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Lung adenocarcinoma (LUAD) is characterized by substantial genetic heterogeneity, posing challenges in identifying reliable biomarkers for improved diagnosis and treatment. Tumor Mutational Burden (TMB) has traditionally been regarded as a predictive biomarker, given its association with immune response and treatment efficacy. In this study, we treated TMB as a response variable to identify genes highly correlated with it, aiming to understand its genetic drivers. We conducted a thorough investigation of recent feature selection methods through extensive simulations, selecting PC-Screen, DC-SIS, and WD-Screen as top performers. These methods handle multi-omics structures effectively, and can accommodate both categorical and continuous data types at the same time for each gene. Using data from The Cancer Genome Atlas (TCGA) via cBioPortal, we combined copy number alteration (CNA), mRNA expression and DNA methylation data as multi-omics predictors and applied these methods, selecting genes consistently identified across all three methods. 13 common genes were identified, including HSD17B4, PCBD2, which show strong associations with TMB. Our multi-omics strategy and robust feature selection approach provide insights into the genetic determinants of TMB, with implications for targeted LUAD therapies.

Infrared small target detection (IRSTD) is widely used in civilian and military applications. However, IRSTD encounters several challenges, including the tendency for small and dim targets to be obscured by complex backgrounds. To address this issue, we propose the Gradient Network (GaNet), which aims to extract and preserve edge and gradient information of small targets. GaNet employs the Gradient Transformer (GradFormer) module, simulating central difference convolutions (CDC) to extract and integrate gradient features with deeper features. Furthermore, we propose a global feature extraction model (GFEM) that offers a comprehensive perspective to prevent the network from focusing solely on details while neglecting the background information. We compare the network with state-of-the-art (SOTA) approaches, and the results demonstrate that our method performs effectively. Our source code is available at https://github.com/greekinRoma/Gradient-Transformer.

Genome-Wide Association Studies (GWAS) face unique challenges in the era of big genomics data, particularly when dealing with ultra-high-dimensional datasets where the number of genetic features significantly exceeds the available samples. This paper introduces an extension to the feature selection methodology proposed by Mirzaei et al. (2020), specifically tailored to tackle the intricacies associated with ultra-high-dimensional GWAS data. Our extended approach enhances the original method by introducing a Frobenius norm penalty into the student network, augmenting its capacity to adapt to scenarios characterized by a multitude of features and limited samples. Operating seamlessly in both supervised and unsupervised settings, our method employs two key neural networks. The first leverages an autoencoder or supervised autoencoder for dimension reduction, extracting salient features from the ultra-high-dimensional genomic data. The second network, a regularized feed-forward model with a single hidden layer, is designed for precise feature selection. The introduction of the Frobenius norm penalty in the student network significantly boosts the method's resilience to the challenges posed by ultra-high-dimensional GWAS datasets. Experimental results showcase the efficacy of our approach in feature selection for GWAS data. The method not only handles the inherent complexities of ultra-high-dimensional settings but also demonstrates superior adaptability to the nuanced structures present in genomics data. The flexibility and versatility of our proposed methodology are underscored by its successful performance across a spectrum of experiments.

This article studies the derivatives in models that flexibly characterize the relationship between a response variable and multiple predictors, with goals of providing both accurate estimation and inference procedures for hypothesis testing. In the setting of tensor product reproducing spaces for nonparametric multivariate functions, we propose a plug-in kernel ridge regression estimator to estimate the derivatives of the underlying multivariate regression function under the smoothing spline ANOVA model. This estimator has an analytical form, making it simple to implement in practice. We first establish $L_\infty$ and $L_2$ convergence rates of the proposed estimator under general random designs. For derivatives with some selected interesting orders, we provide an in-depth analysis establishing the minimax lower bound, which matches the $L_2$ convergence rate. Additionally, motivated by a wide range of applications, we propose a hypothesis testing procedure to examine whether a derivative is zero. Theoretical results demonstrate that the proposed testing procedure achieves the correct size under the null hypothesis and is asymptotically powerful under local alternatives. For ease of use, we also develop an associated bootstrap algorithm to construct the rejection region and calculate the p-value, and the consistency of the proposed algorithm is established. Simulation studies using synthetic data and an application to a real-world dataset confirm the effectiveness of our methods.

