Some Statistical Methods for Random Process Data from Seismology and Neurophysiology
Author(s): David R. Brillinger
Reviewed work(s):
Source: The Annals of Statistics, Vol. 16, No. 1 (Mar., 1988), pp. 1-54
Published by: Institute of Mathematical Statistics
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The Annals of Statistics
1988, Vol. 16, No. 1, 1-54
THE 1983WALD MEMORIAL LECTURES
SOME STATISTICAL METHODS FOR RANDOM PROCESS
DATA FROM SEISMOLOGY AND NEUROPHYSIOLOGY1
BY DAVID R. BRILLINGER
University
ofCalifornia,
Berkeley
To Jeff
AustinBrillinger,
B.A.
Examplesare presentedof statisticaltechniquesforthe analysisof
randomprocessdata andoftheirusesin thesubstantive
fieldsofseismology
and neurophysiology.
The problems
addressedincludefrequency
estimation
fordecayingcosinusoids,
associationmeasurement,
signalestimation,
causal
connectionassessment,
estimationof speed and directionand structural
modeling.
The techniques
includecomplex
nonlinear
employed
demodulation,
regression,
probitanalysis,
maximum
value
deconvolution,
likelihood,
singular
Fourieranalysisand averaging.
decomposition,
Table ofcontents
I. Introduction
II. Seismology
1. The fieldand itsgoals
2. Freeoscillations
oftheEarth
3. Estimationoffault-plane
parameters
4. Quantification
ofearthquakes
5. Arraydata
6. Exploration
seismology
(reflection
seismology)
7. Othertopics
8. Discussion
9. Update
III. Neurophysiology
10. The fieldanditsgoals
11. Neuronalsignaling
12. Assessingconnectivities
13. A structural
stochastic
model
14. Analysisofevokedresponses
15. A confirmed
(Fourier)inference
16. Othertopics
2
3
3
4
10
14
18
21
23
24
24
25
25
26
32
34
36
44
45
ReceivedDecember1986;revisedJuly1987.
'Prepared with the partialsupportof NationalScienceFoundationGrantsCEE-79-01642,
and whiletheauthorwas a Guggenheim
Fellow.
MCS-83-16634
AMS 1980subjectclassifications.
Primary
62M99,62P99.
Key wordsand phrases.Arraydata, averageevokedresponse,
binarydata, Fourierinference,
complexdemodulation,
spatial-temporal
processes,
neurophysiology,
pointprocesses,
probitanalysis,
timeseries.
seismology,
systemidentification,
D. R. BRILLINGER
2
17. Discussion
18. Update
remarks
IV. Concluding
47
48
48
I. Introduction
Thepurposeofstatistics,...,is todescribecertainrealphenomena.
A. Wald(1952)
in timeand/orspace.
The concernof theselecturesis raw data distributed
The basic data are curvesand surfaces.If n denotesthe samplesize and p
thentheconcern
is withthecase of n muchlessthanp.
denotesthedimension,
in
havedevelopedor are developing
thephenomena
In the situationsaddressed,
so thatsubjectmatterplaysessentialrolesin
timeor space. Theyare complex,
drawn.Thereneed
and conclusions
theanalysesmadeand in theinterpretations
Indeed,a principal
reasoning.
ofbothphysicaland statistical
to be combinations
goal ofthe lecturesis to bringout thekeyrolethatsubjectmatterplaysin the
intention
is to showthatthefieldsof
analysisofrandomprocessdata. A further
particuare richin problemsforstatisticians,
seismologyand neurophysiology
The work
larlythose individualswithsome interestin appliedmathematics.
presentedinvolvesa mixtureof data analysisand structuralmodeling.The
employedare broadlyapplibut thetechniques
problemsdiscussedare specific,
cable. The data concernedis of high quality,so that detailedanalysesare
workand a
consistsofpersonal(collaborative)
possible.The materialpresented
methodsthatserveto tiethedevelopment
fewsuccessstoriesofotherparticular
An attemptis madeto presentproblems
froma unifiedpointofview.
together.
results.
ratherthannovelsubstantive
Emphasisis on techniques,
with
ofstatistics
The studyofrandomprocessdata providesa majorinterface
in
collection
of
the
an
there
has
been
scienceand technology.
explosion
Indeed,
of
in
to
much
modern
measurements
part
(corresponding
spatial-temporal
behavingbecomedigital).Some particularissuesand procedures
technology
with
of
statistics
of
the
interaction
technology.
as
a
result
come emphasized
Fourier
systemsanalysis,inverseproblems,
These includesystemidentification,
funcaveraging
(resolutionversusprecision),
bias versusvariability
inference,
run
throughthe
study.These strains
tions,dynamicsand micro-versus-macro
the
broad
rangeof data
examplespresented.Thereis also a desireto display
mustnowbe concemed.
typeswithwhoseanalysisstatisticians
Thereis no claimmade that the analysesare
Some provisosare necessary.
details.Furtheris an overview,
ratherthanspecific
definitive.
Whatis presented
to the
is
of formalism.
The reader referred
more,thereis littlepresentation
forgreaterdetail.
papersreferenced
on somestatisticalmethodsin
Thereare twolectures.The firstconcentrates
It is
methodsin neurophysiology.
thesecondon somecorresponding
seismology,
of
data
in
the
roles
analysis
to see thesamemethodsplayingcentral
interesting
fromtwoquitedisparatefields.Indeed,one oftheprincipalgoalsofthelectures
of statisticaltechniques-byexamplesfrom
was to bringout the universality
thesetwofields.
SEISMOLOGY AND NEUROPHYSIOLOGY
3
II. Seismology
to scientific
methodand statistical
detailhas beenone
Jeffreys...attention
has attaineditspresentlevelof
ofthemainforcesthrough
whichSeismology
precision.
BullenandBolt(1985)
1. The field and its goals. The termseismology
refersto the scientific
of earthquakes
and relatedphenomena.
It has beendefinedas the
investigation
whichare recordsof mechanical
"science based on data called seismograms,
vibrationsofthe Earth"[Akiand Richards(1980)].This latterdefinition
allows
also studyvibrationscaused by the sea, by
the admissionthat seismologists
volcanoesor byman.Onefurther
definition
thathas beengivenis: thescienceof
in theEarth.
strain-wavepropagation
Whateverthedefinition,
thebroadgoalsofseismology
areto learntheEarth's
and a planet'sinteriorcomposition
and to predictthe time,size,locationand
Workersin the fieldseek to
strengthof groundmotionin futureearthquakes.
providevalid explanations
ofearthquake-related
and to understand
phenomena
thesephenomenaso thatlifemaybe madesafer.
locationand quantification
Specificproblemsaddressedincludethedetection,
of earthquakes,the distinguishing
of earthquakesfromnuclearexplosionsand
in theEarth'sinterior
thedetermination
ofwavevelocity
as a function
ofdepth.
in seismology
The accumulationof knowledge
has displayeda steadybackand-forth
thewavesand newinsightconcerning
betweennewinsightconcerning
the mediathroughwhichthewavespropagate.Amongmajor"discoveries"
one
can list are the innercore,theliquidcentralcore,the Mohorovicdiscontinuity,
the movementof tectonicplatescausingearthquakes
themselves
and the locatingofnumerousgas and oil fields.
The fieldis largelyobservational,
withthebasicinstruments
theseismogram
and clock.There are importantexperiments
too, wheretailoredimpulsesare
vibrations
studied.The fieldexperienced
inputto the Earth and the resulting
the "digitalrevolution"in the 1950sand now poses problemsexceedingthe
capabilitiesofeventoday'ssupercomputers.
Statisticalmethodshave playedan important
role in seismology
formany
years-in largepartdue to theefforts
ofHaroldJeffreys
[seeJeffreys
(1977),for
example].Vere-Jones
and Smith(1981)providea reviewofmanycontemporary
instances.Statisticsentersfora varietyof reasons.The data sets are massive.
Thereis substantialinherent
variability
and measurement
error.Modelsneedto
be refined,fittedand revised.Inverseproblemsneed to be addressed.Experimentsneed to be designed.Sometimestheresearcher
mustfallbackon simulations.The basicquantityofconcernis oftena (risk)probability.
In particular,
it
may be pointedout,thatin the construction
ofthe Jeffreys
and Bullen(1940)
traveltime tables,one has an early,perhapsgreatestsuccess,of the use of
methods.[B. A. Bolt's (1976) presidential
robust/resistant
address"Abnormal
is wellworthreadingin thisconnection.]
seismology"
deal withdata of a varietyof types.The important
Seismologists
formsare
fromspatial arraysof seismometers
digitalwaveforms
of variousdimensions
D. R. BRILLINGER
4
in sucha fashionthatan earthquake
havebeenarranged
(wheretheinstruments
listsof
entity)and catalogs(containing
changing
signalmaybe seenas a moving,
forgeographic
regions
an event'stimes,locations,
sizesand othercharacteristics)
ofinterest.
is notwithoutitscontroversies.
ones,such
Thereare fundamental
Seismology
as whetheror notplatetectonics
is a validatedtheory.Thereare practicalones,
such as does the size of the motionof an earthquakeincreasesteadilyas one
data
approachesthefaultordoesit leveloff?As is so oftenthecase,theexisting
and analysismethodsproveinadequateto resolvethesedisputesconclusively.
backseismological
A generalreference
thatprovidesmuchof the pertinent
of somespecific
groundis Bullen and Bolt (1985).We tum to a presentation
problemsand techniques.
2. Free oscillations of the Earth. This subjectis one of the principal
in seismology
overthe last 25 years.Wheneverthereis a great
developments
The seismogramthen
earthquake,the Earth vibratesfor days afterwards.
numberofexponentially
ofa sumofan infinite
decaying
consists,approximately,
ofthecosinusoids
and
cosinusoidsplusnoise;see expression
(2). The frequencies
ratesof decayrelateto the Earth'scomposition.
Measured
the corresponding
The techniques
aboutthatcomposition.
values maybe used to makeinferences
and regularization
may be emof complexdemodulation,
nonlinearregression
willfollow.
Somedetailson thesetechniques
ployedin thisconnection.
thevibratory
motionoftheEarth
As is the case withmanynaturalsystems,
maybe describedby a systemofequationsoftheform
dY(t)
dt
(1)
=
AY(t) + X(t),
withX(.) a (vector-valued)
input.In thecase thattheinputis b8(t), with8(.)
to the earthquakeshock)and initial
the Dirac delta function
(corresponding
as
conditions
Y(O-) = 0, thegeneralsolutionof(1) maybe written
Y(t)
=
exp{At}b
=
Eiexp{t1t}uj
i
,
t > 0,
wheret,uj are the (assumeddistinct)latentsof the matrixA. The spectrum
oftheEarthas a body.Focusingon
is discretebecauseofthefiniteness
occurring
ofY(t) and assumingthepresenceofnoise,onehas
one ofthe coordinates
(2)
Y(t)
=
Eakexp{
k
I3kt)CoS(Ykt
+ Sk) + E(t),
with - /k and yktherealand imaginary
partsofthepj and ?(*) the noise. This
of the seriesY(t) in the
model may be checkedby complexdemodulation
Providedthe
fromtheperiodogram.
offrequencies
neighborhood
Yk,as estimated
is not too great,a singlecosinusoidshouldbe
bandwidthof the demodulation
SEISMOLOGY AND NEUROPHYSIOLOGY
5
1960 Chilean Earthquake
0
L0
0
0
20
40
60
80
timeelapsed (hours)
FIG. 1.
in the
Record of the Chilean great earthquake ofMay 22, 1960, as recordedby the tiltmeter
Grotta Gigante at Trieste. The tides have beenpartially removed.
included,thelogamplitude
shouldfallofflinearly
withtimeand thephaseangle
at (5).
shouldbe approximately
constant.Detailsare givenlater,specifically
recordedat Triesteofthe 1960Chilean
Figure1 is a plotoftheseismogram
thetides.Detailsre the data and the
greatearthquakeafterpartiallyremoving
in
be
tidal removalprocedure
may found Bolt and Marussi(1962).Figure2, a
ofthisdata,suggeststhe
plotofthelower-frequency
portionoftheperiodogram
Periodogram- Chilean Data
co
0
a)
C)
co
0
2
4
6
8
frequency(cycles/hour)
FIG.
2. The periodogramof the data of Figure1 based on 2548 data values. Onlyordinates
correspondingto frequenciesless than 8 cycleslh have beengraphed. The y-axis is loganithmic.
D. R. BRILLINGER
6
presenceof a varietyof periodiccomponents.
The periodogram
of a stretchof
time-series
valuesY(t), t = 0,..., T - 1,is definedas follows.Set
T-1
(3)
dT(X)
=
, Y(t)expt - iXt},
t=o
- x < X < oc.
X is defined
Then theperiodogram
at frequency
as
(4)
Iyy(X\)= (2ZrTT)-IdyT(X)12.
For data fromthemodel(2), IT (X) maybe expectedto showpeaksforX near
the Yk*
The basic ideas ofcomplexdemodulation
are frequency
isolationby narrowband filtering
to focuson a singletermin expression
(2), followed
by frequency
translationto slow the oscillationsdown.The specificsteps are: (i) Y(t) -*
Y(t)exp{iXt}(modulation),
followed
by (ii) localsmoothing
in t of Y(t)exp{ixt}
to obtain Y(t, X), the complexdemodulateat frequency
X. In the case that
Y( t) = a exp -/3t}cos(yt+ 8), onehas
Y(t, A) =1ae8 ePtei(XY-)t, forA neary
0
O, otherwise.
Hence loglY(t,X)I = log(a/2) - /Btand arg{Y(t,A)) = 8 + (A - y)t. Plots of
these quantitiesversust providecheckson modeladequacyand providepreliminaryestimatesof parameters.
