Small Sample Size Situations Research Articles

Two Dimensional Linear Discriminant Analyses for Hyperspectral Data

Most supervised feature extraction methods like linear discriminant analysis (LDA) suffer from the limited number of available training samples. The singularity problem causes LDA to fail in small sample size (SSS) situations. Two dimensional linear discriminant analysis (2DLDA) for feature extraction of hyperspectral images is proposed in this paper which has good efficiency with small training sample size. In this approach, the feature vector of each pixel of hyperspectral image is transformed into a feature matrix. As a result, the data matrices lie in a low-dimensional space. Then, the between-class and within-class scatter matrices are calculated using the matrix form of training samples. The proposed approach has two main advantages: it deals with the SSS problem in hyperspectral data, and also it can extract each number of features (with no limitation) from the original high dimensional data. The proposed method is tested on four widely used hyperspectral datasets. Experimental results confirm that the proposed 2DLDA feature extraction method provides better classification accuracy, with a reasonable computation time, compared to popular supervised feature extraction methods such as generalized discriminant analysis (GDA) and nonparametric weighted feature extraction (NWFE) particularly compared to the 1DLDA in the SSS situation. The experiments show that two dimensional linear discriminant analysis+support vector machine (2DLDA+SVM) is an appropriate choice for feature extraction and classification of hyperspectral images using limited training samples.

Degradation reliability modeling based on an independent increment process with quadratic variance

Degradation testing is an important technique for assessing life time information of complex systems and highly reliable products. Motivated by fatigue crack growth (FCG) data and our previous study, this paper develops a novel degradation modeling approach, in which degradation is represented by an independent increment process with linear mean and general quadratic variance functions of test time or transformed test time if necessary. Based on the constructed degradation model, closed-form expressions of failure time distribution (FTD) and its percentiles can be straightforwardly derived and calculated. A one-stage method is developed to estimate model parameters and FTD. Simulation studies are conducted to validate the proposed approach, and the results illustrate that the approach can provide reasonable estimates even for small sample size situations. Finally, the method is verified by the FCG data set given as the motivating example, and the results show that it can be considered as an effective degradation modeling approach compared with the multivariate normal model and graphic approach.

High-dimensional image data feature extraction by double discriminant embedding

We propose a supervised feature extraction method in this paper that uses two successive transformations to produce the extracted features. The first projection maximizes the difference between spectral features. Thus, produced features have minimum overlap in the new feature space. The second projection maximizes the discrimination between classes. The proposed method, which is called double discriminant embedding (DDE), uses just the first statistics of data. Thus, DDE has good efficiency using limited training samples. The experimental results on four popular hyperspectral images show the better efficiency of DDE in comparison with LDA, GDA, NWFE, and supervised LPP methods in small sample size situation.

Model selection for linear classifiers using Bayesian error estimation

Regularized linear models are important classification methods for high dimensional problems, where regularized linear classifiers are often preferred due to their ability to avoid overfitting. The degree of freedom of the model dis determined by a regularization parameter, which is typically selected using counting based approaches, such as K-fold cross-validation. For large data, this can be very time consuming, and, for small sample sizes, the accuracy of the model selection is limited by the large variance of CV error estimates. In this paper, we study the applicability of a recently proposed Bayesian error estimator for the selection of the best model along the regularization path. We also propose an extension of the estimator that allows model selection in multiclass cases and study its efficiency with L1 regularized logistic regression and L2 regularized linear support vector machine. The model selection by the new Bayesian error estimator is experimentally shown to improve the classification accuracy, especially in small sample-size situations, and is able to avoid the excess variability inherent to traditional cross-validation approaches. Moreover, the method has significantly smaller computational complexity than cross-validation.

Accounting for uncertainty in confounder and effect modifier selection when estimating average causal effects in generalized linear models.

Confounder selection and adjustment are essential elements of assessing the causal effect of an exposure or treatment in observational studies. Building upon work by Wang et al. (2012, Biometrics 68, 661-671) and Lefebvre et al. (2014, Statistics in Medicine 33, 2797-2813), we propose and evaluate a Bayesian method to estimate average causal effects in studies with a large number of potential confounders, relatively few observations, likely interactions between confounders and the exposure of interest, and uncertainty on which confounders and interaction terms should be included. Our method is applicable across all exposures and outcomes that can be handled through generalized linear models. In this general setting, estimation of the average causal effect is different from estimation of the exposure coefficient in the outcome model due to noncollapsibility. We implement a Bayesian bootstrap procedure to integrate over the distribution of potential confounders and to estimate the causal effect. Our method permits estimation of both the overall population causal effect and effects in specified subpopulations, providing clear characterization of heterogeneous exposure effects that may vary considerably across different covariate profiles. Simulation studies demonstrate that the proposed method performs well in small sample size situations with 100-150 observations and 50 covariates. The method is applied to data on 15,060 US Medicare beneficiaries diagnosed with a malignant brain tumor between 2000 and 2009 to evaluate whether surgery reduces hospital readmissions within 30 days of diagnosis.

