Документ взят из кэша поисковой машины. Адрес оригинального документа : http://hea-www.harvard.edu/AstroStat/Stat310_1112/ab_20110906.pdf Дата изменения: Tue Sep 6 19:40:28 2011 Дата индексирования: Tue Oct 2 04:18:39 2012 Кодировка: Поисковые слова: строение земли

Taste of astrostatistics

A taste of astrostatistics: problems, opportunities, & connections
Alexander W Blocker

2011 Sep 06


Taste of astrostatistics Outline

Outline
1 2

Astrostatistics in broad strokes Stacking: statistical challenges can come in small packages Problem Model Computation Data Results Lessons Event detection: massive, messy data Problem Method Results Lessons Connections

3

4


Taste of astrostatistics Astrostatistics in broad strokes

What is astrostatistics?

Applying modern statistical tools to the problems of astronomy & astrophysics Combining scientific and statistical modeling Handling complex instrumental effects Linking theory to data


Taste of astrostatistics Astrostatistics in broad strokes

Where the data comes from

Observations across the entire EM spectrum, from radio to gamma rays Optical observations mostly from ground-based telescopes (e.g. Keck); some from Hubble High-energy observations (x-rays, gamma, etc.) mostly from space telescopes (e.g. Chandra) Each type of data poses distinct challenges


Taste of astrostatistics Astrostatistics in broad strokes

Common types of data

Images Spectra Time series And all combinations of these


Taste of astrostatistics Astrostatistics in broad strokes

Example -- Image

Cold brown-dwarf star from WISE satellite (WISE 1828+2650)


Taste of astrostatistics Astrostatistics in broad strokes

Example -- Spectrum


Taste of astrostatistics Astrostatistics in broad strokes

Example -- Time Series

Supernova SGR 2002 from EROS2 survey


Taste of astrostatistics Astrostatistics in broad strokes

How do we approach problems?

Probabilistic models for the entire data-generating process
Account for instrumental effects, population variation, etc. Framework for inference

Approximate inference via computation
Typically MCMC Can be EM or other methods

Rigorous model checking & validation
Need to establish statistical & scientific validity Value of collaboration -- physical plausibility


Taste of astrostatistics Stacking: statistical challenges can come in small packages Problem

Combining information on faint x-ray sources

Want to understand properties of a given population of sources (e.g. galaxies at a certain distance) For each source, we observe only two counts: one from the background noise & one from a combination of the source & noise Also have telescope's sensitivity etc. for given observation Goal is to combine information from these faint counts to estimate, e.g., mean intensity and variability in intensity among sources


Taste of astrostatistics Stacking: statistical challenges can come in small packages Problem

Example images


Taste of astrostatistics Stacking: statistical challenges can come in small packages Problem

Previous approaches in the astrophysics

Idea: subtract out the background, then average resulting "net" counts Use of background subtraction Gaussian assumption; inappropriate in low count regimes Above manifests as negative individual estimates; for sufficiently faint samples, this can lead to negative aggregate estimates No clean measure of uncertainties on luminosities Solution: model data as Poisson


Taste of astrostatistics Stacking: statistical challenges can come in small packages Model

Assumptions

Same as those for standard stacking analysis For luminosity-based inference, assuming that redshifts are known with no uncertainty
Relatively plausible for spectroscopic; not as much for photometric

Assuming the spectra of sources are know & identical
Typically assume power law with photon index 1.7

Attempting to make inferences only on selected sample, for now; not dealing with selection effects, etc.


Taste of astrostatistics Stacking: statistical challenges can come in small packages Model

Mathematical framework
Modeling source and background counts as Poisson. Assuming background count rates follow log-Normal distribution Assuming log-luminosities (or log-fluxes) follow a log-t distribution
Makes our inferences robust to outliers. More appropriate for modeling distributions with power-law tails.

Using priors to help regularize estimates; only require informative priors on dispersion parameters Need to allow for relationship between distance (redshift) & source intensity; analogous to using a general regression model


Taste of astrostatistics Stacking: statistical challenges can come in small packages Computation

MCMC with finesse

Using MCMC to simulate from posterior of source intensities given prior & observations; can then extract estimands of interest Because our model is Poisson / log-t , we can't use a standard Gibbs sampler Combining independence chain MH, parameter expansion, and data augmentation strategies to obtain an efficient sampler Using numerical optimization (Halley's method, appropriately) to build good proposal distributions


Taste of astrostatistics Stacking: statistical challenges can come in small packages Data

Description of data
We worked with 1546 galaxies from SDSS on which we have spectroscopic redshifts. The distribution of redshift and exposure for these sources can be seen below.


