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Scan statistics is one of the most popular approaches for anomaly detection in spatial and network data. In practice, there are numerous sources of uncertainty in the observed data. However, most prior works have overlooked such uncertainty, which can affect the accuracy and inferences of such meth- ods. In this paper, we develop the first systematic approach to incorporating uncertainty in scan statistics. We study two formulations for robust scan statistics, one based on the sam- ple average approximation and the other using a max-min objective. We show that uncertainty significantly increases the computational complexity of these problems. Rigorous algorithms and efficient heuristics for both formulations are developed with justification of theoretical bounds. We evaluate our proposed methods on synthetic and real datasets, and we observe that our methods give significant improvement in the detection power as well as optimization objective, relative to a baseline. 

ABSTRACT The black hole candidate and X-ray binary MAXI J1535−571 was discovered in 2017 September. During the decay of its discovery outburst, and before returning to quiescence, the source underwent at least four reflaring events, with peak luminosities of ∼1035–36 erg s−1 (d/4.1 kpc)2. To investigate the nature of these flares, we analysed a sample of NICER (Neutron star Interior Composition Explorer) observations taken with almost daily cadence. In this work, we present the detailed spectral and timing analysis of the evolution of the four reflares. The higher sensitivity of NICER at lower energies, in comparison with other X-ray detectors, allowed us to constrain the disc component of the spectrum at ∼0.5 keV. We found that during each reflare the source appears to trace out a q-shaped track in the hardness–intensity diagram similar to those observed in black hole binaries during full outbursts. MAXI J1535−571 transits between the hard state (valleys) and softer states (peaks) during these flares. Moreover, the Comptonized component is undetected at the peak of the first reflare, while the disc component is undetected during the valleys. Assuming the most likely distance of 4.1 kpc, we find that the hard-to-soft transitions take place at the lowest luminosities ever observed in a black hole transient, while the soft-to-hard transitions occur at some of the lowest luminosities ever reported for such systems.