Small Sample Size Scenario Research Articles

Cauchy-Schwarz bounded trade-off weighting for causal inference with small sample sizes

The difficulty of causal inference for small-sample-size data lies in the issue of inefficiency that the variance of the estimators may be large. Some existing weighting methods adopt the idea of bias-variance trade-off, but they require manual specification of the trade-off parameters. To overcome this drawback, in this article, we propose a Cauchy-Schwarz Bounded Trade-off Weighting (CBTW) method, in which the trade-off parameter is theoretically derived to guarantee a small Mean Square Error (MSE) in estimation. We theoretically prove that optimizing the objective function of CBTW, which is the Cauchy-Schwarz upper-bound of the MSE for causal effect estimators, contributes to minimizing the MSE. Moreover, since the upper-bound consists of the variance and the squared ℓ2-norm of covariate differences, CBTW can not only estimate the causal effects efficiently, but also keep the covariates balanced. Experimental results on both simulation data and real-world data show that the CBTW outperforms most existing methods especially under small sample size scenarios.

Military Unmanned Equipment Image Target Recognition Method based on Improved Deep Learning

Abstract Military unmanned equipment target recognition is currently a research hotspot and trend in the field of military intelligence. The small sample size and complex recognition scenarios of military unmanned equipment image datasets result in low recognition accuracy. It is proposed a military unmanned equipment image target recognition method based on improved deep learning, which set Faster R-CNN as the network framework for target recognition, used the Kmeans++ algorithm to label boxes on customized datasets, and then added OHEM to the framework to improve the network’s recognition accuracy for difficult to recognize samples. The accuracy of the algorithm proposed in this article reaches 93.8%, which is 2.8% higher than the YOLOv5 algorithm, providing an improved deep learning method for military unmanned equipment image target recognition.

Parameter Estimation of Exponentiated Half-Logistic Distribution for Left-Truncated and Right-Censored Data

Left-truncated and right-censored data are widely used in lifetime experiments, biomedicine, labor economics, and actuarial science. This article discusses how to resolve the problems of statistical inferences on the unknown parameters of the exponentiated half-logistic distribution based on left-truncated and right-censored data. In the beginning, maximum likelihood estimations are calculated. Then, asymptotic confidence intervals are constructed by using the observed Fisher information matrix. To cope with the small sample size scenario, we employ the percentile bootstrap method and the bootstrap-t method for the establishment of confidence intervals. In addition, Bayesian estimations under both symmetric and asymmetric loss functions are addressed. Point estimates are computed by Tierney–Kadane’s approximation and importance sampling procedure, which is also applied to establishing corresponding highest posterior density credible intervals. Lastly, simulated and real data sets are presented and analyzed to show the effectiveness of the proposed methods.

Conditional Inference in Small Sample Scenarios Using a Resampling Approach

This paper discusses a non-parametric resampling technique in the context of multidimensional or multiparameter hypothesis testing of assumptions of the Rasch model. It is based on conditional distributions and it is suggested in small sample size scenarios as an alternative to the application of asymptotic or large sample theory. The exact sampling distribution of various well-known chi-square test statistics like Wald, likelihood ratio, score, and gradient tests as well as others can be arbitrarily well approximated in this way. A procedure to compute the power function of the tests is also presented. A number of examples of scenarios are discussed in which the power function of the test does not converge to 1 with an increasing deviation of the true values of the parameters of interest from the values specified in the hypothesis to be tested. Finally, an attempt to modify the critical region of the tests is made aiming at improving the power and an R package is provided.

