Small Sample Size Conditions Research Articles

A Subspace Learning Based on a Rank Symmetric Relation for Fuzzy Kernel Discriminant Analysis

Classification of nonlinear high-dimensional data is usually not amenable to standard pattern recognition techniques because of an underlying nonlinear small sample size conditions. To address the problem, a novel kernel fuzzy dual discriminant analysis learning based on a rank symmetric relation is developed in this paper. First, dual subspaces with rank symmetric relation on the discriminant analysis are established, by which a set of integrated subspaces of within-class and between-class scatter matrices are constructed, respectively. Second, a reformative fuzzy LDA algorithm is proposed to achieve the distribution information of each sample represented with fuzzy membership degree, which is incorporated into the redefinition of the scatter matrices. Third, considering the fact that the kernel Fisher discriminant is effective to extract nonlinear discriminative information of the input feature space by using kernel trick, a kernel algorithm based on the new discriminant analysis is presented subsequently, which has the potential to outperform the traditional subspace learning algorithms, especially in the cases of nonlinear small sample sizes. Experimental results conducted on the ORL and Yale face database demonstrate the effectiveness of the proposed method.

Support vector machine based optimal control for minimizing energy consumption of biped walking motions

A kind of support vector machine based optimal control (SVM-OC) approach for minimizing energy consumption of biped walking motions is proposed in this work. Different from existing learning controllers, a SVM controller is incorporated into an optimal controller for biped robots, which aims at minimize an energy-related cost function with three constraints of biped walking robots, including the system dynamics in the single support phase, the system dynamics in the impact phase, and the initial state of the biped. The controller is deduced under the condition of small sample sizes for the SVM. Main contributions of this paper include two aspects: First, a SVM-OC problem for minimizing energy consumption of biped walking motions is defined, which provides new clues to design a kind of optimal controller under the condition of unknown system dynamic model and small sample sizes. Secondly, derivation of the proposed SVM-OC problem is provided in detail, and simulation results demonstrate the advantage of the proposed method compared with conventional SVM control methods and neural network (NN) control methods.

Mitochondrial DNA Content Varies with Pathological Characteristics of Breast Cancer

Changes in mitochondrial DNA (mtDNA) content in cancers have been reported with controversial results, probably due to small sample size and variable pathological conditions. In this study, mtDNA content in 302 breast tumor/surrounding normal tissue pairs were evaluated and correlated with the clinico-pathological characteristics of tumors. Overall, mtDNA content in tumor tissues is significantly lower than that in the surrounding normal tissues, P < 0.00001. MtDNA content in tumor tissues decreased with increasing tumor size. However, when the tumor is very large (>50 cm3), mtDNA content started to increase. Similarly, mtDNA content decreased from grades 0 and I to grade II tumors, but increased from grade II to grade III tumors. Tumors with somatic mtDNA alterations in coding region have significantly higher mtDNA content than tumors without somatic mtDNA alterations (P < 0.001). Tumors with somatic mtDNA alterations in the D-Loop region have significantly lower mtDNA content (P < 0.001). Patients with both low and high mtDNA content in tumor tissue have significantly higher hazard of death than patients with median levels of mtDNA content. mtDNA content in tumor tissues change with tumor size, grade, and ER/PR status; significant deviation from the median level of mtDNA content is associated with poor survival.

Using restricted factor analysis with latent moderated structures to detect uniform and nonuniform measurement bias; a simulation study

Factor analysis is an established technique for the detection of measurement bias. Multigroup factor analysis (MGFA) can detect both uniform and nonuniform bias. Restricted factor analysis (RFA) can also be used to detect measurement bias, albeit only uniform measurement bias. Latent moderated structural equations (LMS) enable the estimation of nonlinear interaction effects in structural equation modelling. By extending the RFA method with LMS, the RFA method should be suited to detect nonuniform bias as well as uniform bias. In a simulation study, the RFA/LMS method and the MGFA method are compared in detecting uniform and nonuniform measurement bias under various conditions, varying the size of uniform bias, the size of nonuniform bias, the sample size, and the ability distribution. For each condition, 100 sets of data were generated and analysed through both detection methods. The RFA/LMS and MGFA methods turned out to perform equally well. Percentages of correctly identified items as biased (true positives) generally varied between 92% and 100%, except in small sample size conditions in which the bias was nonuniform and small. For both methods, the percentages of false positives were generally higher than the nominal levels of significance.