Ridge regression-based feature extraction for hyperspectral data

Feature extraction based on ridge regression (FERR) is proposed in this article. In FERR, a feature vector is defined in each spectral band using the mean of all classes in that dimension. Then, it is modelled using a linear combination of its farthest neighbours from among other defined feature vectors. The representation coefficients obtained by solving the ridge regression model compose the projection matrix for feature extraction. FERR can extract each desired number of features while the other methods such as linear discriminant analysis (LDA) and generalized discriminant analysis (GDA) have limitations in the number of extracted features. Experimental results on four popular real hyperspectral images show that the efficiency of FERR is superior to those of other supervised feature extraction methods in small sample-size situations. For example, for the Indian Pines dataset, the comparison between the highest average classification accuracies achieved by different feature extraction methods using a support vector machine (SVM) classifier and 16 training samples per class shows that FERR is 7% more accurate than nonparametric weighted feature extraction and is also 9% better than GDA. LDA, having the singularity problem in the small sample-size situations, has 40% less accuracy than FERR. The experiments show that generally the performance of FERR using the SVM classifier is better than when using the maximum likelihood classifier.

Feature space discriminant analysis for hyperspectral data feature reduction

Hyperspectral images contain a large number of spectral bands that allows us to distinguish different classes with more details. But, the number of available training samples is limited. Thus, feature reduction is an important step before classification of high dimensional data. Supervised feature extraction methods such as LDA, GDA, NWFE, and MMLDA use two criteria for feature reduction: between-class scatter and within-class scatter. We propose a supervised feature extraction method in this paper that uses a new criterion in addition to two mentioned measures. The proposed method, which is called feature space discriminant analysis (FSDA), at first, maximizes the between-spectral scatter matrix to increase the difference between extracted features. In the second step, FSDA, maximizes the between-class scatter matrix and minimizes the within-class scatter matrix simultaneously. The experimental results on five popular hyperspectral images show the better performance of FSDA in comparison with other supervised feature extraction methods in small sample size situation.

Overlap-based feature weighting: The feature extraction of Hyperspectral remote sensing imagery

Hyperspectral sensors provide a large number of spectral bands. This massive and complex data structure of hyperspectral images presents a challenge to traditional data processing techniques. Therefore, reducing the dimensionality of hyperspectral images without losing important information is a very important issue for the remote sensing community. We propose to use overlap-based feature weighting (OFW) for supervised feature extraction of hyperspectral data. In the OFW method, the feature vector of each pixel of hyperspectral image is divided to some segments. The weighted mean of adjacent spectral bands in each segment is calculated as an extracted feature. The less the overlap between classes is, the more the class discrimination ability will be. Therefore, the inverse of overlap between classes in each band (feature) is considered as a weight for that band. The superiority of OFW, in terms of classification accuracy and computation time, over other supervised feature extraction methods is established on three real hyperspectral images in the small sample size situation.

Feature Extraction Using Attraction Points for Classification of Hyperspectral Images in a Small Sample Size Situation

Hyperspectral images provide a large volume of spectral bands. Feature extraction (FE) is an important preprocessing step for classification of high-dimensional data. Supervised FE methods such as linear discriminant analysis, generalized discriminant analysis, and nonparametric weighted FE use the criteria of class separability. Theses methods maximize the between-class scatter matrix and minimize the within-class scatter matrix. We propose a supervised FE method in this letter, which uses no statistical moments. Thus, it works well using limited training samples. The proposed FE method consists of two important phases. In the first phase, an attraction point for each class is found. In the second phase, by using an appropriate transformation, the samples of each class move toward the attraction point of their class. The experimental results on two real hyperspectral images demonstrate that FE using attraction points has better performance in comparison with some other supervised FE methods in a small sample size situation.

Band Clustering-Based Feature Extraction for Classification of Hyperspectral Images Using Limited Training Samples

Feature extraction plays a central role in classification of hyperspectral data. We propose a clustering-based feature extraction (CBFE) method in this letter. The proposed method is supervised and only needs to calculate the first-order statistics. Thus, CBFE has better performance than some popular supervised feature extraction methods such as linear discriminant analysis, generalized discriminant analysis, and nonparametric weighted feature extraction in small sample size situation. In addition, CBFE works better than unsupervised approaches such as principal component analysis in classification applications. CBFE considers a vector associated with each band that is composed by the mean values of all classes in that band. Then, a clustering method such as k-means is run to group the similar bands in one cluster. The selected number of clusters is equal to the number of extracted features. Experiments carried out on two different hyperspectral data sets demonstrate that the CBFE has better performance in comparison with some conventional feature extraction methods.

