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O NBS TECHNICAL NOTE 1036

*<"?fAU Of "

U.S. DEPARTMENT OF COMMERCE/ National Bureau of Standards
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The Bispectrum and

Higher-Order Spectra:

A Bibliography

JUN 1 5 1981

(-^^v

Peter V. Tryon

Statistical Engineering Division

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303

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U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued April 1981
NATIONAL BUREAU OF STANDARDS TECHNICAL NOTE 1036

Nat. Bur. Stand. (U.S.), Tech. Note 1036, 24 pages (April 1981)

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WASHINGTON: 1981

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402

Price $1.75 (Add 25 percent for other than U.S. mailing)
The Bispectrum and Higher-Order Spectra: A Bibliography

by

Peter V. Tryon

Statistical Engineering Division

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303

The bispectrum or Fourier trans-form of the 3^^*^ order moments of

a time series is useful for the study of nonlinear or non-Gaussian

phenomena. This bibliography cites 134 papers covering both theory

and application. The entries are classified by content with special

effort made to indicate papers that contain material on the computation, display and interpretation of the bispectrum.

Key Words: Bispectrum; cumulant spectra; nonlinear time series;

polyspectra; spectrum analysis; statistics; time series.

INTRODUCTION

The bispectrum or Fourier transform of 3^*^ order moments is useful for

the study of nonlinear structure in time series. It is sensitive to phase

coherence between wave components due, for example, to nonlinear wave interactions or to the harmonic structure of periodic functions that are not

pure sinusoids. The bispectrum is useful both for detecting the presence of

phase coherent structure and measuring the fraction of spectral power due to

coherent components.

The bispectrum was first discussed by Tukey (1953). The general idea of

harmonic analysis of higher order moments was also introduced by

Blanc-Lapierre and Fortet (1953). Magness (1954) used what would now be

called the trispectrum (transform of 4*-" order moments) to compute the

spectral response of a quadratic device to non-Gaussian noise. Mazelsky

(1954) used higher order spectra to determine non-Gaussian probability

functions for the input disturbances and output responses of linear systems.

Tick (1961) showed how the cross-bispectrura could be used to estimate the

transfer function of a quadratic system. Following the suggestion of Tukey

(1963) and Tick (1963), bispectral analysis was first applied to the study of

nonlinear phenomena in ocean waves by Hasselmann, Munk and MacDonald (1963)

The first rather thorough discussion of the bispectrum and its properties

appears in Shaman (1964) which extends Tukey ' s 1953 unpublished manuscript.

Between 1965 and 1967 the general theory of the bispectrum and higher order

cumulant spectra (or polyspectra) was intensively developed by Brillinger,

Rosenblatt, Van Ness, Godfrey, Parzen, and Akaike. Since 1967 the bispectrum

has found a wide variety of applications primarily in the physical sciences.

However, the practice of bispectral analysis has received little attention in

the statistical literature in more than a decade.
The purpose of this bibliography is to make the literature of the last

two decades, now widely scattered in physical sciences journals, more

accessible. There may well be other references to harmonic analysis of higher

order moments in the literature of stochastic processes. In particular, the

vast literature on kernel estimation and identification methods has not been

searched for related papers. However, four such papers that are of special

interest are included. Hung and Stark (1977) is a review of kernel

identification methods with an extensive bibliography (88 entries). Yasul

(1979) discusses the application of kernel methods to nonlinear systems

analysis, and also has an extensive bibliography (49 entries). Both papers

mention the relationship to the bispectrum. Hung and Stark (1979), and Hung,

Brillinger, and Stark (1979) use bispectral analysis to compute higher order

kernels for nonlinear systems.

The bibliography is arranged alphabetically by first author. There is a chronological listing of first authors and an alphabetical cross reference of

second authors with first authors. Finally, the entries are classified by

content. Although this is not a review paper, an effort has been made to

indicate papers that contain material on the computation, display and

interpretation of the bispectrum. Huber, Kleiner, Gasser, and Dummermuth

(1971), and Kim and Powers (1979) give nice reviews of the bispectrum and its

interpretation.

I would like to thank Vicki Schneller of the Department of Commerce

Boulder Library for her assistance with the literature search, and Lorna Buhse

of the Statistical Engineering Division for her help in organizing and typing

the bibliography.
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10
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.

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11
Van Ness, J. W. (1966b). Empirical nonlinear prediction and polyspectra.

Technical Report No. 18, Dept. of Statistics, Stanford University.

Westcott, M. (1970). Identif lability in linear processes. Zeitschrift fuer

Wahrscheinlichkeitstheorie and Verwandte Gebiete, 16, 39-46.

Yamakawa S. (1976). Investigation of peculiarity in some wave-forms through

bispectral analysis. Bulletin of the Japan Society of Mechanical Engineers,

19, 29-36.

