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
)

Arguments

Value

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

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
)

ARL Estimation in CUSUM Control Charts with Gamma Distribution and Cautious Learning for upward 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 progressive update of parameters and optimization of performance in upward detection.

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

Following the methodology presented in the work of Madrid-Alvarez, García-Díaz, and Tercero-Gómez (2024), this implementation utilizes Monte Carlo simulations in C++ for efficient execution, ensuring a dynamic adjustment of parameters based on the evolution of the process.

The values of H_plus, 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 defined by the user.

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

Scenario 2: Both alpha and beta are estimated

• Both alpha and beta are estimated from a dataset or external data source.

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

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

Features:

• Implements Monte Carlo simulations optimized in C++ for ARL estimation.

• Allows dynamic updating of beta0_est to improve the model's adaptability.

• Compatible with scenarios where alpha is known or estimated.

• Ensures stable and reliable performance in detecting changes in processes modeled with Gamma distributions.

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

Usage

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

Arguments

Value

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

Examples

# Option 1: Provide parameters directly
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
)

# Option 2: Use generated data
set.seed(123)
datos_faseI <- rgamma(n = 200, 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_Clplus(
  alpha = 1,
  beta = 1,
  alpha0_est = alpha0_est,
  beta0_est = beta0_est,
  known_alpha = FALSE,
  beta_ratio = 2,
  H_delta = 4.2433,
  H_plus = 8.7434,
  n_I = 200,
  replicates = 1000,
  K_l = 2,
  delay = 25,
  tau = 1
)

ARL Calculation for a Downward CUSUM Control Chart with a Gamma Distribution

Description

This function estimates the average run length (ARL) of a downward CUSUM control chart applied to a Gamma distribution with guaranteed performance (GIC), considering both known and estimated parameters.

This approach follows the methodology described in the work of Madrid‐Alvarez, García-díaz, and Tercero‐Gómez(2024), which provides a detailed analysis of the performance of CUSUM control charts for Gamma distributions with guaranteed efficiency. Specifically, the method implemented in this function enables the precise evaluation of ARL under different parameter settings, ensuring appropriate calibration and monitoring of controlled processes.

Recommendations

For further consultation and to review values of H_delta and H_minus, it is recommended to refer to the following article: 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.

Considerations:

• The control chart is calibrated with beta = 1.

• When alpha and beta are known, it is recommended to use the same values for alpha_est = alpha and beta_est = 1.

• Higher values of m increase the accuracy of the results (⁠m = {10, 100}⁠).

Usage

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

Arguments

Value

A numeric value representing the average run length (ARL) of the downward CUSUM control chart.

Examples

# Example with known parameters
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)

# Example with estimated parameters
GICARL_CUSUM_down(alpha = 1, beta = 1, alpha_est = 1, beta_est = 1.1,
                  beta_ratio = 1/2, H_minus = -4.1497, H_delta = 1.5167,
                  m = 100)

ARL Calculation for an Upward CUSUM Control Chart with a Gamma Distribution

Description

This function estimates the average run length (ARL) of an upward CUSUM control chart applied to a Gamma distribution with guaranteed performance (GIC), considering both known and estimated parameters.

This approach follows the methodology described in the work of Madrid‐Alvarez, García‐Díaz, and Tercero‐Gómez (2024), which provides a detailed analysis of the performance of CUSUM control charts for Gamma distributions with guaranteed efficiency. Specifically, the method implemented in this function enables the precise evaluation of ARL under different parameter settings, ensuring appropriate calibration and monitoring of controlled processes.

Recommendations

For further consultation and to review values of H_delta and H_plus, it is recommended to refer to the following article: 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.

Key Considerations:

• The control chart is calibrated with beta = 1.

• When alpha and beta are known, it is recommended to use the same values for alpha_est = alpha and beta_est = 1.

• Higher values of m increase the accuracy of the results (⁠m = {10, 100}⁠).

Usage

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

Arguments

Value

A numeric value representing the average run length (ARL) of the upward CUSUM control chart.

Examples

# Example with known parameters
GICARL_CUSUM_up(alpha = 0.9, beta = 2.136, alpha_est = 0.9, beta_est = 1,
                beta_ratio = 2.67, H_plus = 25.1592, H_delta = 0, m = 100)

# Example with estimated parameters
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)

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

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)
              

Estimation of the H_delta parameter with learning for upward detection in Gamma CUSUM control 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 higher accuracy and adaptability.

Based on the methodology proposed by Madrid-Alvarez, García-Díaz, and Tercero-Gómez (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 precision.

• Displays the total execution time using tictoc.