Figures3 and 4 presentsuch plots forthe
3.885and 5.6775cycles/h.These frequencies
Chileandata at two frequencies,
weredetermined
thelocationsofpeaksin theperiodogram,
bynoticing
settingA
and thenin somecasesemploying
a nearbyAto get
equal to them,demodulating
in theamplitudeplotcan
a morenearlyhorizontal
phaseplot.The fluctuations
or to splitpeaks
be due to noise,to leakagefromotherfrequency
components
among other things.The rate of decay / is foundto generallyvary with
in thepresentseismological
situation.ResultsfortheChileandata for
frequency
a varietyoffrequencies
maybe foundin Bolt and Brillinger
(1979).
fromthecomplexdemodulate
for
The parameters
couldbe estimated
pictures,
moreeffective
lines.It is generally
to proceedvia
example,by fitting
regression
This has the further
nonlinearregression.
advantageof providingestimated
one
has
a
model
standarderrors.Suppose
Y(t) = S(t; 6) + E(t)
0 and e(*) a noiseseries.
withS(.) knownup to thefinite-dimensional
parameter
In the presentcase, S(t) = aexp{-f3t}cos(yt+ 8) and 0 = {a, /3,y,8). For the
next step, it is convenientto take Xi = 2'Tj/T and to write Yj = dT(Aj),
Ej = dT(Aj) and Sj(6) = dsT(Aj).One willestimateO byminimizing
j(6)
()12
E
jin J
forJ a rangeofsubscripts
withXAnear y.The logicofthisis as follows.There
forempirical
Fouriertransforms
are a varietyofcentrallimittheorems
[see,for
and
example,Brillinger
(1983)].Supposethatthe noiseseriesE(-) is stationary
SEISMOLOGY AND NEUROPHYSIOLOGY
7
at 3.885 cycles/hour
Log Amplitude
C)
LO
O
S
LO
N0
0
10
20
30
40
50
60
70
80
60
70
80
time elapsed (hours)
Phase Angle
LO
0
10
20
30
40
50
timeelapsed (hours)
FIG. 3. The result of complexdemodtdtinzgthe data of Figure 1 at a frequencyof 3.885 cycleslh.
The upper graph gives the logarithmof the runningamplttude.The lowergraph gives the running
phase. The bandwidth
ofthefilter
eAgloyedis 0.594cycleslh.
mixing with power spectrum fee,(A).Then for large T, Ej is approximately
Further the variates
complex normal with mean O and variance 2vrTf,,,(Xj).
It followsthatthe determination
of an
independent.
Ej, Ek are approximately
estimate of 0 to minimizeexpression(6) is approxidmately
the maximumlikelihood procedure. The statistical propertiesof such estimates were indicated in
Bolt and Brillinger(1979) and developedin detail in Hasan (1982). For example,
one findsthe asymptoticvariance of yto be proportionalto
4?Tfe -Y)
3c'2
T
8
D. R. BRILLINGER
at 5.6775 cycles/hour
Log Amplitude
0
co
0
10
20
30
40
50
60
70
80
60
70
80
timeelapsed (hours)
Phase Angle
cm
cn
C:
CZc'
0
10
20
30
40
50
timeelapsed (hours)
as forFigure3, butat thefrequency
FIG. 4. Complexdemoduation
of 5.6775cycles/h.
having considereda limitingprocesswith,B= 4/T as T
-x
o. The inversecubic
It comesfromthe
dependenceon samplesize is on firstglancesurprising.
ofthepeakswhentheyare present.
narrowness
is an exploratory
Complexdemodulation
technique.Hence one has to be
of employing
it at frequencies
of "false"peaks.In
consciousof the possibility
practice,it is foundthatthe nearnessof the phase plot to a straightlineis a
ofthepresenceofa periodiccomponent.
highlysensitiveindicator
Earlier in the paper,it was noted that progressin seismologyshows a
to-and-fro
betweennewknowledge
ofthestructure
ofwavesand newknowledge
of the Earth. This occursin the case of freeoscillations.
Supposeone has an
initialmodelforthe Earthin termsof somephysicalparameters,
e.g.,expres-
SEISMOLOGY AND NEUROPHYSIOLOGY
9
EarthModel CAL8
.~~~~J..
~~-S-wavevelocit
*
P-wavevelocity
CJ
I
I
I
o
t
0
I
I
1000
2000
i
t
I~~~~~~~~~~~~~~~~~
3000
t
4000
t
5000
6000
7000
depth(km)
FIG. 5.
The CAL 8 Earth model. The curvesgive theassumed density(grams per cubic centimeter),
P-wave velocity(kilometersper second) and S-wave velocity(kilometersper second) as a function
of depth assuming a spherical Earth. Given such a model, one can computeimpliedperiods of free
oscillation. Of interestis the inverseproblem,givenperiods what is a correspondingEarth model?
sionsfordensity,
shearwavevelocity
and compression
wavevelocity
as functions
r denotingdepth.Figure5,
of depth,say p(r), c,(r) and cp(r), respectively,
based on the data in Tables 3 and 4 ofBolt (1982),showswhatis meantby an
Earth model.Givensucha model,one can computethe impliedfrequencies
of
freeoscillationYk- How to do thisis describedin Chapter6 of Lapwoodand
Usami (1981),forexample.The relationship
involvedis nonlinear,
butperturbationsmay be expressedlinearlyvia kemels.Specifically,
supposeone perturbs
theparameters
thentheperturbation
byamountsAp,Acs and Acp,respectively,
ofthe frequency
ofthe kthfreeoscillation
is givenby
AYk
jAk(r)
Ap(r) dr + jRBk(r) Acs(r) dr +
Ck(r) Acp(r) dr,
for kemels Ak* Bk and Ck. This expressionis said to lay out the "direct
problem":Given Ap,Ac8 and Acp findAyk.Now supposea greatearthquake
10
D. R. BRILLINGER
occurs.Then new estimatesof the frequencies
Yk are available.One has the
"inverseproblem":GiventheobservedAyk,findAp,Acs and Acp. Becausethe
new frequenciesare just estimates,one seeks a model only approximately
achievingthem.It seemsworthsettingoutthetypeofprobleminvolvedherein
a specificnotation.Let Y and () denotenormedspaces.Let X denotea mapfrom
e to Y, Y = X0. The valuesY and X are given,a value for0 is desired.Let a
denotea scalar.Some,basicallysimilar,
methodsforselecting
a 0 currently
being
employedinclude:(a) regularization,
choose0 to minimizeIIY - X0112
+ allOll2;
< a to minimize
(b) sieve,choose0 subjectto 11011
IIY- X011;
(c) residual,choose
0 subjectto IIY - X0ll< a to minimize11011.
A characteristic
of the solutions
obtainedis that one has to be contentwiththe estimationof some formof
averageof the unknown0. Chapter12 of Aki and Richards(1980) containsa
discussionofinverseproblems
in geophysics.
A characteristic
thatdistinguishes
the presentEarthmodelproblem,
fromtheusualinverseproblems,
is thatthere
are discontinuities
presentin the model-corresponding
to the Earth'slayers.
The aboveperturbation
approachofa nonlinear
problem
to a linearonehas been
employedby geophysicists
formanyyears;see Jeffreys
and Bullen(1940),for
example.
to thestudyoffreeoscillations
Severalotherreferences
maybe noted.Hansen
of Bolt and Brillinger
(1982) extendstheprocedure
(1979)to handlethe case of
fit.Dahlen (1982)
severaleigenfrequencies
presentin the nonlinearregression
sets down the asymptotic
resultsforthe case of tapereddata, that is, when
convergence
factorshave beenintroduced
intothe Fouriertransform
computavia bispectral
tions.Zadroand Caputo(1968)lookfornonlinearities
analysis.
3. Estimation of fault-planeparameters. That thereexistsa see-saw
betweenthe studyoftheEarth'sstructure
and thestudyofearthquakesources
was pointedout earlier.In thissectionit will be indicatedhow a (nonlinear)
ofthesource
probitanalysismaybe employedto estimatebasiccharacteristics
ofan earthquake.
An importantquantityread offthe seismictrace of an earthquakeat a
is the signof the increment
at the arrivalof the first
particularobservatory
to whether
theinitialmotionis a
energyfromthe event.This signcorresponds
compressionor a dilation.In many cases, followingthe observationof an
earthquakeat a numberof stations,if the observedsignsof firstmotionare
plottedon a map centeredat the epicenterof the eventa (radiation)pattern
results.Figure6, takenfromBrilinger,Udias and Bolt (1980),providessucha
plotforone oftheaftershocks
(event4) oftheGoodFriday1964Alaskanevent.
the
due to the locationsof the particularstationsrecording
(Unfortunately,
event,this figuredoes not providea particularly
good exampleof the ideal
Werethe stationswell
radiationpattem,but the data wereofspecialinterest.
one wouldsee mainlysolid circlesin two
scattered,in an ideal circumstance
oppositequadrantsand mainlyopencirclesin theothertwoquadrants.In this
willbe
case onlytwo ofthefourquadrantshavebeencovered.The implication
thatone oftheplaneswillbe poorlydetermined.)
Following
Byerly(1926),plots
such as this have been employedto leam about the source.Beforedescribing
SEISMOLOGY AND NEUROPHYSIOLOGY
11
N
0
0~~~~
0~~~~
~~~/
\
0
0o
FIG. 6. The P-wavefirst-motion
data fortheearthquakeof theAlaskansequencethattookplace
March 30,1964 at 0200. Thesolidcirclesrefertocompressions,
i.e., firstmotionupward,theopen
circlestodilations,i.e., first
motion
downward.
[Reproduced
withpermission
fromBriUlinger,
Udias
and Bolt (1980).]
what may be learned,somedetailsof the earthquakeprocesswillbe set down.
The usual assumption
aredue to
(theelasticreboundtheory)is thatearthquakes
A crackinitiatesat a pointand (in thecase ofpureslip)spreadsout to
faulting.
forma faultplane.As the crackpassesa givenpoint,slip takesplace (on the
in a stressdropand the radiationof seismicwaves.The
faultplane) resulting
radiated(P-) wavesmaybe shownto havea quadrantalpatternwithone ofthe
axes paralleland the otherperpendicular
to the faultplane of the event.It
and thisis whatByerly(1926)contributed,
follows,
thatthedata maybe usedto
the
estimate fault-plane
orientation.
Havingan estimateofthe faultplaneand
the directionof motionon thatplane is important
to geologyand geophysics.
Researchersseek to tie togethersurfaceand subsurfacefeatures,to consider
regionalstressdirections
and to use theresultsto confirm
and extendthetheory
ofplate tectonics.The resultscan be crucialto seismicriskcomputations.
and this has continuedto generallybe the
Byerlyproceededgraphically
workingapproach.However,the resultsso obtainedare subjective,have no
attachedmeasureofuncertainty
and maynotbe easilycombined
withestimates
derivedfromothereventsat thesamesite.
12
D. R. BRILLINGER
The problemmaybe approachedin formalstatisticalfashionas follows.The
data availableconsistof hypocenter
of earthquake,
locationsof observatories,
directions
ofobservedfirstmotions(compressions
or dilations)at theobservatoriesand a storeofknowledge
the Earth'sstructure
concerning
[velocitymodels
as givenin Jeffreys
and Bullen(1940),forexample].It mayfurther
be argued
noiseis approximately
thatseismographic
Gaussian[see Haubrich(1965)].Let a
faultplanebe describedby threeangles(0T' (kT' Op). Let Aij(OT, 4 T, Op) denote
the theoretical
forthewaveamplitude
on thefocalsphereforeventi
expression
at stationj. This expression
maybe foundin Brillinger,
Udias and Bolt (1980).
(The focalsphereis a "little"sphereof unitradiusaroundthe hypocenter.
In
one has to tracethe ray fromthe
carryingout the amplitudecomputation,
hypocenterto the observatory
throughthe focalsphere.)Let Yij denotethe
realizedamplitudeof the seismogram
at the onsetof the event.Then one can
write Yu = aijAij + Eyj,with aij a scale factorand -,j normalmean 0 and
vaxiancea?L variate.Here aij reflects
the attenuation
thesignalexperiences
in
travelingfromthe sourceto the observing
station,whereasEcj represents
noise
caused by disturbances
unrelatedto the earthquakeof concern.Let yj = 1 if
=
>
0
and
It followsthat
O
otherwise.
Yij
Prob{yij = 1}
=
Prob{Yij > 0) =
(pijAij),
writingpij = aj/vaij,forthis signal-to-noise
ratio.The modelmay be further
expandedby includinga term-yijto allowforreaderand recorder
errors,
now
writing
(7)
Prob{yij = 1}
= Yij + (I 2yij)4D(pijAij)A
to y and a small(hencep large)and imprecise
to y near
Precisedata correspond
0.5 or p near0.
The model is seen to take the formof a nonlinearprobit(witha term-y
An exampleofa corresponding
to "naturalmortality").
likelihood
corresponding
is providedby
(8)
HIj(piAij) ij(1 - 0(piA
1-Yi
assumingp to dependon eventalone and y = 0. One can now proceedto
likelihood.
estimatethe unknown
parameters
OT' OT' (P, pi bymaximum
Figure6 includesthe fittedplanes forthe case of event4 of the Alaska
thelikelihood
werecomputed
estimates
sequence.Theseparticular
restricting
(8)
a y termas in (7).
ofeventi = 4 and including
to the observations
Udiasand Bolt(1980),
It is criticalto assessthefitofanymodel.In Brillinger,
functions.
and estimatedprobability
thetheoretical
thiswas doneby comparing
Figure7 is based on a pooledanalysisofsome16 oftheAlaskanevents(labeled
It has beenassumedthatthepi are
thatseemedto go together.
by i previously)
thatthe
all equal in thefitstudied.The figure
providestheempirical
probability
The
ofamplitude.
as a function
observedfirstmotionagreeswiththetheoretical
fittedvalues z = p1Aijhavebeengroupedintocellsofwidth0.1 in theanalysis.
