Garth Tarr

Robust estimation of precision matrices under cellwise contamination

There is a great need for robust techniques in data mining and machine learning contexts where many standard techniques such as principal component analysis and linear discriminant analysis are inherently susceptible to outliers. Furthermore, standard robust procedures assume that less than half the observation rows of a data matrix are contaminated, which may not be a realistic assumption when the number of observed features is large. The problem of estimating covariance and precision matrices under cellwise contamination is investigated. The use of a robust pairwise covariance matrix as an input to various regularisation routines, such as the graphical lasso, QUIC and CLIME is considered. A method that transforms a symmetric matrix of pairwise covariances to the nearest covariance matrix is used to ensure the input covariance matrix is positive semidefinite. The result is a potentially sparse precision matrix that is resilient to moderate levels of cellwise contamination. Since this procedure is not based on subsampling it scales well as the number of variables increases.

Small sample performance of quantile regression confidence intervals

Since the introduction of regression quantiles for estimating conditional quantile functions there has been ongoing research into how best to construct confidence intervals for parameter estimates. The three main methods are direct estimation, rank test inversion and resampling methods. Kocherginsky et al. [Practical confidence intervals for regression quantiles, J. Comput. Graph. Statist. 14 (2005), pp. 41–55] gave an overview of some of the available procedures. Five years on, the aim of this paper is to revisit and extend their analysis, evaluating additional techniques with a focus on smaller sample sizes and more extreme conditional quantiles. In particular, we find the percentile bootstrap (pbs) to be an eminently viable alternative for confidence interval construction. We show that it provides empirical coverage probabilities generally as good as, or better than, the other more complex resampling methods. Furthermore, pbs confidence intervals typically exhibit smaller average lengths across a variety of models than those based on the rank inversion methods which, like the pbs, avoids explicitly estimating asymptotic variances.

A robust scale estimator based on pairwise means

We propose a new robust scale estimator, the pairwise mean scale estimator P n , which in its most basic form is the interquartile range of the pairwise means. The use of pairwise means leads to a surprisingly high efficiency across many distributions of practical interest. The properties of P n are presented under a unified generalised L-statistics framework, which encompasses numerous other scale estimators. Extensions to P n are proposed, including taking the range of the middle τ×100% instead of just the middle 50% of the pairwise means as well as trimming and Winsorising both the original data and the pairwise means. Furthermore, we have implemented a method using adaptive trimming, which achieves a maximal breakdown value. We investigate the efficiency properties of the pairwise mean scale estimator relative to a number of other established robust scale estimators over a broad range of distributions using the corresponding maximum likelihood estimates as a common base for comparison.

QUANTILE BASED ESTIMATION OF SCALE AND DEPENDENCE

The sample quantile has a long history in statistics. The aim of this thesis is to explore some further applications of quantiles as simple, convenient and robust alternatives to classical procedures. The first application we consider is estimating confidence intervals for quantile regression coefficients, however, the core of this thesis is the development of a new, quantile based, robust scale estimator and its extension to autocovariance estimation in the time series setting and precision matrix estimation in the multivariate setting. Chapter 1 addresses the need for reliable confidence intervals for quantile regression coefficients, particularly in small samples. The existing methods for constructing confidence intervals tend to be based on complex asymptotic arguments and little is known about their finite sample performance. We consider taking xy-pair bootstrap samples and calculating the corresponding quantile regression coefficient estimates for each sample. Instead of estimating a covariance matrix based on these bootstrap samples, our approach is to take the appropriate upper and lower quantiles of the bootstrap sample estimates as the bounds of the confidence interval. The resulting confidence interval estimate is not necessarily symmetric, only covers admissible parameter values and is shown to have good coverage properties. This work demonstrates the competitive performance of our quantile based approach in a broad range of model designs with a focus on small and moderate sample sizes. These results were published in [5]. A reliable estimate of the scale of the residuals from a regression model is often of interest, whether it be parametrically estimating confidence intervals, determining a goodness-of-fit measure, performing model selection, or identifying

The effect of packaging on consumer eating quality of beef

This experiment examined 3 packaging systems: overwrap packaging using oxygen permeable film (OWP); vacuum skin packaging (VSP) and modified atmosphere packaging (MAP, 80%O2 and 20%CO2) on consumer sensory. Three primals from 48 carcasses were aged in vacuum packs for 5, 12 or 40 days. Steaks from longissimus lumborum, gluteus medius and psoas major muscles were packed in OWP, VSP and MAP for 9 days. Untrained consumers scored grilled steaks for tenderness, juiciness, liking of flavour and overall acceptability. Steaks in MAP had 10-12 points lower sensory scores (on a 100 point scale) compared to the OWP, or VSP systems (P < 0.001). The packaging effect was independent of days aging and muscle. It was concluded that high oxygen MAP has the potential to be included as an input variable in the Meat Standards Australia beef grading model. This would be contingent upon research into when the MAP effect occurred and the effect of using different gas mixtures on eating quality.

Relationships between marbling measures across principal muscles

As marbling is a principal input into many grading systems it is important to have an accurate and reliable measurement procedure. This paper compares three approaches to measuring marbling: trained personnel, near infrared spectroscopy (NIR) and image analysis. One 25mm slice of meat was utilised from up to 12 cuts from 48 carcasses processed in Poland and France. Each slice was frozen to enable a consistent post-slaughter period then thawed for image analysis. The images were appraised by experienced beef graders and the sample used to determine fat content by NIR. We find that image analysis based marbling measures are capturing something different to trained personnel and that there is a strong relationship between near infrared spectroscopy and trained personnel. Finally, we demonstrate that marbling measures taken on one muscle can be predictive of marbling in other muscles in the same carcase. This is particularly important for cut based models such as the Meat Standards Australia system.

bcGST—an interactive bias-correction method to identify over-represented gene-sets in boutique arrays

Motivation Gene annotation and pathway databases such as Gene Ontology and Kyoto Encyclopaedia of Genes and Genomes are important tools in Gene‐Set Test (GST) that describe gene biological functions and associated pathways. GST aims to establish an association relationship between a gene‐set of interest and an annotation. Importantly, GST tests for over‐representation of genes in an annotation term. One implicit assumption of GST is that the gene expression platform captures the complete or a very large proportion of the genome. However, this assumption is neither satisfied for the increasingly popular boutique array nor the custom designed gene expression profiling platform. Specifically, conventional GST is no longer appropriate due to the gene‐set selection bias induced during the construction of these platforms. Results We propose bcGST, a bias‐corrected GST by introducing bias‐correction terms in the contingency table needed for calculating the Fishers Exact Test. The adjustment method works by estimating the proportion of genes captured on the array with respect to the genome in order to assist filtration of annotation terms that would otherwise be falsely included or excluded. We illustrate the practicality of bcGST and its stability through multiple differential gene expression analyses in melanoma and the Cancer Genome Atlas cancer studies. Availability and implementation The bcGST method is made available as a Shiny web application at http://shiny.maths.usyd.edu.au/bcGST/. Supplementary information Supplementary data are available at Bioinformatics online.