CUSUM

In the statistical process and quality control, the cumulative sum or CUSUM is (of English. "Cumulative sum" ) is a sequential analysis method for the detection of changes in a sequential data set or time series (eg exchange rate changes or turning points ). ES Page 1954 defined a quality number, a parameter of a probability distribution; For example, the expectation value. He developed CUSUM as a method to general changes to the parameter of random noise to filter out and hit a limit criterion ago from should be engaged in the process. A few years later, George Alfred Barnard, the V- Mask chart in front for visual detection of changes.

Method

CUSUM considered the cumulative sums of data values ​​and pre- enclosed values ​​:

It is important to note that the mere CUSUM cumulative sum of the data values ​​is not, but the cumulative sum of the differences between the data values ​​and. Exceeds the value of a pre- enclosed limit, there is a change found. So CUSUM not only detects sharp data value changes but also gradual and continuous over the period under consideration. Most of them are in a likelihood function, although this is not specified in item pages.

Examples

Example 1

In the example will be given, and both positive and considered negative cumulative differences:

Can also be considered:

The mean is the Likelihoodschätzung for the expectation values ​​of normally distributed data.

Example 2

The following graphs show the course of, and in different situations:

• Left: the mean of the process does not change
• Middle: the mean of the process is slowly larger (relative to the scattering)
• Right: the mean abruptly jumps to the top after 60 time units

In the data ( above), these changes are hard to recognize, but not in the course of, and curves (bottom).