SEISMOLOGY AND NEUROPHYSIOLOGY
13
EmpiricalProbability
ofCorrectFirstMotion
0
_-.
0
c'J
0~~~~~~~~~~~~
-4
-2
0
2
4
z
FIG. 7. A plot of thestatistic(9) ofSection3 and ?(z) forthedata of 16 eventsoftheAlaskan
tothevalues
ofthemodel(7). Herez refers
sequenceof 1964. Theplotis meanttoassessthevalidity
pAij.
Whatis plottedare ?(z) and
(9)
(i yj)l sgn Yij
=
sgnAij, Z - h < Aij<Z + h/{(i,
j)z - h < Aij < Z +h,
for h = 0.5. Here A refersto A(OT, (PT,0p) and # refersto the countof the
numberof elementsin theset.The fitseemsadequate.
The resultsof further
ofthistypemaybe foundin Brillinger,
computations
Udias and Bolt (1980)and Buforn(1982).The maximization
VA09Aof
program
the Harwellsubroutine
see Hopper(1980),provedeffective
in determinlibrary,
ingthemaximum
values.The estimates
likelihood
were,however,
nonunique
and
in somecasesofsmalldata sets.
poorlydetermined
An important
ofsuchanalysesis to fonnclusters
oflikefault-plane
by-product
solutionsforeventsin thesameregion,in orderto getat motionsoccurring
on
the same faultplane; see Udias, Munoz and Buforn(1985),forexample.The
maximumlikelihoodstandarderrorsare usefulin thisconnection.
The practical
oftheworkjust reported
implication
is thatfirstmotionsforlargecollections
of
events may be handledroutinelyand that geophysicalconjecturesmay be
checkedformally.
The finalfault-plane
solutionmay be plottedin traditional
fashionallowingexamination
of the data fordifficulties.
What remainsis for
morerealisticseismicsourcemodelsthantheone treatedin thepaperslistedto
D. R. BRILLINGER
14
An elementary
to thesubjectmatterofconcem
reference
be fittedstatistically.
hereis Boore(1977).
quesand difficult
4. Quantificationofearthquakes. Oneoftheimportant
is how to measurethe "size" of an earthquake.Size is an
tionsof seismology
makesuse of in attemptsto deal with
essentialfeaturethat a seismologist
ofconcem.Specifithebasicphenomena
earthquakehazardsand to understand
is not only interestedin estimatingthe directionof
cally, the seismologist
that
in theoveralldeformation
interested
movement
at thesource,he is further
took place and the amountof energythat was released.Amongthe physical
fora givenearthquakeare theseismicmoment(a measure
quantitiesofinterest
of the seismicenergyreleasedfromthe entirefault) and the stressdrop
betweentheinitialand finalstress.)
(difference
haverelatedtheseismic
For a varietyofseismicsourcemodels,seismologists
IS(X)I,the
oftheamplitudespectrum
momentand stressdropto characteristics
is
ofthesignal.Supposethattheseismogram
modulusofthe Fouriertransform
writtenas
Y(t) = S(t; 0) + E(t),
where s(.) is the signal, ( is an unknownparameterand E(*) is a noise
of s(t; 0), thenwhatis
disturbance.If S(X; 0) denotesthe Fouriertransform
formof IS(X,0)1. A reasonfor
given,fromthe sourcemodel,is the functional
is
phase information
workingin the Fourierdomainhereis that distracting
include
measurements)
eliminated.Commonforms(fordisplacement
IS(X; 0)1= a/1/ + (X/Xo)fand a/{1 + (X/Xo) }
ofsuchfunctional
with0 = {a, /B,
X0). The seminalpaperon thedetermination
to the"size" oftheearthquake
oftheirparameters
formsand on therelationship
is Brune(1970/1971).Estimatesoftheseismicmomentand stressdropmaybe
relate
onceestimates
ofa and X0areavailable.That theparameters
determined
in
form
the
discussion
will
seen
for
a
functional
to size and duration be
particular
graphithat follows.The empirical
practicehas beento estimatetheunknowns
of
the
Fourier
of
the
modulus
of
the
empirical
amplitude
cally froma plot
was suggestedin Brillinger
formalprocedure
transform
Idy(X)I.The following
and Ihaka (1982).
of Idy(X)I may be evaluatedin the case of
The asymptoticdistribution
in Section2.
stationarye usinga centrallimittheoremof the typementioned
is foundto dependon IS(X; 0)I and fee(X)alone.
The asymptoticdistribution
Hence one needsan expression
onlyforthemodulusofS, and as statedabove,
provides.Next,withthe modelY(t)=
this is what the seismologist
generally
s(t; 0) + e(t) and small noise,
idy(X)I= IS(X;0)1+ (dT(X) + dT(-X))/2
+
of
whendeviations
on ISI. However,
showingvariationaroundISI notdepending
of
IdTi froma finalfittedformare plottedversusthefittedvalues,dependence
the erroron ISI is apparent.An exampleis providedin Figure8. This is the
SEISMOLOGY AND NEUROPHYSIOLOGY
15
TaiwanEvent- TransverseShearWave- 29 January
6.7
1981,Magnitude
8.0
8.5
9.0
9.5
10.0
10.5
11.0
seconds
ResidualPlot
5
0
500
1000
1500
2000
fitted
values
FIG. 8. The uppergraph gives thetransverseS-wave componentof thevibrationsof theJanuary 29,
1981, magnitude 6.7, Taiwan earthquake as recordedby the central accelerometersof the Smart 1
array. The array is approximately30 km northwestof the epicenterof the event. The lowergraph
plots the differencesbetweenthe amplitudesof theFourier transformof thedata and corresponding
(final) fittedvalues. The data stretchconsistedof 256 points.
resultof computations
foran earthquakeof magnitude6.7 that occurredin
Taiwan on January29, 1981.The data wererecordedby one oftheinstruments
oftheSmart1 array;see Bolt,Tsai,Yeh and Hsu (1982).The uppergraphofthe
S-waveportionoftherecordedaccelerations.
figureprovidesthetransverse
The
lowergraphprovidesthedeviations
plotjust referred
to. This plotsuggeststhat
the noiseis in part"signalgenerated"
in thiscase. Thereare variousphysical
phenomenathat can lead to signal-generated
noise.These includemultipath
reflection
and scattering.
transmission,
The following
is an exampleof a model
thatincludessignal-generated
noise:
(10)
Y(t) = s(t) +
E(YkS(t
k
-
Tk) + akSH(t
-
Tk)) + e(t),
withthe Tk timedelays,withSH the Hilberttransform
of s and withYk, 8k
16
D. R. BRILLINGER
reflecting
the vagariesofthetransmission
process.[The inclusionoftheHilbert
is discussed,
ofphaseshifts.
The Hilberttransform
transform
allowsthepresence
forexample,in Brillinger
(1975a),page32].Withthe Yk, (k' Tk randomand after
thedistribution
evaluatingthe largesamplevariance,oneis led to approximate
of YJ= dykXj)by a complexnormalwithmean S(Xj; 0) and variancerj =
2iTT(p2/S(X1; 0)12 + a2), wherenow E has been assumedto be whitenoise(of
variancea2), and alsoit is assumedthatEyk,E 3k = 0 and thattheprocessTkis
of signal-generated
Poisson.The ratiop2/a2 measuresthe relativeimportance
noise.This varianceis seento dependon the "signal"throughISI and leads to
Taiwan Event- Amplitude
Spectrum
0
O
0
U)
0
1
0.5
0.1
5
50
10
frequency(cycles/second)
FittedPulse
0
0
0
\
O
c'J
CO
0.0
0.5
1.0
1.5
2.0
2.5
3.0
SeCOndS
theamplitudes
FIG. 9. The uppergraphprovides
transformsoftheTaiwandata of
oftheFourietFiure 8 and the correspondinzgfitted
expectedvalues as cornputedfor the model of Section 4. Both
scales of theplot are logarithmic.The lowergraph provides thefittedpulse s( t).
SEISMOLOGY AND NEUROPHYSIOLOGY
17
wedgingof the typepresentin Figure8. One can proceedto estimate0 by
derivingthe marginallikelihoodbased on the IYjl. This likelihoodmay be
evaluatedand foundto be
FH(exp(-
ilr2
S
IO
r;_____
The uppergraphofFigure9 shows
Besselfunction.
whereIo denotesa modified
a fitof the model IS(X)l = all/(l + (X/Xo)4)to the data of Figure8. This
functionalformwas settledon afterthe degreeof fitof two moreelementary
formswas examined.
Details may be foundin Ihaka (1985). We remarkthat this model fit
to a timedomainpulses(t) = aXop(X0t), where
corresponds
p(t) = [sin-i
- tsin(
+
e-
t/
for t > 0 and p(t) = 0 otherwise.
The expressions(t) = aXop(Xot) indicates
how Xo corresponds
to the durationof the eventand how a corre(inversely)
spondsto size. The lowergraphof Figure9 providesa plotof the fittedpulse.
Once estimatesof a, AOare at hand,thesemaybe converted
to estimatesofthe
seismicmomentand stressdrop via theoreticalrelationships
developedby
geophysicists.
The maximumlikelihoodfitof the modelwas carriedout by a computer
programwritten
by Ihaka.Thisprogram
also generates
standarderrorestimates
and standardized
residuals.Theselatermaybe usedto assessthegoodnessoffit
ofthe model.Figure10 providesa plotofthestandardized
residualsagainstthe
Standardized
ResidualPlot
CM
0
1000
1500
2000
fitted
vaues
FIG.10. A standardized
residualplot,basedon themodel(10),corresponding
tothelowergraphof
Figure8. The differences
betweentheamplitudes
of theFouriertransform
valuesand theirfitted
expectedvalues have been dividedby theirfittedstandarddeviationsto obtainstandardized
residuals.
18
D. R. BRILLINGER
fittedvalues of the sameformatas the residualplotof Figure8. The wedging
corresponding
to signal-generated
noisein the laterplot is no longerpresent;
however,thereis a definite
suggestion
that the fitmightbe improvedin the
regionwherethe signalhas low amplitude.Luckily,thisis the regionof least
It awaitsfutureanalysis.It mightbe handledbyallowingtheseries
importance.
to
E(t) have a nonconstant
spectrum.
5. Arraydata. Todayit wouldbe a strangethingindeedforan earthquake
In fact,fromthe veryearliestdays,
to be recordedon just one seismometer.
readingsof the sameeventat geographically
scatteredobservatories
have been
made use of. Since the 1960s,seismometers
have beendeliberately
arrangedin
geometric
designsoverdistancesof the orderof milesto hundredsof milesin
oftraditional
and sometimes
orderto allowextraction
information
ellicitation
of
newinformation.
An importantuse has been the estimationof the directionfromwhicha
withwhichit is moving.
and thevelocity
Onemannerin
seismicsignalis arriving
ofestimatesoffrequency-wavenumber
whichthisis doneis bythecomputation
as follows.
spectra.The procedure
maybe described
Supposeonehas arraydata;
Y(xj, yj, t), j = 0,..., J and t = 0,..., T - 1. Here (xj, yj) denotes the coordi-
natesofthelocationofthejth sensor.The frequency-wavenumber
periodogram
ofthisdata is givenby
(11)
2
|EE
j
t
} ,
Y(xj,yj,t)expt ii(txj+ vyj+ XAt)
< AIv,A.< xo.
is thefollowing.
A motivation
forthisdefinition
Supposeone has a planewave
=
+
+
+
Y(x, y,t) p cos(ax fly yt 8) oftemporalfrequency
y and wavenumber
willhave a peak near(a, 3,y). (Incidentaly,
K = (a, 1). Then the periodogram
azimuth
withapparent
thiswaveis moving
velocity
y/ja2 + /32 from
givenby
ofarraydatais givenbyFigure11.Whatis plotted
tan0 = /3/a.)Anexample
oftheSmart1 arraylocatedin
ofnineoftheseismometers
are thelocations
ofthetracesusedin thecomputations.
Taiwan.Alsoplottedaretheportions
to thevertical
P-wavepart,oftheJanuary
Thesetracescorrespond
29,1981
earthquake.
(Theinitialnear-flat
partis thenoise,savedina buffer,
justbefore
was30 km
ofthisearthquake
theonsetofthewave.)Theestimated
epicenter
ofthefrequency-waveofthearray.
southeast
Figure12givesa central
portion
X
viaformula
number
forthisdata,as computed
periodogram,
(11),at frequency
1.944was pickedon
to 1.944cycles/s.(The temporalfrequency
corresponding
oftheindividual
Thereis seen
thebasisofa times-series
analysis
seismograms.)
at an azimuth
thatturnsoutto
to be a largepeakin thesoutheast
quadrant,
tothatoftheepicenter
oftheevent.Theradialdistance
correspond
corresponds
ofP-waves.
to thevelocity
to employ,
withthistypeofdatahaveoften
Seismologists
working
preferred
or "Capon"statistic
whattheycall,the"high-resolution"
[see Capon(1969)]
shows
statistic
insteadof theperiodogram
(11).The high-resolution
typically
Beforedefining
moredramatic
peaksthantheperiodogram.
it, we introduce
19
SEISMOLOGY AND NEUROPHYSIOLOGY
Taiwan Arrayand Eventof29 January1981
0
0
C')
0
0
0
E
.1
?A
E0
C>
o0
0
o)
C~)0
o
-3000
-2000
I
I
I
-1000
0
1000
2000
3000
distancefromarraycenter(meters)
bulletdenotes sensor location
as recorded
at nineof
FIG. 11. The verticalP-waveportionoftheJanuary29 Taiwanearthquake
" areplottedat thephysicallocationsofthesensors.
thesensorsof theSmart1 array.The " bullets
fromtheeventhad beensavedin a buffer.
