Download Statistics of Bicoherence and Biphase

JOURNAL
OF GEOPHYSICAL
RESEARCH,
VOL. 94, NO. C8, PAGES 10,993-10,998, AUGUST
15, 1989
Statistics of Bicoherence and Biphase
STEVE ELGAR AND GLORIA
$EBERT
Electrical and Computer Engineering Department, Washington State University, Pullman
Statistics of estimates of bicoherence and biphase were obtained from numerical simulations of
nonlinear random (harmonic) processeswith known true bicoherenceand biphase. Expressionsfor the
bias, variance, and probability distributionsof estimatesof bicoherenceand biphaseas functions of the
true bicoherenceand number of degreesof freedom (dof) used in the estimates are presented. The
probability distributions are consistentwith theoretical distributions derived for the limit of infinite dof
and are used to construct confidence limits on estimates of bicoherence and biphase. Maximum
likelihood estimates of true values of bicoherence and biphase given observed values are also
presented.
1.
INTRODUCTION
be used to statistically detect the presence or absence of
Bispectral analysis has been used to study nonlinear
interactions in a variety of ocean processes, including surface gravity waves in intermediate water depths [Hasselman
et al., 1963], perturbations from the mean profiles of temperature, salinity and sound velocity [Roden and Bendiner,
1973], internal waves [Neshyba and Sobey, 1975; McComas
and Briscoe, 1980], shoaling surface gravity waves [Elgar
and Guza, 1985, 1986; Doering and Bowen, 1987], and
temperature fluctuations [MMler, 1987]. A wide range of
phenomenain other physical systemshas also been investigated with the bispectrum (see Nikias and Raghuveer [1987]
for a recent review). In most of these studiesthe bispectrum
was used to determine whether or not the process under
investigationwas consistentwith linear dynamics. Specifically, nonlinear interactions are associated with nonzero
values of the bicoherence, defined here as [Haubrich, 1965;
Kim and Powers, 1979]
nonlinear interactions, the statistics of estimates of bicoher-
ence and biphase,/3(wj, w2), the phaseof B(w•, w2), for the
case of nonzero true bicoherence have not previously been
reported. The purpose of this study is to present such
statistics.Brillinger [1965], Rosenblatt and Van Ness [1965],
Brillinger and Rosenblatt [1967a, b], and others [see Nikias
and Raghuveer, 1987] give some of the statistical properties
of higher-orderspectra, including asymptotic distributions of
the real and imaginary parts of the bispectrum. Haubrich
[1965], Hinich and Clay [1968], Kim and Powers [1979],
Hinich [1982], and Ashley et al. [1986] discussthe estimation
of bicoherence.
For the present study, harmonic random processes with
true valuesof bicoherence
(b2) between0.1 and 1.0 were
numerically simulated (section 2), and the statistics of esti-
matesof bicoherence
(/•2)andbiphase
(•) obtained
fromthe
simulated time series were calculated (section 3). The prob-
abilitydistributions
of/•2 and/3,including
thebiasof/•2 and
the variancesand confidence
limits of /•2 and /3, were
IB(wl, O92)12
asfunctionsof b2 anddof (sections3.1 and3.2).
b2(wl,092)=
(1) determined
E[IA(rol)A(ro2)12]E[IA(rOl
+ ro2)l
2]
These values are useful for experimental design. On the
other hand, once an experiment has been conducted, maximum likelihood techniques may be used to estimate true
where B is the bispectrum,
B(wi, w2)= E[A(o•i)A(o•2)A*(wj + w2)]
(2)
A(w) is the complex Fourier coefficient of the time series at
radian frequency w and E[ ] is the expected, or average,
value. Alternately, the bispectrumcan be calculatedas the
Fourier
transform
of the third-order
correlation
function
of
valuesof b2 and/3basedontheobserved
values.Maximum
likelihood estimates of bicoherence are presented in section
3.3. These results are then briefly applied to estimates of the
bicoherence and biphase of narrow-band surface gravity
waves observed in 9 m water depth (section 3.4). Conclusions follow
the time series [Hasselman et al., 1963]. For a finite length
time series, even a truly linear (e.g., Gaussian)processwill
have nonzero bicoherence. Haubrich [1965] shows that for a
Gaussianprocessthe bicoherence(whose true value equals
in section 4.
