天文数据分析指南：Bayesian模型与Python、R实现

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数据分析

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1 Astrostatistics

1.1 The Nature and Scope of Astrostatistics

Astrostatistics is at the same time one of the oldest disciplines, and one of the youngest. The Ionian

Greek philosopher Thales of Miletus is credited with correctly predicting a total solar eclipse in

central Lydia, which he had claimed would occur in May of 585 BCE. He based this prediction on an

examination of records maintained by priests throughout the Mediterranean and Near East. The fact

that his prediction was apparently well known, and the fact that the Lydians were engaged in a war

with the Medes in Central Lydia during this period, brought his prediction notice and fame. Thales

was forever after regarded as a sage and even today he is named the father of philosophy and the

father of science in books dealing with these subjects.

Thales’ success spurred on others to look for natural relationships governing the motions of

astronomical bodies. Of particular note was Hipparchus (190–120 BCE) who, following on the earlier

work of Aristarchus of Samos (310–230 BCE and Eratosthenes (276–147 BCE), is widely regarded as

the first to clearly apply statistical principles to the analysis of astronomical events. Hipparchus also

is acknowledged to have first developed trigonometry, spherical trigonometry, and trigonometric

tables, applying these to the motions of both the moon and sun. Using the size of the moon’s parallax

and other data from the median percent of the Sun covered by the shadow of the Earth at various

sites in the area, he calculated the distance from the Earth to the Moon as well as from the Earth to

the Sun in terms of the Earth’s radius. His result was that the median value is 60.5 Earth radii. The

true value is 60.3. He also calculated the length of the topical year to within six minutes per year of

its true value.

Others in the ancient world, as well as scientists until the early nineteenth century, also used

descriptive statistical techniques to describe and calculate the movements and relationships between

the Earth and astronomical bodies. Even the first application of a normal or ordinary least squares

regression was to astronomy. In 1801 Hungarian Franz von Zach applied the new least squares

regression algorithm developed by Carl Gauss for predicting the position of Ceres as it came into

view from its orbit behind the Sun.

The development of the first inferential statistical algorithm by Gauss, and its successful

application by von Zach, did not immediately lead to major advances in inferential statistics.

Astronomers by and large seemed satisfied to work with the Gaussian, or normal, model for

predicting astronomical events, and statisticians turned much of their attention to deriving various

probability functions. In the early twentieth century William Gosset, Karl Pearson, and Ronald Fisher

made the most advances in statistical modeling and hypothesis testing. Pearson developed the

mathematics of goodness-of-fit and of hypothesis testing. The Pearson test is still used as an

assessment of frequentist based model fit.

In addition, Pearson developed a number of tests related

to correlation analysis, which is important in both frequentist and Bayesian modeling. Fisher (1890–

1962) is widely regarded as the father of modern statistics. He is basically responsible for the

frequentist interpretation of hypothesis testing and of statistical modeling. He developed the theories

of maximum likelihood estimation and analysis of variance and the standard way that statisticians

understood p-values and confidence intervals until the past 20 years. Frequentists still employ his

definition of the p-value in their research. His influence on twentieth century statistics cannot be

overestimated.

It is not commonly known that Pierre-Simon Laplace (1749–1827) is the person foremost

responsible for bringing attention to the notion of Bayesian analysis, which at the time meant

employing inverse probability to the analysis of various problems. We shall discuss Bayesian

methodology in a bit more detail in Chapter 3. Here we can mention that Thomas Bayes (1702–

1761) developed the notion of inverse probability in unpublished notes he made during his lifetime.

These notes were discovered by Richard Price, Bayes’s literary executor and a mathematician, who

restructured and presented Bayes’ paper to the Royal Society. It had little impact on British

mathematicians at the time, but it did catch the attention of Laplace. Laplace fashioned Bayes’ work

into a full approach to probability, which underlies current Bayesian statistical modeling. It would

probably be more accurate to call Bayesian methodology Bayes–Laplace methodology, but

simplification has given Bayes the nominal credit for this approach to both probability and statistical

modeling.

After enthusiastically promoting inverse probability, Laplace abandoned this work and returned

to researching probability theory from the traditional perspective. He also made major advances in

differential equations, including his discovery of Laplace transforms, which are still very useful in

mathematics. Only a few adherents to the Bayesian approach to probability carried on the tradition

throughout the next century and a half. Mathematicians such as Bruno de Finetti (Italy) and Harold

Jeffreys (UK) promoted inverse probability and Bayesian analysis during the early part of the

twentieth century, while Dennis Lindley (UK), Leonard J Savage (US), and Edwin Jaynes, a US

physicist, were mainstays of the tradition in the early years of the second half of the twentieth

century. But their work went largely unnoticed.