Noise immediately
precedingtheamval ofenergy
somenotation.Let Y(t) denotethej-vector[Y(xj, yj,t)]. Set
T-1
Yk=
T-
Y(t)exp
t=o
-I
L2k
T
for k=0,2,....
Furtherlet B=[exp{-i(pxj +vyy)}]. If X=22d/T, 1 an
to IBTYII2.Nextdefine
thentheperiodogram
(11) is proportional
integer,
M = EykYk,
statisticat
withthe sum overk with2rk/T near X. Now the high-resolution
frequencyX may be definedas 1/B M-'B. If Y(x, y,t) = p cos(ax + fly+
yt+ 8) + noise,thisstatisticmay be expectedto showa peak for(IL,v) near
in part,in orderto be
(a, fi)and X near y. This statistichas beenintroduced,
able to presentthenextexample.
D. R. BRILLINGER
20
Periodogram Taiwan Event
Frequency-Wavenumber
c'LO
C\J
LO)
-0.5 -0.4 -0.3 -0.2 -0.1
0.0
0.1
0.2
0.3
0.4
0.5
wavenumber
(cycles/km)
FIG. 12. The frequency-wavenumber
periodogram of the data of Figure 11. The time series
stretchescontain 720 points. The temporalfrequencyemployedis 1.94 cycles/s.
Figure13 is reproducedfromScheimerand Landers(1974). It showsthe
statisticcomputedfortwo portionsof data recordedby the
high-resolution
a strip-mining
Large ApertureSeismicArray(LASA) in Montanafollowing
blast. These computations
the validityof the high-resolution
confirmed
apof
proach.The statisticforoneportionshowsa singlelargepeakin thedirection
the blast. The statisticforthe following
portionshowsenergyarrivingfrom
variousdirections.This analysisprovidedempiricalproofof the existenceof
of seismicwaves.That thisphenomenon
existedhad beentheorized
scattering
foryears.A frequency-wavenumber
theconfirmation.
data analysishas provided
Spectralanalysesare (too) oftenthoughtof as beingappropriateonlyfor
stationarydata. As thepreceding
exampleshows,the techniquemaybe highly
casesas well.As a secondexamplewemention
usefulin nonstationary
theresults
of Bolt, Tsai, Yeh and Hsu (1982). If, in fact,an earthquakeis caused by
thenthe direction
of thesourceof seismicenergywillbe changingas
faulting,
In thepapercited,Bolt,
thefaultis ripping,
thatis,as thefaulttipis advancing.
timestretches
Tsai, Yeh and Hsu presenthigh-resolution
spectraforsucceeding
SEISMOLOGY AND NEUROPHYSIOLOGY
A~~~~
t4:1he
a
,
49
/~~~~~~TM
19
(mrin.:
21
14 20
sec )
wIth tim.Thehighreworkio
mayhaven
beenthem
firsutexedfrimwaendtal
measurdemen
foflwna
seismicdislocationmovingalonga rupturing
fault.
In each of the preceding
two examples,frequency-wavenumber
analysishas
allowedresearchers
to confirm
thepresenceofsuspectedscientific
phenomena.
6. Exploration seismology (reflection seismology). The problemof
learningthe Earth's crustalstructurecan be approachedas one of system
identification.
The approachto be described
takesadvantageofthefactthatthe
Earth happens to be made up of layeredstrata.Signals,such as powerful
impactsor explosions,
can be deliberately
inputto theEarthand theconsequent
vibrationsrecordedbyan arrayofseismometers
or geophones.
Suchexperiments
may be carriedout in a searchforgas and oil, or in a scientific
studyof the
generalgeologicalmakeupof a regionof interest.The resultsof theseexperimentsmaybe viewedas oneofthegrandsuccessstoriesforstatistical
techniques
generally,
and ofleastsquaresparticularly.
An unusualaspectoftheinferences
made is that in manycases one gets to examinetheirvalidity,by the later
D. R. BRILLINGER
22
in thissectionofthepaper,in
drillingofa well.No workedexampleis presented
largepart becausedata sets are hard to comeby. The materialis presented,
however,because it providesa case wherea rathercompletesolution(design
can be presentedand because the basic experimental
throughconfirmation)
in theneurophysiological
case,whereso muchlessis
techniqueis also employed
known.
propagates
ofan initiatedseismicdisturbance
theenergy
In itssimplestform,
an
interface
it
meets
When
throughthe Earth witha spreadingwavefront.
part
back
and
betweengeologicalstrata,part of the energymay be reflected
at
the
interface.
in acousticimpedance
continueforwarddue to the difference
energyechoes.Knowledgeof subreflected
The sensorsrecordthe returning
ofthe
ofthedepthsand anglesofinclination
surfacevelocitiesallowsestimation
estimaallows
ofthelocationsofreflectors
whereasknowledge
variousreflectors,
In
tionofvelocities.(One notesagaina see-sawin thecollectionofknowledge.)
impactswillbe repeateda numberoftimesat the same
practice,the initiating
is againused.
locationand at pointsofa grid.The powerofaveraging
If the inputsignalis takento be X(t) and if Y(t) denotesthe corresponding
output,thenthe two maybe modeledas related,assuminglinearityand time
invariance,by
(12)
Y(t)
fa(t - s)X(s)
ds.
sinceiftheDiracdeltafunction
The function
a(.) is calledtheimpulseresponse,
a(-)
outputis Y(t) = a(t). The function
8(t) is takenas input,thentheresulting
and velocitiesin the earth beneaththe sourceand
evidencesthe reflectors
as follows.
Suppose
maybe motivated
The modeland itsinterpretation
receiver.
travels
a waveis generated,
a pulseis appliedat timeT. Supposein consequence
back.
a1 is reflected
at distanced, and a proportion
at velocityv1 to a reflector
With X(t) = 8(t - T), then Y(t) = a, 8(t - T - 2d1/v1). (This is actuallythe
that the transmitted
portion
naive modelforradaror sonar.)Supposefurther
at distanced2 and a portionof
at velocityv2to a reflector
continuesdownward
to
by thefirstreflector
back,someofwhichis transmitted
its energyis reflected
has theformY(t) = a18(t - T - 2dJ1v1)+
Nowtheresponse
reachthereceiver.
to the system
a2 8(t - - 2d1/lv- 2d2/v2). This last is seen to correspond
of expression(12) with impulseresponsea(t) = a,8(t - 2dJ1/j)+ a28(t 2d1/v1- 2d2/v2).One can clearlyextendthismodelto situationswithmany
and reflection
transmission
layers,many velocitiesand manycorresponding
to reflecPeaks in the functiona(t) maybe seenas corresponding
coefficients.
is
suchan elementary
interpretation
tors.(It mustbe notedthatunfortunately
phenomenasuch as ghost
likelyto be complicatedin practiceby interfering
havebeendevelopedto handlethese.)The basicsof
reflections.
Some techniques
are discussedin Woodand Treitel(1975),Waters(1978)
seismology
exploration
and Robinson(1983).
as one ofsystemidentification;
given
The problemhas nowbeenformulated
an estimateofthe
stretchesof corresponding
inputX and outputY, determine
impulseresponsea(-). In the case that a pulse close to a Dirac 8 may be
SEISMOLOGY AND NEUROPHYSIOLOGY
23
generatedand that the functiona(-) drops offto 0 reasonablyquickly,a
resultsfromtaking
convenient
procedure
M
X(t)=
E 8(t- MA)
m=1
as input,and the "averageevokedresponse"
1
M
M m=1
to applyingpulsesperiodically.
as an estimateof a(s). This inputcorresponds
to stackingand averaging.
The estimatecorresponds
Suppose one sets myx(t)= Y*X(t) for some convolutionoperation" ."
Then from(12) one has
myx(t)
=
fa(t - s)mxx(s)
ds
and one has a deconvolution
(or inverse)problemto solve.Supposeone decides
to seek an X(*) suchthat
Ja(t - s)mxx(s) ds z a(t),
In termsof Fouriertransforms,
to allow elementary
the left-hand
processing.
side heremay be expressedas Jexp{iXt}A(X)Mxx(X)
dX, withMXXa Fourier
transfornof mxx. Thenwhatis wantedis an X suchthat Mxx(X) - 1 on the
supportof A(-). If A(X) is knownto be near0 for0 < X < Xo and forA > A1,
is the"chirp"signal
thena possiblefunction
X(t) = cos([AO + (A - Ao) ]t)-
for0 <t
< T.
In the seismiccase, the values of o, X, have been determined
in various
in radarworkduringWorldWarII [see
The chirpprobeoriginated
experiments.
Cook and Berenfield
(1967)].It maybe seento attachnearequal powerto the
frequencies
betweenXo and AX.In the seismiccase specialdeviceshave been
developedto inputthechirpsignalto theearth.The signalis inputrepeatedly
and theresultsaveraged.The responseis thenconvolved
withthechirpfunction,
that is, myx is formedto estimatea(.). Structurecan appear dramatically
described
here.
duringthe cross-correlation
processing
In practice,subtlefurther
processing
is employedto handlewavefront
curvaand othernaturalphenomena
thatmaybe present.
ture,ghostreflections
7. Other topics. Thereare otherproblemsarisingin seismology
to which
statisticalmethodology
can be appliedfruitfully.
These includeanalysisof the
coda (i.e.,oftheirregular
trailing
partofthedisturbance),
analysisofscattering,
riskanalysis,nonlinearphenomena,
pointprocessstudies,polarization,
cepstral
of earthquakesfromexplosions[see, e.g., Tjostheim
analysis,discrimination
24
D. R. BRILLINGER
laws,earthattenuation
study,traveltimetableconstruction,
(1981)],seismicity
and
Vere-Jones
quake locationand azimuthaldependenceof characteristics.
In somecasesworkhas begun.
Smith(1981)discussseveraloftheseproblems.
have long been serioususers of statistical
8. Discussion. Seismologists
in theentry,
statement
makingthefollowing
methods.One findsHaroldJeffreys
"Seismology,statisticalmethods,"in the InternationalDictionaryof Geois as important
a partof the resultas the estimate
physics:"The uncertainty
itself. ... An estimate without a standard error is practically meaningless."
traveltimetables
Hudson(1981) remarks:"The successofthe Jeffreys-Bullen
use of soundstatisticalmethods."
consistent
was due in largepartto Jeffreys'
WhenI asked mycolleagueB. A. Bolt whathe saw as the roleof statisticsin
science.... The role
he replied:"Seismology
is largelyan inferential
seismology,
observaforturning
procedure
is to providea rigorous
ofstatisticsin seismology
ofthe
aboutproperties
statements
tionson seismicwaves,etc.,intoprobabilistic
(real) Earth."
is characterized
One maynotethatworkin seismology
by massivedata sets,
models,
error,defining/fitting/refining
inherentvariabilityand measurement
needsforrobust/redescription,
simulation,
probabilistic
designofexperiments,
of
inverseproblemsand combination
situations,
predictive
sistantprocedures,
in all theseconnections.
Statisticshas muchto offer
observations.
in 1983,workhas progressed
9. Update. Sincethe lectureswerepresented
Smart1 data
(1985)has employed
on variousofthetopicscovered.Abrahamson
of the faultrupturetip. Chiu (1986) studiesthe
to bettersee the movement
(1986)
source.Lindberg
ofa movingenergy
theparameters
problemofestimating
ofthefrequencies
offree
develops"optimal"tapersto employin theestimation
data
The approachofKitagawaand Gersch(1985)to nonstationary
oscillations.
The bookbyUdias,Munoz
seemslikelyto proveofbroadpracticalapplicability.
and Buforn(1985) goes into substantialdetail overthe formalestimationof
forthevarianceof
Copas (1983)setsdownan expression
parameters.
fault-plane
a statisticlike that of (9). Brillinger
(1985) developsa maximumlikelihood
ofa planewavegivenarraydata. Donoho,
statisticfordetectionand estimation
procedureforbetter
Chambersand Lamer (1986) developa robust/resistant
aligningthe seismictracesof a section.Mendel(1983,1986)presentsmaximum
seismollikelihoodstatespace-basedmethodsforhandlingthedata ofreflection
to
ogy.Shumwayand Der (1985)indicatehowtheEM methodmaybe employed
of seismograms
deconvolvepulseshiddenin seismictraces.The nongaussianity
obtained in reflection
seismologyis being taken specificadvantageof; see
Giannakisand Mendel(1986).The techniquesof Donoho (1981) and Lii and
case. The
Rosenblatt(1982) seemboundto proveusefulin the seismological
thesis,Ihaka (1985),has been completed.Ogata [e.g.,Ogata (1983) and Ogata
analysesof
and Katsura (1986)] has carriedout a varietyof likelihood-based
on
havebegunworking
earthquaketimesas a pointprocess.Manystatisticians
mention
O'Sullivan(1986).
We specifically
statisticalaspectsofinverseproblems.
SEISMOLOGY AND NEUROPHYSIOLOGY
25
will go in the
One can speculateon wherethe fieldof statisticalseismology
comingyears.It seemsclearthattherewillbe muchconcernwithnon-Gaussian
data and analysiswill
noise and signals,that vector-valued
spatial-temporal
becomethe norm,thatlarge-scale
conceptualmodelswillbe set downand that
strengthin
there will be a varietyof techniquesdevelopedfor borrowing
situationswithscantydata,e.g.,riskestimation.
