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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. 

The aim of this work is to examine the environmental dependence of K‐band luminosity in the Main, Low Redshift (LOWZ), and Constant Mass (CMASS) galaxy samples of the Sloan Digital Sky Survey Data Release 10 (SDSS DR10). Overall, the environmental dependence of K‐band luminosity is very mild. In two volume‐limited Main samples with the luminosity −20.5 ≤M_r≤ −18.5 and −22.5 ≤M_r≤ −20.5, respectively, we find that the environmental dependence of K‐band luminosity in the luminous volume‐limited Main galaxy sample still can be observed, but this dependence in the faint volume‐limited Main galaxy sample is very weak. Considering the disadvantages of the use of volume‐limited samples, we apply the apparent magnitude‐limited Main galaxy sample, divide the Main galaxy sample of the SDSS DR10 into subsamples with a redshift binning size of Δz=0.01, and then analyze the environmental dependence of K‐band luminosity of subsamples in each redshift bin. Such a method also is used in the LOWZ and CMASS galaxy samples. K‐band luminosity of the Main galaxies shows substantial correlation with the local environment in many redshift bins. In the LOWZ galaxy sample, K‐band luminosity of galaxies shows substantial correlation with the local environment only in the redshift region 0.16 ≤z≤ 0.21, while K‐band luminosity of CMASS galaxy sample is only weakly correlated with the local environment. Statistical results also show that in the apparent magnitude‐limited main galaxy sample and the LOWZ galaxy sample, environmental dependence of K‐band luminosity becomes weak with increasing redshift.