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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==
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.
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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