Miguel A. Fernández

Estimation of Parameters Subject to Order Restrictions on a Circle With Application to Estimation of Phase Angles of Cell Cycle Genes

Motivated by a problem encountered in the analysis of cell cycle gene expression data, this article deals with the estimation of parameters subject to order restrictions on a unit circle. A normal eukaryotic cell cycle has four major phases during cell division, and a cell cycle gene has its peak expression (phase angle) during the phase that may correspond to its biological function. Because the phases are ordered along a circle, the phase angles of cell cycle genes are ordered unknown parameters on a unit circle. The problem of interest is to estimate the phase angles using the information regarding the order among them. We address this problem by developing a circular version of the well-known isotonic regression for Euclidean data. Because of the underlying geometry, the standard pool adjacent violator algorithm (PAVA) cannot be used for deriving the circular isotonic regression estimator (CIRE). However, PAVA can be modified to obtain a computationally efficient algorithm for deriving the CIRE. We illustrate the CIRE by estimating the phase angles of some of well-known cell cycle genes using the unrestricted estimators obtained in the literature.

Identification of a core set of signature cell cycle genes whose relative order of time to peak expression is conserved across species

A cell division cycle is a well-coordinated process in eukaryotes with cell cycle genes exhibiting a periodic expression over time. There is considerable interest among cell biologists to determine genes that are periodic in multiple organisms and whether such genes are also evolutionarily conserved in their relative order of time to peak expression. Interestingly, periodicity is not well-conserved evolutionarily. A conservative estimate of a number of periodic genes common to fission yeast (Schizosaccharomyces pombe) and budding yeast (Saccharomyces cerevisiae) (‘core set FB’) is 35, while those common to fission yeast and humans (Homo sapiens) (‘core set FH’) is 24. Using a novel statistical methodology, we discover that the relative order of peak expression is conserved in ∼80% of FB genes and in ∼40% of FH genes. We also discover that the order is evolutionarily conserved in six genes which are potentially the core set of signature cell cycle genes. These include ace2 (a transcription factor) and polo-kinase plo1, which are well-known hubs of early M-phase clusters, cdc18 a key component of pre-replication complexes, mik1 which is critical for the establishment and maintenance of DNA damage check point, and histones hhf1 and hta2.

Soft thresholding for medical image segmentation

A new soft thresholding method is presented. The method is based on relating each pixel in the image to the different regions via a membership function, rather than through hard decisions. The membership function of each of the regions is derived from the histogram of the image. As a consequence, each pixel will belong to different regions with a different level of membership. This feature is exploited through spatial processing to make the thresholding robust to noisy environments.

The Loss of Efficiency Estimating Linear Functions under Restrictions

This article is motivated by the problem of estimating contrast in a one-way ANOVA model with restrictions in the parameter vector. We prove that when the restrictions are given by a tree order or a simple order the MLE of some contrast has greater MSE than the unrestricted estimator. A similar behaviour of the MLE is exhibited in a general restricted setting given by a multivariate normal distribution with mean vector constrained to belong to a circular cone. The approach we use focuses on the central direction of the cones. These directions appear to have the greater MSE when the dimension of the restricted cone is big enough.

Incorporating Additional Information to Normal Linear Discriminant Rules

The most useful and broadly known rule in the classical two-group linear normal discriminant analysis is Andersons rule. In this article we propose some alternative procedures that prove useful when prior constraints on the mean vectors are known. These rules are based on new estimators of the difference of means. We prove under mild conditions that the new rules perform better when the common covariance matrix is known. Simulated experiments show that the misclassification errors are lower for the restricted rules defined here in the general case of an unknown covariance matrix. The prior constraints on the mean vector restrict the parameter space to a cone. A family of estimators indexed by a parameter γ, with 0 ≤ γ ≤ 1, is defined using an iterative procedure in such a way that the estimator with a higher value for γ takes values closer to the center of the cone with a greater probability. When γ = 0, the restricted maximum likelihood estimator is given, although the most interesting rule from a theoretical and practical standpoint is obtained when the estimator chosen is given by γ = 1. The usefulness of the proposed rules with real data is demonstrated by their application to two medical examples, the first dealing with heart attack patients and the second dealing with a diabetes dataset. In the former case, restrictions among surviving and nonsurviving patients are used; in the latter, the restrictions arise from differences between the healthy and diabetic populations.

Assessing trends in Duane plots using robust fits

Abstract The Duane plot is a simple and widely used graphical technique in the analysis of repairable systems. The fitting of a straight line to points in that graph serves to determine the behavior of the system assuming a power law process. However, the classical least squares fitting suffer from lack of robustness and it is specially heavily affected by the more remote and perhaps least interesting points. In order to solve this drawback we propose a robust fit based on Least Median of Squares (LMS) or Least Trimmed Squares (LTS) regression estimators of [1] . This methodology has been applied in a project aimed to study the reliability, availability and maintainability of the aerial contact line of the Spanish railways.

Classification of samples into two or more ordered populations with application to a cancer trial

In many applications, especially in cancer treatment and diagnosis, investigators are interested in classifying patients into various diagnosis groups on the basis of molecular data such as gene expression or proteomic data. Often, some of the diagnosis groups are known to be related to higher or lower values of some of the predictors. The standard methods of classifying patients into various groups do not take into account the underlying order. This could potentially result in high misclassification rates, especially when the number of groups is larger than two. In this article, we develop classification procedures that exploit the underlying order among the mean values of the predictor variables and the diagnostic groups by using ideas from order-restricted inference. We generalize the existing methodology on discrimination under restrictions and provide empirical evidence to demonstrate that the proposed methodology improves over the existing unrestricted methodology. The proposed methodology is applied to a bladder cancer data set where the researchers are interested in classifying patients into various groups.

Criticality Determination Based on Failure Records for Decision-making in the Overhead Contact Line System

Abstract In this article, a new methodology to determine the criticality of the components of the overhead contact line (OCL) system by means of a component criticality index is proposed. The methodology is based on classic failure mode and effects analysis (FMEA) but adapted to its application on this system, with characteristics far from industrial systems where FMEA was developed. The procedure is focused on the analysis of the systems operating history with the important advantage that the subjectivity of the usual procedures is reduced. This methodology has proved to be very useful when applied to decision-making and for establishing maintenance frequencies of the elements of the OCL of the Spanish railway network, and it could also be applied to other kind of large-scale distributed systems.

On the maximum likelihood estimator under order restrictions in uniform probability models

We consider random samples from independent uniform populations U(0, θ i ) where the parameters θ i follow a known partial order. The aim of this paper is to compare the restricted and the unrestricted MLE using the universal domination and the squared error criterions when linear functions of the parameter are estimated. We determine functions for which the unrestricted estimator performs better than the unrestricted estimator and a class of functions for which the reverse is true.

Simultaneous estimation by isotonic regression

Abstract We consider a random vector X with unimodal density, diagonal covariance matrix, Ω −1 , and location parameter θ subject to order restrictions. For this model the maximum likelihood estimator (MLE) X∗ is the so-called isotonic regression. It is well known that, in certain situations, X∗ is not better than X when coordinate estimation is considered. We focus on the comparison between the simultaneous constant length confidence intervals for the coordinates centered at X or at X∗ . We characterize situations where the coverage probability of the intervals is greater if X∗ is used instead of X. Special attention is given to the elliptically symmetric distributions with a total ordering or a tree ordering on θ.