Coherence estimate between a random and a periodic signal: Bias, variance, analytical critical values, and normalizing transforms
The present work deals with recent results on the sampling distribution of the magnitude-squared coherence (also called just coherence) estimate between a random (Gaussian) and a periodic signal, in order to obtain analytical critical values, alternative expressions for the probability density funct...
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| Published in: | Journal of the Franklin Institute Vol. 346; no. 9; pp. 841 - 853 |
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| Abstract | The present work deals with recent results on the sampling distribution of the magnitude-squared coherence (also called just coherence) estimate between a random (Gaussian) and a periodic signal, in order to obtain analytical critical values, alternative expressions for the probability density function (PDF) as well as the variance and bias of the estimate. A comparison with the more general case of coherence estimation when both signals are Gaussian was also provided. The results indicate that the smaller the true coherence (TC) values the closer both distributions become. The behaviour of variance and bias as a function of the number of data segments and the TC is similar for both coherence estimates. Additionally, the effect of a normalizing function (Fisher's z transform) in the coherence estimated between a random and a periodic signal was also evaluated and normality has been nearly achieved. However, the variance was less equalized in comparison with coherence estimate between two Gaussian signals. |
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| AbstractList | The present work deals with recent results on the sampling distribution of the magnitude-squared coherence (also called just coherence) estimate between a random (Gaussian) and a periodic signal, in order to obtain analytical critical values, alternative expressions for the probability density function (PDF) as well as the variance and bias of the estimate. A comparison with the more general case of coherence estimation when both signals are Gaussian was also provided. The results indicate that the smaller the true coherence (TC) values the closer both distributions become. The behaviour of variance and bias as a function of the number of data segments and the TC is similar for both coherence estimates. Additionally, the effect of a normalizing function (Fisher's z transform) in the coherence estimated between a random and a periodic signal was also evaluated and normality has been nearly achieved. However, the variance was less equalized in comparison with coherence estimate between two Gaussian signals. |
| Author | Ferreira, Danton D. Dias, Edson W. Felix, Leonardo B. Mendes, Eduardo M.A.M. Miranda de Sá, Antonio Mauricio F.L. |
| Author_xml | – sequence: 1 givenname: Antonio Mauricio F.L. surname: Miranda de Sá fullname: Miranda de Sá, Antonio Mauricio F.L. email: amflms@peb.ufrj.br organization: Biomedical Engineering Program/COPPE, Federal University of Rio de Janeiro, P.O. Box 68510, CEP 21941-972 Rio de Janeiro, RJ, Brazil – sequence: 2 givenname: Danton D. surname: Ferreira fullname: Ferreira, Danton D. organization: Electrical Engineering Graduation Program, Federal University of Juiz de Fora, CEP 36036-900 Juiz de Fora, MG, Brazil – sequence: 3 givenname: Edson W. surname: Dias fullname: Dias, Edson W. organization: Federal University of São João del Rei – Alto Paraopeba Campus, CEP: 36420-000, Ouro Branco, MG, Brazil – sequence: 4 givenname: Eduardo M.A.M. surname: Mendes fullname: Mendes, Eduardo M.A.M. organization: Electrical Engineering Graduation Program, Federal University of Minas Gerais, CEP 31270-901 Belo Horizonte, MG, Brazil – sequence: 5 givenname: Leonardo B. surname: Felix fullname: Felix, Leonardo B. organization: Department of Electrical and Production Engineering, Federal University of Viçosa, CEP 36570-000 Viçosa, MG, Brazil |
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| Cites_doi | 10.1016/j.jfranklin.2006.05.002 10.1109/48.838988 10.1016/0167-9473(94)90162-7 10.1093/biomet/24.3-4.441 10.1016/j.sigpro.2006.10.003 10.2307/2685369 10.1109/LSP.2005.861596 10.1109/TAU.1969.1162046 10.1109/TAU.1973.1162496 10.21236/AD0620987 10.21236/AD0619993 10.1007/BF02347702 10.1109/LSP.2003.821751 10.1109/TASSP.1981.1163512 10.2307/2684359 10.1007/BF02546511 |
