Selected Works of David Brillinger

David began working on this paper while he held appointments at Princeton and Bell Labs, and completed it at the London School of Economics. He recalls (Panaretos [16]) that his motivation to consider this problem came from Don Fraser's program of

structural probability

, and in particular from the issue of formalising aspects of Fisher's

fiducial probability.

A particular example that David had in mind was that of the correlation coefficient: could Fraser's results be used to show that Fisher's fiducial distribution (Fisher [7]) can be obtained as a Bayesian posterior for some prior - as is often the case when a unique sufficient statistic exists? Lindley [15] had proved that, in the real case, a fiducial distribution would arise as a posterior if and only if the statistical problem were invariant, and so David set out to find conditions for invariance.

Although a great deal has been written concerning the theory of tests, decisions and inference for statistical problems invariant under the action of some group, (see for example [4]–[7], [9], [12]–[14], [16]), no great amount of literature exists concerning the problem of discerning whether or not a given problem is actually invariant under some group. In fact the literature seems to consist of one abstract [8] and one paper [15].

1. Let X1,...,

X

n

be independent observations from the uniform distribution on (0, 1].

We consider stationary. additive. interval functions X(Δ). These are vector valued stochastic processes having real intervals Δ = (α, β] as domain, having finite dimensional distributions invariant under time translation and satisfying

A variety of statistical properties have been developed for the number of solutions of an equation

Finite Fourier transforms of stationary mixing processes have been shown to be asymptotically normal in quite a variety of circumstances. The case of a time series

X(t)

with

t

in

R

is considered in, for example, Leonov and Shiryaev [12], Picinbono [16], Rosenblatt [19], Rozanov [21]. The case

oft

in

Z

is considered in Hannan [8, Chap.IV], in Hannan and Thomson [9], in Brillinger [5].

Consider a particle moving on the surface of the unit sphere in R

3

and heading towards a specific destination with a constant average speed, but subject to random deviations.

Examples are presented of statistical techniques for the analysis of random process data and of their uses in the substantive fields of seismology and neurophysiology. The problems addressed include frequency estimation for decaying cosinusoids, signal estimation, association measurement, causal connection assessment, estimation of speed and direction and structural modeling. The techniques employed include complex demodulation, nonlinear regression, probit analysis, deconvolution, maximum likelihood, singular value decomposition, Fourier analysis and averaging.

This volume honors the work of David Brillinger in several areas, but most notably in the fields of point processes and of time series analysis and applications. It has been a privilege to me to have been his PhD student at Berkeley and then become a friend for the past 40 years. It has been quite a journey. David's work on time series has been influential to many people, especially for his students(over forty), many of them who pursued careers in this field. He is well know in Brazil, for his constant visits and valuable collaboration over the years. His book "Time Series: Data Analysis and Theory" became a classic and I was very fortunate to attend his classes using an earlier draft of the book.

The subject of this paper is the higher-order spectra or polyspectra of multivariate stationary time series. The intent is to derive (i) certain mathematical properties of polyspectra, (ii) estimates of polyspectra based on an observed stretch of time series, (iii) certain statistical properties of the proposed estimates and (iv) several applications of the results obtained.

Notation and Assumptions.—Let X(t)

be a strictly stationary r-vector valued process with real-valued components. All moments are assumed to exist.

Let

X(t) (t

= 0, ± 1,... ) be a zero mean,

r

vector-valued, strictly stationary time series satisfying a particular assumption about the near-independence of widely separated values.

This Paper begins with a description of some of the important Procedures of the Fourier analysis of real-valued stationary discrete time series.

Statistical concepts and techniques are basic to scientific investigation. One concept that enjoys both a theoretical and a physical existence is the spectrum. A spectrum may be described as a display of the intensity or variability of a phenomenon versus period or frequency. Spectra are particularly useful in the study of systems subject to resonance, but have many other uses. This paper begins with some of the historical development of the field, describing a sequence of contributions by Michelson, Schuster, Einstein, Fisher, Bartlett, Tukey, and Whittle. The paper next presents collaborative applications to the study of the free oscillations of the earth, to the dispersion of seismic surface waves and to nuclear-magnetic-resonance spectroscopy. Finally, there is mention of open problems and opinions on future directions.

For a variety of musical pieces the following questions are addressed: Are the power spectra of 1/.f form? Are the processes Gaussian? Are the higher-order spectra of 1/.fform? Are the processes linear? Is long-range dependence present? Both score and acoustical signal representations of music are discussed and considered. Parametric forms are fit to sample spectra. Approximate distributions of the quantities computed are basic to drawing inferences. In summary, 1/f seems to be a reasonable approximation to the overall spectra of a number of pieces selected to be representative of a broad population. The checks for Gaussianity, really for bispectrum 0, in each case reject that hypothesis. The checks for linearity, really for constant bicoherence, reject that hypothesis in the case of the instantaneous power of the acoustical signal but not for the zero crossings of the signal or the score representation.