2.
SIMULATION
PROCEDURE
Time series consisting of triads of sinusoids(with frequen-
zero)isapproximately
chi-square
(X2)distributed
in thelimit cies Wi, tOj,and Wi+j)were numericallysimulatedon the
of large degrees of freedom (dof), and thus significance CRAY XMP at the San Diego Supercomputer Center. By
adjusting the amplitudes of the component sinsuoids, some
levels for zero bicoherence as a function of dof can be
calculated. Elgar and Guza [1988] demonstrated that esti- of whose phases were chosen from a uniform random
mates of bicoherence from a Gaussian process are also distribution (0 - 2rr), the true bicoherence value of the triad
approximately
X2 distributed
forlowvaluesof dof(dof= 32) could be varied. For example, consider the case of a single
and are not sensitive to smoothing procedures used to
increase
dof.
Although the significancelevels for zero bicoherence can
triad:
x(t) = COS(wit + qbl)+ cos (w2t + 4>2)+ a cos (w3t + qb3)
+ COS(ro3t+ qbI + qb2)
Copyright 1989by the American GeophysicalUnion.
(3)
Paper number 89JC01034.
where w3 = wl + w2, &•, &2, and &3 are randomphases,and
0148-0227/89/89JC-01034505.00
a is an adjustable parameter. As a increases from 0 to o•,
10,993
10,994
ELGAR AND SEBERT: STATISTICSOF BICOHERENCEAND BIPHASE
b2(wl,w2)decreases
from1.0to 0.0.Fortheexample
given
0.05
in (3), if a = 0, then/3(w•, w2) = 0. Time seriesconsistingof
the following triads were simulated: a singletriad, (Wl, w2,
w3); two triads with the same sum frequency, (w•, ws, w6)
and (w3, w3, w6); and two triads with a commonfrequency
other than the sum frequency, (w•, w3, w4) and (w3, ws, ws).
True values of/3 = 0 and/3 = rr/4 were used. For each case
0.04
0.03
the true bicoherence of each triad could be adjusted. The
statistics of estimates of bicoherence and biphase reported
0.02
herewerefoundto depend
onlyontruevaluesof b2anddof;
they did not depend on the number of triads, nor on how the
triads were interrelated, nor on /3. Additional time series
were
simulated
with
Gaussian-distributed
Fourier
0.01
coeffi-
cients rather than uniformly distributed random phases
[Elgar et al., 1985, and references therein]. The resulting
0.00
0.00
statistics
of/•2and/3wereessentially
identical
tothoseofthe
random phase simulations.
Each simulation consisted of generating many realizations
of 65,536-point time series, each of which was subdivided
into short sectionsof 256 points. The short sectionswere not
tapered. By ensemble averaging the bispectrum over all 256
of the short records, estimates of bicoherence and biphase
with 512 dof were produced from each 65,536-point time
series. Each 65,536-point record was also subdivided into 16
groups of 16 short records, each group producing estimates
with
32 dof.
Estimates
with
dof between
0.01
0.02
0.05
THEORETICAL
0.04
0.05
BIAS
Fig. 1. Bias of bicoherence observed in simulated data versus
the theoretical bias (equation (4)). Squares, dof = 32' triangles, dof
= 64; diamonds, dof = 128' circles, dof = 256; and asterisks, dof =
512.The bicoherence,
b2 = 0.1, 0.2, ..., 0.9 for eachdof.
(!•)(b2)(v/2)-le-b2/
(6)
32 and 512 were
where F( ) is the gamma function. The mean and variance
randomvariableare av and 2a2v,
point time series were generated, and thus 1000 (dof = 512) of an aX2 distributed
to 16,000 (dof = 32) bicoherenceand biphaseestimateswere respectively.Thus the parametersa and v can be determined
produced for each true value of bicoherence. Results from from the bicoherence as
simulations using 100 realizations were indistinguishable
E[b2]
2E2[b2]
from those where averages of 1000 realizations were used.