Control Chart Pattern Recognition Based on PSO and LS-SVM

For the sake of reducing the cost of process monitoring, least squares support vector machines (LS-SVM) technique is introduced in this research to recognise the six typical patterns of control chart. The parameters of the LS-SVM pattern recogniser are optimised by particle swarm optimisation (PSO) algorithm so as to improve the performance of LS-SVM pattern recogniser. Simulation results show that the optimised LS-SVM pattern recogniser can effectively recognise control chart patterns with high performance. The techniques introduced in this research can meet the requirements of process fault diagnosis under the condition of small sample size.

Small‐Sample Equating Using a Single‐Group Nearly Equivalent Test (SiGNET) Design

A single‐group (SG) equating with nearly equivalent test forms (SiGNET) design was developed by Grant to equate small‐volume tests. Under this design, the scored items for the operational form are divided into testlets or mini tests. 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 and the second form is composed of testlets 2–7. The seven testlets are administered as a single administered form, and when a sufficient number of examinees have taken the administered form, the second form (testlets 2–7) is equated to the first form (testlets 1–6) using an SG equating design. As evident, this design facilitates the use of an SG equating and allows for the accumulation of data, both of which may reduce equating error. This study compared equatings under the SiGNET and common‐item equating designs and found lower equating error for the SiGNET design in very small sample size conditions (e.g., N = 10).

Heteroscedastic one‐factor models and marginal maximum likelihood estimation

In the present paper, a general class of heteroscedastic one-factor models is considered. In these models, the residual variances of the observed scores are explicitly modelled as parametric functions of the one-dimensional factor score. A marginal maximum likelihood procedure for parameter estimation is proposed under both the assumption of multivariate normality of the observed scores conditional on the single common factor score and the assumption of normality of the common factor score. A likelihood ratio test is derived, which can be used to test the usual homoscedastic one-factor model against one of the proposed heteroscedastic models. Simulation studies are carried out to investigate the robustness and the power of this likelihood ratio test. Results show that the asymptotic properties of the test statistic hold under both small test length conditions and small sample size conditions. Results also show under what conditions the power to detect different heteroscedasticity parameter values is either small, medium, or large. Finally, for illustrative purposes, the marginal maximum likelihood estimation procedure and the likelihood ratio test are applied to real data.

Impact and realization of increased class separability on the small sample size problem in hyperspectral classification

A hyperspectral image is often represented by a high-dimensional dataset for analysis. Under the condition of small sample sizes, the Hughes phenomenon often occurs in high-dimensional classification because of poorly estimated class parameters. Solutions to the Hughes phenomenon have been focused on sample sizes and dimensionality to improve parameter estimation. Considering the Hughes phenomenon as an interaction of two counter-effects between degrading parameter estimation and increasing class separability with dimensionality, this study raises class separability to compensate for the degradation of accuracy in parameter estimation. Instead of merely taking the lowpass (LP) filter as a smoothing technique in image processing, this study presents the LP filter as a means of increasing class separability from the standpoint of pattern recognition. An increase in class separability not only mitigates the Hughes phenomenon but also facilitates parameter estimation in joint supervised-unsupervised learning based on the expectation and maximization (EM) algorithm. LP filtering may hurt classification performance if the filtered, high-frequency contents embed discriminant information. This study overcomes the problem by extracting frequency-based features and forming an augmented feature set. Our experiments demonstrate the feasibility of a spatial filter for the improvement of class separability and parameter estimation in classification.

