Getting Started with gpci

The gpci package provides a distribution-agnostic framework for calculating Process Capability Indices (PCIs), performing bootstrap confidence interval estimation, and running bootstrap cross-validation coverage diagnostics.

Setup

First, load the package and ggplot2:

library(gpci)
library(ggplot2)

Simulating Process Data

We simulate a quality characteristic \(X \sim N(10, 1.2^2)\) from a stable process. We set specification limits: * Lower Specification Limit (LSL) = 7 * Upper Specification Limit (USL) = 13 * Target (\(T\)) = 10

set.seed(123)
process_data <- rnorm(100, mean = 9.8, sd = 1.1)

Capability Analysis

We construct a standard normal distribution object and fit it to the data using Maximum Likelihood Estimation (MLE):

# Create standard normal distribution template
dist_norm <- dist_normal()

# Compute capability indices (moment-based and quantile-based)
fit <- capability(
  data = process_data,
  distribution = dist_norm,
  USL = 13,
  LSL = 7,
  target = 10,
  indices = c("Cp", "Cpk", "Cpl", "Cpu", "Cpm", "Cpmk", "Spmk", "Cpc"),
  fit = TRUE,
  fit_method = "mle",
  mode = "moments"
)

# Print results
print(fit)
#> --- Process Capability Analysis (Class: gpcifit) ---
#> Distribution:  normal 
#> Parameters:    mean = 9.8994, sd = 0.9982 
#> Spec Limits:  LSL = 7 , USL = 13 , Target = 10 
#> Mode:          moments 
#> Expected Nonconforming (p_hat):  0.2785 %
#> 
#> Point Estimates of Capability Indices:
#>     Cp    Cpk    Cpl    Cpu    Cpm   Cpmk   Spmk    Cpc 
#> 1.0018 0.9682 0.9682 1.0354 0.9968 0.9634 0.9918 0.9694

Bootstrap Confidence Intervals

Next, we compute bootstrap confidence intervals at multiple significance levels (\(\alpha = 0.10, 0.05, 0.01\)) using the percentile bootstrap:

# Calculate CIs
ci <- boot_ci(
  fit = fit,
  B = 30, # Optimized B for fast vignette generation
  alpha = c(0.10, 0.05, 0.01),
  method = "percentile",
  type = "parametric"
)

# View CI table
print(ci)
#> --- Bootstrap Confidence Intervals ---
#> Bootstrap Type:    parametric 
#> CI Method:         percentile 
#> Replicates (B):    30 
#> 
#>    index estimate     method       type alpha conf_level  lower  upper  width
#> 1     Cp   1.0018 percentile parametric  0.10        90% 0.9113 1.1444 0.2331
#> 2     Cp   1.0018 percentile parametric  0.05        95% 0.9024 1.1452 0.2428
#> 3     Cp   1.0018 percentile parametric  0.01        99% 0.9024 1.1452 0.2428
#> 4    Cpk   0.9682 percentile parametric  0.10        90% 0.8792 1.0990 0.2198
#> 5    Cpk   0.9682 percentile parametric  0.05        95% 0.8689 1.1298 0.2609
#> 6    Cpk   0.9682 percentile parametric  0.01        99% 0.8689 1.1298 0.2609
#> 7    Cpl   0.9682 percentile parametric  0.10        90% 0.8792 1.0990 0.2198
#> 8    Cpl   0.9682 percentile parametric  0.05        95% 0.8689 1.1298 0.2609
#> 9    Cpl   0.9682 percentile parametric  0.01        99% 0.8689 1.1298 0.2609
#> 10   Cpu   1.0354 percentile parametric  0.10        90% 0.9138 1.2011 0.2874
#> 11   Cpu   1.0354 percentile parametric  0.05        95% 0.9042 1.2081 0.3039
#> 12   Cpu   1.0354 percentile parametric  0.01        99% 0.9042 1.2081 0.3039
#> 13   Cpm   0.9968 percentile parametric  0.10        90% 0.9092 1.1312 0.2221
#> 14   Cpm   0.9968 percentile parametric  0.05        95% 0.8979 1.1440 0.2461
#> 15   Cpm   0.9968 percentile parametric  0.01        99% 0.8979 1.1440 0.2461
#> 16  Cpmk   0.9634 percentile parametric  0.10        90% 0.8697 1.0865 0.2168
#> 17  Cpmk   0.9634 percentile parametric  0.05        95% 0.8645 1.1285 0.2640
#> 18  Cpmk   0.9634 percentile parametric  0.01        99% 0.8645 1.1285 0.2640
#> 19  Spmk   0.9918 percentile parametric  0.10        90% 0.9057 1.1189 0.2132
#> 20  Spmk   0.9918 percentile parametric  0.05        95% 0.8934 1.1428 0.2494
#> 21  Spmk   0.9918 percentile parametric  0.01        99% 0.8934 1.1428 0.2494
#> 22   Cpc   0.9694 percentile parametric  0.10        90% 0.4246 3.9652 3.5406
#> 23   Cpc   0.9694 percentile parametric  0.05        95% 0.3821 4.5053 4.1232
#> 24   Cpc   0.9694 percentile parametric  0.01        99% 0.3821 4.5053 4.1232

Plotting Results

The package provides S3 plot methods for visualizing the process capability:

1. Process Density and Specification Limits

plot(fit, type = "density")

2. Empirical CDF vs Fitted CDF

plot(fit, type = "cdf")

3. Quantile-Quantile (Q-Q) Plot

plot(fit, type = "qq")

4. Process Run Chart

plot(fit, type = "run")

5. Bootstrap Sampling Distributions and Confidence Intervals

We can also visualize the bootstrap results:

plot(ci, type = "boot")

6. Forest Plot of CIs

plot(ci, type = "forest")