getDeltaH_up: Estimation of the optimal 'H_delta' value to guarantee...

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

Estimation of the optimal H_delta value to guarantee performance in the upward CUSUM control chart

Description

This function calculates the optimal H_delta value that ensures specific performance in the Gamma CUSUM control chart for upward detection. It relies on Monte Carlo simulations and an iterative adjustment process to determine the appropriate value.

Following the methodology proposed by Madrid-Alvarez, Garcia-Diaz, and Tercero-Gomez (2024), this function allows adjusting H_delta for different sample size scenarios, ensuring that the control chart maintains the expected performance in terms of ARL.

Features:

• Implements Monte Carlo simulations to estimate H_delta.

• Based on parameter estimates obtained in Phase I.

• Iteratively adjusts H_delta until the specified ARL is reached.

• Displays total execution time using tictoc.

Recommendations

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

  Madrid-Alvarez, H. M., Garcia-Diaz, J. C., & Tercero-Gomez, V. G. (2024). A CUSUM control chart for gamma distribution with guaranteed performance. Quality and Reliability Engineering International, 40(3), 1279-1301.

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

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

• For selecting values of H_plus, a, and b, refer to the reference article, which presents specific strategies and recommendations for calibration.

Usage

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

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 estimate.
H_plus	Initial upper 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.
m	Number of states in the Markov matrix.
N_init	Number of initial iterations.
N_final	Number of final iterations.
known_alpha	Indicates whether alpha is known (TRUE) or needs to be estimated (FALSE).

Value

A numeric value corresponding to the optimal H_delta for the upward CUSUM control chart, ensuring the expected performance.

Examples

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)
             

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