Asymptotically Optimal Linear Shrinkage of Sample LMMSE and MVDR Filters

Conventional implementations of the linear minimum mean-square (LMMSE) and minimum variance distortionless response (MVDR) estimators rely on the sample matrix inversion (SMI) technique, i.e., on the sample covariance matrix (SCM). This approach is optimal in the large sample size regime. Nonetheless, in small sample size situations, those sample estimators suffer a large performance degradation. Thus, the aim of this paper is to propose corrections of these sample methods that counteract their performance degradation in the small sample size regime and keep their optimality in large sample size situations. To this aim, a twofold approach is proposed. First, shrinkage estimators are considered, as they are known to be robust to the small sample size regime. Namely, the proposed methods are based on shrinking the sample LMMSE or sample MVDR filters towards a variously called matched filter or conventional (Bartlett) beamformer in array processing. Second, random matrix theory is used to obtain the optimal shrinkage factors for large filters. The simulation results highlight that the proposed methods outperform the sample LMMSE and MVDR. Also, provided that the sample size is higher than the observation dimension, they improve classical diagonal loading (DL) and Ledoit-Wolf (LW) techniques, which counteract the small sample size degradation by regularizing the SCM. Finally, compared to state-of-the-art DL, the proposed methods reduce the computational cost and the proposed shrinkage of the LMMSE obtains performance gains.

Sample size planning for classification models

In biospectroscopy, suitably annotated and statistically independent samples (e.g. patients, batches, etc.) for classifier training and testing are scarce and costly. Learning curves show the model performance as function of the training sample size and can help to determine the sample size needed to train good classifiers. However, building a good model is actually not enough: the performance must also be proven. We discuss learning curves for typical small sample size situations with 5–25 independent samples per class. Although the classification models achieve acceptable performance, the learning curve can be completely masked by the random testing uncertainty due to the equally limited test sample size. In consequence, we determine test sample sizes necessary to achieve reasonable precision in the validation and find that 75–100 samples will usually be needed to test a good but not perfect classifier. Such a data set will then allow refined sample size planning on the basis of the achieved performance. We also demonstrate how to calculate necessary sample sizes in order to show the superiority of one classifier over another: this often requires hundreds of statistically independent test samples or is even theoretically impossible. We demonstrate our findings with a data set of ca. 2550 Raman spectra of single cells (five classes: erythrocytes, leukocytes and three tumour cell lines BT-20, MCF-7 and OCI-AML3) as well as by an extensive simulation that allows precise determination of the actual performance of the models in question.

A Support Vector Machine MUSIC Algorithm

This paper introduces a new Support Vector Machine (SVM) formulation for the direction of arrival (DOA) estimation problem. We establish a theoretical relationship between the Minimum Variance Distortionless Response (MVDR) and the MUltiple SIgnal Characterization (MUSIC) methods. This leads naturally to the derivation of an SVM-MUSIC algorithm, which combines the benefits of subspace methods with those of SVM. Spatially smoothed versions and a recursive form of the algorithms exhibit good performance against coherent signals. We test the method's performance in scenarios with noncoherent and coherent signals, and in small-sample size-situations obtaining an improved performance in comparison with existing standard approaches.

A note on Cochran test for homogeneity in one-way ANOVA and meta-analysis

In this paper we provide a formal yet simple and straightforward proof of the asymptotic χ2 distribution for Cochran test statistic. Then, we show that the general form of this type of test statistics is invariant for the choice of weights. This fact is important since in practice many such test statistics are constructed with more complicated forms which usually require calculating generalized inverse matrices. Based on our results, we can simplify the construction of the test statistics. More importantly, properties such as anti-conservativeness of this type of test statistics can be drawn from Cochran test statistic. Furthermore, one can improve the performance of the tests by using some modified statistics with correction for small sample size situations.

Unsupervised Discriminant Analysis Based on the Local and Non-local Mean

Abstract Considering the performance of unsupervised discriminant projections (UDP) is gravely influenced by outliers, especially in small training sample size situation, a novel method called unsupervised discriminant analysis (UDA) based on the local and non-local mean for feature extraction is proposed in this paper, which is robust to outliers. It utilizes the local and non-local mean to construct the local and non-local scatter, to some extent, overcomes the discriminant difficulty caused by outliers. Besides, LUDA is computationally more efficient than UDP. Experimental results on ORL, YALE and AR face image databases show that the proposed UDA is more efficient and effective than UDP.

