Seminars (SCH)

In this presentation, we review some fundamentals of visualization and then proceed to describe methods and combinations of methods useful for visualizing high dimensional data. Some methods include parallel coordinates, smooth interpolations of parallel coordinates, grand tours including wrapping tours, fractal tours, pseudo-grand tours, and pixel tours.

This talk considers questions of two types concerning high-dimensional inference. First, given a practical (polynomial-time) algorithm, what are the limits of its performance? Second, how do such practical limitations compare to information-theoretic bounds, which apply to the performance of any algorithm regardless of computational complexity?

We analyze these issues in high-dimensional versions of two canonical inference problems: (a) support recovery in sparse regression; and (b) the sparse PCA or eigenvector problem. For the sparse regression problem, we describe a sharp threshold on the sample size n that controls success/failure of \ell_1 constrained quadratic programming (the Lasso), as function of the problem size p, and sparsity index k (number of non-zero entries). Using information-theoretic methods, we prove that the Lasso is order-optimal for sublinear sparsity (vanishing k/p), but sub-optimal for linear sparsity (k/p bounded away from zero). For the sparse eigenvector problem, we analyze a semidefinite programming relaxation due to Aspremont et al., and establish a similar transition in failure/success for triplets (n,p,k) tending to infinity.

Based on joint works with Arash Amini, John Lafferty, and Pradeep Ravikumar.

Dimension reduction in regression, represented primarily by principal components, is ubiquitous in the applied sciences. This is an old idea that has moved to a position of prominence in recent years because technological advances now allow scientists to routinely formulate regressions in which the number p of predictors is considerably larger than in the past. Although "large" p regressions are perhaps mainly responsible for renewed interest, dimension reduction methodology can be useful regardless of the size of p.

Starting with a little history and a definition of "sufficient reductions", we will consider a variety of models for dimension reduction in regression. The models start from one in which maximum likelihood estimation produces principal components, step along a few incremental expansions, and end with forms that have the potential to improve on some standard methodology. This development provides remedies for two concerns that have dogged principal components in regression: principal components are typically computed from the predictors alone and then do not make apparent use of the response, and they are not equivariant under full rank linear transformation of the predictors.

Related Links

• http://www.stat.umn.edu/~dennis

Model selection and classification using high-dimensional features arise frequently in many contemporary statistical studies such as tumor classification using microarray or other high-throughput data. The impact of dimensionality on classifications is largely poorly understood. We first demonstrate that even for the independence classification rule, classification using all the features can be as bad as the random guessing due to noise accumulation in estimating population centroids in high-dimensional feature space. In fact, we demonstrate further that almost all linear discriminants can perform as bad as the random guessing. Thus, it is paramountly important to select a subset of important features for high-dimensional classification, resulting in Features Annealed Independence Rules (FAIR). The connections with the sure independent screeing (SIS) and iterative SIS(ISIS) of Fan and Lv (2007) in model selection will be elucidated and extended. The choice of the optimal number of features, or equivalently, the threshold value of the test statistics are proposed based on an upper bound of the classification error. Simulation studies and real data analysis support our theoretical results and demonstrate convincingly the advantage of our new classification procedure.

Related Links

• http://orfe.princeton.edu/~jqfan/papers/07/FAIR1.pdf - One of the paper
• http://orfe.princeton.edu/~jqfan/papers/06/SIS.pdf - Second paper

An ultrametric topology formalizes the notion of hierarchical structure. An ultrametric embedding, referred to here as ultrametricity, is implied by a hierarchical embedding. Such hierarchical structure can be global in the data set, or local. By quantifying extent or degree of ultrametricity in a data set, we show that ultrametricity becomes pervasive as dimensionality and/or spatial sparsity increases. This leads us to assert that very high dimensional data are of simple structure. We exemplify this finding through a range of simulated data cases. We discuss also application to very high frequency time series segmentation and modeling. Other applications will be described, in particular in the area of textual data mining.

References

[1] F. Murtagh, On ultrametricity, data coding, and computation, Journal of Classification, 21, 167-184, 2004.

[2] F. Murtagh, G. Downs and P. Contreras, "Hierarchical clustering of massive, high dimensional data sets by exploiting ultrametric embedding", SIAM Journal on Scientific Computing, in press, 2007.

[3] F. Murtagh, The remarkable simplicity of very high dimensional data: application of model-based clustering, submitted, 2007.

[4] F. Murtagh, Symmetry in data mining and analysis: a unifying view based on hierarchy, submitted, 2007.

Related Links

• http://www.cs.rhul.ac.uk/home/fionn/papers - Copies of papers

The general goal of clustering is to identify distinct groups in a collection of objects. To cast clustering as a statistical problem we regard the feature vectors characterizing the objects as a sample from some unknown probability density. The premise of nonparametric clustering is that groups correspond to modes of this density. The cluster tree summarizes the connectivity structure of the level sets of a density; leaves of the tree correspond to modes of the density. I will define the cluster tree, present methods for its estimating, show examples, and discuss some open problems.

Related Links

• http://www.stat.washington.edu/wxs/Learning-papers/gsl-techreport-4-24-07.pdf - Technical report

We consider the fundamental problem of estimating the mean of a vector y = X beta + z, where X is an n by p design matrix in which one can have far more variables than observations and z is a stochastic error term---the so-called `p > n' setup. When \beta is sparse, or more generally, when there is a sparse subset of covariates providing a close approximation to the unknown mean response, we ask whether or not it is possible to accurately estimate the mean using a computationally tractable algorithm.

We show that in a surprisingly wide range of situations, the lasso happens to nearly select the best subset of variables. Quantitatively speaking, we prove that solving a simple quadratic program achieves a squared error within a logarithmic factor of the ideal mean squared error one would achieve with an oracle supplying perfect information about which variables should be included in the model and which variables should not. Interestingly, our results describe the average performance of the lasso; that is, the performance one can expect in an overwhelming majority of cases where X\beta is a sparse or nearly sparse superposition of variables, but not in all cases.

Our results are sharp, nonasymptotic and widely applicable since they simply require that pairs of predictor variables be not overly collinear.

We discuss a new method of estimation of parameters in semiparametric and nonparametric models. The method is based on estimating equations that are $U$-statistics in the observations. The $U$-statistics are based on higher order influence functions that extend ordinary linear influence functions of the parameter of interest, and represent higher derivatives of this parameter. For parameters for which the matching cannot be perfect the method leads to a bias-variance trade-off, and results in estimators that converge at a slower than root-n-rate. In a number of examples the resulting rate can be shown to be optimal. We are particularly interested in estimating parameters in models with a nuisance parameter of high dimension or low regularity, where the parameter of interest cannot be estimated at root-n-rate.

