“Astronomy and Statistics”

Daniel Mortlock (Imperial)

   Why "the statistical frontiers of astrophysics"?  Why not "the statistical frontiers of particle physics", or "the statistical frontiers of neuroscience" or "the statistical frontiers of palmistry?"  Any measurement that yields quantitative data inevitably requires statistical analysis, but lack of repeatable experiments in astronomy makes probabilistic methods unusually important, while huge modern astronomical data-sets provide an exciting real-world testing ground for innovative statistical techniques.  This introductory talk will explore astronomical statistics through a number concrete case-studies that range from completely straight-forward applications of Bayes's theorem to parameter estimation, through more ambiguous model-selection problems, to the far more open-ended world of data-mining.  There will also be particular emphasis on cases in which manifestly erroneous conclusions have been reached, and how this could often have been avoided simply by following Laplace's principle that "probability is nothing but common sense reduced to calculus".

“Statistics of TAOS:  the good, the bad, and the ugly”

John Rice (Berkeley)

   The Taiwanese-American Occultation Survey (TAOS) operates four 50 cm telescopes at Lu-Lin Observatory in central Taiwan to search for stellar occultations by small Kuiper belt objects. The robotic telescopes simultaneously monitor several hundred stars at 5 Hz searching for extremely rare occultations which would typically result in flux drops less than 30% in one or two consecutive time points.  (Larger objects could produce longer occultations of more complex shapes,  due to Fresnel diffraction.) To date, more than 10^5 star-hours of data have been recorded. 

   Requiring simultaneous detection by four telescopes guards against false positive events, including events of terrestial origin such as bats, airplanes, and satellites, and events due to “noise.”   The noise distribution of each lightcurve is not of a known form, and in particular is not Gaussian or Poisson.  Typical signal to noise ratios are around 10, and changes in atmospheric transparency introduce further uncertainty into the flux measurements.

   After briefly reviewing how the data are collected and the novel photometry. I will focus on a variety of statistical problems that arise in TAOS.  Although anchored in the context of TAOS, I think some of the issues are of wider spread interest.  As examples, in the context of TAOS, I will discuss the problem of detecting rare events among a large number of measurements and the importance of robust methods.  The talk will also illustrate the tensions inherent at the nexus of scientific data and statistical idealizations of the stochastic nature of that data, reflected in the often quoted phrase of George Box, “all models are wrong, but some are useful.”

“Issues and Directions in Parameter Estimation with Complex Models”

Chad Schafer (CMU)

   Complex simulation models are increasingly-important tools in the research of cosmologists and astronomers. Yet, the use of these models as part of formal statistical analysis presents challenging issues: How does one perform parametric inference when only provided a simulation code for mapping from parameter space to data? This presentation combines elements of recent publications of Schafer and Stark (2009) and Richards, et al. (2009), among others, to provide a basis for approaching parameter estimation in such situations. Specifically, I will explore the use of dimension reduction techniques for approximating the requisite likelihood functions, and describe how this can be naturally incorporated into a Monte Carlo approach to constructing confidence regions of optimal (minimax) expected size.

  This is joint work with Ann Lee, Joseph Richards, Philip Stark, and Peter Freeman.

“Bayesian Analysis of Stellar Populations”

Paul D. Baines (Harvard)

  The highly-structured nature of many problems in Astronomy can provide an excellent opportunity for the development and application of advanced techniques in statistical computation. The analysis of photometric data for stellar clusters provides an example of both the statistical and computational challenges present in many Astronomy applications. Typically, properties of stellar clusters are estimated using Color-Magnitude Diagrams, whereby the observed data are often simply compared to what one would expect under a theoretical mapping from the model parameters to the observed data. This mapping is determined by a set of isochrone tables, listing the expected photometric measurements for a given set of input parameters.