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| Keywords | Periodic input Fisher's z transform Statistics Estimation Coherence Correlation coefficient Random signal Statistical analysis Critical value Probabilistic approach Bias Signal estimation Periodic signal Variance Sampling distribution Fisher information Z transformation Gaussian signal Probability density function |
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| References | Felix, Miranda de Sa, Mendes, Moraes (bib17) 2006; 13 Miranda de Sá, Infantosi (bib19) 2002; 18 Hinich (bib16) 2000; 25 Benignus (bib15) 1969; 17 Fisher (bib1) 1915; 10 Ding (bib21) 1994; 18 Wiener (bib3) 1930; 55 Miranda de Sá, Infantosi, Simpson (bib10) 2002; 40 Hawkins (bib2) 1989; 43 L.D. Enochson, N.R. Goodman, Gaussian approximations to the distribution of sample coherence, Measurement Analysis Corporation, Technical Report AFFDL-TR-65-57, 1965. Otnes, Enochson (bib9) 1978 Carter, Knapp, Nuttall (bib6) 1973; 21 N.R. Goodman, Measurement of matrix frequency response functions and multiple coherence functions, Measurement Analysis Corporation, Technical Report AFFDL-TR-65-56, 1965. Wishart (bib20) 1932; 44 D’Agostino, Belanger, D’Agostino (bib22) 1990; 44 Miranda de Sá (bib13) 2006; 343 N.R. Goodman, On the joint estimation of the spectra, cospectrum and quadrature spectrum of a two-dimensional stationary Gaussian processes, Sci. Paper 10 (AD 134 919), New York University, New York, 1957. Miranda de Sá (bib12) 2004; 11 Bendat, Piersol (bib14) 2000 Bortel, Sovka (bib7) 2007; 87 Nuttall (bib11) 1981; 29 Johnson, Kotz, Balakrishnan (bib18) 1995; vol. 2 Hinich (10.1016/j.jfranklin.2009.07.009_bib16) 2000; 25 Miranda de Sá (10.1016/j.jfranklin.2009.07.009_bib19) 2002; 18 Wishart (10.1016/j.jfranklin.2009.07.009_bib20) 1932; 44 Otnes (10.1016/j.jfranklin.2009.07.009_bib9) 1978 Nuttall (10.1016/j.jfranklin.2009.07.009_bib11) 1981; 29 Miranda de Sá (10.1016/j.jfranklin.2009.07.009_bib13) 2006; 343 D’Agostino (10.1016/j.jfranklin.2009.07.009_bib22) 1990; 44 Wiener (10.1016/j.jfranklin.2009.07.009_bib3) 1930; 55 Felix (10.1016/j.jfranklin.2009.07.009_bib17) 2006; 13 Miranda de Sá (10.1016/j.jfranklin.2009.07.009_bib10) 2002; 40 Benignus (10.1016/j.jfranklin.2009.07.009_bib15) 1969; 17 Ding (10.1016/j.jfranklin.2009.07.009_bib21) 1994; 18 10.1016/j.jfranklin.2009.07.009_bib8 Johnson (10.1016/j.jfranklin.2009.07.009_bib18) 1995; vol. 2 Bendat (10.1016/j.jfranklin.2009.07.009_bib14) 2000 10.1016/j.jfranklin.2009.07.009_bib4 Miranda de Sá (10.1016/j.jfranklin.2009.07.009_bib12) 2004; 11 10.1016/j.jfranklin.2009.07.009_bib5 Fisher (10.1016/j.jfranklin.2009.07.009_bib1) 1915; 10 Hawkins (10.1016/j.jfranklin.2009.07.009_bib2) 1989; 43 Carter (10.1016/j.jfranklin.2009.07.009_bib6) 1973; 21 Bortel (10.1016/j.jfranklin.2009.07.009_bib7) 2007; 87 |
| References_xml | – volume: 87 start-page: 1100 year: 2007 end-page: 1117 ident: bib7 article-title: Approximation of statistical distribution of magnitude squared coherence estimated with segment overlapping publication-title: Signal Processing contributor: fullname: Sovka – year: 1978 ident: bib9 article-title: Applied Time Series Analysis contributor: fullname: Enochson – volume: 18 start-page: 449 year: 1994 end-page: 455 ident: bib21 article-title: On the computation of the noncentral beta distribution publication-title: Computational Statistics and Data Analysis contributor: fullname: Ding – volume: vol. 2 year: 1995 ident: bib18 publication-title: Distributions in Statistics: Continuous Univariate Distributions contributor: fullname: Balakrishnan – volume: 17 start-page: 145 year: 1969 end-page: 150 ident: bib15 article-title: Estimation of the coherence spectrum and its confidence interval using fast Fourier transform publication-title: IEEE Transactions on Audio and Electroacoustics contributor: fullname: Benignus – volume: 55 start-page: 117 year: 1930 end-page: 258 ident: bib3 article-title: Generalized harmonic analysis publication-title: Acta Mathematica contributor: fullname: Wiener – volume: 343 start-page: 688 year: 2006 end-page: 698 ident: bib13 article-title: A note on the coherence-based signal-to-noise ratio estimation in systems with periodic inputs publication-title: Journal of the Franklin