III. Neurophysiology
endeavorto buildstructural
modernbiometry
is the interdisciplinary
stochasticmodelsofbiological
phenomena.
J.Neyman(1974)
is the branchof science
10. The field and its goals. Neurophysiology
concernedwith how the elementsof the nervoussystemfunctionand work
is seen to involvechemicalmechanisms,
electrical
together.The functioning
The studiesextendfromthemovements
mechanisms
and physicalarrangement.
ofthebrain.
to themassbehaviorofthecomponents
ofindividualions,through
rangeto theheroic:howto explainthingslike
The goalsofneurophysiologists
behavior.At a less ambitious
memory,emotion,learning,sleep,expectation,
withhowa singlenervecellrespondsto
level,neurophysiologists
are concerned
oftheenvironment.
information
and changeswithalterations
stimuli,transmits
and structural
unitofthenervoussystem.
The neuronis boththefunctional
The elementsof the
The brain is a multiprocessor
of dramaticcomplexity.
fromthosein theseismiccase,in thatthey
nervoussystemmaybe said to differ
apparentlyhave purposes.
withresearchers
collecting
variedand extenThe fieldis largelyexperimental
sive data sets. The data includephotographs
made via electronmicroscopes,
fluctuating
voltagesand currentlevels withinsinglenervecells and finally
electroencephalograms
(thebrain'selectricalpotentialat pointsnearthe skull.)
The studiesare sometimes
butoftencomplexexperimental
simplyobservational,
designsare employed.
Importanttechniquesthat are made use of includestainingto identify
of microelectrodes
within
individualneurons,insertion
to makemeasurements
ofwholesuitesofresponsesto a stimulusof
individualcells and the averaging
interestin orderto reducewhat can be the dominanteffectsof noise.Many
are computer
controlled
and computer
processed.
experiments
Discoveriesmade by neuroscientists
includethe following.
Nervecells communicatewitheach otherin botha chemicaland electricalfashion,
the voltage
pulse that travelsalonga neuron'soutputfiberis of nearconstantshape and
thereare a broad varietyof nonlinearphenomenathat occur.A numberof
verifiablephysical laws and effectivedeterministic
models (such as the
Hodgkins-Huxley
equations)havebeensetdown.Muchinsighthas beengained,
especiallyat thelevelofsmallgroupsofneurons.At thelevelofthebrainitself,
Herethebrainis viewedas a blackbox
knowledgeis mainlyphenomenological.
and studied by systemidentification
techniques.Whateverthe approach,
discoverieshave been made leadingto lifesavingand life-improving
clinical
diagnoses.
26
D. R. BRILLINGER
Statisticalmethodsenteredwiththe quantification
of the field.No single
individualscientistseemsto have had a dominating
effect,
rathertherehave
beenmanycontributing
workers-researchers
concerned
withelectroencephalograms(EEGs) and researchers
concernedwith small collectionsof neurons.
Statisticalmethodsenteredboth because of highnoise levelsand because a
varietyofphenomena
seemedto be inherently
stochastic.
Evidenceforthislast
is presentedin Burns(1968)and Holden(1976).Pertinent
bookson neurophysiologyincludeFreeman(1975),Aidley(1978)and Segundo(1984).Generalreviews
ofstatisticalmodelsand methodsin neurophysiology
are givenin Moore,Perkel
and Segundo(1966)forthecasesofsingleneuronsand ofsmallgroupsofneurons
and by Glaserand Ruchkin(1976)forEEGs. Statisticalmethodsforclassificationand pattemrecognition,
forhandlingartifacts
and fordata summarization
are in commonuse.
is one ofthemostactivebranchesofscience.The physiological
Neurobiology
phenomenawithwhichit is concerned
arefundamental
and in mostcasesbarely
understood.
11. Neuronal signaling. One oftheimportant
meansby whichnervecells
communicate
is via spiketrains.The inlaysat the tops of the threegraphsof
Figure14 giveexamplesofspiketimesrepresentative
ofthreedifferent
sortsof
neuronalbehavior;pacemaker
occursin bursts)
(near-periodic),
bursting
(activity
and bursting
withacceleration
withinbursts).
(offiring
Suppose that a neuronfiresat timesTn, n = 0, + 1, + 2,.... A convenient
ofits temporalbehavioris providedbywriting
formalrepresentation
Y(t) =
n
a(t
Tn
with8(.) theDirac deltafunction.
Thisrepresentation
leadsto resultsanalogous
to ordinary
time-series
resultsin manycases.In thecase thatthe T, arerandom,
one has a stochasticpointprocess{Tn}. A principaldescriptor
ofa pointprocess
is providedby its ratefunction.
This is givenby
lim Prob{point in (t, t + h }h,
h
as h tendsto 0. In the stationary
of the
case,wherethe stochasticproperties
is constantand so
processdo not dependon the timeorigin,the ratefunction
onlycrudelyusefulthen.
is an important
The autointensity
in thestationary
function
case.
parameter
It is definedas
lim Prob{point in (t, t + h]lpoint at O}/h,
h
as h tendsto 0. It is a pointprocessanalogof the autocovariance
of
function
time-series
be
for
analysisin a generalsense.Thisparameter
may used, example,
to describethe behaviorof spontaneously
firingneurons.Figure14 presents
examplesforthreecases. In the firstcase, the neuronis firingapproximately
The (estimateof)theautointensity
is seento oscillate(withperiod
periodically.
27
SEISMOLOGY AND NEUROPHYSIOLOGY
- L10 as Pacemaker
Function
Autointensity
0
5
10
15
20
30
40
lag(seconds)
co
- L10 Burstinrg
Function
Autointensity
0
10
20
lag (seconds)
u)
*
_ .
.
,
.
~~~~~~~~~~~~~~~~~.
. . . . . . . .. . . ... . . ..... . . .............
cM
Thecellis
FIG. 14. Pointprocessdata (spiketrain)fromthenervecellL10 ofAplysiacalifornica.
of
fashions.The inlaysat thetopsofthethreegraphsgivebriefstretches
behavingin threedifferent
Thefunctions
thedata (butnotonIthesametimescalesas theautointensities).
plottedare estimates
of the autointensityfunctionsbased on 1538, 1019, 1631 spikes, respectively.
28
D. R. BRILLINGER
equal to the intervalbetweenthe points).In the secondcase, the neuronis
evidencing
activityin bursts.The probability
thatthe neuronfiresagainsoon
afterit has firedis high.In the thirdcase,the neuronis also firing
in bursts;
however,now thereis structure
withinthe bursts,the rateof firing
is seen to
increasetherein.The burstshereare at regularintervals.
The autointensity
functions
have been estimated,forthis figure,by the
statistic
ITn -Tm
-
ti < h/2}/Nh,
with N the total numberof points,withh a smallbinwidthand witht lag.
(Here # refersto the countof the numberof pointsin the set.) The data
analyzedare forthe cell L10 of Aplysiacalifornica,
the sea hare.They were
collectedand previously
analyzedby Bryant,Marcosand Segundo(1973).The
experimental
procedures
and detailsof the data preparation
may be foundin
thatreference.
A questionthat arisesin the studyof smallnetworks
of neuronsis which
neuronsare interacting
withwhich?In otherwords,whichspike trainsare
associatedwithwhichothers?A usefulparameter
to employin thestudyofsuch
questionsis providedby the cross-intensity
function.
Supposingone has spike
trainsnamedM and N, thenthecross-intensity
function
of N givenM at lag t
is definedas
limProb{N pointin (t, t + h]M pointat 0)/h,
h
as h tendsto 0. If the M spiketrainconsistsof pointsUrn and the N trainof
pointsT thenthiscross-intensity
maybe estimatedby
# IT,,-,rn -
t <
h/2)/Mh,
with M denotingthe numberof M pointsin the data set, with h a small
binwidthand with t lag. Figure 15 presentsthree examplesof estimated
functions.
The firstgraphrefers
cross-intensity
to data fromcellsL3 and L10 of
Aplysiacalifornica.
The behaviorexhibited
hereis thatofnegativeassociation;
L1O's firingis inhibiting
the firing
of L3 (forapproximately
0.5 s). If one asks
whetherthe values at negativelags differ
fromthe level of no-association
by
morethansamplingfluctuations,
one findstheydo not.This resultis consistent
withthe cell L10 drivingthecell L3. The middlegraphcorresponds
to positive
association.It is fora cellin therightvisceropleural
connective
(RVP) and cell
R15. The firstcell tendsto excitethe secondforabout0.25 s. The finalgraph
a morecomplicated
situation.Thesedata setswerealso
represents
(polyphasic)
detailsmaybe
analyzedin Bryant,Marcosand Segundo(1973),wherefurther
weredeveloped
found.The approximate
ofsuchstatistics
distributions
sampling
in Brillinger
(1975b).It was found,forexample,thatit couldbe moreconvenient
to graphthe squarerootoftheestimatein somecircumstances.
The cross-intensity
function,
beinga pointprocessanalogofcovariance,
may
be expectedto be an inadequatemeasureof relationship
(as usual,correlation
statisticaldata,it is usual
does notimplycausation).In thecase ofelementary
SEISMOLOGY AND NEUROPHYSIOLOGY
29
- L3givenL10
Function
Crossintensity
0
-2.5
-2.0
-1.5
-1.0
.0.5
0.0
0.5
1.0
1.5
2.0
2.5
1.5
2.0
2.5
lag (seconds)
- R15givenRVP
Function
Crossintensity
-2.5
-2.0
-1.0
-1.5
-0.5
0.0
0.5
1.0
lag (seconds)
- L3givenLi0
Function
Crossintensity
-20
-10
0
10
20
lag (seconds)
FIG.15. Estimatesofthecross-intensity
functions
forthree
pairsofAplysianeurons.Theestimates
are basedon (1746,302),(1101,288),(1019,993) spikesin thepairs oftrains, respectively.
30
D. R. BRILLINGER
to turnto regression
In thepointprocesscase it is possible
as a bettertechnique.
to carryout regression-type
analyses.For example,one may fitthe following
fonnofmodel:
lim Prob{N spikein (t, t + h)IM spike train}/h = i + Ea(t
h
- am),
m
to as the
as h tendsto 0. The function
a(t) appearingin thismodelis referred
Set
impulseresponse.Thismodelmaybe fitas follows.
M
dT()=
E
m=1
exp{-iXam,,
Coherence - LI 0 withL3
co
o L\.
C;
*\7-
0
1
_>
,\---
3
2
4
A
5
frequency
(cycles/secnd)
ImpulseResponse
cm
ICM
-0
-10
-
54
-5
A
10
0
0
5
10
lag(seconds)
and impulseresponseforthedata oftheuppergraphofFigure
FIG. 16. The estmted coherence
15. Thehorizontal
linegivesan estimate
ofthelevelexceededbychanceonly5% ofthehmewhenthe
spiketrainsare independent.
SEISMOLOGY AND NEUROPHYSIOLOGY
31
with a similardefinition
fordT(X). These are point processanalogsof the
empiricalFouriertransform
(3) oftime-series
data.The cross-periodogram
ofthe
givendata at frequency
X is definedas
dT(X).
INM(X) = (27TT)1dT()
If the cross-periodogram
is smoothedto obtain fTM(X),then fNMQ(X)is an
in thecase that {M, N) is a bivariatestationary
estimateofthe cross-spectrum
of the impulseresponsea(t),
pointprocess.Now A(X), the Fouriertransform
may be estimatedby f T (X)fMTm(X)-1. The impulseresponseitselfmay be
Coherence - RVP withRl 5
co
c
C;
~AA
0
1
4
3
2
5
frequency(cycles/second)
ImpulseResponse
LO
LOl
-10
-5
0
5
10
lag (seconds)
FIG. 17. The estimated
coherence
and impulseresponseforthedata ofthemiddlegraphofFigure
14. The horizontalline in the uppergraphgives theapproximate
upper95% nullpointof the
distribution
ofthesamplecoherence.
32
D. R. BRILLINGER
estimatedby back Fouriertransforming
AT. The strength
of the relationship
proposedin the model may be measured,at frequencyX, by the sample
coherencyfunctionR'M(X) = fN(A)/ IMM(X)I NN(X). Its modulussquared
is calledthesamplecoherence.
The coherence
liesbetween0 and 1, beingnearer
to 1 the stronger
the relationship.
Moredetailsof thesecomputations
maybe
foundin Brllinger(1975b)and Brllinger,
Bryantand Segundo(1976).[We here
in Wiener(1930).]
followtheuse oftheterms"coherency"
and "coherence"
Figures16 and 17 providetheresultsofsuchan analysisforthefirsttwodata
sets of Figure15. In each case the firstgraphis of the samplecoherence.
The
coherencesare at some distancefromthe value 1.0, but above the 95% null
linesin thefigures).
significance
level(givenbythehorizontal
The relationship
is
inherentlynonlinear,so it could have been anticipatedthat the coherence
estimatewould not be close to 1.0. Furtherdiscussionof these and similar
analysesmaybe foundin Brillinger,
Bryantand Segundo(1976).