The major problem with Bayesian analysis until recent times has been related to the use of

priors. Priors are distributions representing information from outside the model data that is

incorporated into the model. We shall be discussing priors in some detail later in the text. Briefly

though, except for relatively simple models the mathematics of calculating so-called posterior

distributions was far too difficult to do by hand, or even by most computers, until computing speed

and memory became powerful enough. This was particularly the case for Bayesian models, which

are extremely demanding on computer power. The foremost reason why most statisticians and

analysts were not interested in implementing Bayesian methods into their research was that

computing technology was not advanced enough to execute more than fairly simple problems. This

was certainly the case with respect to the use of Bayesian methods in astronomy.

We shall provide a brief overview of Bayesian methodology in Chapter 3. Until such methods

became feasible, however, Fisherian or frequentist methodology was the standard way of doing

statistics. For those who are interested in understanding the history of this era of mathematics and

statistics, we refer you to the book Willful Ignorance: The Mismeasure of Uncertainty

(Weisberg, 2014).

Astronomy and descriptive statistics, i.e., the mathematics of determining mean, median, mode,

range, tabulations, frequency distributions, and so forth, were closely tied together until the early

nineteenth century. Astronomers have continued to employ descriptive statistics, as well as basic

linear regression, to astronomical data. But they did not concern themselves with the work being

done in statistics during most of the twentieth century. Astronomers found advances in telescopes

and the new spectroscope, as well as in calculus and differential equations in particular, to be much

more suited to understanding astrophysical data than hypothesis testing and other frequentist

methods. There was a near schism between astronomers and statisticians until the end of the last

century.

As mentioned in the Preface, astrostatistics can be regarded as the statistical analysis of

astronomical data. Unlike how astronomers utilized statistical methods in the past, primarily focusing

on descriptive measures and to a moderate extent on linear regression from within the frequentist

tradition, astrostatistics now entails the use, by a growing number of astronomers, of the most

advanced methods of statistical analysis that have been developed by members of the statistical

profession. However, we still see astronomers using linear regression on data that to a statistician

should clearly be modeled by other, non-linear, means. Recent texts on the subject have been aimed

at informing astronomers of these new advanced statistical methods.

It should be mentioned that we have incorporated astroinformatics under the general rubric of

astrostatistics. Astroinformatics is the study of the data gathering and computing technology needed

to gather astronomical data. It is essential to statistical analysis. Some have argued that

astroinformatics incorporates astrostatistics, which is the reverse of the manner in which we

envision it. But the truth is that gathering information without an intent to analyze it is a fairly useless

enterprise. Both must be understood together. The International Astronomical Union (IAU) has

established a new commission on astroinformatics and astrostatistics, seeming to give primacy to the

former, but how the order is given depends somewhat on the interests of those who are establishing

such names. Since we are focusing on statistics, although we are also cognizant of the important

role that informatics and information sciences bring to bear on statistical analysis, we will refer to

the dual studies of astrostatistics and astroinformatics as simply astrostatistics.

In the last few decades the size and complexity of the available astronomical information has

closely followed Moore’s law, which states roughly that computing processing power doubles every

two years. However, our ability to acquire data has already surpassed our capacity to analyze it.

The Sloan Digital Sky Survey (SDSS), in operation since 2000, was one of the first surveys to face

the big data challenge: in its first data release it observed 53 million individual objects. The Large

Synoptic Survey Telescope (LSST) survey is aiming to process and store 30 terabytes of data each

night for a period of ten years. Not only must astronomers (astroinformaticists) deal with such

massive amounts of data but also it must be stored in a form in which meaningful statistical analysis

can proceed. In addition, the data being gathered is permeated with environmental noise (due to

clouds, moon, interstellar and intergalactic dust, cosmic rays), instrumental noise (due to distortions

from telescope design, detector quantum efficiency, a border effect in non-central pixels, missing

data), and observational bias (since, for example, brighter objects have a higher probability of being

detected). All these problem areas must be dealt with prior to any attempt to subject the data to

statistical analysis. The concerns are immense, but nevertheless this is a task which is being handled

by astrostatisticians and information scientists.

1.2 The Recent Development of Astrostatistics

Statistical analysis has developed together with the advance in computing speed and storage.

Maximum likelihood and all such regression procedures require that a matrix be inverted. The size

of the matrix is based on the number of predictors and parameters in the model, including the

intercept. Parameter estimates can be determined using regression for only very small models if

done by hand. Beginning in the 1960s, larger regression and multivariate statistical procedures could

be executed on mainframe computers using SAS, SPSS, and other statistical and data management

software designed specifically for mainframe systems. These software packages were ported to the

PC environment when PCs with hard drives became available in 1983. Statistical routines requiring

a large number of iterations could take a long time to converge. But as computer speeds became

ever faster, new regression procedures could be developed and programmed that allowed for the

rapid inversion of large matrices and the solution to complex modeling projects.