Yao, N. C. (1974). Bispectral and cross-bispectral analysis of wind and

currents off the Oregon Coast. Doctoral dissertation, Oregon State

University.

Zhurbenko, G., and Zuev, N. M. (1975). On higher spectral densities of

stationary processes with mixing. Ukrainian Mathematics Journal, 27,

364-375.

Yao, N. C, Neshyba, S., and Crew, H. (1975). Rotary cross-bispectra and

energy transfer functions between non-Gaussian vector processes I.

Development and example. Journal of Physical Oceanography, 5, 164-172.

Yao, N. C, Neshyba, S., and Crew, H. (1977). Rotary cross-bispectra and

energy transfer functions between non-Gaussian vector processes II. Winds and

currents off the Oregon coast. Journal of Physical Oceanography, 7, 892-903.

Yasui, S. (1979). Stochastic functional Fourier series, Volterra series, and

nonlinear systems analysis. IEEE Transactions on Automatic Control, AC-24,

230-242.

Yeh, T. T., and Van Atta, C. W. (1973). Spectral transfer of scalar and

velocity fields in heated-grid turbulence. Journal of Fluid Mechanics, 58,

Part 2, 233-261.

12
First authors by year

1953

Blanc-Lapierre, A.

Tukey, J. W.

1954

Magness, T.A.

Mazelsky, B.

1959

Tukey, J. W.

1960

Shiryaev, A. N.

1961

Tick, L. J.

1967

Aubry, M. P.

Bartlett, M. S.

Brillinger, D. R. (2)

Farzen, E.

1968

Bozzi Zadro, M.

Cartwright, D. E. Hinich, M. J.

Korein, J.

Ma ru s s i , A

.

1970

Alekseyev, V^ G.

Brillinger, D. R. Dumermuth, G.

Van Atta, C. W.

Westcott, M.

1963

Hasselniann, K. MacDonald, G.

Sinai, Ya. G.

Tick, L. J.

Tukey, J. W.

1971

Barnett, T.

Dumermuth, G.

Huber, P. J.

Kleiner, B.

Rosenblatt, M.

1964

Leonov, V, P.

Madden, T.

Rosenblatt, M.

Shaman, P.

1965

Brillinger, D. R.

Godfrey, M. D.

Haubrich, R. A.

Rosenblatt, M.

1966

Akaike, H.

Aubry, M. P.

Hasselraann, K.

Rosenblatt, M.

Van Ness, J. W. (2)

13

1972

Brillinger, D. R.

Dalzell, J. F. (2)

Feuerverger, A.

Gasser, T.

Kiriyama, K.

Mitsuishi, A.

Murty, T. S.

Sasaki, K.

1973

Brillinger, D. R.

Roden, G. I. Tachi, S.

Yeh, T. T.

1974

Brillinger, D. R. (2)

Mager, P. P.

Van Atta, C. W.

Yao, N. C.
1975

Akaike, H.

Brillinger, D. R.

Gasser, T.

Kedem-Kiraelfeld, B. Mager, P. P.

Neshyba, S

Ohta, M.

Sasaki, K

Sato, T.

Tachi, S.

Yao, N. C.

Zhurbenko, G.

1976

Akaike, H.

Briscoe, M. G.

Helland, K. N.

Kim, Y. C.

Lii, K. S.

Lumley , J . L

.

Ten Hoopen, M.

Ueno, T.

Yamakawa, S.

1977

Armstrong, J. W.

Brillinger, D. R. Davies, R. B.

Helland, K. N.

Hung, G.

Kim, Y. C.

Murata, T.

Ohta, M.

Powers, E. J.

Rao, S. T.

Sasaki, K.

Sato, T. (3)

Ten Hoopen, M.

Ueno, T.

Yao, N. C.

1978

Barresen, R.

Dubkov, A. A.

Dumermuth, G.

Gerzon, M. A.

Hasselraann, D. E.

Kim, Y. C.

McComas, C. H. Ill

Ohta, M.

Rosenblatt, M,

Sasaki, K.

Sato, T. (3)

Sclove, S. L.

Shimizu, H.

Tanaka, K.

1979

Fried, D. L.

Gabrielli, C.

Helland, K. N.

Hinich, M. J.

Hung, G. (2)

Kim, Y. C.

Sasaki, K.

Sato, T.

Van Atta, C. W. Yasui, S.

1980 (Partial)

Herring, J.

Kim, Y. C.

McComas, C. H.

Rosenblatt, M.

Sasaki, 0.

14
Second Author's Name First Author's Name

Arahata, E.

Beall, J. M.

Bendiner, D. J.

Bozzi Zadro, M.

Brillinger, D. R.

Briscoe, M. G.

Caputo, M.

Clay, C. S.

Czapskl, U.

Fortet, R.

Gasser, T.

Hatakeyama, K.