. =
v=
(7)
obtained in a similar manner. One thousand of the 65,536-
v
3.
3.1.
SIMULATION
RESULTS
Var [b2]
Combining (5) and (7), and using the biased value of bicoherence,
Bicoherence
(dof)b2
As shownbelow,/•2 is approximately
chi-square
distributed and thus will have a positive bias, similar to the
ordinary coherence between two time series [Jenkins and
Watts, 1968; Beningus, 1969]. Following Beningus [1969],
the bias was empirically found to be approximately given by
Bias[/•2]= (2/dof)(1- b2)2
•'=2(1- b2)3
As b2 --• 1 and/ordofincreases,
theparameter
•,increases,
andfx•(b2/a)
approaches
aGaussian
distribution.
Probability
(4)
Equation (4) is similar to the correspondingexpressionfor
coherence [Beningus, 1969] and is comparedto the observed
bias in Figure 1. Guided by expressionsbased on somewhat
heuristic theoretical arguments [Hinich and Clay, 1968; Kim
10-2
andPowers,1979],thevariance
of/•2 wasempirically
found
Z
to be approximately
Var [/32]= (4b2/dof)(1- b2)3
(8)
•
10 -3
•
10-4
(5)
Equation (5) is similar to the analogous approximate theoretical equation for coherence [Jenkins and Watts, 1968],
and the agreement between (5) and the simulated data is
excellent, as shown in Figure 2.
Haubrich[1965]suggests
thatfor a process
withb2 = 0.0,
/32will asymptotically
(largedof) approach
a chi-square
distributionwith parameter v = 2. Hinich [1982] showsthat
bicoherenceis approximately asymptoticallydistributedas a
noncentral chi-square random variable. The noncentral chi-
10-5 I
IIIII
I
10 -5
I
I
I IIIII
I
10 -•'
THEORETICAL
Fig. 2.
I
I
I I IIII
I
10 -3
I
I I IIII
10 -2
VARIANCE
Variance of bicoherence observed in simulated data
Each symbol represents different dof, as described in the caption to Figure 1.
square
distribution
canbeapproximated
by anaX2 distribu- versus the theoretical variance (equation (5)).
tion [Abramowitz and Stegun, 1972], given by
I
ELGAR AND SEBERT: STATISTICS OF BICOHERENCEAND BIPHASE
3.5
0.030
10,995
-
,•
p,i
0.025
:
.,,
,
2.5
o.oo i
,.
t-•
0.015
ß•
0.010
1.5
o
z
1.0
0.005
0.5
0.000
I
0.0
0.2
0.4
0.6
I
I
I
I
i
I
I
I
I
0.8
b2
Fig. 3. Probability
distribution
of/;2 observed
in thesimulated
I
I
I
I
I
lO 2
DEGREES
Fig. 4.
OF
FREEDOM
The 90% confidence limits for bicoherence. The values
the correspondingtrue value of
shown have been normalized by
datafordof= 32(dashed
curve•)
andtheoretical
aX2 distribution
(solid curve). From left to right, b • = 0.1,0.2, and 0.4. The bin width
is A(b2) = 0.0067.
bicoherence.
Thus90%of thetime,/;2willbein therangegivenby
theappropriate
ordinate
valuestimesb2. Solidcurvesaretheoret-
ical values, and symbols are values observed in the simulated data.
Squares,
b2 = 0.1; triangles,
b2 = 0.2; diamonds,
b2 = 0.4; and
distributions
of the simulateddata are comparedto aX2
circles,b2 = 0.8.
distributionsin Figure 3, with a and v obtained from (7) and
(8),respectively.
