Statistical process control: Difference between revisions

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{{redirect|SQc|other uses|SQC (disambiguation)}}

{{More citations needed section|date=March 2022}}

[[File:Example Control Chart - DSE Si Etch.jpg|alt=plot showing silicon etch rate versus date, over months, with ±5% and mean values shown.|thumb|297x297px|Simple example of a process control chart, tracking the etch (removal) rate of Silicon in an [[Inductively coupled plasma|ICP]] [[Plasma etching|Plasma Etcher]] at a [[microelectronics]] [[Wafer fabrication|waferfab]].<ref>{{Cite web |last1=Dutra |first1=Noah |last2=John |first2=Demis |title=Process Group - Process Control Data - UCSB Nanofab Wiki |url=https://wiki.nanofab.ucsb.edu/wiki/Process_Group_-_Process_Control_Data |access-date=2024-11-08 |website=wiki.nanofab.ucsb.eduUCSB NanoFab Wiki}}</ref>. Time-series data shows the mean value and ±5% bars. A more sophisticated SPC chart may include "control limit" & "spec limit" % lines to indicate whether/what action should be taken.]]

'''Statistical process control''' ('''SPC''') or '''statistical quality control''' ('''SQC''') is the application of [[statistics|statistical methods]] to monitor and control the quality of a production process. This helps to ensure that the process operates efficiently, producing more specification-conforming products with less waste scrap. SPC can be applied to any process where the "conforming product" (product meeting specifications) output can be measured. Key tools used in SPC include [[run chart]]s, [[control chart]]s, a focus on [[Continuous Improvement Process|continuous improvement]], and [[Design of experiments|the design of experiments]]. An example of a process where SPC is applied is manufacturing lines.

SPC must be practiced in two phases: Thethe first phase is the initial establishment of the process, and the second phase is the regular production use of the process. In the second phase, a decision of the period to be examined must be made, depending upon the change in 5M&E conditions (Man, Machine, Material, Method, Movement, Environment) and wear rate of parts used in the manufacturing process (machine parts, jigs, and fixtures).

An advantage of SPC over other methods of quality control, such as "[[inspection]]," is that it emphasizes early detection and prevention of problems, rather than the correction of problems after they have occurred.

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From an SPC perspective, if the weight of each cereal box varies randomly, some higher and some lower, always within an acceptable range, then the process is considered stable. If the cams and pulleys of the machinery start to wear out, the weights of the cereal box might not be random. The degraded functionality of the cams and pulleys may lead to a non-random linear pattern of increasing cereal box weights. We call this common cause variation. If, however, all the cereal boxes suddenly weighed much more than average because of an unexpected malfunction of the cams and pulleys, this would be considered a special cause variation.

==Industry 4.0 and Artificial Intelligence==

The advent of Industry 4.0 has broadened the scope of statistical process control from traditional manufacturing processes to modern cyber-physical and data-driven systems. The review article of Colosimo et al. (2024)<ref>{{cite journal

|last1=Colosimo

|first1=Bianca M.

|last2=Jones-Farmer

|first2=L. Allison

|last3=Megahed

|first3=Fadel M.

|last4=Paynabar

|first4=Kamran

|last5=Ranjan

|first5=Chetan

|last6=Woodall

|first6=William H.

|title=Statistical process monitoring from Industry 2.0 to Industry 4.0: Insights into research and practice

|journal=Technometrics

|date=October 2024

|volume=66

|issue=4

|pages=507–530

|doi=10.1080/00401706.2024.2327341

|doi-access=free}}</ref> note that SPC now plays a role in monitoring complex, high-dimensional, and often automated processes that characterise Industry 4.0 environments, including the use of machine learning and artificial intelligence (AI) models in production settings.

One emerging line of research applies SPC techniques to artificial neural networks and other machine learning models. Instead of directly monitoring product quality, the focus is on the detection of unreliable behavior of AI systems. For example, nonparametric multivariate control charts have been proposed to track shifts in the distribution of neural network embeddings, allowing detection of nonstationarity and concept drift without requiring labelled data. This enables real-time monitoring of deployed AI systems in industrial contexts<ref>{{cite journal

|last1=Malinovskaya

|first1=Anna

|last2=Mozharovskyi

|first2=Pavlo

|last3=Otto

|first3=Philipp

|title=Statistical process monitoring of artificial neural networks

|journal=Technometrics

|date=January 2024

|volume=66

|issue=1

|pages=104–117

|doi=10.1080/00401706.2023.2239886

|doi-access=free}}</ref>.

==Application==

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==Mathematics of control charts==

Digital control charts use logic-based rules that determine "derived values" which signal the need for correction. For example,

Control charts are based on a time-ordered sequence of observations <math>X_1, X_2, \dots, X_t</math> of a process characteristic. The monitored characteristic can be single observations, averages of samples or batches, ranges, variances, or residuals from a fitted model, depending on the application.

:derived value = last value + [[Mean absolute difference|average absolute difference]] between the last N numbers.

A typical chart consists of:

* a center line (CL) representing the in-control mean, often estimated as

<math>\text{CL} = \bar{X} = \tfrac{1}{n}\sum_{i=1}^n X_i ,</math>

* control limits, usually defined as

<math>\text{UCL} = \mu_0 + k\sigma, \quad \text{LCL} = \mu_0 - k\sigma ,</math>

where <math>\mu_0</math> and <math>\sigma</math> denote the in-control mean and standard deviation, and <math>k</math> is commonly chosen as 3 (the "three-sigma rule").

An observation <math>X_t</math> falling outside the interval <math>[\text{LCL}, \text{UCL}]</math> signals a potential out-of-control condition. Variants such as the cumulative sum ([[CUSUM]]) chart and the exponentially weighted moving average charts ([[EWMA chart]]) are used to improve sensitivity to small or persistent shifts.

In many applications, however, the assumption of independent observations is violated, for example in autocorrelated time series. In such cases, the conventional control limits may produce excessive false alarms. A common solution is to fit a time series model (e.g., ARIMA) and construct a residual control chart, where the model residuals

<math>\hat{\varepsilon}_t = X_t - \hat{X}_t</math>

are monitored instead, or to adjust the control limits accordingly. Because the residuals are designed to be approximately independent and identically distributed, standard control chart theory can be applied to them. Adjusted control limits or model-based approaches are therefore required when processes exhibit dependence.

==See also==

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==Bibliography==

{{refbegin|30em}}

*{{cite book |lastlast1=Barlow |firstfirst1=R.E. |last2=Irony |first2=T.Z. |editor-last=Ghosh |editor-first=M. |editor2-last=Pathak |editor2-first=P.K. |chapter=Foundations of statistical quality control |chapter-url={{GBurl|qddbFWqbl4YC|p=99}} |title=Current Issues in Statistical Inference: Essays in Honor of D. Basu |publisher=Institute of Mathematical Statistics |___location=Hayward, CA |date=1992 |isbn=978-0-940600-24-9 |pages=99–112 }}

*{{cite journal |first=B. |last=Bergman |title=Conceptualistic Pragmatism: A framework for Bayesian analysis? |journal=IIE Transactions |volume=41 |issue= |pages=86–93 |date=2009 |doi=10.1080/07408170802322713 |s2cid=119485220 }}

*{{cite journal |author-link=W. Edwards Deming |first=W. E. |last=Deming |title=On probability as a basis for action |journal=The American Statistician |volume=29 |issue=4 |pages=146–152 |date=1975 |doi=10.1080/00031305.1975.10477402 |pmid=1078437 |s2cid=21043630 }}