Taste of astrostatistics Stacking: statistical challenges can come in small packages Results

Summary of results on SDSS data


Taste of astrostatistics Event detection: massive, messy data Problem

Finding events in time series -- lots of time series

Have massive (order of 10-100 million) dataset of time series, possibly spanning multiple spectral bands Goal is to identify and classify time series containing events How do we define an event?
Not interested in isolated outliers Looking for groups of observations that differ significantly from those nearby (ie, "bumps" and "spikes") Also attempting to distinguish periodic and quasi-periodic time series from isolated events


Taste of astrostatistics Event detection: massive, messy data Problem

The data
We used data from the MACHO survey for training, and are actively analyzing the EROS2 survey MACHO data consists of approx. 38 million LMC sources, each observed in two spectral bands
Collected 1992-1999 on 50-inch telescope at Mount Stromlo Observatory, Australia Imaged 94 43x 43 fields in two bands, using eight 2048 x 2048 pixel CCDs Substantial gaps in observations due to seasonality and priorities

EROS2 data consists on approx. 87.2 million sources, each observed in two spectral bands
Imaged with 1m telescope at ESO, La Silla between 1996 and 2003 Each camera consisted of mosaic of eight 2K x 2K LORAL CCDs Typically 800-1000 observations per source


Taste of astrostatistics Event detection: massive, messy data Problem

Exemplar time series from the MACHO project:
A null time series:


Taste of astrostatistics Event detection: massive, messy data Problem

Exemplar time series from the MACHO project:
An isolated event (microlensing):


Taste of astrostatistics Event detection: massive, messy data Problem

Exemplar time series from the MACHO project:
A quasi-periodic time series (LPV):


Taste of astrostatistics Event detection: massive, messy data Problem

Exemplar time series from the MACHO project:
A variable time series (quasar):


Taste of astrostatistics Event detection: massive, messy data Problem

Exemplar time series from the MACHO project:
A variable time series (blue star):


Taste of astrostatistics Event detection: massive, messy data Problem

Notable properties of this data
Fat-tailed measurement errors
Common in astronomical data, especially from ground-based telescopes Need more sophisticated models for the data than standard Gaussian approaches

Quasi-periodic and other variable sources
Changes the problem from binary classification (null vs. event) to k -class Need more complex test statistics and classification techniques

Non-linear, low-frequency trends make less sophisticated approaches far less effective Irregular sampling can create artificial events in naive analyses


Taste of astrostatistics Event detection: massive, messy data Method

Our approach
Use a Bayesian probability model for both initial detection and to reduce the dimensionality of our data (by retaining posterior summaries) Using posterior summaries as features for machine learning classification technique to differentiate between events & variables Our goal is not to perform a final, definitive analysis on these events
Objective to predict which time series are most likely to yield phenomena characterized by events (e.g. microlensing, blue stars, flares, etc.) Allows for use of complex, physically-motivated methods on massive datasets by pruning set of inputs to manageable size Provides assessments of uncertainties at each stage of screening and allows for the incorporation of domain knowledge


Taste of astrostatistics Event detection: massive, messy data Method

Summarized mathematically

Symbolically, let V be the set of all time series with variation at an interesting scale (e.g., the range of lengths for events), and let E be the set of events For a given time series Yi , we are interested in P (Yi E ) We decompose this probability as P (Yi E ) P (Yi V ) · P (Yi E |Yi V ) via the above two steps


Taste of astrostatistics Event detection: massive, messy data Method

Probability model - specification
Linear model for each time series with an incomplete wavelet basis:
k
l

M

y (t ) =
i =1

i i (t ) +
j =kl +1

j j (t ) + (t )

First kl elements contain low-frequency, "trend" components; remainder contain frequencies of interest; highest frequencies are left as noise Idea: compare smooth (trend-only) and complete model fits; if they differ, could have an event Assume residuals (t ) are distributed as iid t (0, 2 ) for robustness ( = 5) -- fat tails Address irregular sampling through regularization -- informative priors on wavelet coefficients smooth undersampled periods Extremely fast estimation via EM -- 0.15 - 0.2 seconds


Taste of astrostatistics Event detection: massive, messy data Method

Examples of model fit
Idea is that, if there is an event at the scale of interest, trend-only and complete fits with differ substantially:


Taste of astrostatistics Event detection: massive, messy data Method

Example of model fit
For null time series, the difference will be small:


Taste of astrostatistics Event detection: massive, messy data Method

Example of model fit
However, for quasi-periodic time series, the difference will be huge:


Taste of astrostatistics Event detection: massive, messy data Method

Probability model - testing

kl

M

y (t ) =
i =1

i i (t ) +
j =kl +1

j j (t ) + (t )

Using LLR statistic to test if coefficients on all non-trend components are zero (H0 : kl +1 = kl +2 = . . . = M = 0) Controlling false discovery rate (FDR) to 10-4 to set the critical region for our test statistic


Taste of astrostatistics Event detection: massive, messy data Method

Feature Selection I

Engineered two features based on fitted values for discrimination between diffuse and isolated variability First is a relatively conventional CUSUM statistic Let {zt } be the normalized fitted values for a given time series, excepting the "trend" components corresponding to 1 , . . . , kl . We then define:
t

St =
k =1 t

2 (zk - 1)