According
to a X2 goodness
of fit test,the hand,it is oftenthe casethat b2 mustbe estimatedfrom a
theoretical and observed distributionswith 32 dof and b 2 --< limited set of data. In this section maximum likelihood
0.4 do not differ significantly at the 95% confidence level. estimates [Jenkins and Watts, 1968] of the true value of
givenan observed
value,/;2,withfinitedofare
For 32 dof the probability distributionsof the simulateddata bicoherence
are slightlyoffsettowardlowervaluesof b2 relativeto the presented. Freilich and Pawka [1987] present a similar
aX2 distributions,
withtheamount
of offsetdecreasing
with analysis for estimates of the cross spectrum, with results
increasing
b2 (Figure3). For highervaluesof doftheoffset analogousto those reported here. Maximum likelihood estiis virtually zero, and, consequently,no attempt was made to mation consistsof determiningparameters of the underlying
account for it in the probability distributions with 32 dof probability distribution function (pdf) from samplesof the
shown in Figure 3. The scatter in the probability distribu- distribution. The pdf is recast as a function of the parameters
tions of the simulateddata shown in Figure 3 is due to finite to be determined, and those values of the parameters that
record lengths and finite bin width. There is even better maximize this likelihood function are considered the maxiagreement
between
simulated
dataandaX2 distributions
as mum likely estimates (mle) of the true parameters of the
b2 --) 1 and/or dof--)
underlyingdistributiongiven the particular samplesat hand.
thelikelihood
function,L(b2),is
Assuming/;2
is aX2 distributed
witha andv givenby (7) Forthecaseof bicoherence
(equation
(6))andis shownas
and(8), respectively,
confidence
limitsfor/•2 canbe con- givenby theaX2 distribution
structed as functions of b 2 and dof. The 90% confidence
limits are comparedto the correspondingvalues observedin
the simulationsin Figure 4.
3.2.
Biphase
10-1
-
10-:•
_
Similar to the phase of the cross spectrum [Jenkins and
Watts, 1968], the biphase is approximately Gaussiandistributed, unbiased, and has variance (in units of radians)
The theoretical variance and confidence limits (calculated
from a Gaussiandistribution with variance given by (9)) for
10-3 -
/3 are compared
to simulated
valuesin Figures5 and6,
respectively.
i
i
i
i i i iii
i
i
i
I i i iii
10 -3
THEORETICAL
3.3.
Maximum
Likelihood
i
i
i
10 -2
I i i iii
I
i
i
i
10 -I
VARIANCE
Estimation
Fig. 5.
Variance of biphase observed in simulated data versus
The results of the previous sectionsdescribethe statistics the theoretical variance (equation (9)). The units are radians
of bicoherence
giventhe true value,b2, thusenablingthe
designof experiments to measurebicoherence. On the other
squared. Each symbol representsdifferent dof, as described in the
caption to Figure 1.
10,996
ELGAR AND SEBERT: STATISTICS OF BICOHERENCE AND BIPHASE
6O
TABLE
5O
-r
20
•2
dof
0.1
0.1
0.1
0.1
0.2
0.2
0.2
0.3
0.3
0.4
32
64
128
256
32
64
128
32
64
32
1. Parameters for the Likelihood
b2mle
0.129
0.116
0.108
0.104
0.217
0.209
0.204
0.309
0.304
0.403
2
bmele
0.146
0.125
0.113
0.107
0.223
0.212
0.206
0.310
0.305
0.404
Functions
s.d.
(7.5:1)
0.047
0.041
0.032
0.024
0.054
0.042
0.031
0.047
0.037
0.045
(0.018, 0.237)
(0.026, 0.201)
(0.039, 0.172)
(0.054, 0.151)
(0.078, 0.314)
(0.109, 0.284)
(0.135, 0.261)
(0.191, 0.387)
(0.224, 0.364)
(0.323, 0.462)
lO
Theparameter/;
2istheunbiased
observed
bicoherence,
dofisthe
degrees
of freedomassociated
withthe measurement,
b2mle
is the
0 -I
I
I
I
I
I
I
I
I
lo :•
DEGREES
OF
FREEDOM
Fig. 6. The 90% confidencelimits for biphase.Solid curves are
theoreticalvalues, and symbolsare values observedin the simulated
maximum
of thelikelihood
function(L), bmele
2 is the mean of L, s.d.
