arXiv subjects

J. S. Marron

Publications and source records attributed to J. S. Marron.

68 records · Page 4Linked to original sources

arXiv2012

New Approaches to Principal Component Analysis for Trees

Object Oriented Data Analysis is a new area in statistics that studies populations of general data objects. In this article we consider populations of tree-structured objects as our focus of interest. We develop improved analysis tools for data lying in a binary tree space analogous to classical Principal Component Analysis methods in Euclidean space. Our extensions of PCA are analogs of one dimensional subspaces that best fit the data. Previous work was based on the notion of tree-lines. In this paper, a generalization of the previous tree-line notion is proposed: k-tree-lines. Previously proposed tree-lines are k-tree-lines where k=1. New sub-cases of k-tree-lines studied in this work are the 2-tree-lines and tree-curves, which explain much more variation per principal component than tree-lines. The optimal principal component tree-lines were computable in linear time. Because 2-tree-lines and tree-curves are more complex, they are computationally more expensive, but yield improved data analysis results. We provide a comparative study of all these methods on a motivating data set consisting of brain vessel structures of 98 subjects.

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arXiv2011

Registration of Functional Data Using Fisher-Rao Metric

We introduce a novel geometric framework for separating the phase and the amplitude variability in functional data of the type frequently studied in growth curve analysis. This framework uses the Fisher-Rao Riemannian metric to derive a proper distance on the quotient space of functions modulo the time-warping group. A convenient square-root velocity function (SRVF) representation transforms the Fisher-Rao metric into the standard $\ltwo$ metric, simplifying the computations. This distance is then used to define a Karcher mean template and warp the individual functions to align them with the Karcher mean template. The strength of this framework is demonstrated by deriving a consistent estimator of a signal observed under random warping, scaling, and vertical translation. These ideas are demonstrated using both simulated and real data from different application domains: the Berkeley growth study, handwritten signature curves, neuroscience spike trains, and gene expression signals. The proposed method is empirically shown to be be superior in performance to several recently published methods for functional alignment.

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arXiv2011

Consistency of Sparse PCA in High Dimension, Low Sample Size Contexts

Sparse Principal Component Analysis (PCA) methods are efficient tools to reduce the dimension (or the number of variables) of complex data. Sparse principal components (PCs) are easier to interpret than conventional PCs, because most loadings are zero. We study the asymptotic properties of these sparse PC directions for scenarios with fixed sample size and increasing dimension (i.e. High Dimension, Low Sample Size (HDLSS)). Under the previously studied spike covariance assumption, we show that Sparse PCA remains consistent under the same large spike condition that was previously established for conventional PCA. Under a broad range of small spike conditions, we find a large set of sparsity assumptions where Sparse PCA is consistent, but PCA is strongly inconsistent. The boundaries of the consistent region are clarified using an oracle result.

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arXiv2011

Principal arc analysis on direct product manifolds

We propose a new approach to analyze data that naturally lie on manifolds. We focus on a special class of manifolds, called direct product manifolds, whose intrinsic dimension could be very high. Our method finds a low-dimensional representation of the manifold that can be used to find and visualize the principal modes of variation of the data, as Principal Component Analysis (PCA) does in linear spaces. The proposed method improves upon earlier manifold extensions of PCA by more concisely capturing important nonlinear modes. For the special case of data on a sphere, variation following nongeodesic arcs is captured in a single mode, compared to the two modes needed by previous methods. Several computational and statistical challenges are resolved. The development on spheres forms the basis of principal arc analysis on more complicated manifolds. The benefits of the method are illustrated by a data example using medial representations in image analysis.

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arXiv2010

Analysis of dependence among size, rate and duration in internet flows

In this paper we examine rigorously the evidence for dependence among data size, transfer rate and duration in Internet flows. We emphasize two statistical approaches for studying dependence, including Pearson's correlation coefficient and the extremal dependence analysis method. We apply these methods to large data sets of packet traces from three networks. Our major results show that Pearson's correlation coefficients between size and duration are much smaller than one might expect. We also find that correlation coefficients between size and rate are generally small and can be strongly affected by applying thresholds to size or duration. Based on Transmission Control Protocol connection startup mechanisms, we argue that thresholds on size should be more useful than thresholds on duration in the analysis of correlations. Using extremal dependence analysis, we draw a similar conclusion, finding remarkable independence for extremal values of size and rate.

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arXiv2010

Visualizing the Structure of Large Trees

This study introduces a new method of visualizing complex tree structured objects. The usefulness of this method is illustrated in the context of detecting unexpected features in a data set of very large trees. The major contribution is a novel two-dimensional graphical representation of each tree, with a covariate coded by color. The motivating data set contains three dimensional representations of brain artery systems of 105 subjects. Due to inaccuracies inherent in the medical imaging techniques, issues with the reconstruction algo- rithms and inconsistencies introduced by manual adjustment, various discrepancies are present in the data. The proposed representation enables quick visual detection of the most common discrepancies. For our driving example, this tool led to the modification of 10% of the artery trees and deletion of 6.7%. The benefits of our cleaning method are demonstrated through a statistical hypothesis test on the effects of aging on vessel structure. The data cleaning resulted in improved significance levels.