What do the fishing net models of self-organizing maps look like in the data space? How do the estimated mean vectors and variance-covariance ellipses from model-based clustering fit to the clusters? How does small n, large p affect the variability in the esimates of the separating hyperplane from support vector machine models? These are a few of the things that we may discuss in this talk. The goal is to calibrate participants' eyes to viewing high-dimensional spaces and stimulate thought about what types of plots might accompany high-dimensional statistical analysis

In high-dimensional data analysis, one is often faced with the problem that real data is noisy and in many cases given in coordinates that are not informative for understanding the data structure itself or for performing later tasks, such as clustering, classification and regression. The combination of noise and high dimensions (>100-1000) presents challenges for data analysis and calls for efficient dimensionality reduction tools that take the inherent geometry of natural data into account. In this talk, I will first describe treelets – an adaptive multi-scale basis inspired by wavelets and hierarchical trees. I will then, in the second half of my talk, describe diffusion maps -- a general framework for dimensionality reduction, data set parameterization and clustering that combines ideas from eigenmaps, spectral graph theory and harmonic analysis. Our construction is based on a Markov random walk on the data, and allows one to define a system of coordinates that is robust to noise, and that reflects the intrinsic geometry or connectivity of the data points in a diffusion process. I will outline where we stand and what problems still remain.

(Part of this work is joint with R.R. Coifman, S. Lafon, B. Nadler and L. Wasserman)

Over the last few years, substantial progress has been achieved on high-dimensional variable selection (and graphical modeling) using L1-penalization methods. Diametrically opposed to penalty-based schemes is the PC-algorithm, a special hierarchical multiple testing procedure, which exploits the so-called faithfulness assumption from graphical modeling. For asymptotic consistency in high-dimensional settings, the different approaches require very different "coherence" conditions, say for the design matrix in a linear model. From a conceptual aspect, the PC-algorithm allows to identify not only regression-type associations but also directed edges in a graph and causal effects (in the sense of Pearl's intervention operator). Thereby, sparsity, faithfulness and stability play a crucial role. We will discuss potential and limitations from a theory and practical point of view.

Related Links

• http://stat.ethz.ch/~buhlmann/bibliog.html

Extracting useful information from high-dimensional data is the focus of today's statistical research and practice. Penalized loss function minimization has been shown to be effective for this task both theoretically and empirically. With the vitues of both regularization and sparsity, the L1-penalized L2 minimization method Lasso has been popular. However, Lasso is often seen as not having enough regularization in the large p case.

In this talk, we propose two methods that take into account side information in the penalized L2 framework, in order to bring the needed extra regularization in the large p case. First, we combine different norms including L1 to to introduce the Composite Absolute Penalties (CAP) family. CAP allows the grouping and hierarchical relationships between the predictors to be expressed. It covers and goes beyond existing works including grouped lasso and elastic nets. Path following algorithms and simulation results will be presented to compare with Lasso in terms of prediction and sparsity. Second, motivated by the problem of predicting fMRI signals from input natural images, we investigate a method that uses side information in the unlabeled data for prediction. We present a theoretical result in the case of p/n -> constant and apply the method to the fMRI data problem. (It is noted that the second part is a report on on-going research.) ~

Related Links

• http://www.stat.berkeley.edu/~binyu - Speaker's website for related

Sources of uncertainty related to model specification are often the single biggest factors in inference. In the predictive context, we demonstrate the effect of varying the model list used for averaging and varying the averaging strategy in computational examples. In addition, by varying the model space while using similar lists and averaging strategies, we demonstrate that the effect of the space itself computationally. Thus, it is reasonable to associate a concept of variance and bias not just to individual models but to other aspects of an overall modeling strategy. Moreover, although difficult to formalize, good prediction is seen to be associated with a sort of complexity matching between the space and the unknown function, and robustness. In some cases, the relationship among complexity, variance-bias, robustness and averaging strategy seems to be dependent on sample size. Taken together, these considerations can be formalized into an overview that may serve as a framework for more general inferential problems in Statistics

We will show that a class of model selection procedures are asymptotically sharp minimax to recover sparse signals over a wide range of parameter spaces. Connections to Bayesian model selection, the MDL principle and wavelet estimation will be discussed.

Spectral methods are of fundamental importance in statistical learning, as they underlie algorithms from classical principal components analysis to more recent approaches that exploit manifold structure. In most cases, the core technical problem can be reduced to computing a low-rank approximation to a positive-definite kernel. Using traditional methods, such an approximation can be obtained with computational complexity that scales as the cube of the number of training examples. For the growing number of applications dealing with very large or high-dimensional data sets, however, these techniques are too costly. A known alternative is the Nystrom extension from finite element methods. While its application to machine learning has previously been suggested in the literature, we introduce here what is, to the best of our knowledge, the first randomized algorithm of this type to yield a relative approximation error bound. Our results follow from a new class of algorithms for the approximation of matrix products, which reveal connections between classical linear algebraic quantities such as Schur complements and techniques from theoretical computer science such as the notion of volume sampling.

We consider the properties of a large class of learning algorithms defined in terms of classical regularization operators for ill-posed problems. This class includes regularized least-squares, Landweber method, $\nu$-methods and truncated singular value decomposition on hypotyesis spaces of vector-valued functions defined in terms of suitable reproducing kernels. In particular universal consistency, minimax rates and statistical adaptation of the methods we will be discussed.

Text mining can be thought of as a synthesis of information retrieval, natural language processing and statistical data mining. The set of documents being considered can scale to hundreds of thousands and the associated lexicon can be a million or more words. Analysis is often done by consideration of a term-document matrix or even a bigram-document matrix. The dimensionality of the term vector can thus easily be a million or more. In this talk I will describe some of the approaches to text mining on which we have been working. This is a joint work with Dr Edward Wegman.

Expectation propagation (EP) is a novel variational method for approximate Bayesian inference, which has given promising results in terms of computational efficiency and accuracy in several machine learning applications. It can readily be applied to inference in linear models with non-Gaussian priors, generalised linear models, or nonparametric Gaussian process models, among others, yet has not been used in Statistics so far to our knowledge. I will give an introduction to this framework. I will then show how to address sequential experimental design for a linear model with non-Gaussian sparsity priors, giving some results in two different machine learning applications. These results indicate that experimental design for these models may have significantly different properties than for linear-Gaussian models, where Bayesian inference is analytically tractable and experimental design seems best understood. EP as a statistical approximation technique, and especially experimental design for models different from linear-Gaussian ones, is not well-understood theoretically. To advance on the understanding, it seems promising to relate it to work in Statistics on multivariate continuous-variable distributions, and I am hoping very much for feedback from the audience in that respect.