is the standarddeviationof L, and (7.5:1)indicatesvaluesof b2
correspondingto likelihood values down by a factor of 7.5 from the
maximumof L (if L is Gaussian,these values correspondto 95%
confidence limits).
data. Each symbolrepresentsdifferentbe as describedin the
caption to Figure 4.
that an observed value comes from the center of the distri-
a functionof dofand/;2in Figure7. For largevaluesof/;2 butionwith high •, (i.e., high b2) than from the uppertail of
and/ordof,L(b2) is symmetrical
aboutits maximum.
Howa distributionwith lower •,. Table 1 and Figure 8 both show
ever,for small/;2and/ordof,L(b2) haslongtailsextendingthat exceptfor small/;2 and/ordof, bmele
2 does not differ
towardhigh valuesof /;2 (Figure7). In order to better substantially
from/;2.
account for the asymmetrical shape of the likelihood funcL(b2)canalsobeusedto calculate
confidence
intervalsfor
tion, Jenkinsand Watts [ 1968]suggestusingthe meanof the the estimated true values of bicoherence. The likelihood
likelihoodfunction(mele)as an estimateof the true paramfunctionspresentedhere (Figure7) are approximatelyGauseter. Both mle and mele estimates of bicoherence are listed
sian,andthusthe meleandthe varianceof L(b2) determine
in Table 1. As L(b2) becomes
moresymmetrical,
2 --> the entire likelihoodfunction [seeJenkinsand Watts, 1968].
bmele
b2mle
__>/;2 (Table1 andFigure8), although
thelikelihoodThe standarddeviationsof L(b2) and the b2 valueswhose
estimates
of b2 arealwaysgreaterthanthecorresponding
likelihoodsare down by a factor of 7.5 from the mle (95%
measured values (Figure 8). This can be understoodby
considering
theX2 pdfsshownin Figure9. It is morelikely
confidence
limitsonb2if L(b2)isGaussian)
arepresented
in
Table
1. From the table it can be seen that the standard
deviation
of L(b2) is nearlyindependent
of/;2 anddepends
primarily on dof.
Since the distributionof biphaseis approximatelyGaus-
0.20
sian,•mele•-' •mle• •'
0.15
O.lO
1.4
0.05
0.00
.•ce
1.2
0.0
0.1
0.2
0.3
0.4
b2
1.1
Fig. 7. LikelihoodfunctionsL(b2) for variousmeasured
bicoherenceanddof values.Solidcurves,unbiased/;2= 0.1; longest
dashes,
/;2 = 0.2; mediumlengthdashes,
/;2 = 0.3; andshortest
dashes,/;2
= 0.4.Foreach/;2theoutermost
curvehas32dof,and
the dof increaseby factorsof 2 for eachsucceeding
innercurve.•The
1.0
I
O. 10
I
0.20
I
I
0.30
I
I
0.40
I
I
0.50
numberof suchcurvesdecreases
forincreasing/;
2. Unbiased
values
are obtainedby subtractingthe bias(equation(4)) from the observed
value, where the observed value of bicoherence is used on the
Fig. 8. Ratioof bmele
2 to/;2 versus/;
2. Eachcurveisfordifferent
right-handside of (4). Replacing/;2 with /;2
melein (4) results in
dof, increasingby factorsof 2 from dof - 32 (top curve) to dof - 256
negligible differences in the bias.
(bottom curve).
ELGARAND SEBERT:STATISTICSOF BICOHERENCEAND BIPHASE
10,997
simulations(Figure 4). The statisticsof biphaseare similar to
those of the phase of the cross spectrum (equation (9) and
Figure 5), and confidencelimits for biphase estimatesbased
0.18
0.16
onb2anddofalsoagreewithcorresponding
valuesobserved
0.14
in the simulations (Figure 6). Maximum likelihood estimates
of the true value of bicoherence given an observed value
•0.12
0.10
wereobtained
fromtheaX2 distribution
(Table1 andFigure
7). The ratio of maximum likelihood estimates of bicoher-
0.08
ence to observedvaluesapproaches1 as /•2 and/ordof
increases,
withlessthan10%differences
if/•2 > 0.25(fordof
-> 32)and/ordof > 150(for/•2 _>0.1) (Figure8).