Recommendations

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

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

• It is recommended to define a proper convergence criterion to optimize execution time without compromising the accuracy of H_delta estimation.

• For selecting values of a, b, k_l, delay, tau, and H_plus, consulting the reference article is recommended, as it provides specific strategies for their calibration in different scenarios.

Usage

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
)

Arguments

Value

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

Examples

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
            )
            

Estimation of the Optimal H_delta Value to Guarantee Performance in the Downward CUSUM Control Chart

Description

This function calculates the optimal value of H_delta that guarantees a specific performance in the Gamma CUSUM control chart for downward detection. It employs a Monte Carlo simulation approach and an iterative adjustment process to determine the appropriate value.

Following the methodology presented by Madrid‐Alvarez, García‐Díaz, and Tercero‐Gómez (2024), this function allows adjusting H_delta for different sample size configurations, ensuring that the control chart maintains the desired performance in terms of expected 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 achieved.

• Displays the 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 paper:

  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, 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_minus, a, and b, it is advisable to consult the reference paper, which provides specific calibration strategies and recommendations.

Usage

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

Arguments

Value

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

Examples

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)
               

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

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)
             

CUSUM Control Chart with Cautious Learning and Guaranteed Performance

Description

This function generates a bidirectional (upward and downward) CUSUM control chart for a Gamma distribution, incorporating a cautious parameter update mechanism with guaranteed performance. Its purpose is to enhance sensitivity and precision in detecting changes in dynamic processes.

Based on the methodology presented by Madrid-Alvarez, García-Díaz, and Tercero-Gómez (2024), this implementation allows control limits to adapt according to the evolution of the process, ensuring early detection of variations while minimizing the risk of false alarms.

Features:

• If the user does not provide Phase I and Phase II data, the function automatically generates them.

• If known_alpha = TRUE, alpha is fixed and not estimated.

• If known_alpha = FALSE, alpha is estimated from Phase I data.

• Includes dynamic control limits and a summary table of parameters.

• Enables the detection of both upward and downward deviations, progressively adjusting the control limits.

Recommendations

• The parameters k_l, delay, and tau are crucial for the learning process in the control chart. They regulate the progressive update of control limits, allowing the dynamic update of beta0_est, H_plus_c, and H_minus_c, ensuring that the control chart gradually adjusts to changes in the process. It is recommended to use reference values presented in:

  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.

• Similar to the parameters above, for proper selection of H_plus, H_minus, H_delta_plus, and H_delta_minus values, it is recommended to review the reference article, where detailed calibration strategies for different scenarios are presented.

Usage

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 = NULL,
  faseII = NULL
)

Arguments

Value

A plot showing the evolution of the CUSUM statistic with cautious learning, including:

• Dynamically adjusted accumulated values of the CUSUM statistic.

• Progressively updated control limits with guaranteed performance.

• A summary of the parameters used in the control chart.

Examples

# Option 1: Automatically generated data
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)

# Option 2: User-provided data
datos_faseI <- rgamma(n = 200, shape = 1, scale = 1)
datos_faseII <- rgamma(n = 700, shape = 1, scale = 1)
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 = FALSE, k_l = 2, delay = 25, tau = 1,
                 n_I = 200, n_II = 700,
                 faseI = datos_faseI, faseII = datos_faseII)

Downward CUSUM Control Chart with Cautious Learning and Guaranteed Performance

Description

This function generates a downward CUSUM control chart for a Gamma distribution, incorporating a cautious parameter updating mechanism based on guaranteed performance.

It enables dynamic process monitoring, ensuring progressive adaptation to changes in the distribution.

This approach follows the methodology presented in the work of Madrid-Alvarez, García-Díaz, and Tercero-Gómez (2024), where a cautious learning scheme for parameter updating in CUSUM control charts applied to Gamma distributions is proposed.

The implementation captures changes in the distribution and adjusts the control limits to enhance the detection of process variations.

Features:

• If the user does not provide Phase I and Phase II data, the function generates them automatically.

• If known_alpha = TRUE, alpha is fixed and not estimated.

• If known_alpha = FALSE, alpha is estimated from Phase I data.

• Dynamic control limits and a summary of parameters are included.

• Integrates a cautious learning scheme using the parameters k_l, delay, and tau.

Recommendations

The parameters k_l, delay, and tau are part of the cautious learning mechanism of the CUSUM chart. These values enable the dynamic updating of beta0_est and H_minus, ensuring that the control chart progressively adapts to process changes, improving sensitivity in detecting deviations.

For proper implementation, it is recommended to reference the values proposed in:

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.

While these parameters have been tested and validated in the referenced article, users can adjust them based on the specific characteristics of their process, considering factors such as system variability and desired update frequency.

