Optimal network online change point localisation
Abstract
We study the problem of online network change point detection. In this setting, a collection of independent Bernoulli networks is collected sequentially, and the underlying distributions change when a change point occurs. The goal is to detect the change point as quickly as possible, if it exists, subject to a constraint on the number or probability of false alarms. In this paper, on the detection delay, we establish a minimax lower bound and two upper bounds based on NPhard algorithms and polynomialtime algorithms, i.e., \[ \mbox{detection delay} \begin{cases} \gtrsim \log(1/\alpha) \frac{\max\{r^2/n, \, 1\}}{\kappa_0^2 n \rho},\\ \lesssim \log(\Delta/\alpha) \frac{\max\{r^2/n, \, \log(r)\}}{\kappa_0^2 n \rho}, & \mbox{with NPhard algorithms},\\ \lesssim \log(\Delta/\alpha) \frac{r}{\kappa_0^2 n \rho}, & \mbox{with polynomialtime algorithms}, \end{cases} \] where $\kappa_0, n, \rho, r$ and $\alpha$ are the normalised jump size, network size, entrywise sparsity, rank sparsity and the overall TypeI error upper bound. All the model parameters are allowed to vary as $\Delta$, the location of the change point, diverges. The polynomialtime algorithms are novel procedures that we propose in this paper, designed for quick detection under two different forms of TypeI error control. The first is based on controlling the overall probability of a false alarm when there are no change points, and the second is based on specifying a lower bound on the expected time of the first false alarm. Extensive experiments show that, under different scenarios and the aforementioned forms of TypeI error control, our proposed approaches outperform stateoftheart methods.
 Publication:

arXiv eprints
 Pub Date:
 January 2021
 arXiv:
 arXiv:2101.05477
 Bibcode:
 2021arXiv210105477Y
 Keywords:

 Mathematics  Statistics Theory;
 Computer Science  Machine Learning