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Monitoring Covariance in Multichannel Profiles via Functional Graphical Models

Christian Capezza, Davide Forcina, Antonio Lepore, Biagio Palumbo

TL;DR

A multichannel profile covariance (MPC) control chart based on functional graphical models that provide an interpretable representation of conditional dependencies between profiles is proposed and compared with state-of-the-art competitors.

Abstract

Most statistical process monitoring methods for multichannel profiles focus solely on the mean and are almost ineffective when changes involve the covariance structure. Although it is known to be crucial, covariance monitoring requires estimating a much larger number of parameters, which may shift in a subtle and sparse fashion. That is, an out-of-control (OC) state may manifest with small deviations and affect only a very limited subset of these parameters. To address these difficulties, we propose a multichannel profile covariance (MPC) control chart based on functional graphical models that provide an interpretable representation of conditional dependencies between profiles. A nonparametric combination of the likelihood-ratio tests corresponding to different sparsity levels is then used to draw an overall inference and signal whether an OC state may have occurred. Between-profile relationships that are likely to have shifted are naturally identified at no additional computational cost. An extensive Monte Carlo simulation study compares the MPC control chart with state-of-the-art competitors, and a case study on monitoring multichannel temperature profiles in a roasting machine illustrates its practical applicability.

Monitoring Covariance in Multichannel Profiles via Functional Graphical Models

TL;DR

A multichannel profile covariance (MPC) control chart based on functional graphical models that provide an interpretable representation of conditional dependencies between profiles is proposed and compared with state-of-the-art competitors.

Abstract

Most statistical process monitoring methods for multichannel profiles focus solely on the mean and are almost ineffective when changes involve the covariance structure. Although it is known to be crucial, covariance monitoring requires estimating a much larger number of parameters, which may shift in a subtle and sparse fashion. That is, an out-of-control (OC) state may manifest with small deviations and affect only a very limited subset of these parameters. To address these difficulties, we propose a multichannel profile covariance (MPC) control chart based on functional graphical models that provide an interpretable representation of conditional dependencies between profiles. A nonparametric combination of the likelihood-ratio tests corresponding to different sparsity levels is then used to draw an overall inference and signal whether an OC state may have occurred. Between-profile relationships that are likely to have shifted are naturally identified at no additional computational cost. An extensive Monte Carlo simulation study compares the MPC control chart with state-of-the-art competitors, and a case study on monitoring multichannel temperature profiles in a roasting machine illustrates its practical applicability.
Paper Structure (9 sections, 30 equations, 5 figures, 1 table, 1 algorithm)

This paper contains 9 sections, 30 equations, 5 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Methodology
  3. Modeling Multichannel Profile Relationships
  4. Monitoring Procedure
  5. Post-signal Diagnostics
  6. Implementation Details
  7. Simulation Study
  8. Case Study
  9. Conclusions

Figures (5)

  • Figure 1: Mean $ARL$ achieved in Phase II by MPC, HGM, and Ren for Model I with $p = 10$ functional variables as a function of SL.
  • Figure 2: Mean $ARL$ achieved in Phase II by MPC, HGM, and Ren for Model I and $p = 30$ functional variables as a function of SL.
  • Figure 3: Estimated IC graph (left) and norms of the blocks of the estimated IC precision matrix (right) in the case study.
  • Figure 4: A sample of the original IC temperature data recorded from 2016-02-06 18:05:00 to 2016-02-07 22:05:00 (left panel). Original OC temperature data recorded from 2017-08-27 09:05:00 to 2017-08-28 13:05:00 (right panel), monitored in Phase II of the case study. The grey dashed vertical lines separate the hourly profiles for each product, while the red and green vertical bands highlight the stopping time and the change point identified by the proposed method, respectively.
  • Figure 5: Graph of the significantly shifted conditional relationships (left) and matrix of the significant p-values (right) obtained with the BH procedure in the case study.