Folded LDA: Extending the Linear Discriminant Analysis Algorithm for Feature Extraction and Data Reduction in Hyperspectral Remote Sensing

The rich spectral information provided by hyperspectral imaging (HSI) has made this technology very useful in the classification of remotely sensed data. However, classification of hyperspectral data is typically affected by noise and the Hughes phenomenon due to the presence of hundreds of spectral bands and correlation among them, with usually a limited number of samples for training. Linear Discriminant Analysis (LDA) is a well-known technique that has been widely used for supervised dimensionality reduction of hyperspectral data. However, the use of LDA in hyperspectral remote sensing is limited due to 1) its poor performance on small training datasets and 2) the limited number of features that can be selected i.e. c-1 where c is the number of classes in the data. To solve these problems, this work presents a Folded LDA (F-LDA) for dimensionality reduction of remotely sensed HSI data in Small Sample Size (SSS) scenarios. The proposed approach allows many more discriminant features to be selected in comparison to the conventional LDA since the selection is no longer bound by the limiting factor, leading to significantly higher accuracy in the classification of pixels under SSS restrictions. The proposed approach is evaluated on five different datasets, where the experimental results demonstrate the superiority of the F-LDA to the conventional LDA in terms of not only higher classification accuracy but also reduced computational complexity, and reduced contiguous memory requirements.

Improved Shrinkage Estimator of Large-Dimensional Covariance Matrix under the Complex Gaussian Distribution

Estimating the covariance matrix of a random vector is essential and challenging in large dimension and small sample size scenarios. The purpose of this paper is to produce an outperformed large-dimensional covariance matrix estimator in the complex domain via the linear shrinkage regularization. Firstly, we develop a necessary moment property of the complex Wishart distribution. Secondly, by minimizing the mean squared error between the real covariance matrix and its shrinkage estimator, we obtain the optimal shrinkage intensity in a closed form for the spherical target matrix under the complex Gaussian distribution. Thirdly, we propose a newly available shrinkage estimator by unbiasedly estimating the unknown scalars involved in the optimal shrinkage intensity. Both the numerical simulations and an example application to array signal processing reveal that the proposed covariance matrix estimator performs well in large dimension and small sample size scenarios.

Number of Source Signal Estimation by the Mean Squared Eigenvalue Error

Detection of the number of source signals (NoSS) in the presence of additive noise is considered. We present a new approach denoted by the mean squared eigenvalue error (MSEE). The MSEE is the mean squared error between the desired noise-free eigenvalues and the available estimated eigenvalues. The approach investigates and analyzes the probabilistic distribution of the available eigenvalue estimates and revisits proper thresholding of these sorted values. The optimum NoSS is provided by minimizing the MSEE. A probabilistic worst-case technique is proposed to estimate the value of the MSEE by using only the available data. It is shown that the proposed method is consistent as the data length increases. It is also shown that the method is consistent as the signal-to-noise ratio (SNR) increases. Simulation results illustrate advantages of the MSEE over competing approaches and confirm effectiveness and robustness of the MSEE even in low-SNR or small sample size scenarios.

Mapping Burn Severity of Forest Fires in Small Sample Size Scenarios

Mapping burn severity of forest fires can contribute significantly to understanding, quantifying and monitoring of forest fire severity and its impacts on ecosystems. In recent years, several remote sensing-based methods for mapping burn severity have been reported in the literature, of which the implementations are mainly dependent on several field plots. Therefore, it is a challenge to develop alternative method of mapping burn severity using limited number of field plots. In this study, we proposed a support vector regression based method using multi-temporal satellite data to map the burn severity, evaluated its performance by calculating correlations between the predicted and the observed Composite Burn Index, and compared the performance with that of the regression analysis method (based on dNBR). The results show that the performance of support vector regression based mapping method is more accurate (RMSE = 0.46–0.57) than that of regression analysis method (RMSE = 0.53–0.68). Even with fewer training sets, it can map the detailed distribution of burn severity of forest fires and can achieve relatively better generalization, compared to regression analysis burn severity mapping methods. It could be concluded that the proposed support vector regression based mapping method is an alternative to the regression analysis method in small sample size scenarios. This method with excellent generalization performance should be recommended for future studies on burn severity of forest fires.