12. Assessing connectivities. Questionsthat can arise with small networksofneuronsinclude;is oneneurondriving
therestand ifoneapparently
is,
whichone is it? The nextdata analysisto be presented
addressesthisquestion
forthreeAplysiacellsL2, L3 and L10. Fromotherexperiments
theneurophysiologistsknewthatcellL10 was driving
cellsL2 and L3. It wasnotknownifthere
wereanydirectconnections
betweenL2 and L3. The firstthreegraphsofFigure
18 presentestimatesofthethreecoherences,
L10 withL2, L2 withL3 and L10
with L3. As mighthave been anticipated,
thesesuggestrelationship
existsin
each case.
ofcellsL2 and
It is possibleto addressthequestionofthe directconnection
L3, in thepresenceofL10,bypartialcoherence
analysis.Supposethat{A, B, C)
is a trivariatestationarypoint process.Let RAB(X) denotethe coherency
of RACand RBC. Then
function
of processesA and B, withsimilardefinitions
oftheprocessesB and C, havingremovedthe(lineartime
the partialcoherency
ofprocessA, is givenby
invariant)effects
(13)
RBCIA
~~V(1-
RBC-
RBARAC
IRBAI12) (1
-_IRcAI12)
suppressingthe dependenceon X. This definition
may be motivatedseveral
when
betweenthe processesresulting
ways. For example,it is the coherency
theirbestlinearpredictors
basedon A are removed.
Or,it is givenby
lim corr(d -
ddT,dT
;
d[T
2
and fBA/fAA' fCA/fAA are
Here corrdenotethe(complex)correlation
coefficient
An estimatemay be determined
coefficients.
approximateregression
by subsideofexpresestimatesforthequantitiesappearingon theright-hand
stituting
betweentheprocessesB and C beyondtheir
sion (13). If thereis no connection
SEISMOLOGY AND NEUROPHYSIOLOGY
Coherence Li 0 and L2
o
Coherence L2 and L3
o
CR~~~~~~~~~~~~c
o
33
.I
,
N
N
14,0
1
2
3
4
Coherence Li 0 and L3
6
1
2
3
4
Partialcoherence L2 and L3
o
6~~~~~~~~~~~
expectedtoo 0e near zero.
6h
fiaorp
celsL,Lan
't,0
fFgr
L10
2hr
8poie
is no sugsto
On
Onrelecton,
relecton, tfrequien
tfisquienatoisin
(Hz)shngth
(Hz)
herslso
of
diec
h
optto
o
h
4oncinbigpeet
2
degrizontal
whine: linea:95%onullsleve
95heul leveloesn
hogrizontalhth
a nticshvor
Dtatistfor
FIG.ra18.
anetwr ofpthreenAlysiapneurons
esimat
hishl
highl baedonln
baedonln
the dpatloerdence
FIG.raticDatatis
dpatloendence esinmat
ofppareentlysiaapturonsthe
taeten
to be
0nd zerg.
the otereample
jorinutidepndencrter
onsA,sthen the
coherence
samperxmlso
mayub
s
coherence
ofapatial
RT coIA
partial
earoinutiopedncuter onscussion
coAmayuearin
sitation. fourthern
othert
patil9ohrececopu
eamplSesunof
driscsiongand
D. R. BRILLINGER
34
so far
13. A struCturalstochasticmodel. The analysesofneuronalfiring,
direct
with
Parameters
type.
and
regression
correlation
presented,are of the
and Segundo
In Brillinger
have not beenintroduced.
biologicalinterpretations
and fittedbythemethodofmaximum
(1979),a conceptualmodelis constructed
elements.
likelihood.The modelinvolvesthefollowing
flowsto a
genesis.Thiscurrent
Inputto a nervecellleadsto electrical-current
in thecourseofitspassage.Whenthevoltagelevelat
zone,beingfiltered
trigger
the currentzone exceedsa thresholdvalue,the nervecell fires.The neuron
Thisprocessmaybe specified
backonlyto thetimeofpreviousfiring.
remembers
potential)at the
as follows.
Let U(t) denotethevoltage(membrane
analytically
triggerzone at timet. Let B(t) denotethe timeelapsedsincethe neuronlast
fired.Let X(t) denotethe(measured)inputto thecell.Then,assuminglinearity
onecan write
and timeinvariance,
U(t) =
f(
B
(s)X(t - s) ds,
(impulseresponse)a(.). The neuronfireswhenthe
function
forsomesummation
level0(t). Dependingon thelevelat whichthe
processU(t) crossesa threshold
ofthenervecell,theinputwilleither
mechanics
is set and theinternal
threshold
and Segundo(1979),
In Brillinger
accelerate(excite)or slow(inhibit)the firing.
as follows.Inputto thecellwas
and discretized
was completed
thismechanism
outputwas Yt, t = O,..., T - 1,
writtenXt, t = 0,..., T - 1. Corresponding
to t and with
in the(small)intervalimmediate
withYt= 1 iftherewas a firing
With Bt denotingthe timeelapsedat t sincethe preceding
Yt= 0 otherwise.
time that Y = 1, theyset
Bt- 1
Ut-
E asXt-s.
s=O
a formoffeedback.
ofintroducing
The presenceofBt in themodelhad theeffect
had theform(t = 0 + Etwith
function
Finally,theyassumedthatthethreshold
distribunormalshavingmean0, variance1 and cumulative
the e's independent
tion functiond(*).
ofthe givendata and themodelthentookthe form
The likelihoodfunction
T-1
H
4(Ut
t=O
-
)Yt(1 - 1(Ut
-
0))l-Yt.
withrespect
thislikelihood
bymaximizing
Parameterestimatesweredetermined
determined
were
errors
by
procedures
0
standard
to and the a.. Approximate
likelihood.
traditionalto maximum
Figure19 presentsthe resultsof one such analysis.In thiscase fluctuating
level
intothecellR2 ofAplysia.The current
currentX(t) was injecteddirectly
uniform
was
that
(but
distribution
approximately
was takento have marginal
The
rate
was
50
The
samples/s.
the
technique). sampling
that is not crucialto
the
and
the
noise
of
a
stretch
injected
of
the
signal
figuregives
uppergraph
are
the
fired.
Details
of
the
neuron
which
at
experiments
times
corresponding
ifnotimpossible,
to see a
givenin Bryantand Segundo(1976).It is verydifficult,
35
SEISMOLOGY AND NEUROPHYSIOLOGY
NeuronR2 - Noise Driven
0
4
3
2
1
5
tky (seconds)
Function
Summation
C
' 0.0
0.2
0.4
0.6
0.8
1.0
lag (seconds)
Probability
ofFiring
Empirical
0.
40
ci ,r ,
,
-300
-200
I I I/,.|
-100
0
X
50 100
200
300
hnearpreditor
with
FIG. 19. The resultsoffittintheneuron
modelofSection13 todata obtainedin an experiment
theceUR2 ofAplysia.Theuppergraphis a segment
ofthedata. Noise(lowertrace)is injectedinto
the cell. The uppertracegies corresponding
observed
firngtimes.The middlegraphgives the
maximum
likelihood
at 25 lags. Thelowergraph
estinateofthesummation
function
a(-), estimated
providesthe statistic(14) of Section13 and the curve4D(U- I) with# the estimatedmean
threshold.
The verticallineis at U = ?.
36
D. R. BRILLINGER
connectionbetweenthesetwo stretchesof data. The middlegraphgivesthe
function
estimatedsummation
a'. The lowergraphis onemeansofassessingthe
(9) ofSection3, and givenby
fitofthemodel.It is analogousto expression
(14) #{Yt=1withU-h<Ut<
U+h}l#{twithU-h<
forsmall h, plottedversusU. Here
U<
Ut
+h},
Bt- 1
Ut
E
s=O
as t-s
The smoothcurveis the corresponding
is the fittedlinearpredictor.
D(U - 0).
werecarriedout by a
The fitmaybe describedas adequate.The computations
data of
variantof the programdevelopedforhandlingthe seismicfirst-motion
and
Section 3. Furtherexamplesand discussionmay be foundin Brillinger
Segundo(1979). Othertypesof inputare employedand alternateestimating
procedures
comparedthere.
The large-sample
ofsuchestimatesmaybe studiedas in Sagalovproperties
approach,ofthissection,is
sky(1982).A greatadvantageofthemodel-building
A
and estimated
havebiologicalinterpretations.
introduced
thattheparameters
furtheradvantageof the maximumlikelihoodapproach,overthat of partial
is thatthespiketrainsinvolvedcan be highlynonstationary.
coherency,
14. Analysis of evoked responses. A traditionalmeansof studyingthe
the
sensory
stimulito a subjectand examining
nervoussysteminvolvesapplying
foran evokedresponse.The stimulusmay be
ongoingelectroencephalogram
somatosensory
olfactory,
pattern),
visual(e.g.,lightflash,checkerboard
auditory,
thestimulusis applied
or a task.Generally,
(e.g.,an electricalshock),gustatory
fora timeintervalthatis briefin comparison
to the durationof the response.
play an essentialrole in quantitativebiology.
experiments
Evoked-response
is able to choosewhichstimulito applyand whento
Because the experimenter
noted,to formalinferences
can pass beyondassociations
applythem,conclusions
thesameas the
are formally
Theseexperiments
causal mechanisms.
concerning
in Section6.
described
reflection
experiments
seismological
One is
Some dramaticsuccessstoriesof the techniquemay be mentioned.
Fra,Gandiglioand Mutani(1967).Siamese
Bergamasco,
presentedin Bergamini,
twinswerejoined in such a way that it was not possibleto determineby
traditionalmeans if the peripheralnervouspathwayswere interconnected.
of the twins.
it was crucialto determine
theinterconnections
Beforeoperating,
OngoingEEGs wererecordedforeach. A seriesof trialswerecarriedout in
in turnby electricalshocks.What
whicheach ofthe twins'legswas stimulated
was
a
of
one
twin
stimulated,
responsewas noted
was foundwas thatwhen leg
the twins were
the
of
this
basis
information,
only in her EEG. On
use
of the evoked
of
the
separated-successfully.A secondnotableexample
infants
for
newbom
exams
is
(including
responsetechnique providedby hearing
Theseare examinedforresponsesafterloud
infantsasleep.)EEGs are recorded.
clicksare made near the infants'ears. Rapin and Graziani(1967) presentan
SEISMOLOGY AND NEUROPHYSIOLOGY
37
bothwearingand notwearinga
exampleforan infantwithhearingdifficulties,
measurableeffect.
hearingaid. The hearingaid is foundto have an objectively
data recordedat a 4 x 4
Figure20 presentsan exampleof evoked-response
in a rabbit.In thiscase thestimuluswas an odorand
arrayofsensorsimplanted
in orderto studytherabbit'solfactory
system.These
thesensorswereimplanted
A secondexampleis givenin Figure21. It
responseswererecordedconcurrently.
gives the 20 successiveresponsesevokedby applyinga currentpulse to the
froma sensorimplantedin the
tractofa rabbitand recording
lateralolfactory
in Figure20. In
cortex.
The
is
of
the
fairly
pronounced
signal
pre-piriform
depth
the
is
was
and
the
weak
of
stimulus
signal notapparent.
Figure21 the strength
in
the
laboratoryof W. J. Freeman,
Both of these data sets were collected
Somedetailsofhisexperiments
ofCalifornia,
maybe found
Berkeley.
University
in Freemanand Schneider(1982).
is the factthatit is generally
Crucialto manyevokedresponseexperiments
to apply a stimulusonce. Ratherit must be applied repeatedly
insufficient
averaged.(Thisis also truein the
(perhapsthousandsoftimes)and theresponses
case of reflection
seismologyas mentionedearlier.)In the twinsand infant
if
M equaled250 and 100,respectively.
Formally,
examplesdiscussedpreviously,
Y(t) denotes the measuredEEG and the stimulusis applied at times am,
m = 1,..., M, thenit is usual to takeas thebasicstatistic,the averageevoked
response
1M m
M m=1
thedata of
The left-hand
columnin Figure22 presentstheresultsofaveraging
averaginga signalis
Figure21 with M = 3, 5, 10, 20 and 38. Withincreasing
slowlyappearingfromthe noise.Some alternateevidenceforthe presenceof a
column.Theseare averagesof
signalis providedbytheresultsoftheright-hand
38 responses,wherethe stimulushas beenappliedat a successionofincreasing
strengths.
arisein thecourse
A varietyofquestions,
whichhavestatisticalformulations,
(2)
ofworkwithevokedresponses.
(1) Does an appliedstimuluselicita response?
stimulielicitthe same response?(3) Is the same response
Do two different
(5) If the
elicitedat twodifferent
sensorlocations?(4) Is theresponsestationary?
responsesaltered?
orderof stimuliapplicationis altered,are the corresponding
stimuliadditive?(7) How does the response
(6) Are the effectsof different
dependon exogenous
(8) Howdo theresponses
intensity?
dependon thestimulus
seekquickefficient
researchers
variables?To go withanswersto thesequestions,
that
Difficulties
ofvariability.
data collection,
preciseestimatesand indications
commonlyarise includesmall response,large noise,variabilityin response,
artifactspresentand superposed
effects.
Nextin thissection,twoformalset-ups
willbe presentedthatmaybe employed
to addressthesituation.
Suppose, to begin,that thereis a singlestimulusand that it is applied
If
experiment.
at times am. Let a(-) denotethe responsein a single-shock
the systemis time invariantand the effectsof the variousshocksadditive
38
D. R. BRILLINGER
RabbitOlfactory
System- Responses at 4 by4 Array
2.
0.0
2.
0.0
.2 00
.6 00
2 .
0.02 0.04 0.06 0.08
.2 00
0.02
o
.6 00
0.04
0.06
0.0
02 .
0.08
0.0
.2 00
.0
2 2.
0.02 0.04 0.06 0.08
0.0
.2 00
0.02
0.04
2.0
0.06
0.08
.2
00
0.0
.6
00
0.0
00
.4
00
.8
0
s
0.02 0.04 0.06 0.08
.2 00
0.02
0.04
FI.2
0.02
0
2wure
in
0ec0nd0
0.04
0.0
.6 00
0.06
.
