Sample Size Situations Research Articles

Robust linear receivers for STBC systems with unknown co-channel interference

Focusing on space-time block code (STBC) systems with unknown co-channel interference, an oblique projection-based robust linear receiver is proposed in this paper. Based on the oblique projection, the desired signal subspace and interference-plus-noise subspace are first identified from the received signal. Then the matched filter receiver is used to decode the STBC encoded signals in the desired signal subspace. Simulation results show that the proposed linear receiver obtains significant performance improvement over conventional Capon-type receivers under finite sample-size situations and in the presence of channel estimation errors.

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.

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>

On the Asymptotic Behavior of the Sample Estimates of Eigenvalues and Eigenvectors of Covariance Matrices

This paper analyzes the asymptotic behavior of the sample estimators of the eigenvalues and eigenvectors of covariance matrices. Rather than considering traditional large sample-size asymptotics, in this paper the focus is on limited sample size situations, whereby the number of available observations is comparable in magnitude to the observation dimension. Using tools from random matrix theory, the asymptotic behavior of the traditional sample estimates is investigated under the assumption that both the sample size and the observation dimension tend to infinity, while their quotient converges to a positive quantity. Assuming that an asymptotic eigenvalue splitting condition is fulfilled, closed form asymptotic expressions of these estimators are derived, proving inconsistency of the traditional sample estimators in these asymptotic conditions. The derived expressions are shown to provide a valuable insight into the behavior of the sample estimators in the small sample size regime.

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.

Modified Subspace Algorithms for DoA Estimation With Large Arrays

This paper proposes the use of a new generalized asymptotic paradigm in order to analyze the performance of subspace-based direction-of-arrival (DoA) estimation in array signal processing applications. Instead of assuming that the number of samples is high whereas the number of sensors/antennas remains fixed, the asymptotic situation analyzed herein assumes that both quantities tend to infinity at the same rate. This asymptotic situation provides a more accurate description of a potential situation where these two quantities are finite and hence comparable in magnitude. It is first shown that both MUSIC and SSMUSIC are inconsistent when the number of antennas/sensors increases without bound at the same rate as the sample size. This is done by analyzing and deriving closed-form expressions for the two corresponding asymptotic cost functions. By examining these asymptotic cost functions, one can establish the minimum number of samples per antenna needed to resolve closely spaced sources in this asymptotic regime. Next, two alternative estimators are constructed, that are strongly consistent in the new asymptotic situation, i.e., they provide consistent DoA estimates, not only when the number of snapshots goes to infinity, but also when the number of sensors/antennas increases without bound at the same rate. These estimators are inspired by the theory of G-estimation and are therefore referred to as G-MUSIC and G-SSMUSIC, respectively. Simulations show that the proposed algorithms outperform their traditional counterparts in finite sample-size situations, although they still present certain limitations.

Assessing and improving the stability of chemometric models in small sample size situations

Small sample sizes are very common in multivariate analysis. Sample sizes of 10-100 statistically independent objects (rejects from processes or loading dock analysis, or patients with a rare disease), each with hundreds of data points, cause unstable models with poor predictive quality. Model stability is assessed by comparing models that were built using slightly varying training data. Iterated k-fold cross-validation is used for this purpose. Aggregation stabilizes models. It is possible to assess the quality of the aggregated model without calculating further models. The validation and aggregation methods investigated in this study apply to regression as well as to classification. These techniques are useful for analyzing data with large numbers of variates, e.g., any spectral data like FT-IR, Raman, UV/VIS, fluorescence, AAS, and MS. FT-IR images of tumor tissue were used in this study. Some tissue types occur frequently, while some are very rare. They are classified using LDA. Initial models were severely unstable. Aggregation stabilizes the predictions. The hit rate increased from 67% to 82%.

FFT/IFFT Based Blind SIMO Channel Identification

This paper presents an FFT/IFFT based blind identification method for estimating the finite impulse response of single-input multiple-output channels driven by an unknown deterministic signal. The proposed algorithm successfully handles a very small size of received data, for which the existing blind channel estimation methods, including the subspace, cross-relation and shifted correlation algorithms, are known to be ineffective. Moreover, 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. Simulations show that the proposed algorithm outper forms the existing methods in small sample size situations.