Substantial progress has recently been made on understanding the behaviour of sparse linear models in the high-dimensional setting, where the number the variables can greatly exceed the number of samples. This problem has attracted the interest of multiple communities, including applied mathematics, signal processing, statistics and machine learning. But linear models often rely on unrealistically strong assumptions, made mainly for convenience. Going beyond parametric models, can we understand the properties of high-dimensional functions that enable them to be estimated accurately from sparse data? In this talk we present some progress on this problem, showing that many of the recent results for sparse linear models can be extended to the infinite-dimensional setting of nonparametric function estimation. In particular, we present some theory for estimating sparse additive models, together with algorithms that are scalable to high dimensions. We illustrate these ideas with an application to functional sparse coding of natural images. This is joint work with Han Liu, Pradeep Ravikumar, and Larry Wasserman.

Consider the canonical regression set-up where one wants to learn about the relationship between y, a variable of interest, and x_1,...,x_p, p potential predictor variables. Although one may ultimately want to build a parametric model to describe and summarize this relationship, preliminary analysis via flexible nonparametric models may provide useful guidance. For this purpose we propose BART (Bayesian Additive Regression Trees), a flexible nonparametric ensemble Bayes approach for estimating f(x_1,...,x_p), which is E(Y|x_1,...,x_p), for obtaining predictive regions for future y, for describing the marginal effects of subsets of x_1,...,x_p and for model-free variable selection. Essentially, BART approximates f by a Bayesian 'sum-of-trees' model where fitting and inference are accomplished via an iterative backfitting MCMC algorithm. By using a large number of trees, which yields a redundant basis for f, BART is seen to be remarkably effective at finding highly nonlinear relationships hidden within a large number of irrelevant potential predictors. BART also provides an omnibus test: the absence of any relationship between y and any subset of x_1,...,x_p is indicated when BART posterior intervals for f reveal no signal. (This is joint work with Hugh Chipman and Robert McCulloch.)

In this talk I will first discuss Gaussian Process Functional Regression (GPFR) model, which is used to model functional response curves with a set of functional covariates (the dimension of the covariates may be very large). There are two main features: modelling nonlinear and nonparametric functional regression relationship and modelling covariance structure and mean structure simultaneously. The method gives very accurate results for curve fitting and prediction but side-steps the problem of heterogeneity. I will then discuss how to define a hierarchical mixture model to model 'spatially' indexed functional data, i.e., the heterogeneity is dependent on factors such as region or individual patient's information. The mixture model has also been used for curve clustering, but focusing on the problem of clustering functional relationships between response curve and covariates, i.e. the clustering is based on the surface shape of the functional response against the set of functional covariates. Some applications based on simulated data and real data will be presented.

We discuss the approximation of data from one- and two-dimensional curves using total-variation-based techniques. Our aim will be to minimise complexity among all functions which satisfy a criterion for approximation. Complexity will be measured by the number of local extreme values or variational properties of the functions. Our criteria for approximation will be based on a multiscale analysis of the residuals.

Everything you were taught about underdetermined systems of linear equations is wrong...

Okay, that's too strong. But you have been taught things in undergraduate linear algebra which, if you are an engineer or scientist, may be holding you back. The main one is that if you have more unknowns than equations, you're lost. Don't believe it. At the moment there are many interesting problems in the information sciences where researchers are currently confounding expectations by turning linear algebra upside down:

• (a) An standard magnetic resonance imaging device can now produce a clinical-quality image using a factor 8 less time than previously thought.
• (b) A Fourier imaging system can observe just the lowest frequencies of a sparse nonnegative signal and perfectly reconstruct all the unmeasured high frequencies of the signal.
• (c) a communications system can transmit a very weak signal perfectly in the presence of intermittent but arbitrarily powerful jamming.

Moreover, in each case the methods are convenient and computationally tractable.

Mathematically, what's going on is a recent explosion of interest in finding the sparsest solution to certain systems of underdetermined linear equations. This problem is known to be NP-Hard in general, and hence the problem sounds intractable. Surprisingly, in some particular cases, it has been found that one can find the sparsest solution by l¹ minimization, which is a convex optimization problem and so tractable. Many researchers are now actively working to explain and exploit this phenomenon. It's responsible for the examples given above.

In my talk, I'll discuss that this curious behavior of l¹ minimization and connect with some pure mathematics -- convex polytope theory and oriented matroid theory.

Ultimately, the pure math behind this phenomenon concerns some accessible but very surprising properties of random point clouds in very high dimensions: each point gets very neighborly!

I'll also explain the connection of this phenomenon to the Newton Institute's ongoing program "Statistical Theory and Methods for Complex, High-Dimensional Data".

Independence graphs give an overview of multivariate dependency. After a brief introduction to information divergence and to conditional independence graphs we show DWIGs fall within the paradigm of design based inference. Bootstrap resampling tests the stability of the DWIG parameters when increasing the dimension of the underlying data set.

The Bernstein-Von Mises theorem provides a detailed relation between frequentist and Bayesian statistical methods in smooth, parametric models. It states that the posterior distribution converges to a normal distibution centred on the maximum-likelihood estimator with covariance proportional to the Fisher information. In this talk we consider conditions under which such an assertion holds for the marginal posterior of a parameter of interest in semiparametric models. From a practical point of view, this enables the use of Bayesian computational techniques (e.g. MCMC simulation) to obtain (hard to compute otherwise) frequentist confidence intervals. (Joint work with P. Bickel.)

We will discuss the 0-dimensional statistical problem of alignment, and its relation to the high-dimensional problem of phylogeny. In particular, we will discuss the relevance of the "space of phylogenetic oranges" and its relation to the above problems. We will also discuss "sequence annealing", which is a new alignment strategy based on these ideas.

An important problem in microarray experiments is the detection of genes that are differentially expressed in a given number of classes. As there are usually thousands of genes to be considered simultaneously, one encounters high-dimensional testing problems. We provide a straightforward and easily implemented method for estimating the posterior probability that an individual gene is null (not differentially expressed). The problem can be expressed in a two-component mixture framework, using an empirical Bayes approach. Current methods of implementing this approach either have some limitations due to the minimal assumptions made or with the computationally intensive nature of more specific assumptions. By converting to a z-score the value of the test statistic used to test the significance of each gene, we propose a simple two-component normal mixture that models adequately the distribution of this score. The approach provides an estimate of the local false discovery rate (FDR) for each gene, which is taken to be the posterior probability that the gene is null. Genes with the local FDR less than a specified threshold C are taken to be differentially expressed. For a given C, this approach also provides estimates of the implied overall errors such as the (global) FDR and the false negative/positive rates.