Second and fourth order statistics-based reduced polynomial rooting direction finding algorithms

Polynomial rooting direction finding (DF) algorithms are a computationally efficient alternative to search-based DF algorithms and are particularly suitable for uniform linear arrays (ULA) of physically identical elements provided mutual interaction among the array elements can be either neglected or compensated for. A popular polynomial rooting algorithm is Root-MUSIC (RM) wherein, for an N-element array, the estimation of the directions of arrivals (DOA) requires the computation of the roots of a 2 N−2-order polynomial for a second order (SO) statistics- and a 4 N−4-order polynomial for a fourth order (FO) statistics-based approach, wherein the DOA are estimated from L pairs of roots closest to the unit circle, when L signals are incident on the array. We derive SO- and FO statistics reduced polynomial rooting (RPR) algorithms capable to estimate L DOA from L roots only. We demonstrate numerically that the RPR algorithms are at least as accurate as the RM algorithms. Simplified algebraic structure of RPR algorithms leads to better performance than afforded by RM algorithms in saturated array environment, especially in the case of FO methods when number of incident signals exceeds number of elements and under low SNR and/or small sample size conditions.

IMPROVED VSF ALGORITHM FOR SMOOTH SURFACE RECONSTRUCTION FROM SPARSE MEDICAL DATA

This paper presents a Modified Variational Splines Fitting (MVSF) algorithm for surface reconstruction using thin plate splines on scattered patches or points of originally smooth surfaces. In particular, a more accurate derivation of the discrete equations for the energy corresponding to the thin plate model is introduced. The results obtained on simulated data show that the proposed algorithm converges faster than the original VSF algorithm. Additionally, we discuss an approach for choosing the algorithm’s parameters using a cross validation technique. Results obtained with the modified algorithm are compared to those using a Frequency Fourier-based 3D Harmonic modelling (3DHM) algorithm and show that the proposed algorithm gives an improved performance under the small sample size condition. The developed model has been successfully applied for real biomedical data; in particular for the reconstruction of left ventricle of human heart.

The Analysis of Repeated Measurements with Mixed-Model Adjusted F Tests

One approach to the analysis of repeated measures data allows researchers to model the covariance structure of their data rather than presume a certain structure, as is the case with conventional univariate and multivariate test statistics. This mixed-model approach, available through SAS PROC MIXED, was compared to a Welch-James type statistic. The Welch-James approach is known to provide generally robust tests of treatment effects in a repeated measures between-by within-subjects design under assumption violations given certain sample size requirements. The mixed-model F tests were based on Kenward-Roger’s adjusted degrees of freedom solution, an approach specifically proposed for small sample settings. The authors investigated Type I error control for repeated measures main and interaction effects in unbalanced designs when normality and covariance homogeneity assumptions did not hold. The mixed-model Kenward-Roger’s adjusted F tests showed superior Type I error control in small sample size conditions in which the Welch-James type statistic was nonrobust; power rates, however, did not favor one approach over the other.

A Comparison of Two General Approaches to Mixed Model Longitudinal Analyses Under Small Sample Size Conditions