   To address many of the substantive questions in a coherent statistical manner, we present a flexible hierarchical Bayesian model for the analysis of stellar populations. The computation for the model is done via Markov Chain Monte Carlo (MCMC), the standard tool of choice for Bayesian computation. Both the complex dependence structure and the peculiar nature of the isochrone mapping, however, present a formidable challenge to standard statistical computation methods. By applying the Ancillary Sufficient Interweaving Scheme (ASIS) of Yu & Meng (presented in a separate talk by Meng), we show how competing parametrizations can be constructed and combined to help overcome the weaknesses of individual schemes, and drastically improve the efficiency and reliability of the computation.

“Robot Astronomy: Model selection and decision-making”

David W. Hogg (NYU)

  Scientific activity is filled with decision-making; traditionally this makes real, complex scientific investigations difficult to automate or repeat.  The most challenging kinds of decisions---and the most common---involve deciding among qualitatively different models or among models of very different complexity.  This is challenging because more complex models almost inevitably perform better. Strictly speaking, no system of inference tells you which model to choose; it only assigns relative likelihoods given the data or posterior probabilities.  Model choice requires making hard (non-probabilistic) decisions, which themselves require additional input about the investigator's utility.  A principled Bayesian doesn't ever decide but rather marginalizes over complexity, but in many cases utility must be expressed and a hard decision must be made.  I discuss these issues in the context of work towards fully automating astronomical imaging surveys, from survey design through to observing strategy, data analysis, and discovery of astrophysically interesting objects.  Fully automated systems are also fully repeatable procedures, so these ideas are important even if robotics per se is not our goal.

“From Science of Methodology to Science of Data”

Ritei Shibata (Keio)

   Statistics is a science of data but expectation of other scientists is often to be a science of methodology or just a collection of methodologies useful for analyzing data or making inferences. It is a unique paradigm which provides us a data driven model. The data driven model is inductive which could lead us to a new view of underlying phenomena, hopefully a discovery. The data driven model is not necessarily unique.  Collaboration with

scientists in each field is indispensable for seeking better data driven models. To smooth the process, I have been advocating a new paradigm “Data Science” in place of statistics.  Data scientist is an expert of data, who knows how to deal with data from its upriver to downriver and how to derive an honest data driven model from given data, reflecting discussion with scientists in each science.  In this talk, I will report my current activities as a data scientist, including construction of infrastructure of data science.

“Correlation Analyses to find the sources of highest energy cosmic rays”

Hajime Takami (IPMU)

  The highest energy cosmic rays (HECRs) are the most energetic particles which human beings have ever observed, but their origin is still unknown. Recent progress of HECR observations has unveiled the importance of the statistical features of the arrival direction distribution of HECRs to investigate the origin of HECRs. In this talk, we briefly review recent progress on the highest energy cosmic rays in the statistical points of view, and then discuss our recent works about statistics on HECRs.

“The critical role of statistics in Cosmic Microwave Background studies”

Hiranya Peiris (Cambridge)

  I will review the physics of the cosmic microwave background (CMB), and how precision observations of the CMB can be used to obtain information about the initial conditions, constituents, and dynamics of the universe. I will then highlight the critical role played by statistics in CMB data analysis, using specific examples that highlight the use of both Bayesian and frequentist techniques. Interestingly, there are some fundamental statistical issues in this field which remain unresolved, and I will highlight the limitations of some of the current methods and describe some exciting open problems.

“Needlet Tools for Data Analysis in Cosmology and Astrophysics”

Domenico Marinucci (Roma)

  Our aim in this talk is to review present and future applications of wavelets/needlets techniques for various areas of astrophysical and cosmological research. We shall also illustrate present and forthcoming applications to experimental data. We shall first discuss rapidly the main features of the needlets construction, with a special focus on uncorrelation and localization properties, and their possible advantages over rival techniques. We will then discuss the application of needlets to CMB temperature data, mentioning in particular the following themes:

1) estimation of (cross-) angular power spectra, cross-correlation CMB-Large Scale Structure data, detection of the Integrated Sachs-Wolfe effect

2) testing for non-Gaussianity, estimation of the nonlinearity parameter f_NL, needlets bispectrum

3) searching for features and anisotropies (Cold Spot)

We will then consider the extension of needlet construction to data which are spin- (or tensor-) valued. This is the environment which is met in the analysis of CMB polarization data (an analogous mathematical setting is relevant for  weak lensing experiments). We will discuss the properties and potential of so-called spin needlets for this framework, firstly for the estimation of polarization angular power spectra and for map reconstruction. Finally, we will discuss the possibility to exploit needlets for density estimation with directional data, such as those connected with ongoing cosmic rays experiments.