Henry, R. F.

Ruber, P. J.

Inoue, T.

Johnson, L. C.

Keddam, M.

Kikkawa, S.

Kim, Y. C.

Kishimoto, T.

Kleiner, B.

Lii, K. S.

Malakhov, A. N.

Munk , W

.

Nakajima, T.

Nakashima, S. N.

Neshyba, S.

Ohara, H.

Powers, E. J.

Rosenblatt, M.

Rosenblatt, M.

Sasaki, K.

Sasaki, 0.

Sato, T.

Sato, T.

Sato, T.

Sobey, E. J. C.

Stark, L.

Tachi, S.

Takeuchi, K.

Tick, L. J.

Van Atta, C. W.

Van Atta, C. W.

Van Ness, J . W.

Yamaguchi, S.

Yeh, T. T.

Zandt, P. A.

Zuev, N. M.

Akaike, H. (2)

Kim, Y. C.

Roden, G. I.

Ma ru s s i , A

.

Hung . G

.

McComas, C. H.

Bozzi Zadro, M.

Hinich, M. J.

Rao. S. T.

Blanc-Lapierre, A.

Dumermuth, G.

Ohta, M.

Murty, T. S.

Dumermuth, G. (2)

Shimizu, H.

Barnett, T.

Gabrielli, C.

Tanaka, K.

Powers, E. J.

Sato, T.

Huber, P. J.

Helland, K. N. (2)

Dubkov, A. A.

Hasselmann, K.

Ueno, T.

Mitsuishi, A.

Yao, N. C. (2)

Murata, T.

Kim, Y. C. (4)

Brillinger, D. R. (2)

Lii, K. S.

Sato, T. (5)

Sato, T. (2)

Kiriyama, K.

Sasaki, K. (5)

Sasaki, 0.

Neshyba, S.

Hung, G. (2)

Ueno, T.

Lumley , J . L

.

Korein, J.

Helland, K. N.

Yeh, T. T.

Rosenblatt, M.

Ohta, M. (2)

Van Atta, C. W.

Ten Hoopen, M. (2).

Zhurbenko, G.

15
Brief Discussion or Close Relationship

Tukey (1959) (1963)

Tick (1963)

Bartlett (1967)

Hinich and Clay (1968)

Westcott (1970)

Dumerrauth, Huber, Kleiner and Gasser (1970)

Rosenblatt (1971)

Brillinger (1974)

Magar (1974) (1975)

Van Atta (1974)

Ohta, Hatakeyama, Hiromitsu and Yamaguchi (1975)

Kedera-Kimelfeld (1975)

Helland and Van Atta (1976)

Hung and Stark (1977)

Ohta, Yamaguchi, and Hiromitsu (1978)

Sclove (1978)

Yasui (1979)

Gabrielli, Keddam, and Raillon (1979)

General Theory

Blanc-Lapierre and Fortet (1953)

Shiryaev (1960)

Tick (1961)

Hasselraann, Munk and MacDonald (1963)

MacDonald (1963)

Sinai (1963)

Shaman (1964)

Brillinger (1965)

Rosenblatt and Van Ness (1965)

Rosenblatt (1966)

Van Ness (1966)

Akaike (1966)

Parzen (1967)

Brillinger and Rosenblatt (1967a, b)

Huber, Kleiner, Gasser, and Dumermuth (1971)

Sasaki, Sato, and Yamahita (1975)

Brillinger (1975)

16
Estimation /Computation

Tick (1961)

Hasselmann, Munk and MacDonald (1963)

Shaman (1964)

Brillinger (1965)

Rosenblatt and Van Ness (1965)

Godfrey (1965)

Haubrich (1965)

Rosenblatt (1966)

Van Ness (1966)

Brillinger and Rosenblatt (1967a, b)

Alekseyev (1970)

Van Atta and Yeh (1970)

Huber, Kleiner, Gasser and Dumermuth (1971)

Sasaki, Sato and Yamashlta (1975)

Akaike, Arahata, Ozaki (1975)

Dumermuth and Gasser (1978)

Kim and Powers (1979)

Display /Interpret at ion

Hasselmann, Munk and MacDonald (1963)

MacDonald (1963)

Shaman (1964)

Godfrey (1965)

Haubrich (1965)

Brillinger and Rosenblatt (1967b)

Cartwright (1968)

Marussi, Bozzi Zadro and Manzoni (1968)

Bozzi Zadro and Caputo (1968)

Huber, Kleiner, Gasser and Dumermuth (1971)

Roden and Bendiner (1973)

Neshyba and Sobey (1975)

Lii, Rosenblatt and Van Atta (1976)

Helland, Lii, and Rosenblatt (1977), (1979)

Dumermuth and Gasser (1978)

Kim and Powers (1979)

Kim, Beall, Powers, and Miksad (1980)

