LGCU (version 0.1.5)

ARL_Clminus: ARL Estimation in CUSUM Control Charts with Gamma Distribution and Cautious Learning for downward detection

Description

This function calculates the Average Run Length (ARL) of a CUSUM control chart based on the Gamma distribution, incorporating a cautious learning scheme for the dynamic update of parameters.

The function allows the evaluation of the CUSUM chart’s performance under different parameterization scenarios, ensuring efficient detection of process changes.

Based on the methodology presented in the work of Madrid-Alvarez, García-Díaz, and Tercero-Gómez (2024), this implementation uses Monte Carlo simulations optimized in C++ for efficient execution and progressive adjustment of the control chart parameters.

The values for H_minus, H_delta, K_l, delay, and tau can be referenced in the tables from the article:

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, 1-23.

Usage Scenarios:

Scenario 1: Known alpha and estimated beta

• The alpha parameter is assumed to be fixed and known in advance.

• beta is estimated from a dataset or provided by the user.

• The user must specify alpha and an initial estimate of beta (beta0_est).

Scenario 2: Both alpha and beta are estimated

• Both alpha and beta are estimated from an external dataset.

• The user must calculate alpha0_est and beta0_est before calling the function.

• beta0_est is dynamically updated during the simulation when a predefined condition is met.

Features:

• Implements Monte Carlo simulations for ARL estimation.

• Allows dynamic updating of beta0_est to improve model adaptation.

• Uses C++ optimization for efficient and precise execution.

• Compatible with scenarios where alpha is either known or estimated.

• Recommended values for H_minus, H_delta, K_l, delay, and tau can be found in the reference article.

This function is ideal for quality control studies where reliable detection of process changes modeled with Gamma distributions is required.

Usage

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

Value

A numeric value corresponding to the ARL estimate for the downward CUSUM control chart with cautious learning.

Arguments

alpha

Shape parameter of the Gamma distribution.

beta

Scale parameter of the Gamma distribution.

alpha0_est

Initial estimate of the shape parameter alpha. If known_alpha is TRUE, this value will be equal to alpha.

beta0_est

Initial estimate of the scale parameter beta. This value is updated dynamically during the simulation.

known_alpha

TRUE if alpha0_est is fixed, FALSE if it must be estimated.

beta_ratio

Ratio between beta and its posterior estimate.

H_delta

Increment of the lower control limit in the CUSUM chart.

H_minus

Initial control limit of the CUSUM chart for downward detection.

n_I

Sample size in Phase I.

replicates

Number of Monte Carlo simulations.

K_l

Secondary control threshold for parameter updating.

delay

Number of observations before updating beta0_est.

tau

Time point at which beta changes.

Examples

# Option 1: Provide parameters directly
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
)

# Option 2: Use generated data
set.seed(123)
datos_faseI <- rgamma(n = 500, shape = 1, scale = 1)
alpha0_est <- mean(datos_faseI)^2 / var(datos_faseI)  # Alpha estimation
beta0_est <- mean(datos_faseI) / alpha0_est  # Beta estimation

ARL_Clminus(
   alpha = 1,
   beta = 1,
   alpha0_est = 1.067,  # alpha = known_alpha
   beta0_est = 0.2760,   # Estimated Beta
   known_alpha = FALSE,
   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
)