0.06
0.04
0.02
Acknowledgments. This research was supportedby the Coastal
SciencesBranch of the Officeof Naval Researchand by the Physical
Oceanography Program of the National Science Foundation. The
computations were performed at the San Diego Supercomputer
Center (funded by NSF). R. T. Guza made helpful comments.
0.00
0.0
1.0
2.0
:.3.0
4.0
X2/v
Fig. 9. Chi-squareprobability distributionfunctions.From outermostto innermostcurves, v = 4, 12, 20, and 28, respectively.The
abscissa has been normalized by v so that the mean of each
distributionis 1, and the scaleof the ordinateis arbitrary.
3.4. Application to Shallow Water
Surface Gravity Waves
REFERENCES
Abramowitz, M., and I. Stegun, Handbook of Mathematical Functions, p. 1046, Dover, New York, 1972.
Ashley, R. A., D. M. Patterson, and M. J. Hinich, A diagnostictest
for nonlinear serial dependence in time series fitting errors, J.
Time Set. Anal., 7, 165-178, 1986.
Beningus, V. A., Estimation of the coherence spectrum and its
confidence interval using the fast Fourier transform, IEEE Trans.
Audio E!ectroacoust., 17, 145-150, 1969.
Using bispectral analysis, Elgar and Guza [1985, p. 432]
concludedthat narrow-bandswell (peak period about 16 s) Brillinger, D. R., An introductionto polyspectra,Ann. Math. Stat.,
•
'•'
30,
10..J 1--10'• / •'t, 1965 .
observedin 9 m water depth was "... qualitatively consistent with Stokes-like
nonlinearities."
Their conclusion was
basedon observedvalues of biphasefor triads consistingof
thepowerspectral
primarypeakfrequency
(fp) andits first
few harmonics.Based on the results presentedabove, bispectral estimatesfrom these data can now be interpreted
quantitatively. For dof - 256 the observed values of bicoherence and biphase for the two lowest-order triads were
Q2(,fp,
fp)= 0.161,
•OCp,
fp)= --7øand/•2OCp,
f2p)= 0.067,
13(fp,
f2p) = -35ø. (Bicoherences
at othertriadswere not
significantlygreater than zero.) Accounting for the bias,
equation
(4),b2meleOCp
,fp2)
160andb2meleOCp,
f2p)= 0.070.
=O.
For dof = 256 and bmeleas given above, 90% confidence
limits for biphase can be constructed. For the self-self
interaction
(fp, fp, f2p)the90%limitsare _+13.5ø.Thusthe
Brillinger, D. R., and M. Rosenblatt, Asymptotic theory of estimates of k-th order spectra, in Advanced Seminar on Spectral
Analysis of Time Series, edited by B. Harris, pp. 153-188, John
Wiley, New York, 1967a.
Brillinger, D. R., and M. Rosenblatt, Computation and interpretation of k-th order spectra, in Advanced Seminar on Spectral
Analysis of Time Series, edited by B. Harris, pp. 189-232, John
Wiley, New York, 1967b.
Doering, J. C., and A. J. Bowen, Skewness in the nearshore zone:
A comparisonof estimatesfrom Marsh-McBirney current meters
and collocated pressure sensors, J. Geophys. Res., 92, 13,17313,183, 1987.
Elgar, S., and R. T. Guza, Observations of bispectra of shoaling
surface gravity waves, J. Fluid Mech., 161,425-448, 1985.
Elgar, S., and R. T. Guza, Nonlinear model predictionsof bispectra
of shoaling surface gravity waves, J. Fluid Mech., 167, 1-18,
1986.
include the observed value (-35ø). Consequently,with a
10% possibility of being incorrect, it is concluded that the
Elgar, S., and R. T. Guza, Statistics of bicoherence, IEEE Acoust.