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arXiv2009

PCA consistency in high dimension, low sample size context

Principal Component Analysis (PCA) is an important tool of dimension reduction especially when the dimension (or the number of variables) is very high. Asymptotic studies where the sample size is fixed, and the dimension grows [i.e., High Dimension, Low Sample Size (HDLSS)] are becoming increasingly relevant. We investigate the asymptotic behavior of the Principal Component (PC) directions. HDLSS asymptotics are used to study consistency, strong inconsistency and subspace consistency. We show that if the first few eigenvalues of a population covariance matrix are large enough compared to the others, then the corresponding estimated PC directions are consistent or converge to the appropriate subspace (subspace consistency) and most other PC directions are strongly inconsistent. Broad sets of sufficient conditions for each of these cases are specified and the main theorem gives a catalogue of possible combinations. In preparation for these results, we show that the geometric representation of HDLSS data holds under general conditions, which includes a $ρ$-mixing condition and a broad range of sphericity measures of the covariance matrix.

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arXiv2008

A Principal Component Analysis for Trees

The active field of Functional Data Analysis (about understanding the variation in a set of curves) has been recently extended to Object Oriented Data Analysis, which considers populations of more general objects. A particularly challenging extension of this set of ideas is to populations of tree-structured objects. We develop an analog of Principal Component Analysis for trees, based on the notion of tree-lines, and propose numerically fast (linear time) algorithms to solve the resulting optimization problems. The solutions we obtain are used in the analysis of a data set of 73 individuals, where each data object is a tree of blood vessels in one person's brain.

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arXiv2008

MultiResolution Anomaly Detection Method for Long Range Dependent Time Series

Driven by network intrusion detection, we propose a MultiResolution Anomaly Detection (MRAD) method, which effectively utilizes the multiscale properties of Internet features and network anomalies. In this paper, several theoretical properties of the MRAD method are explored. A major new result is the mathematical formulation of the notion that a two-scaled MRAD method has larger power than the average power of the detection method based on the given two scales. Test threshold is also developed. Comparisons between MRAD method and other classical outlier detectors in time series are reported as well.

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arXiv2008

SiZer for Censored Density and Hazard Estimation

The SiZer method is extended to nonparametric hazard estimation and also to censored density and hazard estimation. The new method allows quick, visual statistical inference about the important issue of statistically significant increases and decreases in the smooth curve estimate. This extension has required the opening of a new avenue of research on the interface between statistical inference and scale space.

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arXiv2008

A scale-based approach to finding effective dimensionality in manifold learning

The discovering of low-dimensional manifolds in high-dimensional data is one of the main goals in manifold learning. We propose a new approach to identify the effective dimension (intrinsic dimension) of low-dimensional manifolds. The scale space viewpoint is the key to our approach enabling us to meet the challenge of noisy data. Our approach finds the effective dimensionality of the data over all scale without any prior knowledge. It has better performance compared with other methods especially in the presence of relatively large noise and is computationally efficient.

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arXiv2007

Analysis of nonlinear modes of variation for functional data

A set of curves or images of similar shape is an increasingly common functional data set collected in the sciences. Principal Component Analysis (PCA) is the most widely used technique to decompose variation in functional data. However, the linear modes of variation found by PCA are not always interpretable by the experimenters. In addition, the modes of variation of interest to the experimenter are not always linear. We present in this paper a new analysis of variance for Functional Data. Our method was motivated by decomposing the variation in the data into predetermined and interpretable directions (i.e. modes) of interest. Since some of these modes could be nonlinear, we develop a new defined ratio of sums of squares which takes into account the curvature of the space of variation. We discuss, in the general case, consistency of our estimates of variation, using mathematical tools from differential geometry and shape statistics. We successfully applied our method to a motivating example of biological data. This decomposition allows biologists to compare the prevalence of different genetic tradeoffs in a population and to quantify the effect of selection on evolution.

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arXiv2007

Object oriented data analysis: Sets of trees

Object oriented data analysis is the statistical analysis of populations of complex objects. In the special case of functional data analysis, these data objects are curves, where standard Euclidean approaches, such as principal component analysis, have been very successful. Recent developments in medical image analysis motivate the statistical analysis of populations of more complex data objects which are elements of mildly non-Euclidean spaces, such as Lie groups and symmetric spaces, or of strongly non-Euclidean spaces, such as spaces of tree-structured data objects. These new contexts for object oriented data analysis create several potentially large new interfaces between mathematics and statistics. This point is illustrated through the careful development of a novel mathematical framework for statistical analysis of populations of tree-structured objects.

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arXiv2007

SiZer for time series: A new approach to the analysis of trends

Smoothing methods and SiZer are a useful statistical tool for discovering statistically significant structure in data. Based on scale space ideas originally developed in the computer vision literature, SiZer (SIgnificant ZERo crossing of the derivatives) is a graphical device to assess which observed features are `really there' and which are just spurious sampling artifacts. In this paper, we develop SiZer like ideas in time series analysis to address the important issue of significance of trends. This is not a straightforward extension, since one data set does not contain the information needed to distinguish `trend' from `dependence'. A new visualization is proposed, which shows the statistician the range of trade-offs that are available. Simulation and real data results illustrate the effectiveness of the method.

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