Additionally, if detailed guidance on selecting values for H_delta and H_minus is needed, it is recommended to review the referenced article, which presents calibration and adjustment strategies for these limits to ensure optimal control chart performance.

Usage

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

Arguments

Value

A plot showing the evolution of the downward CUSUM statistic with cautious learning, including:

• The dynamically adjusted accumulated values of the CUSUM statistic.

• Progressively updated control limits with guaranteed performance.

• A summary of the parameters used in the control chart.

Examples

# Option 1: Providing Phase I and Phase II data
phaseI_data <- rgamma(n = 200, shape = 1, scale = 1)
phaseII_data <- rgamma(n = 710, shape = 1, scale = 1)
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 = phaseI_data, faseII = phaseII_data)

# Option 2: Without providing data, the function automatically generates them
plot_GICCLdown_Chart(alpha = 1, beta = 1, beta_ratio = 1/2, H_delta = 1.6763,
                     H_minus = -4.8257, known_alpha = FALSE, k_l = 0.739588,
                     delay = 25, tau = 1, n_I = 200,
                     n_II = 710, faseI = NULL, faseII = NULL)

Upward CUSUM Control Chart with Cautious Learning and Guaranteed Performance

Description

This function generates an upward CUSUM control chart for a Gamma distribution, incorporating a cautious parameter update mechanism based on guaranteed performance.

It enables dynamic process monitoring, ensuring progressive adaptation to distribution changes. This approach follows the methodology presented in the work of Madrid-Alvarez, García-Díaz, and Tercero-Gómez (2024), where a cautious learning scheme for parameter updates in CUSUM control charts applied to Gamma distributions is proposed.

The implementation captures distribution changes and adjusts the control limits to improve process variation detection.

Features:

• If the user does not provide Phase I and Phase II data, the function automatically generates them.

• If known_alpha = TRUE, alpha is fixed and not estimated.

• If known_alpha = FALSE, alpha is estimated from Phase I data.

• Includes dynamic control limits and a summary of parameters.

• Integrates a cautious learning scheme using the parameters k_l, delay, and tau.

Recommendations

The parameters k_l, delay, and tau are part of the cautious learning mechanism of the CUSUM chart. These values enable the dynamic updating of beta0_est and H_plus, ensuring that the control chart progressively adapts to process changes, thus improving sensitivity in detecting deviations.

For proper implementation, it is recommended to reference the values proposed in:

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.

While these parameters have been tested and validated in the referenced article, users can adjust them based on the specific characteristics of their process, considering factors such as system variability and desired update frequency.

Additionally, if detailed guidance on selecting values for H_delta and H_plus is required, it is recommended to review the referenced article, which presents calibration and adjustment strategies for these limits to ensure optimal control chart performance.

Usage

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

Arguments

Value

A plot showing the evolution of the upward CUSUM statistic with cautious learning, including:

• The dynamically adjusted accumulated values of the CUSUM statistic.

• Progressively updated control limits with guaranteed performance.

• A summary of the parameters used in the control chart.

Examples

# Option 1: Providing Phase I and Phase II data
phaseI_data <- rgamma(n = 200, shape = 1, scale = 1)
phaseII_data <- rgamma(n = 710, shape = 1, scale = 1)
plot_GICCLup_Chart(alpha = 1, beta = 1, beta_ratio = 2, H_delta = 4.2433,
                   H_plus = 8.9345, known_alpha = FALSE, k_l = 2, delay = 25,
                   tau = 1,n_I = 200, n_II = 700, faseI = phaseI_data,
                   faseII = phaseII_data)

# Option 2: Without providing data, the function automatically generates them
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)

CUSUM Control Chart for Gamma Distribution with Guaranteed Performance

Description

This function generates a bidirectional (upward and downward) CUSUM control chart for a Gamma distribution, allowing the monitoring of the evolution of the CUSUM statistic while ensuring optimal performance in detecting process changes.

Based on the methodology proposed by Madrid‐Alvarez, García‐Díaz, and Tercero‐Gómez (2024), this implementation employs a Monte Carlo-based approach to estimate the Gamma distribution parameters and determine control limits with precise calibration. The function enables visualization of process evolution and the detection of deviations with reduced risk of false alarms.

Features:

• Implements Monte Carlo simulations for control chart calibration.

• Allows the use of known Gamma distribution values or estimation in Phase I.

• Provides a graphical representation of the CUSUM statistic evolution with guaranteed performance.

• Includes control limits and a legend with key configuration details of the control chart.