Related Links

• http://bioinformatics.oxfordjournals.org/cgi/reprint/22/13/1608 - Main reference paper

Over the past three years it has become rapidly appreciated that the human genome varies in its structure as well as its sequence, by virtue of a panoply of different chromosomal rearrangements, some that alter the number of copies of DNA segments, and others that alter orientation but not copy number. Evidence is growing from diverse sources that this source of genomic variation has an appreciable functional impact, and yet we remain far from a complete catalogue of this form of variation let alone its biological consequences. In my talk I will summarise the progress to date and highlight the appreciable statistical challenges that remain, with particular reference to the approaches being adopted in my group towards assaying copy number variation and assessing its impact on complex traits through genetic association studies.

With the advent of new sequencing technologies that reduce the cost of DNA sequence by a factor of a hundred, we have moved into the era of population genomic sequencing, where we sample many individuals from a population to study natural genetic variation genome-wide. However, at this scale sequencing is still costly. I will discuss strategies to use low coverage sequencing on multiple samples from a population, and some of the complications in using the resulting data for population genetic analyses. Examples will be drawn from the Saccharomyces Genome Resequencing Project (SGRP) in which we have collected sequence data from 70 yeast strains, and planning for the 1000 Genomes Project to characterise human genetic variation down to 1% allele frequency.

Related Links

• http://www.sanger.ac.uk/Teams/Team71/durbin/sgrp/ - SGRP: Yeast resequencing project
• http://www.1000genomes.org/ - 1000 Genomes Project

The recent comparative analysis of the human genome has revealed a large fraction of functionally constrained non-coding DNA in mammalian genomes. However, our understanding of the function of non-coding DNA is very limited. In this talk I will present recent analysis in my group and collaborators that aims at the identification of functionally variable regulatory regions in the human genome by correlating SNPs and copy number variants with gene expression data. I will also be presenting some analysis on inference of trans regulatory interactions and evolutionary consequences of gene expression variation.

Motivated by the proposed format of talks, we include the following: (i) a review of statistical facts about L1-regularization for high-dimensional problems; (ii) some adaptations of motif regression (Conlon, Liu, Lieb & Liu, 2003) for scoring potential motifs or for presence/absence of other biological targets of interest (e.g. proteins) by integrating multiple data sources; (iii) using the concepts for analyzing ChIP-on-chip data from human liver cells (with a side remark on signal extraction) for HIF-dependent transcriptional networks.

Issue (i) deals with a general purpose method for variable selection or feature extraction which is potentially useful for a broad variety of (multiple) bio-molecular and high-dimensional data. Issue (ii) is - in our experience - an interesting method to improve upon some chosen "standard" methodology by making use of additional data sources. Finally, issue (iii) is work in progress with the Ricci lab at ETH Zurich: it is an illustration for statisticians and - of course - the "real thing" for biologists.

Related Links

• http://www.stat.ethz.ch/~buhlmann

The diversity of virus populations within single infected hosts presents a major difficulty for the natural immune response, vaccine design, and antiviral drug therapy. Recently developed ultra-deep sequencing technologies can be used for quantifying this diversity by direct sequencing of the mixed virus population. We present statistical and computational methods for the analysis of such sequence data. Inference of the population structure from observed reads is based on error correction, reconstruction of a minimal set of haplotypes that explain the data, and eventually estimation of haplotype frequencies. We demonstrate our approach by analyzing simulated data and by comparison to 165 sequences obtained from clonal Sanger sequencing of four independent, diverse HIV populations.

Related Links

• http://arxiv.org/abs/0707.0114 - preprint

Inference about genomic features faces the particular difficulty that, save for interspecies variation, we have only one copy of any of the genomes that Nature might have produced. We postulate a framework which includes an “ergodic hypothesis” which permits us to compute p values and confidence bounds. These seem to be as conservative as could be hoped for. Out methods in crude form were applied to data for the ENCODE project (Birney et al (2007)). We will discuss our model and refinements of the methods previously proposed.

I will discuss methods we use to identify active gene regulatory elements within the human genome and some of the current obstacles and hurdles we still need to overcome

Embryonic stem (ES) cells are similar to the transient population of self-renewing cells within the inner cell mass of the preimplantation blastocyst (epiblast), capable of pluripotential differentiation to all specialised cell types comprising the adult organism. These cells undergo continuous self-renewal to produce identical daughter cells, or can develop into specialised progenitors and terminally differentiated cells. A variety of molecular pathways involved in embryonic development have been elucidated, including those influencing stem cell differentiation. As a result, we know of a number of key transcriptional regulators and signalling molecules that play essential roles in manifesting nuclear potency and self-renewal capacity of embryo-derived and tissue-specific stem cells. Despite these efforts however, a small number of components have been identified and large-scale characterisation of these processes remains incomplete. While the precise biological niche is believed to direct differentiation and development in vivo, it is now possible to utilise explanted stem cell lines as an in vitro model of cell fate assignment and differentiation. The aim of the studies discussed here is to map the global transcriptomic and proteomic activity of ES cells during various stages of differentiation and lineage commitment in tissue culture. This approach will help characterise the functional roles of key developmental regulators and yield more rational approaches to manipulating stem cell behaviour in vitro. The generation of large-scale data from microarray and functional genomic experiments will help to identify and characterise the regulatory influence of key transcription factors, signaling genes and non-coding RNAs involved in early developmental pathways, leading to a more detailed understanding of the molecular mechanisms of vertebrate embryogenesis.

Small molecule metabolism is the highly coordinated interconversion of chemical substrates through enzyme-catalysed reactions. It is central to the viability of all organisms as it enables the assimilation of nutrients for energy production and the synthesis of precursors for all cellular components. The system is tightly regulated so cells can respond efficiently to environmental changes. This is optimised to minimise the substantial cost of enzyme production and core metabolite depletion, and to maximise the benefit of cell growth and division. It is commonly known that this regulation is achieved by controlling either (i) the availability of enzymes or (ii) their activities. Though the molecular mechanisms behind these two regulatory processes have been elucidated in great detail, and we still lack insight into how they are deployed and complement each other at a global level. Here, I will present a genome-scale analysis of how regulatory feedback by small molecules control the metabolic system, and examine how the two modes of regulation are deployed throughout the system.

