Christopher Genovese - US grants

Proposal: DMS 9505007 PI(s): William Eddy, Chris Genovese Institution: Carnegie Mellon University Title: Statisitcal Methods for the Analysis of Functional Magnetic Resonance Imaging Data Abstract: This research involves the development of new statistical methods for the analysis and interpretation of functional Magnetic Resonance Imaging (fMRI) data. Such data can be viewed as the realization of a spatio-temporal process with a very complicated distributional structure. Models in current use are grossly simplified for both mathematical and computational expediency. The statistical challenges in constructing more realistic models are difficult and manifold. Many revolve around understanding the nature of the noise in the measurements and its effect on successfully detecting regions of neural activation. Noise in the data shows significant spatial and temporal correlations that depend strongly on how the data are collected. Outliers are common, and there are strong sources of systematic variation such as the subject's respiratory and cardiac cycles. Variances in the images depend nonlinearly on the means, and the observed absolute levels of activation tend to shift between sessions because of subject movement. Moreover, all of these difficulties occur for data collected from a single subject; the situation becomes much more complicated if comparisons across subjects are attempted. This research focusses on three general problems in the statistical analysis of fMRI data: 1. The characterization of the response to an activating stimulus in the fMRI signal and the use of this information to build more realistic models and make more precise inferences; 2. The development of robust procedures for identifying active regions that account for the complexity of the underlying spatio-temporal process; and 3. The construction of functional maps within a specified system of the brain (e.g., the visual system) and the use these maps for making predictiv e inference across subjects. Functional Magnetic Resonance Imaging (fMRI) is an exciting new technique that uses advanced technology to obtain images of the active human brain. The technique is of particular interest to cognitive neuropsychologists because of the unique perspective it offers into high-level cognitive processing in humans: areas of the brain that are activated by a stimulus or cognitive task ``light up'' in an fMRI image. This technology will thus play a critical role in understanding how the brain works; however, before this potential can be realized, significant statistical challenges in the interpretation and analysis of fMRI data must be overcome. For example, there is substantial uncertainty in the identification of neural activity from these images and in the attribution of that activity to particular cognitive processes. Moreover, there is a need for new methods of making statistical inferences of scientific interest from these large and complex sets of data. This research focusses on three broad aspects of the general problem: 1. Constructing models for the systematic components of the process that generates the data, 2. Studying and modeling the properties of the noise in the measurements so that analysis and inference can be made more precise, and 3. Developing new methods of inference for addressing interesting scientific questions with massive sets of data that arise from measurements over space and time.

Eddy, Genovese, & Lazar 9705034 Functional Magnetic Resonance Imaging (fMRI) is a powerful new tool for understanding the brain. With fMRI, it is possible to study the human brain in action and trace its processing in unprecedented detail. During an fMRI experiment, a subject performs a carefully planned sequence of cognitive tasks while magnetic resonance images of the brain are acquired. The tasks are designed to exercise specific cognitive processes and the measured signal contains information about the nature and location of the resulting neural activity. Neuroscientists use these data to help identify the neural processes underlying cognition and to build and test theoretical models of cognitive function. This is inherently a problem of statistical inference, yet the statistical methods for fMRI are still undeveloped. In this project, the statistical methodology for these large and complex data sets is advanced on three fronts: dealing with model response variation, developing better registration and acquisition methods, and analyzing spatial activation patterns. Functional Magnetic Resonance Imaging (fMRI) is a new tool that is currently being used to study the brain and the way it functions. Very large amounts of data, with considerable noise, are collected on neural activity while specific cognitive tasks are being performed. In this way, cognitive scientists hope to understand the processes underlying the way humans think. Statistical inference is a natural way of approaching this question. However, the complex nature of the data means that standard methods are not applicable and the methodologies used in fMRI for data analysis are still relatively undeveloped. The current project advances the statistical methodology for fMRI data by working in three directions. Brain response to a given task varies not only by location, but also in different replications of the same experiment. This source of variability is not taken into account by the models now in use. The first direction of the project incorporates this source of variation, resulting in more precise inferences. Subject motion during fMRI scanning is the focus of the second direction, while the third direction involves quantifying how spatial patterns of activation change over time. This allows the comparison of different individuals and groups.

