Sorted L-One Penalized Estimation (SLOPE) has shown the nice theoretical property as well as empirical behavior recently on the false discovery rate (FDR) control of high-dimensional feature selection by adaptively imposing the non-increasing sequence of tuning parameters on the sorted L1 penalties. This paper goes beyond the previous concern limited to the FDR control by considering the stepdown-based SLOPE in order to control the probability of k or more false rejections (k-FWER) and the false discovery proportion (FDP). Two new SLOPEs, called k-SLOPE and F-SLOPE, are proposed to realize k-FWER and FDP control respectively, where the stepdown procedure is injected into the SLOPE scheme. For the proposed stepdown SLOPEs, we establish their theoretical guarantees on controlling k-FWER and FDP under the orthogonal design setting, and also provide an intuitive guideline for the choice of regularization parameter sequence in much general setting. Empirical evaluations on simulated data validate the effectiveness of our approaches on controlled feature selection and support our theoretical findings.

Subway security check systems (SSCS), which play an important role for smooth operation of subways of metropolises, exhibit a high complexity and randomness due to their multiple queues, mixed services, and heterogeneous passengers. Traditional approaches for evaluating configurations of SSCS either assume a single-queueing model or ignore various attributes of passengers for the sake of simplicity of models, leading to unsatisfactory results. In this paper, we propose a robust framework for evaluating the performance of configurations of SSCS. By considering passenger attributes (e.g., with or without bags) and special services (e.g., special check, additional search), we regard SSCS as two subsystems each of which is modeled as an iterative probabilistic formula, and introduce a hybrid queueing model to evaluate the waiting time of each passenger in SSCS. The parameters of the model are calibrated and validated using the field observation data collected in 24 subway stations of Beijing, which serve as the input of simulation studies. We conduct two simulation studies to evaluate three indices of SSCS: passengers’ waiting time in SSCS,$K$-systematic time of SSCS, and density flow map, and draw three conclusions. (a) 1X2D, 2X3D, 2X4D configurations are suitable for small, medium and large passenger flows, respectively, where$m\text{X}n\text{D}$denotes the configuration of$m$X-ray machines and$n$detector doors; (b) acceptable efficiency of SSCS is achieved when the ratio between the numbers of X-ray machines and detector doors is in the range of 1:2-1:1, and the efficiency achieves highest at the ratio 1:2; (c) Increasing additional detector doors for no-bag channels can improve the efficiency of SSCS. However, switching existing doors to no-bag channels leads to opposite results. Our findings are beneficial to improving both the capacity and efficiency of SSCS, as well as allocating security check facilities reasonably.