Bio: Nick Luscombe, Group Leader, EMBL-European Bioinformatics Institute Nick completed his PhD with Professor Janet Thornton at University College London (1996-2000), studying the basis for specificity of DNA-binding proteins. He then moved to Yale University as a post-doctoral fellow with Professor Mark Gerstein (2000-2004). During this time, he shifted his research focus to genomics, with a particular emphasis on transcriptional regulation in yeast. He has been a Group Leader at EMBL-EBI since 2005, examining the control of interesting biological systems.

It has long been recognized that covariate adjustment can increase precision in randomized experiments, even when it is not strictly necessary. Adjustment is often straightforward when a discrete covariate partitions the sample into a handful of strata, but becomes more involved when modern studies collect copious amounts of baseline information on each subject. This dilemma helped motivate locally efficient estimation, in which one attempts to gain efficiency through a (possibly misspecified) working model. However, with complex high-dimensional covariates, where one might have no belief in the working model, misspecification can actually decrease precision. We propose a new method, empirical efficiency maximization, to target the working model element minimizing asymptotic variance for the resulting parameter estimate, whether or not the working model is (approximately) correct. Gains are demonstrated relative to standard locally efficient estimators.

We take a graph theoretic approach to the component ordering problem in the layout of statistical graphics. We use Eulerian tours and Hamiltonian decompositions of complete graphs to ameliorate order effects. Similarly, visual effects of selected salient features in the data are amplified with traversals of edge weighted graphs. Examples of these techniques include improved versions of multiple comparison displays, interaction plots, star glyph displays and parallel coordinate plots. Improved versions of interaction plots and star glyph displays are described based on graph traversals. We present algorithms based on classical graph theory methods. These along with the new graphical displays are available as an R package.

This is joint work with R.W. Oldford (Waterloo).

The talk briefly reviews generalisation bounds for Support Vector Machines and poses the question of whether the spectrum of the empirical covariance matrix can be used to improve the quality of the bounds. Early results in this direction are surveyed before introducing a recent bound on the number of dichotomies of a graph in terms of the spectrum of the graph Laplacian. This result gives a bound on transductive algorithms that minimise the cut size of the classification. The result is then generalised to other bilinear forms and hence applied to Support Vector Classification. In order to obtain an inductive bound the eigenvalues of the true covariance must be estimated from those of a sample covariance matrix. Possible improvements in the quality of the bound are discussed.

The k-nearest-neighbour procedure is a well-known deterministic method used in supervised classification. This paper proposes a reassessment of this approach as a statistical technique derived from a proper probabilistic model; in particular, we modify the assessment made in a previous analysis of this method undertaken by Holmes & Adams (2002,2003), and evaluated by Manocha & Girolami (2007), where the underlying probabilistic model is not completely well-defined. Once a clear probabilistic basis for the $k$-nearest-neighbour procedure is established, we derive computational tools for conducting Bayesian inference on the parameters of the corresponding model. In particular, we assess the difficulties inherent to pseudo-likelihood and to path sampling approximations of an intractable normalising constant, and propose a perfect sampling strategy to implement a correct MCMC sampler associated with our model. If perfect sampling is not available, we suggest using a Gibbs sampling approximation. Illustrations of the performance of the corresponding Bayesian classifier are provided for several benchmark datasets, demonstrating in particular the limitations of the pseudo-likelihood approximation in this set-up.

[Joint work with L. Cucala, J.-M. Marin, and D.M. Titterington]

Donoho and Jin (2004), following work of Ingster (1999), studied the problem of testing for a signal in a sparse normal means model and showed that there is a ``detection boundary'' above which the signal can be detected and below which no test has any power. They showed that Tukey's ``higher criticism'' statistic achieves the detection boundary. I will introduce a new family of test statistics based on phi-divergences (indexed by a real number s with values between -1 and 2)which all achieve the Donoho-Jin-Ingster detection boundary. I will also review recent work on estimating the proportion of non-zero means.

The topic of l_1 regularized or Lasso-type estimation has received considerable attention over the past decade. Recent theoretical advances have been mainly concerned with the risk of the estimators and corresponding sparsity oracle inequalities. In this talk we will imvestigate the quality of the l_1 penalized estimators from a different perspective, shifting the emphasis to non-asymptotic variable selection, which complements the consistent variable selection literature. Our main results are established for regression models, with emphasis on the square and logistic loss. The identification of the tagged SNPs associated with a disease, in genome-wide association studies, provides the principal motivation for this analysis. The performance of the method depends crucially on the choice of the tuning sequence and we discuss non-asymptotic choices for which we can correctly detect sets of variables associated with the response at any pre-specified confidence level. These tuning sequences are different for the two loss functions, but in both cases larger than those required for best risk performance, The stability of the design matrix is another major issue in correct variable selection, especially when the total number of variables exceeds the sample size. A possible solution os provided by further regularization, for instance via an l_1+l_2 or elastic net penalty. We discuss the merits and limitations of this method in the same context as above.

We consider the problem of binary classification where one can, for a particular cost, choose not to classify an observation. We present a simple oracle inequality for the excess risk of structural risk minimizers using a generalized lasso penalty.

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 components 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. Even in situations where Euclidean analysis makes sense, there are statistical challenges because of the High Dimension Low Sample Size problem, which motivates a new type of asymptotics leading to non-standard mathematical statistics.

Having built a probabilistic model, a natural question is: "what probability does my model assign to the data?".

We might fit the model's parameters to avoid having to compute an intractable marginal likelihood. Even then, evaluating a test-set probability with fixed parameters can be difficult. I will discuss recent work on evaluating high-dimensional undirected graphical models and models with many latent variables. This allows direct comparisons of the probabilistic predictions made by graphical models with hundreds of thousands of parameters against simpler alternatives.

Bayes factors provide a means of objectively ranking a number of plausible statistical models based on their evidential support. Computing Bayes factors is far from straightforward and methodology based on thermodynamic integration can provide stable estimates of the integrated likelihood. This talk will consider a stratified sampling strategy in estimating the thermodynamic integral and will consider issues such as optimal paths and the variance of the overall estimator. The main application considered will be the computation of Bayes factors for biochemical pathway models based on systems of nonlinear ordinary differential equations (ODE). A large scale study of the ExtraCellular Regulated Kinase (ERK) pathway will be discussed where recent Small Interfering RNA (siRNA) experimental validation of the predictions made using the computed Bayes factors is presented.