Superpixel-Based Multitask Learning Framework for Hyperspectral Image Classification

Due to the high spectral dimensionality of hyperspectral images as well as the difficult and time-consuming process of collecting sufficient labeled samples in practice, the small sample size scenario is one crucial problem and a challenging issue for hyperspectral image classification. Fortunately, the structure information of materials, reflecting region of homogeneity in the spatial domain, offers an invaluable complement to the spectral information. Assuming some spatial regularity and locality of surface materials, it is reasonable to segment the image into different homogeneous parts in advance, called superpixel, which can be used to improve the classification performance. In this paper, a superpixel-based multitask learning framework has been proposed for hyperspectral image classification. Specifically, a set of 2-D Gabor filters are first applied to hyperspectral images to extract discriminative features. Meanwhile, a superpixel map is generated from the hyperspectral images. Second, a superpixel-based spatial–spectral Schroedinger eigenmaps (S4E) method is adopted to effectively reduce the dimensions of each extracted Gabor cube. Finally, the classification is carried out by a support vector machine (SVM)-based multitask learning framework. The proposed approach is thus termed Gabor S4E and SVM-based multitask learning (GS4E-MTLSVM). A series of experiments is conducted on three real hyperspectral image data sets to demonstrate the effectiveness of the proposed GS4E-MTLSVM approach. The experimental results show that the performance of the proposed GS4E-MTLSVM is better than those of several state-of-the-art methods, while the computational complexity has been greatly reduced, compared with the pixel-based spatial–spectral Schroedinger eigenmaps method.

Detecting Pairwise Interactive Effects of Continuous Random Variables for Biomarker Identification with Small Sample Size.

Aberrant changes to interactions among cellular components have been conjectured to be potential causes of abnormalities in cellular functions. By systematic analysis of high-throughput-omics data, researchers hope to detect potential associations among measured variables for better biomarker identification and phenotype prediction. In this paper, we focus on the methods to measure pairwise interactive effects among continuous random variables, representing molecular expressions, with respect to a given categorical outcome. Together with a comprehensive review on the existing measures, we further propose new measures that better estimate interactive effects, especially in small sample size scenarios. We first evaluate the performance of the existing and new methods for both small and large sample sizes based on simulated datasets that shows our proposed methods outperform previous methods in general. The best performing method for small sample size scenarios suggested by simulation experiments is then implemented to estimate interactive effects among genes with respect to the metastasis outcome in two breast cancer studies based on micro-array gene expression datasets. Our results further demonstrate that integrating detected interactive effects together with individual effects can help in finding more accurate biomarkers for breast cancer metastasis, which are indeed involved in important pathways related to cancer metastasis based on gene set enrichment analysis.

Pooled shrinkage estimator for quadratic discriminant classifier: an analysis for small sample sizes in face recognition

The quadratic discriminant classifier (QDC) is a well-known parametric Bayesian classifier that has been successfully applied to statistical pattern recognition problems. One such application is in automatic face recognition where the number of training images per subject is often found to be much less than the length of the facial features. In such a case, the QDC cannot be used because the class-specific covariance matrix on which it depends is either poorly estimated or singular thereby resulting in unacceptable classifier performance. High dimensional covariance estimation techniques such as shrinkage can alleviate this problem but only to a certain extent. This paper presents a computationally simple yet effective solution for further improving the QDC performance in small sample size scenarios. The proposed technique adopts a strategy of combining the class-specific shrinkage estimates of the covariance matrix to obtain a pooled shrinkage estimate, which is then plugged into the QDC. Experiments indicate that the proposed classifier leads to remarkable improvement in face recognition accuracy as compared to the existing classifiers such as the nearest neighbor, support vector machine and naive Bayes, irrespective of the nature of the database and feature extraction method. Monte Carlo simulations reveal that this improvement is due to the much lower mean squared error of the pooled shrinkage estimator which offers greater stability to the QDC.