The research component of this project will develop new statistical methods for making inferences from a class of data in which measurements over time are associated with several "experimental units" that have a complicated dependence in their structural and/or stochastic properties. Such data arise frequently in diverse applications such as traffic flow, evolutionary biology, growth curve analysis, neuroimaging, and finance. This research will focus on cases where there is a discrete (and often small) set of units whose relationships are scientifically meaningful. Specifically, this project will develop models in three primary areas: (i) spatial models to improve inferences from functional Magnetic Resonance Imaging (fMRI) data, (ii) models of functional connectivity that allow fMRI data to provide information about how the components of brain systems interact, and (iii) new models for risk assessment and dependence analysis in computational finance.

The educational component of this project will focus on teaching statistical methods and reasoning to experimental psychologists. This project will design a course that builds on case studies and new pedagogical approaches to provide such authentic learning for these students. The emphasis will be on applying statistics in real scientific problems drawn from the psychological literature, effective communication and writing skills, and integrating the computer into the course at all levels. The project will produce the following: (i) A suite of interactive software modules written in Java for exploring the key concepts covered in the class, (ii) A complete set of homework projects and computer laboratory exercises based on issues in the psychological literature, (iii) A set of tools for developing interactive lectures, and (iv) An assessment of student performance both in the class and in downstream courses. These products will be packaged to be easily distributable to and useable by other instructors.

ABSTRACT

Project Summary

The department of Statistics at Carnegie Mellon University will purchase a cluster of 32 Dual processors computers which will be used for several research projects, including in particular:

1. Computational Astrostatistics by Larry Wasserman and Chris Genovese

2. Statistical Genetics and Evolutionary Simulations by Kathryn Roeder, Bernie Devlin and Larry Wasserman

3. Data Analytic Approach to Seismic Imaging by William F. Eddy, Mark Schervish and Pantelis Vlachos

4. Parallelized Spatio-temporal Analyses of Functional Magnetic Resonance Data by Chris Genovese, William F. Eddy and Nicole Lazar

Complex Statistical Models: Theory and Methodology
for Scientific Applications

Larry Wasserman, Christopher Genovese, Robert E. Kass
and Kathryn Roeder

ABSTRACT

This project is aimed at developing statistical theory and methodology for highly complex, possibly infinite dimensional models. Although the methodology and theory will be quite general, we will conduct the research
in the context of three scientific collaborations. The first is ``Characterizing Large-Scale Structure in the Universe,'' a joint project with astrophysicists and computer scientists. The main statistical challenges are nonparametric density estimation and clustering, subject to highly non-linear constraints. The second project is ``Locating Disease Genes with Genomic Control.'' We aim to locate regions of the genome with more genetic similarity among cases (subjects with disease) than controls. These regions are candidates for containing disease genes. Finding these regions ina statistically rigorous fashion requires testing a vast number of hypotheses. We will extend and develop recent techniques for multiple hypothesis testing.
The third projects is ``Modeling Neuron Firing Patterns.'' The goal is to construct and fit models for neuron firing patterns, called spike trains. The data consist of simultaneous voltage recordings of numerous neurons which have been subjected to time-varying stimuli. The data are correlated over time and a major effort is to develop a class of models, called inhomogeneous Markov interval (IMI) process models, which can adequately represent the data.

Statistical methods for simple statistical models with a small number of parameters are well established.
These models often do not provide an adequate representation of the phenomenon under investigation.
Currently, scientists are deluged with huge volumes of high quality data. These data afford scientists the opportunity to use very complex models that more faithfully reflect reality. The researchers involved in this proposal are developing methodology and theory for analyzing data from these complex models. The methods are very general but they are being developed for applications in Astrophysics, Genetics and Neuroscience.