The expression of genes as messenger RNA (mRNA) in the cell is regulated by the activity of transcription factor proteins. The measurement of mRNA concentration for essentially all genes can be routinely carried out using high-throughput experimental techniques such as microarrays. It is much less straightforward to measure the concentration of activated transcription factor proteins. Latent variable models have therefore been developed which treat transcription factors as unobserved chemical species who's active concentration and effect can be inferred indirectly from the expression levels of their target genes. We are developing two classes of latent variable models. For small sub-systems, e.g. genes controlled by a single transcription factor, we model the process of transcription using ordinary differential equations with the transcription factor's concentration modeled using a Gaussian process prior distribution over functions. For larger systems, with hundreds of transcription factors controlling thousands of genes, we use simple discrete-time or non-temporal linear models. Bayesian methods provide a natural means for inference of transcription factor concentrations and other model parameters of interest.

Joint work with Neil Lawrence.

We consider the functional linear regression model where the explanatory variable is a random surface and the response is a real random variable, with bounded or normal noise. Bivariate splines over triangulations represent the random surfaces. We use this representation to construct least squares estimators of the regression function with or without a penalization term. Under the assumptions that the regressors in the sample are bounded and span a large enough space of functions, bivariate splines approximation properties yield the consistency of the estimators. Simulations demonstrate the quality of the asymptotic properties on a realistic domain. We also carry out an aplication to ozone forecasting over the US that illustrates the predictive skills of the method.

This is joint work with Ming-Jun Lai.

Global testing is an accepted strategy for 'screening' regression problems and controlling the family-wise error rate. For p>n a global test was introduced by Jelle Goeman based on a random effect imbedding of the regression problem. This talk will first review this global test and discuss its link with goodness of fit tests. Secondly, its use as a screening instrument will be discussed along with its link with crossvalidation error. Finally, an adaptation will be presented that is better suited for screening on predictability. References. Goeman et al. (2006) JRSSB, 68, 477-493. Goeman et al. (2005) Bioinformatics 21, 1950-1957. Goeman et al. (2004) Bioinformatics 20, 93-99. LeCessie and van Houwelingen (1995) Biometrics 51, 600-614.

Many time series in economics and finance are non-Gaussian. In this paper, we propose a Bayesian approach to non-Gaussian autoregressive quantile function time series models where the scale parameter of the models does not depend on the values of the time series. This approach is parametric. So we also compare the proposed parametric approach with the semi-parametric approach (Koenker, 2005). Simulation study and applications to real time series show that the method works very well.

The auxiliary particle filter (APF) is a popular algorithm for the Monte Carlo approximation of the optimal filtering equations of state space models. This talk presents a summary of several recent developments which affect the practical implementation of this algorithm as well as simplifying its theoretical analysis. In particular, an interpretation of the APF, which makes use of an auxiliary sequence of distributions, allows the approach to be extended to more general Sequential Monte Carlo algorithms. The same interpretation allows existing theoretical results for standard particle filters to be applied directly. Several non-standard implementations and applications will also be discussed.

In this paper, we develop set of novel Markov chain Monte Carlo algorithms for Bayesian inference in partially observed non-linear diffusion processes. The Markov chain Monte Carlo algorithms we develop herein use an approximating distribution to the true posterior as the proposal distribution for an independence sampler. The approximating distribution utilises the posterior approximation computed using the recently developed variational Gaussian Process approximation method. Flexible blocking strategies are then introduced to further improve the mixing, and thus the efficiency, of the Markov chain Monte Carlo algorithms. The algorithms are tested on two cases of a double-well potential system. It is shown that the blocked versions of the variational sampling algorithms outperform Hybrid Monte Carlo sampling in terms of computational efficiency, except for cases where multi-modal structure is present in the posterior distribution.

Continuous time Markov processes (such as jump processes and diffusions) play an important role in the modelling of dynamical systems in many scientific areas.

In a variety of applications, the stochastic state of the system as a function of time is not directly observed. One has only access to a set of nolsy observations taken at a discrete set of times. The problem is then to infer the unknown state path as best as possible. In addition, model parameters (like diffusion constants or transition rates) may also be unknown and have to be estimated from the data. While it is fairly straightforward to present a theoretical solution to these estimation problems, a practical solution in terms of PDEs or by Monte Carlo sampling can be time consuming and one is looking for efficient approximations. I will discuss approximate solutions to this problem such as variational approximations to the probability measure over paths and weak noise expansions.

We present a framework for maintaining and updating a time varying distribution over permutations matching tracks to real world objects. Our approach hinges on two insights from the theory of harmonic analysis on noncommutative groups. The first is that it is sufficient to maintain certain “low frequency” Fourier components of this distribution. The second is that marginals and observation updates can be efficiently computed from such components by extensions of Clausen’s FFT for the symmetric group.

Related Links

• http://www.gatsby.ucl.ac.uk/~risi - home page

We propose a powerful prediction algorithm built upon Gaussian processes (GPs). They are particularly useful for their flexibility, facilitating accurate prediction even in the absence of strong physical models. GPs further allow us to work within a completely Bayesian framework. As such, we show how the hyperparameters of our system can be marginalised by use of Bayesian Monte Carlo, a principled method of approximate integration. We employ the error bars of the GP's prediction as a means to select only the most informative observations to store. This allows us to introduce an iterative formulation of the GP to give a dynamic, on-line algorithm. We also show how our error bars can be used to perform active data selection, allowing the GP to select where and when it should next take a measurement. We demonstrate how our methods can be applied to multi-sensor prediction problems where data may be missing, delayed and/or correlated. In particular, we present a real network of weather sensors as a testbed for our algorithm.

We discuss Markov chain Monte Carlo algorithms for sampling functions in Gaussian process models. A first algorithm is a local sampler that iteratively samples each local part of the function by conditioning on the remaining part of the function. The partitioning of the domain of the function into regions is automatically carried out during the burn-in sampling phase. A more advanced algorithm uses control variables which are auxiliary function values that summarize the properties of the function. At each iteration, the algorithm proposes new values for the control variables and then generates the function from the conditional Gaussian process prior. The control input locations are found by minimizing the total variance of the conditional prior. We apply these algorithms to estimate non-linear differential equations in Systems Biology.

In this talk, we present in a common unifying framework several adaptive Monte Carlo Markov chain algorithms (MCMC) that have been recently proposed in the literature. We prove that under a set of verifiable conditions, ergodic averages calculated from the output of a so-called adaptive MCMC sampler converge to the required value and can even, under more stringent assumptions, satisfy a central limit theorem. We prove that the conditions required are satisfied for the Independent Metropolis-Hastings algorithm and the Random Walk Metropolis algorithm with symmetric increments. Finally we propose an application of these results to the case where the proposal distribution of the Metropolis-Hastings update is a mixture of distributions from a curved exponential family. Several illustrations will be provided.