  These works are the outcome of a joint effort with several other people, among which we mention Daryl Geller, Gerard Kerkyacharian, Davide Pietrobon, Frode Hansen, Xiaohong Lan, Oystein Rudjord, Amedeo Balbi, Paolo Baldi, and Dominique Picard.

“Spatial Statistics in Cosmology”

Istvan Szapudi (Hawaii)

  The spatial statistics of Large Scale Structure surveys and Cosmic Microwave Background maps is characterized by correlation functions and their corresponding power spectra. I will focus on two-point correlation functions (angular power spectrum), and three-point correlation functions (bi-spectrum). The estimation of these

quantities has statistical (noise, bias, optimality), as well as computational challanges. Since Moore's law is driving both the accumulation of data and the available computational resources, only those methods will be viable in the long term that scale as $N \log N$ or better, where $N$ is a number characterizing the amount of data. This observation questions the long-term viability of optimal methods, typically scaling as $N^3$, and requires the careful

balancing of optimality with the available resources. I will show examples, where symmetries, multiresolution techniques, and Monte Carlo methods, help achieving scientific goals for large data sets using realistic computational resources, when naive or optimal methods would fail. In addition, I will show how two-point statistics with non-linear maps replacing higher order statistics.

“Non-Gaussian error properties of large-scale structure probes”

Masahiro Takada (IPMU)

   Understanding the nature and origin of hierarchical large-scale structures in the Universe, seen today, is one of the most important issues in cosmology. The current standard scenario is given by the cold dark matter dominated model, where the seed (nearly) Gaussian density perturbations are amplified via gravitational instability to form the present-day nonlinear structures. The properties of large-scale structure can be quantifies by clustering statistics, e.g. correlation functions. Interestingly, measuring the clustering properties of large-scale structure as a function of redshifts and distance scales, e.g. from galaxy surveys, allows us to constrain cosmological model including parameters related to dark matter and dark energy, which is the main scientific target of ongoing and planned massive cosmological surveys.

   However, due to the nature of nonlinear gravitational clustering, the statistical properties of large-scale structure observables are non-Gaussian, therefore, a simple Gaussian likelihood analysis cannot be used. In other words, in order to extract unbiased, robust cosmological constraints from the data, the non-Gaussian errors need to be properly taken into account. In this talk I will review the non-Gaussian aspects of cosmological probes, and show how the non-Gaussian errors affect parameter estimations.

“Statistical Problems in the Study of Other Worlds

using High Precision Radial Velocity Measurements"

Ed Turner (Princeton)

    The first discoveries of planets orbiting ordinary stars other than the Sun (exoplanets) were made almost 15 years ago via ultra-high precision Doppler measurements of the radial velocity of the primary star's reflex motion in response to the planet's orbital motion.  This technique has also led to the discovery of the large majority of the several hundred exoplanets discovered and catalogued in the intervening years.  Four hundred years (almost exactly!) after Galileo revolutionized  our understanding of the Earth's place in the cosmos via the discovery of "other worlds" (as he termed the moons of Jupiter) via the first astronomical use of the telescope, astronomers have begun a (reasonably) systematic exploration of the individual and population properties of exoplanets, literally "other worlds".  Results to-date will be summarized.

   Although the basic physics of the radial velocity technique and the interpretation of the data it produces involves only elementary physics (literally Physics 101) and is understood essentially perfectly, there are non-trivial statistical challenges in both deriving the orbital properties of individual exoplanet systems and in characterizing the ensemble properties of the exoplanet population.  These challenges have been engaged by astronomers with widely varying degrees of statistical sophistication, resulting in considerable confusion and erroneous conclusions in some cases.  Two or three examples of such issues and their current "state of play" will be presented in some detail.