DESCRIPTION (provided by applicant): This project consists of two intertwined components: (a) development of new statistical methods that address recurrent problems in the analysis of functional neuroimaging data and (b) neuroscientific studies of visual remapping in human cortex, which will frame the need for and guide the development of our new methods. The new statistical methods in this proposal apply directly to diverse applications beyond functional neuroimaging, from galaxy clustering to DNA microarrays. The three neurosciences questions in this proposal address fundamental issues regarding neural mechanisms of remapping in humans. Visual remapping is the process that coordinates the visual and eye-movement systems in order to maintain stable perception of the world when the eyes move. This project will achieve the following specific aims, each tied to a specific scientific question. Aim 1. To develop new multiple testing methods for false discovery control. Question 1. Does remapping occur outside parietal cortex? Extending recent work on controlling the False Discovery Rate this project will develop procedures that bound the unobserved proportion (or number) of false discoveries with specified confidence. The method will be applied to investigate Question 1. Aim 2. To develop tools for finding nonlinearly optimal fMRI designs. Question 2. What is the time course of remapping? This project will implement design tools that optimize targeted inferences -- linear and nonlinear. The tools will be applied to design experiments that address Question 2. Aim 3. To develop nonparametric confidence sets for structured function estimation. Question 3. How do the shapes of the visual and remapped responses differ? This project will build on recent work in nonparametric regression to construct confidence sets for an unknown function under shape constraints and with dependent noise. The method will be applied to Question 3.

AST-0434343
Genovese

Recent technological advances have enabled astronomers and cosmologists to collect data of unprecedented quality and quantity. These large data sets can reveal more complex and subtle effects than ever before, but they also demand new statistical approaches. This project consists of two intertwined components: (a) development of new nonparametric statistical methods that address recurrent problems in the analysis of astrophysical and cosmological data and (b) application of the new methods to help answer significant astrophysical and cosmological questions. Specifically, this research will improve inference for the Cosmic Microwave Background spectrum by constructing uniform confidence sets in nonparametric regression, characterize the influence of local environment on galaxy evolution by developing new methods for nonparametric errors-in-variables problems, and estimate the matter density from magnitude limited galaxy surveys by producing accurate density estimators for doubly truncated data.

The research provides interdisciplinary training for postdoctoral fellows and graduate students, and strengthens an interdisciplinary infrastructure between the mathematical and physical sciences.

AST-0709394
Connolly

The current concordance model for the Universe contains, as the most significant contribution to the energy budget, some form of 'dark energy', needed to explain the apparent acceleration of the expansion. Despite this evident importance, there are no compelling theories that explain either the energy density or the properties of the dark energy. The nature of dark energy ranks among the very most compelling of all outstanding problems in physical science. A number of ambitious wide-field optical and infrared imaging surveys will address questions about dark energy and dark matter, but their success depends critically on how well they can detect sources, characterize the photometric properties and shapes of galaxies, identify common and unusual features within images, and control the number of false positive detections. This must be achieved for images covering multiple wavelengths, observed under different conditions, and in almost real time, compounded by the thousand-fold increase in the data rate of this next generation of surveys. This project will develop state-of-the-art statistical and image analysis methods for the next generation of large area astronomical surveys. It will include the co-addition and subtraction of images taken over the period of a year, the identification and classification of sources within these images, and the robust detection of anomalous objects relative to an earlier set of observations.

The work will not only improve the quality of source detection for current and planned astronomical wide-field imaging surveys, but also benefit other fields. The physical and biological sciences are facing an exponential rate of growth of data, and will need fast and efficient image analysis techniques for characterizing the distribution of sources, and for the identification of variations between images.

Nonparametric inference has become an essential tool for studying the cosmos.
This project consists of two intertwined components: (a) development of new theoretical tools and nonparametric methodologies that are inspired by problems in astrophysics but apply more broadly, and (b) application of these tools to two important astrophysical problems, which will frame the need for and guide the development of new statistical theory. Specifically, the investigators will focus on inference for the dark energy equation of state and on identifying filamentary structures from point process data such as that produced by galaxy surveys. The first problem gives rise to a challenging nonlinear inverse problem and demands a nonparametric approach, given what little is known about the dark energy equation of state. The investigators will develop new theory for nonlinear inverse problems that allow for accurate estimates and sharp confidence statements about the unknown function. These techniques will then be applied to Type Ia supernova data, possibly combined with other data sources, to make inferences about dark energy. The second problem gives rise to challenging spatial and inference problems. Current theory in the statistical literature applies to a single filament only, and techniques in the astronomical literature are not supported by theory. The investigators on this project will close that gap, developing theory for defining, identifying, and making inferences about the filamentary structures. The investigators will test this technique and apply it to galaxy survey data.

