Statistical Inference and Prediction in Climatology: A Bayesian Approach

204 Pages·1985·16.969 MB·English

by  Edward S. Epstein (auth.)

STATISTICAL INFERENCE AND
PREDICTION IN CLIMATOLOGY:
A BAYESIAN APPROACH
METEOROLOGICAL MONOGRAPHS
Volume 1
No. 1 Wartime Developments in Applied Climatology, 1947 (Out of Print)
No. 2 The Observations and Photochemistry of Atmospheric Ozone, 1950 (Out of Print)
No. 3 On the Rainfall of Hawaii, 1951 (Out of Print)
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No. 5 Forecasting in Middle Latitudes, 1952 (Out of Print)
Volume 2
No. 6 Thirty-Day Forecasting, 1953. ISBN 0-933876-01-7
No. 7 The Jet Stream, 1954. ISBN 0-933876-02-5
No. 8 Recent Studies in Bioclimatology, 1954. ISBN 0-933876-03-3
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No. 10 Interaction of Sea and Atmosphere, 1957. ISBN 0-933876-05-X
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Volume 3
Nos. 12-20 Meteorological Research Reviews, 1957. Review of Climatology. Meteorological In
struments. Radiometeorology. Weather Observations, Analysis and Forecasting. Applied
Meteorology. Physics of the Upper Atmosphere. Physics of Clouds. Physics of Precip
itation. Atmosphere Electricity
Bound in One Volume. ISBN 0-933876-07-6
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No. 21 Studies of Thermal Convection, 1959. ISBN 0-933876-09-2
No. 22 Topics in Engineering Meteorology, 1960. ISBN 0-933876-10-6
No. 23 Atmospheric Radiation Tables, 1960. ISBN-0933876-11-4
No. 24 Fluctuations in the Atmospheric Inertia, 1961. ISBN 0-933876-12-2
No. 25 Statistical Prediction by Discriminant Analysis, 1962. ISBN 0-933876-13-0
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No. 28 Agricultural Meteorology, 1965. Paperbound, ISBN 0-933876-19-X; Clothbound, ISBN
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No. 29 Scattered Radiation in the Ozone Absorption Bands at Selected Levels of a Terrestrial,
Rayleigh Atmosphere, 1966. Paperbound, ISBN 0-933876-22-X; Clothbound, ISBN 0-933876-
21-1
VolumeS
No. 30 The Causes of Climatic Change, 1968. ISBN 0-933876-28-9
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No. 31 Meteorological Investigations of the Upper Atmosphere, 1968. ISBN 0-933876-29-7
Volume 10
No. 32 On the Distribution and Continuity of Water Substance in Atmospheric Circulations, 1969.
ISBN 0-933876-30-0
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No. 33 Meteorological Observations and Instrumentation, 1970. ISBN 0-933876-31-9
Volume 12
No. 34 Long-Period Global Variations of Incoming Solar Radiation, 1972. ISBN 0-933876-37-8
Volume 13
No. 35 Meteorology of the Southern Hemisphere, 1972. ISBN 0-933876-38-6
Volume 14
No. 36 Alberta Hailstorms, 1973. ISBN 0-933876-39-4
Volume 15
No. 37 The Dynamic Meteorology of the Stratosphere and Mesosphere, 1975. ISBN 0-933876-41-6
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No. 39 Solar Radiation and Clouds, 1980. ISBN 0-933876-49-1
Volume 18
No. 40 METROMEX: A Review and Summary, 1981. ISBN 0-933876-52-1
Volume 19
No. 41 Tropical Cyclones-Their Evolution, Structure and Effects, 1982. ISBN 0-933876-54-8
Volume 20
No. 42 Statistical Inference and Prediction in Climatology: A Bayesian Approach, 1985. ISBN
0-933876-62-9
Orders for the above publications should be sent to:
THE AMERICAN METEOROLOGICAL SOCIETY
45 Beacon St., Boston, Mass. 02108
METEOROLOGICAL MONOGRAPHS
Volume 20 September 1985 Number 42
STATISTICAL INFERENCE AND
PREDICTION IN CLIMATOLOGY:
A BAYESIAN APPROACH
EdwardS. Epstein
CLIMATE ANALYSIS CENTER
NATIONAL METEOROLOGICAL CENTER
NWS/NOAA
WASHINGTON, D.C.