“Parameter Estimation from Time-Series Data with Correlated Errors:

A Wavelet-Based Method and its Application to Exoplanetary Transit Light Curves”

Josh Winn (MIT)

   Exoplanetary transits are the rare and cherished occasions when the planet's orbit is viewed close enough to edge-on for the planet to cross in front of the disk of its parent star. Transits can be detected from the slight diminution in the light received from the star, even when there is no hope of spatially resolving the star or the planetary orbit. Careful observations of transits have provided the best available information about the absolute dimensions of exoplanets, the shape and orientations of their orbits, and the constituents of their atmospheres.  Precise timing of transits offers a means of discovering moons, or additional planets, through their gravitational perturbations of the transiting planet.  However, it has recently been recognized that correlated errors in time-series

photometry is often a limiting factor in the analysis of transit data.

  We consider the problem of fitting a parametric model to time-series data that are afflicted by correlated noise. The noise is represented by a sum of two stationary Gaussian processes: one that is uncorrelated in time, and another that has a power spectral density varying as 1/f^\gamma . We present an accurate and fast [O(N)] algorithm for

parameter estimation based on computing the likelihood in a wavelet basis. The method is illustrated and tested using simulated time-series photometry of exoplanetary transits, with particular attention to estimating the midtransit time. We compare our method to two other methods that have been used in the literature, the time-averaging method and the residual-permutation method. For noise processes that obey our assumptions, the algorithm presented here gives more accurate results for midtransit times and truer estimates of their uncertainties.

“Bayesian inference and experimental design for exoplanet studies”

Thomas J. Loredo (Cornel)

  In the last decade astronomers have discovered over 300 planetary systems around nearby stars; new systems are announced almost weekly and the pace of discovery is accelerating.  Exoplanet science is now shifting from being discovery-oriented to focusing on detailed astrophysical modeling and analysis of the growing catalog of observations.  In this talk I will describe work in progress by a collaboration of astronomers and statisticians developing a suite of Bayesian tools for exoplanet detection, planetary orbit estimation, and adaptive scheduling of observations. This work focuses on analysis of stellar reflex motion data, where a planet is detected by observing the ``wobble'' of its host star as it responds to the gravitational tug of the orbiting planet. Kepler's laws specify an analytical model for the resulting time series, but it is strongly nonlinear, yielding complex, multimodal likelihood functions.  The parameter spaces range in size from few-dimensional to dozens of dimensions, depending on the number of planets in the system, and the type of motion measured (line-of-sight velocity, or position on the sky).  Since orbits are periodic, Bayesian generalizations of periodogram methods facilitate the analysis.  Subsequent analysis uses adaptive MCMC methods and adaptive importance sampling to perform the integrals required for both inference (planet detection and orbit measurement), and information-maximizing sequential design (for adaptive scheduling of observations).

“Designing the next generation of extra-solar planet observations”

Dmitry Savransky (Princeton)

   In the last decade, the discovery and characterization of extra-solar planets (exoplanets) has greatly advanced, with almost 400 known exoplanets and multiple proven detection techniques. As we push the limits of planet sizes and separations that are discoverable from the ground, dedicated orbiting observatories, such as Kepler and COROT, have already been launched and will soon begin to tell us about the structure and diversity of planetary systems in our galaxy. At the same time, technology for planet-finding astrometric and direct imaging space observatories is being steadily advanced towards flight readiness levels.

   Statistical methods play a vital role in exoplanet instrument design, mission planning, and data analysis. In this talk, we will present statistical mission and data analysis tools for both indirect (astrometric) and direct (imaging) planet-finding methods. Further, we will show how the coupling of these tools with autonomous scheduling algorithms results in a generalized framework for the analysis of space based planet-finding missions. This framework, along with an assumed planetary population, is used in a Monte Carlo analysis to produce ensembles of mission timelines—schedules of observations and their simulated results, which are then used to extract distributions of science yield metrics and mission requirements. Along with science returns such as planet detection and spectral characterization, we also consider the application of filtering techniques to orbital fitting using multiple data streams.