CMB data analysis and sparsity

The statistical analysis of the soon to come Planck satellite CMB data will help set tighter bounds on major cosmological parameters. On the way, a number of practical difficulties need to be tackled, notably that several other astrophysical sources emit radiation in the frequency range of CMB observations. Some level of residual contributions, most significantly in the galactic region and at the locations of strong radio point sources will unavoidably contaminate the estimated spherical CMB map. Masking out these regions is common practice but the gaps in the data need proper handling. In order to restore the stationarity of a partly incomplete CMB map and thus lower the impact of the gaps on non-local statistical tests, we developed an inpainting algorithm on the sphere based on a sparse representation of the data, to fill in and interpolate across the masked regions.     

Cross-validation methods in principal component analysis: A comparison

In principal component analysis (PCA), it is crucial to know how many principal components (PCs) should be retained in order to account for most of the data variability. A class of “objective” rules for finding this quantity is the class of cross-validation (CV) methods. In this work we compare three CV techniques showing how the performance of these methods depends on the covariance matrix structure. Finally we propose a rule for the choice of the “best” CV method and give an application to real data.     

DDMA-charts: Nonparametric multivariate moving average control charts based on data depth

Summary: This paper studies the DDMA–chart, a data depth based moving–average control chart for monitoring multivariate data. This chart is nonparametric and it can detect simultaneously location and scale changes in the process. It improves upon the existing r– and Q–chart in the efficiency of detecting location changes. Both theoretical justifications and simulation studies are provided. Comparisons with some existing multivariate control charts via simulation results are also provided. Some applications of the DDMA–chart to the analysis of airline performance data (collected by the FAA) are demonstrated. The results indicate that the DDMA–chart is an effective nonparametric multivariate control chart.*Research supported in part by grants from the National Science Foundation, the National Security Agency, and the Federal Aviation Administration. The discussion on aviation safety in this paper reects the views of the authors, who are solely responsible for the accuracy of the analysis results presented herein, and does not necessarily reect the official view or policy of the FAA. The dataset used in this paper has been partially masked in order to protect confidentiality.     

Exact Tests in Panel Data Using Generalized p-Values

In this article, we consider exact tests in panel data regression model with one-way and two-way error component for which no exact tests are available. Exact inferences using generalized p-values are obtained. When there are several groups of panel data, test for equal coefficients under one-way and two-way error component are derived.     

A superiority problem in misspecified restricted linear models

Abstract

In this article we study the relationship between principal component analysis and a multivariate dependency measure. It is shown, via simulated examples and real data, that the information provided by principal components is compatible with that obtained via the dependency measure δ. Furthermore, we show that in some instances in which principal component analysis fails to give reasonable results due to nonlinearity among the random variables, the dependency statistic δ still provides good results. Finally, we give some ideas about using the statistic δ in order to reduce the dimensionality of a given data set.     

Identification of genomic markers correlated with sensitivity in solid tumors to Dasatinib using sparse principal components

Differential analysis techniques are commonly used to offer scientists a dimension reduction procedure and an interpretable gateway to variable selection, especially when confronting high-dimensional genomic data. Huang et al. used a gene expression profile of breast cancer cell lines to identify genomic markers which are highly correlated with in vitro sensitivity of a drug Dasatinib. They considered three statistical methods to identify differentially expressed genes and finally used the results from the intersection. But the statistical methods that are used in the paper are not sufficient to select the genomic markers. In this paper we used three alternative statistical methods to select a combined list of genomic markers and compared the genes that were proposed by Huang et al. We then proposed to use sparse principal component analysis (Sparse PCA) to identify a final list of genomic markers. The Sparse PCA incorporates correlation into account among the genes and helps to draw a successful genomic markers discovery. We present a new and a small set of genomic markers to separate out the groups of patients effectively who are sensitive to the drug Dasatinib. The analysis procedure will also encourage scientists in identifying genomic markers that can help to separate out two groups.     

Pruning a sufficient dimension reduction with a p-value guided hard-thresholding

Principal fitted component (PFC) models are a class of likelihood-based inverse regression methods that yield a so-called sufficient reduction of the random p-vector of predictors X given the response Y. Assuming that a large number of the predictors has no information about Y, we aimed to obtain an estimate of the sufficient reduction that ‘purges’ these irrelevant predictors, and thus, select the most useful ones. We devised a procedure using observed significance values from the univariate fittings to yield a sparse PFC, a purged estimate of the sufficient reduction. The performance of the method is compared to that of penalized forward linear regression models for variable selection in high-dimensional settings.     