There is no general exact analysis for the class of generalized mixed models, and asymptotic procedures are widely used. Importantly, under small sample conditions equivalent asymptotic procedures can yield conflicting inference when applied to the same data set [Aubin, E. C. Q., Cordeiro, G. M. (2000). Bartlett-corrected tests for normal linear models when the covariance matrix is nonscalar. Commun. Statist.—Theory Methods 29:2405–2426]. For the classical likelihood ratio test (LRT), Bartlett’s [Bartlett, M. S. (1937). Properties of sufficiency and statistical tests. Proc. R. Soc. London. Ser. A, Math. Phys. Sci. 160(901):268–282] correction may be used to yield improved small sample performance. Zucker et al. [Zucker, D. M., Lieberman, O., Manor, O. (2000). Improved small sample inference in the mixed linear model: Bartlett correction and adjusted likelihood. J. R. Statist. Soc., Ser. B 62:827–838] proposed and investigated methods for improved small sample inference in the mixed linear model using refined LRTs. The refinements included the use of a Bartlett correction and the Cox–Reid adjusted likelihood [Cox, D. R., Reid, N. (1987). Approximations to noncentral distributions. Can. J. Statist. 15(2):105–114], which using simulation studies (under a random-line model, and a two-period, four-treatment crossover design) were shown to yield Type I error rates very close to the nominal level. An alternative approach which has also been shown [Kowalchuk, R., Keselman, H. (2001). The analysis of repeated measurements with mixed-model Kenward Roger's adjusted F-tests. Paper presented at the Annual Meeting of the American Educational Research Association, Seattle, Washington] to have improved performance characteristics is a procedure involving t and F statistics for tests of fixed effects with modified degrees of freedom calculations detailed by Kenward and Roger [Kenward, M. G., Roger, J. H. (1997). Small sample inference for fixed effects from restricted maximum likelihood. Biometrics 53:983–997]. However, to date there has been no direct comparison of the Bartlett modified likelihood ratio and Kenward Roger procedures. This paper provides the results from a Monte Carlo simulation study examining the Type I error control and power profiles of modified likelihood ratio procedures including those described in Zucker et al. and proposed by Kenward and Roger for tests of fixed effect parameters in mixed linear models with data simulated from small sample unbalanced repeated measures designs, followed by results from a real data example.

The Effectiveness of Methods for Analyzing Multivariate Factorial Data

A Monte Carlo simulation was used to examine the effectiveness of univariate analysis of variance (ANOVA), multivariate analysis of variance (MANOVA), and multiple indicator structural equation (MISE) modeling to analyze data from multivariate factorial designs. The MISE method yielded downwardly biased standard errors for the univariate parameter estimates in the small sample size conditions. In the large sample size data conditions, the MISE method outperformed MANOVA and ANOVA when the covariate accounted for variation in the dependent variable and variables were unreliable. With multivariate statistical tests, MANOVA outperformed the MISE method in the Type I error conditions and the MISE method outperformed MANOVA in the Type II error conditions. The Bonferroni methods were overly conservative in controlling Type I error rates for univariate tests, but a modified Bonferroni method had higher statistical power than the Bonferroni method. Both the Bonferroni and modified methods adequately controlled multivariate Type I error rates.

Comparing Linear Discriminant Function With Logistic Regression for the Two-Group Classification Problem

The performances of predictive discriminant analysis (PDA) and logistic regression (LR) for the 2-group classification problem were compared. The authors used a fully crossed 3-factor experimental design (sample size, group proportions, and equal or unequal covariance matrices) and 2 data patterns. When the 2 groups had equal covariance matrices, PDA and LR performed comparably for the conditions of both equal and unequal group proportions. When the 2 groups had unequal covariance matrices (4:1, as implemented in this study) and very different group proportions, PDA and LR differed somewhat with regard to the classification error rates of the 2 groups, but the classification error rates of the 2 methods for the total sample remained comparable. Sample size played a relatively minor role in the classification accuracy of the 2 methods, except when LR was used under relatively small sample-size conditions.

An analysis of Kansei structure on shoes using self-organizing neural networks

Kansei engineering is a technology for translating human feelings into product design. Several multivariate analyses are used for analyzing human feelings and building rules. Although these methods are reliable, they require large computing resources. It is difficult for general users to deal with many variables because of small personal computers, and the need for the user to be an expert on statistics. This paper presents an automatic semantic structure analyzer and Kansei expert systems builder using self-organizing neural networks, ART1.5-SSS and PCAnet. ART1.5-SSS is our modified version of ART1.5, a variant of the Adaptive Resonance Theory neural network. It is used as a stable non-hierarchical classifier and a feature extractor, in a small sample size condition. PCAnet performs principal component analysis based on generalized Hebbian algorithm by Sanger (1989). These networks enable quick and automatic rule building in Kansei engineering expert systems. AKSYONN4 system is the automatic builder for Kansei engineering expert systems because it uses self-organizing neural networks. The system enables ‘real-world’ applications of Kansei engineering in product development. Relevance to industry An automatic analysis of human feelings on products and automatic building of Kansei engineering expert systems can increase the prospects of applying Kansei engineering to acceptable product design. Neural networks-based analysis and automatic expert system building enable the on-site analyzing.