American Meteorological Society
ISSN 0065-940 I
ISBN 978-1-935704-27-0 (eBook)
DOI 10.1007/978-1-935704-27-0
American Meteorological Society
45 Beacon Street, Boston, Massachusetts
Table of Contents
l. INTRODUCTION
1.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I
1.2 Probability, the Language of Uncertainty . . . . . . . . . . . . 3
1.3 Stochastic Processes and Climate Prediction . . . . . . . . . 6
1.4 Final Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I 0
2. SOME FUNDAMENTALS OF PROBABILITY
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II
2.2 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Probability Distributions . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 Expectations and Moments . . . . . . . . . . . . . . . . . . . . . . . 16
2.5 Joint, Marginal and Conditional Probabilities . . . . . . . . 18
2.6 Bayes' Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2. 7 Sufficient Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.8 Conjugate Distributions; Prior and Posterior Parameters 25
3. BERNOULLI PROCESSES
3.1 Definition of a Bernoulli Process . . . . . . . . . . . . . . . . . . 29
3.2 Distributions of Sufficient Statistics . . . . . . . . . . . . . . . . 31
3.3 Prior and Posterior Probabilities . . . . . . . . . . . . . . . . . . . 33
3.4 Conjugate Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.5 Selecting Prior Parameters . . . . . . . . . . . . . . . . . . . . . . . . 40
3.6 Predictions of Future Results . . . . . . . . . . . . . . . . . . . . . 47
3. 7 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4. POISSON PROCESSES
4.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2 Distributions of Sufficient Statistics . . . . . . . . . . . . . . . . 54
4.3 Conjugate Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.4 Selection of Prior Parameters . . . . . . . . . . . . . . . . . . . . . 62
4.5 Predictive Distributions and Probabilities . . . . . . . . . . . 67
4.6 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5. NORMAL DATA-GENERATING PROCESSES
5.1 Normal Distributions and the Central Limit Theorem 77
5.2 Sufficient Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.3 Bivariate Prior and Posterior Densities: Prior and Posterior
Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.4 Conjugate Density, Precision Known . . . . . . . . . . . . . . . 82
5.5 Predictive Distribution, Precision Known . . . . . . . . . . . 84
5.6 An Example: Normal Data-Generating Process with Pre-
cision Known . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.7 Conjugate Distribution, Precision Unknown . . . . . . . . . 88
5.8 The Normal-Gamma Distribution: Marginal and Con-
ditional Densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.9 Predictive Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.10 An Example of Inference and Prediction: Normal Data-
Generating Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6. NORMAL LINEAR REGRESSION
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.2 Sufficient Statistics for Simple Linear Regression . . . . . 107
6.3 Diffuse Prior-Simple Linear Regression . . . . . . . . . . . . 109
6.4 Simple Linear Regression with a Nondiffuse Conjugate
Prior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.5 Predictive Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.6 An Example: Normal Simple Linear Regression 123
7. FIRST-ORDER AUTOREGRESSION
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
7.2 First-Order Normal Autoregression . . . . . . . . . . . . . . . . 140
7.3 Inferences and Predictions . . . . . . . . . . . . . . . . . . . . . . . . 142
7.4 A Numerical Example: Annual Streamflow . . . . . . . . . . 146
7.5 Comments on Computational Methods . . . . . . . . . . . . . 151
7.6 Results When the Prior Is Relatively Uninformative . . 155
7. 7 Results When the Prior Is Informative . . . . . . . . . . . . . . 161
Appendix A: SUMMARY OF BASIC INFORMATION ON
PROBABILITY DISTRIBUTIONS ENCOUN
TERED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
Appendix B: SELECfED TABLES OF PROBABILITY DIS
TRIBUTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
Appendix C: FORTRAN PROGRAM TO IMPLEMENT EX
AMPLE GIVEN IN CHAPTER 7 . . . . . . . . . . . . 191
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
Chapter
1
Introduction
1.1 OBJECTIVE
The objective of this monograph is to introduce to the climatological
and meteorological community a set of statistical techniques for making
predictions about events that can at best be described as being the output
of a stochastic process. These techniques are especially useful when one's
knowledge of the system is incomplete and there is only limited empirical
evidence; these are situations where most of the more widely known ap
proaches are oflittle help. The techniques themselves are very general, but
they will be presented here in the context of climatological and meteoro
logical applications, especially the former.
