Implementation of Learning Gamma CUSUM (Cumulative Sum) Control Charts

Implements Cumulative Sum (CUSUM) control charts specifically designed for monitoring processes following a Gamma distribution. Provides functions to estimate distribution parameters, simulate control limits, and apply cautious learning schemes for adaptive thresholding. It supports upward and downward monitoring with guaranteed performance evaluated via Monte Carlo simulations. It is useful for quality control applications in industries where data follows a Gamma distribution. Methods are based on Madrid-Alvarez et al. (2024) and Madrid-Alvarez et al. (2024) .

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LGCU: Learning Gamma CUSUM

Overview

LGCU (Learning Gamma CUSUM) is an R package for designing and analyzing CUSUM control charts applied to Gamma distributions with guaranteed performance. The package provides methods for:

• Estimating Average Run Length (ARL) for both upward and downward CUSUM charts.
• Visualizing CUSUM control charts for process monitoring.
• Calibrating control limits using Monte Carlo simulations.
• Applying cautious learning mechanisms to dynamically update parameters.

The methodologies implemented follow the work of Madrid-Alvarez, García-Díaz, and Tercero-Gómez (2024), ensuring rigorous statistical foundations and practical applications in industrial quality control.

Installation

To install the package from GitHub:

# Install devtools if not already installed
install.packages("devtools")

# Install LGCU from GitHub
devtools::install_github("your_github_username/LGCU")

To load the package:

library(LGCU)

Functions & Usage

1. ARL Estimation for CUSUM Control Charts

Downward CUSUM ARL Calculation (Guaranteed Performance)

GICARL_CUSUM_down(alpha, beta, alpha_est, beta_est, beta_ratio, H_minus, H_delta, m)

# Example:
GICARL_CUSUM_down(alpha = 1, beta = 1, alpha_est = 1, beta_est = 1,
                   beta_ratio = 1/2.5, H_minus = -2.792, H_delta = 0, m = 100)

Upward CUSUM ARL Calculation (Guaranteed Performance)

GICARL_CUSUM_up(alpha, beta, alpha_est, beta_est, beta_ratio, H_plus, H_delta, m)

# Example:
GICARL_CUSUM_up(alpha = 1, beta = 1, alpha_est = 1.2, beta_est = 0.8,
                 beta_ratio = 2, H_plus = 6.5081, H_delta = 2.9693, m = 100)

Downward CUSUM ARL Calculation (With Learning)

ARL_Clminus(alpha, beta, alpha0_est, beta0_est, known_alpha, beta_ratio, H_delta, H_minus, n_I, replicates, K_l, delay, tau)

# Example:
ARL_Clminus(
   alpha = 1,
   beta = 1,
   alpha0_est = 1.067,  # alpha = known_alpha
   beta0_est = 0.2760,   # Estimated Beta
   known_alpha = TRUE,
   beta_ratio = 1/2,
   H_delta = 0.6946,
   H_minus = -4.8272,
   n_I = 500,
   replicates = 1000,
   K_l = 0.5,
   delay = 25,
   tau = 1
)

Upward CUSUM ARL Calculation (With Learning)

ARL_Clplus(alpha, beta, alpha0_est, beta0_est, known_alpha, beta_ratio, H_delta, H_plus, n_I, replicates, K_l, delay, tau)

# Example:
ARL_Clplus(
  alpha = 1,
  beta = 1,
  alpha0_est = 1,  # alpha = known_alpha
  beta0_est = 1.1,   # Estimated Beta
  known_alpha = TRUE,
  beta_ratio = 2,
  H_delta = 4.2433,
  H_plus = 8.7434,
  n_I = 200,
  replicates = 100,
  K_l = 2,
  delay = 25,
  tau = 1
)

2. CUSUM Control Chart Visualization

Downward CUSUM Chart

plot_GICCdown_chart(alpha, beta, beta_ratio, H_delta, H_minus, n_I, n_II, faseI, faseII, known_alpha)

# Example:
plot_GICCdown_chart(alpha = 3, beta = 1, beta_ratio = 1/2, H_delta = 0.9596, H_minus = -4.6901,
                    n_I = 100, n_II = 200, faseI = NULL, faseII = NULL, known_alpha = FALSE)

Upward CUSUM Chart

plot_GICCup_chart(alpha, beta, beta_ratio, H_delta, H_plus, n_I, n_II, faseI, faseII, known_alpha)

# Example:
plot_GICCup_chart(alpha = 1, beta = 1, beta_ratio = 2, H_delta = 0, H_plus = 5.16,
                  n_I = 100, n_II = 200, faseI = NULL, faseII = NULL, known_alpha = TRUE)

Bidirectional CUSUM Chart

plot_GICC_chart2(alpha, beta, beta_ratio_plus, beta_ratio_minus, H_delta_plus, H_plus, H_delta_minus, H_minus, n_I, n_II, faseI, faseII, known_alpha)