Quantifying the predictive accuracy of time-to-event models in the presence of competing risks

Prognostic models for time-to-event data play a prominent role in therapy assignment, risk stratification and inter-hospital quality assurance. The assessment of their prognostic value is vital not only for responsible resource allocation, but also for their widespread acceptance. The additional presence of competing risks to the event of interest requires proper handling not only on the model building side, but also during assessment. Research into methods for the evaluation of the prognostic potential of models accounting for competing risks is still needed, as most proposed methods measure either their discrimination or calibration, but do not examine both simultaneously. We adapt the prediction error proposal of Graf et al. (Statistics in Medicine 1999, 18, 2529–2545) and Gerds and Schumacher (Biometrical Journal 2006, 48, 1029–1040) to handle models with competing risks, i.e. more than one possible event type, and introduce a consistent estimator. A simulation study investigating the behaviour of the estimator in small sample size situations and for different levels of censoring together with a real data application follows.

Shrinkage‐based Diagonal Discriminant Analysis and Its Applications in High‐Dimensional Data

High-dimensional data such as microarrays have brought us new statistical challenges. For example, using a large number of genes to classify samples based on a small number of microarrays remains a difficult problem. Diagonal discriminant analysis, support vector machines, and k-nearest neighbor have been suggested as among the best methods for small sample size situations, but none was found to be superior to others. In this article, we propose an improved diagonal discriminant approach through shrinkage and regularization of the variances. The performance of our new approach along with the existing methods is studied through simulations and applications to real data. These studies show that the proposed shrinkage-based and regularization diagonal discriminant methods have lower misclassification rates than existing methods in many cases.

TRANSFORMAÇÃO DE DADOS E IMPLICAÇÕES DA UTILIZAÇÃO DO TESTE DE KRUSKAL-WALLIS EM PESQUISAS AGROECOLÓGICAS

Neste trabalho foram discutidos aspectos conceituais e práticos (mediante exemplo hipotético com Lycopersicum esculentum) que devem ser levados em consideração na utilização do teste de Kruskal-Wallis (teste estatístico não-paramétrico), enfatizando-se as complicações que podem surgir se esse procedimento estatístico for implementado de maneira arbitrária em estudos agroecológicos. Também foram abordados os equívocos que podem ser cometidos ao serem feitas afirmações muito gerais sobre as transformações de dados, mais especificamente a transformação arco seno da raiz quadrada no caso de dados referentes a proporções. A cuidadosa ponderação sobre as implicações dos resultados de testes paramétricos e não-paramétricos deve estar presente como fator essencial no delineamento experimental de pesquisas agroecológicas. O exemplo hipotético considerado indicou que os resultados desses dois tipos de teste não permitem as mesmas conclusões em situações com tamanho amostral reduzido.

Blind SIMO channel identification using FFT/IFFT

This paper presents an FFT/IFFT-based deterministic blind identification method for estimating the finite impulse response of single-input multiple-output (SIMO) communication channels. The new method delivers a frequency-domain realization of the cross-relation between each channel output pair. With no assumption of the precise knowledge of channel order, the proposed algorithm is capable of estimating channel parameters as well as detecting channel order. In small sample size situations, our method exhibits consistent or better performance compared to the existing methods, e.g., the subspace, cross-relation and shifted correlation algorithms.

AN ALTERNATIVE DATA COLLECTION DESIGN FOR EQUATING WITH VERY SMALL SAMPLES

ABSTRACTA single group (SG) equating design with nearly equivalent test forms (SiGNET) design was developed by Grant (2006) to equate small volume tests. The basis of this design is that examinees take two largely overlapping test forms within a single administration. The scored items for the operational form are divided into mini‐tests called testlets. An additional testlet is created but not scored for the first form. If the scored testlets are Testlets 1‐6 and the unscored testlet is Testlet 7, then the first form is composed of Testlets 1‐6, the second form is composed of Testlets 2‐7, and Testlets 2‐6 are common to both test forms. They are administered as a single administered form, and when a sufficient number of examinees have taken the administered form for an SG equating, the second form (Testlets 2‐7) is equated to the first form (Testlets 1‐6) using SG equating. As evident, there are at least two merits of the SiGNET design over the nonequivalent groups with anchor test (NEAT) design. First, it facilitates the use of an SG equating design, which has the least random equating error, and second, it allows for the accumulation of sufficient data to equate the second form. Since the examinees scores are based on only the first form (i.e., the operational form), the two forms can be administered until sufficient data are collected to equate the second form. This study compared equatings under the SiGNET and NEAT designs and found reduced bias and error for the SiGNET design in very small sample size situations (e.g., N = 10 or 15). Implications for practice using the SiGNET design are also discussed.