Most applications of climatological information involve, in one way
or another, predictions. Climatological predictions are not based on detailed
projections of the evolution of weather events, but rather on knowledge
and empirical evidence of the collective behavior of weather (or climate)
at time scales that extend beyond the limit within which weather events
are predictable in any detail. The ultimate objective of (short-range) me
teorological predictions, however unattainable, is totally accurate forecasts
of weather at specific times and locations. In the usual climatological con
text, the ultimate is a description of what may be expected, and how likely
1.1
2 CHAPTER 1
or unlikely various alternatives are. Whether generated by statistical or
physical methods, climatological predictions are inherently uncertain. Even
if we discovered a "perfect" climate prediction technique, our predictions
would still necessarily be imperfect, because inherently unpredictable
weather events will give rise to a background of "noise" that cannot be
avoided. This concept of noise due to weather has been utilized, especially
by Madden (1976}, to estimate the limits of predictability of monthly and
seasonal climate.
Thus the language of climatological predictions must be probability,
even in the best of circumstances when the "climate" is very well-known.
When the "climate" is not so well-known, either because of our lack of
understanding or because of insufficient empirical information, then there
is even more reason to turn to probabilities to express that uncertainty.
If the empirical evidence is very substantial, and directly relevant to
the situation for which a prediction is desired, then the problem is an easy
one. The necessary frequencies are simply extracted from the data and
interpreted as predictions. For example, if one is interested in the maximum
temperatures to be expected next July on an experimental farm in a remote
rural location for which a long and homogeneous climatological record is
available, the relative frequencies of maximum temperatures in past Julys
are examined and accepted as the probabilities of what will happen in the
next year.
But now consider how to deal with the same situation when the farm
is being moved to a new location, and for purposes of experimental design
one wants to know how the July maximum temperatures at the new location
will differ from those at the old location. If the move has been well planned,
measurements at the new location for the last year or two may be available.
How is this information used to make a credible prediction? How does
one at the same time use more qualitative information about location,
drainage, land use, etc., that tells the trained climatologist a good deal
about the likely differences between the two locations? This is clearly a
more difficult problem than the former one, and it cannot be solved
uniquely.
But there do exist methods that optimize the combination of these
partial sources of information: the climatologist's useful although incom
plete knowledge on the one hand, and the too-short empirical record on
the other. We will describe a series of such methods; while they are not an
exhaustive set, they cover a wide sampling of situations with which the
climatologist must deal. These are not methods that can be mechanically
applied. They do involve rigorous procedures and manipulations, but they
also rely particularly strongly on the judgement and expertise of the prac
titioner. Although there will be some very useful applications to the drawing
INTRODUCTION 3
of inferences when large quantities of relevant data are available, the em
phasis will always be on situations where data are relatively scarce.
The limit of "relatively scarce" data is no data at all. In that limiting
condition, all that is left to the climatologist who is required to make a
prediction is his or her judgement and expertise. The methods we will
describe allow for a smooth transition from no data to ample data. They
indeed are applicable when no data are available. This is the case when
the need for the climatologist to be able to quantify his or her judgement
is most critical. The climatologist can learn from the formal developments
described later how to-we believe better and more systematically-express
his or her judgements when data are absent or limited. The predictions
generated with no or meager data will not warrant as much confidence as
those based on substantial empirical evidence-that will be quite clear.
But it will also be clear how much can be gained from additional obser
vations. We will not extend our analysis in this monograph into the im
portant question of the value of additional observations. Instead we will
limit the scope of our treatment. The goal is to produce useful predictions
that are consistent with one's best judgements, and to allow consistent
revisions of such judgements as data do become available.
1.2 PROBABILITY, THE LANGUAGE OF UNCERTAINTY
We use probability to express and to quantify our uncertainty. For
the most part the concepts we employ correspond quite closely to our
intuitive notions of what we mean when we use the term in our everyday
language. We insist on adhering to certain formal rules for assigning and
manipulating probabilities, but these are in general necessary to ensure
that we will be consistent in applying the basic concepts of probability
under circumstances that are occasionally quite complicated.
Probabilities are real numbers in the interval between 0 and l (inclu
sive) that are associated with "events" or "occurrences". If S represents
the set of all possible events (the sample space), then the probability we
assign to S, P{S} is one. Iftwo events are mutually exclusive (their inter
section cannot occur-it has probability zero), then the probability of at
least one of the two events occurring (the probability of their union) is the
sum of the probabilities of the two individual events. There are numerous
texts that discuss these basic axioms for dealing with probabilities and the
consequent rules for manipulating probabilities of compound and condi
tional events. We will not try to repeat such a development here, but in
order to understand the developments and discussions that follow, the
reader should be quite familiar with the basic rules for manipulating prob
abilities. In Chapter 2 we will review some of the formal concepts and