As a concept and as a tool, the Fourier transform is pervasive in applied mathematics, computing, mathematics, probability and statistics as well as in substantive sciences such as chemistry, geophysics and physics. This chapter presents a review of such applications and then four personal analyses of scientific data based on Fourier transforms. Specific points made include: Fourier analysis is conceptually simple, its concepts often have direct physical interpretations, useful statistical properties are available, and there are various interesting connections between the mathematical and physical concepts.

Data of process type are now routinely collected and analyzed in the environmental sciences. This is a consequence, in part, of today' s general availabilty of sophisticated computing, storage, display and analysis equipment. At the same time stochastic models have been developed that take detailed note of the special characteristics of such data and hence allow more appropriate and efficient analyses to be carried through. The problems can be difficult, but often an aproach is suggested by basic scientific background and the parameters have physical interpretations. Recognizing a process type is an important step along the way to its analysis. The goal of this work is to bring out some basic ideas by presenting a number of elementary examples of random process data analysis.

In addition to statistics, David took care in developing my attitude as a scientist - and he wrote a poem of his own in my draft thesis about whales and statistics. He also cared for us personally. We were invited to use Lorie's and David's house when they went to New Zealand in the summer of 1973. Our newborn child spent her first time out of Alta Bates hospital in their house. David also gave me support in a more touchy matter. I was on a US Navy grant, and felt uneasy when I realized that I had to acknowledge the grant in a publication. Strike it in the last galley, was David's advice - which I in the end did not follow. And there was fun, also outside the soccer field. David suggested the movie "The harder they come". My son, an aspiring reggae musician, was happy to find the Jimmy Cliff LP in my old stock. In the last couple of years we have been lucky to have David as an advisor in our Centre for Ecological and Evolutionary Synthesis in Oslo, and to have David repeatedly visiting.

The total numbers of births and deaths in a population are given at discrete equispaced time intervals. It is assumed that the birth and death rates depend on age, the population size and possibly time. Further it is assumed that the rates fluctuate randomly from individual to individual. The problem is to estimate average birth and death rates and the age structure of the evolving population. Results are presented for a population of sheep blow-flies maintained under stable conditions for a two year period (361 observations) by A. J. Nicholson.

This letter concerns the use of stochastic gradient systems in the modeling of the paths of moving particals and the consequent estimation of a potential function.

There is a growing literature on the statistical analysis of data from association-football/soccer games, seasons or groups of seasons. In contrast this paper is concerned with a single play, that is a sequence of successful passes. The play studied contained 25 passes and ended in a goal for Argentina in World Cup 2006. One question addressed is how to describe analytically the spatialtemporal movement of such a particular sequence of passes. The basic data are points in the plane, successively joined by straight lines. The resulting figure represents the trajectory of the moving soccer ball. The approach of this study is to develop a useful potential function, a concept arising from physics and engineering. In particular the potential function leads to a regression model that may be fit directly by linear least squares. The resulting potential function may be used for simple description, summary, comparison, simulation, prediction, model appraisal, bootstrapping, and employed for estimating quantities of interest. The purpose illustrated here is to simulate play in a game where the ball goes back and forth between two teams each having their own potential function.

Potential functions are a physical science concept often used in modelling the motion of particles and planets. In the work of this paper potential function based models are considered for the movement of free-ranging elk in a large, fenced ex- perimental forest. Equations of motion are set down and the parameters involved are estimated nonparametrically. The question of whether a potential function is plausible for describing the elk motion is considered. The conclusion is that it is not possible to reject this hypothesis for the data set and estimates considered.

Elephant seals migrate over vast areas of the eastern Nonh Pacific Ocean between rookeries in southern California and distant northern foraging areas. Several models of particle movement were evaluated and a model for great-circle motion found to give reasonable results for the movement of an adult female. This model takes specific account of the fact that the movement is on the surface of a sphere and that the animal is apparently heading toward a particular destination. The parameters of the motion were estimated. Such a great-circle path of migration may imply that these seals have the ability to assess their position with respect to some global or celestial cues, allowing them to continually adjust their course and achieve the most direct geodesic route between origin and destination of migration. But the navigational mechanism actually used by these seals to accomplish such feats is as yet unknown.

Random processes are basic to the study of environmental data, particularly data in time and space. This work presents three data analyses based on random process models: (a) a trend analysis, based on tting a monotonic trend to river heights; (b) an analysis of point process data, with ordinal-valued marks, for damage assessment following an earthquake, and (c) an analysis of spatial-temporal meteorological data to estimate the speed of motion of a 500 mbar surface. There is discussion of stochastic processes generally. Environmetrics, 8, 269-281 (1997) No. of Figures: 8. No. of Tables: l. No of References: 22

Jerzy Neyman's life history and some of his contributions to applied statistics are reviewed. In a 1960 article he wrote:

"Currently in the period of dynamic indeterminism in science, there is hardly a serious piece of research which, if treated realistically, does not involve operations on stochastic processes. The time has arrived for the theory of stochastic processes to become an item of usual equipment of every applied statistician."