# Example:
plot_GICC_chart2(alpha = 1, beta = 1, beta_ratio_plus = 2, beta_ratio_minus = 0.5,
                 H_delta_plus = 2.0, H_plus = 5.0, H_delta_minus = 1.5, H_minus = -4.5,
                 n_I = 100, n_II = 200, faseI = NULL, faseII = NULL, known_alpha = TRUE)

3. CUSUM Chart with Learning Mechanism

Downward CUSUM with Learning

plot_GICCLdown_Chart(alpha, beta, beta_ratio, H_delta, H_minus, known_alpha, k_l, delay, tau, n_I, n_II, faseI, faseII)

# Example:
plot_GICCLdown_Chart(alpha = 1, beta = 1, beta_ratio = 1/2, H_delta = 4.2433, H_minus= -4.8257,
                     known_alpha = FALSE, k_l = 0.739588, delay = 25, tau = 1,
                     n_I = 200, n_II = 700, faseI = NULL, faseII = NULL)

Upward CUSUM with Learning

plot_GICCLup_Chart(alpha, beta, beta_ratio, H_delta, H_plus, known_alpha, k_l, delay, tau, n_I, n_II, faseI, faseII)

# Example:
plot_GICCLup_Chart(alpha = 1, beta = 1, beta_ratio = 2, H_delta = 2.9819, H_plus = 6.5081,
                    known_alpha = TRUE, k_l = 2, delay = 25, tau = 1,
                    n_I = 200, n_II = 710, faseI = NULL, faseII = NULL)

Bidirectional CUSUM with Learning

plot_GICCL_chart2(alpha, beta, beta_ratio_plus, beta_ratio_minus, H_delta_plus, H_plus, H_delta_minus, H_minus, known_alpha, k_l, delay, tau, n_I, n_II, faseI, faseII)

# Example:
plot_GICCL_chart2(alpha = 1, beta = 1, beta_ratio_plus = 2, beta_ratio_minus = 0.5,
                  H_delta_plus = 3.0, H_plus = 6.5, H_delta_minus = 2.0, H_minus = -5.0,
                  known_alpha = TRUE, k_l = 2, delay = 25, tau = 1,
                  n_I = 200, n_II = 700, faseI = NULL, faseII = NULL)

4. Calibration of H_delta for Guaranteed Performance

Downward Calibration

getDeltaH_down(n_I, alpha, beta, beta_ratio, H_minus, a, b, ARL_esp, m, N_init, N_final, known_alpha)

# Example:
getDeltaH_down(n_I = 100, alpha = 1, beta = 1, beta_ratio = 1/2, H_minus = -4.1497,
               a = 0.1, b = 0.05, ARL_esp = 370, m = 100, N_init = 10, N_final = 1000, known_alpha = TRUE)

Upward Calibration

getDeltaH_up(n_I, alpha, beta, beta_ratio, H_plus, a, b, ARL_esp, m, N_init, N_final, known_alpha)

# Example:
getDeltaH_up(n_I = 100, alpha = 1, beta = 1, beta_ratio = 2, H_plus = 6.8313,
              a = 0.1, b = 0.05, ARL_esp = 370, m = 100, N_init = 10, N_final = 1000, known_alpha = TRUE)

5. Calibration of H_delta with Learning

Downward Calibration with Learning

getDeltaHL_down(n_I, alpha, beta, beta_ratio, H_minus, a, b, ARL_esp, replicates, N_init, N_final, known_alpha, K_l, delay, tau)

# Example:
getDeltaHL_down(n_I = 200, alpha = 1, beta = 1, beta_ratio = 1/1.5,
                H_minus = -6.2913, a = 0.1, b = 0.05, ARL_esp = 370,
                replicates = 10, N_init = 100, N_final = 1000, known_alpha = TRUE,
                K_l = 0.7, delay = 25, tau = 1)

Upward Calibration with Learning

getDeltaHL_up(n_I, alpha, beta, beta_ratio, H_plus, a, b, ARL_esp, replicates, N_init, N_final, known_alpha, K_l, delay, tau)

# Example:
getDeltaHL_up(n_I = 200, alpha = 1, beta = 1, beta_ratio = 2,
              H_plus = 6.8313, a = 0.1, b = 0.05, ARL_esp = 370,
              replicates = 100, N_init = 100, N_final = 500, known_alpha = TRUE,
              K_l = 2, delay = 25, tau = 1)

References

📖 Madrid-Alvarez, H. M., García-Díaz, J. C., & Tercero-Gómez, V. G. (2024). A CUSUM control chart for gamma distribution with guaranteed performance. Quality and Reliability Engineering International.

📖 Madrid-Alvarez, H. M., García-Díaz, J. C., & Tercero-Gómez, V. G. (2024). A CUSUM control chart for the Gamma distribution with cautious parameter learning. Quality Engineering.

Contributing

If you would like to contribute, please follow these steps: 1. Fork the repository. 2. Create a new branch (feature-newFunction). 3. Commit your changes. 4. Push to the branch and submit a Pull Request.

License

This package is licensed under the MIT License, which means you are free to use, modify, and distribute the software, provided that the original copyright and license notice is included in all copies. This allows both open-source and commercial use.

Contact

For any questions or suggestions, please contact: 📩 [email protected]{.email}