Speech Signal Process., 36, 1667-1668, 1988.
Elgar, S., R. T. Guza, and R. J. Seymour, Wave group statistics
from numerical simulationsof a random sea, Appl. Ocean. Res.,
7, 93-96, 1985.
Freilich, M. H., and R. T. Guza, Nonlinear effects on shoaling
surface gravity waves, Philos. Trans. R. Soc. London, Ser. A,
311, 1-41, 1984.
biphase of this triad is not consistent with Stokes-like
Freilich,M. H., andS.S. Pawka,Statistics
of Sxyestimates,
J.
nonlinearities
Phys. Oceanogr., 17, 1786-1797, 1987.
Hasselman, K., W. Munk, and G. MacDonald, Bispectra of ocean
waves, in Time Series Analysis, edited by M. Rosenblatt, pp.
125-139, John Wiley, New York, 1963.
Haubrich, R. A., Earth noises, 5 to 500 millicycles per second, 1, J.
Geophys. Res., 70, 1415-1427, 1965.
Hinich, M. J., Testing for Gaussianity and linearity of a stationary
time series, J. Time Ser. Anal., 3, 66-74, 1982.
Hinich, M. J., and C. S. Clay, The application of the discrete
Fourier transform in the estimation of power spectra, coherence,
and bispectra of geophysical data, Rev. Geophys., 6, 347-363,
observed value (-7 ø) is within the 90% limits of the Stokes
biphase,/3 = 0ø, and the data are consistentwith Stokes-like
nonlinearities. On the other hand, 90% confidence limits for
thebiphase
of thetriad(fp,f2p,f3p)are +21.6øanddo not
and that
shallow
water
resonance
effects
[Freilich and Guza, 1984]are important.
4.
CONCLUSIONS
Numerical simulationsof random processeswith nonzero
bicoherence were used to investigate the statisticsof estimates of bicoherence and biphase. Bicoherence is biased
(Figure1) andis approximately
aX2 distributed
(Figure3),
with parameter v a function of the true bicoherence and the
1968.
degreesof freedom associatedwith the estimate (equation Jenkins, G. M., and D. G. Watts, Spectral Analysis and Its
(8)). Confidence limits for estimates of bicoherence can be
constructed
fromtheaX2 distribution
(i.e.,fromb2 anddof)
and agree with the correspondingvalues observed in the
Applications, Holden-Day, San Francisco, Calif., 1968.
Kim, Y. C., and E. J. Powers, Digital bispectral analysis and its
application to nonlinear wave interactions, IEEE Trans. Plasma
Sci., 1, 120-131, 1979.
10,998
ELGAR AND SEBERT: STATISTICSOF BICOHERENCEAND BIPHASE
McComas, C. H., and M. G. Briscoe, Bispectra of internal waves,
J. Fluid Mech., 97, 205-213, 1980.
Mfiller, D., Bispectra of sea-surface temperature anomalies, J.
Phys. Oceanogr., 17, 26-36, 1987.
Neshyba, S., and E. J. C. Sobey, Vertical crosscoherenceand cross
bispectrabetween internal waves measuredin a multiple layered
ocean, J. Geophys. Res., 80, 1152-1162, 1975.
Nikias, C. L., and M. R. Raghuveer, Bispectrum estimation: A
digital signal processing framework, Proc. 1EEE, 75, 867-891,
with depth off northeasternJapan, J. Phys. Oceanogr., 3, 308317, 1973.
Rosenblatt, M., and J. W. Van Ness, Estimation of the bispectrum,
Ann. Math. Stat., 36, 1120-1136, 1965.
S. Elgar and G. Sebert, Electrical and Computer Engineering
Department, Washington State University, Pullman, WA 991642752.
1987.
Roden, G. I., and D. J. Bendiner, Bispectra and cross-bispectraof
temperature, salinity, sound velocity, and density fluctuations
(Received December 20, 1988;
accepted March 27, 1989.)