Complex Bayesian modeling, if based on Markov chain Monte Carlo (MCMC) sampling, was

simply not feasible until near the turn of the century. By 2010 efficient Bayesian sampling algorithms

were implemented into statistical software, and the computing power by then available allowed

astronomers to analyze complex data situations using advanced statistical techniques. By the closing

years of the first decade of this century astronomers could take advantage of the advances in

statistical software and of the training and expertise of statisticians.

As mentioned before, from the early to middle years of the nineteenth century until the final

decade of the twentieth century there was little communication between astronomers and

statisticians. We should note, though, that there were a few astronomers who were interested in

applying sophisticated statistical models to their study data. In 1990, for example, Thomas Loredo

wrote his PhD dissertation in astrophysics at the University of Chicago on “From Laplace

Supernova SN 1987A: Bayesian inference in astrophysics” (see the book Feigelson and

Babu, 2012a), which was the first thorough application of Bayesian modeling to astrophysical data.

It was, and still is, the seminal work in the area and can be regarded as one of the founding articles

in the new discipline of astrostatistics.

In 1991 Eric Feigelson and Jogesh Babu, an astronomer and statistician respectively at

Pennsylvania State University, collaboratively initiated a conference entitled Statistical Challenges

in Modern Astronomy. The conference was held at their home institution and brought together

astronomers and a few statisticians for the purpose of collaboration. The goal was also to find a

forum to teach astronomers how to use appropriate statistical analysis for their study projects.

These conferences were held every five years at Penn State until 2016. The conference site has

now shifted to Carnegie Mellon University under the direction of Chad Schafer. During the 1990s

and 2000s a few other conferences, workshops, and collaborations were held that aimed to provide

statistical education to astronomers. But they were relatively rare, and there did not appear to be

much growth in the area.

Until 2008 there were no astrostatistics working groups or committees authorized under the

scope of any statistical or astronomical association or society. Astrostatistics was neither recognized

by the IAU nor recognized as a discipline in its own right by the principal astronomical and statistical

organizations. However, in 2008 the first interest group in astrostatistics was initiated under the

International Statistical Institute (ISI), the statistical equivalent of the IAU for astronomy. This

interest group, founded by the first author of the book, and some 50 other interested astronomers

and statisticians, met together at the 2009 ISI World Statistics Congress in Durban, South Africa.

The attendees voted to apply for the creation of a standing committee on astrostatistics under the

auspices of the ISI. The proposal was approved at the December 2009 meeting of the ISI executive

board. It was the first such astrostatistics committee authorized by an astronomical or statistical

society.

In the following month the ISI committee expanded to become the ISI Astrostatistics Network.

Network members organized and presented two invited sessions and two special theme sessions in

astrostatistics at the 2011 World Statistics Congress in Dublin, Ireland. Then, in 2012, the

International Astrostatistics Association (IAA) was formed from the Network as an independent

professional scientific association for the discipline. Also in 2012 the Astrostatistics and

Astroinformatics Portal (ASAIP) was instituted at Pennsylvania State under the editorship of Eric

Feigelson and the first author of this volume. As of January 2016 the IAA had over 550 members

from 56 nations, and the Portal had some 900 members. Working Groups in astrostatistics and

astroinformatics began under the scope of IAU, the American Astronomical Society, and the

American Statistical Association. In 2014 the IAA sponsored the creation of its first section, the

Cosmostatistics Initiative (COIN), led by the second author of this volume. In 2015 the IAU

working group became the IAU Commission on Astroinformatics and Astrostatistics, with Feigelson

as its initial president. Astrostatistics, and astroinformatics, is now a fully recognized discipline.

Springer has a series on astrostatistics, the IAA has begun agreements with corporate partnerships,

Cambridge University Press is sponsoring IAA awards for contributions to the discipline, and

multiple conferences in both astrostatistics and astroinformatices are being held – all to advance the

discipline. The IAA headquarters is now at Brera Observatory in Milan. The IAA website is

located at: http://iaa.mi.oa-brera.inaf.it and the ASAIP Portal URL is https://asaip.psu.edu. We

recommend the readers of this volume to visit these websites for additional resources on

astrostatistics.

1.3 What is a Statistical Model?

Statistics and statistical modeling have been defined in a variety of ways. We shall define it in a

general manner as follows:

Statistics may be generically understood as the science of collecting and analyzing data for the purpose of classification,

prediction, and of attempting to quantify and understand the uncertainty inherent in phenomena underlying data

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