The emphasis in this article is on stochastic processes and on stochastic process data analysis. A number of data sets and corresponding substantive questions are addressed. The data sets concern sardine depletion, blowfly dynamics, weather modification, elk movement and seal journeying. Three of the examples are from Neyman's work and four from the author's joint work with collaborators.

This part will start with a couple of earth sciences papers. We then proceed via an influential methodology paper to describing some work in neurophysiology. In addition, a paper on latent variables and two on robustness of regression to misspecification of the regression function are discussed.

The axis of instantaneous rotation of the Earth does not remain fixed relative to the body of the Earth, rather, its points of interception with the surface wander about within a region approximately the size of a tenniscourt. This wandering was predicted by Euler -in 1765 and confirmed by observation in 1891. The top graph of Figure I provides the

x

and

y

coordinates of the deviation of the North pole from its mean position for the period 1960-1969. (In units of 0".001 =.101ft.) The motion of the pole produces a variation in the latitude which may be used to deduce the time path of the pole. We mention briefly bow this is done.

The response of many dynamical systems to an impulse is a linear combination of decaying cosines. The frequencies of the cosines have generally been estimated in geophysics by periodogram analysis and little formal indication of uncertainty has been provided. This work presents an estimation procedure by the methods of complex demodulation and nonlinear regression that specifically incorporates in the basic model the decaying aspect of the cosines (periodogram analysis does not). The use of plots of the instantaneous phase as a function of time is shown to greatly enhance resolution. Expressions for the variances of eigenfrequencies, amplitudes, phases and damping constants Q are derived by non-linear least-squares. The results are illustrated, for the problem of the free oscillations of the Earth, by computations with the record made at Trieste of the Chilean earthquake of 1960 May 22.

This work is divided into three principal sections which also correspond to the three lectures given at Bloomington. The topics cover, some useful point process parameters and their properties, estimation of time domain parameters and the estimation of freq1.1ence domain parameters.

A point process system is a random operator assigning a nonnegative integer-valued measure to a random nonnegative integer-valued measure. We define certain parameters for such a system and discuss the problem of estimating these parameters. We also consider the related problem of measuring the degree of association of two point processes.

Modern applied statistics typically involves elements of computation, probability theory, statistical theory and collaboration with specialists in the subject matter of some substantive field. In this article I shall describe part of a continuing experience of collaboration with two neurophysiologists from U.C.L.A., H. L. Bryant Jr. and J. P. Segundo. In formal terms, the problem considered is one of measuring the degree of association of points of two different sorts distributed along a straight line in an irregular manner.

An elementary model of neuronal activity involves temporal and spatial summation of postsynaptic currents that are elicited by presynaptic spikes and that, in turn, elicit postsynaptic potentials at a trigger zone; when the potential at the trigger zone exceeds a "threshold" level, a postsynaptic spike is generated. This paper describes three methods of estimating the "summation function", that is, the function of time that converts the synaptic current into potential at the trigger zone: namely, maximum likelihood, cross-correlation analysis and cross-spectral analysis. All three methods, when applied to inputoutput data collected on various neurons of

Aplysia californica

, give comparable results. As estimated, the summation function involved in the explored cells has an early positive-going swing that is large and brief. In the cell L5, but not in R2, there was also a late negative-going swing of longer duration.

Collections of occurrence times of events taking place irregularly in time provide a fairly common, but not broadly discussed, data type. This article is concerned with the particular circumstance of firing times in nerve cells that interact and form networks. The article reviews a progression of statistical analysis techniques: description, association as measured by moments and correlation, regression, and finally likelihood. The data is point process, but may be seen as that of regression and of multivariate analysis in standard parlance. A simple description of data collected simultaneously for one or more cells is provided.

A model in which the conditional expected value of a response variate is an unknown nonlinear function of an unknown linear combination of regressor variates is considered. It is shown that in the case that the regressors are stochastic and jointly Gaussian, or are deterministic and quasi-Gaussian, the ordinary least squares estimates provide useful estimates of the coefficients of the linear combination up to an arbitrary multiplier. The cases of both conditional and unconditional inference are investigated.

A nonlinear time series system is considered. The system has the property that the output series corresponding to a given input series is the sum of a noise series and the result of applying in turn the operations of linear filtering, instantaneous functional composition and linear filtering to the input series. Given a stretch of Gaussian input series and corresponding output series, estimates are constructed of the transfer functions of the linear filters, up to constant multipliers. The investigation discloses that for such a system, the best linear predictor of the output given Gaussian input, has a broader interpretation than might be suspected. The result is derived from a simple expression for the covariance function of a normal variate with a function of a jointly normal variate.