Institute contributor: fullname: Miranda de Sá – volume: 44 start-page: 316 year: 1990 end-page: 321 ident: bib22 article-title: A suggestion for using powerful and informative tests of normality publication-title: American Statistician contributor: fullname: D’Agostino – volume: 11 start-page: 323 year: 2004 end-page: 325 ident: bib12 article-title: A note on the sampling distribution of coherence estimate for the detection of periodic signals publication-title: IEEE Signal Processing Letters contributor: fullname: Miranda de Sá – year: 2000 ident: bib14 article-title: Random Data: Analysis and Measurement Procedures contributor: fullname: Piersol – volume: 21 start-page: 337 year: 1973 end-page: 344 ident: bib6 article-title: Estimation of the magnitude-squared coherence function via overlapped fast Fourier transform processing publication-title: IEEE Transactions on Audio and Electroacoustics contributor: fullname: Nuttall – volume: 13 start-page: 1 year: 2006 end-page: 4 ident: bib17 article-title: Statistical aspects concerning signal coherence applied to randomly modulated periodic signals publication-title: IEEE Signal Processing Letters contributor: fullname: Moraes – volume: 29 start-page: 120 year: 1981 end-page: 122 ident: bib11 article-title: Invariance of distribution of coherence estimate to second-channel statistics publication-title: IEEE Transactions on Acoustic Speech and Signal Processing contributor: fullname: Nuttall – volume: 18 start-page: 39 year: 2002 end-page: 49 ident: bib19 article-title: A coherence-based technique for evaluating the degree of synchronism in the EEG during sensory stimulation publication-title: Revista Brasileira de Engenharia Biomédica contributor: fullname: Infantosi – volume: 40 start-page: 99 year: 2002 end-page: 104 ident: bib10 article-title: Coherence between one random and one periodic signal for measuring the strength of responses in the electro-encephalogram during sensory stimulation publication-title: Medical and Biological Engineering and Computing contributor: fullname: Simpson – volume: 44 start-page: 441 year: 1932 end-page: 456 ident: bib20 article-title: A note on the distribution of the correlation ratio publication-title: Biometrika contributor: fullname: Wishart – volume: 10 start-page: 507 year: 1915 end-page: 521 ident: bib1 article-title: Frequency distribution of the values of the correlation coefficient in samples from indefinitely large population publication-title: Biometrika contributor: fullname: Fisher – volume: 43 start-page: 235 year: 1989 end-page: 237 ident: bib2 article-title: Using U statistics to derive the asymptotic distribution of Fisher's Z statistic publication-title: American Statistician contributor: fullname: Hawkins – volume: 25 start-page: 256 year: 2000 end-page: 261 ident: bib16 article-title: A statistical theory of signal coherence publication-title: IEEE Journal of Oceanographic Engineering contributor: fullname: Hinich – volume: 343 start-page: 688 year: 2006 ident: 10.1016/j.jfranklin.2009.07.009_bib13 article-title: A note on the coherence-based signal-to-noise ratio estimation in systems with periodic inputs publication-title: Journal of the Franklin Institute doi: 10.1016/j.jfranklin.2006.05.002 contributor: fullname: Miranda de Sá – volume: 25 start-page: 256 year: 2000 ident: 10.1016/j.jfranklin.2009.07.009_bib16 article-title: A statistical theory of signal coherence publication-title: IEEE Journal of Oceanographic Engineering doi: 10.1109/48.838988 contributor: fullname: Hinich – volume: 18 start-page: 449 year: 1994 ident: 10.1016/j.jfranklin.2009.07.009_bib21 article-title: On the computation of the noncentral beta distribution publication-title: Computational Statistics and Data Analysis doi: 10.1016/0167-9473(94)90162-7 contributor: fullname: Ding – ident: 10.1016/j.jfranklin.2009.07.009_bib4 – volume: 44 start-page: 441 year: 1932 ident: 10.1016/j.jfranklin.2009.07.009_bib20 article-title: A note on the distribution of the correlation ratio publication-title: Biometrika doi: 10.1093/biomet/24.3-4.441 contributor: fullname: Wishart – volume: 87 start-page: 1100 year: 2007 ident: 10.1016/j.jfranklin.2009.07.009_bib7 article-title: Approximation of statistical distribution of magnitude squared coherence estimated with segment overlapping publication-title: Signal Processing