Clinical trials often involve the assessment of multiple endpoints to comprehensively evaluate the efficacy and safety of interventions. In the work, we consider a global nonparametric testing procedure based on multivariate rank for the analysis of multiple endpoints in clinical trials. Unlike other existing approaches that rely on pairwise comparisons for each individual endpoint, the proposed method directly incorporates the multivariate ranks of the observations. By considering the joint ranking of all endpoints, the proposed approach provides robustness against diverse data distributions and censoring mechanisms commonly encountered in clinical trials. Through extensive simulations, we demonstrate the superior performance of the multivariate rank-based approach in controlling type I error and achieving higher power compared to existing rank-based methods. The simulations illustrate the advantages of leveraging multivariate ranks and highlight the robustness of the approach in various settings. The proposed method offers an effective tool for the analysis of multiple endpoints in clinical trials, enhancing the reliability and efficiency of outcome evaluations.

A bioequivalence study is a type of clinical trial designed to compare the biological equivalence of two different formulations of a drug. Such studies are typically conducted in controlled clinical settings with human subjects, who are randomly assigned to receive two formulations. The two formulations are then compared with respect to their pharmacokinetic profiles, which encompass the absorption, distribution, metabolism, and elimination of the drug. Under the guidance from Food and Drug Administration (FDA), for a size-$\alpha$ bioequivalence test, the standard approach is to construct a $100(1-2\alpha)\%$ confidence interval and verify if the confidence interval falls with the critical region. In this work, we clarify that $100(1-2\alpha)\%$ confidence interval approach for bioequivalence testing yields a size-$\alpha$ test only when the two one-sided tests in TOST are ``equal-tailed''. Furthermore, a $100(1-\alpha)\%$ confidence interval approach is also discussed in the bioequivalence study.

In this work, we propose a deep learning-based method to perform semiparametric regression analysis for spatially dependent data. To be specific, we use a sparsely connected deep neural network with rectified linear unit (ReLU) activation function to estimate the unknown regression function that describes the relationship between response and covariates in the presence of spatial dependence. Under some mild conditions, the estimator is proven to be consistent, and the rate of convergence is determined by three factors: (1) the architecture of neural network class, (2) the smoothness and (intrinsic) dimension of true mean function, and (3) the magnitude of spatial dependence. Our method can handle well large data set owing to the stochastic gradient descent optimization algorithm. Simulation studies on synthetic data are conducted to assess the finite sample performance, the results of which indicate that the proposed method is capable of picking up the intricate relationship between response and covariates. Finally, a real data analysis is provided to demonstrate the validity and effectiveness of the proposed method.

Variable selection problem for the nonlinear Cox regression model is considered. In survival analysis, one main objective is to identify the covariates that are associated with the risk of experiencing the event of interest. The Cox proportional hazard model is being used extensively in survival analysis in studying the relationship between survival times and covariates, where the model assumes that the covariate has a log-linear effect on the hazard function. However, this linearity assumption may not be satisfied in practice. In order to extract a representative subset of features, various variable selection approaches have been proposed for survival data under the linear Cox model. However, there exists little literature on variable selection for the nonlinear Cox model. To break this gap, we extend the recently developed deep learning-based variable selection model LassoNet to survival data. Simulations are provided to demonstrate the validity and effectiveness of the proposed method. Finally, we apply the proposed methodology to analyze a real data set on diffuse large B-cell lymphoma.

The applications of traditional statistical feature selection methods to high-dimension, low sample-size data often struggle and encounter challenging problems, such as overfitting, curse of dimensionality, computational infeasibility, and strong model assumption. In this paper, we propose a novel two-step nonparametric approach called Deep Feature Screening (DeepFS) that can overcome these problems and identify significant features with high precision for ultra high-dimensional, low-sample-size data. This approach first extracts a low-dimensional representation of input data and then applies feature screening based on multivariate rank distance correlation recently developed by Deb and Sen (2021). This approach combines the strengths of both deep neural networks and feature screening, and thereby has the following appealing features in addition to its ability of handling ultra high-dimensional data with small number of samples: (1) it is model free and distribution free; (2) it can be used for both supervised and unsupervised feature selection; and (3) it is capable of recovering the original input data. The superiority of DeepFS is demonstrated via extensive simulation studies and real data analyses.