One of the most important problems in cosmology is understanding dark energy.
The relationship between observable quantities and dark energy produces a challenging nonlinear inverse problem. With very little strong a priori information about the nature of dark energy, parametric approaches to the problem are limited and suboptimal. And with the promise of much larger data sets in the near future, there will be need and opportunity to extract fine-scale features of the dark energy equation of state. The investigators will develop new theory of inference for such problems, with a focus on estimation under shape constraints, sharp hypothesis testing, and accurate confidence sets. The goal is a substantial improvement in accuracy over the current best techniques. In particular, the investigators will focus on the problem of understanding dark energy and on identifying filamentary structures in distribution of matter. The former is one of the central problems in modern cosmology and demands state of the art statistical techniques to get the most from the data. The investigators will develop new statistical theory and methodologies that substantially improve the precision with which features of dark energy can be estimated from supernova data and other data sources. The latter problem is central to understanding the distribution of matter in the universe. Current statistical theory only applies to a limited version of the problem, and current astronomical methodologies do not have strong theoretical support. The investigators will close that gap and develop a method and corresponding theory that can handle realistic versions of the problem and give optimal or near-optimal performance.

Recent technological improvements in functional Magnetic Resonance Imaging (fMRI) are making it possible to study the brain as more than a collection of volume elements (voxels) but rather as a system of interacting components. Instead of considering individual regions, we can study functional networks. Instead of computing voxels' individual response curves, we can estimate their collective response to a stimulus. Instead of settling for responses averaged over brain regions, we can image fine spatial structure. Such a system-oriented approach requires advances in both imaging and statistical methodology. This project consists of two intertwined components. The first is performing fMRI experiments to address three questions about the representation of space in the human brain. The second is developing and validating three new statistical techniques that allow the system-level inferences needed to answer the neuroscientific questions. These techniques are motivated by and developed for the proposed experimental studies, but with minor adaptation, they will be broadly applicable to other neuroimaging studies. In Aim 1, the project will develop methods for identifying and characterizing distributed functional networks. These methods will be used to study the cortical circuit that underlies visual remapping. In Aim 2, the project wil develop methods for simultaneously estimating fMRI response fields. These methods will be used to test the interaction of visual and eye movement signals. In Aim 3, the project will develop adaptive spatial smoothing techniques for high-resolution fMRI data. These tools will be used to test the fine-scale structure of eye position signals in visual cortex. The experimental protocols and theoretical principles developed in this project will increase understanding of the basic function of the human visual system. The statistical techniques developed in this project will give new ways to understand of functional systems with neuroimaging and will advance broadly applicable methods for making inferences about regions in spatio-temporal data.

Statistics curricula have required excessive up-front investment in statistical theory, which many quantitatively-capable students in ``big science'' fields initially perceive to be unnecessary. A training program at Carnegie Mellon will expose students to cross-disciplinary research early, showing them the scientific importance of ideas from statistics and machine learning, and the intellectual depth of the subject. Graduate students will receive instruction and mentored feedback on cross-disciplinary interaction, communication skills, and teaching. Postdoctoral fellows will become productive researchers who understand the diverse roles and responsibilities they will face as faculty or members of a research laboratory.

The statistical needs of the scientific establishment are huge, and growing rapidly, making the current rate of workforce production dangerously inadequate. The Department of Statistics at Carnegie Mellon University will train undergraduates, graduate students, and postdoctoral fellows in an integrated program that emphasizes the application of statistical and machine learning methods in scientific research. The program will build on existing connections with computational neuroscience, computational biology, and astrophysics.Carnegie Mellon will recruit students from a broad spectrum of quantitative disciplines, with emphasis on computer science. Carnegie Mellon already has an unusually large undergraduate statistics program. New efforts will strengthen the training of these students, and attract additional highly capable students to be part of the pipeline entering the mathematical sciences.

