A Comparative Study of Self-Starting CUSUM Control Charts for Detecting Location Shifts in Normally Distributed Data

Khái niệm cốt lõi

Self-starting CUSUM control charts, specifically the Bayesian Predictive Ratio CUSUM (PRC) and the frequentist Self-Starting CUSUM (SSC), demonstrate comparable performance in detecting location shifts in normally distributed data, with the effectiveness of each method influenced by factors like the magnitude of the shift, the timing of the shift, and the availability of prior information.

Tóm tắt

• Bibliographic Information: Bourazas, K. (2024). A comparative study of self-starting CUSUM control charts for location shifts. arXiv preprint arXiv:2410.12736v1.

• Research Objective: This paper compares the performance of two self-starting control charts, the frequentist Self-Starting CUSUM (SSC) and the Bayesian Predictive Ratio CUSUM (PRC), in detecting location shifts in normally distributed data.

• Methodology: The study employs an extensive simulation study to evaluate the performance of SSC and PRC under various scenarios involving changes in the mean of normally distributed data. The performance is assessed using the Conditional Expected Delay (CED) metric, which measures the average delay in detecting a shift after it occurs. The study considers different magnitudes of shifts, change point positions, and design parameter values for both methods. Additionally, a prior sensitivity analysis is conducted for PRC using a non-informative reference prior and a weakly informative prior.

• Key Findings: Both SSC and PRC exhibit effective detection of larger shifts compared to smaller shifts. Their performance improves with a larger IC data history, meaning when the change point occurs later in the process. The prior information in PRCi positively impacts detection, especially with limited data points early in the process. However, as the IC data volume increases, the performance of PRCi converges with PRCn and SSC, diminishing the impact of the prior. PRCn and SSC demonstrate similar performance overall, with SSC showing a slight advantage for small design parameter values and early change points, while PRCn performs slightly better for larger design parameter values.

• Main Conclusions: The study concludes that both SSC and PRC are viable options for online change point detection in the mean of univariate Normal data without a Phase I calibration phase. The choice between the two methods might depend on factors like the expected magnitude of shifts, the availability of prior information, and the desired sensitivity to early change points.

• Significance: This research contributes to the field of Statistical Process Control and Monitoring (SPC/M) by providing a comparative analysis of two prominent self-starting control chart methods. The findings offer valuable insights for practitioners and researchers in selecting and implementing appropriate methods for online change point detection, particularly in scenarios where a Phase I calibration phase is impractical or infeasible.

• Limitations and Future Research: The study focuses on univariate Normal data and specific types of shifts in the mean. Future research could explore the performance of these methods for other data distributions, shift types, or multivariate data. Additionally, investigating the robustness of these methods to deviations from normality assumptions and exploring adaptive schemes for dynamically adjusting design parameters could be valuable extensions of this research.

Thống kê

The study uses an Average Run Length (ARL0) of 370 for both methods in the in-control state.
Four different shift sizes (δ) are considered: 0.5, 1, 1.5, and 2.
Ten different change point locations (τ) are evaluated, ranging from 11 to 101.
Three different design parameter (k) values are used for each method: kSSC = 0.25, 0.375, 0.5 and kPRC = 0.5, 0.75, 1.

Trích dẫn

Thông tin chi tiết chính được chắt lọc từ

A comparative study of self-starting CUSUM control charts for location shifts

by Konstantinos... lúc arxiv.org 10-17-2024

https://arxiv.org/pdf/2410.12736.pdf

Yêu cầu sâu hơn

How might these self-starting control chart methods be adapted for use in monitoring non-stationary processes, where the process parameters themselves may be subject to gradual drift or change over time?

Adapting self-starting control charts like SSC (Self-Starting CUSUM) and PRC (Predictive Ratio CUSUM) for non-stationary processes requires addressing the challenge of distinguishing between common-cause variation due to process drift and special-cause variation signaling a significant change point. Here are some potential adaptation strategies:

Dynamic Parameter Updating: Instead of assuming fixed parameters, implement mechanisms to update the estimated mean and variance over time. This could involve:

Moving Window Approaches: Use a sliding window of recent observations to calculate parameter estimates, effectively "forgetting" older data that may no longer be representative of the current process state.
Adaptive Forgetting Factors: Incorporate forgetting factors into the parameter estimation process, giving more weight to recent observations and gradually discounting older data. Exponential smoothing methods are commonly used for this purpose.

Trend and Seasonality Adjustments: If the non-stationarity exhibits predictable patterns like trends or seasonality, incorporate these into the control chart calculations. This might involve:

Differencing: Calculate control limits based on the differences between consecutive observations to remove trend components.
Seasonal Decomposition: Decompose the data into trend, seasonal, and residual components, and apply control chart monitoring to the residual component after adjusting for trend and seasonality.

Change Point Detection within Control Limits: Instead of relying solely on exceeding control limits, develop methods to detect significant changes within the control limits themselves. This could involve:

Run Rules: Implement run rules that trigger an alarm based on patterns of consecutive points near the control limits, even if no single point exceeds them.
Cumulative Sum (CUSUM) for Drift: Adapt CUSUM charts to be sensitive to gradual drifts in the process mean, signaling an alarm when the cumulative sum of deviations from a target value exceeds a threshold.