doi: 10.1016/j.sigpro.2006.10.003 contributor: fullname: Bortel – volume: 43 start-page: 235 year: 1989 ident: 10.1016/j.jfranklin.2009.07.009_bib2 article-title: Using U statistics to derive the asymptotic distribution of Fisher's Z statistic publication-title: American Statistician doi: 10.2307/2685369 contributor: fullname: Hawkins – volume: 13 start-page: 1 year: 2006 ident: 10.1016/j.jfranklin.2009.07.009_bib17 article-title: Statistical aspects concerning signal coherence applied to randomly modulated periodic signals publication-title: IEEE Signal Processing Letters doi: 10.1109/LSP.2005.861596 contributor: fullname: Felix – volume: 17 start-page: 145 year: 1969 ident: 10.1016/j.jfranklin.2009.07.009_bib15 article-title: Estimation of the coherence spectrum and its confidence interval using fast Fourier transform publication-title: IEEE Transactions on Audio and Electroacoustics doi: 10.1109/TAU.1969.1162046 contributor: fullname: Benignus – volume: 10 start-page: 507 year: 1915 ident: 10.1016/j.jfranklin.2009.07.009_bib1 article-title: Frequency distribution of the values of the correlation coefficient in samples from indefinitely large population publication-title: Biometrika contributor: fullname: Fisher – volume: 21 start-page: 337 year: 1973 ident: 10.1016/j.jfranklin.2009.07.009_bib6 article-title: Estimation of the magnitude-squared coherence function via overlapped fast Fourier transform processing publication-title: IEEE Transactions on Audio and Electroacoustics doi: 10.1109/TAU.1973.1162496 contributor: fullname: Carter – ident: 10.1016/j.jfranklin.2009.07.009_bib8 doi: 10.21236/AD0620987 – ident: 10.1016/j.jfranklin.2009.07.009_bib5 doi: 10.21236/AD0619993 – year: 1978 ident: 10.1016/j.jfranklin.2009.07.009_bib9 contributor: fullname: Otnes – volume: 40 start-page: 99 year: 2002 ident: 10.1016/j.jfranklin.2009.07.009_bib10 article-title: Coherence between one random and one periodic signal for measuring the strength of responses in the electro-encephalogram during sensory stimulation publication-title: Medical and Biological Engineering and Computing doi: 10.1007/BF02347702 contributor: fullname: Miranda de Sá – volume: 11 start-page: 323 year: 2004 ident: 10.1016/j.jfranklin.2009.07.009_bib12 article-title: A note on the sampling distribution of coherence estimate for the detection of periodic signals publication-title: IEEE Signal Processing Letters doi: 10.1109/LSP.2003.821751 contributor: fullname: Miranda de Sá – volume: 29 start-page: 120 year: 1981 ident: 10.1016/j.jfranklin.2009.07.009_bib11 article-title: Invariance of distribution of coherence estimate to second-channel statistics publication-title: IEEE Transactions on Acoustic Speech and Signal Processing doi: 10.1109/TASSP.1981.1163512 contributor: fullname: Nuttall – volume: 44 start-page: 316 year: 1990 ident: 10.1016/j.jfranklin.2009.07.009_bib22 article-title: A suggestion for using powerful and informative tests of normality publication-title: American Statistician doi: 10.2307/2684359 contributor: fullname: D’Agostino – volume: 18 start-page: 39 year: 2002 ident: 10.1016/j.jfranklin.2009.07.009_bib19 article-title: A coherence-based technique for evaluating the degree of synchronism in the EEG during sensory stimulation publication-title: Revista Brasileira de Engenharia Biomédica contributor: fullname: Miranda de Sá – volume: 55 start-page: 117 issue: 2 year: 1930 ident: 10.1016/j.jfranklin.2009.07.009_bib3 article-title: Generalized harmonic analysis publication-title: Acta Mathematica doi: 10.1007/BF02546511 contributor: fullname: Wiener – volume: vol. 2 year: 1995 ident: 10.1016/j.jfranklin.2009.07.009_bib18 contributor: fullname: Johnson – year: 2000 ident: 10.1016/j.jfranklin.2009.07.009_bib14 contributor: fullname: Bendat |
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| SubjectTerms | Applied sciences Coherence Detection, estimation, filtering, equalization, prediction Estimation Exact sciences and technology Fisher's z transform Information, signal and communications theory Periodic input Signal and communications theory Signal, noise Statistics Telecommunications and information theory |
| Title | Coherence estimate between a random and a periodic signal: Bias, variance, analytical critical values, and normalizing transforms |
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