0.08
0.06
00
00
4
00
Th2usso
.8
2n 2abt
.6 00
0.02 0.04 0.06 0.08
.2 00
0.0
0.02
00
00
.6 00
0.04
0.06
.4
00
0.08
0.0
0
I
0.08
,2 00
0,
inseconds.
0.0
.
.6 00
o0
0t;X
o .
2.
.2 00
0.0
0.02
0.04
lcrecparrpi
onigtesito
0.06
0.08
0.0
0.02
0.04
ciiy
eodda
fterbitb
2
0.06
ot
0.08
h 6snoso
noo.Teuiso
0.0
n
0.02
h
0.04
0.06
-n
0.08
ra
r
SEISMOLOGY AND NEUROPHYSIOLOGY
39
Individual
Responses - RabbitPre-piriform
Cortex
response 1
response 2
response 3
response 4
0
o
o3
0.0
0.04
0;
0.08
0.0
response 5
o.0
0.04
0.08
0.o
0.08
0.04
9
0.0
response 13
0.0
0.04
0.08
0.04
0.08
0.08
0.o
0.04
0.08
0.0
0.04
0.08
V
0.0
0.04
0.08
0.0
0.04
0.04
0.08
=
0.04
0.0
0.04
? E;
0.08
0.04
0.08
0.08
response 12
0.0
::
0.04
0.08
response 16
0.0
response 19
0.o
0.04
0.o
0
0.08
0.08
response 8
response 15
response 18
0.o
0.08
response 11
0
tA>
0.04
response 7
response 14
response 17
o.o
0.0
response 10
0
0.04
0.08
response 6
response 9
0.0
0.04
0.04
0.08
response 20
0.o
0.04
0.08
FIG. 21. Twentysuccessiveresponsesevokedin thepre-piriform
cortexby (electricaey) stimulating
a rabbit. The x-axis unitsare in seconds.
40
D. R. BRILLINGER
Average Evoked Responses - Several M's, Several Intensities
- averageof 3
100% threshold
1
89% threshold
- averageof38
I-
0.0
0.02
0.04
0.06
0.08
0.10
0.0
0.02
0.04
0.06
0.08
0.10
0.0
- averageof 10
100% threshold
0.0
0.02
0.04
0.06
0.08
0.02
0.04
0.06
0.08
0.10
0.0
0.02
0.04
0.06
0.08
0.08
0.10
0.02
0.04
0.06
0.08
0,10
0 02
0.04
0.06
0.08
0.10
- averageof38
133% threshold
0.10
0.0
- averageof 38
100% threshold
0.0
0.06
122% threshold
- averageof38
- averageof 20
100% threshold
0.0
0.04
- averageof38
111% threshold
- averageof 5
100% threshold
0.0
0.02
0.02
0.04
0.06
0.08
0.10
- averageof38
144% threshold
010
0.0
0.02
0.04
0.06
0.08
0.10
FIG. 22. The variousgraphs hereare meantto show theeffects
of changingthenumberof responses
averaged (left column) and thestrengthofstimulusapplied (rightcolumn) fordata such as thatof
Figure 21.
SEISMOLOGY AND NEUROPHYSIOLOGY
41
thena modelforconsideration
is
(superposable),
(15)
Y(t) = ,u+
2a(t- am)+ E(t),
m
noise.In thecase ofthe
withY(.) denotingtheongoingEEG and E(.) denoting
ratherthanbeingan
EEG thismodelseemsto have to be empirically
verified,
case it cameout ofa concepimplicationof basic biology.(In theseismological
For example,the assumptionof superposability
tual framework.)
may be examinedas followsfortheanimalstudied.To begin,carryout somesingle-shock
experiments,
i.e.,applytheshocksat timesfarenoughapartthattheirindividual effectsseem likelyto have died off.Let a&(s) denotethe averageof the
responsesevoked,withs lag sincestimulusapplication.Now carryout some
two-shockexperiments,
i.e., apply shockssay A timeunitsapart. Let b(s, A)
denote the averageof the responsesevoked.To examinethe assumptionof
superposability
compared(s) + a(s - A) withb(s, A). The resultsof carrying
out such a check,in an experimental
situation,are givenin Biedenbachand
Freeman(1965). They forn averagesof M = 150 responsesand do not note
departurefromsuperposability.
We nowturnto one formalanalysisofthemodel(15). If one writes
X(t) = E 8(t
m
rnM),
then(15) takestheform
Y(t) = ,u+ la(t - s)X(s) ds + e(t)q
i.e., it is seen to be the modelof cross-spectral
analysis.TakingFouriertransone has
forms,
dT(X) = A(X) dT(X) + d[(X),
forX > 0, withA(X) denoting
theFouriertransform
of a(-). Considera number
of frequenciesXk = 27rk/Tnear X. Then,assumingA(-) smooth,one has the
linearmodel
approximate
Yk
A(X)Xk + Ek,
with
T-1
Yk=
2,,kt\
E Y(t)exp - i T }
t=0
and similardefinitions
of Xk, Ek. Next,via a centrallimittheorem
forempirical
Fouriertransforms,
thenoisevariatesEk maybe approximated
by independent
All the inference
(complex)normalshavingmean 0 and variance2fTfTee(X).
forthelinearmodelbecomeavailable.Forexample,as an estimateof
procedures
the transfer
function
A(X), one has
A(X)=
D. R. BRILLINGER
42
as complexnormalwithmean
distributed
and thisvariatewillbe approximately
has a varietyofconveniA(X) and variance27TTfee(X)/E4lXkI2.Thisformulation
It directly
extendsto the cases ofmultiplestimuliand multiple
ent properties.
responses.It handlesstimuliof varyingintensity.It allows the individual
suchas
procedures,
responsesoftheseparateshocksto overlap.Formalinference
maybe designedand analyzed-comtests,are available.Complexexperiments
meastructure,
factorialtreatment
rotation,
plexitieshandledsuchas blocking,
are available.Finally,one can
suredcovariates.Formalchecksforinteraction
turnto the questionofoptimaldesign.
viewpointforthe problem.
to adopt a different
It is sometimesconvenient
Suppose that the shocks are applied at times such that ,m,, -,,,
a(s) = 0 fors > V and s < 0. Write
=
Y.(S)
Y(s +
> V with
am.)
Then Yin(S) = ,x+ a(s) + Em(S) for0 < s ? V. The averageevokedresponseis
now conveniently
denotedY(s). As an exampleof the use of thisformulation,
stimuliand thateachare appliedJ times,thenone
supposethereare I different
is led to set downthemodel
Y,j(s)
= Alij+
a(s) + b,(s) + eij(s),
suchas
Othermethodologies,
replicates.
withi indexingstimuliand j indexing
analysis,are seento becomeavailablewiththis
growncurvesand discriminant
formulation.
byartifacts.
data maybe contaminated
thatevoked-response
It wasmentioned
available.
are directly
estimates
It is perhapsworthnotingthatrobust/resistant
such
as
of
a
measure
one
has
distance,
Suppose
[Y(s) - a(s)]
IIY- al2 -
ds
estimatesis
and an estimateof scale p. Then a familyof robust/resistant
providedby
a(s) = ,WmYm(S) F2Wm,
m
m
with WM= W(MIY - ad/,p) and W(.) a univariateset of multipliersfor
An eleThe estimatewillneed to be computedrecursively.
robust/resistance.
mean"
mentaryexampleis providedby the"trimmed
a(s) - E'Ym(S)/13M,
was proposedin
- all. This classofestimates
withE' overthe fBMsmallestIYYm
in Folledo(1983).The uppergraphof
(1979,1981a)and investigated
Brillinger
(/3 = 0.5),in
Figure23 providesan exampleofthisestimatewith50%trimming
122%of
the case of data likethatofFigure21 (butwitha stimulusofstrength
the thresholdstimulus).The solidcurvedenotesthe averageevokedresponse,
the dashed one the trimmedstatistic.The two curvesare nearlyidentical,
arefoundto differ
theindividual
noticeably.
responses
althoughwhenexamined,
a highly
becausethisis usuallyconsidered
was employed,
trimming
Fifty-percent
SEISMOLOGY AND NEUROPHYSIOLOGY
43
Average Evoked Response and Robust/Resistant
Variant
If)
0
cmJ
0.0
0.02
0.04
0.06
0.08
0.10
time(seconds)
122% threshold- 50% trimming
RunningTrimmedMean and Robust/Resistant
Variant
0
LO
0.0
0.02
0.04
0.06
0.08
0.10
time(seconds)
122% threshold- 50% trimming
FIG.23. The uppergraphcomparestheaverageevokedresponsewiththe50% trined meanfor
thedata takenat 122%ofa threshold
stimulation
value.Thelowergraphcontrasts
the50%trimned
meanstatisticwitha valuecomputed
recursively.
resistantlevel in the case of elementary
statistics.The factthat the trimming
had suchlittleeffect
on thefinalanswersuggeststhattherewereno substantial
outlyingcurvesin thedata set.Had a curvebeenfarremoved
fromtherest,then
it wouldhave beenrejectedfromthe average.It is to be remarked
thatin this
case ofpresentconcem,wholecurvesare beingeliminated
fromtheaverage,not
just outlyingpointsthatsomecurvesmighthave.
It is to be notedthata "real-time"
versionofsucha trimmed
meanmaybe
computed;see Brillinger
(1981a). This statisticis givenrecursively
for m =
44
D. R. BRILLINGER
1,2,... by
Pm
m
=
a
L
1
-
-
Pm' and
if IIYm? -amII <
Pm+? = Pm+ L/m
and by
otherwise,
am?i(S)
if IIYm+?
-
Pm,
+
am(S)
(Ym?i(S)
Ain(S)),
and
A
?i(S)
=
am(S)
otherwise.
a workedexample,it was foundmoreconvenient
in the
(In preparing
The lowergraphofFigure23 givesthe
choiceofL to replacep byitslogarithm.)
resultforthe same data as that of the uppergraph.The algorithm
was run
settingad(s) = Yl(s) and L = 0.15.The real-time
estimate,givenby the solid
as wellas thedead-time
estimatein thiscase. One
line,has performed
virtually
can remarkagain that had therebeen somehighlydissimilarcurvespresent,
fromthesampleaverage.Followingthe
thenthisestimatewouldhave differed
advice sometimesgivenin connectionwith resistantregression
estimates,it
wouldseemsensibleto computeboththeordinary
and theresistant
If the
forms.
If thetwodiffer
twoare similar,thenthereis no difficulty.
thenthe
noticeably,
situationshouldbe examinedin somedetail.
Brillinger
(1979)proposedthe preceding
techniquesand variousothers.Brillinger(1981a,b) werebased on that lectureand coversome otherstatistical
methods.Tukey (1978) also addresses
problemsarisingfromevoked-response
statisticalissuesand proposessomeprocedures.
15. A confirmed(Fourier) inference. Muscle cells are electrochemical
is appliedat theneuromuscular
devices.If the chemicalacetylcholine
junction,
releasecauses
result.Specifically,
measurablevoltagefluctuations
acetylcholine
Katz
membrane
channelsto openleadingto voltagefluctuations.
postsynaptic
associatedwith this
and Miledi (1971, 1972) measuredvoltagefluctuations
and foundthatthepowerspectrum
couldbe approximated
phenomenon
by the
to suchdata
forma/(/32+ A2).[Anexampleofthefitofthisfunction
functional
of a fitting
and a description
procedure
maybe foundin Bevan,Kullbergand
Rice (1979).]Theyproposedthemodel
Y(t) = >a(t -am),
m
withthe o(Jpointsof a Poissonprocessand witha(t) = exp{-,t}. This a(-)
to theeffectiveness
of an openchanneldecayingexponenfunction
corresponds
tiallyand leads to a powerspectrumof the indicatedform.Katz and Miledi
but
ofrandomduration,
thatthepulsesmightactuallybe rectangular
mentioned
form.Stevens(1972)proposedthe
to deal withthe exponential
theypreferred
SEISMOLOGY AND NEUROPHYSIOLOGY
45
specificmodel
Y(t) = Yam(t
am),
m
also with (am) Poisson, but now with am(t)
=
1 for0
<
t < Tmand am(t) = 0
The Tmare independent
otherwise.
ofmean1/fBand correspond
exponentials
to
the lengthsof timethatthe channelsare open.Stevensnotedthat thismodel
also led to a powerspectrumof the forma/(,2' + X2). The models were
withthedata collected.
indistinguishable
The problemwas laterresolvedby improvedexperimental
technique.Neher
and Sakman(1976)developeda techniquethatallowedtheopeningand closings
of individualchannelsto be seen. They foundthat the channelsremained
and openfortimeperiodsofvarying
equallyeffective
The twoproposed
lengths.
modelscouldbe distinguished.
Examples of single-channel
data and the corresponding
estimatedpower
spectrummaybe foundin Lecar(1981).Jacksonand Lecar(1979)presentresults
the exponential
confirming
durationoftheopenings.
16. Other topics. Spatial-temporal
data are commonly
collectedby neuroscientists.One formis theelectroencephalogram
recordedby an arrayofsensors
on the scalp. Figure19 presented
an exampleofdata collectedforthe olfactory
systemoftherabbit.The stimulus
wasreleaseoftheodorethylacetate.