17
Applications

Acoustics

Ohta, Yamaguchi, and Iwashlge (1977)

Sasaki, Sato, and Nakaraura (1977)

Gerzon (1978)

Sato and Sasaki (1979)

Astrophysics

Aubry (1966), (1967)

Armstrong (1977)

Fried (1979)

Biomedicine

Korein, Tick, Zeitlin, and Randt (1968)

Dumermuth, Huber, Kleiner and Gasser (1970), (1971)

Huber, Kleiner, Gasser, and Dumermuth (1971)

Barnett, Johnson, Naitoh, Hicks and Nute (1971)

Magar (1974)

Tachi (1975)

Ten Hoopen and Zandt (1976), (1977)

Economics

Godfrey (1965)

Fluid Mechanics - Turbulence

Van Atta and Yeh (1970)

Yeh and Van Atta (1973)

Van Atta (1974), (1979)

Helland and Van Atta (1976)

Lumley and Takeuchi (1976)

Lii, Rosenblatt and Van Atta (1976)

Helland, Lii, and Rosenblatt (1977), (1979)

Rosenblatt (1978)

Herring (1980)

Geophysics

Brillinger (1973)

Haubrich (1965)

Hinich and Clay (1968)

Bozzi Zadro and Caputo (1968)

Madden (1964)

Marussi, Bozzi Zadro and Manzoni (1968)

18
Goodness-of-Flt Tests

Gasser (1975)

Hydromechanics

Hasselmann (1966)

Dalzell (1972a, b)

Image Processing

Sato and Sasaki (1977)

Sato, Sasaki, and Nonaka (1978)

Sasaki, Sato and Nakamura (1978)

Kernel Estimation and Identification

Hung and Stark (1977), (1979)

Hung, Brillinger and Stark (1979)

Yasui (1979)

Mechanical Engineering

Mazelsky (1954)

Yamakawa (1976)

Sato, Sasaki and Nakamura (1977)

Sato and Sasaki (1977)

Ueno and Nakajima (1977)

Sato, Sasaki, and Nonaka (1978)

Shimizu and Inoue (1978)

Meteorology

MacDonald (1963)

Sato, Sasaki, and Mori (1975)

Nonlinear Prediction

Van Ness (1966)

19
Oceanography

Hasselraann, Munk and MacDonald (1963)

Cartwright (1968)

Murty and Henry (1972)

Roden and Bendiner (1973)

Yao (1974)

Neshyba and Sobey (1975)

Yao, Neshyba and Crew (1975), (1977)

Briscoe (1976)

Rao, Czapski, and Sedefian (1977)

McComas (1978)

Hasselmann (1978)

McComas and Briscoe (1980)

Optics

Sato, Kishiraoto and Sasaki (1978)

Sato and Sasaki (1978)

Sasaki, Sato and Oda (1980)

Plasma Physics

i Kim and Powers (1976), (1977), (1978), (1979)

Powers and Kim (1977)

Kim, Beall, Powers, and Miksad (1980)

Transfer Function Estimation

Magness (1954)

Tick (1961)

Brillinger (1970)

Tachi (1975)

Kedem-Kimelfeld (1975)

Brillinger (1977)

Borresen (1978)

Hinich (1979)

Signal Processing

Parzen (1967)

Tachi (1973)

Murata and Ohara (1977)

Tanaka, Kikkawa, and Ohara (1978)

Dubkov and Malakhov (1978)

20
Simulation

Sasaki and Sato (1978)

Stochastic Processes

Rosenblatt (1964), (1980)

Brillinger (1972), (1974)

Zhurbenko and Zuev (1975)

Davies (1977)

21
NB5-n4A (REV. 2-8CI U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO. NBS TN-1036

2. Performing Organ. Report No, 3. Publication Date

April 1981

4. TITLE AND SUBTITLE

The Blspectrum and Higher-Order Spectra: A Bibliography

5. AUTHOR(S)

Peter V. Tryon

Nali

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6. PERFORMING ORGANIZATION (If joint or other tlian NBS. see instructions)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

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10. SUPPLEMENTARY NOTES

QJ Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual summary of most si gnificant information. If document includes a si gnificant

bi bliography or literature survey, mention it here)

I The bispectrum or Fourier transform of the 3rd order moments of a time series is

useful for the study of nonlinear or non-Gaussian phenomena. This bibliography cit(

134 papers covering both theory and application. The entries are classified by

content with special effort made to indicate papers that contain material on the

computation, display and interpretation of the blspectrum.

tiifiii Mm

(oreif

Mono

Vino:

PIMl

tali

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Specji

sored

Hblio

Applie Mit

iiioloj

'^'itioii H

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

Bispectrum; cumulant spectra; nonlinear time series; polyspectra; spectrum analysis

statistics; time series.

13. AVAILABILITY

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