FarmTest: Factor-Adjusted Robust Multiple Testing with Approximate False Discovery Control

Large-scale multiple testing with correlated and heavy-tailed data arises in a wide range of research areas from genomics, medical imaging to finance. Conventional methods for estimating the false discovery proportion (FDP) often ignore the effect of heavy-tailedness and the dependence structure among test statistics, and thus may lead to inefficient or even inconsistent estimation. Also, the commonly imposed joint normality assumption is arguably too stringent for many applications. To address these challenges, in this paper we propose a Factor-Adjusted Robust Multiple Testing (FarmTest) procedure for large-scale simultaneous inference with control of the false discovery proportion. We demonstrate that robust factor adjustments are extremely important in both controlling the FDP and improving the power. We identify general conditions under which the proposed method produces consistent estimate of the FDP. As a byproduct that is of independent interest, we establish an exponential-type deviation inequality for a robust $U$-type covariance estimator under the spectral norm. Extensive numerical experiments demonstrate the advantage of the proposed method over several state-of-the-art methods especially when the data are generated from heavy-tailed distributions. The proposed procedures are implemented in the R-package FarmTest.Comment: 52 pages, 9 figure

User-Friendly Covariance Estimation for Heavy-Tailed Distributions

We offer a survey of recent results on covariance estimation for heavy-tailed distributions. By unifying ideas scattered in the literature, we propose user-friendly methods that facilitate practical implementation. Specifically, we introduce element-wise and spectrum-wise truncation operators, as well as their $M$-estimator counterparts, to robustify the sample covariance matrix. Different from the classical notion of robustness that is characterized by the breakdown property, we focus on the tail robustness which is evidenced by the connection between nonasymptotic deviation and confidence level. The key observation is that the estimators needs to adapt to the sample size, dimensionality of the data and the noise level to achieve optimal tradeoff between bias and robustness. Furthermore, to facilitate their practical use, we propose data-driven procedures that automatically calibrate the tuning parameters. We demonstrate their applications to a series of structured models in high dimensions, including the bandable and low-rank covariance matrices and sparse precision matrices. Numerical studies lend strong support to the proposed methods.Comment: 56 pages, 2 figure

Research on intelligent fault diagnosis of mechanical equipment based on sparse deep neural networks

In the big data background, the accuracy of fault diagnosis and recognition has been difficult to be improved. The deep neural network was used to recognize the diagnosis rate of the bearing with four kinds of conditions and compared with traditional BP neural network, genetic neural network and particle swarm neural network. Results showed that the diagnosis accuracy and convergence rate of the deep neural network were obviously higher than those of other models. Fault diagnosis rates with different sample sizes and training sample proportions were then studied to compare with the latest reported methods. Results showed that fault diagnosis had a good stability using deep neural networks. Vibration accelerations of the bearing with different fault diameters and excitation loads were extracted. The deep neural network was used to recognize these faults. Diagnosis accuracy was very high. In particular, the fault diagnosis rate was 98 % when signal features of vibration accelerations were very obvious, which indicated that using deep neural network was effective in diagnosing and recognizing different types of faults. Finally, the deep neural network was used to conduct fault diagnosis for the gearbox of wind turbines and compared with the other models to present that it would work well in the industrial environment

Dark Matter Spike surrounding Supermassive Black Holes Binary and the nanohertz Stochastic Gravitational Wave Background

Recently, the NANOGrav, PPTA, EPTA and CPTA collaborations reported compelling evidence of the existence of the Stochastic Gravitational-Wave Background (SGWB). The amplitude and spectrum of this inferred gravitational-wave background align closely with the astrophysical predictions for a signal originating from the population of supermassive black-hole binaries. In light of these findings, we explore the possibility to detect dark matter spikes surrounding massive black holes, which could potentially impact the gravitational-wave waveform and modulate the SGWB. We demonstrate that the SMBH binary evolution induced by the combined effects of GW radiation and the dynamical friction of the dark matter spike exhibits detectable manifestations within the nHz frequency range of the SGWB.Comment: 5 pages, 1 figure. arXiv admin note: text overlap with arXiv:1408.3534 by other author