Microstructure noise can substantially bias the estimation of volatility of an Ito process. Such noise is inherently multiscale, causing eventual inconsistency in estimation as the sampling rate becomes more frequent. Methods have been proposed to remove this bias using subsampling mechanisms. We instead take a frequency domain approach and advocate learning the degree of contamination from the data. The volatility can be seen as an aggregation of contributions from many different frequencies. Having learned the degree of contamination allows us to frequency-by-frequency correct these contributions and calculate a bias-corrected estimator. This procedure is fast, robust to different signal to microstructure scenarios, and is also extended to the problem of correlated microstructure noise. Theory can be developed as long as the Ito process has harmonizable increments, and suitable dynamic spectral range.

The properties of L1-penalized regression have been examined in detail in recent years. I will review some of the developments for sparse high-dimensional data, where the number of variables p is potentially very much larger than sample size n. The necessary conditions for convergence are less restrictive if looking for convergence in L2-norm than if looking for convergence in L0-quasi-norm. I will discuss some implications of these results. These promising theoretical developments notwithstanding, it is unfortunately often observed in practice that solutions are highly unstable. If running the same model selection procedure on a new set of samples, or indeed a subsample, results can change drastically. The choice of the proper regularization parameter is also not obvious in practice, especially if one is primarily interested in structure estimation and only secondarily in prediction. Some preliminary results suggest, though, that the stability or instability of results is informative when looking for suitable data-adaptive regularization.

In a recent paper by Bovelstad et al. [1] partial likelihood ridge regression as used in [2] turned out to be the most successful approach to predicting survival with gene expression data.

However the proportional hazard model used in these models is quite simple and might not be realistic if there is a long survival follow-up. Exploring the fit of the model by using a cross-validated prognostic index leads to the conclusion that the effect of the predictor derived in [2] is neither linear nor constant over time.

We will discuss penalized reduced rank models as a way to obtain robust extensions of the Cox model for this type of data. For time varying effects the reduced rank model of [3] can be employed, while nonlinear effects can be introduced by means of bilinear terms. The predictive performance of such models can be regulated by penalization in combination with cross-validation.

References [1] Bovelstad, HM; Nygard, S; Storvold, HL; et al. Predicting survival from microarray data - a comparative study BIOINFORMATICS, 23 (16): 2080-2087 AUG 15 2007 [2] van Houwelingen, HC; Bruinsma, T; Hart, AAM; et al. Cross-validated Cox regression on microarray gene expression data STATISTICS IN MEDICINE, 25 (18): 3201-3216 SEP 30 2006 [3] Perperoglou, A; le Cessie, S; van Houwelingen, HC Reduced-rank hazard regression for modeling non-proportional hazards STATISTICS IN MEDICINE, 25 (16): 2831-2845 AUG 30 2006

The quality of solving several statistical problems, such as adaptive nonparametric estimation, aggregation of estimators, estimation under the sparsity scenario and weak learning can be assessed in terms of sparsity oracle inequalities (SOI) for the prediction risk. One of the challenges is to build estimators that attain the sharpest SOI under minimal assumptions on the dictionary. Methods of sparse estimation are mainly of the two types. Some of them, like the BIC, enjoy nice theoretical properties in terms of SOI without any assumption on the dictionary but are computationally infeasible starting from relatively modest dimensions p. Others, like the Lasso or the Dantzig selector, can be easily realized for very large p but their theoretical performance is conditioned by severe restrictions on the dictionary. We will focus on Sparse Exponential Weighting, a new method of sparse recovery realizing a compromise between theoretical properties and computational efficiency. The theoretical performance of the method in terms of SOI is comparable with that of the BIC. No assumption on the dictionary is required. At the same time, the method is computationally feasible for relatively large dimensions p. It is constructed using an exponential weighting with suitably chosen priors, and its analysis is based on the PAC-Bayesian ideas in statistical learning.

Advances in bioengineering technologies are generating the ability to measure increasingly high-resolution, dynamic data on complex cellular networks at multiple biological and temporal scales. Single-cell molecular studies, in which data is generated on the levels of expression of a small number of proteins within individual cells over time using time-lapse fluorescent microscopy, is one critical emerging area. Single cell experiments have potential to develop centrally in both mechanistic studies of natural biological systems as well as via synthetic biology -- the latter involving engineering of small cellular networks with well-defined function, so providing opportunity for controlled experimentation and bionetwork design. There is a substantial lag, however, in the ability to integrate, understand and utilize data generated from single-cell fluorescent microscopy studies. I will highlight aspects of this area from the perspective of our work in single cell studies in synthetic bacterial systems that emulate key aspects of mammalian gene networks central to all human cancers. I will touch on:

(a) DATA: Raw data come as movies of colonies of cells developing through time, with a need for imaging methods to estimate cell-specific levels of fluorescence measuring mRNA levels of one or several tagged genes within each cell. This is complicated by the progression of cells through multiple cell divisions that raises questions of tracking the lineages of individual cells over time.

(b) MODELS: In the context of our synthetic gene networks engineered into bacterial cells, we have developed discrete-time statistical dynamic models inspired by basic biochemical network modelling of the stochastic regulatory gene network. These models allow the incorporation of multiple components of noise that is "intrinsic" to biological networks as well as approximation and measurement errors, and provide the opportunity to formally evaluate the capacity of single cell data to inform on biochemical parameters and "recover" network structure in contexts of contaminating noise.

(c) INFERENCE & COMPUTATION: Our approaches to model fitting have developed Bayesian methods for inference in non-linear time series. This involves MCMC methods that impute parameter values coupled with novel, effective Metropolis methods for what can be very high-dimensional latent states representing the unobserved levels of mRNA or proteins on nodes in the network as well as contributions from "missing" nodes.

This work is collaborative with Jarad Niemi and Quanli wang (Statistical Science at Duke), Lingchong You and Chee-Meng Tan (Bioengineering at Duke).

Bioinformatics came to the scene when biology started to automate its experiments. Although this would have led to “large n and small p” situations in other sciences, the complex nature of biology meant that it soon started to focus on lots of different variables, resulting in now well-known “small n, large p” situations. One such case is the inference of regulatory networks: the amount of networks is exponential in the number of nodes, whereas the available data is typically just a fraction thereof. We will present a penalized inference method that deals with such problems, that draws on experience with hypothesis testing. It has similarities with Approximate Bayesian Computation and seems to lead to exact inference in a few specific cases.