Adaptive interval estimation in one-way random effects models

Two-stage procedures are introduced to control the width and coverage (validity) of confidence intervals for the estimation of the mean, the between groups variance component and certain ratios of the variance components in one-way random effects models. The procedures use the pilot sample data to estimate an “optimal” group size and then proceed to determine the number of groups by a stopping rule. Such sampling plans give rise to unbalanced data, which are consequently analyzed by the harmonic mean method. Several asymptotic results concerning the proposed procedures are given along with simulation results to assess their performance in moderate sample size situations. The proposed procedures were found to effectively control the width and probability of coverage of the resulting confidence intervals in all cases and were also found to be robust in the presence of missing observations. From a practical point of view, the procedures are illustrated using a real data set and it is shown that the resulting unbalanced designs tend to require smaller sample sizes than is needed in a corresponding balanced design where the group size is arbitrarily pre-specified.

A cluster-adjusted sample size algorithm for proportions was developed using a beta-binomial model

Objective The objective of the paper was to design a computer algorithm to calculate sample sizes for estimating proportions incorporating clustered sampling units using a beta-binomial model when information concerning the intraclass correlation is not available. Study Design and Setting A computer algorithm was written in FORTRAN and evaluated for a hypothetical sample size situation. Results The developed algorithm was able to incorporate clustering in estimated sample sizes through the specification of a beta distribution to account for within-cluster correlation. In a hypothetical example, the usual normal approximation method for estimation of a proportion ignoring the clustered sampling design resulted in a calculated sample size of 107, whereas the developed algorithm suggested that 208 sampling units would be necessary. Conclusion It is important to incorporate cluster adjustment in sample size calculations when designing epidemiologic studies for estimation of disease burden and other population proportions in the situation of correlated data even when information concerning the intraclass correlation is not available. Beta-binomial models can be used to account for clustering, and design effects can be estimated by generating beta distributions that encompass within-cluster correlation.

A new method to help diagnose cancers for small sample size

For many years, scientists have engaged in profiling altered genes to help diagnose related cancers. However, the size of the sample to develop a new profile of cancer genes in the beginning stage is usually small because of costly procedure. Researchers are often disturbed by the analytical method because there has been no effective technique to deal with such small sample size situations in cancer genes diagnosis. The purpose of the study was to employ a new method, mega-trend-diffusion technique, to improve the accuracy of gene diagnosis for bladder cancer on a very limited number of samples. The modeling results showed that when the number of training data increased, the learning accuracy of the bladder cancer diagnosis was enhanced stably, from 82% to 100%. Compared with traditional methods, this study provides a new approach of a reliable model for small dataset analysis. Although the study treats bladder cancer as an example, it is believed that the findings can be generalized to other diseases with limited sample size.

Sequential confidence intervals for variance components in one-way random models

A sequential procedure is introduced to construct controlled width confidence intervals for the among groups variance component, as well as for certain ratios of variance components, in one-way random models. Width is controlled in the sense that it does not exceed a predetermined precision d. Several asymptotic results concerning the proposed procedure are given along with simulation results to assess its performance in moderate sample size situations. The proposed interval is also compared with the confidence interval based on the modified large sample method. In addition, some alternatives are suggested to accelerate the proposed procedure. An expression for the asymptotic expected sample size, which was obtained as a by-product of the sequential procedure, is used to discuss planning considerations in one-way random designs.

Regularized linear discriminant analysis and its application in microarrays

In this paper, we introduce a modified version of linear discriminant analysis, called the "shrunken centroids regularized discriminant analysis" (SCRDA). This method generalizes the idea of the "nearest shrunken centroids" (NSC) (Tibshirani and others, 2003) into the classical discriminant analysis. The SCRDA method is specially designed for classification problems in high dimension low sample size situations, for example, microarray data. Through both simulated data and real life data, it is shown that this method performs very well in multivariate classification problems, often outperforms the PAM method (using the NSC algorithm) and can be as competitive as the support vector machines classifiers. It is also suitable for feature elimination purpose and can be used as gene selection method. The open source R package for this method (named "rda") is available on CRAN (http://www.r-project.org) for download and testing.

A local mean-based nonparametric classifier

A considerable amount of effort has been devoted to design a classifier in practical situations. In this paper, a simple nonparametric classifier based on the local mean vectors is proposed. The proposed classifier is compared with the 1-NN, k-NN, Euclidean distance (ED), Parzen, and artificial neural network (ANN) classifiers in terms of the error rate on the unknown patterns, particularly in small training sample size situations. Experimental results show that the proposed classifier is promising even in practical situations.

