rule` argument:

    shewhart_regression(temperature_drift, value = temp_c, index = minute,
                        phase_rule = "we_seven_same")        # legacy default
    shewhart_regression(temperature_drift, value = temp_c, index = minute,
                        phase_rule = "nelson_2_nine_same")    # new default

## Beyond normal residuals

`shewhart_arl()` simulates from a normal distribution by default. Real residuals are often non-normal — heavy-tailed, skewed, or autocorrelated. The closed-form ARL_0 of 370 for Nelson 1 then overstates how rarely false alarms occur. A bootstrap variant (simulating from your actual residuals) is on the roadmap; until then, a quick check is to call `shewhart_diagnostics(fit)` and inspect the Q-Q and ACF panels.

## References

- Champ, C. W., & Woodall, W. H. (1987). Exact Results for Shewhart Control Charts with Supplementary Runs Rules. Technometrics, 29(4), 393-399.
- Wald, A. (1947). Sequential Analysis. Wiley.
- Page, E. S. (1954). Continuous Inspection Schemes. Biometrika, 41(1-2), 100-115.
- Lucas, J. M., & Saccucci, M. S. (1990). Exponentially Weighted Moving Average Control Schemes: Properties and Enhancements. Technometrics, 32(1), 1-12.