CUSUM = max St - min St
t


Taste of astrostatistics Event detection: massive, messy data Method

Feature Selection II

Second is "directed variation"
Idea is to capture deviation from symmetric, periodic variation Defining zt as before and letting zmed be the median of zt , we define: DV = 1 #{ t : z t > z } zt2 -
t :z t > z
med

med

1 #{ t : z t < z

med

}

z
t :zt med

2 t


Taste of astrostatistics Event detection: massive, messy data Method

Distribution of features on MACHO data


Taste of astrostatistics Event detection: massive, messy data Method

Methods

Tested a wide variety of classifiers on our training data, including k NN, SVM (with radial and linear kernels), LDA, QDA, and others Regularized logistic regression performs best Using weakly informative (Cauchy) prior for regularization


Taste of astrostatistics Event detection: massive, messy data Results

Summary

First stage shows reduction from 87.2 million candidate light curves by approximately 98% (to approximately 1.5 million) in blue band from likelihood-ratio screen Approximately 16,000 of the latter group are likely isolated events, based on analysis from classification stage and filtering for chip-level errors (265 with P (event) 0.80 in both bands) Scientific follow-up on candidates yielded identified 126 known gravitational lensings and 42 known supernovae (via Simbad & VizieR) Several candidates identified for further analysis in multiple categories


Taste of astrostatistics Event detection: massive, messy data Results

Examples of highly-ranked events
Examples from top 10:


Taste of astrostatistics Event detection: massive, messy data Results

Examples of highly-ranked events
Examples from top 10:


Taste of astrostatistics Event detection: massive, messy data Results

Examples of highly-ranked events
Examples from top 10:


Taste of astrostatistics Event detection: massive, messy data Results

Examples of highly-ranked events
Examples from top 10:


Taste of astrostatistics Event detection: massive, messy data Results

Examples of highly-ranked events
Examples from top 10:


Taste of astrostatistics Event detection: massive, messy data Lessons

Lessons from event detection
Massive data presents a new set of challenges to statisticians & astronomers that many of our standard tools are not well-suited to address Machine learning has some valuable ideas and methods to offer, but we should not discard the power of probability modeling Conversely, we can use reasonable probability models with massive datasets without excessive computational burdens It is tremendously important to put each tool in its proper place for these types of analyses
Rigorous modeling of observation processes is particularly crucial; mistakes here can destroy information for any later analyses

Our work on event detection for astronomical data shows the power of this approach by combining both rigorous probability models and standard machine learning approaches


Taste of astrostatistics Connections

Astrophysics & biology

These subjects appear extremely dissimilar on the surface However, they are following a similar path in terms of data, as both address:
An increasing need to address complex instrumental/experimental properties A transition to regimes where non-Gaussian error distributions matter An explosion in the volume of data

The largest difference is that the astrostatistics community has been facing these problems & building high-quality solutions for longer


Taste of astrostatistics Connections

Complex instrumental & experimental properties

Astronomers face extremely complex instrumental & experimental properties:
Inhomogeneous sensitivity (sources look dimmer or brighter depending on where the telescope sees them) Blurring due to detector/telescope properties Subtle, non-ignorable patterns of missing data

All of these are increasingly important in biology as high-throughput methods become more common Physical mechanisms differ between fields, but statistical challenges are analogous


Taste of astrostatistics Connections

Non-Gaussian errors

Optical astronomy dealt almost exclusively with Gaussian errors With high-energy observations (x-ray, gamma, etc.), observations are counts, so errors can be extremely non-Gaussian We deal with these problems constantly in astrostatistics and have built a methodological foundation to address them High-throughput biology must address these problems as, e.g., sequencing (counts) replace micro-arrays (continuous) for analyses of gene expression


Taste of astrostatistics Connections

Methodological arbitrage

Both astrophysics & biology have many more problems than statisticians; many opportunities Astronomy can be an excellent setting to address these problems
Field emphasizes pinning-down & understanding sources of error Have data available to model & analyze complex observation processes Direct physical underpinnings allow us to focus on the core problems


Taste of astrostatistics Connections

Example -- telescopes & enzymes
In astrophysics, need to account for blurring of observations due to instrument
Expected read centers per nucleosome

Typically handled via PSF (point spread function), which describes distribution of observations given location of source Exactly analogous phenomenon occurs with high-throughput sequencing in biology due to enzymatic digestion Methods from astrostatistics formed basis for solution to biological problem

0.040 0.035 0.030 0.025 0.020 0.015 0.010 0.005

Gaussian Nonparametric MLE







0.000 80

60

40

20 0 20 bp from nucleosome center

40

60

80




Taste of astrostatistics Connections

Wrapping up

Astrostatistics is a vibrant, exciting area for research Plenty of open problems Challenges from foundational theory to computation Major opportunity to apply methods across multiple fields And, of course, great collaborators


Taste of astrostatistics Acknowledgements

Acknowledgments

Many thanks to Xiao-Li Meng & Edo Airoldi, my advisors Tom Aldcroft & Paul Green, my collaborators on stacking Pavlos Protopapas, Dae-Won Kim, & Jean-Baptiste Marquette, my collaborators on event detection