[Paper Review] Nonparametric Adaptive CUSUM Chart for Detecting Arbitrary Distributional Changes

Jun Li|arXiv (Cornell University)|Dec 14, 2017

Advanced Statistical Process Monitoring12 references6 citations

TL;DR

This paper proposes a nonparametric adaptive CUSUM chart that detects arbitrary distributional changes without tuning parameters, using data categorization and self-starting sequential ranks. It enables fast computation, automatic post-signal diagnostics, and outperforms existing nonparametric charts across diverse distributional shifts.

ABSTRACT

Nonparametric control charts that can detect arbitrary distributional changes are highly desirable due to their flexibility to adapt to different distributional assumptions and distributional changes. However, most of such control charts in the literature either involve some tuning parameter, which needs to be pre-specified, or involve intensive computation. In this paper, we propose a new nonparametric adaptive CUSUM chart for detecting arbitrary distributional changes. The proposed control chart does not depend on any tuning parameter and is efficient in computation. Its self-starting nature makes the proposed control chart applicable to situations where no sufficiently large reference data are available. Our proposed control chart also has a built-in post-signal diagnostics function that can identify what kind of distributional changes have occurred after an alarm. Our simulation study and real data analysis show that the proposed control chart performs well across a broad range of settings, and compares favorably with existing nonparametric control charts.

Motivation & Objective

• To develop a nonparametric control chart that detects arbitrary distributional changes without requiring pre-specified tuning parameters.
• To overcome the computational intensity of existing change-point detection and CUSUM-based nonparametric charts.
• To enable application in settings with limited or no reference data by ensuring self-starting capability.
• To provide built-in diagnostics to identify the type of distributional change after an alarm is triggered.
• To improve detection efficiency over rank-based and categorized data methods that lose ordering information.

Proposed method

• Uses data categorization to transform continuous process observations into categorical data based on sequential ranks.
• Applies a CUSUM statistic to monitor the resulting categorical data, with separate statistics for location and scale shifts.
• Employs a self-starting mechanism using the sequential rank of each new observation within the current window of size $ m+t $, ensuring no need for large reference data.
• Defines two primary CUSUM statistics: $ ilde{S}^{(1+)}_t $, $ ilde{S}^{(1-)}_t $ for positive and negative location shifts, and $ ilde{S}^{(2+)}_t $, $ ilde{S}^{(2-)}_t $ for scale increases and decreases.
• Uses the probability integral transform to show that the categorized data follow a multinomial distribution with equal probabilities under in-control conditions, ensuring validity and independence across time points.

Experimental results

Research questions

• RQ1 Can a nonparametric CUSUM chart detect arbitrary distributional changes without requiring tuning parameters or large reference datasets?
• RQ2 How does the proposed chart's performance compare to existing nonparametric control charts across various types of distributional shifts?
• RQ3 Can the chart automatically diagnose the type of distributional change (e.g., location vs. scale shift) after triggering an alarm?
• RQ4 Does the self-starting nature of the chart maintain in-control run length stability with minimal initial data?
• RQ5 How does the loss of ordering information in categorization affect detection power compared to rank-based methods?

Key findings

• The proposed control chart is free of tuning parameters and does not require a large reference data set, making it self-starting and practical for real-world applications with limited historical data.
• The chart maintains stable in-control run lengths and exhibits low variability in performance across diverse distributional changes, unlike prior CUSUM methods with small tuning parameters.
• Simulation studies show the proposed chart achieves the best overall detection performance compared to existing nonparametric control charts, including Zou and Tsung’s EWMA and Qiu and Li’s CUSUM with tuning parameters.
• The built-in diagnostics function successfully identifies the type of distributional change (e.g., location shift, scale increase) after an alarm, enhancing interpretability.
• The method is computationally efficient, avoiding the intensive computation of change-point detection frameworks that evaluate all possible change-point scenarios at each time point.
• Theoretical proof confirms that the categorized data follow a multinomial distribution with equal probabilities under in-control conditions, ensuring the validity of the CUSUM statistics.