DRME: Count-based differential RNA methylation analysis at small sample size scenario

Differential methylation, which concerns difference in the degree of epigenetic regulation via methylation between two conditions, has been formulated as a beta or beta-binomial distribution to address the within-group biological variability in sequencing data. However, a beta or beta-binomial model is usually difficult to infer at small sample size scenario with discrete reads count in sequencing data. On the other hand, as an emerging research field, RNA methylation has drawn more and more attention recently, and the differential analysis of RNA methylation is significantly different from that of DNA methylation due to the impact of transcriptional regulation. We developed DRME to better address the differential RNA methylation problem. The proposed model can effectively describe within-group biological variability at small sample size scenario and handles the impact of transcriptional regulation on RNA methylation. We tested the newly developed DRME algorithm on simulated and 4 MeRIP-Seq case–control studies and compared it with Fisher's exact test. It is in principle widely applicable to several other RNA-related data types as well, including RNA Bisulfite sequencing and PAR–CLIP. The code together with an MeRIP-Seq dataset is available online (https://github.com/lzcyzm/DRME) for evaluation and reproduction of the figures shown in this article.

A computationally fast alternative to cross-validation in penalized Gaussian graphical models

We study the problem of selecting a regularization parameter in penalized Gaussian graphical models. When the goal is to obtain a model with good predictive power, cross-validation is the gold standard. We present a new estimator of Kullback–Leibler loss in Gaussian Graphical models which provides a computationally fast alternative to cross-validation. The estimator is obtained by approximating leave-one-out-cross-validation. Our approach is demonstrated on simulated data sets for various types of graphs. The proposed formula exhibits superior performance, especially in the typical small sample size scenario, compared to other available alternatives to cross-validation, such as Akaike's information criterion and Generalized approximate cross-validation. We also show that the estimator can be used to improve the performance of the Bayesian information criterion when the sample size is small.

AGGREGATION OF SPARSE LINEAR DISCRIMINANT ANALYSES FOR EVENT-RELATED POTENTIAL CLASSIFICATION IN BRAIN-COMPUTER INTERFACE

Two main issues for event-related potential (ERP) classification in brain-computer interface (BCI) application are curse-of-dimensionality and bias-variance tradeoff, which may deteriorate classification performance, especially with insufficient training samples resulted from limited calibration time. This study introduces an aggregation of sparse linear discriminant analyses (ASLDA) to overcome these problems. In the ASLDA, multiple sparse discriminant vectors are learned from differently l1-regularized least-squares regressions by exploiting the equivalence between LDA and least-squares regression, and are subsequently aggregated to form an ensemble classifier, which could not only implement automatic feature selection for dimensionality reduction to alleviate curse-of-dimensionality, but also decrease the variance to improve generalization capacity for new test samples. Extensive investigation and comparison are carried out among the ASLDA, the ordinary LDA and other competing ERP classification algorithms, based on different three ERP datasets. Experimental results indicate that the ASLDA yields better overall performance for single-trial ERP classification when insufficient training samples are available. This suggests the proposed ASLDA is promising for ERP classification in small sample size scenario to improve the practicability of BCI.

Edgeworth Expansion of the Moment-Based Test for Homogeneity in an NEF-QVF Mixture Model

In this article, we study the moment-based test procedure for a mixture distribution for the Natural exponential family with quadratic variance functions (NEF-QVF) family proposed by Ning et al. (2009b) in the small sample size scenario. We derive the approximation for the null distribution of the test statistic by the Edgeworth expansion. The simulations are conducted for a binomial mixture distribution, which includes the situation corresponding to the detection of the linkage in the genetic analysis, with different sample sizes and family sizes at various significance levels. The simulation results show that our test performs reasonably well. We also apply the proposed method to the real clinical data to verify the significant difference between two drug treatments. The critical values associated with a binomial mixture distribution are also provided.

Exploratory factor analysis for small samples

Traditionally, two distinct approaches have been employed for exploratory factor analysis: maximum likelihood factor analysis and principal component analysis. A third alternative, called regularized exploratory factor analysis, was introduced recently in the psychometric literature. Small sample size is an important issue that has received considerable discussion in the factor analysis literature. However, little is known about the differential performance of these three approaches to exploratory factor analysis in a small sample size scenario. A simulation study and an empirical example demonstrate that regularized exploratory factor analysis may be recommended over the two traditional approaches, particularly when sample sizes are small (below 50) and the sample covariance matrix is near singular.