A note on the mean residual life of a coherent system with a cold standby component

In this paper, we investigate the effect of a cold standby component on the mean residual life (MRL) of a system. When the system fails, a cold standby component is immediately put in operation. We particularly focus on the coherent systems in which, after putting the standby component into operation, the failure of the system is due to the next component failure. For these systems, we define MRL functions and obtain their explicit expressions. Also some stochastic ordering results are provided. Such systems include k-out-of-n systems. Hence, our results extend some results in literature.     

On the risk performance of extended sequence compound rules for classification between N(–1, 1) and N(1, 1) †

Since Robbins (1951) first introduced the compound decision problem, there has evolved a large literature on the subject for the most part dealing with the construction of compound rules whose excess risk over the simple envelope is no greater than zero in the limit as the number N of component problems goes to infinity. Such rules have compound risk which is asymptotically subminimax. Johns (1967) has introduced more stringent (extended) envelopes and has proposed extended compound rules whose risks achieve these envelopes in the limit. This paper reports some Monte Carlo results on the compound risk behavior of selected unextended and extended rules for moderate N values and certain parameter sequences for Robbins original example. The results show that the extended rules compare favorably with the minimax rule and the unextended rules for moderate N and parameter sequences exhibiting higher order empirical dependencies, for example, those generated by a Markov process.     

GSH Dependence Modeling with an Application to Risk Management

The generalized secant hyperbolic distribution (GSH) can be used to represent financial data with heavy tails as an alternative to the Student-t, because it guarantees the existence of all moments, also with a high kurtosis value. In order to obtain a multivariate extension of the GSH distribution, in this article we present two approaches to model the dependence, the copula approach and independent component analysis. Since the methodologies considered allow to simulate the GSH dependence, we show also the empirical results obtained in the estimation of risk of a financial portfolio by the Monte Carlo method.     

Kurtosis modelling by means of the J-transformation

Summary: The H–family of distributions or H–distributions, introduced by Tukey (1960; 1977), are generated by a single transformation of the standard normal distribution and allow for leptokurtosis represented by the parameter h. Alternatively, Haynes et al. (1997) generated leptokurtic distributions by applying the K–transformation to the normal distribution. In this study we propose a third transformation, the so–called J–transformation, and derive some properties of this transformation. Moreover, so-called elongation generating functions (EGFs) are introduced. By means of EGFs we are able to visualize the strength of tail elongation and to construct new transformations. Finally, we compare the three transformations towards their goodness–of–fit in the context of financial return data.     

Testing concerning the homogeneity of some predator–prey populations

In this paper, we study an inference problem for a stochastic model where k deterministic Lotka–Volterra systems of ordinary differential equations (ODEs) are perturbed with k pairs of random errors. The k deterministic systems describe the ecological interaction between k predator–prey populations. These k deterministic systems depend on unknown parameters. We consider the testing problem concerning the homogeneity between k pairs of the interaction parameters of the ODEs. We assume that the k pairs of random errors are independent and that, each pair follows correlated Ornstein–Uhlenbeck processes. Thus, we extend the stochastic model suggested in Froda and Colavita [2005. Estimating predator–prey systems via ordinary differential equations with closed orbits. Aust. N.Z. J. Stat. 2, 235–254] as well as in Froda and Nkurunziza [2007. Prediction of predator–prey populations modeled by perturbed ODE. J. Math. Biol. 54, 407–451] where k=1. Under this statistical model, we propose a likelihood ratio test and study the asymptotic properties of this test. Finally, we highlight the performance of our method through some simulations studies.     

Principal components of sample estimates: an approach through symbolic data analysis

This paper deals with the analysis of datasets, where the subjects are described by the estimated means of a p-dimensional variable. Classical statistical methods of data analysis do not treat measurements affected by intrinsic variability, as in the case of estimates, so that the heterogeneity induced among subjects by this condition is not taken into account. In this paper a way to solve the problem is suggested in the context of symbolic data analysis, whose specific aim is to handle data tables where single valued measurements are substituted by complex data structures like frequency distributions, intervals, and sets of values. A principal component analysis is carried out according to this proposal, with a significant improvement in the treatment of information.     

