getDeltaHL_down {LGCU} | R Documentation |
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
, andH_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 |
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 |
known_alpha |
|
K_l |
Secondary control threshold for parameter update. |
delay |
Number of observations before updating |
tau |
Time point where |
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)