A one‐class peeling method for multivariate outlier detection with applications in phase I SPC

Abstract

In phase I of statistical process control (SPC), control charts are often used as outlier detection methods to assess process stability. Many of these methods require estimation of the covariance matrix, are computationally infeasible, or have not been studied when the dimension of the data, p, is large. We propose the one-class peeling (OCP) method, a flexible framework that combines statistical and machine learning methods to detect multiple outliers in multivariate data. The OCP method can be applied to phase I of SPC, does not require covariance estimation, and is well suited to high-dimensional data sets with a high percentage of outliers. Our empirical evaluation suggests that the OCP method performs well in high dimensions and is computationally more efficient and robust than existing methodologies. We motivate and illustrate the use of the OCP method in a phase I SPC application on a N = 354, p = 1917 dimensional data set containing Wikipedia search results for National Football League (NFL) players, teams, coaches, and managers. The example data set and R functions, OCP.R and OCPLimit.R, to compute the respective OCP distances and thresholds are available in the supplementary materials.

Robustness of the one‐class Peeling method to the Gaussian Kernel Bandwidth

“…The mean estimator, $$\hat{\bm{\mu }}_{\text{OCP}}$$, is determined by averaging the last remaining observations and is similar to the idea of the multivariate median obtained through convex hull peeling 16 . Martinez et al 8 . investigated the robustness of the OCP method using several remaining values of $$n$$ and showed improved robustness with $$n=2$$; however, the user can specify other values of $$n$$ when using the OCP.…”

“…The RMDP method uses only the diagonal elements of the sample covariance matrix and therefore can be applied efficiently in settings where $$p$$ is large. Martinez et al 8 . compared the OCP method with an improved version of the Ro et al 9 .…”

“…In order to construct more flexible boundaries the inner products in Equation () are replaced with a kernel function. The OCP method can be used with many kernel functions, but we consider the implementation using the Gaussian kernel function 8 defined as $$\begin{equation} \begin{aligned} & \text{KS}_G({\bf x}_i,{\bf x}_j) = \exp {{\left(-\frac{\Vert {\bf x}_i - {\bf x}_j \Vert ^2}{s^2}\right)}}, \end{aligned} \end{equation}$$where the parameter $$s$$ must be pre‐specified by the user. Weese et al 14 .…”

A robust meta‐method for interpreting the out‐of‐control signal of multivariate control charts using artificial neural networks

“…These techniques have been widely used in industry for many decades in order to assess the process stability as well as the final product quality. Nowadays, the MSPM techniques have gained popularity in many non‐industrial fields, such as healthcare, finance, and so forth, providing even more opportunities for research 1–3 …”

Monitoring combined multivariate process approaching hybrid kernel‐CUSUM analysis

“…Healy, 2 Woodal and Ncube, 3 Lowry et al, 4 Mason and Young, 5 Xie and Sun, 6 Haq et al, 7 Huwang et al, 8 and several other researchers focused on monitoring a multivariate process using MCUSUM and MEWMA methods. Some researchers such as Martinez et al, 9 Sabahno et al, 10 Haq and Khoo, 11 Chew et al, 12 and Xiang et al 13 contributed monitoring a multivariate environment using different methods. In some multivariate cases one may monitor a random vector consists of several attribute characteristics.…”