Empirical Bayes testing for guarantee lifetime: Non identical components case

This article considers an empirical Bayes testing problem for the guarantee lifetime in the two-parameter exponential distributions with non identical components. We study a method of constructing empirical Bayes tests under a class of unknown prior distributions for the sequence of the component testing problems. The asymptotic optimality of the sequence of empirical Bayes tests is studied. Under certain regularity conditions on the prior distributions, it is shown that the sequence of the constructed empirical Bayes tests is asymptotically optimal, and the associated sequence of regrets converges to zero at a rate O(n^{? 1 + 1/[2(r + α) + 1]}) for some integer r ? 0 and 0 ? α ? 1 depending on the unknown prior distributions, where n is the number of past data available when the (n + 1)st component testing problem is considered.     

Classical testing in functional linear models

We extend four tests common in classical regression – Wald, score, likelihood ratio and F tests – to functional linear regression, for testing the null hypothesis, that there is no association between a scalar response and a functional covariate. Using functional principal component analysis, we re-express the functional linear model as a standard linear model, where the effect of the functional covariate can be approximated by a finite linear combination of the functional principal component scores. In this setting, we consider application of the four traditional tests. The proposed testing procedures are investigated theoretically for densely observed functional covariates when the number of principal components diverges. Using the theoretical distribution of the tests under the alternative hypothesis, we develop a procedure for sample size calculation in the context of functional linear regression. The four tests are further compared numerically for both densely and sparsely observed noisy functional data in simulation experiments and using two real data applications.     

The specification of vector autoregressive moving average models

In this paper we propose a new identification method based on the residual white noise autoregressive criterion (Pukkila et al., 1990) to select the order of VARMA structures. Results from extensive simulation experiments based on different model structures with varying number of observations and number of component series are used to demonstrate the performance of this new procedure. We also use economic and business data to compare the model structures selected by this order selection method with those identified in other published studies.     

A new futility test approach in clinical interim data monitoring

Sequential monitoring of efficacy and safety data has become a vital component of modern clinical trials. It affords companies the opportunity to stop studies early in cases when it appears as if the primary objective will not be achieved or when there is clear evidence that the primary objective has already been met. This paper introduces a new concept of the backward conditional hypothesis test (BCHT) to evaluate clinical trial success. Unlike the regular conditional power approach that relies on the probability that the final study result will be statistically significant based on the current interim look, the BCHT was constructed based on the hypothesis test framework. The framework comprises a significant test level as opposed to the arbitrary fixed futility index utilized in the conditional power method. Additionally, the BCHT has proven to be a uniformly most powerful test. Noteworthy features of the BCHT method compared with the conditional power method will be presented. Copyright © 2012 John Wiley & Sons, Ltd.     

The eigenstructure of block-structured correlation matrices and its implications for principal component analysis

Block-structured correlation matrices are correlation matrices in which the p variables are subdivided into homogeneous groups, with equal correlations for variables within each group, and equal correlations between any given pair of variables from different groups. Block-structured correlation matrices arise as approximations for certain data sets’ true correlation matrices. A block structure in a correlation matrix entails a certain number of properties regarding its eigendecomposition and, therefore, a principal component analysis of the underlying data. This paper explores these properties, both from an algebraic and a geometric perspective, and discusses their robustness. Suggestions are also made regarding the choice of variables to be subjected to a principal component analysis, when in the presence of (approximately) block-structured variables.     

Compositional biplots including external non-compositional variables

Biplots represent a widely used statistical tool for visualizing the resulting loadings and scores of a dimension reduction technique applied to multivariate data. If the underlying data carry only relative information (i.e. compositional data expressed in proportions, mg/kg, etc.) they have to be pre-processed with a logratio transformation before the dimension reduction is carried out. In the context of principal component analysis, the resulting biplot is called compositional biplot. We introduce an alternative, the ilr biplot, which is based on a special choice of orthonormal coordinates resulting from an isometric logratio (ilr) transformation. This allows to incorporate also external non-compositional variables, and to study the relations to the compositional variables. The methodology is demonstrated on real data sets.     

Exploring the variability of DNA molecules via principal geodesic analysis on the shape space

Most of the linear statistics deal with data lying in a Euclidean space. However, there are many examples, such as DNA molecule topological structures, in which the initial or the transformed data lie in a non-Euclidean space. To get a measure of variability in these situations, the principal component analysis (PCA) is usually performed on a Euclidean tangent space as it cannot be directly implemented on a non-Euclidean space. Instead, principal geodesic analysis (PGA) is a new tool that provides a measure of variability for nonlinear statistics. In this paper, the performance of this new tool is compared with that of the PCA using a real data set representing a DNA molecular structure. It is shown that due to the nonlinearity of space, the PGA explains more variability of the data than the PCA.