I will discuss some procedures for modelling high-dimensional data, based on L1 (lasso) -style penalties. I will describe pathwise coordinate descent algorithms for the lasso, which are remarkably fast and facilitate application of the methods to very large datasets for the first time. I will then give examples of new applications of the methods to microarray classification, undirected graphical models for cell pathways, and the fused lasso for signal detection, including comparative genomic hybridization.

Sample reuse methods like the bootstrap and cross-validation are widely used in statistics and machine learning. They provide measures of accuracy with some face value validity that is not dependent on strong model assumptions.

These methods depend on repeating or omitting cases, while keeping all the variables in those cases. But for many data sets, it is not obvious whether the rows are cases and colunns are variables, or vice versa. For example, with movie ratings organized by movie and customer, both movie and customer IDs can be thought of as variables.

This talk looks at bootstrap and cross-validation methods that treat rows and columns of the matrix symmetrically. We get the same answer on X as on X'. McCullagh has proved that no exact bootstrap exists in a certain framework of this type (crossed random effects). We show that a method based on resampling both rows and columns of the data matrix tracks the true error, for some simple statistics applied to large data matrices.

Similarly we look at a method of cross-validation that leaves out blocks of the data matrix, generalizing a proposal due to Gabriel that is used in the crop science literature. We find empirically that this approach provides a good way to choose the number of terms in a truncated SVD model or a non-negative matrix factorization. We also apply some recent results in random matrix theory to the truncated SVD case.

Related Links

• http://stat.stanford.edu/~owen/reports - Page with research articles
• http://stat.stanford.edu/~owen/reports/cvsvd.pdf - Technical report on bi-cross-validation (in revision)
• http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&page=toc&handle=euclid.aoas/1196438015 - AOAS page with link to Pigeonhole boostrap paper

A number of problems in regression and classification can be stated as penalized empirical risk minimization over a linear span or a convex hull of a given dictionary with convex loss and convex complexity penalty, such as, for instance, $\ell_1$-norm. We will discuss several oracle inequalities showing how the error of the solution of such problems depends on the "sparsity" of the problem and the "geometry" of the dictionary.

It is well known that sample means and sample covariance matrices are independent if the samples are from the Gaussian distribution and are i.i.d.. In this talk, via investigating the random quardratic forms involving sample means and sample covariance matrices, we suggest the conjecture that the sample means and the sample covariance matrices under general distribution functions are asymptotically independent in the large dimensional case when the dimension of the vectors and the sample size both go to infinity with their ratio being a positive constant. As a byproduct, the central limit theorem for the Hotelling $T^2$ statistic under the large dimensional case is established.

Statistical theory for estimating large covariance matrix shows that even for noiseless synchronized high-frequency financial data, the existing realized volatility based estimators of integrated volatility matrix of p assets are inconsistent, for large p (the number of assets and large n (the sample size for high-frequency data). This paper proposes new types of estimators of integrated volatility matrix for noisy non-synchronized high-frequency data. We show that when both n and p go to infinity with p/n approaching to a constant, the proposed estimators are consistent with good convergence rates. Our simulations demonstrate the excellent performance of the proposed estimators under complex stochastic volatility matrices. We have applied the methods to high-frequency data with over 600 stocks.

Estimation of covariance matrices has a number of applications, including principal component analysis, classification by discriminant analysis, and inferring independence and conditional independence between variables, and the sample covariance matrix has many undesirable features in high dimensions unless regularized. Recent research mostly focused on regularization in situations where variables have a natural ordering. When no such ordering exists, regularization must be performed in a way that is invariant under variable permutations. This talk will discuss several new sparse covariance estimators that are invariant to variable permutations. We obtain convergence rates that make explicit the trade-offs between the dimension, the sample size, and the sparsity of the true model, and illustrate the methods on simulations and real data. We will also discuss a method for finding a "good" ordering of the variables when it is not provided, based on the Isomap, a manifold projection algorithm.

The talk includes joint work with Adam Rothman, Amy Wagaman, Ji Zhu (University of Michigan) and Peter Bickel (UC Berkeley).

We consider the problem of selecting the best or most informative dimension for dimension reduction and feature extraction in high-dimensional data. We review current methods, and propose a dimension selector based on Independent Component Analysis which finds the most non-Gaussian lower-dimensional directions in the data. A criterion for choosing the optimal dimension is based on bias-adjusted skewness and kurtosis. We show how this dimension selector can be applied in supervised learning with independent components, both in a regression and classification framework.

Abstract: We present a theory of point and interval estimation for nonlinear functionals in parametric, semi-, and non-parametric models based on higher order influence functions. The theory reproduces many previous results, produces new non-root n results, and opens up the ability to perform optimal non-root n inference in complex high dimensional models. We present novel rate-optimal point and intervals estimators for various functionals of central importance to biostatistics in settings in which estimation at the expected root n rate is not possible, owing to the curse of dimensionality. We also show that our higher order influence functions have a multi-robustness property that extends the double robustness property of first order influence functions. Open questions will be discussed

More than a thousand small planetary bodies with radii >100 km have recently been detected beyond Neptune using large telescopes. The purpose of the TAOS project is to measure directly the number of these Kuiper Belt Objects (KBO’s) down to the typical size of cometary nuclei (a few km). When a KBO moves in between the earth and a distant star it will block the starlight momentarily, for about a quarter of a second. A telescope monitoring the starlight will thus see it blinking. Three small (20 inch) dedicated robotic telescopes equipped with 2,048 x 2,048 CCD cameras are operated in a coincidence so that the sequence and timing of the three separate blinks can be used to distinguish real events from false alarms. A fourth telescope will be added soon. TAOS will increase our knowledge about the Kuiper Belt, the home of most short period comets that return to the inner solar system every few years. This knowledge will help us to understand the formation and evolution of comets in the early solar system as well as to estimate their flux of impacting our home planet.

In this talk I will describe some of the statistical challenges that arise when hundreds or thousands of stars are simultaneously monitored every quarter of a second, every night of the year on which observation is possible, with the aim of detecting a few events. TAOS will produce a databank of the order of 10 terabytes per year, which is small by the standards of recent and future astronomical surveys. My intent in this talk is not to provide definitive methods of analysis but, rather, I hope that this concrete example of high dimensional non-Gaussian data informs the discussion of future directions in high dimensional data analysis to which this meeting is devoted.

Related Links

• http://taos.asiaa.sinica.edu.tw/