Automated Design of Robust Discriminant Analysis Classifier for Foot Pressure Lesions Using Kinematic Data

In the recent years, the use of motion tracking systems for acquisition of functional biomechanical gait data, has received increasing interest due to the richness and accuracy of the measured kinematic information. However, costs frequently restrict the number of subjects employed, and this makes the dimensionality of the collected data far higher than the available samples. This paper applies discriminant analysis algorithms to the classification of patients with different types of foot lesions, in order to establish an association between foot motion and lesion formation. With primary attention to small sample size situations, we compare different types of Bayesian classifiers and evaluate their performance with various dimensionality reduction techniques for feature extraction, as well as search methods for selection of raw kinematic variables. Finally, we propose a novel integrated method which fine-tunes the classifier parameters and selects the most relevant kinematic variables simultaneously. Performance comparisons are using robust resampling techniques such as Bootstrap 632+ and k-fold cross-validation. Results from experimentations with lesion subjects suffering from pathological plantar hyperkeratosis, show that the proposed method can lead to approximately 96% correct classification rates with less than 10% of the original features.

SVM decision boundary based discriminative subspace induction

We study the problem of linear dimension reduction for classification, with a focus on sufficient dimension reduction, i.e., finding subspaces without loss of discrimination power. First, we formulate the concept of sufficient subspace for classification in parallel terms as for regression. Then we present a new method to estimate the smallest sufficient subspace based on an improvement of decision boundary analysis (DBA). The main idea is to combine DBA with support vector machines (SVM) to overcome the inherent difficulty of DBA in small sample size situations while keeping DBA's estimation simplicity. The compact representation of SVM boundary results in a significant gain in both speed and accuracy over previous DBA implementations. Alternatively, this technique can be viewed as a way to reduce the run-time complexity of SVM itself. Comparative experiments on one simulated and four real-world benchmark datasets highlight the superior performance of the proposed approach.

Molecular Diagnosis

We discuss supervised classification techniques applied to medical diagnosis based on gene expression profiles. Our focus lies on strategies of adaptive model selection to avoid overfitting in high-dimensional spaces. We introduce likelihood-based methods, classification trees, support vector machines and regularized binary regression. For regularization by dimension reduction, we describe feature selection methods: feature filtering, feature shrinkage and wrapper approaches. In small sample-size situations efficient methods of data re-use are needed to assess the predictive power of a model. We discuss two issues in using cross-validation: the difference between in-loop and out-of-loop feature selection, and estimating model parameters in nested-loop cross-validation. Gene selection does not reduce the dimensionality of the model. Tuning parameters enable adaptive model selection. The feature selection bias is a common pitfall in performance evaluation. Model selection and performance evaluation can be combined by nested-loop cross-validation. Classification of microarrays is prone to overfitting. A rigorous and unbiased assessment of the predictive power of the model is a must.

Asymptotic Performance Evaluation of Space-Frequency MMSE Filters for OFDM

This paper proposes and analyzes two different linear space-frequency architectures for the reception of OFDM-modulated signals: the classical sample matrix inversion (SMI) algorithm and a new architecture that maximizes the output signal-to-interference-plus noise ratio (MSINR). The performance of these two linear receivers is compared in terms of asymptotic output SINR, taking into account the finite sample size effect through the asymptotic covariance of the filter weights. The analysis is asymptotic in the sense that the performance is analyzed, assuming that both the number of carriers and the prefix length of the OFDM signal increase without bound at the same rate, whereas their quotient remains constant. Assuming that the carrier frequencies become asymptotically close to one another, we are able to derive explicit equations that shed some light on the influence of the frequency selectivity of channel and interference on the relative performance of the two approaches. The results are useful in the sense that they provide first-order approximations to the (asymptotically) optimum number of adjacent carriers to be processed by a single beamformer in a finite sample size situation.

A nonparametric confidence interval for the trimmed mean

We propose a nonparametric confidence interval for the trimmed mean. The confidence interval is based on the empirical likelihood interval for the sample mean. Its coverage probability is bounded below by the desired nominal level. Simulation results on its coverage probability and the power of the associated test show that it works well in small to moderate sample size situations.