Statistics curricula have required excessive up-front investment in statistical theory, which many quantitatively-capable students in ``big science'' fields initially perceive to be unnecessary. A research training program at Carnegie Mellon exposes students to cross-disciplinary research early, showing them the scientific importance of ideas from statistics and machine learning, and the intellectual depth of the subject. Graduate students receive instruction and mentored feedback on cross-disciplinary interaction, communication skills, and teaching. Postdoctoral fellows become productive researchers who understand the diverse roles and responsibilities they will face as faculty or members of a research laboratory.

The statistical needs of the scientific establishment are huge, and growing rapidly, making the current rate of workforce production dangerously inadequate. The research training program in the Department of Statistics at Carnegie Mellon University trains undergraduates, graduate students, and postdoctoral fellows in an integrated environment that emphasizes the application of statistical and machine learning methods in scientific research. The program builds on existing connections with computational neuroscience, computational biology, and astrophysics. Carnegie Mellon is recruiting students from a broad spectrum of quantitative disciplines, with emphasis on computer science. Carnegie Mellon already has an unusually large undergraduate statistics program. New efforts will strengthen the training of these students, and attract additional highly capable students to be part of the pipeline entering the mathematical sciences.

The recent years have seen rapid growth in the depth, richness, and scope of
scientific data, a trend that is likely to accelerate. At the same time,
simulation and analytical models have sharpened to unprecedented detail
the understanding of the processes that generate these data. But what has
advanced more slowly is the methodology to efficiently combine the information
from rich, massive data sets with the detailed, and often nonlinear,
constraints of theory and simulations. This project will bridge that gap.
The investigators develop, implement, and disseminate new statistical methods
that can fully exploit the available data by adhering to the constraints
imposed by current theoretical understanding. The central idea in the
work is constructing sparse, possibly nonlinear, representations of both
the data and the distributions for the data predicted by theory. These
representations can then be transformed onto a common space to allow sharp
inferences that respect the inherent geometry of the model. The methodology
developed in this project will apply to a wide range of scientific problems.
The investigators focus, however, on a critical challenge in astronomy: using
observations of Type Ia supernovae to improve constraints on cosmological
theories explaining the nature of dark energy, a significant, yet little-
understood, component of the Universe.

Crucial scientific fields have enjoyed huge advances in the ability both to
gather high-quality data and to understand the physical systems that generated
these data. Nevertheless, the full societal and scientific value of
this progress will only be realized with new, advanced statistical methods of
analyzing the massive amounts of available data. The investigators develop
statistical methods for combining theoretical modelling and observational evidence into
improved understanding of these physical processes. The analysis of these data will
requirenot only new methods, but also the use of high-performance computing resources. There is a particular need for these tools in cosmology and astronomy, and this project will bring together statisticians and astronomers to combine expertise, but this research is motivated by problems that are present in other fields, such as the climate sciences.

This project will develop computationally efficient estimation methods with accompanying theory for the problem of identifying low-dimensional structure in point-cloud data, both low and high dimensional. A canonical example is a noisy sample from a manifold. The investigators will develop minimax lower bounds for the estimation problem and construct estimators that achieve these lower bounds. They will then implement these methods in a practically useful form nd apply them to several important scientific problems.

Datasets sometimes contain hidden, low-dimensional structure such as clusters, filaments and low dimensional surfaces. The goal of this project to develop rigorously justified, computationally efficient methods for extracting such structure from data. The developed methods will be applied to a diverse set of problems in astrophysics,seismology, biology, and neuroscience. The project will advance knowledge in several fields including computational geometry, machine learning, and statistics.