Multiple Testing in Candidate Gene Situations: A Comparison of Classical, Discrete, and Resampling-Based Procedures

In candidate gene association studies, usually several elementary hypotheses are tested simultaneously using one particular set of data. The data normally consist of partly correlated SNP information. Every SNP can be tested for association with the disease, e.g., using the Cochran-Armitage test for trend. To account for the multiplicity of the test situation, different types of multiple testing procedures have been proposed. The question arises whether procedures taking into account the discreteness of the situation show a benefit especially in case of correlated data. We empirically evaluate several different multiple testing procedures via simulation studies using simulated correlated SNP data. We analyze FDR and FWER controlling procedures, special procedures for discrete situations, and the minP-resampling-based procedure. Within the simulation study, we examine a broad range of different gene data scenarios. We show that the main difference in the varying performance of the procedures is due to sample size. In small sample size scenarios,the minP-resampling procedure though controlling the stricter FWER even had more power than the classical FDR controlling procedures. In contrast, FDR controlling procedures led to more rejections in higher sample size scenarios.

Uncorrelated Multilinear Discriminant Analysis With Regularization and Aggregation for Tensor Object Recognition

This paper proposes an uncorrelated multilinear discriminant analysis (UMLDA) framework for the recognition of multidimensional objects, known as tensor objects. Uncorrelated features are desirable in recognition tasks since they contain minimum redundancy and ensure independence of features. The UMLDA aims to extract uncorrelated discriminative features directly from tensorial data through solving a tensor-to-vector projection. The solution consists of sequential iterative processes based on the alternating projection method, and an adaptive regularization procedure is incorporated to enhance the performance in the small sample size (SSS) scenario. A simple nearest-neighbor classifier is employed for classification. Furthermore, exploiting the complementary information from differently initialized and regularized UMLDA recognizers, an aggregation scheme is adopted to combine them at the matching score level, resulting in enhanced generalization performance while alleviating the regularization parameter selection problem. The UMLDA-based recognition algorithm is then empirically shown on face and gait recognition tasks to outperform four multilinear subspace solutions (MPCA, DATER, GTDA, TR1DA) and four linear subspace solutions (Bayesian, LDA, ULDA, R-JD-LDA).

Gaussian kernel optimization for pattern classification

This paper presents a novel algorithm to optimize the Gaussian kernel for pattern classification tasks, where it is desirable to have well-separated samples in the kernel feature space. We propose to optimize the Gaussian kernel parameters by maximizing a classical class separability criterion, and the problem is solved through a quasi-Newton algorithm by making use of a recently proposed decomposition of the objective criterion. The proposed method is evaluated on five data sets with two kernel-based learning algorithms. The experimental results indicate that it achieves the best overall classification performance, compared with three competing solutions. In particular, the proposed method provides a valuable kernel optimization solution in the severe small sample size scenario.

Improved Estimation of Eigenvalues and Eigenvectors of Covariance Matrices Using Their Sample Estimates

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <?Pub Dtl=""?>The problem of estimating the eigenvalues and eigenvectors of the covariance matrix associated with a multivariate stochastic process is considered. The focus is on finite sample size situations, whereby the number of observations is limited and comparable in magnitude to the observation dimension. Using tools from random matrix theory, and assuming a certain eigenvalue splitting condition, new estimators of the eigenvalues and eigenvectors of the covariance matrix are derived, that are shown to be consistent in a more general asymptotic setting than the traditional one. Indeed, these estimators are proven to be consistent, not only when the sample size increases without bound for a fixed observation dimension, but also when the observation dimension increases to infinity at the same rate as the sample size. Numerical evaluations indicate that the estimators have an excellent performance in small sample size scenarios, where the observation dimension and the sample size are comparable in magnitude. </para>