Optimal network online change point localisation
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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 NP-hard algorithms and polynomial-time algorithms, i.e., ≳log(1/α) max{r^2/n, 1}/κ_0^2 n ρ, ≲log(Δ/α) max{r^2/n, log(r)}/κ_0^2 n ρ, , ≲log(Δ/α) r/κ_0^2 n ρ, , where κ_0, n, ρ, r and α are the normalised jump size, network size, entrywise sparsity, rank sparsity and the overall Type-I error upper bound. All the model parameters are allowed to vary as Δ, the location of the change point, diverges. The polynomial-time algorithms are novel procedures that we propose in this paper, designed for quick detection under two different forms of Type-I 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 Type-I error control, our proposed approaches outperform state-of-the-art methods.
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