For additional details on selecting parameters H_plus, H_minus, H_delta_plus, and H_delta_minus, as well as calibration strategies, it is recommended to consult the reference article:

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, 40(3), 1279-1301.

Usage

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 = NULL,
  faseII = NULL,
  known_alpha
)

Arguments

Value

A plot showing the evolution of the CUSUM statistic for a Gamma distribution with guaranteed performance, including:

• The accumulated values of the CUSUM statistic.

• Control limits with guaranteed performance.

• A summary of the parameters used in the control chart.

Examples

# Option 1: Automatically generate data with predefined sample sizes
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
                     )

# Option 2: Use custom data
phaseI_data <- rgamma(n = 100, shape = 1, scale = 1)
phaseII_data <- rgamma(n = 200, shape = 1, scale = 1)
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 = phaseI_data, faseII = phaseII_data,
                 known_alpha = TRUE
                 )

Downward CUSUM Control Chart for Gamma Distribution with Guaranteed Performance

Description

This function generates a downward CUSUM control chart for a Gamma distribution, displaying the evolution of the CUSUM statistic, control limits, and a summary of the parameters.

Based on the approach presented by Madrid‐Alvarez, García‐Díaz, and Tercero‐Gómez (2024), this implementation allows for the evaluation and visualization of monitored processes using a CUSUM chart adapted to Gamma distributions with guaranteed performance.

In particular, the function incorporates a Monte Carlo model to simulate the behavior of the control chart, allowing for the estimation of the Gamma distribution in Phase I or the use of predefined values. Additionally, it provides a clear graphical representation of the evolution of the CUSUM statistic, ensuring appropriate calibration and process control.

Recommendations

To reference specific values of H_delta and H_minus, it is recommended to consult the following article: 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, 40(3), 1279-1301.

Features:

• Based on a Monte Carlo model.

• The Gamma distribution is either estimated in Phase I or predefined values are used.

• Accumulated values of the CUSUM statistic with guaranteed performance are plotted.

• Control limits and a summary table are included.

Usage

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

Arguments

Value

A plot showing the evolution of the downward CUSUM statistic, including:

• The accumulated values of the CUSUM statistic.

• Control limits with guaranteed performance.

• A summary of the parameters used in the control chart.

Examples

# Option 1: Automatically generate data with defined sample sizes
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
                    )

# Option 2: Use custom data
phaseI_data <- rgamma(n = 100, shape = 1, scale = 1)
phaseII_data <- rgamma(n = 200, shape = 1, scale = 1)
plot_GICCdown_chart(
                    alpha = 1, beta = 1, beta_ratio = 1/2, H_delta = 2.9693,
                    H_minus = -6.5081, n_I = 100, n_II = 200,
                    faseI = phaseI_data, faseII = phaseII_data,
                    known_alpha = TRUE
                    )

Upward CUSUM Control Chart for Gamma Distribution with Guaranteed Performance

Description

This function generates an upward CUSUM control chart for a Gamma distribution, displaying the evolution of the CUSUM statistic, control limits, and a summary of the parameters.

Based on the approach presented by Madrid‐Alvarez, García‐Díaz, and Tercero‐Gómez (2024), this implementation enables the evaluation and visualization of the monitored process using a CUSUM chart adapted to Gamma distributions with guaranteed performance.

Specifically, the library incorporates a Monte Carlo model for simulating the control chart behavior, allowing the Gamma distribution to be estimated in Phase I or using predefined values. Additionally, it provides a clear graphical representation of the CUSUM statistic's evolution, ensuring proper calibration and process control.

Recommendations

To check specific values for H_delta and H_plus, it is recommended to review the reference article: 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, 40(3), 1279-1301.

Features:

• Based on a Monte Carlo model.

• Estimates the Gamma distribution in Phase I or uses predefined values.

• Plots the accumulated values of the CUSUM statistic with guaranteed performance.

• Includes control limits and a summary table.

Usage

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

Arguments

Value

A plot displaying the evolution of the upward CUSUM statistic, including:

• The accumulated values of the CUSUM statistic.

• Control limits with guaranteed performance.

• A summary of the parameters used in the control chart.

Examples

# Option 1: Automatically generate data with defined sample sizes
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
                  )

# Option 2: Use custom data
phaseI_data <- rgamma(n = 100, shape = 1, scale = 1)
phaseII_data <- rgamma(n = 200, shape = 1, scale = 1)
plot_GICCup_chart(
                  alpha = 1, beta = 1, beta_ratio = 2, H_delta = 2.9693,
                  H_plus = 6.5081, n_I = 100, n_II = 200,
                  faseI = phaseI_data, faseII = phaseII_data,
                  known_alpha = TRUE
                  )