This PI team aims to use artificial intelligence to exploit data collected from intelligent tutoring systems to provide feedback both to students and to teachers effectively and at the right times. The team is using a new analytic approach, which introduces hierarchical modeling to learning analytics, to investigate how to better understand students' learning states. Algorithms make valid interpretable and actionable inferences from student-learning data, drawing on cognitive theories and statistics to make it work. As in tutoring systems, analysis is at the level of component skills rather than looking at end performance on a task as a whole. Research is around construction of the algorithms for deducing student learning and student learning states and around learning ways of signaling both to learners and to their teachers what concepts and skills learners understand and are capable of and which they are having trouble with. A learning dashboard will allow teachers to visualize the learning needs of a whole class and adapt activities to student needs. Feedback aimed at learners themselves will help learners recognize activities they need to engage in next to better their skills or understanding. Evaluation will include the degree to which learners development of metacognitive skills when such tools are available.


The proposed work will contribute towards the next generation of intelligent tutoring systems as well as contribute to the data analytics needed to make use of large-scale educational data repositories. Because the Learning Dashboard will be independent of any particular domain, and because metacognition and self-assessment are foregrounded, the Learning Dashboard and what is learned about designing an effective learning dashboard should be applicable across disciplines and classes. The proposal brings together what is known about learning, metacognition, and intelligent tutoring systems to address timely learning analytics issues.

Scientific data increasingly takes the form of networks, but the ability to collect and process graph data has out-paced our ability to analyze them statistically. Most of the critical scientific questions about networks revolve around comparisons between networks (e.g., over time or across experimental conditions). Such problems arise in fields as different as neuroscience, epidemiology, economics, climatology and criminology. Currently, network analysts compare only basic descriptive statistics (e.g., the average distance between nodes in the graph), ignoring issues of global structure and statistical validity. We will develop a rigorous statistical theory of network comparisons. Our approach rests on recent develops in network theory which show how large graphs approximate continuous geometric objects, so that tools for geometric comparisons can be applied to networks.

Our project will develop rigorous statistical methods and efficient algorithms for network comparisons. The first step is the flexible non-parametric estimation of continuous network models, where we will pursue three complementary strategies, using regression smoothing, density estimation in non-Euclidean latent spaces, and ensembles of trees. Having represented networks as continuous stochastic processes, we will develop statistical theory and methods for detecting and characterizing differences between such processes. Interdisciplinary proof-of-concept applications, including those in public health (through online social networks), finance (through financial networks), neuroscience (through brain connectivity networks), genetics (through gene regulatory networks), and proteinomics (through protein interaction networks), will demonstrate the power of the geometric approach in comparing large and disparate sample data.

Data in high dimensional spaces are now very common. This project will develop methods for analyzing these high dimensional data. Such data may contain hidden structures. For example, clusters (which are small regions with a large number of points) can be stretched out like a string forming a structure called a filament. Scientists in a variety of fields need to locate these objects. It is challenging since the data are often very noisy. This project will develop rigorously justified and computationally efficient methods for extracting such structures. The methods will be applied to a diverse set of problems in astrophysics, seismology, biology, and neuroscience. The project will advance knowledge in several fields including computational geometry, astronomy, machine learning, and statistics.

Finding hidden structure is useful for scientific discovery and dimension reduction. Much of the current theory on nonlinear dimension reduction assumes that the hidden structure is a smooth manifold and is very restrictive. The data might be concentrated near a low dimensional but very complicated set, such as a union of intersecting manifolds. Existing algorithms, such as the Subspace Constrained Mean Shift exhibit erratic behavior near intersections. This project will develop improved algorithms for these cases. At the same time, contemporary theory breaks down in these cases and this project will develop new theory to address the aforementioned problem. A complete method (which will be called singular clusters) will be developed for decomposing point clouds of varying dimensions into subsets.

Understanding how matter is distributed in the Universe is key to developing accurate models of how it evolved. The investigators will use mapping of filamentary structures in large samples of galaxies as a new indicator of large scale structure that can then be compared to other data. They seek to trace the cosmic web from these data. Broader impacts of the work include training of a graduate student, and engagement of the broader community through public lectures at the Allegheny Observatory, online educational games, and existing programs for middle school students.

The research will cross correlate the new data with other cosmological observables like Baryon Acoustic Oscillations and study intrinsic galaxy shapes with filaments. The group has developed a method for identifying filaments in purely photometric data and, with this award, will develop a pre-existing prototype into a reconstruction tool. They will then apply the tool to simulated data.