Bayesian Dynamic Linear Models: For more complex non-stationary behaviors, employ Bayesian dynamic linear models (DLMs) to model the time-varying process parameters. DLMs provide a flexible framework for incorporating prior information, handling missing data, and making predictions about future process behavior.

These adaptations aim to make self-starting control charts more robust to the presence of non-stationarity, enabling them to effectively detect significant change points while accommodating gradual drifts or changes in the underlying process parameters.

Could the performance differences between SSC and PRC be attributed to the specific assumptions made about the underlying data distribution, and would these differences persist if alternative distributional assumptions were considered?

Yes, the performance differences between SSC and PRC can be partly attributed to the assumption of normality. Here's why:

SSC's Reliance on Normality: SSC relies on transforming the data to follow a standard normal distribution using the sample mean and variance. This transformation is optimal under normality but might not be efficient for other distributions. Deviations from normality could lead to inaccurate control limits and affect the false alarm rate and detection power.

PRC's Bayesian Framework and Prior Information: PRC, being a Bayesian method, incorporates prior information about the process parameters. This prior information can be particularly beneficial when the data deviates from normality, as it provides additional guidance for parameter estimation. The choice of prior distribution can influence PRC's performance for different data distributions.
Impact of Alternative Distributions:
If alternative distributional assumptions were considered, the performance differences between SSC and PRC might persist or even become more pronounced.

Heavy-Tailed Distributions: For heavy-tailed distributions (e.g., t-distribution), SSC might exhibit an increased rate of false alarms due to the higher probability of extreme values. PRC, with an appropriate heavy-tailed prior, could potentially adapt better and maintain a more accurate false alarm rate.

Skewed Distributions: For skewed distributions (e.g., gamma distribution), both SSC and PRC might require modifications. SSC's reliance on symmetry might lead to biased control limits. PRC might need a prior distribution that reflects the skewness of the data.
Addressing Distributional Assumptions:

Transformations: Data transformations (e.g., Box-Cox) can be applied to make the data approximately normal before applying SSC. However, finding suitable transformations for different distributions can be challenging.

Nonparametric Methods: Consider nonparametric control charts (e.g., EWMA charts based on ranks) that do not rely on specific distributional assumptions.

Generalized Bayesian Models: Explore Bayesian models with more flexible likelihood functions that can accommodate various data distributions.
In conclusion, while both SSC and PRC can be effective for monitoring processes, their performance can be sensitive to deviations from normality. When dealing with non-normal data, it's crucial to consider the distributional assumptions and explore appropriate adaptations or alternative methods to ensure reliable monitoring and change point detection.

Considering the increasing prevalence of high-dimensional data in various fields, how can the principles of self-starting control charts be extended and applied to effectively monitor and detect change points in multivariate or high-dimensional data streams?

Extending self-starting control charts to high-dimensional data presents exciting opportunities and challenges. Here are some key approaches:

Dimensionality Reduction:

Principal Component Analysis (PCA): Project the high-dimensional data onto a lower-dimensional subspace spanned by the principal components capturing the most significant variation. Monitor these principal components using univariate or multivariate control charts.
Partial Least Squares (PLS): Similar to PCA, but PLS considers the relationship between the process variables and a response variable (if available) to identify latent variables for monitoring.

Multivariate Control Charts:

Hotelling's T-squared Chart: Generalizes the univariate Shewhart chart to monitor the mean vector of multivariate data. It considers the covariance structure of the variables.
Multivariate EWMA (MEWMA) Chart: Extends the univariate EWMA chart to monitor shifts in the mean vector, incorporating past information and being sensitive to smaller shifts.

Sparsity and Regularization:

LASSO (Least Absolute Shrinkage and Selection Operator): Incorporate LASSO regularization into the control chart estimation process to encourage sparsity, effectively selecting a subset of relevant variables for monitoring and reducing dimensionality.
Elastic Net: Combines LASSO and Ridge regularization to handle situations where there are highly correlated variables.

Machine Learning-Based Approaches:

One-Class Support Vector Machines (OCSVMs): Train an OCSVM on a dataset representing the in-control state. The OCSVM can then identify deviations from this in-control state as potential change points.
Autoencoders: Use deep learning-based autoencoders to learn a compressed representation of the in-control data. Monitor the reconstruction error of new data points; a significant increase in reconstruction error could indicate a change point.

Bayesian Nonparametrics:

Dirichlet Process Mixtures: Employ Dirichlet process mixtures to model the data distribution nonparametrically, allowing for flexibility in handling high-dimensional data with complex structures.

Challenges and Considerations:

Interpretability: Dimensionality reduction techniques and machine learning models can be complex, making it challenging to interpret the results and identify the specific variables contributing to a change point.
Computational Cost: High-dimensional data analysis can be computationally intensive, especially for complex models and large datasets. Efficient algorithms and computational resources are essential.
Data Sparsity: High-dimensional data often suffers from sparsity, where many variables have missing or zero values. Handling missing data appropriately is crucial for reliable monitoring.
By leveraging these approaches and addressing the associated challenges, the principles of self-starting control charts can be effectively extended to monitor and detect change points in high-dimensional data streams, enabling proactive process monitoring and anomaly detection in various domains.