CUSUM multi-chart for detecting unknown abrupt changes under finite measure space for network observation sequences

Abstract

This paper considers the model-based change-point problem with an unknown abrupt change for large-scale network observation sequences. To avoid the difficulty of calculating the normalization coefficients that let the axioms of probability hold, such as $Z(\mathrm{\Lambda })$ in the Exponential Random Graphical Model (ERGM), we present the measure ratio statistics to replace the likelihood ratio statistics. Since the parameter difference reflecting the abrupt change for the network observation sequences is unknown, we first select the dimensions where the parameter difference exists through the $L_1$-norm penalized maximum likelihood estimation process, then propose a CUSUM multi-chart scheme based on the selected dimensions. Moreover, an optimal design of the CUSUM multi-chart is given when $AR{L}_0$ (in-control Average Run length) is large. Two examples are used to illustrate the related theoretical results.

Keywords:

• CUSUM multi-chart
• optimal design
• finite measure space
• penalized maximum likelihood estimation

MSC 2010 subject classifications:

• Primary 62L10
• Secondary 62L15

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

Note: The first line of each cell is a CUSUM chart with $p_1=0.70,p_2=0.75$, the second line of each cell is the stochastic CUSUM multi-chart, the last line is the asymptotic optimal CUSUM multi-chart.

Note: The first line of each cell is a CUSUM chart with $p_1=0.70,p_2=0.75$, the second line of each cell is the stochastic CUSUM multi-chart, the last line is the asymptotic optimal CUSUM multi-chart.

Note: The first line of each cell is a CUSUM with $p_1=0.70,p_2=0.75$, the second line of each cell is the stochastic CUSUM multi-chart, the last line is the asymptotic optimal CUSUM multi-chart.

Note: The first line of each cell is a CUSUM with $p_1=0.70,p_2=0.75$, the second line of each cell is the stochastic CUSUM multi-chart, the last line is the asymptotic optimal CUSUM multi-chart.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [grant number 11531001].