An8 x 8
arrayof electrodeswas imbeddedin theanimal.The data, alreadypresented
in
Figure19,givetheresponsesforthesensorsat thepositionswithx-coordinates
2, 4, 6 and 8 and y-coordinates
1, 3, 5 and 7 of Figure23. One procedure
that
Freemanhas foundhelpfulforunderstanding
thistypeofdata is thecomputing
of empiricalorthogonalfunctions;see Freeman(1980). Figure 24 gives an
example.These resultsare derivedby stackingthe responsesinto a matrixX
withrowscorresponding
to sensorand columnsto time,and thencomputing
the
singularvalue decomposition
X = UDVT,ofthatmatrix.The U fora particular
component,
are thenplottedversussensorlocationas in theupper
say thefirst,
graphofFigure23. The V valuesaresimilarly
plottedversustimeand appearin
thelowergraph.The resultsofFigure23 arebasedon 64 series,notjust the16 of
Figure19. The contourplot suggeststhe presenceof a focusof activity.The
time-series
elicitedmaybe seenlurking
component
in theindividualresponses
of
Figure 19. (It may be mentionedthat meteorologists
have long computed
empiricalorthogonalfunctionsforspatial-temporaldata and used themin
see Lorentz(1956),forexample.A numberof otherreferences
forecasting;
are
givenin Jolliffe
(1986).]
Childershas also made use of arraydata in studyingthe neuralsystem.In
Childers(1977),he estimates
thefrequency-wavenumber
spectrum
forresponses
evokedby visual stimuli(lightflashes)in the humanEEG. He was concerned
withestimating
thespeedand direction
ofpropagating
waves.In thepapercited
he firstnotes an apparenthigh-velocity
wave. Afterthis wave has been "removed,"he notesthepresenceofa pairofwavesmovingin oppositedirections.
His researchis directedat developing
a diagnosticprocedure
forvariousvisual
disordersand obtaining
insightconcerning
howthevisualsystemfunctions.
D. R. BRILLINGER
46
LatentSpatial Component
N-
(0~~~~~~~~~~
LCI)
cl-)
co
1
0.0
0
2
0.02
3
5
4
0.04
6
7
8
0.06
0.08
time(seconds)
FIG. 24. The results of a singular value decomtposition
of the full set of the data fromwhich the
burstsofFigure20 weretaken.The valuesgraphedare for
thefirst
components.
upper graph give spatial location.
The axes in the
The decayingcosinemodelof Section2 has also founda use in nseurophysiology.In his workwiththeolfactory
system,Freeman(1972,1975,1979)found
that the averageevokedresponsecould be weR fittedby the sunnof a few
He developeda modelinvolving
decayingcosinetertns.
spike-to-wave
conversion,
collections
of
constantcoefficient
second-order
differential
involving
equations,
feedforward
and feedbackand involving
conversion.
He
involving
wave-to-spike
in thetimedomainto estimatetheunknowns.
In
employednonlinear
regression
one case, involving two cosines, he was led to view the strongerwave as
representingintracorticalnegative feedback and the weaker as representinga
SEISMOLOGY AND NEUROPHYSIOLOGY
47
secondfeedbackloop. Of interestin thistypeof workis whathappensto the
conditions
are altered.A
and thedecayrateswhentheexperimental
frequencies
to decayingcosinesis Childersand Pao (1972).They consider
secondreference
the model
t > 0,
Y(t) = EakteXp{
3kt}coS(Ykt + Sk) + e(t),
k
forvisual evokedresponsesmonitored
overthe occipitalregion.In particular,
theystudythe data by complexdemodulation.
Briefreference
will be made to severalothertopics.Dumermuth,
Huber,
Kleinerand Gasser(1971)estimatethe bispectrum
of humanEEGs. de Weerd
and Kap (1981) discuss the computationof some time-varying
quantities.
Marmarelisand Naka (1974)considerthecase ofbiologicalsystemswithseveral
inputs.An extremecase of thisoccurswhenthe inputis varyingin bothtime
and space.This circumstance
is considered
in Yasui,Davis and Naka (1979).The
bookby Marmarelis
and Marmarelis
(1978)goesintogreatdetailconceming
the
identification
of systemsthat are polynomial
and timeinvariantin the input.
Theyemphasizetheadvantagesresulting
fromemploying
a Gaussianwhite-noise
input.The dedicationofthebookis worthmentioning-"To an ambitiousnew
breed:SYSTEMS PHYSIOLOGISTS".
Anotherarea of researchactivityhas been that of control.The worksby
Poggioand Reichardt(1981)and Wehrhahn,
Poggioand Bulthoff
(1982)maybe
noted.They are concemedwithdata thatare three-dimensional
trajectories.
17. Discussion. As theexamplespresented
indicate,a broadrangeofdata
data are collectedat both the
typesarise in the neurosciences.
Furthermore,
microand macrolevel.The procedures
to
developedoftenhavetheopportunity
moveon to directclinicaluse.
It is particularly
to notetheevolution
oftheanalysisin thecase of
interesting
the neuronalsignalinganalysisas presentedin Sections11 and 13. One can
recognizethe stagesof (1) (feature)description;
(2) correlation/association;
(3)
(ad hoc) regression;(4) conceptualmodel. These stages are usual in many
situations.
elementary
The fieldof neurophysiology
has the satisfying
aspect that in manycases
controlledlaboratoryexperiments
are possibleand repeatable.Furthermore,
thereare opportunities
forthedesignofexperiments.
In thefield,statisticshas
beenseento providetechniques
formodelformation
and validation,
formeasurin
conclusions
and
for
of
inguncertainty
addressing
questions causality.Statistical techniqueshave led to insightconcerning
In this
theunderlying
physiology.
connectionit seemsimportant
to notethe following
collaborator
provisoofmy
J. P. Segundo, however,"....
The maximof all of the above is that the powerof
available mathematics(and of the instrumentation
that implementsthem)
should be used exhaustively,
gded by an unflagging
biologicalrealism,misand keepingin mindthattheultimategoalis understandtrustful
and stubborn,
ingin strictly
biologicalterms."[See Segundo(1984),page 294.]
It seems likelythat in the neurosciences,
more oftenthan not, notable
advanceswillcomefromthecarrying
outofnovelexperiments,
ratherthanfrom
48
D. R. BRILLINGER
thingsat new
novelanalyticmethods.
Experiments
willbe carriedoutmeasuring
ordersof smallness.Morecomplexstimuliwillbe invoked.Nonlinearsystems
will be the norn. Neuralnetworkswill be a major concern.Luckily,forus
havebecomecommonin thelaboratory
and this
statisticians,
digitalcomputers
seemsto be bringing
a movetowardquantizationof otheraspectsof the work
ofthedata.
beyondthe simplerecording
data, briefly
referred
to in
18. Update. The analysisof singleion-channel
Section 15, has becomea wholeindustry.
Modelswithseveralstatesare now
routinelyfitted.References
includeColoquhounand Hawkes(1983),Labarca,
Rice,Fredkinand Montal(1986)and Milne,Edesonand Madsen(1986).Extending the workof Section13,Brillinger
(1986)presentsa numberof examplesof
corresponding
spike
themaximumlikelihood
fitting
ofa neuralmodelemploying
traininputand outputdata. Smithand Chen(1986)studya morecomplicated
as beingof substantialimporneuralmodel.The chirpsignalwas propounded
in physiologitancein seismicexploration.
Someuse ofit has beenmaderecently
cal studies.In Norciaand Tyler(1985),a 10-sspatialfrequency
sweepstimulus
is
visualevokedpotentialmeasured.Th. Gasser
employedand the corresponding
and collaborators
have nowcarriedout a substantialnumberof statisticaland
We mentionin particularthe papers
substantiveanalysesof evokedresponses.
by Mocks,Tuan and Gasser(1984),Gasser,Mocks,Kohlerand de Weerd(1986)
and Gasser,Mocksand Kohler(1986).Finally,we notethat Grajski,Breiman,
di Priscoand Freeman(1986) applymodernclassification
to study
procedures
bulb EEGs of rabbits
the effectsof applyingdifferent
odorson the olfactory
and that Gevins,Morgan,Bressler,Cutillo,White,Illes, Greer,Doyle and
accuracyto brainelectricalpatterns
Zeitlin(1987) relatehumanperformance
just beforea task.
IV. Concluding remarks. In thisarticlewe have presenteda numberof
showingthe use of the
examples,drawnmainlyfromour personalexperience,
same statisticaltechniquein the ratherseparatesciencesof seismologyand
have the
to ask what,if anything,
It nowseemsappropriate
neurophysiology.
fromeach other
threesciences-statistics,
neurophysiology-gained
seismology,
even thoughtheyare indirect?Havingin minda
as a resultof connections
broaderclassofexamplesthanthosediscussedin thispaper,onecan saythat:(i)
statisticsis richerforhavingbeen led to developand studyvariousnovel
or neurophysiology;
methodsto handlespecificproblemsarisingin seismology
are the richerforthe other'sfield
(ii) both seismologyand neurophysiology
thatthe
to abstractsufficiently
havinggenerateda problemforthestatistician
or
result'sapplicabilityto theirfieldbecameapparent;(iii) eitherseismology
becausevariousof their
benefitfroma statisticalfornulation
neurophysiology
to need to be statedin termsof probabilities
(e.g.,
problemsseem necessarily
and becausethese
nor earthquakesseem deterministic)
neitherneuronfirings
to validateresultsand to fitconceptualmodels.That the
fieldsneedprocedures
in submethodsof statisticscan lead to important
insightand understanding
stantiveproblemsseemsagreed.
SEISMOLOGY AND NEUROPHYSIOLOGY
49
It may be remarkedthatthe applicability
of statisticalprocedures
to these
two substantivefieldshas further
oftheirmoveto
grownin directconsequence
greaterquantification
and digitaldata collection.
The data setsanalyzedwereof
highquality.The factthattheanalyseswereinformative
to an extentherebodes
in fieldswithdata oflesserquality.I needto
wellfortheuse ofsuchtechniques
remarkhow crucial,in working
withthedata setsdiscussed,I have foundit to
be to plot the data in its originalform.Something
specialseemedto be learned
in each case fromdoingso. Thisis whyforthevariousanalyses,I havesoughtto
providedata plotsas partsofthepresentation.
The readerwillhavenotedthatsomeoftheanalysesweretime-side
and some
werefrequency-side.
Each domainhas its advantages.It seemsworthpointing
out specifically
thatstationarity
was notrequiredforsomeofthefrequency-side
procedures.It wouldseemthat mosttimeseriesand pointprocesssituations
would benefitfromcarryingout simultaneoustime-sideand frequency-side
analyses.
On reviewit maybe seenthatthe techniquesemployedfortime-series
data
and forpointprocessdata in manycasesare notthatdifferent.
Brillinger
(1978)
presentssome comparative
discussionof the techniquesforthe two cases. Our
presentation
is somewhat
remissin theseismological
case in notpresenting
some
workedexamplesof auto-and cross-intensity
estimation.
Examplescouldhave
been provided.
It shouldbe apparentthatmajordata management
and computational
efforts
wererequiredin thederivation
ofall resultspresented.
I havebeenimpressed
by
the way that the neuroscientists
couldturnto theirlab book keptduringthe
and pulloutcrucialdetails,sometimes
experiments
manyyearsaftertheexperimentshad been completed.My analysesalso extendovermanyyearsnow. I
have foundit veryusefulrecently
to maintaina "Readme" filein the various
computerdirectories
forthedata sets,whereinI listwhatthevariousprograms
do, futurewishesconcerning
the programs
and all of the thingsthat I thinkI
willneverforget.
Turningto thoughtsconcerning
to come,it seemsthat the
developments
futurewillsee manyofthetraditional
statistical
techniques
extendedto applyto
datumofmorecomplicated
forms-specifically,
to curves,movingsurfaces,
point
cloudsand thelike.It seemsthattechniques
developedin onefieldwillcontinue
to be transferred
(by statisticians?)
to otherfields.For example,I expectto see
theresultsdevelopedbyneuroscientists
forarrayson a curvedsurface(theskull)
to be takenup by the seismologists
as theyneed to take specificnote of the
Earth's curvature.If I have foundanythinglackingin our currenttoolkitof
statisticalmethodsand devices,it is a collection
oftechniques
thatsuggestwhat
to do nextwhena modelfailsa validationcheck.Perhapsthefuturewillsee such
techniquesdevelopedin an organizedfashion.
Acknowledgments. One wondersif it is possibleto workmeaningfully
on
problemsfromsubstantive
areasunlessoneis in closecontactwithresearchers
of
those areas. I doubtit. I can say that theselectureswouldneverhave come
about,but forthehelpand encouragement
thatmycollaborators
B. A. Bolt and
50
D. R. BRILLINGER
J. P. Segundohave providedthroughthe years.I cannotthankthemenough.
The influence
ofmydoctoralsupervisor,
J.W. Tukey,shouldbe readilyapparent
I remember
hissaying,whenI was a student,
as well.In particular,
"One cannot
unlessone becomesa chemist."This has certainly
consultwitha chemist,
been
withrespectto thepartsofneurophysiology
and seismology
myownexperience
on whichI have worked.The reversehas also provedto be the case, namely,
fromthosefieldshave had to learnpartsofstatisticsas wellas I
collaborators
knewthem.I also mustthankmystudentswhohaveworkedwithmeon various
I mentionM. Folledoand R. Ihaka,
of the problemsdescribed.In particular,
whosetheseshave beendetailed.
Finally,I would like to thankJ. Rice and R. Zuckerforprovidingsome
I wouldlike to thankN. Abrahamsonand R.
neurophysiological
references.
Daraghforproviding
theSmart1 data sets.I wouldliketo thankW. J.Freeman
the rabbitolfactory
systemdata. I wouldliketo
and K. Grajskiforproviding
thankFinbarrO'Sullivan,theAssociateEditor,and thereferees
fortheirhelpful
of
initial
draft
forimprovement the
suggestions
paper.
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DEPARTMENTOF STATISTICS
UNIVERSITYOF CALIFORNIA
BERKELEY,CALIFORNIA94720