getDeltaHL_down: Estimation of the 'H_delta' parameter with learning for...

In LGCU: Implementation of Learning Gamma CUSUM (Cumulative Sum) Control Charts

Estimation of the H_delta parameter with learning for downward detection in CUSUM Gamma charts

Description

This function calculates the optimal value of H_delta using a dynamic learning scheme based on the ARL_Clplus function, iteratively adjusting H_delta to achieve an expected ARL with greater accuracy and adaptability.

Based on the methodology proposed by Madrid-Alvarez, Garcia-Diaz, and Tercero-Gomez (2024), this function allows adjusting H_delta in different sample size scenarios, ensuring that the control chart progressively adapts to changes in the Gamma distribution.

Features:

• Implements Monte Carlo simulations to estimate H_delta.

• Relies on parameter estimates obtained in Phase I.

• Iteratively adjusts H_delta until the specified ARL is reached.

• Incorporates a cautious learning mechanism to improve adjustment accuracy.

• Displays total execution time using tictoc.

Recommendations

• This function is useful for estimating H_delta values when the sample size differs from those reported in the reference article:

  Madrid-Alvarez, H. M., Garcia-Diaz, J. C., & Tercero-Gomez, V. G. (2024). A CUSUM control chart for the Gamma distribution with cautious parameter learning. Quality Engineering, 1-23.

• The adjustment process is iterative and computationally demanding, as execution time depends on the number of iterations (N_init + N_final) and the sample size (n_I).

• It is recommended to define an appropriate convergence criterion to optimize execution time without compromising accuracy in the estimation of H_delta.

• For selecting values of a, b, k_l, delay, tau, and H_minus, refer to the reference article, which presents specific strategies for their calibration in different scenarios.

Usage

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
)

Arguments

n_I	Sample size in Phase I.
alpha	Shape parameter of the Gamma distribution.
beta	Scale parameter of the Gamma distribution.
beta_ratio	Ratio between beta and its posterior estimate.
H_minus	Lower limit of the CUSUM chart.
a	Tolerance level for the expected ARL (0 <= a < 1).
b	Tolerance level for the expected ARL (0 < b < 1).
ARL_esp	Desired expected ARL value.
replicates	Number of replications in the Monte Carlo simulation.
N_init	Initial iterations for adjustment.
N_final	Final iterations for averaging H_delta.
known_alpha	TRUE if alpha is fixed, FALSE if it must be estimated.
K_l	Secondary control threshold for parameter update.
delay	Number of observations before updating beta0_est.
tau	Time point where beta changes.

Value

A numeric value corresponding to the optimal H_delta estimated with learning for the downward CUSUM control chart.

Examples

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)
              

LGCU documentation built on April 12, 2025, 1:59 a.m.