Comparability of Jackknife and Bootstrap Results: An Investigation for a Case of Canonical Correlation Analysis

Abstract The jackknife and bootstrap methods are becoming increasingly popular in research. Although the two approaches have similar goals and use similar strategies, information is lacking with regard to the comparability of their results. In the present study, this issue was systematically investigated for a case of canonical correlation analysis. Bootstrap, jackknife, and Monte Carlo experiments were carried out for 4 sample sizes (n = 200, 100, 50, 20). The jackknife analyses were also varied as regards the number of jackknife observations deleted in each analysis. Some meaningful discrepancies were observed between the bootstrap and jackknife results, especially under small sample-size conditions. Based on the comparisons made with Monte Carlo estimates, the empirical results suggest that the bootstrap technique provides less biased and more consistent results than the jackknife technique does.

An automatic builder for a Kansei Engineering expert system using self-organizing neural networks

Kansei Engineering is a technology for translating human feelings into a product design. Linear multiple regression analysis is used as a tool to analyze the feelings-design relations and building rules for Kansei Engineering expert systems. Although the method is reliable, it is nevertheless, time and resource consuming and requires statistical expertise in relation to its mathematical constraints. In this paper, we introduce an automatic builder of Kansei expert systems using a self-organizing neural network ART1.5-SSS. ART1.5-SSS is our modified version of ART1.5, a variant of the Adaptive Resonance Theory neural network. Improvement on learning rule makes ART1.5-SSS a stable non-hierarchical cluster analyzer and feature extractor, even in a small sample size condition. The network enables quick, automatic rule building in Kansei Engineering expert systems. The categorization and feature selection performance of our new learning rule is compared to multivariate analyses and to the original ART1.5. Relevance to industry The automatic building of Kansei Engineering expert systems based on a self-organizing neural network technique presented here will assist designers in creating products fit for consumers' underlying needs, with minimal effort.

A Model for Events Occurring at Random Points in Time and an Example Application to Casing Failures in Cedar Creek Anticline Wells

Abstract A familiar example of the "Poisson Pure Death Process" is the phenomenon of radioactive decay. It contains a single parameter, λ, which assumes specific values for each distinct unstable nucleus. Rather surprisingly, data on casing failures caused by salt-flow loading in Cedar Creek Anticline wells are also consistent with this model. This analysis is presented in the hope that other complex failure processes, such as breakdown of sand consolidation treatments, might also exhibit the same statistical behavior. In that case, this model could provide a useful tool for evaluating the economic consequences of such failures. Data for one homogeneous group of wells are used to test the model and to determine λ for the group. The fit is quite good, and the data analysis techniques illustrated should prove useful to anyone attempting to apply the model under the usual condition of small sample size. An analysis and example show how the model provides a basis for deciding when to exercise preventive maintenance and which remedial alternative is most economical. Introduction Many types of complex physical phenomena are described quite accurately by a probability model known to mathematical statisticians as the "Poisson Pure Death Process." A familiar example is the phenomenon of radioactive decay. One cannot predict the precise moment that a given atom of a radioactive element will disintegrate. However, the Poisson model does allow one to calculate the probability that disintegration will occur within any time period of arbitrary duration. This model contains a single parameter, λ, which may be regarded as the average number of disintegrations per unit time when a large number of individual atoms are observed. This parameter has a different value for each radioactive element. Thus, its value depends upon physical details such as the chemical species involved, but the mathematical form of the probability law is the same for all radioactive elements. The model also applies to other problems in which events occur at random times, especially when the physical phenomenon depends on many variables. In this paper, the model is applied to part of the data available on casing failures in Cedar Creek Anticline wells. The failures show up as complete or partial collapse of casing within salt beds at depths of 5,000 to 6,000 ft in most of Shell Oil Co.’s Cedar Creek Anticline fields.