CUSUM change-point detection — how does it work?

hard

Answer

$S_{t}\; = \;\operatorname{max}(0, \;S_{t - 1}\; + \;(x_{t}\; - \;{\mu})\; - \;k)$
$S_{t}\; > \;\mathrm{threshold}$
$Quickly detects small persistent shifts.$
$Bidirectional variant.$
$Tune k, h to control false-alarm rate + detection latency.$
$Standard in manufacturing SPC, network monitoring, sensor drift.$
$Bayesian alternatives (BOCPD) provide posterior over change points.$

Check yourself — multiple choice

Random
$S_{t}\; = \;\operatorname{max}(0, \;S_{t - 1}\; + \;(x_{t}\; - \;{\mu})\; - \;k)$ ; alarm when $S_{t}\; > \;h$ ; fast detection of small persistent shifts; standard in SPC / network monitoring
Same as z-score
Not real

CUSUM: cumulative sum with reset; fast change-point detection.

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