CUSUM

In statistical process and quality control , the cumulative sum or CUSUM (from English cumulative sum ) is a sequential analysis method to discover changes in a sequential data series or time series (e.g. change of course or turning points ). In 1954, ES Page defined a quality number , a parameter of a probability distribution; z. B. the expected value. He developed CUSUM as a method to filter out general changes in the parameter from random noise and proposed a limit criterion from which the process should be intervened. A few years later, George Alfred Barnard introduced the V-Mask diagram for the visual detection of changes in . ${\displaystyle \theta }$ ${\displaystyle \theta }$

method

CUSUM considers the cumulative sums of data values and given values : ${\displaystyle x_{n}}$ ${\displaystyle \omega _{n}}$

${\displaystyle S_{0}=0}$

${\displaystyle S_{n}=\max(0,S_{n-1}+x_{n}-\omega _{n})}$

It is important to note that CUSUM is not the mere cumulative sum of the data values, but the cumulative sum of the differences between the data values and . If the value exceeds a specified limit value, then a change has been found. CUSUM not only recognizes sharp changes in data values, but also gradually and continuously over the observation period. Most of the time it is a likelihood function , although this is not specified in Pages articles. ${\displaystyle x_{n}}$ ${\displaystyle \omega _{n}}$ ${\displaystyle S_{n+1}}$ ${\displaystyle \omega }$

Examples

example 1

In the example, both positive and negative cumulative deviations are specified and considered: ${\displaystyle \omega _{n}=5}$

${\displaystyle S_{0}=S_{0}^{+}=S_{0}^{-}=0}$

${\displaystyle S_{n}^{+}=\max(0,S_{n-1}^{+}+x_{n}-\omega _{n})}$

${\displaystyle S_{n}^{-}=-\min(0,-(S_{n-1}^{-}-x_{n}+\omega _{n}))}$

${\displaystyle S_{n}=S_{n-1}-\omega _{n}+x_{n}}$

Graphic representation of the CUSUM calculation.

n	Data value ${\displaystyle x_{n}}$	${\displaystyle x_{n}-\omega _{n}}$	${\displaystyle S_{n}^{+}}$	${\displaystyle S_{n}^{-}}$	CUSUM ${\displaystyle S_{n}}$
0			0	0	0
1	2	−3	0	3	−3
2	4th	−1	0	4th	−4
3	7th	+2	2	2	−2
4th	3	−2	0	4th	−4
5	9	+4	4th	0	0

${\displaystyle S_{n}}$ can also be understood:

1. All data points are mean-adjusted ( ) and ${\displaystyle x_{n}-\omega _{n}}$
2. For each newly created value, all previous mean-adjusted differences are added.

The mean is the likelihood estimate for the expected value of normally distributed data values.

Example 2

The following graphics show the course of , and in different situations: ${\displaystyle S_{n}\,}$ ${\displaystyle S_{n}^{+}}$ ${\displaystyle S_{n}^{-}}$

• left: the mean value of the process does not change
• middle: the mean value of the process slowly increases (in relation to the variation)
• right : the mean value jumps up abruptly after 60 time units

These changes can hardly be seen in the data (above), but not in the course of the , and curves (below). ${\displaystyle S_{n}\,}$ ${\displaystyle S_{n}^{+}}$ ${\displaystyle S_{n}^{-}}$

literature

• Michèle Basseville, Igor V. Nikiforov: Detection of Abrupt Changes: Theory and Application . Prentice-Hall, Englewood Cliffs, NJ 1993, ISBN 0-13-126780-9 ( irisa.fr ).
• Douglas M. Hawkins , David H. Olwell : Cumulative sum charts and charting for quality improvement Springer Verlag , 1998, ISBN 0-387-98365-1 .

Web links

• http://www.medialabinc.net/keyword-details.asp?keyword=CUSUM&courseid=1026

Individual evidence

1. ↑ ES Page: Continuous Inspection Schemes . In: Biometrika . Vol. 41, No. 1/2 (June 1954), ISSN  0006-3444 , pp. 100-115.
2. ^ GA Barnard: Control Charts and Stochastic Processes . In: Journal of the Royal Statistical Society. Series B (Methodological) . Vol. 21, No. 2 (1959), ISSN  0035-9246 , pp. 239-271.