Ordinary differential equations (ODEs) are widely used to model complex dynamics that arises in biology, chemistry, engineering, finance, physics, etc. Calibration of a complicated ODE system using noisy data is generally very difficult. In this work, we propose a two-stage nonparametric approach to address this problem. We first extract the de-noised data and their higher order derivatives using boundary kernel method, and then feed them into a sparsely connected deep neural network with ReLU activation function. Our method is able to recover the ODE system without being subject to the curse of dimensionality and complicated ODE structure. When the ODE possesses a general modular structure, with each modular component involving only a few input variables, and the network architecture is properly chosen, our method is proven to be consistent. Theoretical properties are corroborated by an extensive simulation study that demonstrates the validity and effectiveness of the proposed method. Finally, we use our method to simultaneously characterize the growth rate of Covid-19 infection cases from 50 states of the USA.

In massive data analysis, training and testing data often come from very different sources, and their probability distributions are not necessarily identical. A feature example is nonparametric classification in posterior drift model where the conditional distributions of the label given the covariates are possibly different. In this paper, we derive minimax rate of the excess risk for nonparametric classification in posterior drift model in the setting that both training and testing data have smooth distributions, extending a recent work by Cai and Wei (2019) who only impose smoothness condition on the distribution of testing data. The minimax rate demonstrates a phase transition characterized by the mutual relationship between the smoothness orders of the training and testing data distributions. We also propose a computationally efficient and data-driven nearest neighbor classifier which achieves the minimax excess risk (up to a logarithm factor). Simulation studies and a real-world application are conducted to demonstrate our approach.

For the classical Shiryaev--Roberts martingale diffusion considered on the interval $[0,A]$, where $A>0$ is a given absorbing boundary, it is shown that the rate of convergence of the diffusion's quasi-stationary cumulative distribution function (cdf), $Q_{A}(x)$, to its stationary cdf, $H(x)$, as $A\to+\infty$, is no worse than $O(\log(A)/A)$, uniformly in $x\ge0$. The result is established explicitly, by constructing new tight lower- and upper-bounds for $Q_{A}(x)$ using certain latest monotonicity properties of the modified Bessel $K$ function involved in the exact closed-form formula for $Q_{A}(x)$ recently obtained by Polunchenko (2017)., Comment: 15 pages

We consider the quasi-stationary distribution of the classical Shiryaev diffusion restricted to the interval $[0,A]$ with absorption at a fixed $A>0$. We derive analytically a closed-form formula for the distribution's fractional moment of an {\em arbitrary} given order $s\in\mathbb{R}$; the formula is consistent with that previously found by Polunchenko and Pepelyshev (2018) for the case of $s\in\mathbb{N}$. We also show by virtue of the formula that, if $s<1$, then the $s$-th fractional moment of the quasi-stationary distribution becomes that of the exponential distribution (with mean $1/2$) in the limit as $A\to+\infty$; the limiting exponential distribution is the stationary distribution of the reciprocal of the Shiryaev diffusion., Comment: Accepted for publication in Communications in Statistics - Simulation and Computation. arXiv admin note: text overlap with arXiv:1805.07580

Statistical inference based on lossy or incomplete samples is often needed in research areas such as signal/image processing, medical image storage, remote sensing, signal transmission. In this paper, we propose a nonparametric testing procedure based on samples quantized to $B$ bits through a computationally efficient algorithm. Under mild technical conditions, we establish the asymptotic properties of the proposed test statistic and investigate how the testing power changes as $B$ increases. In particular, we show that if $B$ exceeds a certain threshold, the proposed nonparametric testing procedure achieves the classical minimax rate of testing (Shang and Cheng, 2015) for spline models. We further extend our theoretical investigations to a nonparametric linearity test and an adaptive nonparametric test, expanding the applicability of the proposed methods. Extensive simulation studies {together with a real-data analysis} are used to demonstrate the validity and effectiveness of the proposed tests.